problem solving heuristics for primary school mathematics

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Some Helpful Problem-Solving Heuristics

A  heuristic  is a thinking strategy, something that can be used to tease out further information about a problem and thus help you figure out what to do when you don’t know what to do. Here are 25 heuristics that can be useful in solving problems. They help you monitor your thought processes, to step back and watch yourself at work, and thus keep your cool in a challenging situation.

• Ask somebody else  how to do the problem. This strategy is probably the most used world-wide, though it is not one we encourage our students to use, at least not initially.
• Guess and try  (guess, check, and revise). Your first guess might be right! But incorrect guesses can often suggest a direction toward a solution. (N.B. A spreadsheet is a powerful aid in guessing and trying. Set up the relationships and plug in a number to see if you get what you want. If you don’t, it is easy to try another number. And another.)
• Restate the problem  using words that make sense to you. One way to do this is to explain the problem to someone else. Often this is all it takes for the light to dawn.
• Organize information  into a table or chart. Having it laid out clearly in front of you frees up your mind for thinking. And perhaps you can use the organized data to generate more information.
• Draw a picture  of the problem. Translate problem information into pictures, diagrams, sketches, glyphs, arrows, or some other kind of representation.
• Make a model  of the problem. The model might be a physical or mental model, perhaps using a computer. You might vary the problem information to see whether and how the model may be affected.
• Look for patterns , any kind of patterns: number patterns, verbal patterns, spatial/visual patterns, patterns in time, patterns in sound. (Some people define mathematics as the science of patterns.)
• Act out the problem , if it is stated in a narrative form. Acting it out can have the same effect as drawing a picture. What’s more, acting out the problem might disclose incorrect assumptions you are making.
• Invent notation . Name things in the problem (known or unknown) using words or symbols, including relationships between problem components.
• Write equations . An equation is simply the same thing named two different ways.
• Check all possibilities  in a systematic way. A table or chart may help you to be systematic.
• Work backwards  from the end condition to the beginning condition. Working backwards is particularly helpful when letting a variable (letter) represent an unknown.
• Identify subgoals  in the problem. Break up the problem into a sequence of smaller problems (“If I knew this, then I could get that”).
• Simplify the problem . Use easier or smaller numbers, or look at extreme cases (e.g., use the minimum or maximum value of one of the varying quantities).
• Restate the problem again . After working on the problem for a time, back off a bit and put it into your own words in still a different way, since now you know more about it.
• Change your point of view . Use your imagination to change the way you are looking at the problem. Turn it upside down, or pull it inside out.
• Check for hidden assumptions  you may be making (you might be making the problem harder than it really is). These assumptions are often found by changing the given numbers or conditions and looking to see what happens.
• Identify needed and given information clearly . You may not need to find everything you think you need to find, for instance.
• Make up your own technique . It is your mind, after all; use mental actions that make sense to you. The key is to do something that engages you with the problem.
• Try combinations of the above heuristics .

These heuristics can be readily pointed out to students as they engage problems in the classroom. However, real-world problems are often confronted many times over or on increasingly complex levels. For those kinds of problems, George Polya, the father of modern problem-solving heuristics, identified a fifth class (E) of looking-back heuristics. We include these here for completeness, but also with the teaching caveat that solutions often improve and insights grow deeper after the initial pressure to produce a solution has been resolved. Subsequent considerations of a problem situation are invariably deeper than the first attempt.

• Check your solution . Substitute your answer or results back into the problem. Are all of the conditions satisfied?
• Find another solution . There may be more than one answer. Make sure you have them all.
• Solve the problem a different way . Your first solution will seldom be the best solution. Now that the pressure is off, you may readily find other ways to solve the problem.
• Solve a related problem . Steve Brown and Marion Walter in their book,  The Art of Problem Posing , suggest the “What if not?” technique. What if the train goes at a different speed? What if there are 8 children, instead of 9? What if . . .? Fascinating discoveries can be made in this way, leading to:
• Generalize the solution . Can you glean from your solution how it can be made to fit a whole class of related situations? Can you prove your result?

Ever tried to help your child with primary math homework and got stumped? Today’s math questions can be challenging – even for adults.

Math education is changing. While many parents spent time memorising procedures and formulas, today’s students are expected to not only understand and master the concepts, but also to have strong thinking skills and problem-solving skills to solve complex math questions.

As new concepts and strategies are being taught, and homework turns from arithmetic exercises to using multiple ways to solve a math word problem, you may feel unsure or have no idea how to help your child.

In this series, the curriculum team at Seriously Addictive Mathematics (S.A.M) shares expert tips on math heuristics and how to use them to solve math word problems.

What are Math Heuristics?

Heuristics  – a word that baffles many primary school students and their parents.

To define it simply, math heuristics are strategies that students can use to solve complex word problems.

Word problems can be solved in several ways using different heuristics, while some word problems are solved using a combination of heuristics.

To solve word problems efficiently, students must be familiar with both the problem-solving methods (heuristics) and the problem-solving process.

How many Heuristics are there?

In Singapore Math, there are 12 heuristics in the primary math syllabus that can be grouped into four main categories:

• To give a representation: Draw a diagram/model, draw a table, make a systematic list
• To make a calculated guess: Look for pattern(s), guess and check, make suppositions
• To go through the process: Act it out, work backwards, use before-after concept
• To change the problem: Restate the problem in another way, simplify the problem, solve part of the problem

Another important point to note is that Singapore Math adopts Polya’s four-step problem-solving process:

1.  Understand the problem : What to find? What is known and unknown? 2.  Devise a plan : Choose the most suitable heuristic 3.  Carry out the plan : Solve the problem 4.  Look back : Check the answer

So, how can we use heuristics to solve math word problems? Here are examples of word problems with solutions provided by the curriculum team at S.A.M.

_______________________________________________________________________________________________________

Heuristic: Act it out

Word Problem (Grade 1):

Alan, Ben and Carol are in the school’s Art Club. Their teacher, Mr Tan, wants two of them to join a contest. How many ways can Mr Tan choose two pupils?

Alan and Ben, Alan and Carol, Ben and Carol.

Mr Tan can choose two pupils in 3 ways.

Heuristic: Draw a diagram/model

Word Problem (Grade 3):

The smaller of two numbers is 1217. The greater number is 859 more than the smaller number. (a) What is the greater number? (b) What is the sum of the numbers?

The greater number is 1217 + 859 = 2076.

The sum of the numbers is 2076 + 1217 = 3293.

Heuristic: Look for pattern(s)

Draw the shape that comes next.

Label all shapes, the pattern is in repeating blocks of A, B, C.

Heuristic: Make a systematic list

Word Problem (Grade 2):

A shop sells apples in bags of 3. It sells lemons in bags of 4. Paul buys some bags of apples and lemons. He buys the same number of each fruit. He buys more than 20 and fewer than 30 pieces of each fruit. How many apples does Paul buy?

Heuristic: Guess and check

Word Problem (Grade 5):

Vijay is presented with the equations below. Insert one pair of brackets in each equation to make it true. 4 × 11 + 18 ÷ 3 + 6 = 46

Heuristic: Restate the problem in another way

There are some identical pens and erasers. 2 pens and 3 erasers are 45 centimetres long altogether. 6 erasers and 2 pens are 60 centimetres long altogether. What is the length of 3 erasers?

If we subtract the total length of 2 pens and 6 erasers from that of 2 pens and 3 erasers, we get the length of 3 erasers.

60 cm – 45 cm = 15 cm

The length of 3 erasers is 15 cm.

Heuristic: Solve part of the problem

Word Problem (Grade 6):

The diagram below shows 4 shaded triangles in Triangle ABC. All the triangles in the diagram are equilateral triangles. If the area of Triangle ABC is 64 cm 2 , find the total area of the shaded triangles.

Heuristic: Simplify the problem

The shaded figure below shows a semicircle and two quarter circles. Find the area of the shaded figure. (Take pi = 3.14)

30 × 15 = 450

The area of the shaded figure is 450 cm 2 .

Heuristic: Work backwards

Darren had some stickers in his collection. He bought 20 more stickers and gave 33 stickers to his sister. He had 46 stickers left. How many stickers did Darren have in his collection at first?

Label the changes as C1 and C2.

Heuristic: Draw a table

Word Problem (Grade 4):

Janice wanted to distribute stickers equally among some children. If each child received 8 stickers, she would have 3 stickers left. If each child received 11 stickers, she would need another 9 stickers. How many children were there?

Heuristic: Make suppositions

Farmer James has some ducks, horses and cows on his farm. He has 30 ducks and cows altogether. The total number of legs the ducks and cows have is 82. The total number of legs the horses have is 28. How many ducks and how many cows are there on the farm?

Suppose that James has 30 ducks. 30 × 2 = 60 30 ducks have 60 legs altogether. 82 – 60 = 22 The total number of legs is 22 less than the actual total number. A cow has 2 more legs than a duck. 22 ÷ 2 = 11 James has 11 cows. 30 – 11 = 19

There are 19 ducks and 11 cows on the farm.

Heuristic: Use before-after concept

Ray and Sam each brought some money for shopping. The ratio of the amount of Ray’s money to the amount of Sam’s money was 3 : 4. After each of them bought a laptop for $1250, the ratio of the amount of Ray’s money to the amount of Sam’s money became 1 : 3. How much money did Sam bring for shopping?

This is the first part to S.A.M Heuristics series. Look out for part two where we will share more expert tips on math heuristics.

Established in 2010, Seriously Addictive Mathematics (S.A.M) is the world’s largest Singapore Math enrichment program for children aged four to 12. The award-winning S.A.M program is based on the global top-ranking Singapore Math curriculum with a focus on developing problem solving and thinking skills.

The curriculum is complemented with S.A.M’s two-pillared approach of Classroom Engagement and Worksheet Reinforcement, with an individual learning plan tailored to each child at their own skill level and pace, because no two children learn alike.

Singapore Math Heuristics: Draw a Table, Make Suppositions and Use Before-After Concept

Heuristics, in the context of problem-solving, are a set of strategies to help students solve mathematical problems. Although problem-solving is by and large the process of working towards a goal to which a solution may not be immediately present, it is important that problem solvers (or students) are not only aware of what they are […]

Singapore Math Heuristics: Solve Part of the Problem, Simplify the Problem and Work Backwards

Problem-solving in mathematics helps children develop reasoning and communication skills that are transferrable and important life skills. Reasoning is required on three levels when children solve word problems. First, they use reasoning to recognise what information is provided or missing. Then, they use reasoning to figure out what information they need to find. Finally, they […]

Singapore Math Heuristics: Make A Systematic List, Guess And Check, Restate The Problem In Another Way

The skills children pick up in math are indispensable; they can be applied to other academic subjects and to solve real-world problems in their daily lives and future work. The Singapore Math curriculum focuses on problem solving. Through problem solving, children develop thinking skills such as creative thinking and critical thinking. When children analyse math […]

Singapore Math Heuristics: Act It Out, Draw A Diagram, Look For Patterns

In part one of our Singapore Math Heuristics series, we gave an overview of the 12 heuristics in Singapore Primary Math syllabus, with tips from the curriculum team at Seriously Addictive Mathematics (S.A.M) on how to solve various math word problems using them. To recap, heuristics are methods or strategies students can use to solve complex […]

Heuristics in Mathematics Education

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• Bharath Sriraman 3  

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In this entry we examine Polya’s contribution to the role of heuristics in problem solving, in attempting to propose a model for enhancing students’ problem-solving skills in mathematics and its implications in the mathematics education.

Characteristics

Research studies in the area of problem solving, a central issue in mathematics education during the past four decades, have placed a major focus on the role of heuristics and its impact on students’ abilities in problem solving. The groundwork for explorations in heuristics was established by the Hungarian Jewish mathematician George Polya in his famous book “ How to Solve It ” (1945) and was given a much more extended treatment in his Mathematical Discovery books (1962, 1965). In “ How to Solve It ,” Polya ( 1945 ) initiated the discussion on heuristics by tracing their study back to Pappus, one of the commentators of Euclid, and other great mathematicians and philosophers like Descartes and Leibniz, who attempted to build a...

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Begle EG (1979) Critical variables in mathematics education. MAA & NCTM, Washington, DC

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Burkhardt H (1988) Teaching problem solving. In: Burkhardt H, Groves S, Schoenfeld A, Stacey K (eds) Problem solving – a world view (Proceedings of the problem solving theme group, ICME 5). Shell Centre, Nottingham, pp 17–42

English L, Sriraman B (2010) Problem solving for the 21st century. In: Sriraman B, English L (eds) Theories of mathematics education: seeking new frontiers. Springer, Berlin, pp 263–290

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Goldin G (2010) Problem solving heuristics, affect, and discrete mathematics: a representational discussion. In: Sriraman B, English L (eds) Theories of mathematics education: seeking new frontiers. Springer, Berlin, pp 241–250

Polya G (1945) How to solve it. Princeton University Press, Princeton

Polya G (1962) Mathematical discovery, vol 1. Wiley, New York

Polya G (1965) Mathematical discovery, vol 2. Wiley, New York

Schoenfeld A (1992) Learning to think mathematically: problem solving, metacognition, and sense making in mathematics. In: Grouws DA (ed) Handbook of research on mathematics teaching and learning. Macmillan, New York, pp 334–370

Sriraman B, English L (eds) (2010) Theories of mathematics education: seeking new frontiers (Advances in mathematics education). Springer, Berlin

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Department of Mathematical Sciences, The University of Montana, Missoula, MT, USA

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Mousoulides, N., Sriraman, B. (2018). Heuristics in Mathematics Education. In: Lerman, S. (eds) Encyclopedia of Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-319-77487-9_172-4

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DOI : https://doi.org/10.1007/978-3-319-77487-9_172-4

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Published : 30 July 2018

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Heuristics For Problem Solvers

Adapted from Meiring, S. P. (1980). Problem solving – A basic mathematics goal . Columbus: Ohio Department of Education.

A heuristic is a thinking strategy, something that can be used to tease out further information about a problem and that can thus help you figure out what to do when you don't know what to do. Here are twenty heuristics that can be useful when you are facing what seems intractable. They help you to monitor your thought processes: to step back and watch yourself at work, thus keeping your cool.

• Ask somebody else how to do it. This is probably the most-used strategy, world-wide, though it's not one we encourage our students to use, at least not initially. (Google it goes here too, and is never encouraged.)
• Guess and try (guess, check, and revise). Your guess might be right! But incorrect guesses can often suggest a direction toward a solution. (N.B. a spreadsheet is a powerful aid in guessing and trying: set up the relationships and plug in a number to see if you get what you want. If you don't, it's easy to try another one. And another. And another...)
• Restate the problem using words that make sense to you. One way to do this is to explain the problem to someone else. Often this is all it takes for the light to dawn.
• Organize information into a table or chart. Having it laid out clearly in front of you frees your mind up for thinking, and perhaps you can use the organized data to generate more information.
• Draw a picture of problem information. Translate problem information into pictures, diagrams, sketches, glyphs, arrows, or...?
• Make a model of the problem. The model might be a physical or mental model, perhaps using a computer. You might vary the problem information to see how or whether it changes the model.
• Look for patterns – any kind of patterns: number patterns, verbal patterns, spatial/visual patterns, patterns in time, patterns in sound. (Some people define mathematics as the science of patterns.)
• Act the problem out, if it's stated in a narrative form. This can have the same effect as drawing a picture. What's more, doing the problem might disclose incorrect assumptions you are making.
• Invent notation. Name things in the problem (known or unknown) using words or symbols, including relationships between problem components.
• Write equations. An equation is simply the same thing named two different ways.
• Check all possibilities in a systematic way. A table or chart may help you to be systematic.
• Work backwards from the end condition to the beginning condition. This is particularly helpful when letting a variable (letter) represent an unknown.
• Identify subgoals in the problem. Break up the problem into a sequence of smaller problems ("if I knew this, then I could get that").
• Make the problem simpler. Use easier or smaller numbers; or look at extreme cases (for example, assuming that the maximum amount of one of the varying quantities is used). Often you can use what you learn from the mini-version to help unlock the big one.
• Restate the problem yet again. After working on the problem for a time, back off a bit and put it into your own words in still a different way, since now you know more about it.
• Change your point of view. Use your imagination to change the way you are looking at the problem – turn it upside down, or pull it inside out.
• Check for hidden assumptions that you may be making (you may be making the problem harder than it really is). These assumptions are often found by changing the given numbers or conditions and looking to see what happens.
• Identify needed and given information clearly. You may not need to find everything you think you need to find, for instance.
• Make up your own technique. It is your mind, after all; use mental actions that make sense to you. The key is to do something that engages you with the problem.
• Try combinations of these heuristics.

The above heuristics are those which are easily pointed out to students as they engage with problems in the classroom. However, real world problems are often those which are confronted many times over or on increasingly complex levels. For those, George Polya, the father of modern problem solving heuristics, identified a fifth class (E) called looking back heuristics. We include those here for completeness, but also with the teaching caveat that solutions often improve and insights grow deeper after the initial "pressure" to produce a solution has been resolved. Subsequent looks at a problem situation are invariably deeper and can lead to wonderful surprises.

• Check your solution. Substitute your answer or results back into the problem. Are all of the conditions satisfied?
• Find another solution. There may be more than one answer. Make sure you have them all.
• Solve the problem a different way. Your first solution will seldom be the best solution. Now that the pressure is off, you may readily find other ways to solve the problem.
• Solve a related problem. Steve Brown and Marion Walter in their book, The Art of Problem Posing , suggest the "What if not?" technique. What if the train goes at a different speed? What if there are 8 children, instead of 9? What if...? Fascinating discoveries can be made in this way, leading to:
• Generalize the solution. Can you glean from your solution how it can be made to fit a whole class of related situations? Can you prove your result?

Heuristic Approach to Problem-solving: Examples

Related Topics: More Lessons for Singapore Math Math Worksheets

Videos, worksheets, solutions, and activities to help students learn how to use the heuristic approach to solve word problems in Singapore Math.

Use A Picture / Diagram / Model Example: The total cost of 2 similar bags, 3 wallets and 4 belts is $1188. A bag cost thrice as much as a wallet and a wallet costs twice as much as a belt. How much will Ted have to pay for a bag, a wallet and a belt?

Heuristic Approach to problem-solving Example: 7/10 of the boys who participated in a marathon race were Chinese. The rest of the boys were made up of Eurasians and Malays in the ratio 5:7 respectively. There were 756 more Chinese than Malay boys. Find the total number of boys who participated in the marathon race.

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Math Heuristics: Solve Part of the Problem, Simplify the Problem and Work Backwards

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Problem-solving in mathematics helps children develop reasoning and communication skills that are transferrable and important life skills.

Reasoning is required on three levels when children solve word problems. First, they use reasoning to recognise what information is provided or missing.

Then, they use reasoning to figure out what information they need to find. Finally, they use reasoning to draw on relevant prior knowledge and choose the most suitable heuristic to work out the solution.

Communication is required for comprehension and expression during problem-solving. Children need to read and understand word problems and then write and express their solutions.

When children are encouraged to explain their thinking verbally, visually and in written form, they gain a better understanding of math concepts and develop stronger communication skills.

Singapore Math Heuristics

In part one of our Math Heuristics series , we gave an overview of the 12 heuristics in Singapore Primary Math syllabus, with tips from the curriculum team at Seriously Addictive Mathematics (S.A.M) on how to solve various math word problems using them.

In part two of the Math Heuristics series , we expanded on the heuristics – Act It Out, Draw A Diagram and Look For Patterns, and also demonstrated how to apply the Polya’s 4-step problem-solving process in sample word problems.

In the third part of the Math Heuristics series , we focused on the heuristics: Make a Systematic List, Guess and Check and Restate the Problem in Another Way.

In part 4 of this series, we will zoom in on these 3 heuristics: Solve Part of the Problem , Simplify the Problem and Work Backwards.

Heuristic: Solve Part of the Problem

Word Problem (Primary 3):

At a school library, each student could borrow up to 4 books. The bar graph below shows how many books students borrowed from the school library in one week. What was the total number of books borrowed from the library that week?

1. Understand: What to find: total number of books borrowed What is known: 19 students borrowed 1 book. 27 students borrowed 2 books. 16 students borrowed 3 books. 11 students borrowed 4 books.

2. Choose: Solve part of the problem

19 students borrowed 1 book = 19 × 1 = 19 books 27 students borrowed 2 books = 27 × 2 = 54 books 16 students borrowed 3 books = 16 × 3 = 48 books 11 students borrowed 4 books = 11 × 4 = 44 books

19 + 54 + 48 + 44 = 165 books

165 books were borrowed from the library that week.

4. Check: Did I read the information from the bar graph correctly? Yes Did I calculate the number of books for each bar correctly? Yes Did I add the number of books for all bars? Yes Try solving the following word problem using Polya’s 4-step process.

Word Problem (Primary 6):

The diagram below shows 4 shaded triangles in Triangle ABC. All the triangles in the diagram are equilateral triangles. If the area of Triangle ABC is 64 cm 2 , find the total area of the shaded triangles.

Answer: The total area of the shaded triangles is 28 cm 2 . See the solution in part one of our Singapore Math Heuristics series .

Heuristic: Simplify the Problem

A bakery sold cupcakes at $4 each. For every 6 cupcakes bought, a discount of $2 was given. Ling bought 50 cupcakes. How much did Ling pay for the cupcakes?

1. Understand: What to find: How much Ling paid for the cupcakes. What is known: Each cupcake cost $4. A discount of $2 was given for every 6 cupcakes bought. Ling bought 50 cupcakes.

2. Choose: Simplify the problem

6 cupcakes form 1 set. 50 ÷ 6 = 8 remainder 2 Ling bought 8 sets of 6 cupcakes and 2 more cupcakes.

Each cupcake costs $4. 6 × $4 = $24 $24 – $2 = $22. Each set of 6 cupcakes cost $22.

Cost of 8 sets of 6 cupcakes = 8 × $22 = $176 Cost of 2 more cupcakes = 2 × $4 = $8 Total cost = $176 + $8 = $184

Ling paid $184 for the cupcakes.

4. Check: How many cupcakes are there in 8 sets? 8 × 6 = 48 Were there 50 cupcakes? 48 + 2 = 50. Yes

Word Problem (Primary 4):

Square EFGH is made up of 4 rectangles. The perimeter of Square EFGH is 32 centimetres. Find the perimeter of each rectangle.

1. Understand: What to find: The perimeter of each rectangle. What is known: Square EFGH is made up of 4 rectangles. The perimeter of EFGH is 32cm. EF is made up of the length of a rectangle. FG is made up of the breadths of 4 rectangles.

A square has 4 equal sides. 32 ÷ 4 = 8 EF is 8cm. The length of a rectangle is 8cm. FG is also 8cm. 8 ÷ 4 = 2 The breadth of a rectangle is 2cm.

8 + 2 + 8 + 2 = 20 The perimeter of each rectangle is 20 centimetres.

4. Check: Is the perimeter of EFGH 32cm? 8 + 2 + 2 + 2 + 2 + 8 + 2 + 2 + 2 + 2 = 32. Yes. Is EF the same length as FG? EF = 8. FG = 4 × 2 = 8. Yes.

Try solving the following word problem using Polya’s 4-step process.

The shaded figure below shows a semicircle and two quarter circles. Find the area of the shaded figure. (Take pi = 3.14)

Answer: The area of the shaded figure is 450 cm 2 . See the solution in part one of our Singapore Math Heuristics series .

Heuristic: Work Backwards

Mr Adam had some pens in his stationery store. He sold 318 of the pens at $1 each. He then put the remaining pens into packs of 5 and sold each pack for $3. He made $249 from selling all the packs. How many pens did Mr Adam have at first?

1. Understand: What to find: How many pens did Mr Adam have at first? What is known: He sold 318 pens. He packed the remaining pens into packs of 5. He sold the packs at $3 each and made $249.

2. Choose: Work backwards

Let’s label the changes C1, C2 and C3 C1: He sold 318 pens. C2: He packed the remaining pens into packs of 5 C3: He sold the packs at $3 each.

We can organise the information like this:

C3 is he sold the packs at $3 each.

Before C3: 249 ÷ 3 = 83 He had 83 packs.

C2 is he packed the remaining pens into packs of 5.

Before C2: 83 × 5 = 415 He had 415 pens remaining.

C1 is he sold 318 pens.

Before C1: 415 + 318 = 733 Mr Adam had 733 pens at first.

4. Check: How many remaining pens did he have? 733 – 318 = 415 How many packs of 5 pens did he have? 415 ÷ 5 = 83 How much did he made from selling the packs? 83 × 3 = $249

Darren had some stickers in his collection. He bought 20 more stickers and gave 33 stickers to his sister. He had 46 stickers left. How many stickers did Darren have in his collection at first?

Answer: Darren has 59 stickers in his collection at first. See the solution in part one of our Singapore Math Heuristics series .

Look out for the final part of this series for the remaining Math Heuristics and word problems with step-by-step worked solutions.

This is the fourth part to S.A.M Math Heuristics series for expert tips on math heuristics.

Read the rest of the “S.A.M Math Heuristics” five-part series below :

Part 1: What Are Heuristics? Part 2: Math Heuristics: Act It Out, Draw A Diagram, Look For Patterns Part 3: Math Heuristics: Make A Systematic List, Guess And Check, And Restate The Problem In Another Way Part 5: Math Heuristics: Draw a Table, Make Suppositions and Use Before-After Concept

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Tackle math word problems with greater confidence: The parent’s guide to heuristics

What are math heuristics.

Heuristics are simple rules or mental shortcuts that help us understand a problem or arrive at a decision quickly.

If you’ve followed a “rule of thumb”, you’ve used a heuristic.

If I’m sending an important e-mail, I will always review it twice, at different timings during the day, to minimise the possibility of making an embarrassing mistake!

Likewise, math heuristics are proven tactics we can use to solve problems effectively — by being more strategic, systematic, and conscientious.

Let’s say you had to tighten a screw. You could go through the screwdrivers in your toolbox one by one, or you could approximate the size of the screw and narrow down your selection to the last 3 screwdrivers. Which method gets the job done faster?

How do math heuristics help my child with problem-solving?

As your child approaches upper primary (P5, P6), they’ll find that word problems become more complex — the approaches and solutions are less obvious.

Hence, knowing common math heuristics will give them the tools they need to tackle challenging problem sums that come up in their homework or exams.

Not only can it dramatically increase your chances of solving any Math problem and help you get started, it can also guide you along your thinking processes to reduce the effort and time needed when problem solving.

To prepare your child for PSLE questions, it helps to know a variety of ways to tackle any problem that comes their way. PSLE questions are complex, so mastering heuristics is like having every tool you could ever need in your toolbox!

We teach the heuristics that your child will learn in school in a way that is fun and easy to understand — just check out some of our videos below if you’re curious! We have many more of such videos in our system.

Why do parents need to know about math heuristics too?

Practicle’s math content covers all heuristics that are tested by MOE, but we recommend that parents have foundational knowledge to supervise their children, especially when preparing for exams.

Want to know how your child is performing? We’ll send you reports about how well they’ve mastered their math concepts and heuristics.

You’ll no longer need to rely on teachers for guidance and feedback all the time, or spend exorbitant amounts of money on assessment books.

What heuristics does MOE test for primary school mathematics?

According to the Singapore Mathematics framework developed by the Curriculum Planning and Development Division (CPDD) team at the Ministry of Education Singapore (MOE), the types of heuristics in Mathematics that can be applied to primary school math problems can be grouped as follow:

1. Visualise a problem 2. Make a calculated guess 3. Walk through the process 4. Simplify the problem 5. Consider special cases

Hence, it is crucial for your child to learn how to use them.

Here’s an example of a heuristic that your child will learn from P4-P6:

We have more of such math videos on our YouTube channel , and if you’d like to try to out our questions and receive question-specific video explanations, how about signing up for a free trial ? No minimum commitment, cancel anytime.

How does Practicle teach math heuristics?

As a team of former teachers and game developers, we put a great deal of thought into making the learning experience in Practicle engaging for your kids using a 2-pronged approach:

1. It needs to be educationally sound and aligned with the school’s syllabus 2. It needs to stir interest and encourage kids to learn more, making it truly effective

#2 is where many math learning solutions fall short, especially traditional methods like tuition and assessment books. Students usually tune out quickly.

Practicle makes learning math heuristics fun, while ensuring your kids learn the proper skills needed to do well in their tests and exams.

Experience Practicle free, no commitment no hidden cost no lock-in

Try our learning platform free with a 7 day trial and see if your child likes it.

Practicle is an online gamified Singapore Math adaptive learning platform that helps primary school children master Math through understanding and fun.

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Heuristic Problem Solving: A comprehensive guide with 5 Examples

What are heuristics, advantages of using heuristic problem solving, disadvantages of using heuristic problem solving, heuristic problem solving examples, frequently asked questions.

• Speed: Heuristics are designed to find solutions quickly, saving time in problem solving tasks. Rather than spending a lot of time analyzing every possible solution, heuristics help to narrow down the options and focus on the most promising ones.
• Flexibility: Heuristics are not rigid, step-by-step procedures. They allow for flexibility and creativity in problem solving, leading to innovative solutions. They encourage thinking outside the box and can generate unexpected and valuable ideas.
• Simplicity: Heuristics are often easy to understand and apply, making them accessible to anyone regardless of their expertise or background. They don’t require specialized knowledge or training, which means they can be used in various contexts and by different people.
• Cost-effective: Because heuristics are simple and efficient, they can save time, money, and effort in finding solutions. They also don’t require expensive software or equipment, making them a cost-effective approach to problem solving.
• Real-world applicability: Heuristics are often based on practical experience and knowledge, making them relevant to real-world situations. They can help solve complex, messy, or ill-defined problems where other problem solving methods may not be practical.
• Potential for errors: Heuristic problem solving relies on generalizations and assumptions, which may lead to errors or incorrect conclusions. This is especially true if the heuristic is not based on a solid understanding of the problem or the underlying principles.
• Limited scope: Heuristic problem solving may only consider a limited number of potential solutions and may not identify the most optimal or effective solution.
• Lack of creativity: Heuristic problem solving may rely on pre-existing solutions or approaches, limiting creativity and innovation in problem-solving.
• Over-reliance: Heuristic problem solving may lead to over-reliance on a specific approach or heuristic, which can be problematic if the heuristic is flawed or ineffective.
• Lack of transparency: Heuristic problem solving may not be transparent or explainable, as the decision-making process may not be explicitly articulated or understood.
• Trial and error: This heuristic involves trying different solutions to a problem and learning from mistakes until a successful solution is found. A software developer encountering a bug in their code may try other solutions and test each one until they find the one that solves the issue.
• Working backward: This heuristic involves starting at the goal and then figuring out what steps are needed to reach that goal. For example, a project manager may begin by setting a project deadline and then work backward to determine the necessary steps and deadlines for each team member to ensure the project is completed on time.
• Breaking a problem into smaller parts: This heuristic involves breaking down a complex problem into smaller, more manageable pieces that can be tackled individually. For example, an HR manager tasked with implementing a new employee benefits program may break the project into smaller parts, such as researching options, getting quotes from vendors, and communicating the unique benefits to employees.
• Using analogies: This heuristic involves finding similarities between a current problem and a similar problem that has been solved before and using the solution to the previous issue to help solve the current one. For example, a salesperson struggling to close a deal may use an analogy to a successful sales pitch they made to help guide their approach to the current pitch.
• Simplifying the problem: This heuristic involves simplifying a complex problem by ignoring details that are not necessary for solving it. This allows the problem solver to focus on the most critical aspects of the problem. For example, a customer service representative dealing with a complex issue may simplify it by breaking it down into smaller components and addressing them individually rather than simultaneously trying to solve the entire problem.

Test your problem-solving skills for free in just a few minutes.

The free problem-solving skills for managers and team leaders helps you understand mistakes that hold you back.

What are the three types of heuristics?

What are the four stages of heuristics in problem solving.

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Developing mathematical problem-solving skills in primary school by using visual representations on heuristics

LUMAT: International Journal on Math, Science and Technology Education

Developing students’ skills in solving mathematical problems and supporting creative mathematical thinking have been important topics of Finnish National Core Curricula 2004 and 2014. To foster these skills, students should be provided with rich, meaningful problem-solving tasks already in primary school. Teachers have a crucial role in equipping students with a variety of tools for solving diverse mathematical problems. This can be challenging if the instruction is based solely on tasks presented in mathematics textbooks. The aim of this study was to map whether a teaching approach, which focuses on teaching general heuristics for mathematical problem-solving by providing visual tools called Problem-solving Keys, would improve students’ performance in tasks and skills in justifying their reasoning. To map students' problem-solving skills and strategies, data from 25 fifth graders’ pre-tests and post-tests with non-routine mathematical tasks were analysed. The results indicate t...

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Jiřina Ondrušová

The paper describes one of the ways of developing pupils’ creative approach to problem solving. The described experiment is a part of a longitudinal research focusing on improvement of culture of problem solving by pupils. It deals with solving of problems using the following heuristic strategies: Analogy, Guess – check – revise, Systematic experimentation, Problem reformulation, Solution drawing, Way back and Use of graphs of functions. Most attention is paid to the question whether short-term work, in this case only over the period of three months, can result in improvement of pupils’ abilities to solve problems whose solving algorithms are easily accessible. It also answers the question which strategies pupils will prefer and with what results. The experiment shows that even short-term work can bear positive results as far as pupils’ approach to problem solving is concerned.

PEOPLE: International Journal of Social Sciences

Attakan Vongyai

Journal on Efficiency and Responsibility in Education and Science

Petr Eisenmann

International Group for the Psychology of Mathematics Education

Azita Manouchehri

In this work we investigated the problem solving behaviors of 3 highschool students as each solved four common non-routine problems with the goal to trace performance constancy across different subject areas and problem types. Additionally, we aimed to identify possible factors that influenced children's choices of heuristics in different problem contexts. The results suggested the individual's confidence and preference for the use of certain strategies. Inconsistency in the same individual's mathematics problem solving behaviors across different subject areas was revealed.

List of Figures 3 List of Tables 5 Introduction 7 Overview 7 Background 8 The Mathematical Problem Solving Suite of Studies 8 Literature Review 9 Mathematics Problem Solving 9 Metacognition and Problem Solving 11 Mathematical tasks as developed by the teachers 14 Methodology 15 Design of the Study 15 Explanation of changes in work program due to shifts in Principal Investigators ...... 17 Data collection 18 Data analysis approach 21 Findings 25 Overall findings 25 The Teachers‘ Classroom Practices 31 Student Pair Work 44 Student Problem Solving Exercises 67 Discussion 83 Pedagogical implications 83 Limitations of the current study 91 Conclusions and Further Research 92 Acknowledgement 94 References 95 Appendices 101 Appendix 1: Information for participants 102 Appendix 2: Coding Scheme Manual 105 Appendix 3: Pair work exercise 115 Appendix 4: An Adapted Artzt & Armour-Thomas Cognitive-Metacognitive Framework 117 Appendix 5: Problem solving exercise – Pretest 120 FINAL RESEARCH REPOR...

Melanie Gurat

Zenodo (CERN European Organization for Nuclear Research)

Asmae BAHBAH

Konstantinos Zacharos

May problem solving be the object of teaching in early education? May appropriate teaching interventions develop as to scaffold children’s efforts to solve mathematical problems? These are the central questions of this paper. The sample consisted of 18 children of a Cyprus public pre school classroom and the problem they were asked to solve was to find all solutions of the pentomino. Graphically representing the solutions on squared paper sheet supported the children’s efforts. The findings show that children responded positively to the problem and were successful in finding all solutions for the specific problem. Graphically representing the solutions as well as the forms of teacher-children interaction played an important role for the positive outcome of the activity.

Al-Jabar: Jurnal Pendidikan Matematika

Munaya Nikma Rosyada

Problem solving is an essential aspect of students' mathematical activities. This ability could practice by using heuristic strategies in learning. Besides, these are assured to be able to promote metacognitive skills. In the implementation, teachers faced several challenges. This research aims to describe the challenges of teachers in implementing learning with heuristic strategies. This research is a descriptive qualitative. Participants of this research were 12 junior high school mathematics teachers from 12 high schools in the Special Region of Yogyakarta and Central Java. Data collection was taken by questionnaire and added with documentation. Data were analyzed using the Miles & Huberman stage-data reduction, data display, and drawing conclusion/verification. The data then validated using triangulation technique. The results revealed that some of teacher has already implement heuristic strategy in the learning process, but unable to define the heuristic strategy correctly. In its implementation, teachers experience several obstacles. These obstacles were found in providing nonroutine problems to students, solving problems by students, and in discussions conducted to solve problems.

Marc Schafer

Mathematics education research argues that mathematical problem solving relies heavily on visualisation in its different forms and at different levels, far beyond the obvious field of geometry. Mathematics educators are thus encouraged and inspired to ‘see’ not only what comes ‘within sight’ but also what we are unable to see when reviewing their students’ work. The qualitative case study described in this paper speaks to this research as it examines the use of visualisation processes (as called visual imagery) in word problem solving. In our study, 17 Grade 11 learners participated in one-on-one task-based interviews. They answered 10 word problems, which we compiled in a worksheet, whose aim was to analyse the evidence of visual imagery in the participants’ solutions and problem solving strategies. To analyse this evidence, we developed a visual imagery analytical framework that facilitated the analysis of the participants’ responses in the worksheet, their interview transcripts a...

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Developing mathematical problem-solving skills in primary school by using visual representations on heuristics

• Susanna Kaitera Faculty of Education, University of Oulu, Finland https://orcid.org/0000-0001-8726-4284

Developing students’ skills in solving mathematical problems and supporting creative mathematical thinking have been important topics of Finnish National Core Curricula 2004 and 2014. To foster these skills, students should be provided with rich, meaningful problem-solving tasks already in primary school. Teachers have a crucial role in equipping students with a variety of tools for solving diverse mathematical problems. This can be challenging if the instruction is based solely on tasks presented in mathematics textbooks. The aim of this study was to map whether a teaching approach, which focuses on teaching general heuristics for mathematical problem-solving by providing visual tools called Problem-solving Keys, would improve students’ performance in tasks and skills in justifying their reasoning. To map students' problem-solving skills and strategies, data from 25 fifth graders’ pre-tests and post-tests with non-routine mathematical tasks were analysed. The results indicate that the teaching approach, which emphasized finding different approaches to solve mathematical problems had the potential for improving students’ performance in a problem-solving test and skills, but also in explaining their thinking in tasks. The findings of this research suggest that teachers could support the development of problem-solving strategies by fostering classroom discussions and using for example a visual heuristics tool called Problem-solving Keys.

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Problem solving heuristics, affect, and discrete mathematics

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It has been suggested that activities in discrete mathematics allow a kind of new beginning for students and teachers. Students who have been “turned off” by traditional school mathematics, and teachers who have long ago routinized their instruction, can find in the domain of discrete mathematics opportunities for mathematical discovery and interesting, nonroutine problem solving. Sometimes formerly low-achieving students demonstrate mathematical abilities their teachers did not know they had. To take maximum advantage of these possibilities, it is important to know what kinds of thinking during problem solving can be naturally evoked by discrete mathematical situations—so that in developing a curriculum, the objectives can include pathways to desired mathematical reasoning processes. This article discusses some of these ways of thinking, with special attention to the idea of “modeling the general on the particular.” Some comments are also offered about students' possible affective pathways and structures.

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Vielfach wird vorgeschlagen, dass Inhalte der Diskreten Mathematik geeignet sind, Lehrern und Schülern ein neues Bild der Mathematik zu vermitteln. Mathematische Entdeckungen an Problemen, die nicht zur Unterrichtsroutine gehören, sind hier leichter möglich als in vielen anderen Gebieten der Mathematik. Das gilt selbst für Schülerinnen und Schüler, die eher als weniger leistungsstark angesehen werden können. Damit Lehrerinnen und Lehrer allerdings die möglichen Vorteile optimal nutzen können, sollten sie wissen, welche Art von Denken und mathematischer Argumentation bei solchen Aufgaben gefordert ist. Der Artikel diskutiert das Thema und geht dabei insbesondere auf das Modellieren des allgemeinen Falls auf der Basis eines speziellen Falls ein. Einige Bemerkungen befassen sich mit der affektiven Komponente des Problemlösens.

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Part 1 Reaction: Problem Posing and Solving Today

Discrete Mathematics and the Affective Dimension of Mathematical Learning and Engagement

Broadening Research on Mathematical Problem-Solving: An Introduction

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Goldin, G.A. Problem solving heuristics, affect, and discrete mathematics. Zentralblatt für Didaktik der Mathematik 36 , 56–60 (2004). https://doi.org/10.1007/BF02655759

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