Extending High Expectations to ALL Students

Glenda Anthony

Massey University

New goals, new challenges

Raising the floor by expanding achievement for all.

Lifting the ceiling of achievement to better prepare future leaders in mathematics.

Mathematics is a key resource for building a socially just and diverse democracy (Ball, 2005)

Critical lever for social and educational progress (Moses & Cobb, 2001)

Changing discourse

Raising teacher expectations Closing the gap Shortening the tail

Extending high expectations to all students Inclusive pedagogies, responsive pedagogies Integrating the multiples voices of the classroom to

orchestrate/occasion learning spaces/opportunities Teacher agency

Complexity of occasioning learning

Providing Appropriate challenge

Establishing participation

rights and understandings

Taskcognitive demands

Participation

Attention to the rights and obligations of mathematical participation

Inclusiveness, positioning (positive mathematical identity)

Culture of respect and care Valuing of students’ contributions

We all felt like a family in maths. Does that make sense? Even if we weren’t always sending out brotherly/sisterly vibes. Well we got used to each other… so we all worked…We all knew how to work with each other…it was a big group…more like a neighbourhood with loads of different houses. (BES p. 58)

Who can participate?

Top and bottom sets (BES p. 120) Immigrants and locals (BES p.65) Fast and slow kids (BES, p. 123) Busy versus challenging work (BES, p. 127)

See Empsons fraction CASE 2

Differential access to the curriculum

Do students in low streamed classes/groups have poorer access to mathematics.

They follow a protracted curriculum.

Their reduced social obligations and lesser cognitive demands placed on low streamed students had the effect of excluding them from full engagement in mathematics.

Culture of mathematical proficiency

Okay to make mistakes More than a climate of politeness Caring about the development of

mathematical proficiency

Mathematical thinking

Annie and Sam, in Year 1 both know that

4 + 2 = 6 and 3 + 3 = 6.

Is that good?

Michael: In a CGI class

How is he positioned and included in the classroom community?

What sorts of understandings is he forming about learning mathematics?

How is his mathematical identity developing?

What are the key pedagogical practices?

Tasks need to be purposeful and provide an appropriate challenge It’s safer—children feel more comfortable if

they’re not made to think. I realise this is cynical—but for many children with low IQs and poor/non existent English language skills, the concept of problem solving is alien. Also it takes up too much time and there is great pressure to “get through” the curricula. So whilst in theory I acknowledge the potential of problem solving, in reality with some clientele it’s too hard. (Anderson, 2003, p. 76)

Focus on what is learnt rather than what is completed Ms Summers: You’ve finished! Doesn’t it feel

good when you’ve done it? (Late in Y 3)

Mrs Kyle: How many finished? (Looking around at the show of hands) Most of you didn’t finish. You must learn to put ‘DNF’—did not finish, at the bottom. (Early in Y 4)

Ms Torrance:We have some amazing speedsters who have got on their rollerblades and got their two sheets done already. (p. 206)

Classroom Tasks

Analysing Mathematical Tasks

Tasks lead to different opportunities

Low levels of cognitive demands

Memorisation tasks Procedures without connections

High level of cognitive demand

Procedures with connections Doing mathematics

Low-level demand tasks Involve reproducing previously learned facts, rules,

formulae, or definitions OR committing facts, rules, formulae, or definitions to memory.

Cannot be solved using procedures because a procedure does not exist or because the time frame in which the task is being completed is too short to use a procedure.

Involve exact reproduction of previously seen material and what is to be reproduced is clearly and directly stated.

Have no connection to the concepts that underlie the facts, rules, formulae, or definitions being learned and reproduced.

Memorisation task

What are the decimal and percent equivalents for the fractions ½ and ¼ ?

Expected Student Response:

½ = .5 = 50%

¼ = .25 = 25%

Procedures without connections

Are algorithmic. Procedure is either specifically called for or its use is evident based on prior instruction, experience, or placement of the task.

Little ambiguity about what needs to be done and how to do it.

Have no connection to the concepts or meaning that underlie the procedure being used.

Focus on producing correct answers.

If required, explanations focus solely on describing the procedure that was used.

Procedures without connections task

Convert the fraction 3/8 to a decimal and a percent.

Expected Student Response:

Fraction Decimal Percent 3 0.375 .375 = 37.5% 8

8 3.000 24 60 56 40 40

Procedures with connections

Focus attention on the use of procedures for the purpose of developing deeper levels of understanding

Suggest pathways (explicitly or implicitly) that are broad general procedures that connect to underlying conceptual ideas as opposed to algorithms.

Usually are represented in multiple ways (e.g., diagrams, manipulatives, symbols, problem situations).

Require cognitive effort. Although general procedures may be followed, they cannot be followed mindlessly. Students need to engage with the conceptual ideas that underlie the procedures in order to successfully complete the task and develop understanding.

Procedures with connections task

Using a 10 x 10 grid, identify the decimal and percent equivalents of 3/5.

Expected Student Response:

‘Doing’ mathematics tasks

Require complex and nonalgorithmic thinking (i.e., there is not a predictable, well-rehearsed approach explicitly suggested by the task.

Require exploration and understanding of the mathematical concepts, processes, or relationships.

Demand self-monitoring of one’s own cognitive processes.

Require students to access relevant knowledge and experiences and make appropriate use of them.

Require students to analyse the task and actively examine task constraints that may limit possible solution strategies and solutions.

Require cognitive effort and may involve some anxiety due to the unpredictable nature of the solution process.

‘Doing mathematics’ task

Shade 6 small squares in a 4 x 10 rectangle.

Using the rectangle, explain how to determine each of the following:

a) the percent of area that is shaded,b) the decimal part of area that is shaded, andc) the fractional part of area that is shaded.

One possible student response

(a) One column will be 10%, since there are 10 columns. So four squares is 10%. Then 2 squares is half a column and half of 10%, which is 5%. So the 6 shaded blocks equal 10% plus 5%, or 15%.

(b) One column will be 0.10, since there are 10 columns. The second column has only 2 squares shaded, so that would be one-half of 0.10, which is 0.05, which equals 0.15.

(c) Six shaded squares out of 40 squares is 6/40, which reduces to 3/20.

What is the Task Cognitive Level?

The cost of a sweater at I.M Wolly’s was $45.

At the Waitangi day sale it was marked 30% off the original price. What was the price of the sweater during the sale? Explain the process you used to find the sale price?

Beware of superficial task features

Requires the use of a calculator or diagram Involve multiple steps to complete Requires a written explanation Has a real-world context Is the task worthwhile just because students

find it difficult?

Decline of high-cognitive demands

The higher the demands that a task placed on students at the task-set-up phase, the less likely it was for the task to have been carried out faithfully during the implementation phase.

Stein, M., Grover, B., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: an analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455-488.

Slip in demands

Students not held accountable for high-level products or processes.

Shift in emphasis from meaning to completion

Students press teacher to reduce complexity

Time: too little or too much.

To maintain high-level cognitive demands

Provide scaffolding of student thinking Provide means for students’ to monitor their

progress Yourself or your students model high

performance Require justification and explanation and

meaning Build on students’ prior knowledge Provide conceptual connections Allow sufficient time: not too much or too little!