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Mathematical Thinking CATs || Fault Finding and Fixing || Plausible Estimation
Creating Measures || Convincing and Proving ||
Reasoning from Evidence
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Classroom Assessment Techniques
'Convincing and Proving' Tasks
(Screen 2 of 4)
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Description
In the first collection of tasks, students are asked to evaluate a series of statements. These typically concern mathematical results or hypotheses, such as "The square of a number is greater than the number". Students are invited to classify each statement as "always true," "sometimes true," or "never true" and offer reasons for their decision. (In this case, for example, they should decide that the statement is sometimes true - when the number is negative or greater than one.) The best responses will contain convincing explanations and proofs; the weaker responses will typically contain just a few examples and counter-examples. These tasks vary in difficulty, according to the statements being considered and the difficulty of providing a convincing or rigorous explanation. These tasks can also diagnose student misconceptions, which often arise from over-generalizing from limited domains.
In the second collection of tasks, students are asked to evaluate "proofs," some of which are correct and others which are flawed. (For example, in one question, three 'proofs' of the Pythagorean theorem are given). The flawed "proofs" may be:
- inductive rather than deductive arguments which only work with special cases;
- arguments which assume the result to be proved; or
- arguments which contain invalid assumptions.
There are also some partially correct proofs that contain large unjustified jumps in reasoning. In these tasks, students are expected to identify the most convincing proof and provide critiques for the remaining attempts.
Examples of the Two Types of Tasks
1. "Always, Sometimes or Never True"
The aim of this assessment is to provide the opportunity for you to:
- test statements to see how far they are true;
- provide examples or counterexamples to support your conclusions; and
- provide convincing arguments or proofs to support your conclusions.
For each statement, say whether it is always, sometimes or never true. You must provide several examples or counterexamples to support your decision. Try also to provide convincing reasons for your decision. You may even be able to provide a proof in some cases.
The more digits a number has, then the larger is its value.
Is this always, sometimes or never true? ..........
Reasons or examples:
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2. Critiquing 'Proofs'
The aim of this assessment is to provide the opportunity for you to:
- evaluate 'proofs' of given statements and identify which are correct; and
- identify errors in 'proofs.'
1. Consecutive Addends
Here are three attempts at proving the following statement:
When you add three consecutive numbers, your answer is always a multiple of three.
Look carefully at each attempt. Which is the best 'proof'? Explain your reasoning as fully as possible.
Attempt 1:
Attempt 2:
3 + 4 + 5
The two outside numbers (3 and 5) add up to give twice the middle number (4).
So all three numbers add to give three times the middle number.
So it must be true
Attempt 3.
Let the numbers be:
Since
n + n + 1 + n + 2 = 3n + 3 = 3(n + 1)
It is clearly true.
The best proof is attempt number ..........
This is because ..........
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Assessment Purposes
The purposes underlying these tasks are twofold:
- to probe the conceptual understanding of students concerning the given statements, and
- to assess how well students are able to justify and prove.
Limitations
'Convincing and Proving' tasks are rather pure and mathematical in flavor. They have been constructed in this manner so that the situations are well-defined and unambiguous. Some students may not like this esoteric approach. They also require the use and analysis of algebra.
Tell me more about this technique:
Mathematical Thinking CATs || Fault Finding and Fixing || Plausible Estimation
Creating Measures || Convincing and Proving ||
Reasoning from Evidence
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