Assessing pupils’ understanding through rich classroom activity 21st June 2012 Alan Easterbrook

Starter:  Write down five terms of a linear sequence

Starter:  Write down five terms of a linear sequence  Pair up with a partner and decide who is x and who is y  Plot the set of 5 coordinates

Starter:  Write down five terms of a linear sequence  Pair up with a partner and decide who is x and who is y  Plot the set of 5 coordinates  What do you notice? Why?  What if you swapped x and y?

Starter:  Which topics might we have strayed into?

Starter: Transforming graphs? Inverse functions? Symmetry? n th term? Graphs? Ratio? Gradient? Reciprocals? Sequences

Starter: Relational understanding

Ofsted: Made to measure “few of the schools surveyed successfully planned and taught for strong mathematical progression over time, reflected in depth of conceptual understanding... This is a key area for development for any school wishing to improve pupils’ achievement.” Para 181

 In the very best schools, all lessons had a clear focus on thinking and understanding... Para 51 Ofsted: Made to measure

 In the very best schools, all lessons had a clear focus on thinking and understanding...  Whole-class teaching was dynamic with pupils collaborating extensively with each other. Para 51 Ofsted: Made to measure

 In the very best schools, all lessons had a clear focus on thinking and understanding...  Whole-class teaching was dynamic with pupils collaborating extensively with each other.  It challenged them to think for themselves, for instance by suggesting how to tackle a new problem or comparing alternative approaches. Para 51 Ofsted: Made to measure

 In the very best schools, all lessons had a clear focus on thinking and understanding...  Whole-class teaching was dynamic with pupils collaborating extensively with each other.  It challenged them to think for themselves, for instance by suggesting how to tackle a new problem or comparing alternative approaches.  Teachers’ explanations were kept suitably brief and focused on the underlying concepts, how the work linked with previous learning and other topics and, where appropriate, an efficient standard method. Para 51 Ofsted: Made to measure

 In the very best schools, all lessons had a clear focus on thinking and understanding...  Whole-class teaching was dynamic with pupils collaborating extensively with each other.  It challenged them to think for themselves, for instance by suggesting how to tackle a new problem or comparing alternative approaches.  Teachers’ explanations were kept suitably brief and focused on the underlying concepts, how the work linked with previous learning and other topics and, where appropriate, an efficient standard method.  Their questions were designed to encourage pupils to give reasoned answers. Para 51 Ofsted: Made to measure

 In the very best schools, all lessons had a clear focus on thinking and understanding...  Whole-class teaching was dynamic with pupils collaborating extensively with each other.  It challenged them to think for themselves, for instance by suggesting how to tackle a new problem or comparing alternative approaches.  Teachers’ explanations were kept suitably brief and focused on the underlying concepts, how the work linked with previous learning and other topics and, where appropriate, an efficient standard method.  Their questions were designed to encourage pupils to give reasoned answers. Para 51 Ofsted: Made to measure Sounds like rich tasks are essential....

 In the very best schools, all lessons had a clear focus on thinking and understanding...  Whole-class teaching was dynamic with pupils collaborating extensively with each other.  It challenged them to think for themselves, for instance by suggesting how to tackle a new problem or comparing alternative approaches.  Teachers’ explanations were kept suitably brief and focused on the underlying concepts, how the work linked with previous learning and other topics and, where appropriate, an efficient standard method.  Their questions were designed to encourage pupils to give reasoned answers. Para 51 Ofsted: Made to measure.... and they also provide a natural vehicle for assessment

Depth of conceptual understanding Making connections across topics Topic mats

Making connections across topics Topic mats What other topics could we make a mat for? Depth of conceptual understanding

Exposing the misconceptions  Adding and subtracting negative numbers What might the pupils do wrong? Where are the potential misconceptions?

The gap between them is 5, so the answer is 5 You just take two from seven so it’s - 9 Two negatives make a plus so it’s 9 Adding makes the number get bigger so it’s - 5 When you add a negative it’s the same as taking away so it’s

Concept cartoons  Alternative ideas represented ... alternatives given equal status  All ideas look plausible  Minimal text in dialogue form

Exposing the thinking – demonstrating progress ‘Talking is central to our view of teaching mathematics formatively... Providing opportunities for students to express, discuss and argue about ideas is particularly important in mathematics. Through exploring and unpacking mathematics, students can begin to see for themselves what they know and how well they know it.’ Hodgen & Wiliam (2006): Mathematics inside the black box

Exposing the thinking – demonstrating progress ‘Talking is central to our view of teaching mathematics formatively... Providing opportunities for students to express, discuss and argue about ideas is particularly important in mathematics. Through exploring and unpacking mathematics, students can begin to see for themselves what they know and how well they know it.’ Classroom talk is the key to assessing the progress in students’ understanding

Find all positive integer solutions of the inequality 3x – 6 < 9 It’s x < 5 I think x is 1, 2, 3 or 4 It’s the same as x < 3, so x is 1 or 2 I think x can only be 1 It’s like an equation so x is 5

Concept cartoons  How might you use them with your students?  How will they help you reach a judgement about your students’ understanding?  What do you need to be wary of?

Concept cartoons  Now let’s make one......

Always – sometimes - never Is this statement always true, sometimes true, or never true?  Can you prove it?  If sometimes, under what conditions is the statement true or false? Why is this a good assessment activity? If you multiply two numbers, then the answer is larger than each of them

A bit of maths...  Two numbers add to 1.  Which will be larger: The square of the larger plus the smaller... Or The square of the smaller plus the larger [from ‘Designing & using mathematical tasks’ John Mason & Sue Johnston-Wilder Tarquin / OUP]

Cognitive conflict  Why did that happen?????

Other activities that reveal thinking  Concept cartoons  Always – sometimes – never Always – sometimes – never  I like – I don’t like I like – I don’t like  Odd One Out Odd One Out  Hard and easy – and difficulty sort Hard and easy – and difficulty sort  Same and different Same and different  Agree or disagree Agree or disagree  Dear Doctor Dear Doctor

A bit more maths.... What fraction is the dotted line of the width of the rectangle?

A bit more maths.... What topics might this touch on?

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I likeI don’t like Back

Odd one out y = x + 2y = x + 3y = 2 - x Which is the odd one out, and why? Make up a fourth equation so you have two pairs of cards with the same properties Back

Always – sometimes - never Are these statements always true, sometimes true, or never true?  n 2 > 5n  A set of whole numbers has a median which is not a whole number Can you prove it? If sometimes, under what conditions is the statement true or false? Back

Same and different What is the same and what is different about:  sin x and cos x ?  a square and a rhombus ?  a reflection and a translation ? Back

Hard and easy Think of an easy example and a hard example of…  A linear equation to solve  A calculation with percentages What makes one harder than the other? … and now extend the idea to a difficulty sort Back

Agree or disagree Do you agree or disagree with these statements…  Between every pair of numbers there is a rational number  The mean of five numbers is always larger than the median Can you convince your partner? Back

Dear Doctor Back Dear Doctor I know that to multiply fractions together you times the tops then times the bottoms. So why don ’ t you add the tops then add the bottoms if you need to add two fractions? Dee Nominator Dear Dee,

The gap between them is 5, so the answer is 5 You just take two from seven so it’s -9 Two negatives make a plus so it’s 9 Adding makes the number get bigger so it’s -5 When you add a negative it’s the same as taking away so it’s Back