Transcript Extending High Expectations to ALL Students

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
Task
cognitive demands
Providing
Appropriate challenge
Establishing
participation
rights
and understandings
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
You’ve finished! Doesn’t it feel good
when you’ve done it? (Late in Y 3)
 Ms Summers:
 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
3
8
Decimal
0.375
8 3.000
24
60
56
40
40
Percent
.375 = 37.5%
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, and
c) the fractional part of area that is shaded.
One possible student response
(a)
(b)
(c)
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%.
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.
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!