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In the previous chapter we saw that math is very essential in developing critical, logical and structured thinking and problem solving
These skills can be collectively called Skills of Mathematical Thinking.
Before delving into the topic of word problems let us look briefly at mathematical thinking.
Mathematical thinking is the ability to use mathematics for various purposes.
• Everyday living
• Work (profession)
• Further Education) &
• To get a taste of the intellectual adventure that mathematics can be (intellectual satisfaction)
A successful mathematical thinker has the following 5 characteristics, arranged roughly in increasing order of complexity & difficulty.
We provide a brief description of each. In later chapters, we will deal with them in detail.
Making Connections
Connecting is “sense-making”. It is relating ideas in math to experiences in your life and things and events happening around you in the world around you.
This is the first step while learning math.
Here are some examples.
1. Connecting numbers with quantity. Like the parts of your body and the number of friends you have.
2. Connecting various properties of numbers with geometrical figures. Composite numbers are rectangles & prime numbers are just lines.
3. Connecting operations in math with real life events. Relating addition to when you collect toys and subtraction with giving away some of your toys. Connecting division with sharing your lunch with your friends. Ratios are connected with recipes.
4. Connecting various topics in math. That fractions are related to division. Multiplication is related both the addition & division. Place value is related to algebra.
5. Connect the units of measurement of length, weight & money with the decimal place value system.
6. In general connecting any new knowledge with what you already know. This is one of the fundamental ways in which humans learn.
Representing/ Modelling Math Ideas & Processes
After understanding math, the second step in “mathematical thinking” is to “represent” your idea with materials & other tools so as to communicate your understanding to another person.
1. Representing a 2-digit number with bundles & sticks and a 3-digit number with FLPs (Flats, Longs & Pieces.)
2. Representing a fraction with various geometrical shapes like circle, rectangle & square.
3. Represent a multiplication fact with “repeated addition”, array and area models.
4. Represent the idea of a ratio by mixing a juice of a particular taste.
5. Represent how any triangle can be “transformed” into a rectangle.
6. Represent the idea of two triangles that are similar using paper models.
Communicating Mathematics
The third step in “mathematical thinking” is to communicate your math ideas in spoken & written form. This requires a deep understanding of the vocabulary & syntax of math.
1. Understand that many words used in math are also used in daily-life with a different meaning. Many words sound similar but have different spellings. Example of such words are interest, sum, imaginary, improper etc.
2. Many words in the math classroom are almost never used outside it. Examples are hypotenuse, scalene etc.
3. Ability to understand someone else’s exposition of math ideas, evaluate them and reply accordingly.
Mathematical Reasoning & Proof
The fourth step in “mathematical thinking” is understanding logical reasoning & the idea of proof.
1. Understand the difference between a fact, convention and concept. That 3 X5 =15 is a fact. The rule that multiplication & division are to be done before addition & subtraction is a convention. Understanding what is a prime number is a concept.
2. Have a good understanding of the various properties of & relations between numbers, shapes & operations & laws of arithmetic since these would be used in the logical arguments.
3. Structure your argument in logical and easy to understand arguments using appropriate vocabulary.
4. Keep the "mental abilities" of the other person so that the arguments & examples would make sense to him/ her.
5. Ability to follow a logical argument from the initial statement to the final statement, checking whether each step follows logically from the previous step.
6. Understand various kinds of reasoning & methods of proofs – additive, multiplicative, proportional & functional.
Problem Solving
The final step in “mathematical Thinking” is to develop skills of problem-solving.
1. A problem-solving exercise is one that does not have any “readily available” algorithm.
2. It needs a good mastery of the above 4 skills.
3. In addition to the above, it needs certain “attitudes and beliefs” – self-confidence, persistence & not being discouraged by failures.
4. Gather a collection of problem-solving strategies and apply them to new problems. Some of these are:
a. Specialising – trying special cases, looking at examples
b. Generalising - looking for patterns and relationships
c. Conjecturing – predicting relationships and results
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