Ways to View Problem Openness

1
Ways to View 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)
• 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)

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
• __ __ __ __ __ __ __
  
• __ __ __ __ __ __
• __ __ __ __ __
• __ __ __ __
• __ __ __
• __ __

3

• 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

4

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

5

• 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. 5.
So, 2x208 should be so the number to start
with is 48. 16. (after filling the table) Now I
notice that the numbers all add, like 347 in
row 2, 5260112 in row 5. I could have done
that also! So there are two ways. I like that!
It would be good for my kids to solve!
6

• 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. So
it works both ways. 30. (fills out the table) So,
I found the first number by adding the two
numbers in the previous row that pattern
hold throughout the list. I am not sure why, it
not seem legitimate to me, but it does work.
7

• 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!
8

• 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.
9
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!