Comparing representations and reasoning in children and adolescents with two-year college students

Journal of Mathematical Behavior 23 (2004) 429–442

Comparing representations and reasoning in children andadolescents with two-year college students

Barbara Glass

Rutgers University and Sussex County Community College, 207 Washington Ave, Dover, NJ 07801, USA

Available online 5 October 2004

Abstract

Problem solving and justification of a diversified group of two-year college students was compared with approachesof younger elementary and secondary school students working on the same tasks. The students in this study wereengaged in thoughtful mathematics. Both groups found patterns, justified that their patterns were reasonable and,utilized similar strategies for their solutions and methods of justification. They were also able to make connectionsand build isomorphisms among the various problems.© 2004 Elsevier Inc. All rights reserved.

Keywords:Problem solving; Liberal arts mathematics; Justification; Reasoning

Considerable data have been collected showing elementary and secondary school students’ success insolving open-ended problems, over time, under conditions that encouraged critical thinking (Kiczek &Maher, 1998; Maher, 1998; Maher & Martino, 1996, 1998, 2000; Maher & Speiser, 1997; Muter, 1999;Muter & Maher, 1998). These studies with younger students raise the question if similar results areachievable by liberal-arts college students within a well-implemented curriculum that includes a strandof connected problems to be solved over the course of the semester. Students enrolled in college levelmathematics are expected to have developed effective reasoning skills. Unfortunately this is often notthe case. This may be explained, in part, by a history of mathematics instruction in settings that devaluethinking and focus on rote and procedural learning. From a perspective of conceptualizing reasoning interms of solving open-ended problems, it was of interest to learn whether students in a liberal-arts collegemathematics course could be successful in providing arguments to support their reasoning and in makingconnections in a problem-solving based curriculum.

Specifically, this paper reports on a study of two-year college students enrolled in liberal arts math-ematics. It will describe, in the context of combinatorics: (1) how college students solve non-routinemathematical investigations; (2) how college students’ representations and level of reasoning contrast

0732-3123/$ – see front matter © 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jmathb.2004.09.004

430 B. Glass / Journal of Mathematical Behavior 23 (2004) 429–442

with those of younger students from a longitudinal study engaged in the same investigations; (3) whatconnections, if any, are made to isomorphic problems and previous mathematical knowledge; and (4) towhat extent are justifications made and results generalized.

1. Theoretical framework

The growth of mathematical knowledge is the process whereby a student constructs internal repre-sentations and connects these representations to each other. Understanding is the process of makingconnections between different pieces of knowledge that are already internally represented or betweenexisting internal connections and new knowledge (Hiebert & Carpenter, 1992). A learner confrontedwith a new mathematics situation builds mental representations of input data from the given problemand relevant retrieved knowledge. A mapping between the new data and the existing knowledge is thenmade, checked, and modified until there is a match between new and existing information (Davis, 1984).Understanding occurs “when a new idea can be fitted into a larger framework of previously assembledideas” (Davis, 1992, p. 228). Students build their understanding of concepts by building upon previ-ous experience, not by imitating the actions of a teacher or being told what to do (Maher, Davis, &Alston, 1991). Learners who first learn procedures without attaching meaning to them are less likely todevelop well-connected conceptual knowledge. When students encounter new problems they are morelikely to retrieve prior knowledge that is well connected than to retrieve loosely connected information(Hiebert & Carpenter, 1992). “The most successful students construct and organize new knowledge andrestructure their existing conceptual structures in ways that are qualitatively different from those of theleast successful students” (McGowan, 1998, p. 10). Successful students construct a basic framework towhich they add new knowledge. This knowledge is then integrated to form complex cognitive networks.In contrast, the least successful students do not show the existence of a basic framework into whichthey integrate new concepts but rather replace their cognitive framework with a new one as they learnnew concepts (McGowan, 1998). Even though instruction will have an effect on what a student learnsit will not determine what the student learns. Traditional instruction is geared toward the absorption ofknowledge by students. However, students don’t passively receive knowledge but rather put structure toit and assimilate it in the light of personal mental frameworks that they have constructed for themselves(Romberg & Carpenter, 1986).

Understanding is a personal process that is unique, in ways that are important for understanding theteaching-learning process, for each student. In order to reach conclusions about a student’s level ofunderstanding a teacher must encourage students to justify what they say and do to reveal their thinkingand logic. Just giving answers will not reveal a student’s level of understanding or the way that the studentis thinking about a problem (Pirie & Kieren, 1992). Too often, in traditional mathematics classrooms, theanswer key or the teacher is the source of authority about the correctness of answers, and unfortunately,quick, right answers are often valued more than the thinking that leads to the answer. Requests to explaintheir thinking are often posed to students only when errors have been made.Sanchez and Sacristan(2003)concluded from studying students’ written work that students are not accustomed to expressingmathematical ideas, and offer the explanation that the emphasis in schools is mainly on producing correctsolutions. As a result, students develop the belief that all problems can be solved in a short amount oftime and often stop trying if a problem cannot be solved quickly. In a survey, high school students wereasked to respond to the question “What is a reasonable amount of time to work on a problem before you

B. Glass / Journal of Mathematical Behavior 23 (2004) 429–442 431

know it’s impossible?” The largest response was 20 min and the average was 12 min (Schoenfeld, 1989).Moreover, since students view school mathematics as a process of mastering formal procedures that areremoved from real life so that answers don’t have to make sense, they learn to accept and memorize whattheir teachers tell them without making any attempt to make sense of it (Schoenfeld, 1987).

Since many of the students in this study have been taught mathematics in this fashion, it would notbe entirely surprising if they were unable to apply knowledge that they have been taught in previousmathematics courses to novel situations. Moreover, since a student’s willingness to think about a problemwill be influenced by preconceived notions about what mathematics is and what should be expected ofthem, which they have formed in previous mathematics classes, the possibility exists that they will notdisplay the level of reasoning of which they are capable.

2. Background for elementary and secondary student results

A Rutgers University longitudinal study was initiated in 1989 with a class of 18 first grade children ata public school in a working class community. The children remained together for their first three yearsof elementary school. In grade four new classes were formed. The study continued with a smaller subsetof the original class and several other students who joined. Seven of the children from the original groupwere followed for 14 years and seven others participated for approximately eight years. The children inthe study were invited to think about mathematical situations over long periods of time, present their ideaswith convincing justifications, and consider generalizations and extensions of the problems. The childrenengaged in tasks from several areas of mathematics, including algebra, combinatorics and probabilityand revisited tasks over months and years. During the first eight years the study was classroom basedwith Rutgers researchers conducting sessions four to six times a year. Follow-up individual and smallgroup interviews were also conducted. During high school, students participated in small group afterschool sessions. All sessions were videotaped; groups of researchers observed the children and tooknotes; and the students’ work was collected and analyzed (Maher, 2002). The Rutgers study also includeda cross-sectional classroom based component from two other pubic schools during the years from 1994 to1998. Throughout all phases of the study, the researchers have documented the student’s thinking as theyinvestigate problems to determine how they think about the problems, build mathematical ideas, createmodels, justify their solutions, and generalize their ideas (Kiczek & Maher, 1998; Maher, 1998; Maher &Martino, 1996, 1998, 2000; Maher & Speiser, 1997; Martino, 1992; Muter, 1999; Muter & Maher, 1998).

One of the problems from the area of combinatorics is the Towers Problem, which invites a student todetermine how many different towers of a specified height can be built when selecting from two differentcolor cubes, and to justify that all possibilities have been found. A second isomorphic problem, the PizzaProblem, invites a student to determine how many different pizzas can be created from a given numberof toppings and to justify that all possibilities have been found.

In the Rutgers University longitudinal study students first encountered the Towers Problem in the thirdgrade when they worked on towers that were four tall. They were initially given the Pizza Problem aswell as the five-tall Towers Problem in fourth grade. Researchers found that these third and fourth gradestudents were able to invent strategies that they used to solve the problems.

Most of the third and fourth grade students started the Towers Problem by using guess and checkmethods to create towers, and check for duplicates. They then became more organized and created localorganization strategies This resulted in the use of patterns such as “opposites”, two towers with the colors


reversed, “cousins”, two towers that were vertical inversions, staircase patterns, and elevator patterns.They later developed global organization strategies as they realized that their local organization strategiescreated duplicates and could not be used to account for all possibilities (Maher & Martino, 1999, 2000;Muter, 1999; Martino, 1992).

The initial method of justification of many of the third and fourth grade elementary school studentsthat they had accounted for all possible combinations was to state that they could not find any moretowers (Martino, 1992; Maher & Martino, 1996). For example, fourth grade Milin justified that he andhis partner had found all possibilities because they had gone a long period of time without finding anotherone (Alston & Maher, 1993). Over time these students were able to create convincing arguments thatthey had accounted for all possible combinations (Maher & Martino, 1996, 1998, 1999, 2000; Muter,1999; Muter & Maher, 1998). Some of the children developed a proof by cases and others developed aninductive argument. For example, Milin started to build a proof by cases, but then developed an inductiveargument when he looked at simpler cases of the Towers Problem (one-tall, two-tall and three-tall) andnoticed the doubling pattern. He was able to explain how he could build up the larger towers from thesmaller towers by placing either one or the other of the available colors on the top of the smaller towers(Alston & Maher, 1993). Within this same time period, fourth grade Stephanie was able to present a proofby cases for the towers that were five tall. During a small group assessment, Stephanie presented a proofby cases for towers that were three tall and Milin presented his argument by induction (Alston & Maher,1993; Maher & Martino, 1996). About a year later, in grade five, Stephanie also developed an inductiveargument (Maher & Martino, 1996).

Muter (1999)studied a group of five of the students from the Rutgers longitudinal study who revisitedthe problems that they had done in the third, fourth, and fifth grades during tenth grade, after school,problem-solving sessions. Among the problems that they explored were the Pizza Problem and the TowersProblem. While working on the Pizza Problem, one of the students, Michael, connected the solution ofthe Pizza Problem with the binary system that he had learned about in eighth grade. Michael and Ankur,another student in the group, used the same binary coding system to produce two different justificationsfor the number of towers five-tall with two red cubes and three yellow cubes. Another student, Romina,used a variation of Michael’s binary coding scheme as she solved a problem suggested by Ankur, to findall towers four-tall if three colors were available and at least one of each color must be used.

The five students in the group constructed connections between the Towers Problem and the PizzaProblem by using the binary coding scheme. This was not an isolated instance of students in the Rutgersstudy recognizing the connection between the Towers Problem and the Pizza Problem. There are threeother documented cases nine-year-old Brandon, 12-year-old Lewis, and eight-year-old Meridith, whenchildren demonstrated their understanding of the connection between these two problems. The studentsin Mutter’s study also recognized the connection between the two problems and the numbers of Pascal’striangle and the binomial expansion (Muter, 1999).

3. Data collection and analysis for college students

Nine classes ranging in size from 6 to 25 students were studied from 1998 to 2000. Two groups fromeach class were videotaped as they worked on the Towers Problem. Following the class sessions, eachstudent was required to submit a write-up of the problem. In addition, videotaped, task-based interviewswere conducted.


Ten students, representative of the larger population, were selected for a case study analysis. Thecriteria for selection included student willingness: to be videotaped, fully participate in regular classproblem investigation sessions and participate in follow-up interviews. Videotapes from class sessionsand interviews were transcribed, coded, and analyzed for methods of problem solving and justification.The written work of students was also coded and analyzed (Glass, 2001). The coding scheme for methodsof problem solving and justification were adapted from a coding scheme that had been used with youngerchildren who were working on the same problems (Wier, 1999). Categories needed to be added becausesome of the strategies and method of justification used by the college students had not been used by theyounger students and therefore were not in the coding scheme. The coding scheme for connections wasdeveloped to highlight the level of understanding displayed by the students when making connections aswell as the difference between connections made with previous mathematics and those made with earlierproblem-solving activities.

The study of college students was conducted at a small community college in a mathematics course forliberal arts majors. Sections of the course met for two 75-minute classes each week or three 50-minuteclasses each week for 15 weeks. The students spent approximately half of all class time working onvarious non-routine problems in a small group setting. The Towers Problem was given during the eighthor ninth week of the semester. By this point class enrollment had stabilized and the class had becomeaccustomed to working on problems and justifying their solutions. Two groups from each class werevideotaped as they worked on the problem. The students began by working on towers that were fourcubes tall. They then were asked to consider towers that were five cubes tall. Some groups also workedwith towers where three different color cubes were available.

During the 13th or 14th week of each semester the Pizza Problem was given to the classes. The studentsfirst worked on finding the number of pizzas that could be made if four toppings were available followedby the problem with five toppings. After solving the basic problem some groups were asked to considerthe Pizza with Halves Problem in which a topping could be placed on either a whole pizza or a half pizza.Videotapes were again made of two groups from each class as they worked on the problems. The groupsin the second videotape session were as close in composition to the groups in the first videotape sessionas possible.

4. Results for college students

4.1. Towers Problem

Most of the college students used patterns or some other form of local organization immediately andsome immediately imposed a global organization scheme. One group started the problem by randomlygenerating towers using a build and check method. A student in the group, Jeff, soon suggested, however,that they should organize the towers that they had built by cases so that they could check more easily formissing combinations.

Jeff: Here, put the ones that have three yellows and a red all together. (Danielle rearranges the towers.)OK. So now we do three yellows and a red at the bottom, cause you don’t have that. (Jeff buildsYYYR and hands it to Danielle.) And the ones that have two and two, put those together. (Daniellerearranges the towers.) Now the ones that have three reds and the other.


After another pair of students, Stephanie and Tracy, built all 16 towers using opposite pairs, they stated thatthey had to impose some type of organization to convince themselves that they had found all combinations.They first used staircase patterns, but rejected this because it did not account for all towers. They thenarranged the towers by cases.

Six of the profiled college students justified that they had found all possible towers four tall by usinga cases approach. Each of these students used an elevator pattern to justify the towers with three cubesof one color and one cube of the other color. The students used a variety of methods to justify that theyhad found all towers with two cubes of each color. Two groups, Melinda’s group and Donna’s group,justified that they had found all towers because they couldn’t find anymore. As these groups spoke tothe instructor, they began to organize their towers and moved toward a proof by cases. Both groups stilljustified that they had all towers with two of each color because they could not find anymore.

Three of the profiled students, Lisa, Errol, and Wesley, tried to use the argument that the number oftowers is 16 because 4 times 4 is 16. When the students used this argument, the instructor told themthat they needed a reason why the answer should be four times four. Lisa then did a proof by cases,although she had difficulty justifying the case of two cubes of each color. During her interview sevenweeks later, Lisa did account for all of the towers with two cubes of each color. Wesley rearranged histowers, but offered no explanation for why his arrangement produced all possible towers. About sevenweeks later, during an interview, Wesley produced a similar arrangement and used it to account for allpossible combinations with a proof by cases where his cases were towers with four cubes of the samecolor together, three cubes of the same color together, two cubes of the same color together, and no cubesof the same color together. Errol’s partner, Mary, offered a proof by cases. However, Errol wanted toproduce a proof that his numerical argument worked. As he continued to think about the problem, herearranged the towers in a way that he thought showed that four times four was correct. This arrangementgrouped towers with a red on top together and towers with a yellow on top together. When the instructorcontinued to question him why this showed that the answer should be four times four, Errol workedon simpler cases in an attempt to verify his numerical argument. He noticed the doubling pattern anddeveloped an argument by induction.

Five of the profiled college students did a proof by cases for the five-tall towers. Each of these proofsby cases referred to opposites. Also they all used an elevator pattern to account for the towers with onecube of one color and four cubes of the other color. They used a variety of methods to justify that theyhad all towers with two cubes of one color and three cubes of the other color. After one student, Rob,and his group had built their towers by organizing them by cases they noticed the doubling pattern anddeveloped a proof by induction, which Rob used in his written assignment.

Wesley built his towers five tall by adding a red cube to the top of each of his towers of four. Hethen built the opposites of these towers to find all towers with a yellow cube on top. He justified thathe had found all towers five tall with an inductive argument. He was unable at this time to extend thisreasoning to predict how many towers that he would get six tall. However, during an interview sevenweeks later, Wesley correctly extended the doubling pattern beyond the case that went from four tall tofive tall. Another student, Jeff applied the fundamental counting principle to predict that there wouldbe 32 towers that were five tall. After Jeff’s group had produced the 32 towers he used an inductiveargument to show that they had found all possible towers by pairing each of the five-tall towers withthe corresponding towers that were four tall. Errol used the inductive argument that he had developedwhile working with towers four tall to predict that there would be 32 towers of five. Even though he didnot build the five-tall towers in class, he produced a list of all towers five tall on his written assignment


Fig. 1. Errol’s written justification of towers four-tall.

using an inductive method.Fig. 1 illustrates Errol’s method with his written work for towers that arefour-tall.

Penny, a student in Errol’s group who was absent the day that the class worked on the Towers Problem,completed the problem at home. She invented a tree diagram strategy to produce an inductive argumentfor the towers four and five tall. Penny wrote, “The answer is 24 = 16 because my first cube will be eitherblue or brown (2 choices). My second cube will be either blue or brown matched to a blue or brownfirst cube (4 choices). For each of those combinations I can use either a blue or brown cube doublingmy possibilities to 8, and for each of those eight combinations I can add a blue or a brown cube whichfinally doubles my answer giving me 16 possibilities.” Her written work for the four tall towers is shownin Fig. 2.

Another student, Tim, used a binary coding system to justify that he had found all possible combinations.This is the same method that tenth grade Michael from the Rutgers longitudinal study used to justify thatthere were 32 different pizzas with five available toppings (Muter, 1999).

Several groups also had time to work on the Towers Problem with three available colors. Mike’s groupand Rob’s group worked on four-tall towers, while Jeff’s group worked on three-tall towers. Rob and hispartners applied the inductive method that they had developed for towers with two colors to quickly solvethe problem. Jeff used the fundamental counting principle to calculate the number of towers, but did notuse an inductive method to build the towers. Mike’s group divided the problem into the three problems offinding towers with two available colors and the problem of finding towers that contained all three colors.They then built all the towers in a systematic fashion.


Fig. 2. Penny’s written justification of towers four-tall.

4.2. Pizza Problem

For the Pizza Problem, all of the college students used a cases approach. The students were systematicas they created their two-topping lists. They held one topping fixed and paired it with each of the othertoppings. They then moved to the next topping on the list. Some students only paired each additionaltopping with toppings that were below it on the list, reasoning that they had already accounted for theother combinations. Other students considered all pairs and eliminated the ones that they already had.For the three-topping pizzas, some of the students were again very systematic as they went down the list,while others failed to exhaust both major and minor items. For example, after they would pair pepperoniand green peppers with all possibilities, they would move to green pepper and mushrooms instead ofexhausting all possibilities that contained pepperoni.

Two students used a chart that was similar to the chart that fourth grade Brandon had created when hedid the Pizza Problem (Maher & Martino, 1998). Instead of ones and zeros they made a check when atopping was on a pizza and left a blank space for toppings that were not on the pizza. Interestingly, thisis the method that fourth grade Brandon’s partner, Colin, used to record his pizzas.

Stephanie, who was enrolled in a statistics class during the same semester that she was enrolled in theliberal arts mathematics course, hypothesized that she could calculate the number of pizzas using com-binations. She and her partner, Tracy, systematically created a list of pizzas and compared the numbersto those that Stephanie had conjectured. Several other students who had previously learned about com-binations tried to calculate the number of pizzas using combinations. However, they did not understandcombinations well enough to apply them to the problem correctly. These students attempted to do a singlecalculation to find the answer instead of doing separate calculations for each number of toppings that theycould have on the pizza. One student, Melinda, stated that the problem could be done by either combina-tions or permutations, but she could not remember how to apply these concepts. Besides Stephanie, onlyone other student, Carl, who was not profiled, was able to successfully use combinations to calculate thenumber of pizzas.


4.3. Connections between problems

Several of the students noticed a relationship between the Towers Problem and the Pizza Problemduring the class session of the Pizza Problem. Three of these students, Rob, Mike and Errol, were able toexplain the isomorphism during the class session. For example, Rob explained:

Rob: So we decided the toppings are the block positions.Instructor: OK. So you have pepperoni at the top.

Rob: Right. Because it was convenient. So onion would be the second block, sausage would bethe third block, and mushroom would be the bottom block.

Instructor: OK. What would your colors be?Samantha: Orange and yellow.

Rob: Yeah.Instructor: What would orange be?

Rob: Orange means it’s on the pizza.Instructor: And yellow?

Rob: It’s off.Samantha: Off the pizza.

Another student, Jeff could not explain the isomorphism during class, but did explain it a week laterduring an interview.

Jeff: What you could do to relate that to the topping problem. Is that, you could say, you coulddesignate each spot for a topping. This is the pepperoni. This is the green pepper spot, andthis is the sausage spot. (Jeff writes toppings by drawing of 1st tower.)

Instructor: OK.Jeff: Or onion, or whatever you want to have it.

Instructor: Whatever.Jeff: So with nothing on it, with no toppings, there’s only one possibility. (Jeff points to 1st

tower he drew.) Now for one topping there’s three possibilities. for a one topping pizza.There’s pepperoni, (Jeff marks 1st block of 2nd tower he drew.) there’s green pepper, (Jeffmarks 2nd block of 3rd tower.) and there’s sausage. (Jeff marks bottom block of 4th tower.)

Instructor: OK.Jeff: Then for a two topping pizza, (Jeff draws 3 more towers.) there’re three possibilities. (Jeff

moves the actual towers with 1 white cube.) Pepperoni and green pepper, (Jeff marks 1stand 2nd block on 1st tower he just drew.) pepperoni and sausage, (Jeff marks 1st and 3rdblock.) and green pepper and sausage. (Jeff marks 2nd and 3rd block.) OK. All right.Which is where the blue, the blue blocks are. (Jeff points to actual towers with 1 whitecube.) So in other words, if you took, you couldn’t flip that one around, (Jeff takes WBBand turns it upside down and puts it next to BBW.) because then you’d have two of thesame combination. You’d have two pizzas with pepperoni and green pepper.

Instructor: OK.Jeff: So those are the three possibilities. The ones with 2 toppings. And then for 3 toppings,

there’s only 1 possibility, pepperoni, green pepper, and sausage, (Jeff draws anothertower on paper and marks all the blocks.) or blue, blue, and blue. (Jeff moves the solidblue tower slightly.)


Fig. 3. Lisa’s chart for the Pizza Problem.

Robert’s group tried to relate the pizza toppings to colors in the Towers Problem. This is thesame explanation that third grade Meredith had originally given for the relationship between thetwo problems. Meredith later revised her explanation to make the blocks in the towers correspondto the toppings and the colors in the towers correspond to the presence or absence of each top-ping (Maher & Martino, 1999). Unlike Meredith, however, the students in Robert’s group did notrevise their explanation because they did not explore any further about how the problems wererelated.

Melinda, Stephanie, Wesley, and Lisa mentioned the relationship between the two problems during theirinterviews, approximately one week after the class session on the Pizza Problem. Each student displayedan understanding of the isomorphism between the two problems at this time. During her interview, Lisafirst built the towers four-tall and then produced a chart for the Pizza Problem that was similar to the onethat she had made in class. As she completed the chart she discovered that the rows of the chart lookedlike the towers. As a consequence, she was able to match the towers with the rows in her chart. A portionof Lisa’s chart is shown inFig. 3.

This was the same connection that nine-year old Brandon had made. While working on the PizzaProblem, Brandon used a chart, similar to Lisa’s chart, to organize his combinations. He recorded a one toindicate that a topping was present and a zero to indicate that it was not. After explaining his organization,Brandon was able to map each pizza onto a tower by making one color correspond to zero and the othercolor correspond to one (Maher & Martino, 1998). Lisa also remarked that the entries in her chart werelike the binary system that her class had studied earlier in the semester.

Several of the college students also mentioned the Sandwich Problem as they started to work on thePizza Problem. They said that it was the same type of problem as the Pizza Problem. The SandwichProblem was a three-dimensional counting problem that the students had done on an assignment earlierin the semester. Some of the children in the longitudinal study had noticed the similarity between the Shirts


and Pants Problem, a two-dimensional counting problem, and the Towers Problem, which is isomorphicto the Pizza Problem (Martino, 1992).

4.4. Connections with Pascal’s triangle

While working on the Pizza Problem Mike’s group noticed the doubling pattern and verified that itcontinued to hold for smaller numbers of available toppings. As Mike thought about why the numberof pizzas should double, he discovered that the numbers from the Pizza Problem matched the rows ofPascal’s triangle. He was unable to explain, however, why the addition rule of Pascal’s triangle appliedto pizzas.

After completing the four topping Pizza Problem, Rob noticed that the pizzas from the four-toppingproblem formed a row of Pascal’s triangle when Steve, a member of his group remarked that they shouldlook for a pattern before working on the five-topping problem. After figuring out how the addition ruleof Pascal’s triangle worked with pizzas, Rob used an inductive method and Pascal’s triangle to create alist of pizzas for the five-topping problem. Rob’s explanation of the addition rule for the Pizza Problemwas similar to that used by Michael from Kenilworth in eleventh grade (Kiczek, 2000).

Rob: Two toppings. We take this list of two toppers, which is this one, this one, and this one. And thenwe take this list (of one toppers) and add our sausage to the end of that. So put sausages there andadd it to those. And that’s this.

Rob wrote in his assignment:

The reason this (Pascal’s triangle) works is because every time we add another topping we are increas-ing the possibility of choice, without losing the old ones. In other words, all the two topping pies in the

Fig. 4. Rob’s drawing of Pascal’s triangle.


two topping total still apply when there are three total toppings. Also all those that had two toppings, byadding a third topping, are now three topping pizzas. In this way we can see absolutely, positively, withouta doubt, that we have all possible combinations—each new row is built by adding the old columns withthe new topping. This once again is the Pascal’s triangle principle of adding old combinations with newpossibilities to find new combinations.Rob’s drawing of Pascal’s triangle is shown inFig. 4.

5. Conclusions, implications and limitations

The college students used many of the same strategies for solution and methods of justification that theelementary school students from the longitudinal study had used when they solved the same problems.However, unlike the third and fourth grade children they did not generally rely on random checking asa primary strategy when they began to solve the problems. In general, the college students were able tosolve the problem more quickly than the third and fourth grade children working on the same tasks. Allwere able to solve the four-tall Towers Problem and start the five-tall Towers Problem within a 50-minuteor a 75-minute period. Most also finished the five-tall problem within the same period. Several studentsalso had time to work on extensions of the problem within the class period. The college students alsoshowed less inclination to think about problems for extended periods of time. Many stopped thinkingabout the problem after they had arrived at an answer, even when the instructor asked them to think moreabout the problem in order to reveal their thinking. This is not at all surprising because most of thesestudents had previously been taught mathematics in an atmosphere in which quick right answers werevalued more than explanation, and justification of results.

After college-aged Rob and his group had built their towers by organizing them by cases they noticedthe doubling pattern and developed a proof by induction. This is similar to what fourth grade Milinhad done. College-aged Mike also noticed that the number of towers and pizzas was doubling, but wasunable to think of a reason for the doubling pattern. It is interesting to note that both Stephanie and Milinfrom the Rutgers longitudinal study had noticed the doubling pattern in the Towers Problem as fourthgraders. Stephanie’s progression from pattern recognition to development of an inductive argument tookabout eight months while Milin’s understanding developed more quickly (Alston & Maher, 1993; Maher& Martino, 2000). Perhaps college-aged Mike would also have recognized the reason for the doublingpattern if, like fourth grade Stephanie, he had been given an extended time frame in which to develop hisideas. Mike, however, was limited to about eight weeks to develop his ideas about the problem.

The college students made very few connections with previous mathematical knowledge that hadbeen acquired in a traditional lecture class, and most of these connections were trivial connections.Several students recognized that the problems could be solved by permutations or combinations, but mostdemonstrated that they did not understand these concepts well enough to correctly apply them to thesolution of novel problems. This would indicate that learning about mathematics in an atmosphere inwhich students are told what to do does not enable these students to develop genuine understanding. Incontrast, the students in this study did demonstrate a high level of reasoning as they thought about theproblems, justified their solutions, and made connections between problems and the mathematics theylearned within the course.

Although some of the conditions of the Rutgers University longitudinal study such as extended class-room sessions and revisiting the same problem several times within an extended time frame, could not bereplicated because of time constraints within a college classroom, many of the conditions that enabled the


elementary and secondary students to become thoughtful problem solvers were duplicated. Both groupswere given rich mathematical tasks and were encouraged to explain their reasoning and methods of so-lution and justify their solutions to the problems. Both groups of students were engaged in thoughtfulmathematics. They found patterns, justified that their patterns were reasonable, and developed methodsof justification. While it cannot be disputed that the students in the Rutgers longitudinal study benefitedfrom exposure to rich mathematical experiences over an extended period of time, the students in thisstudy, who had previously experienced a variety of traditional mathematics instruction, demonstratedthat it is not too late to introduce rich mathematical experiences in a collegiate level mathematics class.The level of reasoning that these students demonstrated provides evidence that it is possible to experiencethoughtful mathematics within a traditional 15-week semester.

References

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Comparing representations and reasoning in children and adolescents with two-year college students