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Research into practice: What we can learn from research into good tasks Peter Sullivan AISNSW 2010 What are the challenges you are experiencing in teaching mathematics your school? AISNSW 2010 • • • • • • • • • • • • Timeframe (too much curriculum) Kids did not problem based teaching Making it relevant, especially for low achievers Buildinhg confidence Diverse ability range Retention Busy lives of kids Extension External influences incluidng naplan, parent expectations n… Levels of concentration Gaps in prior knowledge Disruptions to school routine AISNSW 2010 Overview • Findings from research • The Australian mathematics curriculum • 5 principles for improving teaching AISNSW 2010 TASKS AND TEACHER ACTIONS • We investigated ways that particular types of mathematics classroom tasks create different opportunities for students and different challenges for teachers. • the type of task influences the nature of the learning (e.g., Christiansen & Walther, 1986; Hiebert & Wearne, 1997) AISNSW 2010 Task processing model from task to lesson (Stein, 1996) • Mathematical task as presented in instructional materials – which, influenced by the teacher goals, their subject matter knowledge, and their knowledge of students, informs … • … mathematical task as set up by the teacher in the classroom – which, influenced by classroom norms, task conditions, teacher instructional habits and dispositions, and students learning habits and dispositions, influences … • … mathematical task as implemented by students – which creates the potential for … • … students learning. AISNSW 2010 Teachers transform tasks • Stein et al. (1996) noted the tendency of teachers to reduce the level of demand of tasks. • Doyle (1986) and Desforges and Cockburn (1988) attribute this to complicity between teacher and students to reduce risk • Tzur (2008) … two key ways that teachers modify tasks: – at the planning stage; – if responses are not as intended. AISNSW 2010 Our goals were to describe • how the tasks respectively contribute to mathematics learning • the features of successful exemplars of each type • constraints which might be experienced by teachers • teacher actions which can best support students’ learning Teachers Teacher knowledge, attitudes Tasks Cluster meetings Classroom implementation Student survey Teacher responses to tasks Planning units of work Teaching units of work Reflection on teaching Students Student responses to units Planning 3 lessons Teaching 3 lessons AISNSW 2010 Self ratings of perception of mathematics in % (n = 930) Rating 0 1 2 3 4 5 6 7 How good are you at maths? (Q1) 0 1 3 11 21 33 24 8 How happy are you in maths class? (Q2) 1 4 9 18 22 24 15 8 AISNSW 2010 AISNSW 2010 AISNSW 2010 While there was not much difference overall between levels, there was a big difference (mean 3.5 to 5.5) between classes AISNSW 2010 AISNSW 2010 In summary • At each of these middle years levels there is a range of satisfaction and confidence, and teachers should be aware of this • Teachers make a difference and they need support to both find out the students levels of satisfaction and confidence, and to do something about it if they are low Qu 9 In this table there are three maths questions that are pretty much the same type of mathematics content asked in different ways. Put a 1 next to the type of question you like to do (learn) most, 2 next to the one you like (learn) next best and 3 next to the type of question you like (learn) least. We don’t want you to work out the answers. Movies tickets are $13 for adults and $7 for children. How much does it cost for 2 adults and 4 children to go to the movies? (word problem) 2 adults and 4 children went to the movies. They spent $120 on tickets. How much might the adult and children’s tickets cost? (open-ended) (2 ×13) + (4 × 7)= (number work) AISNSW 2010 Preferences for liking, and learning from, the task types as a % (Q9) Task Like most Learn most Number work 54 40 Word problem 35 23 Open-ended 12 37 AISNSW 2010 11. ... Put a 1 next to the type of question you like to do (learn from) most, 2 next to the one you like (learn from) next best, and 3 for the type of question you like (learn from) the least: Find the area inside the following shape: A shape has an area of 10 square units. What might the shape look like? A running track has straights that are 100 m long with half circles at the end. The inside is all grass. What is the area of the grass? AISNSW 2010 Preferences for the Task Types as a percentage Type Like most Learn most Area by counting 39 33 Practical calculation 20 44 Open-ended 42 22 AISNSW 2010 From a different survey after a unit of work AISNSW 2010 Student Preferences (%) Task Type Task favourite 2nd favourite Task best for learning 2nd best for learning Finding similar graphs 1 0.0% 0.0% 2.1% 2.1% Clues on cards 1 0.0% 3.1% 18.0% 10.4% Using excel to present data 1 14.3% 6.0% 12.2% 9.2% Matching graphs 1 2.0% 0.0% 2.0% 10.4% Weather project 2 8.0% 6.0% 2.0% 2.1% Two way tables 2 2.0% 8.0% 6.0% 6.3% Average height of class 2 0.0% 6.0% 8.0% 2.1% This goes with this 2 4.0% 2.0% 4.0% 14.6% Most commonly used letters 2 6.0% 2.0% 8.0% 2.1% Average height in school 2 26.0% 18.0% 8.0% 17.6% A sentence with 5 words 3 4.0% 12.0% 8.0% 4.2% Rock paper scissors 3 22.0% 27.8% 0.0% 4.2% Seven people went fishing 3 10.0% 4.0% 18.0% 12.5% Conducting a survey 3 2.0% 2.0% 2.0% 2.1% Total: 50 50 50 48 Task AISNSW 2010 Student opinions about their mathematics classes: Some free format responses AISNSW 2010 • In summary, it seems that the students were extraordinarily articulate about what they wanted in their maths lessons • In synthesising the responses, students like lessons that – – – – – – – – used materials (although these were not structured materials), were connected to their lives, involved games, were practical with some emphasis on measurement, in which they worked outside, The see “like” and “learn” as different with the method of grouping being important, and over half of the students claim to like to be challenged. AISNSW 2010 What does this mean? AISNSW 2010 In summary • The students were extraordinarily articulate about what they wanted in their maths lessons • The first characteristic of the responses is the diversity of aspects on which the individual students commented, suggesting that there is no commonly agreed ideal lesson, and there are many ways to teach well. AISNSW 2010 • The ways of working in class are clearly important for students, and however the teacher intends that the student work, the reason for this needs to be clarified for the students.. AISNSW 2010 A preliminary task • In 5 words or less, write down an aspect of teaching mathematics that you would advise beginning teachers to ensure they think about in all of their lessons AISNSW 2010 But first ... • An underlying assumption is that at least some learning should come • from engagement of individuals • with – tasks – each other AISNSW 2010 Some of the key decisions • Mathematics success creates opportunities and all should have access to those opportunities • The curriculum should prioritise teacher decision making • The curriculum should foster depth and important ideas rather than breadth • Students can be challenged within basic topics, including the advanced students AISNSW 2010 There are 3 content strands • Number and algebra • Measurement and geometry • Statistics and probability AISNSW 2010 … and 4 proficiency strands • • • • Understanding Fluency Problem solving Reasoning AISNSW 2010 Key teaching idea 1: • Identify big ideas that underpin the concepts you are seeking to teach, and communicate to students that these are the goals of your teaching, including explaining how you hope they will learn AISNSW 2010 An example for us to discuss • Write a sentence that has5 words, with an average of four letters per word (no 4 letter words) AISNSW 2010 Some questions • What is the mathematical point of that task? • What is the pedagogical point of that task? • How do you make these points explicit to students? AISNSW 2010 AISNSW 2010 Some questions • What mathematical actions can be addressed by working on that task? AISNSW 2010 Which card is better value? Please explain your thinking. AISNSW 2010 Some questions • What is the mathematical point of that task? • What is the pedagogical point of that task? • How do you make these points explicit to students? AISNSW 2010 year 7 • Determine mean, median, and range and use these measures to compare data sets explaining reasoning including use of ICT AIZ Zone 2 & 3 Day 1 year 7 • to understand and become fluent with written, mental and calculator strategies for all four operations with fractions, decimals and percentages AIZ Zone 2 & 3 Day 1 year 8 • Generalise from the formulas for perimeter and area of triangles and rectangles to investigate relationships between the perimeter and area of special quadrilaterals and volumes of triangular prisms and use these to solve problems AIZ Zone 2 & 3 Day 1 year 9 • Work fluently with index laws in both numeric and algebraic expressions and use scientific notation, significant figures and approximations in practical situations AIZ Zone 2 & 3 Day 1 year 9 • Solve problems involving linear simultaneous equations, using algebraic and graphical techniques including using ICT AIZ Zone 2 & 3 Day 1 year 10 • Understand and use graphical and analytical methods of finding distance, midpoint and gradient of an interval on a number plane AIZ Zone 2 & 3 Day 1 Key teaching idea 4: • Interact with students while they engage in the experiences, and specifically planning to support students who need it, and challenge those who are ready AISNSW 2010 Establish classroom ways of working • Examples of “norms” – errors are part of learning – all students must persist – all students must be willing to justify their thinking – working as a community of learners benefits everyone AISNSW 2010 An idea we can discuss • 5 people went fishing. The mean number of fish caught was 4, and the median was 3. How many fish might each person have caught? AISNSW 2010 Some questions • What mathematical actions can be addressed by working on that task? • What might be the challenges in turning this into a lesson? AISNSW 2010 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. AISNSW 2010 Factors contributing to difficulty • It may not be clear which aspects may be contributing to a particular student’s difficulty, but by anticipating some of the factors, and preparing prompts that, for example, – – – – reduce the required number of steps, simplify the modes of representing results, make the task more concrete, or reduce the size of the numbers involved, • the teacher can explore ways to give the student access to the task without the students being directed towards a particular solution strategy for the original task. AISNSW 2010 How might you adapt that task for students experiencing difficulty? AISNSW 2010 How might you adapt that task for students who finish quickly? AISNSW 2010 Sample Task 2 • Seven people went fishing. The mean number of fish caught was 4, and the median was 3. How many fish might each person have caught? Considering the quality of response • Basic – “The number of fish could be 1, 2, 2, 3, 4, 5, 11” • Developed – “The following are some possibilities: • • • • 0, 0, 0, 3, 4, 5, 16 0, 0, 1, 3, 4, 5, 15 0, 0, 0, 3, 4, 5, 16 etc • Advanced – “The total fish caught is 28. The middle number is 3, so the first 3 numbers of fish caught must have a numbers that are 3 or less. For each set, the latter three numbers make up the total. So if the first 3 numbers total 5, then the latter 3 numbers must total 20.” Key teaching idea 5: • Adopt pedagogies that foster communication, mutual responsibilities, and encourage students to work in small groups, and using reporting to the class by students as a learning opportunity AISNSW 2010 How do we turn this into lessons? • • • • Launch Explore Summarise review AISNSW 2010 The Japanese have better words • Hatsumon – The initial problem – Kizuki - what you want them to learn • Kikanjyuski – Individual or group work on the problem – Kikan shido – thoughtful walking around the desks • Nerige – Carefully managed whole class discussion seeking the students’ insights • Matome – Teacher summary of the key ideas AISNSW 2010 My suggested words • • • • Task introduction Facilitation of student engagement Whole class discussion Teacher summary AISNSW 2010 Key teaching idea 3 • Engage students by utilising a variety of rich and challenging tasks, that allow students opportunities to make decisions, and which use a variety of forms of representation AISNSW 2010 A story from Mr T (sent last week) • In year 9 revision, the students were working on this problem – You earn $12 per hour for 22.5 hours. You pay 26% of your earnings in tax. How much tax will you pay? • • • • • A girl, Emma, wanted help. Mr T: Do you have a job? Emma : Yes Mr T: How much an hour do they pay you? Emma : I don’t know I just started MAWA National issues Mr T: Let’s say you ear $12 an hour and you work for 3 hours. How much is that? Emma : I don’t know. Do you divide? Mr T: No. Think about earning $12 an hour. You work one hour, and then another and then another. How much have you earned? Emma : I don’t know MAWA National issues • Kylie (sitting nearby): Is it $36? • Mr T: Yes. Good. • Emma (to Kylie): God, you’re smart. MAWA National issues One of the key issues with the Australian Curriculum is teacher decision making • Some examples of adapting AISNSW 2010 Some activities • Relationships AISNSW 2010 The average height of 3 people in this room is 1.7 m. You are one of those people. Who are the three people? AISNSW 2010 Making questions open Method 1: • Write down a question and work out the answer. • Make up a new question that includes the answer as part of the question. Method 2. • Write down a complete question including the answer. • Remove some of the question parts. Race to 10 AISNSW 2010 What questions can we answer? 1 1 2 1 1 2 3 1 4 + 1 4 + 3 4 - 1 2 Using empty number lines • 37 + 45 40 3 2 40 37 80 10 10 37 47 10 10 57 AISNSW 2010 67 82 3 77 80 2 82 • Use an empty number line to work out 1 22 4 - 1 7 2 5 1 2 3 14 4 2 1 15 4 1 20 4 1 22 4 The first Key teaching idea 2:part of this • Build on what the students know, both mathematically and experientially, including creating and connecting students with stories that both contextualise and establish a rationale for the learning AISNSW 2010 One perspective on building on what they know • The idea of the self- contained “lesson” AISNSW 2010 A sample lesson AISNSW 2010 Using isometric paper, draw the shapes made by sticking 3 cubes together (whole faces touching) AISNSW 2010 2 cubes might look like AISNSW 2010 Draw a shape made from 10 cubes • Go beyond the straight line The next task • A block of city buildings is 3 cubes wide and 3 cubes long • It looks like this from the front On isometric paper, draw what the set of buildings might look like The task • A block of city buildings looks like this from the side • And like this from the front • On isometric paper, draw what the set of buildings might look like AISNSW 2010 Side Front Front view AISNSW 2010 In what ways was that different from a conventional mathematics lesson? AISNSW 2010 “Enabling prompts” AISNSW 2010 A different destination • A block of city buildings looks like this from the side • Draw what the set of buildings might look like AISNSW 2010 For students experiencing difficulty: • Have some isometric paper with part of the building already drawn. • Have some cubes. Ask them to model what the building might look like. AISNSW 2010 How might you extend this for students who finish quickly? • How many different designs that fit the directions can you make? • Draw a set of buildings on the isometric paper, and draw the front and side view as well. AISNSW 2010 • Draw what this representation might mean 1 3 1 2 3 3 1 1 2 AISNSW 2010 Key teaching idea 2: the second part • Build on what the students know, both mathematically and experientially, including creating and connecting students with stories that both contextualise and establish a rationale for the learning AISNSW 2010 Getting to work Josie Loreto 20 km How much does it cost Josie to get to work and back home again? Assume that it costs $2 per km for the full costs AISNSW 2010 Getting to work Josie Loreto 73 km How much does it cost Josie to get to work and back home again? Assume that it costs $1.37 per km for the full costs AISNSW 2010 Getting to work Josie Loreto x km How much does it cost Josie to get to work and back home again? Assume that it costs $z per km for the full costs AISNSW 2010 Getting to work 5 km Josie Assume that it costs $2 per km for the full costs Loreto Carmen 20 km How much should Carmen give Josie if she picks her up and takes her home? AISNSW 2010 Getting to work 13 km Josie Assume that it costs $1.37 per km for the full costs Loreto Carmen 59 km How much should Carmen give Josie if she picks her up and takes her home? AISNSW 2010 Getting to work x km Josie Assume that it costs $z per km for the full costs Loreto Carmen y km How much should Carmen give Josie if she picks her up and takes her home? AISNSW 2010 Assume that it costs $2 per km for the full costs Getting to work 5 km 5 km Josie Carmen Susan Loreto 20 km How much should Susan and Carmen give Josie if she picks them up and takes them home? AISNSW 2010 A preliminary task • In 5 words or less, write down an aspect of teaching mathematics that you would advise beginning teachers to ensure they think about in all of their lessons AISNSW 2010