Some “what” strategies to engage students in reasoning and communication in the primary mathematics classroom Professor Berinderjeet Kaur

Why must we emphasize reasoning and communication in our classroom discourse?

School Mathematics Framework

Typical Textbook Questions Bala’s father left for work at 7.45 a.m. and came home 9h 45 min later. What time did he come home? There are 44 chickens and ducks on a farm. 28 of them are chickens. What is the ratio of the total number of chickens and ducks to the number of ducks. Give your anser in its simplest form.

 What number makes sense?  What’s wrong?  What if?  What’s the question if you know the answer?

What number makes sense?

Mathematics version of a cloze passage. A set of numbers is provided and pupils determine where to place each number so the situation makes sense. As pupils work through tasks of this nature, they practice computation and increase their repertoire of problem-solving skills. Reasoning skills are improved by being exposed to a variety of ways to solve a problem

What number makes sense? - An example Read the problem. Look at the numbers in the box. Put the numbers in the blanks where you think they fit best. Read the problem again, do the numbers make sense? Apples in a box Mary bought a box of red and green apples. The box has ______ apples. There are more red apples than green. There are ______ red apples and ________ green apples. The ratio of the red apples to the green apples is __: __

What’s wrong?

The pupils are provided with an opportunity to use their critical thinking skills. They are presented with a problem and its incorrect solution (error may be conceptual or computational). The pupil’s task is to discover the error, correct it and then explain what was wrong, why it was wrong and what was done to correct the error.

What’s wrong? – An Example John and Henry won a prize of $500 at a Charity Fair. With the money, John bought a bicycle for $140. On their way home they decided to share the prize money equally. John’s thinking:- $500 - $140 = $360 $360 ÷ 2 = $180 Each person gets $180 There is something wrong with John’s thinking. Show how you would find the answer to the problem. Explain the mistake in John’s thinking.

What if?

These tasks make two kinds of demand on the pupils’ cognition. The first is when the given information is changed. This modification permits pupils to reexamine the task and see what effect these changes have on the solution process as well as the answer. In this way pupils are reinforcing their critical thinking as they analyze what is taking place. The second is the generation of “what if” questions after they have solved a given task. This draws on the creative thinking skills of the pupil and engages him or her in problem posing

What if ? – An Example Cookies and Boxes Mrs Tan baked 24 cookies. Each box holds 4 cookies. At least how many boxes are needed to hold all the cookies? What if Mrs Tan baked 30 cookies? What if each box can hold 5 cookies? What if each box can hold up to 4 cookies? Generate another 3 “What if” tasks and answer them. Look out for any interesting observation/pattern.

What’s the question if you know the answer?

Pupils are presented with the context and data of a problem but with the question/s missing. They are given a solution and asked to write a question that matches it. Such tasks provide an opportunity for pupils to engage in critical thinking skills.

What’s the question if you know the answer – An Example Red & White Chalk Mr Lee had 3 boxes of red chalk and 8 boxes of white chalk. Each box contained 5 pieces of chalk. 1.What’s the question if the answer is 40 ? 2.What’s the question if the answer is 15 ? 3.What’s the question if the answer is 11 ? 4. What’s the question if the answer is 3:8 ?

Primary 1 Lesson What’s the Q if you know the answer?

19 Reasoning and Communication in a Grade 1 Mathematics Classroom

20 Background Marymount Convent School was one of the schools that participated in the EPMT project (Jan 2007-Dec 2008). The school has as of 2010 adopted a school wide infusion of the strategies for reasoning and communication in their instruction at all levels. Since the beginning of the year 2010, teacher has been infusing strategies for reasoning and communication into her grade 1 math lessons periodically.

21 The Lesson The Lesson for Year 1 (grade 1) pupils was based on the strategy “What’s the question if you know the answer? -Students are presented with the context and data of a problem but with the question/s missing. -They are given a solution and asked to write a question that matches it. -Such tasks provide an opportunity for students to engage in critical thinking skills. The content focus was numbers less than 40 and the four number operations. ( +, -, X, ÷ )

22 Development of the Lesson Warm-up activity -Teacher reviews number sentences and the four operations through Story time Introduction to strategy: What the Question if you know the answer? -Task 1 – Mrs Lee and eggs from her farm -Teacher leads the whole class in doing the task on the board Seat work (Think-Pair-Share) -Task 2 – Old MacDonald had a farm -Teacher monitors pupils at work (in pairs) at their desks & provides guidance and feedback -Teacher draws the attention of the whole class to common difficulties Lesson closure -Teacher highlights diversity of questions and also creative ones -Teacher assigns homework

Warm-up Activity - The students 23

Warm-up Activity - The teacher 24

25 First Task Teacher leads the whole class through the task What’s the question if you know the answer? Eggs-cellent? Mrs Lee has a farm. She collects eggs from her farm every day. Day Number of eggs collected Monday 23 Tuesday 17 Wednesday 8 Thursday 5 Friday 10 What’s the question if the answer is 5? What’s the question if the answer is 25? What’s the question if the answer is 6? What’s the question if the answer is 30?

Whole Class Activity 26

Whole Class Activity 27

Instructions for Task 2 28

Task 2 What’s the question if you know the answer? 29 Old MacDonald had a Farm There are 5 cows and 12 ducks at a farm. Each cow has 4 legs and each duck has 2 legs. What is the question if the answer is 17? What is the question if the answer is 7? What is the question if the answer is 20? What is the question if the answer is 44?

Seatwork – students working in pairs 30

Seatwork – students working in pairs 31

Scaffolding 32

Teacher’s instruction about the next activity 33

Students doing sharing in groups 34

Closure of Lesson 35

36 Thank You! Wish you a fruitful time at the conference