Students connecting mathematical ideas: possibilities in a liberal arts mathematics class

Journal of Mathematical Behavior21 (2002) 75–85

Students connecting mathematical ideas: possibilitiesin a liberal arts mathematics class

Barbara Glass

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

Abstract

Problem solving and justification for a diversified group of 2-year college students enrolled in a mathematicscourse for liberal arts majors were studied as the students met in small groups to work on various non-routineproblems. Several problems from the area of combinatorics were given during these problem-solving sessions.This paper focuses on one small group of students from this study. These students were engaged in thoughtfulmathematics. They identified patterns, justified that their patterns were reasonable and recognized isomorphismsbetween the different problems. The findings support the importance of introducing rich problems to college studentsand encouraging them to explore solutions, explain their reasoning and justify their solutions.© 2002 Elsevier Science Inc. All rights reserved.

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

By the time students study mathematics at the college level, it is hoped that they would have developedan ability to reason effectively. At the same time, if previous mathematical learning were durable thecollege students would be able to apply the specific mathematical content that they had learned up to thispoint to new situations that they encounter. Unfortunately, many of the students have passed through their10 years of mathematics courses primarily in settings that emphasize rote and procedural learning. Froma perspective of conceptualizing reasoning in terms of solving open-ended problems, it was of interestto learn whether students in a liberal-arts college mathematics course could be successful in providingarguments to support their reasoning and making connections in a problem-solving-based curriculum.

Considerable data have been collected showing pre-college students’ success in solving open endedproblems, over time, under conditions that encouraged critical thinking (Kiczek & Maher, 1998; Maher &Martino, 1996, 1998, 2000; Maher, 1998; Maher & Speiser, 1997; Muter, 1999; Muter & Maher, 1998).These studies with younger students raise the question if similar results are achievable by liberal-artscollege students within a well-implemented curriculum that includes a strand of connected problems tobe solved over the course of the semester. Specifically, this paper reports on one aspect of a larger studyof 2-year college students enrolled in liberal arts mathematics.

E-mail address:[email protected] (B. Glass).

0732-3123/02/$ – see front matter © 2002 Elsevier Science Inc. All rights reserved.PII: S0732-3123(02)00104-9

76 B. Glass / Journal of Mathematical Behavior 21 (2002) 75–85

1. Background

Researchers have documented children’s thinking as they investigate problems in the area of com-binatorics to determine how they think about the problems and justify their solutions (Adleman, 1999;English, 1988, 1993, 1996; Kiczek & Maher, 1998; Maher & Martino, 1996, 1997, 1998; Maher, 1998;Maher & Speiser, 1997; Muter, 1999; Muter & Maher, 1998; Wier, 1999). One of these problems is theTowers Problem, which invites a student to determine how many different towers of a specified heightcan be built when selecting from cubes of two different colors, and to justify that all possibilities havebeen found. A second isomorphic problem, the Pizza Problem, invites a student to determine how manydifferent pizzas can be created from a given number of toppings and to justify that all possibilities havebeen found. These tasks were posed to the college students in this study during the problem-solvingcomponent of the course.

2. Objectives

The purpose of this study is to examine how a small group of community college students, enrolled in aliberal arts mathematics class, solve the Towers and Pizza Problems as well as modifications and extensionsof these problems. Some questions that are investigated for those students include: (1) How do they solvenon-routine mathematical investigations; (2) What connections, if any, do they make to isomorphicproblems and previous mathematical knowledge; and (3) To what extent do they make justifications andgeneralize results.

3. Setting

The study was conducted in a mathematics course for liberal arts majors at a small New Jersey com-munity college. The curriculum for the course includes algebra and problem solving. The sections of thecourse that are included in the study met for two 75 min classes each week for 15 weeks. They spent oneclass each week working on various non-routine problems in a small group setting.

This paper will focus on one student, Rob, who worked in one of these small groups with his partners,Steven and Samantha. Rob attended the community college for one semester and took MathematicalConcepts to fulfill his mathematics requirement at another college. On a questionnaire he described himselfas good at mathematics and “somewhat” good at problem solving. Rob had learned about combinationsand permutations in a previous mathematics class, but had not studied mathematics since his junior yearin high school.

The Towers Problem was given during the ninth week of the semester. By this time the students hadbecome accustomed to working on non-routine problems and justifying their solutions. The studentsbegan by working on towers that were four cubes tall. They then were asked to consider towers that werefive cubes tall. Rob’s group also worked with towers where three different color cubes were available.

The Pizza Problem was given during the 14th week of the semester. The students first worked on findingthe number of pizzas that could be made if four toppings were available followed by the problem withfive toppings. After solving the basic problem they were asked to consider the Pizza with Halves Problemin which a topping could be placed on either a whole pizza or a half pizza.

B. Glass / Journal of Mathematical Behavior 21 (2002) 75–85 77

4. 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, 1984,1985). Understanding occurs “when a new idea can be fitted into a larger framework of previously as-sembled ideas” (Davis, 1992, p. 228). Students build their understanding of concepts by building uponprevious 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 likelyto develop well connected conceptual knowledge. When students encounter new problems, they aremore likely to retrieve prior knowledge that is well connected than to retrieve loosely connected in-formation (Hiebert & Carpenter, 1992). Even though instruction will have an effect on what a studentlearns it will not determine what the student learns. Traditional instruction is geared toward the ab-sorption of knowledge by students. However, students don’t passively receive knowledge but rather putstructure to it and assimilate it in the light of personal mental frameworks that they have constructedfor themselves (Romberg & Carpenter, 1986). Understanding is a personal process that is unique, inways that are important for understanding the teaching–learning process, for each student. In orderto reach conclusions about a student’s level of understanding a teacher must encourage students tojustify what they say and do to reveal their thinking and logic. Just giving answers will not reveal astudent’s level of understanding or the way that the student is thinking about a problem (Pirie & Kieren,1992).

In traditional mathematics classrooms, the answer key or the teacher is the source of authority aboutthe correctness of answers. Quick right answers are often valued more than the thinking that leads tothe answer. Teachers generally only question students when an error has been made. They do not askstudents with correct answers to explain their approaches and results (Burns, 1985). As a result, stu-dents develop the belief that all problems can be solved in a short amount of time and will often stoptrying if a problem cannot be solved quickly. Moreover, since students view school mathematics as aprocess of mastering formal procedures that are removed from real life they learn to accept and mem-orize what their teachers tell them without making any attempt to make sense of it (Schoenfeld, 1987).In contrast, the students in this study were encouraged to think about their solutions to the problemsthat they were given and to justify their answers rather than relying on the instructor as the soleauthority.

5. Data collection

Data were collected from nine classes during the period from 1998 to 2000. Two groups from eachclass were videotaped as they worked on the Towers and Pizza Problems. Following the class sessionseach student was required to submit a write-up of the problems. Videotaped task-based interviews werealso conducted.


Table 1Code of problem-solving strategies

Sr: random checking Sp: looked for patterns Sb: worked backwardsSa: thought of a similar problem Ss: thought of a simpler problem Sal: used an algebraic

equation or formulaSi : used inductive method Sc: conjectured Sv: controlled for variablesSsp: divided the problem into sub-problems Sf : applied a previously learned procedure

6. Methodology

Rob was one of 11 students who were chosen for a case study analysis. These students were chosento represent the range of the population that is included in the study. The data has been analyzed todetermine problem-solving behaviors and methods of justification that were used by the students as wellas connections to previous problems and other mathematical knowledge that were made by the students.Videotapes of the 11 selected students have been transcribed, coded, and analyzed for methods of problemsolving, methods of justification and connections that were made. This includes videotapes from bothclass sessions as well as individual interviews. The written work of the students has also been coded andanalyzed. The coding scheme for methods of problem solving and justification were adapted from a codingscheme that had been used with younger children who were working on the same problems (Wier, 1999).Categories needed to be added because some of the strategies and methods of justification for the collegestudents had not been used by younger students and therefore were not in the coding scheme. The codingscheme for connections was developed to highlight the level of understanding displayed by the studentswhen making connections as well as the difference between connections made with previous mathematics

Table 2Code of methods of justification

Ao: none Ac: cases approach Ai: inductive approachAa: checked with another member of the class Ae: everyone in the group agreed Af : can’t find any moreAs: same answer as a previous problem Al: applied previously learned

formula or procedureAn: numerical argument

Av: created a visually appealing pattern Aci: challenges instructor toshow missing combination

Ag: geometric argument

Ap: discovered a pattern Ape: explained why pattern worked

Table 3Code of connections made

Ctm: trivial or meaningless connection to previousmathematical knowledge

Ctp: trivial or meaningless connection to previous problem

Cpm: partial understanding of connection to previousmathematical knowledge

Cpp: partial understanding of connection to previous problem

Ccm: meaningful connection to previous mathematicalknowledge

Ccp: meaningful connection to previous problem

Ccd: connection to material from another discipline


and those made with earlier problem-solving activities. The coding scheme for problem-solving strategiescan be found inTable 1, the coding scheme for methods of justification can be found inTable 2and thecoding scheme for connections can be found inTable 3.

7. Data and analysis

Rob’s group began the Towers Problem by organizing their towers by the number of red cubes andyellow cubes that were in each tower and used a modified “proof by cases” to justify their solution. Theyshowed that they had produced all towers which contained only one color, all towers with a single red ora single yellow cube, all towers with two of each color in which the two yellow cubes were kept together,and then all towers in which the two yellow cubes were separated. After producing the towers five tall,they noticed that the number of towers had doubled from the towers of four to the towers of five. Theyextended the doubling pattern to make a prediction about how many towers they would get if the towerswere three tall, two tall or one tall. They then built the towers one tall and two tall to test their theory.While justifying their number of towers five tall to the instructor, they referred to both their doublingpattern and their proof by cases (Glass, 2001). The instructor asked the students to think of a reason whythe number of towers was doubling. After a few minutes of thought Rob explained to Steve that it doubledbecause you could add either a red or a yellow cube to the bottom of each tower.

Rob: Okay lets say the top of our tower is X, X (Rob writes an X on his paper). Then we’re putting oneon the bottom. For every X we can have a Y downhere, or for every X we can have a red downhere. So for each block we have there is now two more things it could be. So before we just hadX. This is X (Rob picks up the solid red tower of four as an example). Now we have XR and XYderived from this (Rob continues to talk as he looks for the corresponding towers of five) XY andXR (Rob holds up RRRRY and RRRRR).

Steve demonstrated that he understood Rob’s explanation by starting with the towers that were one talland showing how the towers that were two tall could be created. Rob’s written work for the two-colorproblem is shown inFig. 1. Rob wrote:

If you have two colors, you have two possibilities of 1 cube high towers, 1 or 2. Now for 2 cube hightowers you can either place a 1 or 2 on thebottom of each of the previous tower, thus giving a1

1 or 12

from the “1” tower and21 or 2

2 from the “2” tower. To each of these four new towers we can either adda 1 or 2 to the bottom for a 3 cube tower, giving eight possibilities. To those eight we can either addonly a 1 or 2 to thebottom to get a possibility of 16 4 cube towers. (Note: You can add to the top ofthe towers also. To keep the paper work neat, you should pick one way or the other, however, it willultimately yield the same answers whichever end you add the extra cube to, because there are still onlytwo possibilities for each new combination.)

It is interesting to note that Rob’s written justification referred only to the proof by induction and madeno reference at all to the original proof by cases.

Rob’s group was next presented with the task of selecting from three colors. They solved it immediatelyby using the same inductive method that they had developed for the towers of two colors. While workingon the three-color problem, Rob’s group built all the towers that were one-tall and two-tall. They produceda list of all towers that were three-tall, but only built a few of them. They next determined that the number


Fig. 1. Rob’s written work for the four tall Towers Problem.

four-tall would be 81 and built three of the towers as a representative case. They neither built nor listedthe entire set (Glass, 2001).

After they had explained to the instructor what they had done, she gave them the task of determininghow many towers could be built four-tall if three colors were available and at least one cube of each colorhad to be used. They started by using their method of induction that they had developed in the last twoproblems. They built the six towers three-tall that had one of each color. They then added a red cube, ablue cube, and a yellow cube to the bottom of each tower. After they had completed this, Rob stated thatthey did not have all possible towers because “we can’t have two blues up the top anywhere.” They thenadded a red cube, a blue cube, and a yellow cube to the top of each of the six original towers eliminatingany duplicates that this created. This gave them a total of 30 towers, but they were still missing the towersthat had two of the same color in the middle of the tower. When the instructor asked the group to explainwhat they had done and how they knew that they had all possibilities, Rob abandoned the method ofinduction and returned to the method of a proof by cases. It was during his explanation that he noticedthe missing towers (Glass, 2001).

Rob: Where our downfall came is we realized, because the rules changed, when you go from thisstep to this step. Because this step has three colors and that means we cannot have duplicates


of any. This step has four blocks with three colors. So you can have. You have to have two ofone color.

Instructor: You have to have two of one color.Rob: OK. So that means that what you end up happening is, you have to have a place with two blues

on the top. But because this is, you only have one blue on the top, then it messes everythingup. So actually a better way to. Can I rearrange them?

Steve:Go ahead.Rob: OK. So we got two blues, two blues, two blues, two blues, two blues. (Rob finds all the towers

with two blues and puts them together.)Instructor: Are you looking for all the ones with two blues?

Rob: This, yeah. That has two blues. (Instructor moves a tower with two blues, that Rob had missed.)OK. I think I have all the ones with two blues.

Samantha:How about we get rid of these. (Samantha moves other towers out of the way.)Rob: OK. So we start off with two blues together.

Instructor: You have two blues together on the top.Rob: Two blues together.

Instructor: Two blues together on the bottom.Rob: Now, see, we’re missing.

Instructor: Is that the only way you can have two blues together?Rob: No.

Steve:Two blues together in the middle. Like that. (Steve builds tower and hands it to Rob. Stevestarts to build another tower.)

Five weeks later, the group was given the Pizza Problem. They completed the problem with fourtoppings by listing the various pizzas by the number of toppings each contained. Before moving on tothe problem with five toppings Steve commented: “I think it’s good we find a pattern. . . . It’s going to bethe same as the last time we did this.” After a brief period of time Rob noticed a row of Pascal’s trianglein the number of pizzas with the various number of toppings, although he couldn’t remember the name.He referred to it first as “Pythagorean’s triangle” then as “what’s his face with the triangle.”

When the group told the instructor about their discovery, she challenged them to find a reason why theaddition rule of Pascal’s triangle should work for the Pizza Problem. After the group worked for a whilewithout results the instructor’s restatement of the question appeared to give Rob an insight (Glass, 2001).He listed the various pizzas on a copy of Pascal’s triangle and then explained his thinking to the rest ofthe group.

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

Steve:All right. I got it. It’s just. It’s tough, because it’s very crammed in there.

Rob then proceeded to make a list of the five topping pizzas by filling in the fifth row of Pascal’s triangleas shown inFig. 2. Rob wrote in his assignment:

The reason this (Pascal’s triangle) works is because every time we add another topping we are increasingthe possibility of choice, without losing the old ones. In other words, all the two topping pies in thetwo topping total still apply when there are three total toppings. Also all those that had two toppings,


Fig. 2. Rob’s drawing of Pascal’s triangle for the Pizza Problem.

by adding a third topping, are now three topping pizzas. In this way we can see absolutely, positively,without a doubt, that we have all possible combinations — each new row is built by adding theold columns with the new topping. This once again is the Pascal’s triangle principle of adding oldcombinations with new possibilities to find new combinations.

After completing work on the basic Pizza Problem, the group was given the Pizza with Halves Problem.This time the group decided to start by working on an easier problem. They started with three toppingsand found 27 pizzas. (There actually are 36.) They then considered two toppings and found nine pizzas.(There should be 10.) Convinced that they had found a pattern they checked the one topping case andfound three pizzas. They now said that they were sure that the number of pizzas with halves is 3n , wheren is the number of toppings available. When asked why the answer should be three to a power Robanswered “when we did it before it was binary. It’s either on or off. Now it’s off, on one or on two.”(Glass, 2001).

The Instructor then asked the group about the relationship between the Pizza and Towers Problems.Steve and Samantha tried to relate the toppings to the different color blocks, but Rob stated that it is notthe colors that represent the toppings, but rather the position of the blocks.

Rob: Your colors are. See, if you do it like the binary. It’s not that your colors are the topping,it’s the topping is the position. So if you have a tower 4 blocks high, your first block is plainor pepperoni, onion, sausage, mushroom. If it’s orange, it’s on. If it’s yellow, it’s off. (Robwrites as he talks.) So if this is all yellow, that means there’s no toppings, and it’s plain. OK.Now.

Steve:Got it.Rob: If this is orange, that means it’s one topping pizza with pepperoni on it.

Steve:Got it.Samantha:Hold on.

Rob: Yeah.


Samantha:Are we doing the half pizzas?Rob: Not yet, that would be three colors.

Steve:OK. You need three colors to do that.Samantha:OK. Well, I thought she was asking on this problem that. That’s why I didn’t understand why

we only had two colors.Rob: So then it? Yeah.

Samantha:That’s why it’s two raised to the something power.Rob: Right. So then?

Samantha:So it’s three raised to something power.Rob: If it’s blue, it’s half on.

After the group explained the relationship to the instructor, she drew a YBBO tower on Rob’s paperand asked which pizza that this would represent. At this point Rob recognized that you could not tellwhether the half toppings were on the same half or opposite halves of the pizza. Samantha and Stevedescribed the pizza to the instructor During the description Steve also stated that there was a problemwith the two halves.

Steve:I see a problem, though.Rob: The problem is, you can’t tell.

Samantha:No, because it’s.Steve:Which if it’s two on one side. If two have one side. Like pepperoni, onion, on the other side

plain. So maybe we could make it a bigger tower.Rob: Or should we add another color? Blue is right side, red is left side.

The class ended with the instructor asking the students to think about the problem.About a week later Rob was interviewed by the instructor about the problems. After Rob built the 27

towers of three with three colors the instructor asked about the relationship between the Towers Problemwith three colors and the Pizza Problem with halves. Rob related each tower to a pizza using white foron, blue for off, and red for half on. Rob arrived at the first tower that had two red cubes. The instructorpointed out that this was the case that Rob though might cause problems.

Rob: Right, because we know we’ve got half green peppers, and we know we’ve got half mush-rooms,

Instructor: Uh, huh.Rob: but we don’t know which half.

Rob then proceeded to find all possible pizzas for each tower that had more than one red cube to arriveat an answer of 36 pizzas.

8. Conclusions/implications

Rob, Steve and Samantha were engaged in thoughtful mathematics. They identified patterns and justifiedthat their patterns were reasonable. The students not only related the Pizza Problem to Pascal’s triangle byrecognizing the number pattern, but they also explained how the addition rule of Pascal’s triangle relatedto the Pizza Problem. In addition they explained the isomorphism between the Towers Problem and


the Pizza Problem, and related the Towers Problem with three colors to the Pizza with Halves Problem.The students’ reasoning about the Pizza with Halves Problem resulted in an incorrect conclusion, however,the instructor intervened by asking them to describe a pizza with two half toppings. This enabled thestudents to realize the mistake that they had made without being told that they were wrong. Rob laterused the towers with three colors to correctly find all pizza with halves. The three color Towers Problemalso demonstrates how the students applied the methods they had discovered in one problem to anotherproblem. The findings support the importance of introducing rich problems and encouraging students toexplore solutions, explain their reasoning and justify their solutions.

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Students connecting mathematical ideas: possibilities in a liberal arts mathematics class