Explain reasoning

Welcome to "Explain Reasoning"

In the world of Thinking Skills, finding the right answer is only half the battle. The other half—and often the most important part for your grade—is being able to explain how you got there. This chapter focuses on the skill of explaining reasoning within problem-solving scenarios.

Whether you are sitting for the 2027 or 2028 exams, the ability to communicate your logic clearly is a core requirement, especially in Paper 3 (Problem Analysis and Solution). Don't worry if this feels a bit like "writing an essay in a math paper"; it’s actually just about making your thoughts visible to the examiner!

What does "Explain" actually mean?

According to the official syllabus, to Explain means to:

1. Set out purposes or reasons.
2. Make the relationships between different pieces of information clear.
3. Say why and/or how something happened and support it with evidence.

In a problem-solving context, you aren't just showing a calculation; you are telling the "story" of the calculation. You are showing how \( \text{Fact A} \) and \( \text{Fact B} \) lead logically to \( \text{Conclusion C} \).

Why is explaining so important?

The examiners use your explanations for two main reasons:

1. Partial Credit: In Paper 3, credit is often awarded for correct steps towards a solution, even if your final answer is wrong. If you don't explain your steps, the examiner can't give you those "method marks."

2. Required Evidence: For some questions (like those asking for a "Yes" or "No" answer), credit might not be awarded at all if the explanation is missing. You can't just guess; you have to prove you know why.

Step-by-Step: How to Explain Your Reasoning

If a question asks you to "Explain why..." or "Explain how...", follow these three steps:

Step 1: Identify the starting pieces of information

Start by clearly identifying the data you are using.
Example: "The train departs at \( 14:00 \) and the journey takes \( 45 \) minutes."

Step 2: Show the logical connection (The "How")

Connect those pieces of information using a calculation or a logical rule.
Example: "Therefore, the arrival time is \( 14:00 + 45 \text{ minutes} = 14:45 \)."

Step 3: State the relationship to the goal (The "Why")

Finish by showing how this meets (or fails to meet) the criteria in the problem.
Example: "Since the meeting starts at \( 14:40 \), the traveler will be \( 5 \) minutes late because \( 14:45 \) is later than \( 14:40 \)."

Real-World Example: The Discount Dilemma

The Scenario: A shop offers a "Buy one, get the second half-price" deal. A student says this is the same as a \( 25\% \) discount on both items. Explain why the student is correct.

How to explain it:

1. Identify: Let the price of one item be \( \$P \). Two items at full price would cost \( \$P + \$P = \$2P \).
2. Process: Under the deal, the total cost is \( \$P + (\$P \div 2) = \$1.5P \).
3. Connect: The saving is \( \$2P - \$1.5P = \$0.5P \).
4. Conclude: As a fraction of the original cost, this saving is \( \frac{\$0.5P}{\$2P} \), which equals \( \frac{1}{4} \) or \( 25\% \). This shows the student is correct because the total price paid is \( 75\% \) of the original total.

Common Mistakes to Avoid

1. The "Naked Number" Error: Writing down an answer (e.g., "\( 42 \)") without explaining what it represents. Is it \( 42 \) dollars? \( 42 \) minutes? Is it the difference between two values? Always label your numbers.

2. Circular Logic: Saying "It's correct because the numbers work out." You must show how they work out.

3. Assuming the Examiner is a Mind-Reader: The examiner cannot see your scratchpad. If you made a deduction in your head (e.g., "I realized that the shop must be closed on Sundays"), you must write it down as part of your explanation.

Quick Tip: Use connecting words like "Therefore," "Because," "Since," and "This means that..." to make your logic easy to follow.

The 2028 Syllabus Update: A Note on "Communicating Reasoning"

If you are taking exams in 2028 or later, there is an even greater focus on this skill. Paper 3 has been restructured to specifically test "AO2: Create and communicate reasoning." In these newer exams, the syllabus explicitly asks for values and calculations to be labelled with units and words (for example, instead of just writing \( 10 \times 5 \), write "Total income = \( 10 \text{ toys} \times \$5 \text{ per toy} = \$50 \)").

Section Summary: Key Takeaways

1. Explain = Why + How: Don't just provide the "What" (the answer). Show the relationship between the facts.

2. Be Explicit: State every step of your logic, even the parts that seem "obvious" to you.

3. Use Labels: Give your numbers meaning by using units (\( \$, \text{kg, hours} \)) and descriptive words.

4. Secure the Marks: In complex Paper 3 problems, your explanation is your safety net. It allows you to earn points even if you make a small calculation error at the end.

To learn more about related skills, check out the chapters on "Justifying the solution to a problem" and "Showing and labelling working clearly."