Assessment for Learning in the Mathematics classroom

Enrichment workshop Sessions 2 + 3

Assessment for Learning in the Mathematics classroom

"The most important single factor influencing learning is what the learner already knows. Ascertain this and teach him accordingly.”

Ausubel (1968)

Educational Psychology: A Cognitive View

The challenge:

To use evidence about learning to adapt teaching and learning to meet student needs.

Five Key Considerations…

• Assessing learners’ prior knowledge • Adapting lessons in response to what learners do • Looking for evidence of learning • Offering constructive feedback • Questioning for depth of understanding

1. Assessing learners’ prior knowledge, identifying misconceptions

Teacher’s role: watching, listening, questioning

Factors and Multiples Puzzle

Completing Quadrilaterals

I started drawing some quadrilaterals - can you complete them?

Isosceles Triangles

Draw some isosceles triangles with an area of 9 cm2 and a vertex at (20, 20). If all the vertices have whole number coordinates, how many is it possible to draw?

2. Starting with big ideas, sharing learning intentions

Teacher’s role: adapting the lesson in response to what learners do

Offer accessible starting points and opportunities for interesting mathematical journeys.

Low Threshold – High Ceiling tasks

Careful use of questions and prompts

• What have you found out so far? • Do you notice anything? • Is that always true? • Can you convince us? • Can anyone think of a counter example? • What if...? • What might you try next? • Is there a way you could organise your findings?

Tilted Squares

Can you find a quick and easy method to work out the areas of tilted squares?

Odds and Evens To play the game, mix up the balls and randomly pick two balls out together. If the total is even, you win. If the total is odd, you lose. Is this a fair game?

How many odd and even balls do you need for a fair game?

Cuboid Challenge

Cut a square from each corner and fold up the flaps to make a box without a lid. What is the maximum possible volume of the box?

What’s Possible?

Give me a whole number… And another… And another… What questions does this prompt?

What’s Possible?

Can you find a way to write each number from 1 to 30 as the difference of two squares? Can you write any of them in more than one way? If I give you a number, can you tell me all the possible ways to write it as the difference of two squares?

3. Choosing activities that promote discussion and collaboration

Teacher’s role: looking for evidence of learning, offering opportunities to refine thinking

• Creating a safe environment for learners to take risks • Promoting a conjecturing atmosphere • Encouraging discussion and collaboration • Valuing half-formed ideas and learning from mistakes • Accepting 'messy' work • Valuing a variety of approaches • Providing thinking time

Building a community of mathematicians

Dialogic Teaching

“Children, we now know, need to talk, and to experience a rich diet of spoken language, in order to think and learn. Reading, writing and number may be acknowledged curriculum basics, but talk is arguably the true foundation of learning.”

Robin Alexander, 2004

“…language provides us with a means for

thinking together, for jointly creating knowledge and understanding.”

Neil Mercer, Words and Minds (2000)

M, M and M Can you find five positive whole numbers that satisfy the following properties:

Mean = 4 Mode = 3

Median = 3

Can you find all the different sets of five positive whole numbers that satisfy these conditions?

Dicey operations Find a partner and a 1-6 dice, or preferably a 0-9 dice. Each of you draw an addition grid. Take turns to throw the dice and decide which of your cells to fill - either fill in each cell as you throw the dice or collect all your numbers and then decide where to place them.

Throw the dice nine times each until all the cells are full. Whoever has the sum closest to 1000 wins.

Dicey operations

Roll a dice 5 times. Fill in the grid after each roll. Who can get closest to 10 000?

What will your strategy be?

Kite in a Square What fraction of the total area is shaded? Printable cards: Coordinates Similar figures Pythagoras

Reflection

• Choose one activity that we have discussed this morning. • Explain what makes it an Assessment for Learning activity. • Describe how you will apply/adapt it for your own practice. • Be prepared to share.

Lunch!

4. Providing feedback that moves learners forward

Teacher’s role: offering constructive feedback

Encouraging a growth mindset

Carol Dweck has shown that learners and teachers who believe that intelligence is

flexible, and the goal is to learn as much as they can, are more successful than those who

believe in finishing tasks and passing tests.

An expert learner?

WD-40 is the trademark of a widely used penetrating oil spray, developed in 1953 by Norm Larsen. WD-40 stands for “Water Displacement, 40th attempt", a name which came from Larsen's laboratory notebook – he was attempting to concoct a formula to prevent corrosion by displacing water, and arrived at the formula on his 40th attempt!

Square It

Players take it in turns to click on a dot on the grid. The winner is the first to have four dots that can be joined by straight lines to form a square.

Got It - a game for two players

• Decide who will go first. • Imagine there are 18 biscuits on a plate. • The first player can take 1, 2, 3 or 4 biscuits. • The second player can then take 1, 2, 3 or 4 biscuits. • Keep taking turns until all the biscuits are gone. • The person to take the last biscuit wins.

Play the game several times. Can you find a winning strategy?

What will you do:

– if there are a different number of biscuits?

– if you can take a different number of biscuits?

Got It continued…

Dozens Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

Route to Infinity

Can you list the coordinates of the points in the order in which they're visited? Can you develop a strategy for doing this?

5. Focusing on solutions, not answers

Teacher’s role: question for depth of understanding

If you want to build higher, dig deeper

“You can know the name of a bird in all the languages of the world, but when you're finished, you'll know absolutely nothing whatever about the bird... So let's look at the bird and see what it's doing – that's what counts.” Richard P. Feynman

Valuing higher order mathematical thinking • Thinking strategically • Looking for connections • Exploring and conjecturing • Reasoning and generalising • Justifying and proving

Valuing mathematical Habits of Mind

• Natural curiosity • Thinking mathematically • Working collaboratively • Being determined

“A focus on learning can enhance performance. A focus on performance can depress performance.”

Chris Watkins

Take the numbers 1, 2, 3, 4, 5, 6 and choose one to wipe out…

Wipeout

Pair products

Choose four consecutive whole numbers. Multiply the first and last numbers

together. Multiply the middle pair together. What do you notice?...

Warmsnug Double Glazing

Which window has been given the incorrect price?

Mind Reader?

Choose any two digit number, add together both digits and then subtract the total from your original number… Followed by Always a Multiple?

Five strands of mathematical proficiency

NRC (2001) Adding it up: Helping children learn mathematics

Reflection

Identify three ways that assessment for learning could enhance learning in your classroom. What changes will you make to your classroom practice as a result of these sessions?

Further resources from NRICH Maths

Enriching the Secondary Curriculum: http://nrich.maths.org/enriching

What we think and why we think it:

http://nrich.maths.org/whatwethink

All Cambridge endorsed, and suggested support materials can be found in the Resource Centre on the public website www.cie.org.uk/i-want-to/resource-centre/

Support materials and resources

http://cie.org.uk/teaching-and-learning/

We’ve produced a series of resources to support the teaching and learning in your school. They explore different aspects of educational practice, from designing a curriculum to improving the quality of classroom activity.

Question papers, mark schemes and principal examiner reports

PDF copies of these can be found on:

• Teacher Support http://teachers.cie.org.uk

• www.cie.org.uk

• Syllabus Support CD-ROM

Also available in hard copy via publications list.

Core documents

Break time!