difficulties in solving math problems

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Mathematical learning difficulties

Learning mathematics

Executive Summary

• Selective cognitive Mathematics Learning Difficulty (MLD) is diagnosed when children are unable to progress in mathematics but their performance is adequate in other domains.
• There is no reliable gender difference in MLD.
• MLD likely has various subtypes.
• At least some types of MLD are associated with short-term and working memory weaknesses.
• On type of MLD is characterized by weak visuo-spatial working memory and is not associated with reading problems. Another type of MLD is associated with reading problems and is associated with weak verbal working memory.
• Memory weaknesses are likely associated with abnormal structure and function of the Intraparietal Sulci (IPS) of the human brain.

Numerous behavioral and neuroimaging studies suggest that mathematical learning difficulties (MLD), are related to weak performance on short-term memory (STM) and working memory (WM) tasks. This document reviews evidence for this conclusion, details theoretical background and points to educational implications.

Terminology and definition of Mathematical Learning Difficulties

An important note regards terminology. In the cognitive neuroscience and psychological literature mathematical learning difficulties are often called Mathematical Learning Disabilities (also abbreviated MLD) or developmental dyscalculia (DD). Here I deliberately avoid using ‘Disability’ as label as in my opinion there is not much evidence to suggest that learning weaknesses are associated with irreversible conditions whereas the label ‘Disability’ often conveys this implication. The irreversible, ‘disability view’ of mathematical weaknesses is linked to ideas that assume that biologically based cognitive skills may be irreversibly impaired in some children. However, I do not think that much evidence supports the biologically based irreversibility of mathematical weaknesses (see e.g. Brief 2 about problems with the biologically based ‘number sense’ theory of mathematical development). Hence, in this document I will use the more neutral ‘Mathematical Learning Difficulty’ term. Coincident with a large portion of the literature this also abbreviates to MLD.

To date researchers do not have an agreed upon definition of MLD. Some researchers consider ‘mathematical learning disabilities’ and DD to be distinct mathematical learning problems whereas others consider them as different labels for the same underlying mathematical learning problem. The terminological confusion appears because currently there is no generally accepted nomenclature of developmental mathematical problems.

Here, the MLD term will be used in a sense, as we defined DD previously: MLD will stand for ‘persistently weak mathematical performance of developmental origin, related to the weakness of some kind(s) of cognitive function(s) and/or representation(s); appearing when concurrent motivation to study mathematics and access to appropriate mathematics education is normal’ [1; p1].

In practice, in a theory free manner, children are typically considered to have MLD if they show mathematical performance weaker than a certain criterion level under the mean performance of a standardized mathematics achievement test. The criterion level is often one standard deviation or one and half standard deviation under the mean level. Importantly, there is no a priori reason to use a certain criterion level so they may differ widely from study to study.

Further, some studies have tested whether mathematical performance is specifically weak in children by using a control criterion test. For example, researchers may also test reading performance in children and they only categorize children to have MLD if their mathematics performance is weak, but their reading performance is in the normal range of age appropriate achievement [2]. However, many researchers do not explicitly test for the specificity of mathematics problems. Importantly, researchers do not agree in what measures should constitute adequate control variables. Some used reading performance, other used various intelligence (IQ) measures.

It is important to realize that some variability of findings in the MLD literature appears because different researchers use different parameters for the above variables: they may use different control variables and cutoff scores in the diagnosis of MLD. Hence, they select slightly or very different participant groups for study and this results in diverse findings. Hence, it is important to carefully evaluate and compare the exact operational definitions of MLD used in each particular study.

The simplest disagreement concerns prevalence estimates of MLD. For example, a thorough review of many MLD prevalence studies found that prevalence estimates ranged between 1.3-10.3% and their mean was about 5-6% [2].  In view of the above, the large variation in prevalence estimates likely depends on the exact diagnostic measures and thresholds used in studies.

There is no gender difference in MLD

Because of its practical relevance it is important to comment on potential gender differences in MLD. Some investigators suggested that MLD may be more prevalent in girls. This suggestion gave rise to theories linking MLD to X chromosome-based dysfunctions [3-6] (women have two X chromosomes while males have an X and an Y chromosome). However, these studies were based on specially selected clinical populations and cannot be considered representative of the whole population. For example, many studies were based on observations from individuals with fragile X syndrome and Turner syndrome that are rare genetic disorders [3-6].

In contrast to the above-mentioned small studies based on rare populations, large population based empirical studies using standardized mathematics and control (reading) measurements could not find evidence in support of a gender discrepancy in MLD [3,7]. These studies investigated the use of multiple different cutoff points for the diagnosis of MLD and concluded that no gender difference exist independent of the cutoff points used. These results are also in line with a large genetics study of 10-year-old children that concluded that there were no genetically based gender differences in mathematical ability in 10-year-old children [8].

Importantly, gender differences may appear if MLD is defined as a discrepancy between standardized reading and mathematics performance. However, in this case gender differences typically appear not because girls are worse in mathematics than boys but because girls tend to be better readers than boys whereas their mathematics performance is similar [3,7,9-10]. Hence, a reading minus math discrepancy score will show larger discrepancy in girls than in boys penalizing girls with relatively good reading skills. Therefore, reading vs. mathematics discrepancy measures are not adequate for diagnosing MLD.

Short-term and working memory and its neural substrates

The most popular working memory model in MLD research is the classical model of Baddeley [11]. This model assumes that memory function involves modality specific, verbal and visual short-term memory (STM) stores and a domain-general central executive (CE) processing unit. STM is the ability to maintain information in unchanged format for a short while. Working memory would refer to the ability to maintain information while simultaneously carrying out some operation on the maintained information (see Brief 2 for examples). That is, STM tasks only require the mere maintenance of information. In contrast, in WM tasks a secondary, so-called processing task is also carried out besides maintenance. Baddeley’s model assumes that the processing task relies on STM function and on the involvement of the so-called ‘central executive’ function. A version of the model supposed that CE function relies on a limited capacity attentional control system [12].

Central Executive function can be defined in more detail. Many researches assume that various, so-called Executive functions contribute to CE performance. First, it is important to emphasize that there is still no clear definition of executive functions. Hence, ‘executive functions’ is a large umbrella term and its exact definition may vary from study to study. Nevertheless, there seems to be agreement that the followings are important executive functions that also play a role in memory performance: Inhibition; attentional focus shifting and information updating and monitoring [13]. Inhibition typically refers to the suppression of unwanted inference from processed items and it has been emphasized by several investigators [14]. Updating refers to the function when items initially in the focus of attention must be overwritten after becoming irrelevant and a new item should enter the focus of attention. Shifting is typically assessed in non-memory tasks that require volitional control [15].  

Working memory and MLD

Various aspects of working memory function can plausibly be thought to be important for mathematical function and development. Solving even a trivial equation with very small numbers, for example, ((3+4)-(2-1))/(2×2), requires a substantial amount of planning. In fact, even adults are unlikely to be able to solve the above simple equation ‘in an instant’, they have to carefully decompose it, make a plan of how they want to proceed with solving the entire equation, direct their attention to individual parts, solve parts one by one, keep partial results in mind as long as they solve parts of the equation and finally integrate all parts in a final mathematical operation and keep the result in mind as long as they communicate it in some form. Indeed, several studies have reported working memory problems in MLD. Here we have space only to review a few representative studies.

One series of experiments investigated verbal STM and working memory problems in MLD. A very early study [16] concluded that children with MLD were only impaired in remembering numbers but not in general working memory. However, later studies [17] found that while MLD children may indeed have specific weaknesses in so-called forward and backward digit span STM tasks (in these tasks children have to remember series of numbers and have to recite them in the original or reversed order) they also had a general working memory deficit reflected in poor performance in many other verbal working memory tasks.

A follow-up study [18] tested 22 MLD children and 27 control children. The paper replicated the general verbal working memory deficit in MLD children and also reported preserved verbal STM in MLD. That is, only working memory tasks requiring executive function contributions were impaired. In addition, children committed many so-called intrusion errors in memory tasks. This means that irrelevant information influenced their task performance. Based on the intrusion errors investigators concluded that the primary deficit in MLD was the impairment of the inhibitory component of central executive function contributing to working memory performance. Various other papers also concluded about the importance of inhibitory function in MLD [19-20].

While many studies used verbal measures of memory, relatively few studies measured visual memory in MLD children. One such study [21] tested both visual and spatial STM in DD, dyslexic, MLD+dyslexic and normal populations and found only visual STM impairment in MLD and only verbal STM impairment in dyslexics. The results of this study suggested that when reading and verbal function was preserved a crucial problem in MLD may concern visuo-spatial memory processes.

Overall, the above results suggest that various aspects of working memory function may be impaired in MLD. A difficulty in integrating results is that many studies did not use an extended battery of working memory tests. In order to gain a comprehensive view of possible memory function deficit in MLD Szucs et al. [22]. have tested both verbal and visual STM and verbal and visual working memory in a very carefully tested and selected population. All together, researches used 18 standardized tests, 9 experimental tasks and various other measures with each child. MLD children were tightly matched to control children in that only the mathematical performance of children but not their reading performance or intelligence differed. Findings showed that MLD children had weaker visual STM and working memory than control participants (The paper used so-called visual matrix span tasks where children are shown a grid with dots occupying grid positions. Children have to recall the position of dots. In another version the dots appear one by one and children also have to recall the order in which dots appeared [24].) In addition, MLD children also had weaker inhibition performance than control participants in various different experimental tasks measuring inhibition [25-26]. It is important to note that this paper could not find any evidence to support claims that an impairment of a putatively biologically based number sense (see Brief 2 and [27]) plays any role in MLD.  

MLD and working memory: Subtypes of MLD

As described above, a very large number of studies reported working memory weaknesses in MLD. Overall, a confusing array of results was obtained.  Specifically, different studies reporting weaknesses in different memory systems. For example, some studies found verbal but not visual working memory problems and vice versa. Yet, other studies found weaknesses in certain executive functions supporting working memory performance while different studies could not confirm the results. First, it is possible that different participant selection criteria affected results (see above). Second, it is possible to find some variables that help organize the results.

A review of 32 often cited MLD studies investigating short-term and working memory performance determined that an important organizing factor to consider was whether the reading performance of children with MLD and that of control children were similar [28]. Importantly, in many studies children with MLD were not only weak in mathematics but they were also poor readers. That is, they were likely to have additional dyslexia and/or other reading comprehension deficits on top of their mathematical difficulties. In contrast, other studies examined children with MLD who were reading at an age appropriate level. The literature review also considered the effect of whether MLD and control groups had similar intelligence scores or not.

Typically, MLD groups had worse working memory performance in at least some memory related variables than control groups. It turned out that the pattern of memory performance weakness depended on whether reading and/or intelligence was or was not similar across groups. On the one hand, when MLD children had poorer reading performance than control children then the MLD children showed substantially worse visuo-spatial working memory performance than control participants. In contrast, the discrepancy between MLD children and control children in verbal memory tasks was not as pronounced as on visual memory tasks. On the other hand, when the MLD and control groups had similar reading performance scores the pattern of results was the opposite: The MLD children showed much weaker verbal working memory performance than control participants but their visuo-spatial working memory performance differed relatively less from control participants. When both reading and intelligence measures were considered at the same time the results had the same pattern and they were even slightly more expressed.

The above outcomes suggest that there are at least two different types of MLD: In one type MLD appears with associated reading deficits. In this group of children we can also detect verbal short-term and working memory weakness and to a lesser extent visual short-term and working memory weakness. In another type of MLD, reading ability and intelligence is at normal level (note that the MLD children in [22] were in this category). In these children the primary cognitive signature is visuo-spatial short-term and working memory weakness and also a lesser extent they also have some verbal memory weakness. The results are consistent with various studies examining verbal memory function in low reading ability samples. These studies have shown that low readers have verbal STM and verbal WM deficits [29-30].

Overall, the above data are reminiscent of a double dissociation between two different types of MLD: One with associated reading problems and one without associated reading problems [21-22]. However, it is important to note that the data does not represent a full dissociation. This is because there is some visual memory weakness in the MLD children with reading problems and there is some verbal memory weakness in the MLD children with no reading problems. This ‘subtype-independent memory weakness’ may be the consequence of at least two factors. First, it is possible that there is a ‘baseline’ memory impairment in both subtypes of MLD, perhaps because some parts of short-term and working memory function draw on the same processes that are weak in both types of children. Second, the baseline memory weakness may be the consequence of having children with different profiles in groups. Such children may be borderline from the point of view that they may have slight reading problems but not strong enough problems so that they would be excluded from matched experimental groups. Including such borderline children could affect group level data.

MLD and working memory and the intraparietal sulcus of the human brain  

A frequent observation is that children with MLD show abnormal structure and function in the intra-parietal cortex (IPS). This abnormality is often explained by assuming that an approximate magnitude representation residing in the IPS is impaired in MLD. However, a link between a putative deficient magnitude representation and altered IPS activity in MLD children is far from certain [22]. In fact, IPS abnormality can plausibly also be interpreted as a marker of altered working memory function in MLD.

First, the IPS is involved in various cognitive functions frequently implicated in numerical tasks, like working memory [31-36], attention [34; 37-39], inhibitory function [40-41] and spatial processing [42] and the proposed number sense [43]. Therefore, weaknesses in any of these functions could plausibly explain IPS abnormalities in MLD [22].

Second, as described above, several of the IPS related cognitive functions may actually contribute to verbal and visual working memory performance in the form of executive functions and/or as subcomponents of visual-spatial processes. Therefore, IPS related dysfunction in MLD may plausibly be related to a broad domain of working memory related cognitive processes rather than to a dysfunction or underperformance of a magnitude representation [43]. In fact, an overall view of the literature seems to suggest that there is more evidence for a working memory related explanation of IPS abnormalities in MLD than for a magnitude representation-based explanation [28; 44].

Educational implications

It is important to consider what tests were used to diagnose MLD. A practical definition of MLD is that students perform persistently weakly on mathematics but perform adequately on unrelated academic disciplines. MLD should not be due to low motivation to learn or to poor teaching methods. Depending on diagnostic tests and their score criteria there may be substantial variation in which students are diagnosed to have MLD. Boys and girls are equally likely to have MLD. It is not yet clear what exact subtypes of MLD exist, but it is likely that some forms of MLD are linked to reading problems (e.g. dyslexia) while other forms are not. The weakness of visual and verbal memory has often been linked to MLD. However, simple memory training is unlikely to improve MLD. As of today it is unclear what cognitive interventions can improve MLD.

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Students' Mathematics Problem Solving Difficulties and Coping and Strategies: A Model Building Study

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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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June 27, 2011

Insights into Math Learning Difficulties

The innate ability to estimate quantities is impaired in children who have a math learning disability, according to a new report. The study also found that those who do poorly in math but aren’t considered learning disabled struggle with math for different reasons.

People with math learning disability, also known as dyscalculia, have difficulty understanding math concepts and solving even simple math problems despite adequate education. About 10% of school-age children have persistent and significant difficulties with math, while many more fail to reach basic levels of mathematics achievement. The causes of dyscalculia, however, remain poorly understood.

To learn more, researchers decided to explore the relationship between children’s mathematics achievement and their innate ability to estimate and compare quantities without counting. This capability, referred to as the approximate number system (ANS), is normally present in infants and improves with age. We rely on ANS skills in daily life, such as when we estimate which line will move more quickly at the grocery store.

The researchers gave 71 ninth graders 2 series of tests designed to measure their ANS skills. For the first series, the children viewed groups of dots and were asked to say whether there were more blue or yellow dots. In the second, 9 to 15 dots of one color appeared, and the children were asked how many dots they saw. Each screen was visible for only a fraction of a second, so the children didn't have time to count the dots. Each series of tests consisted of dozens of screens.

The students’ math abilities had been tested at regular intervals since kindergarten. The scientists classified the children into 4 groups based on these math achievement scores: high achieving (above the 95th percentile), typically achieving (25th to 95th percentile), low achieving (11th to 25th percentile) and math learning disabled (10th percentile and below). The research was conducted by Dr. Michèle Mazzocco at the Kennedy Krieger Institute and Johns Hopkins University and her colleagues Drs. Lisa Feigenson and Justin Halberda of Johns Hopkins University. It was funded in part by NIH’s Eunice Kennedy Shriver National Institute of Child Health and Human Development (NICHD).

In the advanced online edition of Child Development on June 16, 2011, the researchers reported that math learning disabled students had the poorest ANS scores. This finding suggests that problems with the ANS may underlie math difficulties for children in this group. However, low-achieving children were no more likely to have poor ANS scores than children in the higher achieving groups. Math difficulties in low-achieving children, then, likely stem from a cause or causes distinct from the ANS.

“Children with mathematical learning difficulties are often viewed as a uniform group of students, for whom a single type of special instruction or math curriculum is appropriate,” Mazzocco says. “Our findings suggest, however, that children have difficulty with math for different reasons.”

Research to identify these reasons may now lead to new ways of identifying children at risk and tailoring teaching methods to help them.

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20 Effective Math Strategies To Approach Problem-Solving 

Katie Keeton

Math strategies for problem-solving help students use a range of approaches to solve many different types of problems. It involves identifying the problem and carrying out a plan of action to find the answer to mathematical problems.  

Problem-solving skills are essential to math in the general classroom and real-life. They require logical reasoning and critical thinking skills. Students must be equipped with strategies to help them find solutions to problems.

This article explores mathematical problem solving strategies, logical reasoning and critical thinking skills to help learners with solving math word problems independently in real-life situations. 

What are problem-solving strategies?

Problem-solving strategies in math are methods students can use to figure out solutions to math problems. Some problem-solving strategies: 

• Draw a model
• Use different approaches
• Check the inverse to make sure the answer is correct

Students need to have a toolkit of math problem-solving strategies at their disposal to provide different ways to approach math problems. This makes it easier to find solutions and understand math better. 

Strategies can help guide students to the solution when it is difficult ot know when to start.

The ultimate guide to problem solving techniques

Download these ready-to-go problem solving techniques that every student should know. Includes printable tasks for students including challenges, short explanations for teachers with questioning prompts.

20 Math Strategies For Problem-Solving

Different problem-solving math strategies are required for different parts of the problem. It is unlikely that students will use the same strategy to understand and solve the problem. 

Here are 20 strategies to help students develop their problem-solving skills. 

Strategies to understand the problem

Strategies that help students understand the problem before solving it helps ensure they understand: 

• The context
• What the key information is
• How to form a plan to solve it

Following these steps leads students to the correct solution and makes the math word problem easier .

Here are five strategies to help students understand the content of the problem and identify key information. 

1. Read the problem aloud

Read a word problem aloud to help understand it. Hearing the words engages auditory processing. This can make it easier to process and comprehend the context of the situation.

2. Highlight keywords 

When keywords are highlighted in a word problem, it helps the student focus on the essential information needed to solve it. Some important keywords help determine which operation is needed.  For example, if the word problem asks how many are left, the problem likely requires subtraction.  Ensure students highlight the keywords carefully and do not highlight every number or keyword. There is likely irrelevant information in the word problem.

3. Summarize the information

Read the problem aloud, highlight the key information and then summarize the information. Students can do this in their heads or write down a quick summary.  Summaries should include only the important information and be in simple terms that help contextualize the problem.

4. Determine the unknown

A common problem that students have when solving a word problem is misunderstanding what they are solving. Determine what the unknown information is before finding the answer.  Often, a word problem contains a question where you can find the unknown information you need to solve. For example, in the question ‘How many apples are left?’ students need to find the number of apples left over.

5. Make a plan

Once students understand the context of the word problem, have dentified the important information and determined the unknown, they can make a plan to solve it.  The plan will depend on the type of problem. Some problems involve more than one step to solve them as some require more than one answer.  Encourage students to make a list of each step they need to take to solve the problem before getting started.

Strategies for solving the problem 

1. draw a model or diagram.

Students may find it useful to draw a model, picture, diagram, or other visual aid to help with the problem solving process.  It can help to visualize the problem to understand the relationships between the numbers in the problem. In turn, this helps students see the solution.

Similarly, you could draw a model to represent the objects in the problem:

2. Act it out

This particular strategy is applicable at any grade level but is especially helpful in math investigation in elementary school . It involves a physical demonstration or students acting out the problem using movements, concrete resources and math manipulatives .  When students act out a problem, they can visualize and contectualize the word problem in another way and secure an understanding of the math concepts.  The examples below show how 1st-grade students could “act out” an addition and subtraction problem:

3. Work backwards

Working backwards is a popular problem-solving strategy. It involves starting with a possible solution and deciding what steps to take to arrive at that solution.  This strategy can be particularly helpful when students solve math word problems involving multiple steps. They can start at the end and think carefully about each step taken as opposed to jumping to the end of the problem and missing steps in between.

For example,

To solve this problem working backwards, start with the final condition, which is Sam’s grandmother’s age (71) and work backwards to find Sam’s age. Subtract 20 from the grandmother’s age, which is 71.  Then, divide the result by 3 to get Sam’s age. 71 – 20 = 51 51 ÷ 3 = 17 Sam is 17 years old.

4. Write a number sentence

When faced with a word problem, encourage students to write a number sentence based on the information. This helps translate the information in the word problem into a math equation or expression, which is more easily solved.  It is important to fully understand the context of the word problem and what students need to solve before writing an equation to represent it.

5. Use a formula

Specific formulas help solve many math problems. For example, if a problem asks students to find the area of a rug, they would use the area formula (area = length × width) to solve.   Make sure students know the important mathematical formulas they will need in tests and real-life. It can help to display these around the classroom or, for those who need more support, on students’ desks.

Strategies for checking the solution 

Once the problem is solved using an appropriate strategy, it is equally important to check the solution to ensure it is correct and makes sense. 

There are many strategies to check the solution. The strategy for a specific problem is dependent on the problem type and math content involved.

Here are five strategies to help students check their solutions. 

1. Use the Inverse Operation

For simpler problems, a quick and easy problem solving strategy is to use the inverse operation. For example, if the operation to solve a word problem is 56 ÷ 8 = 7 students can check the answer is correct by multiplying 8 × 7. As good practice, encourage students to use the inverse operation routinely to check their work. 

2. Estimate to check for reasonableness

Once students reach an answer, they can use estimation or rounding to see if the answer is reasonable.  Round each number in the equation to a number that’s close and easy to work with, usually a multiple of ten.  For example, if the question was 216 ÷ 18 and the quotient was 12, students might round 216 to 200 and round 18 to 20. Then use mental math to solve 200 ÷ 20, which is 10.  When the estimate is clear the two numbers are close. This means your answer is reasonable. 

3. Plug-In Method

This method is particularly useful for algebraic equations. Specifically when working with variables.  To use the plug-in method, students solve the problem as asked and arrive at an answer. They can then plug the answer into the original equation to see if it works. If it does, the answer is correct.

If students use the equation 20m+80=300 to solve this problem and find that m = 11, they can plug that value back into the equation to see if it is correct. 20m + 80 = 300 20 (11) + 80 = 300 220 + 80 = 300 300 = 300 ✓

4. Peer Review

Peer review is a great tool to use at any grade level as it promotes critical thinking and collaboration between students. The reviewers can look at the problem from a different view as they check to see if the problem was solved correctly.   Problem solvers receive immediate feedback and the opportunity to discuss their thinking with their peers. This strategy is effective with mixed-ability partners or similar-ability partners. In mixed-ability groups, the partner with stronger skills provides guidance and support to the partner with weaker skills, while reinforcing their own understanding of the content and communication skills.  If partners have comparable ability levels and problem-solving skills, they may find that they approach problems differently or have unique insights to offer each other about the problem-solving process.

5. Use a Calculator

A calculator can be introduced at any grade level but may be best for older students who already have a foundational understanding of basic math operations. Provide students with a calculator to allow them to check their solutions independently, accurately, and quickly. Since calculators are so readily available on smartphones and tablets, they allow students to develop practical skills that apply to real-world situations.  

Step-by-step problem-solving processes for your classroom

In his book, How to Solve It , published in 1945, mathematician George Polya introduced a 4-step process to solve problems. 

Polya’s 4 steps include:

• Understand the problem
• Devise a plan
• Carry out the plan

Today, in the style of George Polya, many problem-solving strategies use various acronyms and steps to help students recall. 

Many teachers create posters and anchor charts of their chosen process to display in their classrooms. They can be implemented in any elementary, middle school or high school classroom. 

Here are 5 problem-solving strategies to introduce to students and use in the classroom.

How Third Space Learning improves problem-solving 

Resources .

Third Space Learning offers a free resource library is filled with hundreds of high-quality resources. A team of experienced math experts carefully created each resource to develop students mental arithmetic, problem solving and critical thinking. 

Explore the range of problem solving resources for 2nd to 8th grade students. 

One-on-one tutoring 

Third Space Learning offers one-on-one math tutoring to help students improve their math skills. Highly qualified tutors deliver high-quality lessons aligned to state standards. 

Former teachers and math experts write all of Third Space Learning’s tutoring lessons. Expertly designed lessons follow a “my turn, follow me, your turn” pedagogy to help students move from guided instruction and problem-solving to independent practice. 

Throughout each lesson, tutors ask higher-level thinking questions to promote critical thinking and ensure students are developing a deep understanding of the content and problem-solving skills.

Problem-solving

Educators can use many different strategies to teach problem-solving and help students develop and carry out a plan when solving math problems. Incorporate these math strategies into any math program and use them with a variety of math concepts, from whole numbers and fractions to algebra. 

Teaching students how to choose and implement problem-solving strategies helps them develop mathematical reasoning skills and critical thinking they can apply to real-life problem-solving.

READ MORE : 8 Common Core math examples

There are many different strategies for problem-solving; Here are 5 problem-solving strategies: • draw a model  • act it out  • work backwards  • write a number sentence • use a formula

Here are 10 strategies of problem-solving: • Read the problem aloud • Highlight keywords • Summarize the information • Determine the unknown • Make a plan • Draw a model  • Act it out  • Work backwards  • Write a number sentence • Use a formula

1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back

Some strategies you can use to solve challenging math problems are: breaking the problem into smaller parts, using diagrams or models, applying logical reasoning, and trying different approaches.

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Differentiated Instruction: 9 Differentiated Curriculum And Instruction Strategies For Teachers 

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Why do students struggle with math word problems? (And What to Try)

Word problems can be a real challenge for students of all ages. While some learners quickly grasp the concepts and transfer these skills to multi-step word problems, others struggle with even the most straightforward, basic word problems. As teachers, we must understand why this is so to help students succeed.

In this blog post, we’ll explore common issues that cause difficulty when solving word problems and potential solutions that can assist learners in becoming more proficient problem-solvers. So, let’s dive into what makes word problems so tricky and how you can help your students master them!

How to Help Learners Conquer Word Problems: Common Challenges & Solutions

Problem #1: students have difficulty reading & understanding the problems..

Word problems can be a daunting task for students of all ages. Solving math problems demands students to comprehend mathematical terms and have solid decoding abilities. If either of these skills is lacking, students may need help understanding the meaning behind certain words and phrases.

Considering that only a few sentences can determine the solution to a problem, it is essential to comprehend the language used in word problems. Yet, only  32% of 4th graders are proficient readers, according to the National Assessment of Educational Progress .

The challenge of comprehending the language in word problems is not only difficult for students who struggle with reading but can also be an obstacle for high-achieving math students. Often, these students know how to solve a problem but need help understanding what the problem is asking.

Word problems further complicate matters due to their use of language that’s different from how we communicate.

For example, students may read a problem that says, “Sarah is baking a pie for her grandmother’s birthday. She needs 7 apples for the recipe. At the store, apples are $2 a piece. If she has $11, will she have enough money to make the pie?”

Students must decode the words and phrases used to understand what the problem is asking them.

Solution: Provide word problems in audio formats & consider how you can incorporate explicit teaching into your math problem-solving routine.

One common strategy for addressing this is to read the problems aloud. Technology can help with this. Recording and storing problems where students can listen to them repeatedly can be helpful. However, you will need to teach your students to use this technology purposefully to help them better understand the word problems they are tackling. Without proper instruction, these recorded problems are no more helpful than reading the problems themselves.

However, this only addresses issues with decoding. It is essential to explain to students the meaning of words and math terms used in questions. A  Problem of the Day format  offers an excellent opportunity to deeply discuss a single problem with students without taking over your entire math lesson.

Explaining these concepts helps students build a stronger foundation for understanding word problems and increases their math comprehension.

Problem #2 :  Students have gaps in vocabulary that would help with math word problems.

Sometimes, story problems require students to have an understanding of math vocabulary. When students don’t wholly understand math vocabulary, they struggle to understand what the problems are asking.

This is more than just decoding!

Even if they can read these words, they may need help understanding how to solve the problems. A strong foundation in math vocabulary is integral to any math classroom.

Solution: Explicitly teach and review math vocabulary regularly.

Ensure that students have a strong foundation in math vocabulary by explicitly teaching terms and concepts. This can be done through direct instruction, visual representations, and activities reinforcing the concepts.

Review these terms regularly throughout the year to ensure they stay fresh in students’ minds.

Problem #3:  Students lack efficient & effective strategies.

Often, students are taught to use keywords early on. However, as problems become more complex, this quickly becomes an ineffective and inefficient strategy For addressing multi-step word problems.

Research has shown keywords often misdirect students’ efforts and derail problem-solving with math word problems.

As a result, many state tests now purposefully include tricky problems designed to fool students who have been taught keywords as a problem-solving strategy.

Solution :  Teach a problem-solving strategy, like CUBES, that helps students break the problem down efficiently.

While keywords are ineffective, giving students a framework for breaking down word problems and identifying the information that CAN help them is a great way to support their problem-solving efforts.

The CUBES strategy (Circle, Underline, Box, Evaluate, Solve) can help older students with math word problems . This strategy helps them break down problems into manageable steps that make sense to them.

Problem #4:  Difficulty mapping out and visualizing the story behind each problem can lead to confusion in solving for an answer.

Another familiar struggle students face when solving word problems is difficulty mapping out and visualizing the story behind each problem. This can lead to confusion in solving for an answer because students may be unable to see how all the pieces fit together. In other words, they don’t have a complete understanding of the context of the problem.

Solution:  Give students an active way to create a picture of what the problem is asking them.

Diagrams with labels, breaking the problem into simpler parts, and making a step-by-step plan with math word problems can help students understand the situation. Having them explain the story in their own words helps them clarify what they’re trying to solve.

Encourage students who automatically add all the numbers to slow down and process the question with numberless word problems.

A numberless word problem is a story problem that does not include numbers . Instead, students are asked to analyze the problem without numbers before they are given the numbers to solve. This can help students notice patterns in the problem and determine what operations will be necessary for solving it. Adding these types of word problems to your instructional routine can be a great way to help students slow down and focus on understanding the scenario being presented in the problem. 

By providing students with different ways to visualize word problems, we can increase their chances of success and provide meaningful math instruction. Equipping them with the right tools and strategies gives them a better chance of tackling any difficult word problem they may encounter.

Problem #5:  Those with poor numeracy skills are disadvantaged when attempting to solve math word problems.

Computational fluency  is a common buzzword in math circles these days. We often discuss whether students know their math facts. However, math fact fluency becomes even more critical when students dive into more challenging word problems.

According to cognitive load theory, students focusing on rote processes such as basic facts have fewer mental resources left for higher-level thinking and processing.

In other words, the more mental energy it takes to work through the first step of a two-step problem, the less likely the student will have the resources to persist in accurately making it through the rest of the problem.

Solution: Build fact fluency practice into your routine in fun, engaging ways. 

Fact fluency practice doesn’t have to be boring, but it is integral to being an effective mathematician. Therefore, finding ways to build it into your math class is essential.

Here are some of my favorite online games that students love:  30+ Awesome Online Games for Math Fact Practice .

Problem #6:  Students lack experience or are only provided with structured word problem practice.

Some curricula only include problems that follow a specific pattern or directly connect to the skill learned in a given lesson. However, formulaic word problems, where students follow a specific set of steps repeatedly, promote complacency.

Students begin to approach every word problem with the same steps. Soon they are grabbing numbers instead of taking the time to comprehend the problem and how best to address it.

Additionally, many word problems require students to apply knowledge from multiple different units to solve the problem. This can be challenging for students still working on mastering previously taught skills. It overwhelms those who have missed chunks of their instruction due to illness or being pulled from instruction.

As a result, these word problems often begin to feel impossible. 

Solution:  Incorporate variety into your problem-solving and allow for productive struggle.

Students need to be provided with an opportunity to approach a variety of different problems across time. They need to see problems that come in various formats. They need uniquely worded problems. This novelty prevents them from sticking with a rote set of strategies. The goal is to get them critically thinking about the problem at hand .

Offering variety builds confidence, competence, and the ability to address any problem they are given. Many students lack confidence in word problems. Varied experience reduces fears and helps students develop a bank of strategies to overcome barriers when complex problems arise.

To help foster independence, you can also support students through the gradual release process. Provide learners with a step-by-step guide to ensure they have completed the problem-solving process’s critical steps when you aren’t doing problems with them. This can help boost their confidence and reduce the risk of careless mistakes.

I’ve created a free mini-book for students with guiding questions and steps to help them independently complete word problems.

Get it here.

Building the math problem solver’s toolbox

Word problems can be difficult for learners, but with the right strategies and resources, teachers can help their students learn to approach word problems confidently. By providing a variety of word problems that come in various formats and require different steps to solve, teachers can allow their students to develop problem-solving skills and build confidence when addressing any problem they are given.

Don’t forget to grab the free problem solver’s guide!

I hope you found this post helpful. Problem-solving is an essential skill for learners. Learn more about word problems or check out my Daily Problem Solving for engaging and meaningful word problem practice. 

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Self-Esteem

It’s ok you can’t solve every problem, trying to “fix" everything can leave you feeling like a failure..

Updated May 10, 2024 | Reviewed by Ray Parker

• What Is Self-Esteem?
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• Your intrinsic value is more than what you can do for other people.

You are still worthwhile and can be successful, even if you don’t have all the solutions.

• Consider which decision will make you feel you’ve stayed true to your values.

In coaching others, I often discuss problem-solving strategies to help individuals think creatively and consider many options when they are faced with challenging situations.

Problem solving 1-2 includes the following:

• Define the problem, identify obstacles, and set realistic goals .
• Generate a variety of alternative solutions to overcome obstacles identified.
• Choose which idea has the highest likelihood to achieve the goal.
• Try out the solution in real-life and see if it worked or not.

Problem-solving strategies can be helpful in many situations. Thinking creatively and testing out different potential solutions can help you come up with alternative ways of solving your problems.

While many problems can be solved, there are also situations in which there is no “perfect” solution or in which what seems to be the best solution still leaves you feeling unsatisfied or like you’re not doing enough.

I encourage you to increase your comfort around the following three truths:

1. You can’t always solve everyone else’s problems.

2. You can’t always solve all of your own problems.

3. You are not a failure if you can’t solve every problem.

You can’t always solve everyone else’s problems.

When someone around you needs help, do you feel compelled to find solutions to their problem?

Are you seen as the problem solver at your job or in your close relationships?

Does it feel uncomfortable for you to listen to someone tell you about a problem and not offer solutions?

There are times when others come to you because they know you can help them solve a problem. There are also times when the other person is coming to you not for a solution to their problem, but for support, empathy, and a listening ear.

Your relationships may be negatively impacted if others feel that you don’t fully listen and only try to “fix” everything for them. While this may feel like a noble act, it may lead the other person to feel like they have failed or that you think they are unable to solve their own problems.

Consider approaching such situations with curiosity by saying to the other person:

• As you share this information with me, tell me how I can best support you.
• What would be most helpful right now? Are you looking for an empathetic ear or want to brainstorm potential next steps?
• I want to be sure I am as helpful as I can be right now; what are you hoping to get out of our conversation?

You can’t always solve all of your own problems.

We are taught from a young age that problems have a solution. For example, while solving word problems in math class may not have been your favorite thing to do, you knew there was ultimately a “right” answer. Many times, the real world is much more complex, and many of the problems that you face do not have clear or “right” answers.

You may often be faced with finding solutions that do the most good for the most amount of people, but you know that others may still be left out or feel unsatisfied with the result.

Your beliefs about yourself, other people, and the world can sometimes help you make decisions in such circumstances. You may ask for help from others. Some may consider their faith or spirituality for guidance. While others may consider philosophical theories.

Knowing that there often isn’t a “perfect” solution, you may consider asking yourself some of the following questions:

• What’s the healthiest decision I can make? The healthiest decision for yourself and for those who will be impacted.
• Imagine yourself 10 years in the future, looking back on the situation: What do you think the future-you would encourage you to do?
• What would a wise person do?
• What decision will allow you to feel like you’ve stayed true to your values?

You are not a failure if you can’t solve all of the problems.

If you have internalized feeling like you need to be able to solve every problem that comes across your path, you may feel like a failure each time you don’t.

It’s impossible to solve every problem.

Your intrinsic value is more than what you can do for other people. You have value because you are you.

Consider creating more realistic and adaptive thoughts around your ability to help others and solve problems.

Some examples include:

• I am capable, even without solving all of the problems.
• I am worthwhile, even if I’m not perfect.
• What I do for others does not define my worth.
• In living my values, I know I’ve done my best.

I hope you utilize the information above to consider how you can coach yourself the next time you:

• Start to solve someone else’s problem without being asked.
• Feel stuck in deciding the best next steps.
• Judge yourself negatively.

1. D'zurilla, T. J., & Goldfried, M. R. (1971). Problem solving and behavior modification. Journal of abnormal psychology, 78(1), 107.

2. D’Zurilla, T. J., & Nezu, A. M. (2010). Problem-solving therapy. Handbook of cognitive-behavioral therapies, 3(1), 197-225.

Julie Radico, Psy.D. ABPP, is a board-certified clinical psychologist and coauthor of You Will Get Through This: A Mental Health First-Aid Kit.

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Students' Difficulty in Solving Mathematical Problems

This study investigates the difficulties experienced by the third year college students in solving Mathematics problem. A total of twenty students are instructed to solve problems in the form of a questionnaire. The data gathered were analyzed to explore difficulties faced by students when solving problems. The major results of the study showed that the students’ difficulties are on the inability to translate problem into mathematical form and inability to use correct mathematics.

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Problem solving is one of the mathematical competencies that students must achieve from elementary school through college. However, in reality problem solving is still a competency that is difficult for students to achieve with various kinds of difficulties in solving mathematical problems. This study has a goal to be achieved in the form of identifying and describing problem solving difficulties based on the theory of Newman to students in fraction material . This research was conducted using descriptive qualitative research methods . The research sample consisted of 34 5th grade elementary school students and 3 students as interview samples with a background of different cognitive abilities. The data of this study were collected from test questions of problem solving skills and interviews. The results showed that students' difficulties in solving problem solving problems were students do not understand the questions in the problem, students are still lacking in understanding m...

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Frontiers in Education

jonni sitorus

Problem-solving is the ability to understand issues and steps that will affect students’ mathematics learning results. The research goals are: describing students’ problem-solving understanding and mathematics learning results; testing the influence of gender and grade level on students’ problem-solving understanding and mathematics learning results; and testing the influence of students’ problem-solving understanding on mathematics learning results. The population is all students in the fourth through sixth grades at two primary schools (263 respondents). The sample consists of 244 respondents, with female and male respondents of 123 and 121, and low- and high-grade students of 74 and 170, respectively. The researchers calculated and converted scores into values. Statistical tests were carried out by means of analysis of linear regression and the t-test. The Me of students’ mathematics learning results and problem-solving understanding are 69.679 and 66.889. The female students’ pr...

Purpose of this study is to determine problem solving skills of primary mathematics preservice teachers in mathematics teaching. This research was carried out with the 3rd year students studying in the department of elementary mathematics teaching at Samsun Ondokuz Mayıs University. Research method was determined as case study, one of the qualitative methods. In the study, the students were taught for 13 weeks (39 hours) Polya’s (1945) problem solving stages that are composed of 4 stages and the problem solving stages were introduced in order to improve their problem solving skills. In the research, two problems developed by Posamentier and Krulik (1998) and semi-structured interview form developed by the researcher were used as data collection tools. In the analysis of the data, solutions of the problems applied were examined considering Polya’s (1945) problem solving steps. The findings obtained via the solutions of applied problems and via the semi-structured interview form were ...

International Journal of Research Publications

Nick Panares

Mathematics helps people to look forward, plan, and decide properly to solve each problem in daily life. The study focused on finding (1) difficulties in solving mathematical problems, (2) the pupils' performance, and (3) the relationship between the difficulties in solving mathematical problems and their Mathematics performance. The respondents were the Grades lV pupils of the three (3) schools in West 2 District, Division of Gingoog City. The instrument used was a researcher-made questionnaire. It employed the descriptive-correlational research design. The statistical tools were the Mean, Standard Deviation, and Pearson Correlation Coefficient. The study led to the findings that the difficulties in solving mathematical problems in terms of comprehension, mathematical skills, and attitude toward Mathematics were Frequent while the pupils' performance in Mathematics was at a Very Satisfactory level. The level of comprehension was the highest. Comprehension, mathematical skills, and attitude towards math were Significant in pupils' Mathematics performance. The encouragement and guidance of the teachers and parents are important for the pupils to love and have a positive attitude toward Mathematics. Teachers may apply varied math learning interventions for the pupils, and they should also provide numeracy materials. School administrators should also emphasize the importance of learning mathematics. Lastly, pupils should inculcate in their minds that Mathematics is part of the curriculum and in their lives. Thus, it must be given importance.

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