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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,UKPRIMUS: Problems, Resources,and Issues in MathematicsUndergraduate StudiesPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/upri20Teaching 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, 473-484To link to this article: http://dx.doi.org/10.1080/10511970903296106PLEASE SCROLL DOWN FOR ARTICLETaylor & Francis makes every effort to ensure the accuracy of all theinformation (the Content) contained in the publications on our platform.However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness,or suitability for any purpose of the Content. 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Terms & Conditions of access and use can be found athttp://www.tandfonline.com/page/terms-and-conditionsDownloaded by [Portland State University] at 22:44 16 October 2014 http://www.tandfonline.com/page/terms-and-conditionsPRIMUS, 21(5): 473484, 2011Copyright Taylor & Francis Group, LLCISSN: 1051-1970 print / 1935-4053 onlineDOI: 10.1080/10511970903296106Teaching Mathematics for Prospective ElementarySchool Teachers: What Textbooks Dont TellShlomo LibeskindAbstract: 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. INTRODUCTIONMost 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 TIMMSAddress correspondence to Shlomo Libeskind, Mathematics Department, Universityof Oregon, Eugene, OR 97403, USA. E-mail: shlomo@uoregon.eduDownloaded by [Portland State University] at 22:44 16 October 2014 474 Libeskindresearchers, 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 MORECAPABLEThe following problem was given as a part of a weekly assignment in a chapteron Rational Numbers:Example 1In a condo 23 of the men are married to34 of the women. What is the ratioof 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:23x = 34y. (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 obtain23x + 23xx + y or43xx + y . (3)Downloaded by [Portland State University] at 22:44 16 October 2014 What Textbooks Dont Tell 475Here 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 numerator23 = 69 and 34 = 68 . Now that I have 69 and 68 , I add the numerators and thedenominators and get 6+69+8 or1217 , 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 or1217 .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 2On 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.Downloaded by [Portland State University] at 22:44 16 October 2014 476 LibeskindA BFigure 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 SOLUTIONSIn 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 3In 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. Consider FAE formed by the angle bisectors AF and AE. Measurethis angle for different positions of point C in the half pane above lineDB and make a conjecture about the measure of FAE.b. Prove your conjecture in part (a).For part (a) I instructed,In GSP select points E, A, and F and then select angle from theMeasure menu. GSP will display the measure of EAF. When youDownloaded by [Portland State University] at 22:44 16 October 2014 What Textbooks Dont Tell 477xyxyCBD AEFFigure 2. Angle Bisectors (color figure available online).drag C to different positions, the measure of the new EAF will bedisplayed.Students observed that for different positions of C the value m (EAF) wasalways 90, and hence conjectured that EAF is a right angle, no matter wherepoint C is located. For part (b) I gave the following proof:Because we labeled the measures of the congruent angles by x and y asshown in Figure 2 we get2x + 2y = 1802 (x + y) = 180x + y = 90.Since m (EAF) = x + y, it follows that m (EAF) = 90.I pointed out that the proof is short and the algebra very simple.At the beginning of the next class meeting, Nicole (a good student, but notoutstanding) said that she had a different proof of the conjecture in Example 3,a proof that is much more meaningful to her than my proof. Nicole explained:I extend ray AE to form a line (Figure 3). I mark the measure of DAGby x, since DAG and BAE are vertical angles and hence congruent,and the measure of DAG was labeled x in the proof that you showed us.Now I see that because GE is a line, GAF and FAE form a linear pairand are congruent, since the measure of each is x + y. Angles that forma linear pair and are congruent were defined as right angles. Thus EAFis a right angle.Downloaded by [Portland State University] at 22:44 16 October 2014 478 LibeskindGxyy xxD BCAEFFigure 3. Two congruent angles forming a linear pair (color figure available online).Nicole pointed out that she liked her approach better because she could actuallysee that EAF is a 90 angle. The class liked Nicoles approach as well, andI pointed out that a proof or a solution that one finds on her own is the best.Moreover, I labeled Nicoles proof beautiful.4. AS OFTEN AS POSSIBLE, EXPLAIN (SOLVE OR PROVE)IN MORE THAN ONE WAY, INCLUDING A WAY APPROPRIATEFOR CHILDRENThe following is an example of a type of problem solved in the text, usingalgebra. I gave a second approach without using algebra.Example 4Jillian received a 10% raise last year. If her present salary is $66,000what was her salary last year?In the non-algebraic approach we decided to set up Table 1 in which we foundhow much $1 of Jillians salary is increased after a year.At this point, several students suggested to replace the ? by x and set upa proportion: x1 = 660001.1 and hence x = 60, 000. Others wanted to know how toproceed without setting a proportion. I said that if we knew the amount in theleft column, the problem would almost be solved. Then we would only needto multiply what is in the left column by 66,000 (this was discussed in greaterdetail than presented here). Students did not know how to proceed. I told themthat a useful strategy is to imagine nicer numbers: suppose instead of 1.1, wehad 2. Then it would be clear that the corresponding amount in the left columnDownloaded by [Portland State University] at 22:44 16 October 2014 What Textbooks Dont Tell 479Table 1. Changes in Jillians EarningsLast years salaryin dollarsPresent salary (after 10%raise) in dollars? 66,0001 (1+ 10% of 1)= 1.1? 111.1 111.1 66, 000 66000is 1 2 or 0.5. If instead of 2 we have 1.1, we perform the same operationof dividing, and get 11.1 in the left column. We wrote this in the table and got11.1 66, 000 for Jillians original salary. The class liked the fact that the solutioncan be explained to children who have not been exposed to algebra.I assigned other problems to solve using two or more approaches; one withalgebra and one without. A very good such problem is the following [1]:Example 5A recipe calls for 1 tsp. of mustard seeds, 3 cups of tomato sauce, 11/2 c.of chopped scallions, and 31/4 c. of beans. If one ingredient is altered, howmust the other ingredients be changed to keep the proportions the same?Explain your reasoning for each of the following changes.a. 2 c. of tomato sauceb. 1 c. of chopped scallionsc. 1 3/4 c. of beansThe common way to approach the problem is to use ratio and proportion.Students feel comfortable with the ratio and proportion approach because itis somewhat algorithmic. In Table 2 we illustrate an alternative approach forpart (a), similar to the approach used in Example 4 (an analogous approach canbe used for parts (b) and (c).)Table 2. Recipe ProportionsMustard seedsin teaspoons Tomatoes in cups Scallions in cups Beans in cups1 3 1 12 31413 1 32 3 = 12 3 14 3 = 1 1122 13 = 23 2 2 12 = 1 2 1 112 = 2 16Downloaded by [Portland State University] at 22:44 16 October 2014 480 LibeskindStudents found the second approach to be more difficult, but agreed that itis important for them to understand it and be able to use it.5. IN EACH LECTURE, OR AS OFTEN AS POSSIBLE, DOSOMETHING THAT YOU AND THE STUDENTS FINDESPECIALLY INTERESTING OR EVEN EXCITINGThere are, of course, plenty of examples, and no doubt each instructor will havedifferent ones. Here are two of my favorites that students have liked a lot. In thefirst (Example 6), the conjecture is surprising, and its justification is especiallyinteresting. I introduced the next example in class as follows:Example 6 Pascals Triangle and one of its many propertiesIn 1653, Blaise Pascal constructed his arithmetic triangle as shown inFigure 4. The terms in the first row are 1 followed by 1. Each row startswith 1 and ends with 1, and any other element in the second or followingrow is obtained as the sum of the two elements in the preceding row lyingjust above, one to the left and the other to the right. Thus, the first 15 inthe sixth row in Figure 4 is obtained as 5+ 10, the number 20 as 10+ 10,and the second as 15 as 10+ 5.The Pascal Triangle has many interesting properties, one of which we willnow explore.If we add the numbers in each row, we find that the sums exhibit a patternas shown in Table 3.We conjecture that in row n, the sum of the numbers is 2n.We can now give a plausible argument showing that the above conjectureis true. In Table 3, we checked that the conjecture is true for n= 1, 2, 3, and4. To show that it is true for n= 5, we could simply add all the numbers in the1 11 2 11 3 3 11 4 6 4 11 5 10 10 5 11 6 15 20 15 6 11 7 21 35 35 21 7 1Figure 4. Pascal Triangle.Downloaded by [Portland State University] at 22:44 16 October 2014 What Textbooks Dont Tell 481Table 3. Sum of Entries in a Row of the Pascal TriangleRow number Sum of the numbers in the row1 1+ 1 = 22 1+ 2+ 1 = 4 = 223 1+ 3+ 3+ 1 = 8 = 234 1+ 4+ 6+ 4+ 1 = 16 = 24fifth row. However, this simplistic approach would not allow us to show that theconjecture is true for all n, since the number of rows is infinite. We thereforeuse a different approachan approach to show that whenever the conjecture istrue for some n, it is also true for the next n. For example, we know that thesum of the terms in the fourth row is the following:1+ 4+ 6+ 4+ 1 = 16.By definition of the Pascal Triangle, the numbers in the fifth row, other than thefirst and the last, are obtained using the terms from the fourth row. Hence, thesum of the terms in the fifth row is the following:1+ (1+ 4)+ (4+ 6)+ (6+ 4)+ (4+ 1)+ 1 = 32.We notice that in this sum, every number from the fourth row appearstwice, and hence the sum of the numbers in the fifth row is twice thesum in the fourth row. To be sure that the sum of the numbers in any rowis twice the sum in the previous row, we search for that phenomenon.We notice that, in the fifth row, every number other than the first andthe last (which are ones) is the sum of the two neighboring numbersjust above (see Figure 4). For example, 5= 1+ 4, 10= 4+ 6, 10= 6+ 4,and 5= 4+ 1. Thus, each number in a row appears in two sums andcontributes twice to the sum of the numbers in the following row. Eveneach of the 1s from the endpoints of the 4th row appears twice in thefifth rowonce as an endpoint and once in a sum. The same argumentworks from any row to the next. Thus, sum of the numbers in the fifthrow is 2 24 or 25. The sum in the sixth row is 2 25 or 26 and so on.Consequently, it is plausible that our conjecture is true for all n.I ask students to notice that we have actually justified the conjecture thatthe sum of the numbers in row n in the Pascal Triangle is 2n for all n= 1, 2, 3,. . . . I also point out that the above approach, when done somewhat more for-mally, is called Mathematical Induction. I tell students that the numbers in thePascal Triangle also represent combinations that will be studied in a later chap-ter. (Of course, Pascals Triangle can be first introduced when combinationsDownloaded by [Portland State University] at 22:44 16 October 2014 482 Libeskindare studied.) I also show them how an expansion of (a + b)n can be quicklyperformed with the use of the triangle.The following problem ([2], p. 2) I introduce to engage students in anactivity which yields a surprising solution.Example 7 The Treasure Island ProblemAmong his great-grandfathers papers, Marco found a parchmentdescribing the location of a hidden pirate treasure buried on a desertedisland. The island contained a coconut tree, a banana tree, and a gallows() where traitors were hanged. A reproduction of the map appears inFigure 5 and was accompanied by the following directions:Walk from the gallows () to the coconut tree, counting the number ofsteps. At the coconut tree, turn 90 to the left, and walk the same numberof steps and put a spike in the ground. Return to the gallows, and walk tothe banana tree counting your steps. At the banana tree, turn 90 to theleft, walk the same number of steps, and put another spike in the ground.The treasure is halfway between the spikes.Marco found the island and the two trees, but no trace of the gallows orthe spikes, which had probably rotted. In desperation, Marco began todig at random but soon gave up because the island was too large. Howcould Marco have found the exact location of the treasure?I introduce this problem in our computer lab and suggest that the studentschoose two fixed locations for the trees and some location for the gallows (),and use GSP (and rotations from the Transform menu) to locate the treasure.I then ask students to change the position of by dragging the point. Theynotice that for many of positions of , the corresponding treasure is always atthe same spot. To dramatize that the location of the treasure is fixed, I ask stu-dents to change one of the rotations, or both, to angles that are not 90. If eachrotation is, for example, 80, students notice that dragging changes theCoconut treeBanana treeSpikeSpikeTreasureFigure 5. Treasure map (color figure available online).Downloaded by [Portland State University] at 22:44 16 October 2014 What Textbooks Dont Tell 483position of the corresponding treasure. (Several proofs that show the treasureis independent of the gallows can be found in [2].)6. CONNECT MATHEMATICS WITH ITS HISTORYAND WITH THE LIVES OF MATHEMATICIANSElementary school teacher are generalists by profession, teaching all subjects.Thus, connecting mathematics with history can be especially important andmeaningful for prospective elementary school teachers. Almost all textbookshave historical notes, where some event or person from history is refer-enced as a way to make further connections to what is being learned. But, theinstructor can also bring additional historical information to enhance studentslearning by providing a more meaningful connection to the material. My stu-dents found the following quote, from the Autobiography of Bertrand Russell[4], and the historical note ([2], p. 9), especially interesting.At the age of eleven, I began Euclid, with my brother as my tutor. Thiswas one of the great events of my life, as dazzling as first love. I hadnot imagined there was anything so delicious in the world . . . From thatmoment until . . . I was thirty-eight, mathematics was my chief interestand my chief source of happiness. (Russell, 1968)HISTORICAL NOTELord Bertrand Arthur William Russell (18721970) was a British philoso-pher, logician, and mathematician who made important contributions tofoundations of mathematics. In 1910 he became a lecturer at CambridgeUniversity but was dismissed and later jailed for making pacifist speechesduring World War I. He abandoned pacifism during World War II in theface of the Nazi threat to Great Britain and Nazi atrocities in Europe.After the war he reverted back to pacifism, and became a leader in theanti-nuclear and anti-Vietnam War movements. Later Russell taught atseveral United States universities including Harvard and the Universityof Chicago. He won the Noble Prize for Literature in 1950. Russell diedat the age of 98.7. CONCLUSIONWhile teaching content courses in mathematics for prospective elementaryschool teachers, we have the opportunity to significantly impact the futureDownloaded by [Portland State University] at 22:44 16 October 2014 484 Libeskindteaching and learning of mathematics in elementary schools. If we model thekind of teaching we would like prospective teachers to engage in, we captureand spark their interest in learning mathematics and in teaching mathematicsin a way that does the same for their students.REFERENCES1. Billstein, R., S. Libeskind, and J. Lott. 2009. A Problem Solving Approachto Mathematics for Elementary School Teachers (10th edition). New York,NY: Addison-Wesley.2. Libeskind, S. 2008. Euclidean and Transformational Geometry. Sudbury,MA: Jones and Bartlett Publishers.3. Russell, B. 2000. The Autobiography. London, UK: Taylor and Francis.4. Stigler, J.W., and Hiebert, J. 1999. The Teaching Gap: Best Ideas from theWorlds Teachers for Improving Education in the Classroom. New York,NY: The Free Press.5. TIMMS 1999 Technical Report. 1999. Michael D. Martin et al., eds. Boston,MA: International Study Center, Boston College.BIOGRAPHICAL SKETCHShlomo Libeskind received his B.Sc. and M.Sc. in mathematics from theTechnionIsrael Institute of Technologyand his Ph.D. from University ofWisconsin, Madison, in 1971. He is a professor of mathematics, specializingin mathematics education, at the University of Oregon. His primary researchinterests include elementary and secondary teacher education, and issues inhigh school mathematics curricula.Downloaded by [Portland State University] at 22:44 16 October 2014