Dealing with difference Peter Sullivan NB September 12 difference
Overview • The Australian context • Articulating the challenge of diversity of readiness • All groups are mixed ability • Nine strategies for teaching in mixed ability classrooms NB September 12 difference
The Australian school context • Decline in specialist mathematics study in senior years • Widespread dissatisfaction (from teachers, students and parents) with the way mathematics is taught • Emphasis on “telling” followed by practice • Overuse of texts designed for practice rather than learning (and certainly not for fostering creativity and student decision making) • Disconnected from meaning and relevance • Belief by teachers (and parents and students) that some (many?) cannot learn mathematics
There is also • Difficulty in recruiting secondary mathematics teachers … • … meaning that some junior mathematics teachers may lack confidence in their own mathematical knowledge
Of course, it is difficult to teach when there are … • … fast learners in the class who shout answers and criticise others who are thinking through problems that the fast learners have already solved, and who complain to their parents about being under-extended • … some other learners who have more or less given up believing that they cannot learn, and who prefer to interrupt others • … extensive and exhaustive lists of content to cover that pressure teachers to skim from topic to topic • … routines in schools that leave teachers with limited time for collaboration, sharing ideas, innovating, …
Nature of challenge • Teachers stressed by kids who cannot do it • Attendance • Wilful indifference to work • Lots of feeder schools (78?) • How to cater for the dfferences (inc in the top class) • Teacher headsets • VET • Pitching the lessons • How to “hold back”
Cognitive load • Classrooms are complex settings and there is usually too much going on for those students who are falling behind to know what to focus on
Sensory register Working Memory Selection Attention Long term memory
Social rationale • Classrooms are also social settings ... and no one wants to feel that they cannot cope or contribute • Performance avoidance, identity and other issues
What classroom organisational factors might restrict opportunities for high achieving students?
What is the issue with difference? • ACARA says there is a 5 year gap: Cockcroft reported a 7 year gap • While we know that there are factors contributing to differences in readiness (Indigenous, SES, rural, gender), even within these subgroups there is the same degree of diversity • Differences in readiness refer not only to achievement, but aspirations, expectations, resilience, mindsets, confidence, satisfaction, etc
But these groups are also diverse PISA and socio economic background
But my school has very weak students … Between-school differences account for approximately 22 per cent of the variation in students’ tertiary entrance scores. About half of this between school variation can be accounted for by differences between schools in individual student characteristics. About half of this variation can be accounted for by differences in the academic and socioeconomic mix of students and school sector.
On NAPLAN • Approximately one-sixth of the variation in achievement scores on both the reading comprehension tests and the mathematics tests could be attributed to differences between schools. • This finding is similar to findings for Australian students who participated in TIMSS and PISA, two recent international studies of student achievement. • A little more than one half of this between-schools variance could be explained by differences in the student composition—school socioeconomic status (SES) and the proportion of students from language backgrounds other than English in the school—and the school climate.
What is the Australian mathematics curriculum hoping for? • development of expert mathematicians • expert users of mathematics in the professions • a workforce capable of meeting all numeracy requirements • citizens able to use the mathematics they need
This is even described as an entitlement … • of each student to knowledge, skills and understandings that provide a foundation for successful and lifelong learning and participation in the Australian community. (p.10) The document also makes the explicit assumption … • that each student can learn and the needs of every student are important. It enables high expectations to be set for each student as teachers account for the current levels of learning of individual students and the different rates at which students develop. (p.10)
What are you hoping for? • Creativity, imagination, adaptability, willingness to think, make decisions, persist, … • … and life long learners (note concern about following CFC models) … • … or correct answers, compliance, acceptance of place in life, …
Some fundamental principles • All can learn • Effort increases ability as well as achievement • We do not learn by listening and teachers do not foster creativity, insight, etc by telling • Much learning is social, so experiences in which the whole class participates contribute to building a class community
Some connections with “curriculum” • In what ways should the learning experiences of 8 year olds who create mathematics easily differ from those of a 13 year old who experiences difficulty in learning? • Is mathematics a hierarchy of micro skills that need to be taught sequentially or a network of mainly connected ideas? • How might the ways that teachers address diversity have an impact on the ways students experience the curriculum? • Which is easier to learn: • comparison of fractions or co-ordinate geometry? • division of fractions or index laws? • reading analogue clocks or vectors?
Heterogeneous grouping can have negative impact if … • … teachers set expectations and starting points based on low achieving students • … teachers over direct the learning (assuming low achieving students cannot cope) which has the effect of encouraging a fixed mind set in the students • … there is negative peer pressure on hard working students • … teachers ignore the diversity of readiness and instead treat everyone as the same (possibly by giving routine tasks that everyone can and is willing to do) • … teachers teach different content to different groups • … low achieving students “performance avoid” either by • misbehaving or • being a group work passenger or • pretending to work (they are good at it)
Homogeneous grouping can restrict student opportunities if … • … teachers teach different content to different groups, thereby narrowing options of students in some groups • … there is limited or no movement between groups (if there is no chance of “promotion” why would students try hard?) • … teachers are not conscious of the impact of “self-fulfilling prophesy” effects • … steps are not taken to avoid development of poor self concept of some members of the upper sets (Big fish little pond effect) • … the top set are taught (in a routine way) the content from the following year (for example) rather than fostering mathematical creativity (for example) using the current content • … the grouping fosters a sense of “entitlement” in top stream students
Self-fulfilling prophesy Step 1: Teachers form early differential expectations for students Step 2: As a result the teachers behave differently to different students and this differential behaviour communicates the teachers’ expectations to the students. If such treatment of the students is consistent, and if the students do not resist, it will have an effect on their self- concept, achievement, motivation, aspirations and classroom conduct. Step 3: The student responses will actually reinforce the teacher’s original expectations. Ultimately there will be a difference in student achievement and outcomes.
The ways that self fulfilling prophesy works • Teachers “need for control” • Attribution • Challenge
Brophy argued that it is common for teachers, when interacting with “low expectation” students, … • - wait less time for them to answer questions; • - give them the answer or call on someone else rather than waiting; • - use inappropriate reinforcements; • - criticise them more for failure and praise them less frequently; • - do not give them public feedback on their responses; • - call on them less to respond; • - demand less; and • - have less friendly verbal and non-verbal contact.
But it is hard … • … if the curriculum is based on stratification to ensure that all students have the same opportunities to learn • … if you see the need of low stream students is development of skills in isolation to communicate connections, meaning and relevance • … to communicate to students in low streams that you think they can learn well (especially if they have a restricted curriculum)
You have heard about the importance of “evidence” • Stratification, streaming, tracking, setting has “… minimal effect on learning outcomes and profound negative equity effects. (Hattie, 2009, p. 90)
But every group is mixed so lesson planning needs to anticipate differences in readiness
Nine task based strategies for dealing with diversity while offering experiences covering common content
Common to all 9 approaches • Building classroom community • Task based, considering the trajectory of tasks (what comes next!) • Explicit pedagogies • Different ways of solving the tasks, and the different approaches are themselves educative • Representing solutions in different ways is both engaging and important mathematically
A possible way of structuring… • Lappan used the structure • Launch, • Explore, • Summarise • Better is (from Walqui) • Preparing learners • Interacting with the concept • Extending understanding • But that process is cyclical and might happen multiple times in a lesson (or learning sequence)
What these approaches are not! • Asking questions that are so easy that everyone can do them • Setting up groups that allow some students to hide • Excessive repetition (of course, some is needed) • …
Strategy 1 • Asking questions with multiple entry points and multiple exit points • Such questions will nearly always be open-ended
Write a sentence with 5 words, with the mean of the number of letters in the words being 4. Do not use any words of 4 letters.
Draw some rectangles with a perimeter of 20 cm. Work out the area of each of your rectangles.
A set of 36 cubes is arranged to form a rectangular prism. What might the rectangular prism look like? What is the surface area of your prisms?
Strategy 2 • Using enabling and extending prompts • These apply to any type of challenging task
Suppose we posed this task: Seven people went fishing. The mean number of fish they caught was 5, the median was 4 and the mode was 3. How many fish might each of the people have caught? (Give at least 3 answers)
Some enabling prompts • Seven people went fishing. The median number of fish caught was 4. How many fish might each of the people have caught? • Seven people went fishing. The mode number of fish they caught was 3. How many fish might each of the people have caught?
What are enabling prompts? • Enabling prompts can involve slightly varying an aspect of the task demand, such as • the form of representation, • the size of the numbers, or • the number of steps, so that a student experiencing difficulty, if successful, can proceed with the original task. • This approach can be contrasted with the more common requirement that such students • listen to additional explanations; or • pursue goals substantially different from the rest of the class.
Extending prompt • Seven people went fishing. The mean number of fish they caught was 5, the median was 4, the mode was 3, and the range is 6. How many fish might each of the people have caught? (Give all possible answers)
A brick weighs the same as 3 kg plus half a brick. What does the brick weigh? Represent your answer in twodifferent ways, one of which involves drawings
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A brick weighs the same as 3 kg plus half a brick. What does the brick weigh? 3 kg SA coaches day 5 September 12
3 kg SA coaches day 5 September 12
I used 1 metre of string to tie up this box. The bow takes 300 mm. What might be the dimensions of the box?