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Structuring numeracy lessons to engage all students: 5-10 Peter Sullivan Overview • We will work through three lessons I have taught this year as part of classroom modelling in years 5-10. • The lessons are structured to maximise engagement of all students, especially those who experience difficulty and those who complete the work quickly. • I will ask you to examine the commonalities and differences between the lessons and identify key teacher actions in supporting this lesson structure. • I will ask you to reflect upon what implications for leading whole school Numeracy improvement. Assumptions • We do not want to tell the students what to do before they have had a chance to explore their own strategy • We want to step back to allow ALL students to engage with the task for themselves • We want them to see new ways of thinking about the mathematics • There is no need to hurry • We want them to know they can learn (as distinct from knowing they can be taught) Patterns with remainders Years 5 - 6 • Some people came for a sports day. • When the people were put into groups of 3 there was 1 person left over. • When they were lined up in rows of 4 there were two people left over. • How many people might have come to the sports day? OLOM Coburg 2013 Multiplication content descriptions • Year 4: Develop efficient mental and written strategies and use appropriate digital technologies for multiplication and for division where there is no remainder • Year 5: Solve problems involving division by a one digit number, including those that result in a remainder • Year 6: Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers OLOM Coburg 2013 Patterns • Explore and describe number patterns resulting from performing multiplication (ACMNA081) • Solve word problems by using number sentences involving multiplication or division where there is no remainder (ACMNA082) OLOM Coburg 2013 Some “enabling” prompts • Some people came for a sports day. When they were lined up in rows of 4 there were two people left over. How many people might have come to the sports day? • • Some people came for a sports day. When the people were put into groups of 3 there was noone left over. When they were lined up in rows of 4 there was no-one left over. How many people might have come to the sports day? OLOM Coburg 2013 An extending prompt • Some people came for a sports day. When the people were put into groups of 3 there was 1 person left over. • When they were lined up in rows of 4 there was 1 person left over. • When they were lined up in columns of 5 there was 1 person left over. • How many people might have come to the sports day? OLOM Coburg 2013 The “consolidating” task • I have some counters. • When I put them into groups of 5 there was 2 left over. • When they were lined up in rows of 6 there was the same number in each column and none left over. • How many counters might I have? OLOM Coburg 2013 How many fish? Year 7 In this lesson, I need you to • show how you get your answers • keep trying even if it is difficult (it is meant to be) • explain your thinking • listen to other students Our goal • The meaning of mean, median and mode • To explain our thinking clearly To start • Write a sentence with 5 words, with the mean of the number of letters in the words being 4. To start • Write a sentence with 5 words, with the mean of the number of letters in the words being 4. These sets of scores each have a mean of 5 5, 5, 5 4, 5, 6 3, 5, 7 1, 1, 13 To start • Write a sentence with 5 words, with the mean of the number of letters in the words being 4. Next • Seven people went fishing. • The mean number of fish the people caught was 5, and the median was 4. • How many fish might each person have caught? Next • Seven people went fishing. • The mean number of fish the people caught was 5, and the median was 4. • How many fish might each person have caught? These sets of scores have a median of 10 10, 10, 10 8, 10, 12 1, 10, 11 9, 10, 200 8, 12, 10 And now • Seven people went fishing. • The mean number of fish the people caught was 5, the median was 4 • How many fish might each person have caught? • Seven people went fishing. • The mean number of fish the people caught was 5, the median was 4 and the mode was 3. • How many fish might each person have caught? • Seven people went fishing. • The mean number of fish the people caught was 5, the median was 4 and the mode was 3. • How many fish might each person have caught? If you are stuck • A family of 5 people has a mean age of 20. What might be the ages of the people in the family? If you are finished • How many different answers are there? • What is the highest number of fish that anyone might have caught? Now try this • The mean age of a family of 5 people is 24. The median age is 15. What might be the ages of the people in the family? Our goal • To see the meaning of mean, median and mode • To explain our thinking clearly Co-ordinates of squares Year 8 - 9 Assumptions • They have had an introduction to placing coordinates The key task Four lines meet in such a way as to create a square. One of the points of intersection is (-3, 2) What might be the co-ordinates of the other points of intersection? Give the equations of the four lines. How might you run that class? • How much would you tell the students? • What approach do you recommend to doing this task? • How much confusion can you cope with? • When is challenge and uncertainty productive? • What is meant by “cognitive activation”? Quotes from PISA in Focus 37 • When students believe that investing effort in learning will make a difference, they score significantly higher in mathematics. • Teachers’ use of cognitive-activation strategies, such as giving students problems that require them to think for an extended time, presenting problems for which there is no immediately obvious way of arriving at a solution, and helping students to learn from their mistakes, is associated with students’ drive. Numeracy keynote SA Where is the point (-2,3)? Where is the point (-2,3)? Show all the points which have an x value of 1 Show all the points which have an x value of 1 Show all the points which have a y value of -2. • What is the equation? Responses Four lines meet in such a way as to create a square. One of the points of intersection is (-3, 2) What might be the co-ordinates of the other points of intersection? Give the equations of the four lines. On this sheet draw the letter of your name and give the co-ordinates of the points at the ends of each line. Mark all the points where y is bigger than x What is your reaction to those lesson? What might make it difficult to teach like that in your school? In what ways were those lessons similar? What actions might you take to encourage teachers to adopt such approaches, at least sometimes?