Emphasizing Critical Thinking When Teaching Mathematics to

Future Elementary School Teachers

By Dr. Renan Sezer

LaGuardia Community College

Traits I wanted to emphasize 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 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

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:

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?

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.

THINKING OUTSIDE OF THE BOX

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?

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?

3) What problem solving methods did you utilize in this problem?

CREATIVE SOLUTIONS 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

In the number 325,647 written in base 10, what does 7, 4, 6, 5, 2, and 3 represent?

How many single digit numbers are there in base 10? What is the largest single digit number you can write in base 10?

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

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 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? 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

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.)

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 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.)