What are students really thinking as they solve two types of problems

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What are students really thinking as they solve
two types of problems?
SSMA Conference - October 2006 - Missoula, Montana

• Victor Cifarelli (UNC Charlotte),
• Mary Margaret and Robert Capraro (Texas A M),
• Linda Zientek (Blinn Community College)

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Problem Solving

• Solving problems is a complex task that is
  essential to the teaching and learning of
  mathematics (NCTM, 2000)

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Developing Good Problem Solvers

• Provide students with rich problem solving
  experiences that include
• both well structured problems (where the problem
  conditions and goals are explicitly stated within
  the problem statement) and
• open-ended problems (Becker Shimada, 1997),
  goals are not explicitly stated within the
  problem statement and the solver must develop
  particular goals for her/his actions.

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Closed Problems

• One and only one correct answer is predetermined
• Answers are either correct or incorrect
• All mathematical conditions are provided for the
  solution
• Students retrieve their learned knowledge
• Evaluation checks students knowledge of applying
  concepts or rules.

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Open-ended Problems

• Multiple correct answers
• Multiple solution strategies
• Many correct answers can be given
• Write down as many findings as possible
• Higher-order thinking
• Students must use more than their
  school-developed repertoire

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Five Advantages to Open-ended Problem Solving
(Sawada,1997)

• Students participate more actively in lessons and
  express their ideas more frequently.
• Open-ended problem solving provides a free,
  responsive, and supportive learning environment
  since so many different correct solutions, so
  that each student has opportunities to get own
  unique solutions.

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Advantage 2

• Every student can respond to the problem in some
  significant ways.
• Important for every student to be involved in
  classroom activities
• Lessons should be understandable for every
  student
• Open-ended problems provide every student with
  the opportunities to find own answers.

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Advantage 3

• Students have more opportunities to make
  comprehensive use of their mathematical knowledge
  and skills.
• Many different solutions, students can choose
  their favorite strategies to obtain the answers
  and create their unique solutions.

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Advantages 4 5

• Provide students with a reasoning experience.
  Through comparing and discussing, students are
  intrinsically motivated to give other students
  reasons for their solutions (great opportunity
  for students to develop their mathematical
  thinking).
• Rich experiences for students to have pleasure of
  discovery receive approval from fellow students.

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Views of Problem Open-ness

• Objective
• Focus on explicit structure of tasks that can be
  manipulated and see how the group copes
• Implications
• 1. expect much inductive exploration at the start
  
• (self-generate the data and then reflect,
  trial and error)
• 2. expect specific patterns to be found ( we have
  an idea of what they will come up with)

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Subjective Implications

• Subjective
• Focus on the sense-making of individual solvers
  and look for areas of fit and compatibility in
  the solvers actions
• Implications
• 1. try to see the solvers point of view
  important to monitor the questions he/she sees
  fit to investigate (may not align directly with
  the task questions)
• 2. expect the unexpected (inductive explorations
  but also be vigilant for deductive and abductive
  acts of reasoning)

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Number Array 1
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Number Array 1 Directions

• The following table of numbers was produced in
  accordance with a certain rule and has many rich
  relationships. Study the arrangement of numbers
  in the table and find as many relationships as
  possible among the numbers.
• Please record on the paper provided the various
  relationships that you see and tell me what you
  are thinking of as you explore the relationships.
  

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Number Array 2

• The following table of numbers was produced in
  accordance with a certain rule. Fill in the rest
  of the table according to the pattern.
• Line 1 2 3 4 5
  6 7 8 9 10
• 1 1 2 3 4 5 6
  7 8 9 10
• 2 3 5 7 9 11
  13 15 17 19
• 3 8 12 16 20 24
  28 32 36
• 4 20 28 36 44 52 60
  68
• __ __ __ __ __ __ __
  
• __ __ __ __ __ __
• __ __ __ __ __
• __ __ __ __
• __ __ __
• __ __

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Number Array 2 Solved

• The following table of numbers was produced in
  accordance with a certain rule. Fill in the rest
  of the table according to the pattern.
• Line 1 2 3 4 5
  6 7 8 9 10
• 1 1 2 3 4 5 6
  7 8 9 10
• 2 3 5 7 9 11
  13 15 17 19
• 3 8 12 16 20 24
  28 32 36
• 4 20 28 36 44 52 60
  68
• 5 48 64 80 96 112 128
• 6 112 144 176 208 240
• 7 256 320 384 448
• 8 576 704 832
• 9 1280 1536
• 10 2816

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M.A in Math Education students (N4)
Summary of Results
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Valerie

• Line 1 2 3 4 5
  6 7 8 9 10
• 1 1 2 3 4 5 6
  7 8 9 10
• 2 3 5 7 9 11
  13 15 17 19
• 3 8 12 16 20 24
  28 32 36
• 4 20 28 36 44 52 60
  68
• 5 48 64 80 96 112 128
• 6 112 144 176 208 240
• 7 256 320 384 448
  
• 8 576 704 832
• 9 1280 1536
• 10 2816
• Valerie
• 1. It is same difference each time, 2, 22,23 and
  so on each row.
• 2. How about 1x123, that may work. So,
  2x328, 3x82 no it doesnt work to
• get the 20.
• 3. But it does seem to involve doubling and
  adding a power of 2 (reflection)
• 4. The line numbers confused me. (reflection) I
  can double each number in the first
• column and then the 2n comes in, so 2x32
  gets 8, yes, 2x8420, , that works.

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Danielle

• Line 1 2 3 4 5
  6 7 8 9 10
• 1 1 2 3 4 5 6
  7 8 9 10
• 2 3 5 7 9 11
  13 15 17 19
• 3 8 12 16 20 24
  28 32 36
• 4 20 28 36 44 52 60
  68
• 5 48 64 80 96 112 128
• Danielle
• 4. So then we add 4 each time, then 8 or 23, then
  16, so I know we go up by each time.
• 7. But where do we get the first number from?
  (reflection)
• 12. 3 and 8 are one less than perfect squares
  (reflection) but thats not it!
• 13. I know I saw that 21, 22, but I am thinking
  too much in terms of exponents.
• 17. (points to first column) That has a
  difference of 5 (3 8), that has a difference of
  12
• (8 20), and (points to second column),
  that has a difference of 7 ( 5 12)
• 20. So (reflection) how about we add those! So
  235 but theres nothing else.
• 21. What about 24, it is 53 is 8 times 3 equals
  24 but it doesnt work anywhere else
• 25. 5 is 41 and 235, oh wait a second!
  (reflection)
• 26. If I do 213 and 538, 128, thats 20, so
  so if that works, first number should be 48.
• 27. (reflection) So the other numbers okay I
  see, 48 24, that is 64 and also 283664.

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Jennifer

• Line 1 2 3 4 5
  6 7 8 9 10
• 1 1 2 3 4 5 6
  7 8 9 10
• 2 3 5 7 9 11
  13 15 17 19
• 3 8 12 16 20 24
  28 32 36
• 4 20 28 36 44 52 60
  68
• 5 48 64 80 96 112 128
• Jennifer
• 3. The row goes up by 1, then by 2, then by 4,
  and the rest go up the same by powers of 2.
• 4. So the problem is finding the first number
  over here. If I can find the start number, I just
  add everywhere these powers of 2.
• 6. (reflection) I guess that I can look over here
  to see what happens (points to 10 and 19 on the
  right side) . These still go up by 1, 2, 4, 8 and
  like that.
• 8. So it goes from 10 to 19, thats 9, 9 to 17,
  thats 8, and 19 to 36, to get 17, so we get
  differences of 9, 8, and 17, and then 32 to 68 to
  get 32.
• 9. Not sure what I am doing. I would have
  thought they would be like powers of 2 over here.
  (long reflection)
• 10. Oh wait 91019. These add to the 19 below
  in the next row. So, maybe I need to be looking
  at these triangles. Yes, that might work.
• 13. Lets look at the first one over here,
  323668, 6068128 the adding seems to work.
• 15. Now I can use that powers of 2 I saw. So,
  128-2, its one row less for the power, so 128-24
  112 in this row. Interesting, I can find all of
  the numbers, down to the first number.
• 16. (after filling the table) Okay, these all
  work and I am done!

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Kassey

• Line 1 2 3 4 5
  6 7 8 9 10
• 1 1 2 3 4 5 6
  7 8 9 10
• 2 3 5 7 9 11
  13 15 17 19
• 3 8 12 16 20 24
  28 32 36
• 4 20 28 36 44 52 60
  68
• 5 48 64 80 96 112 128
• 6 112 144 176 208 240
• 7 256 320 384 448
  
• 8 576 704 832
• 9 1280 1536
• 10 2816
• Kassy
• 1. (reads the problem, reflects)
• 2. Okay it looks like 123, 347. 358.
• 3. Yep, it works in all of the other rows so I
  just keep adding.
• 4. (fills out the entire table) So, I get 2816
  down here and all the other numbers.
• 5. Thats about it.

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Concluding Thoughts

• Solvers determine what is problematic about the
  task (find the first number and then add 2n vs
  extend the pattern of sums)
• Implications Teachers need to be prepared for a
  range of possibilities for discussion with our
  students (to deal with levels of mathematical
  sophistication)
• Tasks may not be problems for all students -- if
  not, are they still beneficial for students to
  complete?
• Different levels of problem solving to find the
  first number
• 1. Danielle and Jennifer exploring to identify
  and extend a pattern.
• 2. Valeries problem solving Making the idea
  work for her!

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Community College Study

• Mathematics content course for elementary
  teachers
• 2nd week of semester - completed problem solving
  section
• Textbook had no similar problems
• Self-selected groups 5 groups (N13)
• Number Array Problem

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Results of Number Array 1

• All 5 groups noticed the diagonal terms were
  squared numbers.
• 4 of the 5 noted that the columns and rows formed
  arithmetic sequences.
• 3 of the 5 noted that column 2 and row 2 were
  multiples of two, column 3 and row 3 were
  multiples of 3 and that this pattern continued.
• Two groups noted that if you take the entry in
  each row by each column then you would get the
  corresponding entry. They had difficulty
  expressing this in a written statement. For
  example, all lines across multiple w/lines down
  to create answers in the box.
• Two groups noticed that the diagonal formed an
  arithmetic sequence.

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Results of Number ARRAY 2

• Groups had no difficulty completing the charts.
• All 5 groups noticed rows formed arithmetic
  sequences.
• 4 of the 5 groups noticed that if a number
  equaled the the sum of the two numbers directly
  above it.
• 2 groups noticed that if you took the difference
  between the columns then the values formed a
  geometric sequence.

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On-line problem solving class

• Middle school pre-service teachers
• MASC 351 Problem Solving - Polyas Four Steps
• Two sections (N 57)
• Solved Number Array 1 and 2
• Results of Number Array 1 - all responses sent
  through WebCt
• No feedback from instructor or classmates
• Chat room was open - no visitors

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On-line class responses

• The number in the mth row and nth column is mn
  (37 responses)
• Numbers in each column constitute an arithmetic
  progression (35 responses)
• Number in each column and row are multiples of
  1,2,3.( 25 responses)
• All numbers on the main diagonal are square
  numbers
• (28 responses)

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Responses continued

• The numbers are symmetrically arranged with
  respect to the main diagonal (18 responses)
• Row 1 identical to Column 1 (13 responses)
• Diagonal from top left to bottom right
  contains a consistent pattern of increasing by an
  incremental even or odd amount (9 responses)
• Each diagonal begins with an even or odd
  amount (5 responses)

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Single Responses

• From the 2nd number to the 3rd multiply by 3/2
  3rd to 4th number multiply by 4/3 4th number to
  5th number by 5/4
• (2x2)/1 4 3x4/264x6/38 6x10/512
• Take a square of 4 numbers, the numbers in the
  bottom left and top right corners can be
  multiples together, then divided by the number in
  the top left corner to give the number in the
  bottom right corner (45x48)/40 54
• As you get closer to the bottom and
• the the right the numbers get larger.

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Results and Conclusions

• Average student found 4 relationships
• Responses ranged from 1 to 8 relationships
• Lower achieving students found less relationships
• Some of the higher achieving students found more
  relationships
• Online format does not lend itself to generating
  a variety of creative responses in its present
  format
• Next semester..still thinking about this