Presentation on theme: "Developing a reasoning classroom"— Presentation transcript:
1Developing a reasoning classroom Written by Tim Handley(Mathematics Specialist Teacher, NCETM Professional DevelopmentLead and author of Problem Solving and Reasoning)
2Aims 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.
3Running order Introduction: What do our children already do? Taking a slice: a non-numerical activityWhat is reasoning and why is it important?Reasoning in the past weekGraffiti Maths with Chicken Nuggets: a numerical activityBreakChildren’s mathematical superpowersKey strategies to make reasoning part of day-to-day life in our classroomsPlanning a reasoning activityNext steps
4What do children in our school already do? Have a look at the statements on the progression cards fromHighlight 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.The NRICH enriching mathematics website provides free professional development activities. This activity is based on the ‘How do pupils progress in their problem solving?’ article by Jennifer Piggott using the progression cards which can be found here:
5What can children do independently? What do theyfind difficult?What can children do independently?
6Take 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?
7Take a slice: a non-numerical activity Things to think aboutSo, what were you doing?What key phrases were you using?What were you thinking?We were all reasoning!
8So, what is reasoning? Some things reasoning can be seen as: Thinking about mathematicsMaking connectionsApplication of facts and knowledge – either explicitly or implicitlyJustifyingConvincing yourself and others
11Academics“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…
12ExaminersJust some of the questions that require children to reason from the 2009 KS KS2 SATs tests.
13OfstedThe 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.
14Ofsted Maths subject-specific criteria OutstandingPupils 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 improvementChildren 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.
15The DfE: 2014 National Curriculum AimsIt is expected that these aims are applied throughout the programmes of study.
16How have our children been reasoning over the past week? Share with your table!
17Graffiti mathsGraffiti 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.Graffiti maths was developed almost simultaneously by a number of teachers, including Claire Lotriet and Geoff Barton in 2012 (http://clairelotriet.com/blog/2012/12/15/graffiti-maths/).
20Conjecturing and Convincing Mathematical superpowersJohn Mason identified a set of 8 ‘Mathematical Powers’ that all children posses. They come in pairs:Conjecturing and ConvincingSpecialising and GeneralisingImagining and ExpressingOrganising and ClassifyingMason, J. and Johnston-Wilder,S (eds) (2004 a) Learners Powers in: Fundamental Constructs in Education, London:Routledge Falmer ppSee page 8 of the Introduction in the Problem Solving and Reasoning books for more details.
21Watch the Maths Stories clip on the DVD. The Story of a NumberAn easy way to get children reasoning, pattern spotting and generalising.Watch the Maths Stories clip on the DVD.
22Money boxes Add in questions to draw out the reasoning… 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?
23Key strategies Watch the clips on the DVD. 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.
24Always, 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?
25Other 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.
26Another, another, another… An even number…Number with the product of 60Factors of 36A triangleA prime numberDraw out the links and generalisations between the sets generated.
27Convince 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.
28Hard and easy A calculation whose answer is 7 A three digit subtractionA word problemA region made up of rectangles whose area and perimeter are to be foundA calculation to be carried out on a calculator
29If this is the answer, what’s the question? 24ImpossibleNoSquare numbers15.3
30Odd 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 =1/4 , 8/36, 6/32, 10/4011, 101, 1001, 101
31POGs: Peculiar, Obvious, General A quadrilateralA multiple of 2A factor of 36A fraction equivalent to ¾A numberA set of 3 numbers with a product of 10
32What do you notice?2, 4, 6, 8, 10, 12, 14, 162, 3, 5, 7, 11,13 (Can you give me more in this set?)1/2 , 2/4 , 3/6 , 4/84, 9, 16, 25, 36, …
33What else do we know? (and why?) 5 x 8 = 403 + 4 =71 cm = 10 mm, 2 cm = 20 mmAll 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
34What’s the same? What’s different? One and 19 and 18Multiples of 3 and Multiples of 61/2, 0.5, 50%, 4:8, 3/6, 1:2Square numbers and numbers that are not square
35What’s the same? What’s different ? Choose any two of the three. In what way are theythe same as each other and different from the third?
36Zooming 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.
37Is 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?
38Planning 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.
39Next 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.
40Resources to help14 key strategies with example questions for each year18 activities and investigations per year level