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Developing a Whole School Approach to Problem Solving and Word Problems Day 1 Dr Paul Swan and David Dunstan Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 1 Proficiency and Content Strands Dr Paul

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Developing a Whole School Approach to Problem Solving and Word Problems Day 1

Dr Paul Swan and David Dunstan

Dr Paul Swan and David Dunstan Developing a Whole School Approach 1

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Proficiency and Content Strands

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What is a problem?

  • A problem is a situation where there is no obvious method of solution.
  • There is no basic routine to follow in order to solve the problem.
  • Students need to think about it.

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Four Stages of Problem Solving

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  • (See)
  • (Plan)
  • (Do)
  • (Check)
  • Understand the problem
  • Devise a plan
  • Carry out the plan
  • Look back

Polya It is better to solve one problem many ways, than many problems one way.

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Download the 16 page KenKen booklet from www.drpaulswan.com.au Go to the Resources section

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A 4 by 4 grid. The digits 1, 2, 3 and 4 are used. The operation is Addition.

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Always fill the ‘Freebie’ cells. The ‘Target Number (TN)’ is given. For example, in the bottom LHC, the TN is 1.

3 1

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Look at the Green horizontal 2-cell Cage. A Cage is defined by the heavy outline and includes the TN and generally, the operation. Why must the top LHC be ‘2’? Each row and column must use the digits 1, 2, 3 and 4, with NO REPEATS. For a sum of 3, only 1 and 2 are used. In the first column, 1 cannot be repeated.

3 1 2

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Another Strategy! Why must the top LHC be ‘2’? Each row and column add to 10. So in the first column, 1 + 7 = 8 and 2 more makes 10 (or, 10 – 8 = 2).

3 1 2

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In the RED cage, the sum of 7 can only be made with 3 and 4,

  • r 4 and 3.

Due to the NO REPEAT rule, in row 3, the red cell must be a 4.

3 1 2 1 3 4

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Why is the bottom RHC = 2? 2 = 10 – (5 + 3) [each row and column adds to 10], or For two digits to make a sum of 5 (PINK cage), the possible number sets are (1,4), (4,1), (2,3) and (3,2). Thus the 2 must go in the bottom RHC.

3 1 2 1 3 4 2 4

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Remember, No Repeats; so ‘4’ goes in the Top RHC (1 cannot go there).

3 1 2 1 3 4 2 4 1 4

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Complete the top and bottom rows.

3 1 2 1 3 4 2 4 1 4 3 3

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Complete the remaining cells. Kengratulations!

3 1 2 1 3 4 2 4 1 4 3 3 1 2 4 2

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A 4 by 4 Grid. The digits 1, 2, 3 and 4 are used. The operation is Addition. No Freebies.

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To make a sum of 3, use 1 and 2. To make a sum of 4, use 1 and 3. So why must 3 be in the top LHC?

3 1

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Why MUST ‘4’ go in the bottom LHC? First column, 10 – (3 + 3) = 4

4 3 1

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Complete the PINK cage.

4 3 1 2 1 2

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Complete the RED cage. Remember, No Repeats of ‘1’ in the second row.

4 3 1 2 1 2 1 2

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Complete the YELLOW cage.

4 3 1 2 1 3 4 2 1 2

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Complete the GREY cage.

4 3 1 2 1 1 3 3 4 2 1 2

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Complete the remaining cages.

2 4 3 1 2 1 1 3 3 4 2 4 3 4 1 2

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Expe pert! Wher ere e is a s a st strateg egic st starting c cel ell?

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“Thinking Clouds” can be used to assess student reasoning. Students can also add in their

  • wn “Thinking Clouds” to

demonstrate their reasoning

  • ability. Dating these records will

assist teachers to monitor student progress.

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Partitions of 10

How many different ways can you partition (split) 10? 3 and 7 3 and 3 and 2 and 1 and 1

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42 Partitions of 10

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Decide what digits go in the boxes?

  • Can you find a solution?
  • How many solutions do you think there might be?
  • How will you know when you have found them all?

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Hurst, C. (2011). Connecting with the Australian Curriculum: Integrating learning through the proficiency strands. In J. Clark, B. Kissane, J. Mousley, T. Spencer, &

  • S. Thornton (Eds.). Mathematics: Traditions and [New] Practices (Proceedings of the 34th annual conference of the Mathematics Education Research Group of

Australasia and the Australian Association of Mathematics Teachers), pp. 973-980. Adelaide: AAMT and MERGA.

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Questions

  • What sort of a number sentence do we have?
  • How big are the numbers?
  • What do we know about the numbers?
  • What digits could be in the ones place?
  • What numbers can we multiply together to make a number ending in 6?

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Hints

  • 1, 6 what about 6, 1?
  • 2, 3
  • 6, 6 Can you use a digit more than once?
  • 2, 8
  • 7, 8
  • 4, 4
  • 4, 9

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Open – Ended Tasks

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Every school should buy at least one copy

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One example

  • I have three coins in my hand, How much money might I have?

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  • From Open – Ended Maths Activities, Sullivan and Lilburn
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I have three coins in my hand. How much money do I have?

Some Prompts:

What is the lowest total? What is the highest total? Can any totals be made in more than one way? What are all the possible totals? How will you know when you have all of the solutions? (Parallel Problems)

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Solve a simpler – related problem: 2 Coins

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Is it the Maths or Literacies that is the Problem?

Newman Analysis 5 Pr Prom

  • mpts
  • 1. Please read the question to me. If you don't know a word, leave it out

(Decoding)

  • 2. Tell me what the question is asking you to do (Comprehending)
  • 3. Tell me how you are going to find the answer (Transforming)

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Newman Analysis

cont

  • 4. Show me what to do to get the answer. "Talk aloud" as you do it, so that I

can understand how you are thinking (Applying)

  • 5. Now, write down your answer to the question (Encoding into required

format)

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Word Problems

  • “70% of errors made by students on standard

word problems could be attributed to lack of comprehension”(Clements, 2004)

  • However,

“There is significant research evidence that reveals that recognising words in isolation does not necessarily mean they will be recognised as quickly in connected text.”

(Ehri, 1997)

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Maths Literacies – Jigsaw

  • What do we mean by this term?
  • 1. Vocabulary and Comprehension;
  • 2. The Structure of Word Problems;
  • 3. Graphics in Mathematics; and
  • 4. Symbols in Mathematics.

Text organisation is also very significant in problem solving.

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Literacy and Mathematics

  • Classroom Discussion
  • Effect Size 0.82
  • Vocabulary Programs
  • Effect Size 0.67

Hattie et al (2017). Visible Learning for Mathematics

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Vocabulary

  • Vocabulary knowledge begets comprehension (and vice versa)
  • Students need 90-95% automatic word recognition to provide necessary

context to allow for comprehension to occur

Hi Hirs rsch an and Nat ation 19 1992

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Vocabulary

Worth Paying Attention To? Marzano (2004) found that teaching academic vocabulary could positively influence standardized test scores by as much as 33%.

  • Dr. Madeline Kovarik, Building Mathematics Vocabulary,

International Journal for Mathematics Teaching and Learning, Oct 2010

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Definitions: Help or Hindrance

Read

Annul nulus us

Explain Define

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Swan, P ., (2018). The impact of vocabulary on numeracy. The Australian Primary Mathematics Classroom. 23, 4.

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Structure of Word Questions

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First steps in Mathematics Number – Book 2 Pages 89 and 90. http://det.wa.edu.au/stepsresources/detcms/navigati

  • n/first-steps-mathematics/

First Steps in Number - Book 2: Understand Operations Calculate Reason About Number Patterns

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Graphic Help or Hinderance?

  • NAPLAN 2016 Yr5 q19

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Symbols

  • Simply recalling what a symbol says is often not enough. Students need

to understand what it actually means and the conceptual implications.

  • Consider the equal sign ( = ) as an example here.

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Critical Thinking Puzzles (rods)

  • Three rods
  • All different
  • Train is as long as blue
  • Longest rod is shorter than yellow

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Reasoning with Rods

  • For further puzzles see

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Can you make it an inquiry task?

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Four Fours Using only four 4’s and any operation, make the numbers 1 to 20, inclusive. Extension: Can you find the numbers 21 to 50? Can you find the numbers 1 to 20 using Five 5’s?

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Can you make it an inquiry task?

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Mystery Spinner Pages 141 – 144 Years 5 - 8

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Leadership: Adopt a Menu Approach

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Menus

  • It is one thing to know a word
  • It is quite another thing to use it
  • Menu provides the opportunity for children to use words
  • Leadership
  • Teachers can try and report
  • Everyone can own/lead something

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E.L.P.S

Experience Language Picture Symbols

Liebeck, P . (1991). How children learn mathematics. Penguin

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Mystery Bag

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9 Card Bingo

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