Math Problem Solving Tips

An Experienced Math Tutor's Tips for Solving Math Word Problems

As a teacher and math tutor, I have seen many students struggle with solving word problems. Who of us doesn't remember being asked questions about 2 trains leaving different stations at different speeds, and being asked to figure out when they will meet? While some of us love these types of problems, many others have great difficulty tackling these sorts of questions. Often we just don't know where to begin, or what strategies we should be using to try to find an answer.

This article is intended to help people who struggle with word problems, by providing them with a strategic approach and a set of tools to use when encountering questions of this sort. Like most writings on Problem solving, this article leans heavily on the ideas and writings of George Polya (1887-1985) a mathematician who refined many of the ideas for problem solving that are in common use today. His classic book 'Solve It' is a must read for the advanced student - a simplified system is provided below:

An Approach to Problem Solving:

Start by being sure that you understand the problem - I can't tell you the number of times that teachers see students turn in copious amounts of work that fails to answer the question that was actually asked. This is often heartbreaking for both the student and the teacher.

Always try to restate the problem in your own words. If you can't rephrase the problem, then there is a good chance that you may not fully understand it.

Be clear on what you need to accomplish. Think about what information you will need. Is there any information missing? Can it be found by using the information that is present?

Make a plan - once you are confident that you know what the question is asking and what data will be needed to answer it, start thinking about how you will tackle the problem

Think about if you have seen a similar question before. If so, how did you solve it?

Solve a simpler problem - if some parts of the problem are not clear try simplifying it and solving what you know. This is a form of 'divide and conquer' - once you have solved a part of the problem it is not uncommon for the answer to the mysterious part to suddenly jump out at you. You can also try simplifying the data - if the numbers are too big and awkward then make them smaller and simpler. This allows you to get clear on the steps and processes involved in solving the problem

Draw a picture - often using a diagram will help to organize the data in a way that allows your brain to create additional connections between the pieces and possibly spark some ideas.

List your data - can you put it in a table? Can you organize it?

Can you spot a pattern?

Do you know a formula? Are there any formulas that you know that could be relevant to the problem? Think about how you could possibly use them in this particular instance.

Can you work backwards? Sometimes approaching a problem in the other direction can cause your mental light bulb to turn on!

Guessing is great! Take a guess if you can and then check if your guess is close. Many of us adults were taught never to guess when we were in school - this is unfortunate as guessing and estimation are powerful problem solving tools.

Carry out your plan - you've got your plan so now it's time to execute it. Use one of the approaches above that you think fit the situation best. Don't be afraid to drop one approach and try another if you don't seem to be making headway.

Look back - is there another method? Could you solve it a different way? Is there a general principle that can be applied?

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What Are Some Examples of the Singapore Math Method?

What are Some Examples of the Singapore Math Method?

Mathematics can be one of the greatest academic challenges a child can face. By introducing Singapore Maths, Our School has integrated into their curriculum a way to reach all students at all learning levels. Singapore Math is a visual method, using pictorials to describe concepts.

1st Grade Addition

Instead of using numbers to describe a problem, Singapore Math uses pictures. These pictures are presented in stories that can aid in the learning process. A child may see a picture of four birds on a branch and two birds flying towards it, then they will be presented with the problem 4 + 2 =? Another addition problem may illustrate two children with two toys in front of each of them. They will be asked how many toys the children have together and be presented with the problem 2 + 2 =?

2nd Grade Multiplication

A child may see a picture of four birds nests, and each nest may contain four birds. They will be asked to solve the problem: there are (blank) birds in four nests. Being able to see the amount of parts in each whole as an illustration, the child can easily count and solve the problem.

2nd Grade Mental Math

Having experienced this visual style of math, students are then introduced to mental math. The pictures are less descriptive and the problems are more complex, encouraging a student to use their imaginative power to solve the problems. The student may receive three possible answers; three groups of circles with a numerical value in each of them. The student will then be asked to solve what 40 less than 578 is and asked to pick from the three answers available. In order to solve this problem, a mental strategy must be used. Students are encouraged to develop their own mental strategies and share it with others.

3rd Grade Word Problems

As the education process moves on, they will be required to rely on their own mental abilities more and more. The pictures will be replaced with word problems, and ask to solve. A student may see the following problem in words:
125 children participated in a math competition. 54 of them were girls. 

How many more boys than girls were there? 

Instead of being shown the two steps used to complete the problem, the student must determine what they are. Students are encouraged to refrain from drawing pictorial models and use equations to solve the problem. 

The country of Singapore has placed in the top three of when tested in mathematics for years for various. This method has been proven by the test of time to provide a strategy for math that works for all students.

Help With How to Solve Math Problems

How to solve math problems can be a particularly daunting task for many students. Are huge number of people have a lot of difficulty with math although there are a number of ways which you can get round the main problems and if you keep positive and dedicated, you should certainly be able to do quite well.

Firstly, it is important not to be afraid to ask questions and write down anything that you're not sure of. The difficulty that many people have with mathematics, is that they have to be a hundred percent focused. This can be difficult and if you lose your concentration for a moment, you will often have to start over again.

This can be really frustrating and can severely hinder the learning process. This is where it is extremely important to keep notes at all times. As soon as something comes to your head, write it down. It is always a big mistake to only right down the answers when they come to you, if they come to you. You need to be able to study the problem intensely and when you have the first clue of how you might be able to solve it, you should write down their clue as clearly as possible.

You should also spend a considerable amount of time practicing. This does not mean that you have to spend several hours in one time; quite the contrary. You should spend maybe half an hour a day or even less but as long as it is on a regular basis, revising over what you have learned.

Collect as many of the objects needed for the math problem 4 + 3 = ___ For this skill building exercise we'll use toy cars. Line the cars up across the top of your child's homework page. (As the parent, you need to remember that linear thought is nearly impossible at this stage, so the cars will act as a visual cue to whatever the math problem might be.) Have your child read the problem out loud in its entirety while you verbally provide reassurance that s/he CAN solve the problem before, during, and after the reading. Next, have your child count out the appropriate number of cars for the largest number (4) and put the cars in the middle of the homework page (or wherever the child prefers the cars be placed, as long as the cars are front and center, because the object here is to give the child a tangible focus for what s/he perceives as intangible, i.e., the math problem). Now have your child count out the next number (3) one car at a time to add to the first 4. When your child reaches the correct answer of seven, s/he should be beaming, because s/he'll have accomplished something s/he'd originally perceived as impossible, and now should see math as something that s/he is very capable of. As you can see, this method will work for a long time for addition, subtraction, multiplication and division, although you may have to switch your tool to pennies to manage the larger numbers!

This will help program it into your mind and you will better be able to cope with the problems that arise in tests and exams. Having regular that short revision sessions are essential to any learning process.

A word of caution here. Don't punish your child when they initially struggle with the activity, nor should you or others say to them, "That's wrong." Instead, try using phrases like, "Are you sure? Let's look at that again." Keep trying until the child gets it right, then clap and make a big deal out of the correct identification just like you did (or may be doing) with potty training. The idea is to build links in the brain, not create more barriers, which negative language can cause. Soon your little one will be marching around telling everyone they know that they are a big girl/boy because they can put on their own shoes! As a parent, you may never know which small accomplishment will provide the breakthrough that your child needs to stop complaining and start getting excited about learning.
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Teaching Kindergarten and Preschool Math

The way math is presented to children makes a tremendous difference in their success as learners, as well as your success as an educator. Children need to take part in activities that encourage them to experiment, to investigate, and to record their observations.

Preschool and kindergarten students always need to have things to move around or manipulate in order to make sense of math concepts. In the education world these are called "manipulatives" and there are a great assortment of these available, such as blocks, counters and pattern blocks. Give children ample time to play with the manipulatives in order to satisfy their curiosity about the materials before attempting to use them to teach a math concept. Introduce new math vocabulary as the children play, as this will help them when they participate in teacher led experiences.

Keep structured lessons short to begin with and do not assume the children understand your expectations. Spend a week teaching proper use of materials and proper cleanup. Teach the children to use mats to identify and define their work area.

The following steps work well when teaching young children. First, demonstrate the math activity two or three times before you give the children materials. You will quickly lose the children's attention if you pass the materials out first. Second, give materials to the children and ask them to try the activity. Check to see all have understood the concept and are experiencing success. Assist children that are having difficulty.

After a few days of the same or similar lessons, record your math experience as the children observe. Keep it simple. For example, after a lesson making repeating patterns print the words, I made a pattern. Say, "I used a red block, a blue block, a red block, a blue block." Draw the pattern and color the blocks. Pass out paper and have the children draw what they did and record words using their knowledge of letters and letter sounds. Recording the activity gives children an opportunity to share and solidify their knowledge.

Visit my kindergarten site for ideas, activities, tips, games and skills lists about teaching preschool and kindergarten maths.

Teaching Problem Solving Heuristics in Mathematics

What are heuristics?

Heuristics are techniques students can use to tackle a problem when the solution to the problem is not obvious. Some examples of heuristics are listed below and grouped into four categories according to how they are used: To give a representation e.g. draw a diagram, make a list, use equations To make a calculated guess e.g. guess and check, look for patterns, make suppositions To go through the process e.g. act it out, work backwards, before-after To change the problem e.g. restate the problem, simplify the problem, solve part of the problem Here is an example of one that involves guess and check: The perimeter of a rectangle is 42 cm. Its area is 108 cm2. Find its length and breadth. Since the perimeter of the rectangle is 42 cm, the sum of its length and breadth will be 21 cm. Make a list of the possible lengths and breadths.



Singapore mathematics syllabuses

The Singapore mathematics syllabuses, developed by Curriculum Planning and Developing Division (CPDD), Ministry of Education Singapore (MOE), have identified thirteen heuristics that are applicable to mathematical problem solving.
1. Act it out
2. Use a diagram/model
3. Use guess-and-check
4. Make a systematic list
5. Look for patterns
6. Work backwards
7. Use before-after concept
8. Make suppositions
9. Restate the problem in another way
10. Simplify the problem
11. Solve part of the problem
12. Think of a related problem
13. Use equations (Heuristics 12 and 13 are not in the primary syllabus.)

We can also treat these four ideas as four general heuristics, “use different representations”, “simplify your problem”, “approach your problem from different directions”, and “bring in solutions”.


Figure 1: Model for problem solving in mathematics (Tiong, Hedberg, & Lioe, 2005)

By definition, all heuristics have the following two characteristics: 1. Heuristics do not guarantee a solution. All heuristics do is pointing us towards possible ways in which we might be able to find our solution. 2. Heuristics do not come with specific procedures. When we use heuristics, we are required to make some judgments of our own regarding what we should do. Heuristics help us to deal with difficult problems or problems that we are not familiar with. Other than that, heuristics usually enables us to find solutions with less time and effort as compared to when we use algorithms to find solutions. Not all heuristics are the same in term of specificity. There are ones that give very general and ambiguous instruction, while the others have more specific procedures that we as problem solvers would like to follow. For example, heuristic “represent you problem differently” only gives us a very general direction to what we should do, where as heuristic “draw a diagram” tells how we can represent our problem, visually. Heuristic “draw a histogram” on the other hand gives a much more specific instruction than the previous two. Representation Simplification Pathway Bring in solution P R O B L E M S O L U T I O N To follow this heuristic “draw a histogram”, we will need to know first the procedures in which we can draw a histogram, whereas to follow the heuristic “draw a diagram” we can invent our own diagrams and the rules or procedures in which we can manipulate them; here we have the freedom to choose and be creative. Of course, we can still end using the histogram, since it might be the best representation for our problems. Here we can construct hierarchy of heuristics according to their specificity. We propose to put the four “general heuristics” from the model above on top of the hierarchy. Using the example in the previous paragraph, we have Figure 2. We should note that Figure 2 is not an exhaustive hierarchy for “representations”, we can still add in more representation into the hierarchy, such as “manipulative” into the second level, “pie chart” into the third level, and so on. Here we are just trying to get a rough idea of how a hierarchy of heuristics might look like. Since the number of heuristics are only limited by our creativity and imagination, it is not possible to construct a hierarchy that contain all possible heuristics.



Figure 2: An example of hierarchy of heuristics

Here we need to clarify first that diagram, symbol or histogram, matrix themselves are not heuristics, but “use diagram”, “use equation” and “draw table” are heuristics. Representations themselves are not heuristics, however suggestions to use representations are. Of course Figure 2 is only part of the whole hierarchy of heuristics for representations, and we can have similar hierarchy for “simplification”, “pathway”, and “bring in solution”. Heuristics on the top of the hierarchy are just very basic and general ideas that can be applied to most problems. These ideas can be further broken downward to make them more specific, making them easier to follow and apply to problems. However by doing so, we have restricted the applicability of the heuristics to only a specific few types of problems.

Heuristics Maths with specific procedures are usually less applicable than those with fewer procedures. Besides that, specific heuristics require less of problem solvers’ interpretation and intuition or creativity. As we can see in Figure 2, the “heuristics” at the bottom of the General Specific Representation Diagram Symbol Histogram Bar chart Matrix Equation Table hierarchy are pointing us to topics that we learn in mathematics lessons that come with a lot of procedures and rules governing how to create and manipulate them. After all, we can see mathematics as a big collection of tools or “representations” in which we use to solve problems.