Assessing mathematical thinking for KS1, KS2 and KS3 teachers
Education Show 2016 An ATM presentation Heather Davis
Why do we assess? To find out what learners know for teaching them so they know more and giving ‘others’ information so …
Assessing whilst learning A rich task: • avoids dead time on tests • allows learners to show process skills • encourages independent use of content • gives opportunities to enhance progress • differentiates • develops mathematical thinking • allows teachers to assess thinking
What do you notice?
Write down, in order, some consecutive numbers. Choose either addition or subtraction to put between each number and the next to make a calculation. Work out the result. Repeat for a different arrangement of subtraction and addition, and for different sets of consecutive numbers.
Why does that happen?
Investigate further
Consecutive Integers For your Key Stage/Year: • What outcomes would you anticipate? • What content is likely to be involved?
• What process skills are likely to be used? • How can you move children on?
Consecutive Integers KS1 Y1 Read, write and interpret mathematical statements involving addition (+), subtraction (–) and equals (=) signs. Y1 Add and subtract one-digit and two-digit numbers to 20. Why does that happen?
What happens if you add two consecutive numbers?
What happens if you put a subtraction sign between the numbers?
Consecutive Integers KS1 Y2 Recall and use addition and subtraction facts to 20 fluently. WM Recognise patterns in numbers e.g. odds, evens.
Why does that happen?
What if you try different sets of three consecutive numbers?
What do you notice?
Consecutive Integers KS2 WM Reason mathematically by following a line of enquiry, conjecturing relationships and generalisations.
Notices that the answers are always even and that the - - + answer is always zero, and attempts to explain why.
Y4 Count backwards through zero to include negative numbers.
Calculates the remaining three answers by counting back through zero.
What do you notice about your sets of answers for each set of numbers? Why is that?
Consecutive Integers KS2 Y6 Use simple formulae.
Writes the numbers as N, N+1, N+2, N+3.
WM Make, test and justify conjectures in respect of patterns and relationships. Y6 Enumerate possibilities of combinations of two variables.
Call the first number of the four consecutive numbers N, and write formulae for the others using N. Develops an argument that may involve some manipulation of the algebraic expressions. Lists all 16 of the arrangements of + and – signs.
Consecutive Integers KS3 WM Begin to reason deductively in algebra. Use and interpret algebraic notation.
Why do some of the arrangements give the same answer each time?
Identifies that the expression for the arrangements with two – operations do not involve n so always give the same answer. WM Look for proofs or counterexamples.
What else do you notice?
Notices that the expressions for the other calculations are identical to the nth terms.
In the classroom Be realistic about what learners can reach without support but prompt rather than hint.
Learners may not think of calling the first number N but they can work out an expression for the next number (N+1)
In the classroom Hints remove the opportunity for learners to do it for themselves.
Work done in response to hints does not suggest confidence and competence (mastery) with the material. Without any intervention (prompts) learners may make no progress.
In the classroom How can you best use what the learners have done in this task for assessment?
Prompts follow immediate assessment of the need for them. What could you record?
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