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Developing a reasoning classroom Written by Tim Handley (Mathematics Specialist Teacher, NCETM Professional Development Lead and author of Problem Solving and Reasoning) Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Aims Understand what reasoning ‘is’ and why it is important. Explore some reasoning activities. Explore ways in which reasoning can become part of day-to-day teaching. Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Running order Introduction: What do our children already do? Taking a slice: a non-numerical activity What is reasoning and why is it important? Reasoning in the past week Graffiti Maths with Chicken Nuggets: a numerical activity Break Children’s mathematical superpowers Key strategies to make reasoning part of day-to-day life in our classrooms Planning a reasoning activity Next steps Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom What do children in our school already do? Have a look at the statements on the progression cards from http://nrich.maths.org/6105. Highlight them green if children in your year group do these regularly and independently. Highlight them orange if children in your year group do these regularly but require support. Leave them blank if children in your year group struggle to do these, or only do them infrequently. Then identify the 4 statements which you consider to be age appropriate, but which children in your year group struggle with the most and mark these with a star. Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom What can children do independently? Problem Solving and Reasoning What do they find difficult? CDP DVD For Years 1 to 6 Developing a reasoning classroom Take a slice: a non-numerical activity Here is one way of splitting a 4 x 4 board into 2 identical pieces: How many other ways can you find? Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Take a slice: a non-numerical activity Things to think about So, what were you doing? What key phrases were you using? What were you thinking? We were all reasoning! Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom So, what is reasoning? Some things reasoning can be seen as: Thinking about mathematics Making connections Application of facts and knowledge – either explicitly or implicitly Justifying Convincing yourself and others Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Why is reasoning important …? What do you think? Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom What OTHERS say ... Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Academics “In our understanding, we have two components – the range of mental representations and the reasoning linking them together. Therefore we must also develop the reasoning that we carry out between representations.” Barnaby et al, Primary Mathematics, Teaching for understanding. And lots, lots more… Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Examiners Just some of the questions that require children to reason from the 2009 KS1 + 2012 KS2 SATs tests. Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Ofsted The reports Mathematics: made to measure (Ofsted, 2012) and Good practice in primary mathematics: evidence from 20 successful schools (Ofsted, 2011) make clear that: Real mathematical thinking lies at the heart of learning mathematics. It involves making sense of mathematics through engagement, reasoning and making connections. Sometimes this may involve significant challenge in terms of the requirement to think. But it is only through their own thinking that children will develop real depth of learning. Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Ofsted Maths subject-specific criteria Outstanding Pupils understand core concepts and make connections. They think for themselves. They show exceptional independence when solving problems in a range of contexts. They think for themselves and persevere with challenges, knowing they will succeed. They embrace the value of learning from mistakes and false starts. They reason, generalise and make sense of solutions. Requires improvement Children use techniques correctly. They have an insecure understanding of concepts. They solve routine problems in known contexts. They follow the teacher’s example. They avoid making mistakes. Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom The DfE: 2014 National Curriculum Aims It is expected that these aims are applied throughout the programmes of study. Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom How have our children been reasoning over the past week? Share with your table! Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Graffiti maths Graffiti maths is an approach to problem solving and reasoning tasks which encourages children to think and work ‘big’. See page 12 of the Introduction in the Problem Solving and Reasoning books for an explanation of how it works. See the Key Strategies Maths Stories section on the DVD for footage of Graffiti maths in action. Now use the graffiti maths approach to try out the Chicken nuggets problem from Problem Solving and Reasoning Year 5. Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom BREAK! Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Mathematical Superpowers All children have them! Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Mathematical superpowers John Mason identified a set of 8 ‘Mathematical Powers’ that all children posses. They come in pairs: Conjecturing and Convincing Imagining and Expressing Specialising and Generalising Organising and Classifying See page 8 of the Introduction in the Problem Solving and Reasoning books for more details. Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom The Story of a Number An easy way to get children reasoning, pattern spotting and generalising. Watch the Maths Stories clip on the DVD. Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Money boxes • Try the Money boxes problem from Problem Solving and Reasoning Year 3. • Can you work out how much money could be in each money box? Add in questions to draw out the reasoning… If we know that half of the coins are £1, what else do we know? Which of these clues are hard/easy to solve? Why? Which of the clues is the odd one out why? Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Key strategies To help include reasoning as part of our day-to-day teaching. The Problem Solving and Reasoning books include 14 key strategies. Watch the clips on the DVD. Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Always, sometimes, never • Look at the statements from the Number Knowledge problem Resource Sheet 13.1 from Problem Solving and Reasoning Year 6. • Discuss with your partner. Are they always, sometimes or never true? • Make sure you give examples. Always, sometimes, never can also be used as part of shared learning. Can you come up with some ‘always, sometimes, never’ statements for your year group? Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Other examples for always, sometimes, never… The picture on a pictogram represents one piece of data. If the answer is in cm2 you are measuring area. The diagonals of a quadrilateral bisect each other. Two negatives make a positive. All multiples of 3 have a digital root of 3, 6 or 9. Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Another, another, another… An even number… Number with the product of 60 Factors of 36 A triangle A prime number Draw out the links and generalisations between the sets generated. Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Convince me ... All angles in a triangle always add up to 180 degrees. That the next whole number after 119 will be 120. That coordinates are useful for identifying points. An odd number plus an odd number always equals an even number. That the area of a rectangle divided by the length of one side gives the length of the other side. Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Hard and easy A calculation whose answer is 7 A three digit subtraction A word problem A region made up of rectangles whose area and perimeter are to be found A calculation to be carried out on a calculator Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom If this is the answer, what’s the question? 24 Impossible No Square numbers 15.3 Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Odd one out (and why?) 3, 6, 9 … 3 x 10 = 30, 31 x 10 = 310, 423 x 10 = 4230, 0.3 x 10 = 3, 1111 x 10 = 11 110 1/4 , 8/36, 6/ 32, 10/ 40 11, 101, 1001, 101 Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom POGs: Peculiar, Obvious, General A quadrilateral A multiple of 2 A factor of 36 A fraction equivalent to ¾ A number A set of 3 numbers with a product of 10 Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom What do you notice? 2, 4, 6, 8, 10, 12, 14, 16 2, 3, 5, 7, 11,13 (Can you give me more in this set?) 1 / 2 , 2 / 4 , 3 / 6 , 4 / 8 4, 9, 16, 25, 36, … Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom What else do we know? (and why?) 5 x 8 = 40 3 + 4 =7 1 cm = 10 mm, 2 cm = 20 mm All even numbers end in an even digit: give me an even number over 1000. 50 is 5 lots of 10, 40 is 4 lots of 10, … 1/2 = 2/4 = 4/8 = 3/6 Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom What’s the same? What’s different? One and 1 9 and 18 Multiples of 3 and Multiples of 6 1/2, 0.5, 50%, 4:8, 3/6, 1:2 Square numbers and numbers that are not square Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom What’s the same? What’s different ? Choose any two of the three. In what way are they the same as each other and different from the third? Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Zooming in Write down a: Multiple of 4... ... that is greater than 30... ... that is also square... ... where both digits are even. Draw a: Quadrilateral... ... that has at least one line of symmetry... ... and only one pair of parallel sides... ... and at least one acute angle. Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Is it a good explanation of…? Is “having three sides” a good explanation of a triangle? Is “adding a zero” a good explanation of multiplying by 10?” Is “a shape with 4 sides” a good explanation of a square? Is “moving the digits 3 places to the left” a good explanation of multiplying 1.3 by 1000? Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Planning a reasoning activity In your year group, plan a reasoning activity that you could use next week. This could be a key question, it could use one of the strategies we’ve just looked at, or be a full lesson activity. Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Next steps Encourage and support our children to reason! Over the next 2 weeks, ensure there are at least 6 opportunities for your children to reason. Try also to include reasoning and some of the key strategies in other subjects. Problem Solving and Reasoning CDP DVD For Years 1 to 6 Developing a reasoning classroom Resources to help 14 key strategies with example questions for each year 18 activities and investigations per year level Problem Solving and Reasoning CDP DVD For Years 1 to 6