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This article was downloaded by: [Portland State University]On: 16 October 2014, At: 22:44Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH,UK

PRIMUS: Problems, Resources,and Issues in MathematicsUndergraduate StudiesPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/upri20

Teaching Mathematics forProspective Elementary SchoolTeachers: What TextbooksDon't TellShlomo LibeskindPublished online: 30 Jun 2011.

To cite this article: Shlomo Libeskind (2011) Teaching Mathematics for ProspectiveElementary School Teachers: What Textbooks Don't Tell, PRIMUS: Problems, Resources,and Issues in Mathematics Undergraduate Studies, 21:5, 473-484

To link to this article: http://dx.doi.org/10.1080/10511970903296106

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PRIMUS, 21(5): 473484, 2011Copyright Taylor & Francis Group, LLCISSN: 1051-1970 print / 1935-4053 onlineDOI: 10.1080/10511970903296106

Teaching Mathematics for Prospective ElementarySchool Teachers: What Textbooks Dont Tell

Shlomo Libeskind

Abstract: In this article, the author suggests that to improve the teaching of mathe-matics in elementary schools, we must, when teaching content courses to prospectiveteachers, model the kind of teaching we want them to engage in. The author suggestsways to model this kind of teaching, which include the following:

1. challenge all students;2. encourage different solutions;3. explain in more than one way, including a way appropriate for elementary chil-

dren;4. include in each lecture something that students find exciting, and5. connect mathematics with its history and with mathematicians.

Keywords: Elementary mathematics, teachers, challenge, different, solutions, children,history, mathematicians.

1. INTRODUCTION

Most competent educators eventually experience the dilemma: Do I coverevery topic in the text, or delve more deeply into a few of them? Each yearI teach content courses for prospective elementary school teachers, I am tornbetween covering the material in the textbook, and the desire to focus insteadon in-depth study of important areas. Study of the findings of the Third Trendsin International Mathematics and Science Study (TIMMS) [3], sharpened mystrong belief in the importance of modeling the kind of teaching that I want mystudents to emulate, and the kind of learning that I would like them to promotewhen they become teachers.

According to TIMMS findings, American mathematics instruction isan inch deep and a mile wide [3]. Stigler, one of the principal TIMMS

Address correspondence to Shlomo Libeskind, Mathematics Department, Universityof Oregon, Eugene, OR 97403, USA. E-mail: shlomo@uoregon.edu

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474 Libeskind

researchers, points out that what characterized American lessons was a realemphasis on procedures and skills over understanding [4].

What follows are a few suggestions (not an exhaustive list) for the instruc-tor that address the TIMMS findings and to which my students have respondedenthusiastically. These suggestions are arranged in themes and illustrated withclassroom examples.

2. CHALLENGE ALL STUDENTS, INCLUDING THE MORECAPABLE

The following problem was given as a part of a weekly assignment in a chapteron Rational Numbers:

Example 1

In a condo 23 of the men are married to34 of the women. What is the ratio

of married people to the total adult population of the condo?

Students knew that some problem in every assignment would be challenging.From a class of 35, only 5 students were able to solve the problem. Amongthose who could not find the answer, several approached the problem in a sim-ilar way, and wanted to know if there is a way to complete the solution usingtheir approach. They designated the number of men in the condo by x and thenumber of women in the condo by y and obtained the following:

Number of married men is 23 xNumber of married women is 34 y.

Since the number of married men equals the number of married women:

2

3x = 3

4y. (1)

The ratio of married people to the total adult condo population is:

23x + 34yx + y . (2)

Some students used equation (1) to simplify the ratio in (2) to obtain

23x + 23xx + y or

43x

x + y . (3)

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What Textbooks Dont Tell 475

Here students reported being stuck and wanted to know if there was a wayto solve the problem using one of the ratios in (3). A student who solved theproblem successfully suggested to further simplify (3) by using (1) to find yin terms of x (or vice versa), and then substitute the expression for y in (3).Another student pointed out that in that approach, the ratio would still containthe unknown x. I suggested using the approach anyway and seeing what can bedone after the required ratio is expressed only in terms of x. I asked the classto complete the solution using this approach for extra credit. This proved tobe a good practice with computations involving fractions. (About half the classsubmitted correct solutions.)

Debbie offered a completely different approach. She said:

I look at the fractions 23 and34 and write each with a common numerator

23 = 69 and 34 = 68 . Now that I have 69 and 68 , I add the numerators and thedenominators and get 6+69+8 or

1217 , which is the correct answer.

The class was amazed. We asked Debbie for an explanation. Debbieexplained,

I know that out of every 3 men, 2 are married, and out of every 4 women,3 are married. Since the number of married men equals the number ofmarried women, and the married ones appear in the numerators, I writeeach fraction with a common numerator. Because 23 = 69 and 34 = 68 , Iknow that out of every 9 men, 6 are married and out of every 8 women,6 are married; so out of 9+ 8 men and women 6+ 6 are married. Hencethe required ratio is 6+69+8 or

1217 .

The class liked Debbies approach and appreciated its simplicity. I pointed outthat the approach is original and creative, and called it brilliant.

The following is an example of a challenging geometry problem that onlya few of the best students were able to tackle. Nevertheless, the problem gen-erated a lot of interest, curiosity, and appreciation when one of the studentswho had a correct proof presented it on the board. The problem is given herewithout a solution or further comments.

Example 2

On a geoboard, rubber bands were used to create the angles A and B inFigure 1.

a. Use Geometers Sketchpad (GSP) to find the sum of the measures ofthe two angles.

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476 Libeskind

A B

Figure 1. Angles on a geoboard.

b. Prove what you found in part (a) by creating, on the geoboard or ona grid created by GSP, a triangle with an angle whose measure is thesum of the measures of the angles A and B.

3. ENCOURAGE DIFFERENT SOLUTIONS AND WHENAPPROPRIATE, ADMIRE STUDENTS SOLUTIONS

In a lesson introducing the concept of an angle and its measure, we proved thatvertical angles are congruent, and defined a right angle and a straight angle.I introduced the following problem (this lesson took place in a computer labwhere students had access to GSP). I let students work with GSP to answerpart (a) and demonstrated on the board a proof of part (b).

Example 3

In Figure 2, BAC and DAC form a linear pair (the angles share a ver-tex and a common side and their non-common sides make a straight line).Each angle has been bisected. Measures of pairs of congruent angles arelabeled by x and y as shown in Figure 2.

a. Co

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