Emphasizing Critical Thinking
When Teaching Mathematics to
Future Elementary
School Teachers
By Dr. Renan Sezer
LaGuardia Community
College
Traits I wanted to emphasize
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Problem solving
Thinking out side of the box
Creative thinking
Adoptability
Making connections between ideas
Communicating our thinking
Being able to see valuable points in opposing view
points
Pre-Test
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Solve the problem below:
“A farmer raises only chickens and pigs. When he
counts the heads of all his animals he finds 58 heads,
when he counts the legs he finds 188. How many
chickens and how many pigs does he have?” Hint:
Chickens have 2 legs and pigs have 4 legs.
Outline how you would approach the problem.
Carry out the steps outlined above i.e. solve the
problem.
 When
you divide fractions, you
take the reciprocal of the second
number and multiply the first
number with it. Explain the
reason why it works. (You are
not asked to give numerical
examples.)
POLYA’S STEPS in PROBLEM SOLVING
GENERALIZING SOLUTIONS
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Please solve the problem below using
Polya’s Steps in Problem Solving (1) Read
and understand the problem, 2) Devise a
Plan, 3) Carry Out the Plan, 4) Look Back)
Be sure that you indicate how you are
fulfilling each step. Make sure that your
approach to this problem is accessible to
elementary school students.
 “A farmer
raises only chickens
and pigs. When he counts the
heads of all his animals he finds
67 heads, when he counts the legs
he finds 214. How many
chickens and how many pigs does
he have?”
Hints for the “Look Back” stage:
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Is the answer to the problem unique? Is
there a shorter approach to solve the
problem?
Could you solve this type of problem (ie
with different numbers) if instead of
chickens and pigs the farmer raised chickens
and spiders? (Spiders have 8 legs.) Would
there be a solution? Would it be unique?
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Could you solve this type of problem (ie with
different numbers) if instead of chickens and pigs
the farmer raised pigs and horses? Would there be
a solution? Would it be unique?
Could you solve this type of problem (ie with
different numbers) if instead of chickens and pigs
the farmer raised chickens, pigs and spiders?
Would there be a solution? Would it be unique?
(This part will require mathematics that is beyond
elementary school, unless you use trial and error.)
Looking at your answers to hints 1-4, under
what conditions does the problem have a
solution? Under what conditions is the
solution unique?
 Write a generalization to solve this type of
problem (do not use particular numbers, but
rather what those numbers stand for.) Make
sure you indicate under what conditions this
problem has a solution/ a unique solution.
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THINKING OUTSIDE OF THE
BOX
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You are on one side of a bank with a live
fox, a live rabbit and a head of lettuce. You
have a small boat with which you can cross
to the other side, but the boat can
accommodate only one more thing other
than yourself. How many times do you
need to cross the river to take all three of
them safely across?
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Read all the questions below before you
start working on the above problem.
1) Solve the problem above. If you know
the solution, please do not tell it to others
in the class, so they have a chance to
think for themselves.
2) While you are working on the problem,
step back and monitor your own thinking.
What did you do when facing an unknown
problem?
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3) What problem solving methods did you
utilize in this problem?
CREATIVE SOLUTIONS
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You are given a candle that takes 1 hour to burn completely.
But the speed with which the candle burns is not uniform. It
burns slower on one end and faster on the other end. You
are not given these rates. You are given the candle, a box of
matches and put in an empty room that has no clock and you
do not have a watch or anything you can tell the time with (no
cell phone, computer etc.). Moreover there is no ruler (or
anything that you can use as a ruler, say strings etc.)
How can you tell exactly if half an hour has passed?
MAKING CONNECTIONS
RE-LEARNING BASE 10 CONCEPT
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In the number 325,647 written in base 10,
what does 7, 4, 6, 5, 2, and 3
represent?
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How many single digit numbers are there
in base 10? What is the largest single
digit number you can write in base 10?
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How many single digit numbers are there
in base 4? What is the largest single digit
number you can write in base 4?
The following numbers are written in base
4. What numbers do they represent in
base 10?
a) 21
b)323
c) 3012
Write following base 10 numbers in base
4.
a) 8 b) 12
c) 38
d) 69
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Do the following operations on base 4.
a) 312+323 b) 301-233
c) 123 x 2
Digits
How many single digit numbers are there in base
12?
What is the largest single digit number you can
write in base 12?
Make suggestions on how you might write the
number 10 in base 12 notation.
QUESTIONING ALGORITHMS
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1)Divide 1462 by 34 using long division.
2) Long division is the only operation among the
basic four operations that start with the biggest
digit i.e. we start with the left most digit of the
dividend (the number being divided). Can you do
long division starting with the ones digit and going
left rather than starting on the left and going right?
Try it with the exercise in #1.
 3)
If you can do the division as
indicated in #2, how do you need
to adjust your record keeping?
 4) a) Is it possible to do division
either way correctly?
 b) If so, which is easier and
why?
WHY DO WE DO WHAT WE DO?
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Solve the following problem:
John painted 2/5 of a room. The next day
Anne came and painted ¼ of the remaining
part of the room. What part of the room
remains unpainted?
Explain the problem using a picture or in
words as you would to an elementary school
student.
 Solving
this problem you had to use
subtraction and multiplication.
(Depending on your approach you
might have used even addition.) In
your explanation include why you
need to put fractions on common
denominator when you subtract/add
but you do not need to when you
multiply.
READING CRITICALLY
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NCTM STANDARDS
This assignment is related to the assigned reading of the
following websites:
NCTM:
http://standards.nctm.org/document/appendix/numb.htm
In the above document click on Principles or go to
http://standards.nctm.org/document/chapter2/index.htm
Then Standards:
http://standards.nctm.org/document/chapter3/index.htm
Then Pre K-2:
http://standards.nctm.org/document/chapter4/numb.htm
Then 3-5 (Overview of Standards 3-5):
http://standards.nctm.org/document/chapter5/numb.htm
W. G. Quirk’s website:
 http://www.wgquirk.com/TruthK12.html
http://www.wgquirk.com/Genmath.html
http://www.wgquirk.com/HMathStd.html
http://www.wgquirk.com/chap3.html
http://www.wgquirk.com/chap4.html
 1)
Please use VERBS that indicate the
kind of mathematical activity that takes
place in a classroom that is described by
NCTM Standards. Shortly describe the
kind of activities that takes place in such
a classroom. (You are NOT asked to list
the Standards but synthesize your
reading of the Standards and visualize a
classroom based on NCTM Standards.)
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2)Please use VERBS that indicate the kind of
mathematical activity that takes place in a
classroom that is described by W. G. Quirk.
Shortly describe the kind of activities that
takes place in such a classroom. (Try to
visualize a classroom based on W. G. Quirk’s
website.)
3) If you were an elementary school student,
from which of the two classes would you have
benefited more? Explain in detail why? (I do
not want TWO WORD answers.)
 4)
a) What are some of the advantages of
being in an NCTM friendly mathematics
classroom?
b) What are some of the advantages in
being in a mathematics classroom
described by W. G. Quirk?
POST-TEST
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1) Solve the problem below:
“A farmer raises only spiders and horses. When
he counts the heads of all his animals he finds
68 heads, when he counts the legs he finds 404.
How many ants and how many horses does he
have?” Hint: Spiders have 8 legs and horses
have 4 legs.
Outline how you would approach the problem.
Carry out the steps outlined above i.e. solve the
problem.
 2)
When you add or subtract
fractions, you must find the
common denominator, but when
you multiply or divide fractions
you need not. Explain why. (You
are not asked to give numerical
examples.)