What are mathematical processes?
Facts and formulas are the what of maths. Processes are the how: the habits you use while working. Most school systems name about seven:
- Problem solving
- Reasoning and proving
- Reflecting
- Connecting
- Representing
- Selecting tools and strategies
- Communicating
They are not a separate chapter. You use them in every topic: fractions, algebra, geometry, data.
Problem solving: the four-step loop
- Understand: What is asked? What do I know? Underline numbers and units.
- Plan: pick a strategy. Draw a picture, make a table, look for a pattern, work backwards, try a simpler case, guess and check.
- Do: carry out the plan carefully.
- Look back: does the answer fit the question? Is there another way?
If the check fails, go round the loop again. Being stuck is a normal part of the loop, not a sign you are bad at maths.
Reasoning and proving
Reasoning means explaining why. A proof is a reason that works for every case.
- Checking examples (3 + 5 = 8, 7 + 9 = 16) gives a conjecture (a smart guess): odd + odd is even.
- A proof explains it: each odd number is pairs plus one spare; two spares make one more pair, so nothing is left over.
- One counterexample is enough to show a claim is false. "All primes are odd" fails because 2 is prime and even.
Representing and connecting
One idea can be shown in many forms:
- Concrete: real objects, like cubes or coins.
- Pictorial: drawings, arrays, number lines.
- Symbolic: numbers and equations, like 3 × 4 = 12.
- Tables and graphs: to see patterns.
Moving between forms deepens understanding. Connecting means linking ideas: ½ = 0.5 = 50%; area of a rectangle links to multiplication; a speed graph links to a real journey.
Tools, reflecting and communicating
Selecting tools and strategies
Use mental maths for friendly numbers (25 × 4), paper for many steps, a calculator for messy numbers, and graphing software or a spreadsheet to explore patterns.
Reflecting
Ask: is the answer sensible? Estimate first. 9.9 × 20 ≈ 10 × 20 = 200, so 198 is believable and 19.8 is not. Check units and signs.
Communicating
Write one step per line, use = signs correctly, give units, and finish with a sentence that answers the question. Use maths words: sum, product, factor, equation.
Try it: the array game
In the 3D, move the sliders to make rows and columns. Predict the total and whether it is odd or even before you let go. Then check. Can you find a rule for when rows × columns is odd? (Hint: both must be odd.) At home, do the same with coins or buttons on a table.
Key formulas and definitions
- Problem-solving loop: understand → plan → do → look back
- Conjecture = smart guess from examples; proof = reason for every case
- Counterexample: one case that breaks a claim
- Representations: concrete → pictorial → symbolic → table/graph
- Reflect: estimate first, then compare with your answer
- Communicate: one step per line, units, answer sentence
Worked examples
1. A pencil costs ₹12 and a notebook costs ₹35. How much do 3 pencils and 2 notebooks cost? Use the four-step loop.
Understand: total cost of 3 pencils and 2 notebooks. Plan: multiply then add. Do: 3 × 12 = 36; 2 × 35 = 70; 36 + 70 = 106. Look back: estimate 3 × 10 + 2 × 35 = 100, close to 106. Answer: ₹106.
2. Show that the sum of two even numbers is always even.
An even number is made only of pairs (2a). Another even number is also pairs (2b). Together they are 2a + 2b = 2(a + b): still only pairs, nothing left over. So the sum is always even.
3. Represent 'a taxi costs ₹50 to start plus ₹15 per km' in three ways.
Words: as above. Table: 0 km → ₹50, 1 km → ₹65, 2 km → ₹80, 4 km → ₹110. Equation: C = 50 + 15d. Graph: a straight line starting at 50 and rising 15 for each km.
4. Riya says 4.8 × 0.5 = 24. Reflect: is she right?
Multiplying by 0.5 means taking half, so the answer must be smaller than 4.8. Half of 4.8 is 2.4. Riya misplaced the decimal point. Correct answer: 2.4.
5. Is the claim 'every number multiplied by itself is bigger than the number' true?
Try cases: 3 × 3 = 9 (bigger). But 0.5 × 0.5 = 0.25 (smaller) and 1 × 1 = 1 (equal). These counterexamples show the claim is false.
6. 42 people go on a trip. Each bus has 30 seats. How many buses?
42 ÷ 30 = 1.4. Looking back, buses come in whole numbers and nobody can be left behind, so round up: 2 buses.
Common mistakes
- Jumping to calculation before understanding what the question asks.
- Thinking a few examples prove a rule. Examples suggest; a reason proves.
- Never checking whether the answer is sensible (e.g. 1.4 buses, a negative length).
- Writing only the final number with no steps or units, so no one can follow the thinking.