Go to Collaborative Learning Go to FLAG Home Go to Search
Go to Learning Through Technology Go to Site Map
Go to Who We Are
Go to College Level One Home
Go to Introduction Go to Assessment Primer Go to Matching CATs to Goals Go to Classroom Assessment Techniques Go To Tools Go to Resources

Go to CATs overview
Go to Attitude survey
Go to ConcepTests
Go to Concept mapping
Go to Conceptual diagnostic tests
Go to Interviews
Go to Mathematical thinking
Go to Fault finding and fixing CAT
Go to Plausible estimation CAT
Go to Creating measures CAT
Go to Convincing and proving CAT
Go to Reasoning from evidence CAT
Go to Performance assessment
Go to Portfolios
Go to Scoring rubrics
Go to Student assessment of learning gains (SALG)
Go to Weekly reports

Mathematical Thinking CATs || Fault Finding and Fixing || Plausible Estimation
Creating Measures || Convincing and Proving || Reasoning from Evidence

Go to previous page

Classroom Assessment Techniques
'Convincing and Proving' Tasks

(Screen 2 of 4)
Go to next page

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:

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:

    1 + 2 + 3 = 6
    2 + 3 + 4 = 9
    3 + 4 + 5 = 12
    4 + 5 + 6 = 15
    5 + 6 + 7 = 18
    3 x 2 = 6
    3 x 3 = 9
    3 x 4 = 12
    3 x 5 = 15
    3 x 6 = 18

    And so on.
    So it must be true.

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:
      n, n + 1 and n + 2
      n + n + 1 + n + 2 = 3n + 3 = 3(n + 1)
    It is clearly true.

The best proof is attempt number ..........

This is because ..........

Assessment Purposes
The purposes underlying these tasks are twofold:

'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.

Go to previous page Go to next page

Tell me more about this technique:

Mathematical Thinking CATs || Fault Finding and Fixing || Plausible Estimation
Creating Measures || Convincing and Proving || Reasoning from Evidence

Got to the top of the page.

Introduction || Assessment Primer || Matching Goals to CATs || CATs || Tools || Resources

Search || Who We Are || Site Map || Meet the CL-1 Team || WebMaster || Copyright || Download
College Level One (CL-1) Home || Collaborative Learning || FLAG || Learning Through Technology || NISE