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Unlocking the Power of Math Learning: Strategies and Tools for Success

posted on September 20, 2023

Mathematics, the foundation of all sciences and technology, plays a fundamental role in our everyday lives. Yet many students find the subject challenging, causing them to shy away from it altogether. This reluctance is often due to a lack of confidence, a misunderstanding of unclear concepts, a move ahead to more advanced skills before they are ready, and ineffective learning methods. However, with the right approach, math learning can be both rewarding and empowering. This post will explore different approaches to learning math, strategies for success, and cutting-edge tools to help you achieve your goals.

Math Learning

Math learning can take many forms, including traditional classroom instruction, online courses, and self-directed learning. A multifaceted approach to math learning can improve understanding, engage students, and promote subject mastery. A 2014 study by the National Council of Teachers of Mathematics found that the use of multiple representations, such as visual aids, graphs, and real-world examples, supports the development of mathematical connections, reasoning, and problem-solving skills.

Moreover, the importance of math learning goes beyond solving equations and formulas. Advanced math skills are essential for success in many fields, including science, engineering, finance, health care, and technology. In fact, a report by Burning Glass Technologies found that 71% of high-salary, entry-level positions require advanced math skills.

Benefits of Math Learning

In today’s 21st-century world, having a broad knowledge base and strong reading and math skills is essential. Mathematical literacy plays a crucial role in this success. It empowers individuals to comprehend the world around them and make well-informed decisions based on data-driven understanding. More than just earning good grades in math, mathematical literacy is a vital life skill that can open doors to economic opportunities, improve financial management, and foster critical thinking. We’re not the only ones who say so:

• Math learning enhances problem-solving skills, critical thinking, and logical reasoning abilities. (Source: National Council of Teachers of Mathematics )
• It improves analytical skills that can be applied in various real-life situations, such as budgeting or analyzing data. (Source: Southern New Hampshire University )
• Math learning promotes creativity and innovation by fostering a deep understanding of patterns and relationships. (Source: Purdue University )
• It provides a strong foundation for careers in fields such as engineering, finance, computer science, and more. These careers generally correlate to high wages. (Source: U.S. Bureau of Labor Statistics )
• Math skills are transferable and can be applied across different academic disciplines. (Source: Sydney School of Education and Social Work )

How to Know What Math You Need to Learn

Often students will find gaps in their math knowledge; this can occur at any age or skill level. As math learning is generally iterative, a solid foundation and understanding of the math skills that preceded current learning are key to success. The solution to these gaps is called mastery learning, the philosophy that underpins Khan Academy’s approach to education .

Mastery learning is an educational philosophy that emphasizes the importance of a student fully understanding a concept before moving on to the next one. Rather than rushing students through a curriculum, mastery learning asks educators to ensure that learners have “mastered” a topic or skill, showing a high level of proficiency and understanding, before progressing. This approach is rooted in the belief that all students can learn given the appropriate learning conditions and enough time, making it a markedly student-centered method. It promotes thoroughness over speed and encourages individualized learning paths, thus catering to the unique learning needs of each student.

Students will encounter mastery learning passively as they go through Khan Academy coursework, as our platform identifies gaps and systematically adjusts to support student learning outcomes. More details can be found in our Educators Hub . 

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How to learn math.

Learning at School

One of the most common methods of math instruction is classroom learning. In-class instruction provides students with real-time feedback, practical application, and a peer-learning environment. Teachers can personalize instruction by assessing students’ strengths and weaknesses, providing remediation when necessary, and offering advanced instruction to students who need it.

Learning at Home

Supplemental learning at home can complement traditional classroom instruction. For example, using online resources that provide additional practice opportunities, interactive games, and demonstrations, can help students consolidate learning outside of class. E-learning has become increasingly popular, with a wealth of online resources available to learners of all ages. The benefits of online learning include flexibility, customization, and the ability to work at one’s own pace. One excellent online learning platform is Khan Academy, which offers free video tutorials, interactive practice exercises, and a wealth of resources across a range of mathematical topics.

Moreover, parents can encourage and monitor progress, answer questions, and demonstrate practical applications of math in everyday life. For example, when at the grocery store, parents can ask their children to help calculate the price per ounce of two items to discover which one is the better deal. Cooking and baking with your children also provides a lot of opportunities to use math skills, like dividing a recipe in half or doubling the ingredients. 

Learning Math with the Help of Artificial Intelligence (AI) 

AI-powered tools are changing the way students learn math. Personalized feedback and adaptive practice help target individual needs. Virtual tutors offer real-time help with math concepts while AI algorithms identify areas for improvement. Custom math problems provide tailored practice, and natural language processing allows for instant question-and-answer sessions. 

Using Khan Academy’s AI Tutor, Khanmigo

Transform your child’s grasp of mathematics with Khanmigo , the 24/7 AI-powered tutor that specializes in tailored, one-on-one math instruction. Available at any time, Khanmigo provides personalized support that goes beyond mere answers to nurture genuine mathematical understanding and critical thinking. Khanmigo can track progress, identify strengths and weaknesses, and offer real-time feedback to help students stay on the right track. Within a secure and ethical AI framework, your child can tackle everything from basic arithmetic to complex calculus, all while you maintain oversight using robust parental controls.

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Math learning is essential for success in the modern world, and with the right approach, it can also be enjoyable and rewarding. Learning math requires curiosity, diligence, and the ability to connect abstract concepts with real-world applications. Strategies for effective math learning include a multifaceted approach, including classroom instruction, online courses, homework, tutoring, and personalized AI support. 

So, don’t let math anxiety hold you back; take advantage of available resources and technology to enhance your knowledge base and enjoy the benefits of math learning.

National Council of Teachers of Mathematics, “Principles to Actions: Ensuring Mathematical Success for All” , April 2014

Project Lead The Way Research Report, “The Power of Transportable Skills: Assessing the Demand and Value of the Skills of the Future” , 2020

Page. M, “Why Develop Quantitative and Qualitative Data Analysis Skills?” , 2016

Mann. EL, Creativity: The Essence of Mathematics, Journal for the Education of the Gifted. Vol. 30, No. 2, 2006, pp. 236–260, http://www.prufrock.com ’

Nakakoji Y, Wilson R.” Interdisciplinary Learning in Mathematics and Science: Transfer of Learning for 21st Century Problem Solving at University ”. J Intell. 2020 Sep 1;8(3):32. doi: 10.3390/jintelligence8030032. PMID: 32882908; PMCID: PMC7555771.

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6 Tips for Teaching Math Problem-Solving Skills

Solving word problems is tougher than computing with numbers, but elementary teachers can guide students to do the deep thinking involved.

A growing concern with students is the ability to problem-solve, especially with complex, multistep problems. Data shows that students struggle more when solving word problems than they do with computation , and so problem-solving should be considered separately from computation. Why?

Consider this. When we’re on the way to a new destination and we plug in our location to a map on our phone, it tells us what lane to be in and takes us around any detours or collisions, sometimes even buzzing our watch to remind us to turn. When I experience this as a driver, I don’t have to do the thinking. I can think about what I’m going to cook for dinner, not paying much attention to my surroundings other than to follow those directions. If I were to be asked to go there again, I wouldn’t be able to remember, and I would again seek help.

If we can switch to giving students strategies that require them to think instead of giving them too much support throughout the journey to the answer, we may be able to give them the ability to learn the skills to read a map and have several ways to get there.

Here are six ways we can start letting students do this thinking so that they can go through rigorous problem-solving again and again, paving their own way to the solution. 

1. Link problem-solving to reading

When we can remind students that they already have many comprehension skills and strategies they can easily use in math problem-solving, it can ease the anxiety surrounding the math problem. For example, providing them with strategies to practice, such as visualizing, acting out the problem with math tools like counters or base 10 blocks, drawing a quick sketch of the problem, retelling the story in their own words, etc., can really help them to utilize the skills they already have to make the task less daunting.

We can break these skills into specific short lessons so students have a bank of strategies to try on their own. Here's an example of an anchor chart that they can use for visualizing . Breaking up comprehension into specific skills can increase student independence and help teachers to be much more targeted in their problem-solving instruction. This allows students to build confidence and break down the barriers between reading and math to see they already have so many strengths that are transferable to all problems.

2. Avoid boxing students into choosing a specific operation

It can be so tempting to tell students to look for certain words that might mean a certain operation. This might even be thoroughly successful in kindergarten and first grade, but just like when our map tells us where to go, that limits students from becoming deep thinkers. It also expires once they get into the upper grades, where those words could be in a problem multiple times, creating more confusion when students are trying to follow a rule that may not exist in every problem.

We can encourage a variety of ways to solve problems instead of choosing the operation first. In first grade, a problem might say, “Joceline has 13 stuffed animals and Jordan has 17. How many more does Jordan have?” Some students might choose to subtract, but a lot of students might just count to find the amount in between. If we tell them that “how many more” means to subtract, we’re taking the thinking out of the problem altogether, allowing them to go on autopilot without truly solving the problem or using their comprehension skills to visualize it. 

3. Revisit ‘representation’

The word “representation” can be misleading. It seems like something to do after the process of solving. When students think they have to go straight to solving, they may not realize that they need a step in between to be able to support their understanding of what’s actually happening in the problem first.

Using an anchor chart like one of these ( lower grade , upper grade ) can help students to choose a representation that most closely matches what they’re visualizing in their mind. Once they sketch it out, it can give them a clearer picture of different ways they could solve the problem.

Think about this problem: “Varush went on a trip with his family to his grandmother’s house. It was 710 miles away. On the way there, three people took turns driving. His mom drove 214 miles. His dad drove 358 miles. His older sister drove the rest. How many miles did his sister drive?”

If we were to show this student the anchor chart, they would probably choose a number line or a strip diagram to help them understand what’s happening.

If we tell students they must always draw base 10 blocks in a place value chart, that doesn’t necessarily match the concept of this problem. When we ask students to match our way of thinking, we rob them of critical thinking practice and sometimes confuse them in the process. 

4. Give time to process

Sometimes as educators, we can feel rushed to get to everyone and everything that’s required. When solving a complex problem, students need time to just sit with a problem and wrestle with it, maybe even leaving it and coming back to it after a period of time.

This might mean we need to give them fewer problems but go deeper with those problems we give them. We can also speed up processing time when we allow for collaboration and talk time with peers on problem-solving tasks. 

5. Ask questions that let Students do the thinking

Questions or prompts during problem-solving should be very open-ended to promote thinking. Telling a student to reread the problem or to think about what tools or resources would help them solve it is a way to get them to try something new but not take over their thinking.

These skills are also transferable across content, and students will be reminded, “Good readers and mathematicians reread.” 

6. Spiral concepts so students frequently use problem-solving skills

When students don’t have to switch gears in between concepts, they’re not truly using deep problem-solving skills. They already kind of know what operation it might be or that it’s something they have at the forefront of their mind from recent learning. Being intentional within their learning stations and assessments about having a variety of rigorous problem-solving skills will refine their critical thinking abilities while building more and more resilience throughout the school year as they retain content learning in the process. 

Problem-solving skills are so abstract, and it can be tough to pinpoint exactly what students need. Sometimes we have to go slow to go fast. Slowing down and helping students have tools when they get stuck and enabling them to be critical thinkers will prepare them for life and allow them multiple ways to get to their own destination.

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Maths Problem Solving: Engaging Your Students And Strengthening Their Mathematical Skills

Meriel Willatt

Maths problem solving can be challenging for pupils. There’s no ‘one size fits all’ approach or strategy and questions often combine different topic areas. Pupils often don’t know where to start. It’s no surprise that problem solving is a common topic teachers struggle to teach effectively to their pupils.

In this blog, we consider the importance of problem solving and share with you some ideas and resources for you to tackle problem solving in your maths classroom, from KS2 up to GCSE.

What is maths problem solving?

Why is maths problem solving so difficult, how to develop problem solving skills in maths, maths problem solving ks2, maths problem solving ks3, maths problem solving gcse.

Maths problem solving is when a mathematical task challenges pupils to apply their knowledge, logic and reasoning in unfamiliar contexts. Problem solving questions often combine several elements of maths.

We know from talking to the hundreds of school leaders and maths teachers that we work with as one to one online maths tutoring providers that this is one of their biggest challenges: equipping pupils with the skills and confidence necessary to approach problem solving questions.

The Ultimate Guide to Problem Solving Techniques

Download these 9 ready-to-go problem solving techniques that every pupil should know

The challenge with problem solving in maths is that there is no generic problem solving skill that can be taught in an isolated maths lesson. It’s a skill that teachers must explicitly teach to pupils, embed into their learning and revisit often.

When pupils are first introduced to a topic, they cannot start problem solving straight away using it. Problem solving relies on deep knowledge of concepts. Pupils need to become familiar with it and practice using it in different contexts before they can make connections, reason and problem solve with it. In fact, some researchers suggest that it could take up to two years to do this (Burkhardt, 2017). 

At Third Space Learning, we specialise in online one to one maths tutoring for schools, from KS1 all the way up to GCSE. Our lessons are designed by maths teachers and pedagogy experts to break down complex problems into their constituent parts. Our specialist tutors then carefully scaffold learning to build students’ confidence in key skills before combining them to tackle problem solving questions.

In order to develop problem solving skills in maths, pupils need lots of different contexts and word problems in which to practise them and the opportunity to engage in mathematical talk that draws on their metacognitive skills. 

The EEF suggests that to develop problem solving skills in maths, teachers need to teach pupils:

• To use different approaches to problem solving
• Use worked examples
• To use metacognition to plan, monitor and reflect on their approaches to problem solving

Below, we take a closer look at problem solving at each stage, from primary school all the way to GCSEs. We’ve also included links to maths resources and CPD to support you and your team’s classroom teaching.

At lower KS2, the National Curriculum states that pupils should develop their ability to solve a range of problems. However, these will involve simple calculations as pupils develop their numeracy skills. As pupils progress to upper KS2, the demand for problem solving skills increases. 

“At this stage, pupils should develop their ability to solve a wider range of problems, including increasingly complex properties of numbers and arithmetic, and problems demanding efficient written and mental methods of calculation. With this foundation in arithmetic, pupils are introduced to the language of algebra as a means for solving a variety of problems.” National curriculum in England: mathematics programmes of study (Upper key stage 2 – years 5 and 6)

KS2 problem solving can often fall into the trap of relying on acronyms, such as RICE, RIDE or even QUACK. The most popular is RUCSAC (Read, Underline, Calculate, Solve, Answer, Check). While these do aim to simplify the process for young minds, it encourages a superficial, formulaic approach to problem solving, rather than deep mathematical thinking. Also, consider how much is wrapped up within the word ‘solve’ – is this helpful?

We teach thousands of pupils KS2 maths problem solving skills every week through our one to one online tutoring programme for maths. In our interventions, we encourage deep mathematical thinking by using a simplified version of George Polya’s four stages of problem solving. Here are the four stages:

Understand the problem

• Devise a strategy for solving it
• Carry out the problem solving strategy
• Check the result

We use UCR as a simplified model: Understand, Communicate & Reflect. You may choose to adapt this depending on the age and ability of your class.

For example:

Maisy, Heidi and Freddie are children in the same family. The product of their ages is a score. How old might they be?

There are three people.

There are three numbers that multiply together to make twenty (a score is equal to 20). There will be lots of answers, but no ‘right’ answer.

Communicate

To solve the word problem we need to find the numbers that will go into 20 without a remainder (the factors).

The factors of 20 are 1, 2, 4, 5, 10 and 20.

Combinations of numbers that could work are: 1, 1, 20 1, 2, 10 1, 4, 5 2, 2, 5.

The question says children, which means ‘under 18 years’, so that would mean we could remove 1, 1, 20 from our list of possibilities. 

In our sessions, we create a nurturing learning environment where pupils feel safe to make mistakes. This is so important in the context of problem solving as the best problem solvers will be resilient and able to overcome challenges in the ‘Reflect’ stage. Read more: What is a growth mindset

Looking for more support teaching KS2 problem solving? We’ve developed a powerpoint on problem solving, reasoning and planning for depth that is designed to be used as CPD by primary school teachers, maths leads and SLT. 

The resource reflects on how metacognition can enhance reasoning and problem solving abilities, the ‘curse’ of real life maths (think ‘Carl buys 60 watermelons…) and how teachers can practically implement and teach strategies in the classroom.

You may also be interested in: 

• Developing Thinking Skills At KS2
• KS2 Maths Investigations
• Word problems for Year 6

At KS3, the importance of seeing mathematical concepts as interconnected with other skills, including problem solving, is foregrounded. The National Curriculum also stresses the importance of a strong foundation in maths before moving on to complex problem solving.

“Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas. The programme of study for key stage 3 is organised into apparently distinct domains, but pupils should build on key stage 2 and connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems” National curriculum in England: mathematics programmes of study (Key stage 3)

“Decisions about progression should be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content in preparation for key stage 4.” National curriculum in England: mathematics programmes of study (Key stage 3)

For many students, the transition from primary to secondary school can be a huge challenge.

Especially in the aftermath of the Covid-19 pandemic and the resultant school closures, students may arrive into Year 7 with various learning gaps and misconceptions that will hold them. Some students may need focused support to plug these gaps and grow in confidence.

You can give pupils a smoother transition from KS2 to KS3 with personalised one to one online tuition with specialist tutors with Third Space Learning. Our lessons cover content from Years 5-7 and build a solid foundation for pupils to develop their problem solving skills. Pupils are supported towards independent practice through worked examples, questioning and support slides.

The challenge for KS3 maths problem solving activities is that learners may struggle to get invested unless you start with a convincing hook. Engage your young mathematicians on topics you know well or you know they’ll be invested in and try your hand at designing your own mathematical problems. Alternatively, get some inspiration from our crossover ability and fun maths problems .

Since the new GCSE specification began in 2015, there has been an increased focus on non-routine problem solving questions. These questions demand students to make sense of lots of new information at once before they even move on to selecting the strategies they’ll use to find the correct answer. This is where many learners get stuck.

In recent years, teachers and researchers in pedagogy (including Ofsted) have recognised that open ended problem solving tasks do not in fact lead to improved student understanding. While they may be enjoyable and engage learners, they may not lead to improved results.

SSDD problems (Same Surface Different Depth) can offer a solution that develops students’ critical thinking skills, while ensuring they engage fully with the information they’re provided. The idea behind them is to provide a set of questions that look the same and use the same mathematical hook but each question requires a different mathematical process to be solved.

Read more about SSDD problems , tips on writing your own questions and download free printable examples. There are also plenty of more examples on the NRICH website.

Worked examples, careful questioning and constructing visual representations can help students to convert the information embedded in a maths challenge into mathematical notations. Read our blog on problem solving maths questions for Foundation, Crossover & Higher examples, worked solutions and strategies.

Remember that students can only move on to mathematics problem solving once they have secure knowledge in a topic. If you know there are areas your students need extra support, check our Secondary Maths Resources library for revision guides, teaching resources and worksheets for KS3 and GCSE topics.

DO YOU HAVE STUDENTS WHO NEED MORE SUPPORT IN MATHS?

Every week Third Space Learning’s specialist online maths tutors support thousands of students across hundreds of schools with weekly online 1 to 1 maths lessons designed to plug gaps and boost progress.

Since 2013 these personalised one to 1 lessons have helped over 150,000 primary and secondary students become more confident, able mathematicians.

Learn how the programmes are aligned to maths mastery teaching or request a personalised quote for your school to speak to us about your school’s needs and how we can help.

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Development and differences in mathematical problem-solving skills: A cross-sectional study of differences in demographic backgrounds

Ijtihadi kamilia amalina.

a Doctoral School of Education, University of Szeged, Hungary

Tibor Vidákovich

b Institute of Education, University of Szeged, Hungary

Associated Data

Data will be made available on request.

Problem-solving skills are the most applicable cognitive tool in mathematics, and improving the problem-solving skills of students is a primary aim of education. However, teachers need to know the best period of development and the differences among students to determine the best teaching and learning methods. This study aims to investigate the development and differences in mathematical problem-solving skills of students based on their grades, gender, and school locations. A scenario-based mathematical essay test was administered to 1067 students in grades 7–9 from schools in east Java, Indonesia, and their scores were converted into a logit scale for statistical analysis. The results of a one-way analysis of variance and an independent sample t -test showed that the students had an average level of mathematical problem-solving skills. The number of students who failed increased with the problem-solving phase. The students showed development of problem-solving skills from grade 7 to grade 8 but not in grade 9. A similar pattern of development was observed in the subsample of urban students, both male and female. The demographic background had a significant effect, as students from urban schools outperformed students from rural schools, and female students outperformed male students. The development of problem-solving skills in each phase as well as the effects of the demographic background of the participants were thoroughly examined. Further studies are needed with participants of more varied backgrounds.

1. Introduction

Problem-solving skills are a complex set of cognitive, behavioral, and attitudinal components that are situational and dependent on thorough knowledge and experience [ 1 , 2 ]. Problem-solving skills are acquired over time and are the most widely applicable cognitive tool [ 3 ]. Problem-solving skills are particularly important in mathematics education [ 3 , 4 ]. The development of mathematical problem-solving skills can differ based on age, gender stereotypes, and school locations [ [5] , [6] , [7] , [8] , [9] , [10] ]. Fostering the development of mathematical problem-solving skills is a major goal of educational systems because they provide a tool for success [ 3 , 11 ]. Mathematical problem-solving skills are developed through explicit training and enriching materials [ 12 ]. Teachers must understand how student profiles influence the development of mathematical problem-solving skills to optimize their teaching methods.

Various studies on the development of mathematical problem-solving skills have yielded mixed results. Grissom [ 13 ] concluded that problem-solving skills were fixed and immutable. Meanwhile, other researchers argued that problem-solving skills developed over time and were modifiable, providing an opportunity for their enhancement through targeted educational intervention when problem-solving skills developed quickly [ 3 , 4 , 12 ]. Tracing the development of mathematical problem-solving skills is crucial. Further, the results of previous studies are debatable, necessitating a comprehensive study in the development of students’ mathematical problem-solving skills.

Differences in mathematical problem-solving skills have been identified based on gender and school location [ [6] , [7] , [8] , [9] , [10] ]. School location affects school segregation and school quality [ 9 , 14 ]. The socioeconomic and sociocultural characteristics of a residential area where a school is located are the factors affecting academic achievement [ 14 ]. Studies in several countries have shown that students in urban schools demonstrated better performance and problem-solving skills in mathematics [ 9 , 10 , 15 ]. However, contradictory results have been obtained for other countries [ 6 , 10 ].

Studies on gender differences have shown that male students outperform female students in mathematics, which has piqued the interest of psychologists, sociologists, and educators [ 7 , 16 , 17 ]. The differences appear to be because of brain structure; however, sociologists argue that gender equality can be achieved by providing equal educational opportunities [ 8 , 16 , 18 , 19 ]. Because the results are debatable and no studies on gender differences across grades in schools have been conducted, it would be interesting to investigate gender differences in mathematical problem-solving skills.

Based on the previous explanations, teachers need to understand the best time for students to develop mathematical problem-solving skills because problem-solving is an obligatory mathematics skill to be mastered. However, no relevant studies focused on Indonesia have been conducted regarding the mathematical problem-solving skill development of students in middle school that can provide the necessary information for teachers. Further, middle school is the important first phase of developing critical thinking skills; thus relevant studies are required in this case [ 3 , 4 ]. In addition, a municipal policy-making system can raise differences in problem-solving skills based on different demographic backgrounds [ 10 ]. Moreover, the results of previous studies regarding the development and differences in mathematical problem-solving skills are debatable. Thus, the present study has been conducted to meet these gaps. This study investigated the development of mathematical problem-solving skills in students and the differences owing demographic backgrounds. Three aspects were considered: (1) student profiles of mathematical problem-solving skills, (2) development of their mathematical problem-solving skills across grades, and (3) significant differences in mathematical problem-solving skills based on gender and school location. The results of the present study will provide detailed information regarding the subsample that contributes to the development of mathematical problem-solving skills in students based on their demographic backgrounds. In addition, the description of the score is in the form of a logit scale from large-scale data providing consistent meaning and confident generalization. This study can be used to determine appropriate teaching and learning in the best period of students’ development in mathematical problem-solving skills as well as policies to achieve educational equality.

2. Theoretical background

2.1. mathematical problem-solving skills and their development.

Solving mathematical problems is a complex cognitive ability that requires students to understand the problem as well as apply mathematical concepts to them [ 20 ]. Researchers have described the phases of solving a mathematical problem as understanding the problem, devising a plan, conducting out the plan, and looking back [ [20] , [24] , [21] , [22] , [23] ]. Because mathematical problems are complex, students may struggle with several phases, including applying mathematical knowledge, determining the concepts to use, and stating mathematical sentences (e.g., arithmetic) [ 20 ]. Studies have concluded that more students fail at later stages of the solution process [ 25 , 26 ]. In other words, fewer students fail in the phase of understanding a problem than during the plan implementation phase. Different studies have stated that students face difficulties in understanding the problem, determining what to assume, and investigating relevant information [ 27 ]. This makes them unable to translate the problem into a mathematical form.

Age or grade is viewed as one factor that influences mathematical problem-solving skills because the skills of the students improve over time as a result of the teaching and learning processes [ 28 ]. Neuroscience research has shown that older students have fewer problems with arithmetic than younger students; however, the hemispheric asymmetry is reduced [ 29 ]. In other words, older students are more proficient, but their flexibility to switch among different strategies is less. Ameer & Sigh [ 28 ] obtained similar results and found a considerable difference in mathematical achievement; specifically, older students performed better than younger students in number sense and computation using one-way analysis of variance (ANOVA) ( F ) of F (2,411) = 4.82, p  < 0.01. Molnár et al. [ 3 ] found that the student grade affects domain-specific and complex problem-solving skills. They observed that the development of problem-solving skills was noticeable across grades in elementary school but stopped in secondary school. The fastest development of domain-specific problem-solving occurred in grades 7 and 8 [ 3 ], but the fastest development of complex problem-solving occurred in grades 5–7 [ 3 ]. No development was detected between grades 4 and 5 as well as grades 6 and 7 for domain-specific and complex problem-solving skills, respectively. Similarly, Greiff et al. [ 4 ] concluded that students developed problem-solving skills across grades 5–11 with older students being more skilled. However, the grade 9 students deviated from the development pattern, and their problem-solving skills dropped. The theories from Molnár et al. [ 3 ] and Greiff et al. [ 4 ] are the benchmark cases herein.

The above studies showed that problem-solving skills mostly developed during compulsory schooling and developed most quickly in specific grades. This indicates that specific development times can be targeted to enhance the problem-solving skills [ 3 ]. However, Jabor et al. [ 30 ] observed contradictory results showing statistically significant differences with small effects in mathematical performance between age groups: those under the age of 19 outperformed those over the age of 19 years old. Grissom [ 13 ] observed a negative correlation between age and school achievement that remained constant over time.

2.2. Effects of school location and gender on mathematical problem-solving skills

School location has been shown to affect mathematical achievement [ 9 , 14 ]. In 15 countries, students in rural schools performed considerably worse than students in urban schools in mathematics [ 9 , 10 ], science and reading [ 9 ]. In addition, Nepal [ 15 ] discovered that urban students significantly outperformed rural students in mathematical problem-solving skills ( t  = −5.11, p  < 0.001) and achievement ( t  = −4.45, p  < 0.001) using the results of an independent sample t -test (t). However, other countries have found that rural students outperformed urban students in mathematics [ 6 , 10 ]. These variations may be attributed to a lack of instructional resources (e.g., facilities, materials, and programs), professional training (e.g., poorly trained teachers), and progressive instruction [ 6 ]. The results of Williams's study [ 10 ] serve as the basis for the current study.

Gender differences in mathematics have received attention because studies show that male students outperform female students on higher-level cognitive tasks [ 31 ]. This is a shift from a meta-analysis study that found gender differences in mathematics to be insignificant and favored female students [ 32 ]. At the college level, female students slightly outperform male students in computation while male students outperform female students in problem solving. However, no gender differences have been observed among elementary and middle school students. This result was strengthened by other meta-analysis studies [ 7 , 8 ], which concluded that there was no gender difference in mathematical performance and problem-solving skills [ 15 , [33] , [35] , [34] ]. Gender similarity in mathematics is achieved when equal learning opportunities and educational choices are provided and the curriculum is expanded to include the needs and interests of the students [ 16 , 18 , 31 ].

From a sociological perspective, gender similarity in mathematics makes sense. If there is a gender difference in mathematics, this has been attributed to science, technology, engineering, and mathematics (STEM) being stereotyped as a male domain [ 8 ]. Stereotypes influence beliefs and self-efficacy of students and perceptions of their own abilities [ 8 , 19 ]. This is the reason for the low interest of female students in advanced mathematics courses [ 18 , 19 ]. However, Halpern et al. [ 16 ] found that more female students are entering many occupations that require a high level of mathematical knowledge. Moreover, Anjum [ 36 ] found that female students outperformed male students in mathematics. This may be because female students prepared better than the male students before the test and were more thorough [ 36 , 37 ]. The study of Anjum [ 36 ] is one of the basis cases of the current study.

Differences in brain structure support the argument that there are gender differences in mathematical performance [ 16 , 17 ]. Females have less brain lateralization (i.e., symmetric left and right hemispheres), which helps them perform better verbally. Meanwhile, males have more brain lateralization, which is important for spatial tasks [ 17 ]. In addition, the male hormone testosterone slows the development of the left hemisphere [ 16 ], which improves the performance of right brain-dominant mathematical reasoning and spatial tasks.

3.1. Instrumentation

In this study, a science-related mathematical problem-solving test was used. This is a mathematics essay test where the problems are in the form of scenarios related to environmental management. Problems are solved by using technology as a tool (e.g., calculator, grid paper). The test was developed in an interdisciplinary STEM framework, and it is targeted toward grades 7–9. There were six scenarios in total: some were given to multiple grades, and others were specific to a grade. They included ecofriendly packaging (grade 7), school park (grade 7), calorie vs. greenhouse gas emissions (grades 7–9), floodwater reservoir (grade 8), city park (grades 8–9), and infiltration well (grade 9). These scenarios cover topics such as number and measurement, ratio and proportion, geometry, and statistics. Every scenario had a challenge, and students were provided with eight metacognitive prompt items to help them explore their problem-solving skills.

The test was administered by using paper and pencils for a 3-h period with a break every hour. At the end of the test, students were asked to fill in their demographic information. Each prompt item had a maximum score of 5 points: a complete and correct answer (5 points), a complete answer with a minor error (4 points), an incomplete answer with a minor error (3 points), an incomplete answer with a major error (2 points), and a completely wrong and irrelevant answer (1 point). Each scenario had a maximum total score of 40 points.

The test was validated to determine whether it contained good and acceptable psychometric evidence. It had an acceptable content validity index (CVI >0.67), moderate intraclass correlation coefficient (ICC) (rxx = 0.63), and acceptable Cronbach's alpha (α = 0.84). The construct validity indicated all scenarios and prompt items were fit (0.77 ≤ weighted mean square ≤1.59) with an acceptable discrimination value (0.48 ≤ discrimination value ≤ 0.93), acceptable behavior of the rating score, and good reliability (scenario reliability = 0.86; prompt item reliability = 0.94).

3.2. Participants

The test was administered to grades 7–9 students in east Java, Indonesia (n = 1067). The students were selected from A-accreditation schools in urban and rural areas; random classes were selected for each grade. The majority of the students were Javanese (95.01%), with the remainder being Madurese (3.3%) and other ethnicities. Table 1 describes the demographics of the participants.

Demographic characteristics of participants.

3.3. Data analysis

Data were collected between July and September 2022. Prior to data collection, ethical approval was sought from the institutional review board (IRB) of the Doctoral School of Education, University of Szeged and was granted with the ethical approval number of 7/2022. In addition, permission letters were sent to several schools to request permission and confirm their participation. The test answers of the students were scored by two raters – the first author of this study and a rater with master's degree in mathematics education – to ensure that the rating scale was consistently implemented. The results showed good consistency with an ICC of 0.992 and Cronbach's alpha of 0.996.

The scores from one of the raters were converted to a logit scale by weighted likelihood estimation (WLE) using the ConQuest software. A logit scale provides a consistent value or meaning in the form of intervals. The logit scale represents the unit interval between locations on the person–item map. WLE was chosen rather than maximum likelihood estimation (MLE) because WLE is more central than MLE, which helps to correct for bias [ 38 ]. The WLE scale was represented by using descriptive statistics to profile the students' mathematical problem-solving skills in terms of the percentage, mean score ( M ) and standard deviation ( SD ) for each phase. The WLE scale was also used to describe common difficulties for each phase. The development of students’ mathematical problem-solving skills across grades was presented by a pirate plot, which is used in R to visualize the relationship between 1 and 3 categorical independent variables and 1 continuous dependent variable. It was chosen because it displays raw data, descriptive statistics, and inferential statistics at the same time. The data analysis was performed using R studio version 4.1.3 software with the YaRrr package. A one-way ANOVA was performed to find significant differences across grades. An independent sample t -test was used to analyze significant differences based on gender and school location. The descriptive statistics, one-way ANOVA test, and independent sample t -test were performed using the IBM SPSS Statistics 25 software.

4.1. Student profiles

The scores of students were converted to the WLE scale, where a score of zero represented a student with average ability, a positive score indicated above-average ability, and a negative score indicated below-average ability. A higher score indicated higher ability. The mean score represented a student with average mathematical problem-solving skills ( M  = 0.001, SD  = 0.39). Overall, 52.1% of students had a score below zero. The distribution of scores among students was predominantly in the interval between −1 and 0. When the problem-solving process was analyzed by phase, the results showed that exploring and understanding were the most mastered problem-solving skills ( M  = 0.24, SD  = 0.51). Only 27.9% of students had below-average scores for the exploring and understanding phases, which indicates that they mostly understood the given problem and recognized the important information. However, the problem-solving skills decreased with higher phases. The students had below-average abilities in the phases of representing and formulating ( M  = −0.01, SD  = 0.36), planning and executing ( M  = −0.15, SD  = 0.41), and monitoring and reflecting ( M  = −0.16, SD  = 0.36). About 57.9% of the students had below-average scores for the representing and formulating phase, which indicates that they had problems making hypotheses regarding science phenomena, representing problems in mathematical form, and designing a prototype. The obvious reason for their difficulty with making hypotheses was that they did not understand simple concepts of science (e.g., CO 2 vs. O 2 ). In the planning and executing phase, 66.8% of the students failed to achieve a score greater than zero. This happened because they failed to apply mathematical concepts and procedures. Because they were unable to plan and execute a strategy, this affected the next phase of the problem-solving process. In the monitoring and reflecting phase, 68.0% of the students had a below-average score.

4.2. Development of mathematical problem-solving skills across grades

The development of the mathematical problem-solving skills of the students across grades was observed based on the increase in the mean score. The problem-solving skills developed from grade 7 to grade 8. The students of grade 7 had a mean score of −0.04 while grade 8 students had the highest mean score of 0.03. The students in grades 7 and 8 also showed more varied problem-solving skills than the grade 9 students did. In contrast, the grade 9 students showed a different pattern of development, and their mean score dropped to 0.01. Although the difference was not large, further analysis was needed to determine its significance.

Fig. 1 displays the development of the mathematical problem-solving skills of the students. The dots represent raw data or WLE scores. The middle line shows the mean score. The beans represent a smoothed density curve showing the full data distribution. The scores of the students in grades 7 and 9 were concentrated in the interval between −0.5 and 0. However, the scores of the grade 8 students were concentrated in the interval between 0 and 0.5. The scores of the students in grades 7 and 8 showed a wider distribution than those of the grade 9 students. The bands which overlap with the line representing the mean score, define the inference around the mean (i.e., 95% of the data are in this interval). The inference of the WLE score was close to the mean.

Differences in students' mathematical problem-solving skills across grades.

Note : PS: Problem-Solving Skills of Students.

The one-way ANOVA results indicated a significant difference among the problem-solving skills of the students of grades 7–9 ( F (1,066) = 3.01, p  = 0.046). The students of grade 8 showed a significant difference in problem-solving skills and outperformed the other students. The students of grades 7 and 9 showed no significant difference in their mathematical problem-solving skills. Table 2 presents the one-way ANOVA results of the mathematical problem-solving skills across grades.

One-way ANOVA results of the mathematical problem-solving across grades.

Note. Post hoc test: Dunnett's T3. 7, 8, and 9: subsample grade. <: direction of significant difference ( p  < 0.05).

Fig. 2 shows the development of the mathematical problem-solving skills of the students across grades based on school location and gender. The problem-solving skills of the urban students increased from a mean score of 0.07 in grade 7 to 0.14 in grade 8. However, the mean score of urban students in grade 9 dropped. In contrast, the mean scores of the rural students increased continuously with grade. The improvements were significant for both the rural ( F (426) = 10.10, p  < 0.001) and urban ( F (639) = 6.10, p  < 0.01) students. For the rural students, grade 9 students showed a significant difference in problem-solving skills. In contrast, urban students in grades 8 and 9 showed significant differences in problem-solving skills but not in grade 7.

Differences in students' mathematical problem-solving skills across grades and different demographic backgrounds.

(a) Differences in students grade 7 of mathematical problem-solving skills across grades and different demographic backgrounds

(b) Differences in students grade 8 of mathematical problem-solving skills across grades and different demographic backgrounds

(c) Differences in students grade 9 of mathematical problem-solving skills across grades and different demographic backgrounds

Note: WLE_PS: The students' problem-solving skills in WLE scale; F: Female; M: Male; ScLoc: School location; R: Rural; U: Urban.

When divided by gender, both female and male students showed improvements in their problem-solving skills from grades 7 and 8. However, female students in grade 9 showed a stable score while the scores of male students in grade 9 declined. Only male students in grade 7 showed a significant difference in the mean score. In urban schools, the scores of male and female students increased and decreased, respectively, from grade 7 to grade 8. Male students in rural schools showed an increase in score from grade 7 to grade 9. However, the scores of female students in rural schools decreased from grade 7 to grade 8. Table 3 presents the one-way ANOVA results for the mathematical problem-solving skills of the students considering gender and school location.

One-way ANOVA results for mathematical problem-solving skills across grades and different demographic backgrounds.

Fig. 2 shows that the distributions of the male and female scores of students were similar for every grade except rural grade 9 students. The scores of the rural female students were concentrated in the interval between 0 and 0.5 while the scores of the rural male students were mostly below 0. The scores of rural students in grade 7 and urban students in grade 9 (both male and female) were concentrated in the interval between −0.5 and 0. The scores of urban students in grades 7 and 8 were concentrated in the interval between −0.5 and 0.5.

Fig. 3 shows a detailed analysis of the development of mathematical problem-solving skills across grades for each phase of the problem-solving process. Similar patterns were observed in the exploring and understanding and the representing and formulating phases: the mean score increased from grade 7 to grade 8 but decreased from grade 8 to grade 9. Grade 8 students had the highest mean score and differed significantly from the scores of students in other grades.

Differences in students' mathematical problem-solving skills in every phase across grades: (1) Exploring & understanding, (2) Representing & formulating, (3) Planning & executing, (4) Monitoring & reflecting.

(a) Differences in students' mathematical problem-solving skills in exploring and understanding phase

(b) Differences in students' mathematical problem-solving skills in representing and formulating phase

(c) Differences in students' mathematical problem-solving skills in planning and executing phase

(d) Differences in students' mathematical problem-solving skills in monitoring and reflecting phase

Note: WLE_Exp_Un: The WLE score in exploring and understanding; WLE_Rep_For: The WLE score in representing and formulating; WLE_Plan_Ex: The WLE score in planning and executing; WLE_Mon_Ref: The WLE score in monitoring and reflecting.

The scores of the students for the planning and executing phase increased with grade. However, the difference was only significant at grade 9. Grades 7 and 8 students showed an increase in score, but the improvement was not significant. There was no pattern detected in the monitoring and reflecting phase. The score was stable for grades 7 and 8 students but improved for grade 9 students. The mean score for each phase and the one-way ANOVA results are presented in Table 4 .

One-way ANOVA results for every phase of problem-solving across grades.

Fig. 3 shows that the distributions of the problem-solving skills of the students were similar across grades and phases. However, the distributions were different for grade 9 students in the representing and formulating, planning and executing, and monitoring and reflecting phases, where 95% of the data were in the interval between −0.5 and 0.5.

4.3. Effects of demographic background

4.3.1. school location.

The mathematical problem-solving skills of the students differed significantly based on school location. Urban students scored higher than rural students. The results of the t -test for mathematical problem-solving skills based on school location are presented in Table 5 .

T-test results for mathematical problem-solving skills based on school location.

The effects of the school's location on the performances of male and female students were analyzed. The results showed that the scores of the female students differed significantly based on school location ( t (613) = −6.09, p  < 0.001). Female students in urban schools ( M  = 0.18, SD  = 0.39) outperformed female students in rural schools ( M  = −0.08, SD  = 0.37). Similar results were observed for male students with urban students ( M  = −0.01, SD  = 0.35) outperforming rural students ( M  = −0.12, SD  = 0.39) by a significant margin ( t (382.764) = −3.25, p  < 0.01).

When analyzed by grade, grades 7 and 8 students contributed to the difference based on school location with t (377.952) = −6.34, p  < 0.001 and t (300.070) = −5.04, p  < 0.001, respectively. Urban students in grades 7 and 8 performed significantly better than their rural counterparts did. However, there was no significant difference between rural and urban students in grade 9 ( t (354) = 0.71, p  = 0.447).

4.3.2. Gender

Male and female students showed a significant difference in their mathematical problem-solving skills. Overall, female students outperformed male students. The detailed results of the independent sample t -test for mathematical problem-solving skills based on gender are presented in Table 6 .

T-test results for mathematical problem-solving skills based on gender.

The results were analyzed to determine whether the school location contributed to the gender difference. The gender difference was most significant among urban students ( t (596.796) = −4.36, p  < 0.001). Female students from urban schools ( M  = 0.12, SD  = 0.39) outperformed male students from urban schools ( M  = −0.01, SD  = 0.35). There was no significant difference between female and male students from rural schools ( t (425) = −1.31, p  = 0.191).

Grades 7 and 9 students contributed to the gender difference with t (372.996) = −3.90, p  < 0.001 and t (354) = −2.73, p  < 0.01, respectively. Female students in grades 7 and 9 outperformed their male counterparts. However, there was no significant gender difference among grade 8 students ( t (329) = −0.10, p  = 0.323).

5. Discussion

The mathematical problem-solving skills of the students were categorized as average. In addition, the difficulties of students increased in line with the problem-solving phase. Fewer students failed the exploring and understanding phase than the subsequent phases. This confirms the results of previous studies indicating that more students failed further along the problem-solving process [ 25 , 26 ]. Because the problem-solving process is sequential, students who have difficulty understanding a problem will fail the subsequent phases [ 27 ].

The development of mathematical problem-solving skills was evaluated according to the mean WLE score. The mathematical problem-solving skills of the students developed from grade 7 to grade 8 based on the increase in their mean scores. However, the development dropped in grade 9. This agrees with previous results that concluded that higher grades had the highest problem-solving skills, but the fastest skill development took place in grades 7–8 after which it dropped [ 3 , 4 ]. These results indicate that the mathematical problem-solving skills of the students should improve and be strengthened in grades 7–8, which will help them perform better in grade 9.

In this study, the effects of the demographic background of the students were analyzed in detail, which is an aspect missing from previous studies. The results showed that the mathematical problem-solving skills of urban students increased from grade 7 to grade 8 but decreased in grade 9. The same pattern was found among male and female students. However, a different pattern was observed for rural students, where the skills of grade 9 students continued to increase. The different patterns may be attributed to a structural reorganization of cognitive processes at a particular age [ 3 ]. However, more research is needed on the effects of the demographic backgrounds of students on mathematical problem-solving skills. These results were different from previous results because the previous studies only analyzed the development in general, without focusing on their demographic background. Hence, different patterns of development were observed when it was thoroughly examined.

Because solving problems is a cognitive process, the development of problem-solving skills for particular phases and processes needed to be analyzed. The students showed the same pattern for knowledge acquisition (i.e., exploring and understanding, and representing and formulating phases), with an increase in skill from grade 7 to grade 8 but a decrease in grade 9. However, the students showed increasing skill in knowledge application (i.e., planning and executing, as well as monitoring and reflecting phases) across grades. This means that the difference between the mean scores in grade 9 was not significant across phases. Grade 9 students had lower scores than grade 8 students for the knowledge acquisition phase but higher scores for the knowledge application phase. In contrast, the gap between the mean scores of grades 7 and 8 was large across phases.

These results proved that there is a significant difference in the mathematical problem-solving skills of students based on their demographic backgrounds. The urban students outperformed rural students, which confirms the results of previous studies [ 9 , 10 , 15 ]. The difference can be attributed to the availability of facilities, teacher quality, and interactive teaching and learning instruction [ 6 ]. In Indonesia, the policies for the public educational system for middle schools are set at the municipal level. This means that each city has its own policies for teacher training, teacher recruitment, teaching and learning processes, facilities, etc. Urban schools mostly have stricter policies as well as various programs to help students improve their knowledge and skills. In addition, they have supportive facilities for teaching and learning. This unequal environment is the strongest reason for the difference in mathematical problem-solving skills.

The results were analyzed in detail to observe which groups in the rural and urban schools contributed to the difference. Both male and female students in urban schools performed better than their counterparts in rural schools did. In addition, urban students in grades 7 and 8 outperformed their rural counterparts. There was no significant difference between urban and rural students in grade 9. This may be because grade 9 is the last grade in middle school, so students have to prepare for high school entrance requirements, including exam and/or grade point average scores. Hence, both rural and urban schools focus much effort on the teaching and learning process in this grade.

In this study, the female students surprisingly had better mathematical problem-solving skills than the male students did. This confirmed the results of the meta-analysis by Hyde et al. [ 32 ] and study by Anjum [ 36 ], which found that female students slightly outperformed male students in mathematics. This difference may be because of motivation and attitude [ 39 , 40 ]. Female Indonesian students are typically more diligent, thorough, responsible, persistent, and serious with their tasks.

A detailed analysis was performed to evaluate which group of students contributed to the gender differences. The results showed that female students outperformed male students in urban schools. This may be because male students at urban schools typically display an unserious attitude toward low-stake tests. In addition, female students outperformed their male counterparts in grades 7 and 9. The reason for this difference requires further analysis.

6. Conclusion

Studying the problem-solving skills of students is crucial to facilitating their development. In this study, the conclusions are presented as follows:

• • The mathematical problem-solving skills of the students were categorized as average. More students failed at higher phases of the problem-solving process.
• • Students showed development of their mathematical problem-solving skills from grade 7 to grade 8 but a decline in grade 9. The same pattern was detected across grades for urban students, both female and male. However, the problem-solving skills of rural students increased with the grade.
• • A similar development was observed for the individual problem-solving phases. In the knowledge acquisition phase, the problem-solving skills of the students developed from grade 7 to grade 8 but decreased in grade 9. However, problem-solving skills increased across grades in the knowledge application phase.
• • The school location was shown to have a significant effect on the mathematical problem-solving skills of the students. Urban students generally outperform students in rural schools. However, gender and grade contributed to differences in mathematical problem-solving skills based on school location. Female and male urban students in grades 7 and 8 outperformed their rural counterparts.
• • In general, female students outperformed male students in mathematical problem-solving skills, particularly those from urban schools and in grades 7 and 9.

The sampling method and the number of demographic backgrounds limited the scope of this study. Only students from A-accreditation schools were selected because higher-order problem-solving skills were considered assets. Moreover, the study only included three demographic factors: grade, gender, and school location. More demographic information, such as school type, can be added (public or private schools). Hence, future studies will need to broaden the sample size and consider more demographic factors. Despite these limitations, this study can help teachers determine the best period for enhancing the development of mathematical problem-solving skills. Moreover, the differences in mathematical problem-solving skills due to demographic background can be used as a basis for educational policymakers and teachers to provide equal opportunity and equitable education to students.

Author contribution statement

Ijtihadi Kamilia Amalina: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Tibor Vidákovich: Conceived and designed the experiments; Contributed reagents, materials, analysis tools or data.

Funding statement

This work was supported by University of Szeged Open Access Fund with the grant number of 6020.

Data availability statement

Additional information.

No additional information is available for this paper.

Declaration of competing interest

No potential conflict of interest was reported by the authors.

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Original research article, mathematical problem-solving through cooperative learning—the importance of peer acceptance and friendships.

• 1 Department of Education, Uppsala University, Uppsala, Sweden
• 2 Department of Education, Culture and Communication, Malardalen University, Vasteras, Sweden
• 3 School of Natural Sciences, Technology and Environmental Studies, Sodertorn University, Huddinge, Sweden
• 4 Faculty of Education, Gothenburg University, Gothenburg, Sweden

Mathematical problem-solving constitutes an important area of mathematics instruction, and there is a need for research on instructional approaches supporting student learning in this area. This study aims to contribute to previous research by studying the effects of an instructional approach of cooperative learning on students’ mathematical problem-solving in heterogeneous classrooms in grade five, in which students with special needs are educated alongside with their peers. The intervention combined a cooperative learning approach with instruction in problem-solving strategies including mathematical models of multiplication/division, proportionality, and geometry. The teachers in the experimental group received training in cooperative learning and mathematical problem-solving, and implemented the intervention for 15 weeks. The teachers in the control group received training in mathematical problem-solving and provided instruction as they would usually. Students (269 in the intervention and 312 in the control group) participated in tests of mathematical problem-solving in the areas of multiplication/division, proportionality, and geometry before and after the intervention. The results revealed significant effects of the intervention on student performance in overall problem-solving and problem-solving in geometry. The students who received higher scores on social acceptance and friendships for the pre-test also received higher scores on the selected tests of mathematical problem-solving. Thus, the cooperative learning approach may lead to gains in mathematical problem-solving in heterogeneous classrooms, but social acceptance and friendships may also greatly impact students’ results.

Introduction

The research on instruction in mathematical problem-solving has progressed considerably during recent decades. Yet, there is still a need to advance our knowledge on how teachers can support their students in carrying out this complex activity ( Lester and Cai, 2016 ). Results from the Program for International Student Assessment (PISA) show that only 53% of students from the participating countries could solve problems requiring more than direct inference and using representations from different information sources ( OECD, 2019 ). In addition, OECD (2019) reported a large variation in achievement with regard to students’ diverse backgrounds. Thus, there is a need for instructional approaches to promote students’ problem-solving in mathematics, especially in heterogeneous classrooms in which students with diverse backgrounds and needs are educated together. Small group instructional approaches have been suggested as important to promote learning of low-achieving students and students with special needs ( Kunsch et al., 2007 ). One such approach is cooperative learning (CL), which involves structured collaboration in heterogeneous groups, guided by five principles to enhance group cohesion ( Johnson et al., 1993 ; Johnson et al., 2009 ; Gillies, 2016 ). While CL has been well-researched in whole classroom approaches ( Capar and Tarim, 2015 ), few studies of the approach exist with regard to students with special educational needs (SEN; McMaster and Fuchs, 2002 ). This study contributes to previous research by studying the effects of the CL approach on students’ mathematical problem-solving in heterogeneous classrooms, in which students with special needs are educated alongside with their peers.

Group collaboration through the CL approach is structured in accordance with five principles of collaboration: positive interdependence, individual accountability, explicit instruction in social skills, promotive interaction, and group processing ( Johnson et al., 1993 ). First, the group tasks need to be structured so that all group members feel dependent on each other in the completion of the task, thus promoting positive interdependence. Second, for individual accountability, the teacher needs to assure that each group member feels responsible for his or her share of work, by providing opportunities for individual reports or evaluations. Third, the students need explicit instruction in social skills that are necessary for collaboration. Fourth, the tasks and seat arrangements should be designed to promote interaction among group members. Fifth, time needs to be allocated to group processing, through which group members can evaluate their collaborative work to plan future actions. Using these principles for cooperation leads to gains in mathematics, according to Capar and Tarim (2015) , who conducted a meta-analysis on studies of cooperative learning and mathematics, and found an increase of .59 on students’ mathematics achievement scores in general. However, the number of reviewed studies was limited, and researchers suggested a need for more research. In the current study, we focused on the effect of CL approach in a specific area of mathematics: problem-solving.

Mathematical problem-solving is a central area of mathematics instruction, constituting an important part of preparing students to function in modern society ( Gravemeijer et al., 2017 ). In fact, problem-solving instruction creates opportunities for students to apply their knowledge of mathematical concepts, integrate and connect isolated pieces of mathematical knowledge, and attain a deeper conceptual understanding of mathematics as a subject ( Lester and Cai, 2016 ). Some researchers suggest that mathematics itself is a science of problem-solving and of developing theories and methods for problem-solving ( Hamilton, 2007 ; Davydov, 2008 ).

Problem-solving processes have been studied from different perspectives ( Lesh and Zawojewski, 2007 ). Problem-solving heuristics Pólya, (1948) has largely influenced our perceptions of problem-solving, including four principles: understanding the problem, devising a plan, carrying out the plan, and looking back and reflecting upon the suggested solution. Schoenfield, (2016) suggested the use of specific problem-solving strategies for different types of problems, which take into consideration metacognitive processes and students’ beliefs about problem-solving. Further, models and modelling perspectives on mathematics ( Lesh and Doerr, 2003 ; Lesh and Zawojewski, 2007 ) emphasize the importance of engaging students in model-eliciting activities in which problem situations are interpreted mathematically, as students make connections between problem information and knowledge of mathematical operations, patterns, and rules ( Mousoulides et al., 2010 ; Stohlmann and Albarracín, 2016 ).

Not all students, however, find it easy to solve complex mathematical problems. Students may experience difficulties in identifying solution-relevant elements in a problem or visualizing appropriate solution to a problem situation. Furthermore, students may need help recognizing the underlying model in problems. For example, in two studies by Degrande et al. (2016) , students in grades four to six were presented with mathematical problems in the context of proportional reasoning. The authors found that the students, when presented with a word problem, could not identify an underlying model, but rather focused on superficial characteristics of the problem. Although the students in the study showed more success when presented with a problem formulated in symbols, the authors pointed out a need for activities that help students distinguish between different proportional problem types. Furthermore, students exhibiting specific learning difficulties may need additional support in both general problem-solving strategies ( Lein et al., 2020 ; Montague et al., 2014 ) and specific strategies pertaining to underlying models in problems. The CL intervention in the present study focused on supporting students in problem-solving, through instruction in problem-solving principles ( Pólya, 1948 ), specifically applied to three models of mathematical problem-solving—multiplication/division, geometry, and proportionality.

Students’ problem-solving may be enhanced through participation in small group discussions. In a small group setting, all the students have the opportunity to explain their solutions, clarify their thinking, and enhance understanding of a problem at hand ( Yackel et al., 1991 ; Webb and Mastergeorge, 2003 ). In fact, small group instruction promotes students’ learning in mathematics by providing students with opportunities to use language for reasoning and conceptual understanding ( Mercer and Sams, 2006 ), to exchange different representations of the problem at hand ( Fujita et al., 2019 ), and to become aware of and understand groupmates’ perspectives in thinking ( Kazak et al., 2015 ). These opportunities for learning are created through dialogic spaces characterized by openness to each other’s perspectives and solutions to mathematical problems ( Wegerif, 2011 ).

However, group collaboration is not only associated with positive experiences. In fact, studies show that some students may not be given equal opportunities to voice their opinions, due to academic status differences ( Langer-Osuna, 2016 ). Indeed, problem-solvers struggling with complex tasks may experience negative emotions, leading to uncertainty of not knowing the definite answer, which places demands on peer support ( Jordan and McDaniel, 2014 ; Hannula, 2015 ). Thus, especially in heterogeneous groups, students may need additional support to promote group interaction. Therefore, in this study, we used a cooperative learning approach, which, in contrast to collaborative learning approaches, puts greater focus on supporting group cohesion through instruction in social skills and time for reflection on group work ( Davidson and Major, 2014 ).

Although cooperative learning approach is intended to promote cohesion and peer acceptance in heterogeneous groups ( Rzoska and Ward, 1991 ), previous studies indicate that challenges in group dynamics may lead to unequal participation ( Mulryan, 1992 ; Cohen, 1994 ). Peer-learning behaviours may impact students’ problem-solving ( Hwang and Hu, 2013 ) and working in groups with peers who are seen as friends may enhance students’ motivation to learn mathematics ( Deacon and Edwards, 2012 ). With the importance of peer support in mind, this study set out to investigate whether the results of the intervention using the CL approach are associated with students’ peer acceptance and friendships.

The Present Study

In previous research, the CL approach has shown to be a promising approach in teaching and learning mathematics ( Capar and Tarim, 2015 ), but fewer studies have been conducted in whole-class approaches in general and students with SEN in particular ( McMaster and Fuchs, 2002 ). This study aims to contribute to previous research by investigating the effect of CL intervention on students’ mathematical problem-solving in grade 5. With regard to the complexity of mathematical problem-solving ( Lesh and Zawojewski, 2007 ; Degrande et al., 2016 ; Stohlmann and Albarracín, 2016 ), the CL approach in this study was combined with problem-solving principles pertaining to three underlying models of problem-solving—multiplication/division, geometry, and proportionality. Furthermore, considering the importance of peer support in problem-solving in small groups ( Mulryan, 1992 ; Cohen, 1994 ; Hwang and Hu, 2013 ), the study investigated how peer acceptance and friendships were associated with the effect of the CL approach on students’ problem-solving abilities. The study aimed to find answers to the following research questions:

a) What is the effect of CL approach on students’ problem-solving in mathematics?

b) Are social acceptance and friendship associated with the effect of CL on students’ problem-solving in mathematics?

Participants

The participants were 958 students in grade 5 and their teachers. According to power analyses prior to the start of the study, 1,020 students and 51 classes were required, with an expected effect size of 0.30 and power of 80%, provided that there are 20 students per class and intraclass correlation is 0.10. An invitation to participate in the project was sent to teachers in five municipalities via e-mail. Furthermore, the information was posted on the website of Uppsala university and distributed via Facebook interest groups. As shown in Figure 1 , teachers of 1,165 students agreed to participate in the study, but informed consent was obtained only for 958 students (463 in the intervention and 495 in the control group). Further attrition occurred at pre- and post-measurement, resulting in 581 students’ tests as a basis for analyses (269 in the intervention and 312 in the control group). Fewer students (n = 493) were finally included in the analyses of the association of students’ social acceptance and friendships and the effect of CL on students’ mathematical problem-solving (219 in the intervention and 274 in the control group). The reasons for attrition included teacher drop out due to sick leave or personal circumstances (two teachers in the control group and five teachers in the intervention group). Furthermore, some students were sick on the day of data collection and some teachers did not send the test results to the researchers.

FIGURE 1 . Flow chart for participants included in data collection and data analysis.

As seen in Table 1 , classes in both intervention and control groups included 27 students on average. For 75% of the classes, there were 33–36% of students with SEN. In Sweden, no formal medical diagnosis is required for the identification of students with SEN. It is teachers and school welfare teams who decide students’ need for extra adaptations or special support ( Swedish National Educational Agency, 2014 ). The information on individual students’ type of SEN could not be obtained due to regulations on the protection of information about individuals ( SFS 2009 ). Therefore, the information on the number of students with SEN on class level was obtained through teacher reports.

TABLE 1 . Background characteristics of classes and teachers in intervention and control groups.

Intervention

The intervention using the CL approach lasted for 15 weeks and the teachers worked with the CL approach three to four lessons per week. First, the teachers participated in two-days training on the CL approach, using an especially elaborated CL manual ( Klang et al., 2018 ). The training focused on the five principles of the CL approach (positive interdependence, individual accountability, explicit instruction in social skills, promotive interaction, and group processing). Following the training, the teachers introduced the CL approach in their classes and focused on group-building activities for 7 weeks. Then, 2 days of training were provided to teachers, in which the CL approach was embedded in activities in mathematical problem-solving and reading comprehension. Educational materials containing mathematical problems in the areas of multiplication and division, geometry, and proportionality were distributed to the teachers ( Karlsson and Kilborn, 2018a ). In addition to the specific problems, adapted for the CL approach, the educational materials contained guidance for the teachers, in which problem-solving principles ( Pólya, 1948 ) were presented as steps in problem-solving. Following the training, the teachers applied the CL approach in mathematical problem-solving lessons for 8 weeks.

Solving a problem is a matter of goal-oriented reasoning, starting from the understanding of the problem to devising its solution by using known mathematical models. This presupposes that the current problem is chosen from a known context ( Stillman et al., 2008 ; Zawojewski, 2010 ). This differs from the problem-solving of the textbooks, which is based on an aim to train already known formulas and procedures ( Hamilton, 2007 ). Moreover, it is important that students learn modelling according to their current abilities and conditions ( Russel, 1991 ).

In order to create similar conditions in the experiment group and the control group, the teachers were supposed to use the same educational material ( Karlsson and Kilborn, 2018a ; Karlsson and Kilborn, 2018b ), written in light of the specified view of problem-solving. The educational material is divided into three areas—multiplication/division, geometry, and proportionality—and begins with a short teachers’ guide, where a view of problem solving is presented, which is based on the work of Polya (1948) and Lester and Cai (2016) . The tasks are constructed in such a way that conceptual knowledge was in focus, not formulas and procedural knowledge.

Implementation of the Intervention

To ensure the implementation of the intervention, the researchers visited each teachers’ classroom twice during the two phases of the intervention period, as described above. During each visit, the researchers observed the lesson, using a checklist comprising the five principles of the CL approach. After the lesson, the researchers gave written and oral feedback to each teacher. As seen in Table 1 , in 18 of the 23 classes, the teachers implemented the intervention in accordance with the principles of CL. In addition, the teachers were asked to report on the use of the CL approach in their teaching and the use of problem-solving activities embedding CL during the intervention period. As shown in Table 1 , teachers in only 11 of 23 classes reported using the CL approach and problem-solving activities embedded in the CL approach at least once a week.

Control Group

The teachers in the control group received 2 days of instruction in enhancing students’ problem-solving and reading comprehension. The teachers were also supported with educational materials including mathematical problems Karlsson and Kilborn (2018b) and problem-solving principles ( Pólya, 1948 ). However, none of the activities during training or in educational materials included the CL approach. As seen in Table 1 , only 10 of 25 teachers reported devoting at least one lesson per week to mathematical problem-solving.

Tests of Mathematical Problem-Solving

Tests of mathematical problem-solving were administered before and after the intervention, which lasted for 15 weeks. The tests were focused on the models of multiplication/division, geometry, and proportionality. The three models were chosen based on the syllabus of the subject of mathematics in grades 4 to 6 in the Swedish National Curriculum ( Swedish National Educational Agency, 2018 ). In addition, the intention was to create a variation of types of problems to solve. For each of these three models, there were two tests, a pre-test and a post-test. Each test contained three tasks with increasing difficulty ( Supplementary Appendix SA ).

The tests of multiplication and division (Ma1) were chosen from different contexts and began with a one-step problem, while the following two tasks were multi-step problems. Concerning multiplication, many students in grade 5 still understand multiplication as repeated addition, causing significant problems, as this conception is not applicable to multiplication beyond natural numbers ( Verschaffel et al., 2007 ). This might be a hindrance in developing multiplicative reasoning ( Barmby et al., 2009 ). The multi-step problems in this study were constructed to support the students in multiplicative reasoning.

Concerning the geometry tests (Ma2), it was important to consider a paradigm shift concerning geometry in education that occurred in the mid-20th century, when strict Euclidean geometry gave way to other aspects of geometry like symmetry, transformation, and patterns. van Hiele (1986) prepared a new taxonomy for geometry in five steps, from a visual to a logical level. Therefore, in the tests there was a focus on properties of quadrangles and triangles, and how to determine areas by reorganising figures into new patterns. This means that structure was more important than formulas.

The construction of tests of proportionality (M3) was more complicated. Firstly, tasks on proportionality can be found in many different contexts, such as prescriptions, scales, speeds, discounts, interest, etc. Secondly, the mathematical model is complex and requires good knowledge of rational numbers and ratios ( Lesh et al., 1988 ). It also requires a developed view of multiplication, useful in operations with real numbers, not only as repeated addition, an operation limited to natural numbers ( Lybeck, 1981 ; Degrande et al., 2016 ). A linear structure of multiplication as repeated addition leads to limitations in terms of generalization and development of the concept of multiplication. This became evident in a study carried out in a Swedish context ( Karlsson and Kilborn, 2018c ). Proportionality can be expressed as a/b = c/d or as a/b = k. The latter can also be expressed as a = b∙k, where k is a constant that determines the relationship between a and b. Common examples of k are speed (km/h), scale, and interest (%). An important pre-knowledge in order to deal with proportions is to master fractions as equivalence classes like 1/3 = 2/6 = 3/9 = 4/12 = 5/15 = 6/18 = 7/21 = 8/24 … ( Karlsson and Kilborn, 2020 ). It was important to take all these aspects into account when constructing and assessing the solutions of the tasks.

The tests were graded by an experienced teacher of mathematics (4 th author) and two students in their final year of teacher training. Prior to grading, acceptable levels of inter-rater reliability were achieved by independent rating of students’ solutions and discussions in which differences between the graders were resolved. Each student response was to be assigned one point when it contained a correct answer and two points when the student provided argumentation for the correct answer and elaborated on explanation of his or her solution. The assessment was thus based on quality aspects with a focus on conceptual knowledge. As each subtest contained three questions, it generated three student solutions. So, scores for each subtest ranged from 0 to 6 points and for the total scores from 0 to 18 points. To ascertain that pre- and post-tests were equivalent in degree of difficulty, the tests were administered to an additional sample of 169 students in grade 5. Test for each model was conducted separately, as students participated in pre- and post-test for each model during the same lesson. The order of tests was switched for half of the students in order to avoid the effect of the order in which the pre- and post-tests were presented. Correlation between students’ performance on pre- and post-test was .39 ( p < 0.000) for tests of multiplication/division; .48 ( p < 0.000) for tests of geometry; and .56 ( p < 0.000) for tests of proportionality. Thus, the degree of difficulty may have differed between pre- and post-test.

Measures of Peer Acceptance and Friendships

To investigate students’ peer acceptance and friendships, peer nominations rated pre- and post-intervention were used. Students were asked to nominate peers who they preferred to work in groups with and who they preferred to be friends with. Negative peer nominations were avoided due to ethical considerations raised by teachers and parents ( Child and Nind, 2013 ). Unlimited nominations were used, as these are considered to have high ecological validity ( Cillessen and Marks, 2017 ). Peer nominations were used as a measure of social acceptance, and reciprocated nominations were used as a measure of friendship. The number of nominations for each student were aggregated and divided by the number of nominators to create a proportion of nominations for each student ( Velásquez et al., 2013 ).

Statistical Analyses

Multilevel regression analyses were conducted in R, lme4 package Bates et al. (2015) to account for nestedness in the data. Students’ classroom belonging was considered as a level 2 variable. First, we used a model in which students’ results on tests of problem-solving were studied as a function of time (pre- and post) and group belonging (intervention and control group). Second, the same model was applied to subgroups of students who performed above and below median at pre-test, to explore whether the CL intervention had a differential effect on student performance. In this second model, the results for subgroups of students could not be obtained for geometry tests for subgroup below median and for tests of proportionality for subgroup above median. A possible reason for this must have been the skewed distribution of the students in these subgroups. Therefore, another model was applied that investigated students’ performances in math at both pre- and post-test as a function of group belonging. Third, the students’ scores on social acceptance and friendships were added as an interaction term to the first model. In our previous study, students’ social acceptance changed as a result of the same CL intervention ( Klang et al., 2020 ).

The assumptions for the multilevel regression were assured during the analyses ( Snijders and Bosker, 2012 ). The assumption of normality of residuals were met, as controlled by visual inspection of quantile-quantile plots. For subgroups, however, the plotted residuals deviated somewhat from the straight line. The number of outliers, which had a studentized residual value greater than ±3, varied from 0 to 5, but none of the outliers had a Cook’s distance value larger than 1. The assumption of multicollinearity was met, as the variance inflation factors (VIF) did not exceed a value of 10. Before the analyses, the cases with missing data were deleted listwise.

What Is the Effect of the CL Approach on Students’ Problem-Solving in Mathematics?

As seen in the regression coefficients in Table 2 , the CL intervention had a significant effect on students’ mathematical problem-solving total scores and students’ scores in problem solving in geometry (Ma2). Judging by mean values, students in the intervention group appeared to have low scores on problem-solving in geometry but reached the levels of problem-solving of the control group by the end of the intervention. The intervention did not have a significant effect on students’ performance in problem-solving related to models of multiplication/division and proportionality.

TABLE 2 . Mean scores (standard deviation in parentheses) and unstandardized multilevel regression estimates for tests of mathematical problem-solving.

The question is, however, whether CL intervention affected students with different pre-test scores differently. Table 2 includes the regression coefficients for subgroups of students who performed below and above median at pre-test. As seen in the table, the CL approach did not have a significant effect on students’ problem-solving, when the sample was divided into these subgroups. A small negative effect was found for intervention group in comparison to control group, but confidence intervals (CI) for the effect indicate that it was not significant.

Is Social Acceptance and Friendships Associated With the Effect of CL on Students’ Problem-Solving in Mathematics?

As seen in Table 3 , students’ peer acceptance and friendship at pre-test were significantly associated with the effect of the CL approach on students’ mathematical problem-solving scores. Changes in students’ peer acceptance and friendships were not significantly associated with the effect of the CL approach on students’ mathematical problem-solving. Consequently, it can be concluded that being nominated by one’s peers and having friends at the start of the intervention may be an important factor when participation in group work, structured in accordance with the CL approach, leads to gains in mathematical problem-solving.

TABLE 3 . Mean scores (standard deviation in parentheses) and unstandardized multilevel regression estimates for tests of mathematical problem-solving, including scores of social acceptance and friendship in the model.

In light of the limited number of studies on the effects of CL on students’ problem-solving in whole classrooms ( Capar and Tarim, 2015 ), and for students with SEN in particular ( McMaster and Fuchs, 2002 ), this study sought to investigate whether the CL approach embedded in problem-solving activities has an effect on students’ problem-solving in heterogeneous classrooms. The need for the study was justified by the challenge of providing equitable mathematics instruction to heterogeneous student populations ( OECD, 2019 ). Small group instructional approaches as CL are considered as promising approaches in this regard ( Kunsch et al., 2007 ). The results showed a significant effect of the CL approach on students’ problem-solving in geometry and total problem-solving scores. In addition, with regard to the importance of peer support in problem-solving ( Deacon and Edwards, 2012 ; Hwang and Hu, 2013 ), the study explored whether the effect of CL on students’ problem-solving was associated with students’ social acceptance and friendships. The results showed that students’ peer acceptance and friendships at pre-test were significantly associated with the effect of the CL approach, while change in students’ peer acceptance and friendships from pre- to post-test was not.

The results of the study confirm previous research on the effect of the CL approach on students’ mathematical achievement ( Capar and Tarim, 2015 ). The specific contribution of the study is that it was conducted in classrooms, 75% of which were composed of 33–36% of students with SEN. Thus, while a previous review revealed inconclusive findings on the effects of CL on student achievement ( McMaster and Fuchs, 2002 ), the current study adds to the evidence of the effect of the CL approach in heterogeneous classrooms, in which students with special needs are educated alongside with their peers. In a small group setting, the students have opportunities to discuss their ideas of solutions to the problem at hand, providing explanations and clarifications, thus enhancing their understanding of problem-solving ( Yackel et al., 1991 ; Webb and Mastergeorge, 2003 ).

In this study, in accordance with previous research on mathematical problem-solving ( Lesh and Zawojewski, 2007 ; Degrande et al., 2016 ; Stohlmann and Albarracín, 2016 ), the CL approach was combined with training in problem-solving principles Pólya (1948) and educational materials, providing support in instruction in underlying mathematical models. The intention of the study was to provide evidence for the effectiveness of the CL approach above instruction in problem-solving, as problem-solving materials were accessible to teachers of both the intervention and control groups. However, due to implementation challenges, not all teachers in the intervention and control groups reported using educational materials and training as expected. Thus, it is not possible to draw conclusions of the effectiveness of the CL approach alone. However, in everyday classroom instruction it may be difficult to separate the content of instruction from the activities that are used to mediate this content ( Doerr and Tripp, 1999 ; Gravemeijer, 1999 ).

Furthermore, for successful instruction in mathematical problem-solving, scaffolding for content needs to be combined with scaffolding for dialogue ( Kazak et al., 2015 ). From a dialogical perspective ( Wegerif, 2011 ), students may need scaffolding in new ways of thinking, involving questioning their understandings and providing arguments for their solutions, in order to create dialogic spaces in which different solutions are voiced and negotiated. In this study, small group instruction through CL approach aimed to support discussions in small groups, but the study relies solely on quantitative measures of students’ mathematical performance. Video-recordings of students’ discussions may have yielded important insights into the dialogic relationships that arose in group discussions.

Despite the positive findings of the CL approach on students’ problem-solving, it is important to note that the intervention did not have an effect on students’ problem-solving pertaining to models of multiplication/division and proportionality. Although CL is assumed to be a promising instructional approach, the number of studies on its effect on students’ mathematical achievement is still limited ( Capar and Tarim, 2015 ). Thus, further research is needed on how CL intervention can be designed to promote students’ problem-solving in other areas of mathematics.

The results of this study show that the effect of the CL intervention on students’ problem-solving was associated with students’ initial scores of social acceptance and friendships. Thus, it is possible to assume that students who were popular among their classmates and had friends at the start of the intervention also made greater gains in mathematical problem-solving as a result of the CL intervention. This finding is in line with Deacon and Edwards’ study of the importance of friendships for students’ motivation to learn mathematics in small groups ( Deacon and Edwards, 2012 ). However, the effect of the CL intervention was not associated with change in students’ social acceptance and friendship scores. These results indicate that students who were nominated by a greater number of students and who received a greater number of friends did not benefit to a great extent from the CL intervention. With regard to previously reported inequalities in cooperation in heterogeneous groups ( Cohen, 1994 ; Mulryan, 1992 ; Langer Osuna, 2016 ) and the importance of peer behaviours for problem-solving ( Hwang and Hu, 2013 ), teachers should consider creating inclusive norms and supportive peer relationships when using the CL approach. The demands of solving complex problems may create negative emotions and uncertainty ( Hannula, 2015 ; Jordan and McDaniel, 2014 ), and peer support may be essential in such situations.

Limitations

The conclusions from the study must be interpreted with caution, due to a number of limitations. First, due to the regulation of protection of individuals ( SFS 2009 ), the researchers could not get information on type of SEN for individual students, which limited the possibilities of the study for investigating the effects of the CL approach for these students. Second, not all teachers in the intervention group implemented the CL approach embedded in problem-solving activities and not all teachers in the control group reported using educational materials on problem-solving. The insufficient levels of implementation pose a significant challenge to the internal validity of the study. Third, the additional investigation to explore the equivalence in difficulty between pre- and post-test, including 169 students, revealed weak to moderate correlation in students’ performance scores, which may indicate challenges to the internal validity of the study.

Implications

The results of the study have some implications for practice. Based on the results of the significant effect of the CL intervention on students’ problem-solving, the CL approach appears to be a promising instructional approach in promoting students’ problem-solving. However, as the results of the CL approach were not significant for all subtests of problem-solving, and due to insufficient levels of implementation, it is not possible to conclude on the importance of the CL intervention for students’ problem-solving. Furthermore, it appears to be important to create opportunities for peer contacts and friendships when the CL approach is used in mathematical problem-solving activities.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Ethics Statement

The studies involving human participants were reviewed and approved by the Uppsala Ethical Regional Committee, Dnr. 2017/372. Written informed consent to participate in this study was provided by the participants’ legal guardian/next of kin.

Author Contributions

NiK was responsible for the project, and participated in data collection and data analyses. NaK and WK were responsible for intervention with special focus on the educational materials and tests in mathematical problem-solving. PE participated in the planning of the study and the data analyses, including coordinating analyses of students’ tests. MK participated in the designing and planning the study as well as data collection and data analyses.

The project was funded by the Swedish Research Council under Grant 2016-04,679.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Acknowledgments

We would like to express our gratitude to teachers who participated in the project.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/feduc.2021.710296/full#supplementary-material

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Keywords: cooperative learning, mathematical problem-solving, intervention, heterogeneous classrooms, hierarchical linear regression analysis

Citation: Klang N, Karlsson N, Kilborn W, Eriksson P and Karlberg M (2021) Mathematical Problem-Solving Through Cooperative Learning—The Importance of Peer Acceptance and Friendships. Front. Educ. 6:710296. doi: 10.3389/feduc.2021.710296

Received: 15 May 2021; Accepted: 09 August 2021; Published: 24 August 2021.

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Copyright © 2021 Klang, Karlsson, Kilborn, Eriksson and Karlberg. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Nina Klang, [email protected]

Problem-Solving Skills, Memory Power, and Early Childhood Mathematics: Understanding the Significance of the Early Childhood Mathematics in an Individual’s Life

• Published: 09 January 2024

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• Wensheng Chen 1  

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Early childhood mathematics is the process of teaching mathematics and learning mathematics in a playful manner to the interests and questions of children. Early childhood mathematics includes counting and the application of counting, which includes mathematical activities such as sorting, matching, and pattern recognition. Deeper understanding of basic math concepts that include number sense, shapes and spatial awareness, measurement, patterns, and basic operations such as addition and subtraction. This study aims to investigate the relationship between children’s mathematical knowledge and skills developed in early stages and their later achievements in mathematics. Research data were collected from 53 children’s parents and 36 experts through telephonic interview questionnaire methods. The data collected was analyzed using SPSS software. The validity and reliability of the variables in the questionnaire have been tested. The results of this study reveal a significant positive correlation between children’s early mathematical knowledge and skills and later achievement in mathematics. Specifically, children who demonstrated higher levels of mathematical competencies during their early years exhibited greater success in later academic performance in mathematics. Additionally, the study identified several factors that influence this relationship, including parental involvement, socioeconomic status, and quality of early mathematics instruction. The novelty of this study is highlighting the importance of early math development; it leads to an understanding of educational practices that improve children’s math learning and promote future success in math. Based on the study findings, it is suggested that educators and policymakers emphasize the importance of early mathematical education and provide targeted interventions and support to enhance children’s mathematical knowledge and skills during their formative years.

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Chen, W. Problem-Solving Skills, Memory Power, and Early Childhood Mathematics: Understanding the Significance of the Early Childhood Mathematics in an Individual’s Life. J Knowl Econ (2024). https://doi.org/10.1007/s13132-023-01557-6

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Mathematics as a Complex Problem-Solving Activity

By jacob klerlein and sheena hervey, generation ready.

By the time young children enter school they are already well along the pathway to becoming problem solvers. From birth, children are learning how to learn: they respond to their environment and the reactions of others. This making sense of experience is an ongoing, recursive process. We have known for a long time that reading is a complex problem-solving activity. More recently, teachers have come to understand that becoming mathematically literate is also a complex problem-solving activity that increases in power and flexibility when practiced more often. A problem in mathematics is any situation that must be resolved using mathematical tools but for which there is no immediately obvious strategy. If the way forward is obvious, it’s not a problem—it is a straightforward application.

Mathematicians have always understood that problem-solving is central to their discipline because without a problem there is no mathematics. Problem-solving has played a central role in the thinking of educational theorists ever since the publication of Pólya’s book “How to Solve It,” in 1945. The National Council of Teachers of Mathematics (NCTM) has been consistently advocating for problem-solving for nearly 40 years, while international trends in mathematics teaching have shown an increased focus on problem-solving and mathematical modeling beginning in the early 1990s. As educators internationally became increasingly aware that providing problem-solving experiences is critical if students are to be able to use and apply mathematical knowledge in meaningful ways (Wu and Zhang 2006) little changed at the school level in the United States.

“Problem-solving is not only a goal of learning mathematics, but also a major means of doing so.”

(NCTM, 2000, p. 52)

In 2011 the Common Core State Standards incorporated the NCTM Process Standards of problem-solving, reasoning and proof, communication, representation, and connections into the Standards for Mathematical Practice. For many teachers of mathematics this was the first time they had been expected to incorporate student collaboration and discourse with problem-solving. This practice requires teaching in profoundly different ways as schools moved from a teacher-directed to a more dialogic approach to teaching and learning. The challenge for teachers is to teach students not only to solve problems but also to learn about mathematics through problem-solving. While many students may develop procedural fluency, they often lack the deep conceptual understanding necessary to solve new problems or make connections between mathematical ideas.

“A problem-solving curriculum, however, requires a different role from the teacher. Rather than directing a lesson, the teacher needs to provide time for students to grapple with problems, search for strategies and solutions on their own, and learn to evaluate their own results. Although the teacher needs to be very much present, the primary focus in the class needs to be on the students’ thinking processes.”

(Burns, 2000, p. 29)

Learning to problem solve

To understand how students become problem solvers we need to look at the theories that underpin learning in mathematics. These include recognition of the developmental aspects of learning and the essential fact that students actively engage in learning mathematics through “doing, talking, reflecting, discussing, observing, investigating, listening, and reasoning” (Copley, 2000, p. 29). The concept of co-construction of learning is the basis for the theory. Moreover, we know that each student is on their unique path of development.

Beliefs underpinning effective teaching of mathematics

• Every student’s identity, language, and culture need to be respected and valued.
• Every student has the right to access effective mathematics education.
• Every student can become a successful learner of mathematics.

Children arrive at school with intuitive mathematical understandings. A teacher needs to connect with and build on those understandings through experiences that allow students to explore mathematics and to communicate their ideas in a meaningful dialogue with the teacher and their peers.

Learning takes place within social settings (Vygotsky, 1978). Students construct understandings through engagement with problems and interaction with others in these activities. Through these social interactions, students feel that they can take risks, try new strategies, and give and receive feedback. They learn cooperatively as they share a range of points of view or discuss ways of solving a problem. It is through talking about problems and discussing their ideas that children construct knowledge and acquire the language to make sense of experiences.

Students acquire their understanding of mathematics and develop problem-solving skills as a result of solving problems, rather than being taught something directly (Hiebert1997). The teacher’s role is to construct problems and present situations that provide a forum in which problem-solving can occur.

Why is problem-solving important?

Our students live in an information and technology-based society where they need to be able to think critically about complex issues, and “analyze and think logically about new situations, devise unspecified solution procedures, and communicate their solution clearly and convincingly to others” (Baroody, 1998). Mathematics education is important not only because of the “gatekeeping role that mathematics plays in students’ access to educational and economic opportunities,” but also because the problem-solving processes and the acquisition of problem-solving strategies equips students for life beyond school (Cobb, & Hodge, 2002).

The importance of problem-solving in learning mathematics comes from the belief that mathematics is primarily about reasoning, not memorization. Problem-solving allows students to develop understanding and explain the processes used to arrive at solutions, rather than remembering and applying a set of procedures. It is through problem-solving that students develop a deeper understanding of mathematical concepts, become more engaged, and appreciate the relevance and usefulness of mathematics (Wu and Zhang 2006). Problem-solving in mathematics supports the development of:

• The ability to think creatively, critically, and logically
• The ability to structure and organize
• The ability to process information
• Enjoyment of an intellectual challenge
• The skills to solve problems that help them to investigate and understand the world

Problem-solving should underlie all aspects of mathematics teaching in order to give students the experience of the power of mathematics in the world around them. This method allows students to see problem-solving as a vehicle to construct, evaluate, and refine their theories about mathematics and the theories of others.

Problems that are “Problematic”

The teacher’s expectations of the students are essential. Students only learn to handle complex problems by being exposed to them. Students need to have opportunities to work on complex tasks rather than a series of simple tasks devolved from a complex task. This is important for stimulating the students’ mathematical reasoning and building durable mathematical knowledge (Anthony and Walshaw, 2007). The challenge for teachers is ensuring the problems they set are designed to support mathematics learning and are appropriate and challenging for all students.  The problems need to be difficult enough to provide a challenge but not so difficult that students can’t succeed. Teachers who get this right create resilient problem solvers who know that with perseverance they can succeed. Problems need to be within the students’ “Zone of Proximal Development” (Vygotsky 1968). These types of complex problems will provide opportunities for discussion and learning.

Students will have opportunities to explain their ideas, respond to the ideas of others, and challenge their thinking. Those students who think math is all about the “correct” answer will need support and encouragement to take risks. Tolerance of difficulty is essential in a problem-solving disposition because being “stuck” is an inevitable stage in resolving just about any problem. Getting unstuck typically takes time and involves trying a variety of approaches. Students need to learn this experientially. Effective problems:

• Are accessible and extendable
• Allow individuals to make decisions
• Promote discussion and communication
• Encourage originality and invention
• Encourage “what if?” and “what if not?” questions
• Contain an element of surprise (Adapted from Ahmed, 1987)

“Students learn to problem solve in mathematics primarily through ‘doing, talking, reflecting, discussing, observing, investigating, listening, and reasoning.”

(Copley, 2000, p. 29)

“…as learners investigate together. It becomes a mini- society – a community of learners engaged in mathematical activity, discourse and reflection. Learners must be given the opportunity to act as mathematicians by allowing, supporting and challenging their ‘mathematizing’ of particular situations. The community provides an environment in which individual mathematical ideas can be expressed and tested against others’ ideas.…This enables learners to become clearer and more confident about what they know and understand.”

(Fosnot, 2005, p. 10)

Research shows that ‘classrooms where the orientation consistently defines task outcomes in terms of the answers rather than the thinking processes entailed in reaching the answers negatively affects the thinking processes and mathematical identities of learners’ (Anthony and Walshaw, 2007, page 122).

Effective teachers model good problem-solving habits for their students. Their questions are designed to help children use a variety of strategies and materials to solve problems. Students often want to begin without a plan in mind. Through appropriate questions, the teacher gives students some structure for beginning the problem without telling them exactly what to do. In 1945 Pólya published the following four principles of problem-solving to support teachers with helping their students.

• Understand and explore the problem
• Find a strategy
• Use the strategy to solve the problem
• Look back and reflect on the solution

Problem-solving is not linear but rather a complex, interactive process. Students move backward and forward between and across Pólya’s phases. The Common Core State Standards describe the process as follows:

“Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary”. (New York State Next Generation Mathematics Learning Standards 2017).

Pólya’s Principals of Problem-Solving

Students move forward and backward as they move through the problem-solving process.

The goal is for students to have a range of strategies they use to solve problems and understand that there may be more than one solution. It is important to realize that the process is just as important, if not more important, than arriving at a solution, for it is in the solution process that students uncover the mathematics. Arriving at an answer isn’t the end of the process. Reflecting on the strategies used to solve the problem provides additional learning experiences. Studying the approach used for one problem helps students become more comfortable with using that strategy in a variety of other situations.

When making sense of ideas, students need opportunities to work both independently and collaboratively. There will be times when students need to be able to work independently and other times when they will need to be able to work in small groups so that they can share ideas and learn with and from others.

Getting real

Effective teachers of mathematics create purposeful learning experiences for students through solving problems in relevant and meaningful contexts. While word problems are a way of putting mathematics into contexts, it doesn’t automatically make them real. The challenge for teachers is to provide students with problems that draw on their experience of reality, rather than asking them to suspend it. Realistic does not mean that problems necessarily involve real contexts, but rather they make students think in “real” ways.

Planning for talk

By planning for and promoting discourse, teachers can actively engage students in mathematical thinking. In discourse-rich mathematics classes, students explain and discuss the strategies and processes they use in solving mathematical problems, thereby connecting their everyday language with the specialized vocabulary of mathematics.

Students need to understand how to communicate mathematically, give sound mathematical explanations, and justify their solutions. Effective teachers encourage their students to communicate their ideas orally, in writing, and by using a variety of representations. Through listening to students, teachers can better understand what their students know and misconceptions they may have. It is the misconceptions that provide a window into the students’ learning process. Effective teachers view thinking as “the process of understanding,” they can use their students’ thinking as a resource for further learning. Such teachers are responsive both to their students and to the discipline of mathematics.

“Mathematics today requires not only computational skills but also the ability
to think and reason mathematically in order to solve the new problems and learn the new ideas that students will face in the future. Learning is enhanced in classrooms where students are required to evaluate their own ideas and those of others, are encouraged to make mathematical conjectures and test them, and are helped to develop their reasoning skills.”

(John Van De Walle)

“Equity. Excellence in mathematics education requires equity—high expectations and strong support for all students.”

How teachers organize classroom instruction is very much dependent on what they know and believe about mathematics and on what they understand about mathematics teaching and learning. Teachers need to recognize that problem-solving processes develop over time and are significantly improved by effective teaching practices. The teacher’s role begins with selecting rich problem-solving tasks that focus on the mathematics the teacher wants their students to explore. A problem-solving approach is not only a way for developing students’ thinking, but it also provides a context for learning mathematical concepts. Problem-solving allows students to transfer what they have already learned to unfamiliar situations. A problem-solving approach provides a way for students to actively construct their ideas about mathematics and to take responsibility for their learning. The challenge for mathematics teachers is to develop the students’ mathematical thinking process alongside the knowledge and to create opportunities to present even routine mathematics tasks in problem-solving contexts.

Given the efforts to date to include problem-solving as an integral component of the mathematics curriculum and the limited implementation in classrooms, it will take more than rhetoric to achieve this goal. While providing valuable professional learning, resources, and more time are essential steps, it is possible that problem-solving in mathematics will only become valued when high-stakes assessment reflects the importance of students’ solving of complex problems.

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Mathematical Skills: What They Are And Examples

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Find a Job You Really Want In

Mathematical skills are important to improve if you want to increase your chances for professional success, no matter what career path you pursue. Many jobs use mathematical skills regularly, and even for the rare jobs that never directly deal with numbers and figures, you’ll often need the same problem-solving and critical thinking abilities used in math to succeed.

So, if you’re a job seeker who wants to know more about how to make your mathematical skills shine, stay tuned. In this article, we’ll cover the most important mathematical skills to master for the workplace and discuss how to improve and highlight your math skills during the job-search process .

Key Takeaways:

10 mathematical skills that are useful in the workplace are time management, mental arithmetic, constructing logical arguments, abstract thinking, data analysis, research, visualization, creativity, forecasting, and attention to detail.

Improve your mathematical skills by acquiring conceptual understandings of the skills and solving practice problems.

A mathematical skill should be listed on a resume when the job listing states the skill as a requirement.

Most mathematical skills are transferable and help you stand out in a crowd of applicants.

What are mathematical skills?

How to improve your mathematical skills, how to highlight mathematical skills on a resume, mathematical skills resume example, mathematical skills faq.

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The term “mathematical skills” doesn’t just refer to nebulous topics taught in school, such as calculus. They’re the practical abilities that are useful no matter the industry or size of business you work in.

This includes skills such as:

Time management. Being able to manage your time efficiently is critical for your day-to-day activities, in addition to long-term planning success.

The average person wastes three entire hours each day due to inefficient time management.

Not only does that immediately translate to wasted money, but to wasted time that could be devoted to your non-work passions and activities.

Mental arithmetic. Being able to do mental math quickly will serve you well in a variety of professions.

Retail workers may need to quickly and accurately figure out a customer’s change when given a large sum of money.

Constructing logical arguments. Many careers demand the same precise, logical reasoning that’s used to solve math problems.

An attorney needs to ensure that their legal argument logically follows from the facts and evidence provided.

Abstract thinking. Abstract thinking is the ability to understand and compare non-physical concepts, such as freedom or honesty.

Improving your abstract thinking skills is useful for any career that involves creativity or navigating through complex rules.

Data analysis. A large variety of professions will require you to interpret and analyze data at some point.

Any scientific career will involve heavy interpretation of complex sets of data.

Research. Knowing how to effectively research information is crucial for developing solutions for the many problems you’ll face in your career.

In the age of the internet, there is a nearly infinite wealth of information about any topic you could wish to learn about.

Visualization. The same ability to visualize problems and outcomes is critical for finding solutions in the workplace.

Any problem you face during your career will present a variety of possible solutions with which to tackle it.

Creativity. Improving your creativity skills allows you to come up with new ideas and innovations.

Presenting fresh ideas and solutions will also help you stand out among the competition in the workplace.

Forecasting. Forecasting is the ability to extrapolate events into the future based on available data and knowledge.

This skill is critical for any job that involves planning for the future.

Attention to detail. Some jobs require more attention to small details than others.

Mathematical skills can be improved in the same way that you would improve any other skill – through consistent practice.

More specifically, there are two key actions you should follow:

Acquire conceptual understanding. You can’t improve a mathematical skill if you don’t even know what that math skill entails.

For example, suppose that you wish to improve your data analysis skills.

A quick Google search reveals that the main elements of data analysis include understanding statistics, visuals such as charts and graphs, and how to apply the data to practical problems.

Solve practice problems. It’s not enough to understand a concept to master it; you must practice practically applying it.

This piece of advice applies to certain math skills more than others. You can find plenty of online games to help you improve your hard skills , such as mental arithmetic, but maybe not your creativity.

Mathematical skills positively effect your work performance, especially when you improve them.

However, we still want to find a way to highlight them to recruiters, so they know that we’ve mastered them.

There are a few important guidelines to follow:

List or prove on your resume. The skills section of your resume can be an okay place to mention your math skills.

The job listing states the skill as a requirement. If your resume doesn’t contain the specific term, some companies’ applicant tracking systems (ATS) may automatically filter you out.

Not significant enough to waste additional resume space. Despite being required, some skills may not be essential enough to waste more than a single bullet point talking about.

For example, a job may require basic clerical skills such as multiplying and dividing small figures.

“Critical-thinking skills” and “ problem-solving skills ” are generically added to so many resumes that the terms often become meaningless.

A better way to highlight your math skills on a resume is to prove it through the results you’ve achieved. Use numbers to emphasize the positive value you created for a past employer.

Prove them in your cover letter. You want to give examples of when you used mathematical skills to create value for a past employer.

This differs from the resume strategy in that cover letters are narrower in scope.

Your resume needs to fit many examples on a single page , while your cover letter can target a few key skills to demonstrate with greater detail.

To figure out which math skills to focus on, pay attention to the essential requirements and duties listed in the job listing. Make your best judgment on the most important skills to highlight.

Explain in-depth during your job interview. Job interviews allow you the time to dive much deeper into examples of how you’ve utilized mathematical skills.

Consider the previous resume example about developing a new marketing strategy using data-analysis skills.

During the interview, you could expand on the specific technical skills and tools you used. Explain the initial problem and the thought process you employed to tackle it.

Mathematical skills can be more tangible when you can see them on a resume. Luckily, we’ve provided an mathematical skills resume for you:

Finnegan Bennett 117 Melrose Ave., Austin, TX , 73301 (662)-280-0092 [email protected] Detail-orientated and organized mathematics teacher with over 10 years of experience working in high schools. Possesses a Masters in Education from Austin University. Strong skills in problem-solving and time management. Professional Experience Austin Independent School District , Austin, TX Geometry Teacher , September 2016 — Present Presented HSPA specific lessons to various classes within the mathematics department. Assist with teaching students as an entire class or in small groups as the lessons are planned. Designed, developed, and implemented courses in coordination with science curriculum Designed and implement classroom management strategies at a school wide level. Educate , Austin, TX Math Teacher , September 2013 — August 2016 Delivered and graded assessments in multiple subject areas meeting educational standards. Provide tutoring services by facilitating small groups or individual students for children in grades kindergarten through high school. Skills Pre-Calculus Classroom Management Student Learning Mathematics Special Education State Standards Clear Objectives Test Scores In-Service Training Small Groups Education University of Austin , Austin, TX Masters Degree Education , May 2016 Baker University , Austin, TX Bachelor’s Degree Business , May 2012 Graduated with honors

What are the most important math skills?

The most important math skills in the workplace depend on your needs. The four fundamental arithmetic operations of adding, subtracting, multiplying and dividing are very important for all adults to have a basic understanding because they appear in many of our lives daily.

Time management, logic, and abstract thinking are also very important for most adults to know, regardless of profession, because they help provide structure to your life and prepare you for critical thinking.

Why are mathematical skills important?

Mathematical skills are important because they provide structure to solving problems rationally. Mathematical skills can be used everyday to make sense of a chaotic world. Recognizing patterns, using logic, building on abstract concepts all are what help keep society moving.

How do you list math skills on a resume?

List math skills under the skills section of a resume. In order to be efficient with space, make sure to only list relevant skills that are found in the job description and avoid general terms.

MAA – 10 Skills and Abilities Every Math Major Should Include on Their Resume

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Chris Kolmar is a co-founder of Zippia and the editor-in-chief of the Zippia career advice blog. He has hired over 50 people in his career, been hired five times, and wants to help you land your next job. His research has been featured on the New York Times, Thrillist, VOX, The Atlantic, and a host of local news. More recently, he's been quoted on USA Today, BusinessInsider, and CNBC.

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Mathematical Skills

Mathematical skills: a comprehensive guide.

Mathematical skills are an essential set of abilities that enable individuals to understand, analyze, and solve complex mathematical problems. These skills encompass a wide range of mathematical concepts and techniques, allowing individuals to interpret and manipulate numerical data with accuracy and efficiency.

At its core, mathematical skills involve numeracy, which is the ability to understand and work with numbers. This includes basic arithmetic operations such as addition, subtraction, multiplication, and division, as well as more advanced concepts like fractions, decimals, and percentages. Proficiency in numeracy forms the foundation for higher-level mathematical skills.

One of the key aspects of mathematical skills is the ability to think critically and solve problems. It involves logical reasoning, pattern recognition, and the application of mathematical concepts to real-life situations. Mathematical problem-solving requires individuals to break down complex problems into simpler steps, identify relevant information, and formulate effective solutions.

Another crucial component of mathematical skills is mathematical reasoning. This involves the ability to understand and apply mathematical principles, formulas, and theorems while making logical connections between different mathematical concepts. Mathematical reasoning allows individuals to evaluate the validity of mathematical statements, arguments, and proofs, ensuring the accuracy and coherence of their mathematical solutions.

Furthermore, mathematical skills encompass spatial and geometric understanding. This involves the ability to visualize, manipulate, and analyze shapes, sizes, and spatial relationships. Spatial reasoning allows individuals to solve problems related to measurement, area, volume, and geometric transformations. It also enables individuals to interpret graphs, charts, and other visual representations of data.

In addition to these fundamental aspects, mathematical skills involve data analysis and interpretation. This includes the ability to collect, organize, and represent data in various formats such as tables, graphs, and charts. It also involves statistical analysis, including measures of central tendency, variability, probability, and correlation. Proficiency in data analysis enables individuals to draw meaningful conclusions from data and make informed decisions.

Mathematical skills are highly valued across a wide range of fields and industries. They are essential for success in academic pursuits, including science, technology, engineering, and mathematics (STEM) disciplines. Furthermore, mathematical skills are increasingly sought after in business, finance, economics, computer science, and many other professional domains. They provide individuals with the ability to analyze and solve complex problems, make data-driven decisions, and think critically.

Why Assess a Candidate's Mathematical Skills?

Assessing a candidate's Mathematical Skills skill level is crucial for organizations seeking to make informed hiring decisions. Here are some compelling reasons why evaluating Mathematical Skills is essential in the recruitment process:

1. Accuracy and Efficiency:

Mathematical Skills are the cornerstone of accuracy and efficiency in decision-making. Having employees who possess strong Mathematical Skills ensures that numerical data is comprehended, analyzed, and interpreted correctly. This skill set allows individuals to perform calculations accurately and efficiently, minimizing errors and enabling precise outcomes in various tasks.

2. Problem Solving and Critical Thinking:

Mathematical Skills empower individuals to think critically and solve complex problems. Candidates who demonstrate proficiency in these skills possess the ability to break down intricate problems into manageable components, apply mathematical concepts, and devise effective solutions. By assessing a candidate's Mathematical Skills, organizations can identify individuals with the aptitude to tackle challenges creatively and analytically.

3. Data Analysis and Decision Making:

In today's data-driven world, the ability to analyze and interpret numerical information is of utmost importance. Mathematical Skills equip candidates with the necessary tools to collect, organize, and process data accurately. Assessing a candidate's Mathematical Skills allows organizations to identify individuals capable of extracting meaningful insights from data, making informed decisions, and driving strategic initiatives.

4. Adaptability and Versatility:

Mathematical Skills are highly transferable and applicable across various industries and roles. Candidates with a strong foundation in Mathematics can adapt to different scenarios, whether it's managing budgets, forecasting sales, conducting research, or analyzing trends. By assessing a candidate's Mathematical Skills, organizations can ensure they are hiring individuals who have the versatility to excel in diverse professional environments.

5. Future Growth and Innovation:

Mathematical Skills are essential for technological advancements, scientific research, and innovation. Organizations seeking to thrive and stay competitive in the rapidly evolving landscape should assess candidates for their Mathematical Skills. By identifying individuals with a solid mathematical foundation, organizations can foster a culture of continuous learning, promote technological innovation, and drive future growth.

In conclusion, assessing a candidate's Mathematical Skills is crucial for organizations looking to hire individuals who possess numerical proficiency, problem-solving capabilities, data analysis skills, adaptability, and the potential to drive innovation. Alooba's assessment platform provides a comprehensive solution to evaluate and identify candidates with the required Mathematical Skills, enabling organizations to make informed hiring decisions and build high-performing teams.

Assessing a Candidate's Mathematical Skills with Alooba

Alooba's online assessment platform provides a seamless method to assess a candidate's Mathematical Skills and determine their skill level accurately. With our comprehensive range of assessment tools and features, organizations can confidently evaluate candidates' numerical proficiency and make data-driven hiring decisions.

1. Concept-Based Assessments

Alooba offers customizable concept-based assessments to evaluate candidates' foundational Mathematical Skills. These assessments consist of carefully designed multiple-choice questions that cover various mathematical topics and concepts. Candidates can demonstrate their understanding of basic arithmetic, algebra, geometry, statistics, and more. These assessments are auto-graded, providing instant results and allowing recruiters to efficiently evaluate candidates' proficiency.

2. Data Analysis Challenges

Assessing a candidate's ability to analyze and interpret numerical data is vital. Alooba provides multi-choice assessments where candidates are presented with datasets and required to analyze and draw insights using their own tools. This helps evaluate candidates' skills in data analysis, statistical reasoning, and problem-solving. These assessments are auto-graded, providing objective and accurate results.

3. SQL Proficiency Tests

Proficiency in SQL is highly sought after in many industries. Alooba offers SQL assessments where candidates are tasked with writing SQL statements to query, insert, or update data. These assessments evaluate candidates' knowledge and practical application of SQL, ensuring they possess the necessary skills for database management and data manipulation.

4. Analytics Coding Challenges

Alooba's analytics coding assessments assess candidates' ability to inspect and manipulate data using programming languages such as Python or R. These challenges evaluate candidates' skills in data manipulation, analysis, and visualization. By assessing candidates' analytics coding abilities, organizations can identify individuals capable of leveraging programming skills to solve data-related problems effectively.

5. Comprehensive In-Depth Assessments

For a more thorough evaluation of Mathematical Skills, Alooba offers in-depth assessments that require candidates to demonstrate their abilities through various formats such as written response, diagramming, asynchronous interviews, and file uploads. These assessments provide a holistic understanding of candidates' Mathematical Skills and their ability to apply mathematical concepts in real-world scenarios.

With Alooba's assessment platform, organizations can effortlessly invite candidates via email, bulk upload, ATS integration, or self-registration links. The platform's user-friendly interface allows recruiters to administer assessments smoothly and efficiently, ensuring a seamless evaluation process.

Furthermore, Alooba enables organizations to leverage their existing question bank or customize assessments to align with their specific requirements. This flexibility empowers recruiters to tailor assessments to their specific needs and industry standards.

After assessments are completed, Alooba provides high-level overviews and improvement insights, aiding recruiters in understanding candidates' performance and identifying potential skill gaps. This valuable feedback helps inform hiring decisions and facilitates ongoing candidate development.

In summary, Alooba's assessment platform offers a comprehensive set of tools and features to assess a candidate's Mathematical Skills effectively. With customizable assessments, advanced coding challenges, and in-depth evaluations, organizations can confidently evaluate candidates and make informed hiring decisions based on their numerical proficiency. Partner with Alooba and unlock the potential of Mathematical Skills in your recruitment process.

Topics Covered in Mathematical Skills

Mathematical Skills cover a broad range of topics and subtopics that enable individuals to excel in numerical analysis, problem-solving, and critical thinking. Here are some key areas included within Mathematical Skills:

1. Numeracy and Basic Operations

Numeracy forms the foundation of Mathematical Skills. It encompasses proficiency in basic mathematical operations such as addition, subtraction, multiplication, and division. Candidates with strong numeracy skills can perform these operations accurately and efficiently, ensuring a solid understanding of numbers and their properties.

2. Fractions, Decimals, and Percentages

Proficiency in working with fractions, decimals, and percentages is vital within Mathematical Skills. Candidates must be capable of converting between these forms, performing operations, and applying them to various real-life scenarios. Mastery of these topics enables individuals to solve problems related to proportions, ratios, and percentages.

3. Algebraic Concepts

Algebra is a fundamental component of Mathematical Skills. Candidates should have a sound understanding of variables, equations, inequalities, and functions. This includes simplifying expressions, solving linear and quadratic equations, and graphing mathematical relationships. Proficiency in algebraic concepts allows individuals to apply mathematical modeling and solve complex equations.

4. Geometry and Spatial Reasoning

Geometry and spatial reasoning are integral aspects of Mathematical Skills. Candidates should comprehend geometric properties, angles, lines, shapes, and their relationships. This also includes an understanding of concepts such as area, volume, symmetry, and geometric transformations. Proficiency in geometry enables individuals to analyze and solve problems involving spatial relationships and visual representations.

5. Data Analysis and Statistics

Data analysis and statistics are vital within Mathematical Skills for making informed decisions based on numerical data. Candidates should be proficient in collecting, organizing, and interpreting data using statistical measures such as mean, median, mode, and standard deviation. Understanding concepts such as probability and correlation allows individuals to make accurate predictions and draw meaningful conclusions from data.

6. Mathematical Reasoning and Problem-Solving

Mathematical reasoning and problem-solving play a crucial role in Mathematical Skills. It involves the ability to think critically, analyze complex problems, and apply mathematical principles and concepts to find effective solutions. Candidates should be capable of breaking down problems, employing logic and reasoning, and applying mathematical methods to reach accurate conclusions.

7. Mathematical Models and Applications

Mathematical Skills also encompass the ability to apply mathematical models and concepts to practical real-world situations. Candidates should be adept at using mathematical frameworks to solve problems in areas such as finance, economics, engineering, and scientific research. This includes understanding mathematical modeling techniques, optimization, and statistical analysis.

By assessing candidates' knowledge and proficiency in these topics, organizations can evaluate their Mathematical Skills effectively and make informed hiring decisions. Alooba's comprehensive assessment platform provides the means to evaluate candidates' capabilities in each of these subtopics, ensuring that organizations can identify individuals with a strong foundation in Mathematical Skills and the ability to apply mathematical concepts in practical scenarios.

Applications of Mathematical Skills

Mathematical Skills find wide-ranging applications across various industries and disciplines. Here are some key areas where Mathematical Skills are used:

1. Science and Research

Mathematical Skills are essential in scientific research, enabling individuals to analyze data, perform experiments, and develop mathematical models to explain natural phenomena. In fields such as physics, chemistry, biology, and environmental science, mathematical concepts and techniques are utilized to understand complex systems, make predictions, and validate scientific theories.

2. Engineering and Technology

Mathematical Skills play a crucial role in engineering and technology disciplines. Engineers rely on mathematical principles to design structures, develop algorithms, optimize processes, and solve complex engineering problems. Mathematical concepts such as calculus, differential equations, and linear algebra are used to model systems, simulate behavior, and ensure the efficiency and safety of engineering projects.

3. Finance and Economics

In the finance and economics sectors, Mathematical Skills are applied extensively for financial analysis, risk assessment, investment strategies, and economic forecasting. Professionals in these fields use mathematical models to evaluate market trends, calculate risk and return on investments, and make informed decisions regarding pricing, budgeting, and portfolio management.

4. Data Science and Analytics

Mathematical Skills are foundational for data science and analytics. Professionals in these fields use mathematical techniques such as statistical analysis, regression modeling, machine learning algorithms, and data visualization to extract insights from large datasets. Mathematical Skills enable them to discover patterns, uncover correlations, and derive meaningful conclusions for informed decision-making.

5. Computer Science and Programming

Mathematical Skills are integral to computer science and programming. From algorithms and data structures to cryptography and artificial intelligence, mathematical concepts provide the necessary foundation for developing efficient and secure software. Mathematical Skills enable computer scientists and programmers to solve problems, optimize code, and create innovative solutions in the digital domain.

6. Education and Academia

Mathematical Skills are essential for teaching and learning mathematics at all levels of education. Educators rely on their Mathematical Skills to design lesson plans, develop curriculum, and assess students' understanding. Mathematical Skills are also fundamental for researchers and academics in the field of mathematics, where they explore new mathematical theories, develop proofs, and contribute to advancing mathematical knowledge.

7. Everyday Life

Mathematical Skills are applied in everyday life, often without individuals realizing it. From managing personal finances and calculating measurements to understanding probabilities and interpreting data, Mathematical Skills enable individuals to make informed decisions and solve practical problems in their daily lives.

By recognizing the wide-ranging applications of Mathematical Skills, organizations can appreciate the value of assessing candidates' proficiency in this critical skill set. Alooba's assessment platform provides a comprehensive solution to evaluate Mathematical Skills effectively, ensuring that organizations can identify individuals with the necessary numerical proficiency for success in various industries and professional pursuits.

Roles that Require Strong Mathematical Skills

Strong Mathematical Skills are a necessity for success in several roles that require the ability to analyze data, solve complex problems, and make data-driven decisions. The following roles benefit greatly from individuals with good Mathematical Skills:

Data Analyst : Data Analysts utilize mathematical techniques to extract meaningful insights from data, identify trends, and create data visualizations. Strong Mathematical Skills enable them to effectively analyze and interpret numerical data, perform statistical analysis, and generate accurate reports and recommendations.

Data Scientist : Data Scientists rely on Mathematical Skills to build and deploy advanced analytical models and algorithms. They apply mathematical concepts such as calculus, linear algebra, and statistical methods to develop predictive models, perform data mining, and make accurate forecasts.

Data Engineer : Data Engineers leverage Mathematical Skills to design and optimize data systems and infrastructure. They utilize mathematical concepts to ensure the efficiency, scalability, and reliability of data pipelines, databases, and data architectures.

Insights Analyst : Insights Analysts use Mathematical Skills to analyze market data, customer behavior, and performance metrics in various industries. They apply statistical analysis, data modeling, and forecasting techniques to derive actionable insights and help drive strategic decision-making.

Marketing Analyst : Marketing Analysts rely on Mathematical Skills to measure marketing campaign effectiveness, analyze consumer behavior, and forecast market trends. They use statistical analysis, regression models, and other mathematical tools to interpret marketing data and optimize marketing strategies.

Product Analyst : Product Analysts utilize Mathematical Skills to gather and analyze product data, conduct A/B testing, and make data-driven product recommendations. They apply statistical methods, data modeling, and visualization techniques to guide product development and improve customer experiences.

Analytics Engineer : Analytics Engineers leverage Mathematical Skills to design and build data analytics solutions. They utilize mathematical algorithms, statistical models, and coding skills to develop complex data processing workflows and implement data-driven applications.

Artificial Intelligence Engineer : Artificial Intelligence Engineers require Mathematical Skills to develop machine learning algorithms and create intelligent systems. They utilize mathematical concepts such as linear algebra, probability theory, and optimization methods to train models, optimize parameters, and solve complex AI problems.

Data Architect : Data Architects use Mathematical Skills to design and create efficient data structures, database schemas, and data integration strategies. They ensure data integrity, optimize data storage, and implement mathematical algorithms for data transformation and aggregation.

Data Pipeline Engineer : Data Pipeline Engineers leverage Mathematical Skills to build and maintain robust data pipelines. They utilize mathematical concepts to extract, transform, and load (ETL) data efficiently, ensuring accuracy and reliability throughout the data processing lifecycle.

Fraud Analyst : Fraud Analysts utilize Mathematical Skills to analyze patterns, anomalies, and statistical deviations in data to detect fraudulent activities. They apply mathematical algorithms and statistical models to identify potential fraud cases and implement strategies to mitigate risks.

In these roles and many others, strong Mathematical Skills are vital for making informed decisions, uncovering insights, and driving data-centric initiatives. By assessing candidates for their Mathematical Skills through Alooba's comprehensive assessment platform, organizations can identify individuals with the necessary numerical proficiency and ensure optimal performance in these specialized roles.

Associated Roles

Data analyst.

Data Analysts draw meaningful insights from complex datasets with the goal of making better decisions. Data Analysts work wherever an organization has data - these days that could be in any function, such as product, sales, marketing, HR, operations, and more.

Data Architect

Data Architects are responsible for designing, creating, deploying, and managing an organization's data architecture. They define how data is stored, consumed, integrated, and managed by different data entities and IT systems, as well as any applications using or processing that data. Data Architects ensure data solutions are built for performance and design analytics applications for various platforms. Their role is pivotal in aligning data management and digital transformation initiatives with business objectives.

Data Engineer

Data Engineers are responsible for moving data from A to B, ensuring data is always quickly accessible, correct and in the hands of those who need it. Data Engineers are the data pipeline builders and maintainers.

Data Governance Analyst

Data Governance Analysts play a crucial role in managing and protecting an organization's data assets. They establish and enforce policies and standards that govern data usage, quality, and security. These analysts collaborate with various departments to ensure data compliance and integrity, and they work with data management tools to maintain the organization's data framework. Their goal is to optimize data practices for accuracy, security, and efficiency.

Data Pipeline Engineer

Data Pipeline Engineers are responsible for developing and maintaining the systems that allow for the smooth and efficient movement of data within an organization. They work with large and complex data sets, building scalable and reliable pipelines that facilitate data collection, storage, processing, and analysis. Proficient in a range of programming languages and tools, they collaborate with data scientists and analysts to ensure that data is accessible and usable for business insights. Key technologies often include cloud platforms, big data processing frameworks, and ETL (Extract, Transform, Load) tools.

Data Scientist

Data Scientists are experts in statistical analysis and use their skills to interpret and extract meaning from data. They operate across various domains, including finance, healthcare, and technology, developing models to predict future trends, identify patterns, and provide actionable insights. Data Scientists typically have proficiency in programming languages like Python or R and are skilled in using machine learning techniques, statistical modeling, and data visualization tools such as Tableau or PowerBI.

Fraud Analyst

The Fraud Analyst role involves deep analysis of financial transactions and behaviors to identify and mitigate risks of fraud and financial crime. This position requires a blend of data analysis skills, expertise in fraud detection methodologies, and the ability to work with complex datasets. The role is critical in safeguarding against fraudulent activities and ensuring secure financial operations, making it suitable for those with a keen eye for detail and a strong analytical mindset.

Insights Analyst

Insights Analysts play a pivotal role in transforming complex data sets into actionable insights, driving business growth and efficiency. They specialize in analyzing customer behavior, market trends, and operational data, utilizing advanced tools such as SQL, Python, and BI platforms like Tableau and Power BI. Their expertise aids in decision-making across multiple channels, ensuring data-driven strategies align with business objectives.

Other names for Mathematical Skills include Mathematics , Maths , and Math .

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I wouldn't dream of hiring somebody in a technical role without doing that technical assessment because the number of times where I've had candidates either on paper on the CV, say, I'm a SQL expert or in an interview, saying, I'm brilliant at Excel, I'm brilliant at this. And you actually put them in front of a computer, say, do this task. And some people really struggle. So you have to have that technical assessment.

Mike Yates , The British Psychological Society ( Head of Data & Analytics )

We were very quickly quite surprised with the quality of candidates we would get from Alooba. We ended up hiring eight different analysts via Alooba in about a year's time, which is quite extraordinary for us because we actually have almost never used a recruitment agency for any role. It has been our best outsourcing solution by far.

Oz Har Adir , Vio.com ( Founder & CEO )

For data engineering & analytics these take-home assignments we were doing ourselves are a bit time consuming so we wanted to automate that and also reduce the time candidates were spending on the assessment.

Sharin Fritz , Personio ( Tech Talent Acquisition )

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Softening the sharp edges in mathematics.

For everyone whose relationship with mathematics is distant or broken, Jo Boaler , a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start — to approach the subject with playfulness and curiosity, not anxiety or dread.

“Most people have only ever experienced what I call narrow mathematics — a set of procedures they need to follow, at speed,” Boaler says. “Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience.”

“Mathematics should be flexible, conceptual, a place where we play with ideas and make connections," says Professor Jo Boaler. (Photo: Robert Houser Photography)

Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed , a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced “How to Learn Math,” the first massive open online course (MOOC) on mathematics education. She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users. 

In her new book, Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics , Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls “ishing” a math problem can help students make better sense of the answer. 

What do you mean by “math-ish” thinking?

It’s a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport – these are generally answered with what I call “ish” numbers, and that’s very different from the way we use and learn numbers in school.

In the book I share an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They’re given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2 — but the most common answer 13-year-olds gave was 19. The second most common was 21. 

I’m not surprised, because when students learn fractions, they often don’t learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.

But don’t you also risk sending the message that mathematical precision isn’t important? 

I’m not saying precision isn’t important. What I’m suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they’ll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.

Some people ask me, “Isn’t ‘ishing’ just estimating?” It is, but when we ask students to estimate, they often groan, thinking it’s yet another mathematical method. But when we ask them to “ish” a number, they're more willing to offer their thinking.

Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mind-set. I think we all need a little more ish in our lives. 

You also argue that mathematics should be taught in more visual ways. What do you mean by that? 

For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things — all of that contributes to our understanding of how it works. 

There’s an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik’s Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking. 

Some years back we were interviewing students a year after they’d done that activity in our summer camp and asked what had stayed with them. One student said, ‘I’m in geometry class now, and I still remember that  sugar cube, what it looked like and felt like.’ His class had been asked to estimate the volume of their shoes, and he said he’d imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.

When we learn about cubes, most of us don’t get to see and manipulate them. When we learn about square roots, we don’t take squares and look at their diagonals. We just manipulate numbers.

I wonder if people consider the physical representations more appropriate for younger kids.

That’s the thing — elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it’s all symbolic. There’s a myth that there’s a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they're drawing, they're sketching, they're building models, and nobody says that's elementary mathematics.

Click to enlarge: A depiction of various ways to calculate 38 x 5, numerically and visually. (Image: Courtesy of Jo Boaler)

There’s an example in the book where you’ve asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn’t it be easier to teach students one standard method?

That narrow, rigid version of mathematics where there’s only one right approach is what most students experience, and it’s a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they’re not developing number sense. They don’t learn how to use numbers flexibly in different situations. It also makes students who think differently believe there’s something wrong with them. 

When we open mathematics to acknowledge the different ways a concept or problem can be viewed, we also open the subject to many more students. Mathematical diversity, to me, is a concept that includes both the value of diversity in people and the diverse ways we can see and learn mathematics. When we bring those forms of diversity together, it’s powerful. If we want to value different ways of thinking and problem-solving in the world, we need to embrace mathematical diversity.

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Need to Boost Your Math Skills? Try Gaming.

Just like math, gamification is rooted in problem-solving and perseverance.

If you are not a math person, consider boosting your skills for your career’s sake. The market for jobs requiring advanced math skills is expected to grow faster than other occupations through 2032, according to the U.S. Bureau of Labor Statistics . 

6 Games to Sharpen Your Math Skills

• Khan Academy  
• Buzzmath  

If you’re not a math person and would like to be, consider gaming to boost your skills. Educational games for adults integrate game mechanics into the teaching-learning process, with the intrinsic structure of the game promoting problem-solving skills .

A real plus Why Math Is Vital to Thrive in Your AI Career

How Games Help With Math

In essence, games are built around a core narrative or a mission where the gamer must conquer challenges or puzzles that require logical reasoning, strategy formation and decision making. These abilities are indispensable when learning math. 

Just like math, gamification is rooted in problem-solving and perseverance. Gamers confront a connected flow of challenges based on their real-time decisions rather than engaging in isolated problems.

Gamified environments present players with obstacles that require them to make quick assessments and judgments. Their decisions result in immediate feedback and consequences, but they play on whether they fail or succeed because the game doesn’t stop. The game presents the next stimulus that requires the next response, and as that stimulus-response loop repeats, learners build problem-solving skills and perseverance. 

A well-designed game is challenging and also encourages creative and strategic thinking challenges learners while encouraging them to think creatively and strategically, enabling them to apply, analyze, synthesize and evaluate various problems. 

Many educational games also encourage effective teamwork, bolstering adults’ collaborative problem-solving skills. This practice deepens their understanding of mathematical concepts by permitting them to explore different perspectives and solutions.

6 Games That Boost Math Skills

Educational gamification makes acquiring and mastering essential math skills an engaging and even addictive experience. Gamified math tools make complex concepts more accessible and learning math more enjoyable.  Here are six examples of educational gamification platforms designed for adults.

While not a traditional game, Photomath gamifies learning by offering instant problem-solving gratification. Users simply take a picture of a handwritten or printed math problem, and Photomath instantly provides a step-by-step solution. For working adults grappling with calculus, algebra or trigonometry, this app can demystify complex problems and reinforce learning through real-time solutions and explanations.

Brilliant  caters to a wide range of ages and skill levels, making it particularly useful for employed professionals seeking to sharpen their math skills. Brilliant uses interactive explorations and a problem-based approach to teach mathematical concepts, offering an array of courses including algebra, number theory and calculus. Users build knowledge through story-like narratives and active problem solving, which can be both challenging and rewarding.

Khan Academy

Khan Academy remains a standout resource for self-paced learning, with its extensive library of practice exercises, instructional videos, and a personalized learning dashboard. Adults can specifically benefit from its advanced math courses through gamification elements such as earning badges and points as they achieve goals, making the experience more interactive and motivating.

Buzzmath provides more than 7,000 activities that align with common mathematical skill standards. With a focus on guiding learners through a storyline and offering immediate detailed feedback, Buzzmath adds a game-like structure to math practice. The goal is to help learners build confidence and proficiency in key math areas, with content continually being updated to challenge and stimulate the user.

Mangahigh takes a unique approach to learning, blending anime-style narratives with rigorous math challenges that appeal to adult learners. Offering a wide range of games and activities that cover everything from basic algebra to advanced calculus , this platform boasts a competitive edge that can hook users as they can compete against peers in math duels, pushing them to improve their skills.

DragonBox Elements & Algebra

While typically associated with younger audiences, DragonBox offers advanced apps such as DragonBox Algebra 12+ and DragonBox Elements that are suitable for adult learners. These apps introduce mathematical concepts in a game environment, allowing learners to solve puzzles that indirectly teach algebra and geometry principles.

related reading Why You Need Critical Thinking Skills

More Tips for Boosting Your Math Skills

Games that boost math skills can be fun and perhaps frustrating as you develop skills. Here are two extra tips for boosting your math knowledge.

Revisit the Basics

If you feel your education didn’t give you a proper foundation in essential math skills, revisit the basics — addition, subtraction, multiplication and division. Math skills build off one another, so it is always best to start at the beginning. 

Ground yourself in a solid understanding of the basics before moving into more advanced skills like algebra or geometry . Even the higher relevant adult math skills center around understanding addition, subtraction, multiplication, division, fractions, negative numbers and basic algebra.

The path to acquiring essential math skills for an adult can be a long and laborious road. Attempting to learn everything all at once can be overwhelming, so approach the task with patience. For working professionals looking to improve their math skills, a little bit of learning each day is better than a five-hour study session once a month. 

Math doesn’t have to be mundane or intimidating. Adult learners have a range of innovative tools at their disposal to help them engage with and master essential math skills. Each of the platforms mentioned provides a unique take on gamification, using the inherent drives of competition, achievement and immediate feedback to redefine the math learning experience. Whether these tools are used inside or outside the classroom, they offer dynamic avenues for anyone looking to level up their math abilities.

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Marking a Milestone: Four Years of Daily Study Groups

From data to discovery: studying computational biology with wolfram, navigating quantum computing: accelerating next-generation innovation, unlock innovative problem-solving skills with creative computation.

As computers continue to perform an increasing number of tasks for us, it’s never been more important to learn how to use computers in creative ways. Creative computing, an interdisciplinary subject combining coding with artistic expression, allows us to blend technology with human experiences. Learning to create in this way can help you unlock your innovative problem-solving skills. By mastering creative computation, you can create interactive artwork, design immersive experiences and develop creative solutions to real-world challenges.

Wolfram U ’s new Creative Computation course combines an introduction to Wolfram Language coding with a project-based exploration of various art forms, like visual art, poetry, audio and video game design. If you’ve never coded in Wolfram Language before, this course is a fantastic introduction to applied computing and will help you learn the language for any project. If you’ve already mastered the basics of coding, this course will help you apply your skills to fascinating new problems and projects.

We would love for you to join us in this interactive course as we explore what it means to work creatively with coding.

Motivation from History

Creative computing is a relatively new subject, but people have been using technology to make art for centuries. From the loom to the printing press or Walkman to Atari, technology has been part of art for as long as both have existed.

We now have a variety of exciting and creative ways to engage with computers, from AI-generated images to immersive virtual realities.

In this course, you will learn how to use Wolfram Language to create various forms of art. There are four main sections to the course: Computational Art, Computational Strings, Sound and Game Development. In each section, there are lessons teaching Wolfram Language skills, with associated exercises, and at the end of each section, there is a larger project. The projects are designed for you to stretch your creative muscles and use your new coding skills to create art. You’ll learn how to create visual art using images, how to write poetry using string manipulation, how to visualize audio and how to make text-based and graphics-based video games, all while learning how to code in Wolfram Language.

Here is a sneak peek at some of the topics in the course (shown in the left-hand column):

With 16 lessons, five quizzes and four projects, this course should take around five hours to complete. We recommend doing all the activities and projects to maximize your understanding and explore your new skills.

There is no background required to participate in this course. We will teach you all the coding skills you need to make the projects, so all that is required is your excitement and creativity.

Let’s explore what’s in the course.

There are 16 lessons in this course spread out over the five total sections (Computational Thinking and Coding, Computational Art, Computational Strings, Sound and Game Development). In each lesson, you will explore a different aspect of coding through a short video. You’ll start off by exploring the concept of computational thinking: how to translate your thoughts and your creativity into something the computer can understand and how to work with a computer to build creative artifacts. Here is a short excerpt from the video for this lesson:

Each lesson teaches a specific coding skill, with lots of examples and exploration of key concepts. In the Computational Art section, the goal is to use images and graphics to create a piece of art. In order to do that, we need to learn skills like variables, functions, lists, the Table and Map functions, colors, graphics and randomness, and image manipulation. Each skill is taught with an interactive video lesson in conjunction with exercises, before you use the project to test your knowledge.

The video lessons range from 5–13 minutes in length, and each video is accompanied by a transcript notebook displayed on the right-hand side of the screen. You can copy and paste Wolfram Language input directly from the transcript notebook to the embedded scratch notebook to try the examples for yourself.

Each lesson has a set of exercises to review the concepts covered during the lesson. Since this course is designed for independent study, a detailed solution is given for all exercises. Each exercise will help you practice a specific skill you’ve learned so that you are ready to use that skill in the project. Here is an example of an exercise from lesson 6 on image manipulation:

The exercise notebooks are interactive, so you can try variations of each problem in the Wolfram Cloud . You’re encouraged to blend skills together as you learn them. For example, for the aforementioned exercise, you could use the skills you just learned about randomness to replace the dominant colors in the image of the wolf with random colors, or you could import images to do the same exercise with a different image. When you’ve gotten further in the course, you could come back and build your own function that can do this to any two images.

Each section of the course includes a short project, and the Game Development section has two longer projects. In each case, you’ll use the skills you learned in that section to build something creative. In the first three sections, we provide detailed solutions and walk you though our processes, but in the Game Development section, we encourage you to build something unique.

In the Computational Art section, you’ll make art using images and shapes. In Computational Strings, you’ll write a Mad Libs haiku. In Sound, you’ll make an audio visualizer. In Game Development, you’ll make a text adventure game and a graphics-based Pac-Man –style game.

These projects will allow you to celebrate your successes and practice your new coding skills while cementing your understanding of creative computation.

Each section of the course ends with a short quiz, which allows you to demonstrate your understanding:

You will get instant feedback on your solutions, and you’re encouraged to try out the code.

Course Certificate

You are encouraged to watch all the lessons and attempt the projects and quizzes in the recommended sequence, since each topic in the course relies on earlier concepts and techniques. When you watch all 16 lesson videos and pass the five course quizzes, you will earn a certificate of course completion. The Track My Progress status bar in the course helps you to chart your progress, showing you where you left off from your previous course session. While you don’t have to submit projects to earn a certificate, they are a fundamental part of gaining computational skills, and we look forward to connecting with course users about their projects on Wolfram Community . Your course certificate represents completion of the basic course requirements, demonstrates your interest in exploring the latest technology and in building new computational skills, and it will add value to your resume or social media profile.

You are also encouraged to use the skills you learn in this course to go on to earn Level 1 certification for Wolfram Language proficiency . While the course does not require the same level of mathematics as the Level 1 certification exam, it will prepare you well for accomplishing the range of computational tasks that are required for Level 1 certification.

A Building Block for Success

A mastery of the fundamental concepts of creative computing will prepare you for working with computers to innovatively solve problems. Whether you’re interested in creating art or you’re interested in developing your coding skills, this course will provide a detailed foundation in both. Learning Wolfram Language is a valuable pursuit regardless of your career aspirations, as you can use the skills you learn in this course in any field.

Acknowledgements

I would like to thank my coauthor Eryn Gillam for their major contributions to the development of this course, as well as others who helped this course come together, including (but not limited to) Anisha Basil, Abrita Chakravarty, Cassidy Hinkle, Joyce Tracewell, Arben Kalziqi, Isabel Skidmore, Zach Shelton, Simeon Buttery, Ryan Domier and Eder Ordonez.

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