A major focus on current mathematics education is "problem solving." But "problem solving" means something very different from "Doing the exercises at the end of the chapter." An explanation of what problem solving is, and how it can be implemented.

1. Problem Solving in Mathematics Education
Jeff Suzuki
Department of Mathematics
Brooklyn College
Brooklyn NY 11210
jeff suzuki@yahoo.com
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2. Problems and Exercises
Mathematics education standards now emphasize problem solving as an important
goal.
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3. Problems and Exercises
Mathematics education standards now emphasize problem solving as an important
goal.
Wait a minute, isn’t that what we’ve been doing with all those things at the end
of each section of a math book?
J. Suzuki (CUNY) Problem Based Learning 2 / 10

4. Problems and Exercises
Mathematics education standards now emphasize problem solving as an important
goal.
Wait a minute, isn’t that what we’ve been doing with all those things at the end
of each section of a math book?
The quick answer:
J. Suzuki (CUNY) Problem Based Learning 2 / 10

5. Problems and Exercises
Mathematics education standards now emphasize problem solving as an important
goal.
Wait a minute, isn’t that what we’ve been doing with all those things at the end
of each section of a math book?
The quick answer: Probably not.
J. Suzuki (CUNY) Problem Based Learning 2 / 10

6. A Lesson on Exponents
Consider the rules of exponents, as presented in a traditional math course.
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7. A Lesson on Exponents
Consider the rules of exponents, as
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8. A Lesson on Exponents
Consider the rules of exponents, as I’ve taught them in the past:
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9. A Lesson on Exponents
Consider the rules of exponents, as I’ve taught them in the past:
Define an
as the product of n as,
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10. A Lesson on Exponents
Consider the rules of exponents, as I’ve taught them in the past:
Define an
as the product of n as,
Examples: 23
= 2 × 2 × 2.
J. Suzuki (CUNY) Problem Based Learning 3 / 10

11. A Lesson on Exponents
Consider the rules of exponents, as I’ve taught them in the past:
Define an
as the product of n as,
Examples: 23
= 2 × 2 × 2.
Problem: 23
× 22
?
J. Suzuki (CUNY) Problem Based Learning 3 / 10

12. A Lesson on Exponents
Consider the rules of exponents, as I’ve taught them in the past:
Define an
as the product of n as,
Examples: 23
= 2 × 2 × 2.
Problem: 23
× 22
?
Solution: 23
× 22
= (2 × 2 × 2) × (2 × 2) = 25
.
J. Suzuki (CUNY) Problem Based Learning 3 / 10

13. A Lesson on Exponents
Consider the rules of exponents, as I’ve taught them in the past:
Define an
as the product of n as,
Examples: 23
= 2 × 2 × 2.
Problem: 23
× 22
?
Solution: 23
× 22
= (2 × 2 × 2) × (2 × 2) = 25
.
Generalization: am
an
= am+n
J. Suzuki (CUNY) Problem Based Learning 3 / 10

14. A Lesson on Exponents
Consider the rules of exponents, as I’ve taught them in the past:
Define an
as the product of n as,
Examples: 23
= 2 × 2 × 2.
Problem: 23
× 22
?
Solution: 23
× 22
= (2 × 2 × 2) × (2 × 2) = 25
.
Generalization: am
an
= am+n
Example: 510
53
= 510+3
.
J. Suzuki (CUNY) Problem Based Learning 3 / 10

15. A Lesson on Exponents
Consider the rules of exponents, as I’ve taught them in the past:
Define an
as the product of n as,
Examples: 23
= 2 × 2 × 2.
Problem: 23
× 22
?
Solution: 23
× 22
= (2 × 2 × 2) × (2 × 2) = 25
.
Generalization: am
an
= am+n
Example: 510
53
= 510+3
.
Homework: Find 35
32
, x5
x8
, etc.
J. Suzuki (CUNY) Problem Based Learning 3 / 10

16. Following Examples
Once you’ve been shown how to solve am
an
, finding am
an
is a matter of following
an example.
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17. Following Examples
Once you’ve been shown how to solve am
an
, finding am
an
is a matter of following
an example.
When mathematics is presented this way, students are trained to look for
examples where similar questions have been solved, then follow the examples to an
answer. But:
J. Suzuki (CUNY) Problem Based Learning 4 / 10

18. Following Examples
Once you’ve been shown how to solve am
an
, finding am
an
is a matter of following
an example.
When mathematics is presented this way, students are trained to look for
examples where similar questions have been solved, then follow the examples to an
answer. But:
It’s impossible to give examples of every type of question that could appear,
so students will often encounter questions for which they have no examples.
J. Suzuki (CUNY) Problem Based Learning 4 / 10

19. Following Examples
Once you’ve been shown how to solve am
an
, finding am
an
is a matter of following
an example.
When mathematics is presented this way, students are trained to look for
examples where similar questions have been solved, then follow the examples to an
answer. But:
It’s impossible to give examples of every type of question that could appear,
so students will often encounter questions for which they have no examples.
Judging similarity requires experience and sophistication:
J. Suzuki (CUNY) Problem Based Learning 4 / 10

20. Following Examples
Once you’ve been shown how to solve am
an
, finding am
an
is a matter of following
an example.
When mathematics is presented this way, students are trained to look for
examples where similar questions have been solved, then follow the examples to an
answer. But:
It’s impossible to give examples of every type of question that could appear,
so students will often encounter questions for which they have no examples.
Judging similarity requires experience and sophistication: 3x + 5 = 2x and
3x + 5 = x2
are similar . . .
J. Suzuki (CUNY) Problem Based Learning 4 / 10

21. Following Examples
Once you’ve been shown how to solve am
an
, finding am
an
is a matter of following
an example.
When mathematics is presented this way, students are trained to look for
examples where similar questions have been solved, then follow the examples to an
answer. But:
It’s impossible to give examples of every type of question that could appear,
so students will often encounter questions for which they have no examples.
Judging similarity requires experience and sophistication: 3x + 5 = 2x and
3x + 5 = x2
are similar . . . but they’re not solved the same way.
J. Suzuki (CUNY) Problem Based Learning 4 / 10

22. Solving Problems
Instead of being given examples, students can solve problems:
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23. Solving Problems
Instead of being given examples, students can solve problems:
Define an
as the product of n as,
J. Suzuki (CUNY) Problem Based Learning 5 / 10

24. Solving Problems
Instead of being given examples, students can solve problems:
Define an
as the product of n as,
Examples: 23
= 2 × 2 × 2.
J. Suzuki (CUNY) Problem Based Learning 5 / 10

25. Solving Problems
Instead of being given examples, students can solve problems:
Define an
as the product of n as,
Examples: 23
= 2 × 2 × 2.
Problem: 23
× 22
?
J. Suzuki (CUNY) Problem Based Learning 5 / 10

26. Solving Problems
Instead of being given examples, students can solve problems:
Define an
as the product of n as,
Examples: 23
= 2 × 2 × 2.
Problem: 23
× 22
?
At this point, the focus shifts to the student.
J. Suzuki (CUNY) Problem Based Learning 5 / 10

27. Solving Problems
Instead of being given examples, students can solve problems:
Define an
as the product of n as,
Examples: 23
= 2 × 2 × 2.
Problem: 23
× 22
?
At this point, the focus shifts to the student.
Because the student hasn’t been shown how to solve 23
22
, this is a real problem.
J. Suzuki (CUNY) Problem Based Learning 5 / 10

28. Solving Problems
Instead of being given examples, students can solve problems:
Define an
as the product of n as,
Examples: 23
= 2 × 2 × 2.
Problem: 23
× 22
?
At this point, the focus shifts to the student.
Because the student hasn’t been shown how to solve 23
22
, this is a real problem.
But if they understand the concept of exponents, they can solve this easily, and
with some guidance, go on to the problems:
J. Suzuki (CUNY) Problem Based Learning 5 / 10

29. Solving Problems
Instead of being given examples, students can solve problems:
Define an
as the product of n as,
Examples: 23
= 2 × 2 × 2.
Problem: 23
× 22
?
At this point, the focus shifts to the student.
Because the student hasn’t been shown how to solve 23
22
, this is a real problem.
But if they understand the concept of exponents, they can solve this easily, and
with some guidance, go on to the problems:
Find 58
512
J. Suzuki (CUNY) Problem Based Learning 5 / 10

30. Solving Problems
Instead of being given examples, students can solve problems:
Define an
as the product of n as,
Examples: 23
= 2 × 2 × 2.
Problem: 23
× 22
?
At this point, the focus shifts to the student.
Because the student hasn’t been shown how to solve 23
22
, this is a real problem.
But if they understand the concept of exponents, they can solve this easily, and
with some guidance, go on to the problems:
Find 58
512
Find (xy3
)2
and (x + 3)2
J. Suzuki (CUNY) Problem Based Learning 5 / 10

31. Solving Problems
Instead of being given examples, students can solve problems:
Define an
as the product of n as,
Examples: 23
= 2 × 2 × 2.
Problem: 23
× 22
?
At this point, the focus shifts to the student.
Because the student hasn’t been shown how to solve 23
22
, this is a real problem.
But if they understand the concept of exponents, they can solve this easily, and
with some guidance, go on to the problems:
Find 58
512
Find (xy3
)2
and (x + 3)2
Find x5
x2
J. Suzuki (CUNY) Problem Based Learning 5 / 10

32. Keys to Incorporating Problem Solving
There are three keys to incorporating problem solving:
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33. Keys to Incorporating Problem Solving
There are three keys to incorporating problem solving:
Practice.
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34. Keys to Incorporating Problem Solving
There are three keys to incorporating problem solving:
Practice.
Patience.
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35. Keys to Incorporating Problem Solving
There are three keys to incorporating problem solving:
Practice.
Patience.
Preparation.
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36. Practice
Problem solving is a skill: you get better at it the more often you do it.
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37. Practice
Problem solving is a skill: you get better at it the more often you do it.
But problem solving is like first impressions: you NEVER get a second chance to
solve a problem for the first time.
J. Suzuki (CUNY) Problem Based Learning 7 / 10

38. Practice
Problem solving is a skill: you get better at it the more often you do it.
But problem solving is like first impressions: you NEVER get a second chance to
solve a problem for the first time. The instant someone shows you how to solve a
problem, the opportunity to solve that problem is gone forever.
J. Suzuki (CUNY) Problem Based Learning 7 / 10

39. Practice
Problem solving is a skill: you get better at it the more often you do it.
But problem solving is like first impressions: you NEVER get a second chance to
solve a problem for the first time. The instant someone shows you how to solve a
problem, the opportunity to solve that problem is gone forever.
This means:
J. Suzuki (CUNY) Problem Based Learning 7 / 10

40. Practice
Problem solving is a skill: you get better at it the more often you do it.
But problem solving is like first impressions: you NEVER get a second chance to
solve a problem for the first time. The instant someone shows you how to solve a
problem, the opportunity to solve that problem is gone forever.
This means:
AVOID giving examples of solved problems.
J. Suzuki (CUNY) Problem Based Learning 7 / 10

41. Practice
Problem solving is a skill: you get better at it the more often you do it.
But problem solving is like first impressions: you NEVER get a second chance to
solve a problem for the first time. The instant someone shows you how to solve a
problem, the opportunity to solve that problem is gone forever.
This means:
AVOID giving examples of solved problems. Instead, emphasize the
underlying concepts.
J. Suzuki (CUNY) Problem Based Learning 7 / 10

42. Practice
Problem solving is a skill: you get better at it the more often you do it.
But problem solving is like first impressions: you NEVER get a second chance to
solve a problem for the first time. The instant someone shows you how to solve a
problem, the opportunity to solve that problem is gone forever.
This means:
AVOID giving examples of solved problems. Instead, emphasize the
underlying concepts.
DISCOURAGE looking up the answer.
J. Suzuki (CUNY) Problem Based Learning 7 / 10

43. Practice
Problem solving is a skill: you get better at it the more often you do it.
But problem solving is like first impressions: you NEVER get a second chance to
solve a problem for the first time. The instant someone shows you how to solve a
problem, the opportunity to solve that problem is gone forever.
This means:
AVOID giving examples of solved problems. Instead, emphasize the
underlying concepts.
DISCOURAGE looking up the answer. This is the age of Google and
MathBFF, and if you don’t show the students “how to solve a problem,”
they’ll look for someone who will.
J. Suzuki (CUNY) Problem Based Learning 7 / 10

44. Practice
Problem solving is a skill: you get better at it the more often you do it.
But problem solving is like first impressions: you NEVER get a second chance to
solve a problem for the first time. The instant someone shows you how to solve a
problem, the opportunity to solve that problem is gone forever.
This means:
AVOID giving examples of solved problems. Instead, emphasize the
underlying concepts.
DISCOURAGE looking up the answer. This is the age of Google and
MathBFF, and if you don’t show the students “how to solve a problem,”
they’ll look for someone who will. Emphasize the once-in-a-lifetime
opportunity to solve a problem.
J. Suzuki (CUNY) Problem Based Learning 7 / 10

45. Patience
Problem solving requires students create solutions . . . but they will probably need
guidance.
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46. Patience
Problem solving requires students create solutions . . . but they will probably need
guidance.
Group work:
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47. Patience
Problem solving requires students create solutions . . . but they will probably need
guidance.
Group work: Real world problems usually require collaboration by hundreds or
thousands of people.
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48. Patience
Problem solving requires students create solutions . . . but they will probably need
guidance.
Group work: Real world problems usually require collaboration by hundreds or
thousands of people.
Time management:
J. Suzuki (CUNY) Problem Based Learning 8 / 10

49. Patience
Problem solving requires students create solutions . . . but they will probably need
guidance.
Group work: Real world problems usually require collaboration by hundreds or
thousands of people.
Time management: You don’t have enough time for a lot of examples and
then problem solving,
J. Suzuki (CUNY) Problem Based Learning 8 / 10

50. Patience
Problem solving requires students create solutions . . . but they will probably need
guidance.
Group work: Real world problems usually require collaboration by hundreds or
thousands of people.
Time management: You don’t have enough time for a lot of examples and
then problem solving, but presenting a lot of examples defeats the problem
solving.
J. Suzuki (CUNY) Problem Based Learning 8 / 10

51. Patience
Problem solving requires students create solutions . . . but they will probably need
guidance.
Group work: Real world problems usually require collaboration by hundreds or
thousands of people.
Time management: You don’t have enough time for a lot of examples and
then problem solving, but presenting a lot of examples defeats the problem
solving.
A flipped/inverted class structure works extremely well for problem based learning:
J. Suzuki (CUNY) Problem Based Learning 8 / 10

52. Patience
Problem solving requires students create solutions . . . but they will probably need
guidance.
Group work: Real world problems usually require collaboration by hundreds or
thousands of people.
Time management: You don’t have enough time for a lot of examples and
then problem solving, but presenting a lot of examples defeats the problem
solving.
A flipped/inverted class structure works extremely well for problem based learning:
students read about/watch videos on basic concepts outside of class, then come
to class to work problems.
J. Suzuki (CUNY) Problem Based Learning 8 / 10

53. Preparation
Given five minutes, most of us could prepare an hour-long lecture on an
introductory math topic (solving linear equations, differentiation, Gauss-Jordan
reduction).
J. Suzuki (CUNY) Problem Based Learning 9 / 10

54. Preparation
Given five minutes, most of us could prepare an hour-long lecture on an
introductory math topic (solving linear equations, differentiation, Gauss-Jordan
reduction).
Classes based around problem solving require significantly more preparation:
J. Suzuki (CUNY) Problem Based Learning 9 / 10

55. Preparation
Given five minutes, most of us could prepare an hour-long lecture on an
introductory math topic (solving linear equations, differentiation, Gauss-Jordan
reduction).
Classes based around problem solving require significantly more preparation:
Know your students:
J. Suzuki (CUNY) Problem Based Learning 9 / 10

56. Preparation
Given five minutes, most of us could prepare an hour-long lecture on an
introductory math topic (solving linear equations, differentiation, Gauss-Jordan
reduction).
Classes based around problem solving require significantly more preparation:
Know your students: Can your students go from the definition of exponents
to finding (xy3
)2
in one set of problems, or will it take several?
J. Suzuki (CUNY) Problem Based Learning 9 / 10

57. Preparation
Given five minutes, most of us could prepare an hour-long lecture on an
introductory math topic (solving linear equations, differentiation, Gauss-Jordan
reduction).
Classes based around problem solving require significantly more preparation:
Know your students: Can your students go from the definition of exponents
to finding (xy3
)2
in one set of problems, or will it take several?
Block the shortcuts:
J. Suzuki (CUNY) Problem Based Learning 9 / 10

58. Preparation
Given five minutes, most of us could prepare an hour-long lecture on an
introductory math topic (solving linear equations, differentiation, Gauss-Jordan
reduction).
Classes based around problem solving require significantly more preparation:
Know your students: Can your students go from the definition of exponents
to finding (xy3
)2
in one set of problems, or will it take several?
Block the shortcuts: Some will already know the rule, so how do you make
this question a problem?
J. Suzuki (CUNY) Problem Based Learning 9 / 10

59. Is It Worth It?
“Mathematics is a collection of rules and algorithms to follow,”
J. Suzuki (CUNY) Problem Based Learning 10 / 10

60. Is It Worth It?
“Mathematics is a collection of rules and algorithms to follow,” said no
mathematician ever.
J. Suzuki (CUNY) Problem Based Learning 10 / 10

61. Is It Worth It?
“Mathematics is a collection of rules and algorithms to follow,” said no
mathematician ever.
OK, technically I just said it, but we’ll ignore the paradox of Epimenides.
J. Suzuki (CUNY) Problem Based Learning 10 / 10

62. Is It Worth It?
“Mathematics is a collection of rules and algorithms to follow,” said no
mathematician ever.
Problem Based Learning:
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63. Is It Worth It?
“Mathematics is a collection of rules and algorithms to follow,” said no
mathematician ever.
Problem Based Learning:
Reinforces conceptual understanding:
J. Suzuki (CUNY) Problem Based Learning 10 / 10

64. Is It Worth It?
“Mathematics is a collection of rules and algorithms to follow,” said no
mathematician ever.
Problem Based Learning:
Reinforces conceptual understanding: If you don’t know what an
means, you
can’t find an
am
;
J. Suzuki (CUNY) Problem Based Learning 10 / 10

65. Is It Worth It?
“Mathematics is a collection of rules and algorithms to follow,” said no
mathematician ever.
Problem Based Learning:
Reinforces conceptual understanding: If you don’t know what an
means, you
can’t find an
am
; and you shouldn’t: it’s like giving a chainsaw to a toddler.
J. Suzuki (CUNY) Problem Based Learning 10 / 10

66. Is It Worth It?
“Mathematics is a collection of rules and algorithms to follow,” said no
mathematician ever.
Problem Based Learning:
Reinforces conceptual understanding: If you don’t know what an
means, you
can’t find an
am
; and you shouldn’t: it’s like giving a chainsaw to a toddler.
Trains students to thinking about mathematics the way that mathematicians
do:
J. Suzuki (CUNY) Problem Based Learning 10 / 10

67. Is It Worth It?
“Mathematics is a collection of rules and algorithms to follow,” said no
mathematician ever.
Problem Based Learning:
Reinforces conceptual understanding: If you don’t know what an
means, you
can’t find an
am
; and you shouldn’t: it’s like giving a chainsaw to a toddler.
Trains students to thinking about mathematics the way that mathematicians
do: “Mathematics is science of necessary consequences” (Peirce).
J. Suzuki (CUNY) Problem Based Learning 10 / 10

68. Is It Worth It?
“Mathematics is a collection of rules and algorithms to follow,” said no
mathematician ever.
Problem Based Learning:
Reinforces conceptual understanding: If you don’t know what an
means, you
can’t find an
am
; and you shouldn’t: it’s like giving a chainsaw to a toddler.
Trains students to thinking about mathematics the way that mathematicians
do: “Mathematics is science of necessary consequences” (Peirce).
Humanizes mathematics:
J. Suzuki (CUNY) Problem Based Learning 10 / 10

69. Is It Worth It?
“Mathematics is a collection of rules and algorithms to follow,” said no
mathematician ever.
Problem Based Learning:
Reinforces conceptual understanding: If you don’t know what an
means, you
can’t find an
am
; and you shouldn’t: it’s like giving a chainsaw to a toddler.
Trains students to thinking about mathematics the way that mathematicians
do: “Mathematics is science of necessary consequences” (Peirce).
Humanizes mathematics: Anything that can be solved by following an
example can be done faster, more accurately, and less expensively by a
computer.
J. Suzuki (CUNY) Problem Based Learning 10 / 10

70. Is It Worth It?
“Mathematics is a collection of rules and algorithms to follow,” said no
mathematician ever.
Problem Based Learning:
Reinforces conceptual understanding: If you don’t know what an
means, you
can’t find an
am
; and you shouldn’t: it’s like giving a chainsaw to a toddler.
Trains students to thinking about mathematics the way that mathematicians
do: “Mathematics is science of necessary consequences” (Peirce).
Humanizes mathematics: Anything that can be solved by following an
example can be done faster, more accurately, and less expensively by a
computer. The real lesson of John Henry: Don’t try to beat the machine; try
to transcend the machine.
J. Suzuki (CUNY) Problem Based Learning 10 / 10