Benny Goes to College: Long-Term Consequences Of

8/14/2019 Benny Goes to College: Long-Term Consequences Of

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Benny Goes to College: Long-Term Consequences of Procedurally-Focused Mathematics Instruction

Our contention in this paper is that focusing mathematics instruction on procedures has long-term, negative consequences on students learning. Instruction of this kind leads to fragile

mathematical understandingsunderstandings that may be adequate in the short-term but thatare readily forgotten over time. Our study examines the mathematical knowledge of students

who are likely the product a procedurally-focused math education. Specifically, they are agroup of community college students enrolled in developmental math classes. Throughinterviews, we seek to answer the questions (1) What do these students think it means to domathematics? (2) What do they understand about basic mathematics concepts? and (3) Ifgiven the chance, can they reason mathematically?

The conception that mathematics is a set of procedures to be memorized and speedily applied is manifest in U.S.textbooks, on tests and in test preparation programs, in classroom teaching, and in the minds of teachers. If

students are taught by teachers who believe math is a set of procedures, in a way that highlights thoseprocedures without equal attention to meaning, and are tested (and therefore rewarded) for their procedural skill,it should come as no surprise that those students come to share the belief that procedures should take center

stage in the study of mathematics. The result though, is not only that students lack conceptual understanding butalso that students procedural understandings are fragile. The procedures theyve been taught have noconceptual grounding and can therefore be easily forgotten. Although we certainly acknowledge that an abilityto follow mathematical rules is critical to solving problems, we suggest that conceiving of math as a set of

procedures and constructing instruction that focuses primarily (if not exclusively) on them has long-term,negative consequences on students understanding of mathematics. In this paper we set out to explore those

misunderstandings and their source. We begin by revisiting a classic case of a student and his misconceptions.We then consider what happens in the broader population of students in the U.S. as they continue in their studiesof mathematics.

Revisiting Benny: Examining a Sixth-Graders Mathematical UnderstandingsMore than 35 years ago we were introduced to Benny, a 6

th-grader receiving remedial instruction in

mathematics (Erlwanger, 1973). From interviews with him, it was evident that Benny had examined problems

for patterns and had, on his own, constructed from them a large number of rules. He was able to explain thoserules and he applied them consistently when solving problems. The trouble with Bennys approach tomathematics was his lack of awareness of the mathematical relationships that underlie rules. As a result of

deriving the rules from patterns of answers (rather than as generalizations derived from concepts), much of histhinking was erroneous. An example demonstrates the point. Benny was given the following exercise:

Circle the fraction that has the same value as the digit underlined in the small box:

.542 4/10 4/100 4/1000

3.20 2/100 2/2 2/10

From these and other exercises, he was able to derive rules about conversion. Applying those rules enabled himto correctly identify 29/10 as 2.9 and 8/100 as .08. When asked to write 4/11 as a decimal, though, he stated

that you cant. You can only work with 10 (p. 21). In this case, the rule he constructed had limitedapplications and his lack of conceptual understanding prevented him from searching for a rule that satisfied thenew condition.

Bennys collection of rules, because they were created without an understanding of underlyingconcepts, sometimes led to different answers to the same problem. Perhaps even more telling was that theinconsistency was not a concern to Benny. He knew, for example, that 3/10 + 4/10 = 7/10 or .7. However, he

argued that .3 + .4 = .07, claiming simply that you are using decimals and that makes the difference (p. 23).He was similarly unbothered by the belief that converting a decimal to a fraction could lead to any number ofsolutions, provided that the sum of the numerator and denominator was the value in the decimal representation.

That is, .5 could be expressed as 3/2, 2/3, 1/4, or 4/1. It was possible for Benny to hold fast to the answers atwhich he arrived because he knew that there are multiple representations of the same value. When his answerwas marked incorrect, he attributed it to him having chosen an alternative (and equally correct) representation.

It wasnt that Benny wasnt thinking. In fact, he recognized the importance of patterns in mathematicsand, with very limited guidance, sought them out. What he failed to discover on his own however, were theinterconnections among those patterns and their links to mathematical principles. He also failed to develop (orrely on) a sense of the values with which we was working. The instruction he received was not designed withthose as central goals. Benny was following his curriculum as it was intended, was making better than average


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progress compared to his peers, and was regarded by his teacher as one of her better students. It was only as a

result of participating in the study that his weaknesses were identified and remediation provided.

Procedurally-Focused Instruction and Where it Might Have Led UsThe particular form of instruction received by Benny was atypical. So what do we know of the mathematicalexperiences of most U.S. students and the understandings that result? The emphasis on procedures in American

classrooms is well-documented. For instance, in national samples of eighth-grade math lessons in the U.S., theTIMSS video studies (Stigler & Hiebert, 1999; Hiebert et al., 2003) showed that the most common teachingmethods used in the U.S. focus almost entirely on practicing routine procedures, with virtually no emphasis onunderstanding of core mathematics concepts that might help students forge connections among the numerous

mathematical procedures that make up the mathematics curriculum in the U.S. The high-achieving countries inTIMSS, in contrast, use instructional methods that focus on actively engaging students with understandingmathematical concepts. Procedures are taught, of course, but are connected with the concepts on which they are

based. In the U.S., procedures are more often presented as step-by-step actions that students must memorize, notas moves that make sense mathematically.

If our hypothesis is true and procedurally-focused instruction has cumulative negative effects, we

should expect to see students understandings fail to advance at an desirable rate over time. That is, as theyproceed through their school years, the growth in students knowledge should slow. International comparisonssuggest that thats happening. The Trends in International Mathematics and Science Study (TIMSS) sets out to

measure over time the mathematics performance of fourth- and eighth-graders. Encouragingly, the results of themost recent TIMSS test administration indicate that from 1995 to 2007, U.S. students saw significant gains intheir scores at both grade levels (Gonzales et al., 2008). In spite of this though, and in spite of reasonably high

scores at fourth-grade, the average scaled score of U.S. eighth-graders was barely above the average across the48 participating countries. Something that occurs between fourth- and eighth-gradeand that perhaps began

before thencauses a decline. We propose that the mechanism of that decline is the emphasis in classrooms onperforming procedures. The 2006 Programme for International Student Assessment (PISA) survey, because itmeasures student performance only among 15-year-olds, cannot inform the discussion related to change overtime. It can, however, be used as additional measure of how U.S. students fare in mathematics in comparison to

their peers in other countries. There, the picture is also disappointing. Among the 57 participating countries,U.S. students scored statistically significantly below average.

If what happens from fourth- to eighth-grade is the embryo of concern, that concern is in full bloom by

the time students near high school graduation and move onto higher education. Although TIMSS hasnt

included twelfth-graders in recent test administrations, they were included in the 1995 data collection effort. Atthat time, U.S. students scored below the international average in mathematics and among the lowest of the 21

nations included in the study. Their standing was lower at twelfth grade than at eighth (U.S. Department ofEducation, 1998). Those results are consistent with assessments we have of the mathematical understandings ofstudents who enter our community colleges. More and more students in the U.S. are attending those institutionsand most of them are not prepared for college-level work. By most accounts, the majority of students enteringcommunity colleges are placed (based on placement test performance) into "developmental" (or remedial)mathematics courses (e.g., Adelman, 1985; Bailey et al., 2005).

The implications for students who are placed in the lower-level developmental courses can be quitesevere. Those students may face two full years of mathematics classes before they can take a college-levelcourse. This might not be so bad if they succeed in the two-year endeavor, but the data show that most do not.

Students either get discouraged and drop out all together, or they get weeded out at each articulation point,failing to pass from one course to the next (Bailey, 2009). In this way, developmental mathematics becomes a

primary barrier for students ever being able to complete a post-secondary degree, which has significant

consequences for their future employment. The fact that community college students, most of whom graduatefrom U.S. high schools, are not able to perform basic arithmetic, pre-algebra, and algebra, shows the real cost ofour failure to teach mathematics in a deep and meaningful way in our elementary, middle, and high schools.

One might hope that the identification of community college students who are in need remedial helpwould serve a turning pointa point at which conceptual deficiencies are remedied. Sadly, all the evidence wehave (which is not much) shows that although community college faculty are far more knowledgeable about

mathematics than are their K-12 counterparts (Lutzer et al., 2007), their teaching methods may not differ muchfrom those used in K-12 schools (Grubb, 1999). "Drill-and-skill" is still thought to dominate most instruction(Goldrick-Rab, 2007). Thus, students who failed to learn how to divide fractions in elementary school, and who

also probably did not benefit from attempts to re-teach the algorithm in middle and high school, are basicallypresented the same material in the same way yet again. Students lack of conceptual understanding is met withmore of the same kind of instruction, and developmental classes at community colleges perpetuate the system

that led to their need.


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The Current StudyThe research weve reviewed suggests that the story of mathematics education for many students in the U.S.may go something like this. The instruction they receive is procedurally focused. The focus on procedures

serves them wellfor a time. Beyond elementary school, the number of procedures they must memorize andmechanically apply mounts. Procedures that carry no meaning become confused with one another anderroneously interchanged or forgotten altogether. Test scores slip at an ever-growing rate and by the time they

graduate from high school, they are unprepared for higher mathematics because they lack an understanding ofthe math on which it is built. Those students who enter community colleges lacking conceptual understandingsare provided little support for acquiring them and the understandings with which they arrive persist. We would

expect from this that students in community college remedial classes would have mathematical understandingsnot radically different from those held by Benny. They might have a larger repertoire of procedures from whichto draw when searching for a way to solve a problem, but that could be as much a hindrance as a help.

This line of reasoning led us to inquire into what we actually know about the mathematics knowledgeand understanding of students who are placed into developmental math courses at community colleges. Anextensive search of the literature revealed that we know surprisingly little. Most of what we know about the

mathematical knowledge of community college students we learn from placement tests (Accuplacer, Compass,MDTP). But placement test data is almost impossible to come by due to the high-stakes nature of the tests andthe need to keep items protected. Further, the most commonly used tests (Accuplacer and Compass) are

adaptive tests, meaning that students take only the minimal items needed to determine their final score, and so

don't take items that might give a fuller picture of their mathematical knowledge. Finally, most of the items onthe placement tests are procedural in nature: they are designed to assess what students are able to do, but not

what students understand about fundamental mathematical concepts.In the present study, more than a generation after becoming acquainted with Benny, we use interviews

to examine the mathematical understandings of a group of community college students who, like Benny, arereceiving remedial math instruction. With up to six additional years of middle- and high school math education

behind them (and, for some, more courses in college), how have their conceptual understandings advancedbeyond those held by Benny? What do these students think it means to do mathematics? Is it just remembering

rules, or is reasoning also required? What do these students understand about basic mathematics concepts? Andif we give them the chance, can they reason? Can they discover some new mathematical fact based only onmaking effective use of other facts they know?

Methodology

ParticipantsNineteen community college students (10 female, 9 male) enrolled in developmental mathematics courses (i.e.,

Arithmetic, Pre-Algebra, and Elementary Algebra) participated in the study.1

Students ranged in age from 17 to51 (M = 22.2, SD = 7.9). Eight were Hispanic, six were white, three were African-American, and two were ofmixed ethnicity. For four students, Spanish was the primary language spoken at home. For two students,Spanish and English shared the primary role, and for the remainder of the students the primary language athome was English. All received $50 for their participation.

ProcedureThe one-on-one interviews took place on the community college campus at which students were enrolled andwere scheduled at students convenience. Guided by a protocol that was refined through extensive pilot testing,the interview set out to address our central research questions: 1) What do students think it means to domathematics? (2) What do students understand about mathematics concepts? and 3) Can we get students to

reason about mathematics or are they stuck with just remembering procedures? We opened each interview withquestions about what it means to do mathematics, asking in a variety of ways what students think about theusefulness of math and what it takes to be good at it. That was followed by questions about seven mathematical

problems (e.g., comparisons and operations with decimals, comparisons of fractions and placement on a numberline, solving equations with one variable, and equivalence). For each line of questioning we anticipated possibleresponses and created structured follow-ups. The general pattern was to begin each line of questioning at the

most abstract level and to become progressively more concrete, especially when students struggled. Each of theseven questions concluded with prompts that pressed for reasoning. Interviews ranged in length from 54 to 101minutes (M= 75.0, SD = 10.6).

1Ultimately, the study will include 30 students10 from each of the three classes from which participants have

been recruited.


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Results

What do Students Think it Means to Do Mathematics?One way we assessed students beliefs about what it means to do math was with the question If

someone is good at math, what exactly is s/he good at? We then suggested that some people think math is

about rules and procedures and others think its about understanding and reasoning. More than half of students

claimed that math means knowing and executing procedures and following rules. A thread that ran through theirresponses was that to be good at math necessitates memorizing many steps that one must follow. One student

put it like this, Math is just all these steps and you have to understand why this is this... Its all about being veryprecise in the way you do it. Understanding, in her case, appeared related not to conceptual understanding but rather to understanding how to successfully apply rules. Another said that being good at math is about

repetition and having to feel the pain of it. Still others responded in a way that discounted the role ofconceptual understanding. One stated that in math, sometimes you have to just accept that thats the way it isand theres no reason behind it and another responded that "I don't think [being good at math] has anything to

do with reasoning. It's all memorization."In all, only three students in the sample discussed conceptual understanding as desirable. One said that

it helps if you actually understand what youre doing and youre not just regurgitating what the teacher says.

That you actually understand the concept. Another said that its cool when it all links together and statedthat when she sees the big picture, shell remember it forever. One astute student spoke at length on the topic.

He observed thata lot of times I find that people can do the mathematics on paper. They cando the order of operations and come up with the right answer fairly often,

but I feel like they're sort of missing the concept that what these numbers

are is really just representations of reality... So there'd be a question like'What is 3% of 600?' And they'll come up with 700 and stand by that answerand argue with me. I don't know that I have the right answer, but I can tell

you that's not right. There's no way... I feel like I see that happen a lot."

His observation about his classmates lack of number sense suggests that he sees students focused on

performing procedures without thought to the concepts that underlie them. It is interesting to note that thisstudent wasnt required to take developmental math courses. He placed into a credit-bearing course, but becausehe had no formal math education before attending the college, opted to take the lowest level course and work

his way up.Another of his comments suggested that he perceived that it wasnt just his peers who were focused on

procedures. His teachers were as well. At the close of the interview, students were asked what advice theywould give to teachers with respect to how to teach in a way that would help them better understandmathematics and he lamented that he only needs to complete one problem to understand a conceptthat he

doesnt need to repeat it 50 times. He stood alone in that view, however. The vast majority of students, whenoffering advice to teachers, thought that they should be careful to include all steps when writing a problemssolution on the board and/or that they should go through the steps of a solution more slowly.

What do students understand about mathematics concepts?The first math problems presented in the interview were to be completed mentally. They included a series of

multiplication problems designed so that students could rely on decomposition and found answers when theysolved. That series of problems was as follows:

10 x 3 = ___10 x 13 = ___20 x 13 = ___22 x 13 = ___30 x 13 = ___31 x 13 = ___

29 x 13 = ___So, for instance, when calculating 22 x 13, students could use the answer to 20 x 13 and add to it the result of 2x 13. Almost no one did that. Nearly all instead put the second number below the first and used the standard

algorithm to solve the problem in their head. Although the method may be efficient when written, it becomesdifficult in the absence of paper. Students clearly had trouble keeping track of partial products as theycomputed. After completing the series of problems, the more efficient method was demonstrated to studentsand they were given an opportunity to use it on the problems theyd solved. Nearly all could do so correctly and

yet, when a new problem was presented, they resorted to the familiar algorithm almost without fail. What welearned from those questions is that even if students have the tools to complete problems, they are unable todraw upon them flexibly to make their task efficient.


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In another problem, students were provided with two, three-digit addends as well as their sum and were

asked to check for correctness. Some did so simply by re-working the addition. Others used subtraction tocheck, nearly always by subtracting the second addend from the sum. When asked if the first addend could beused similarly, many were skeptical and a few claimed it could not. This finding was startling, given how

fundamental a concept it is and how far-reaching the misunderstanding of it.Two questions that asked students to set up subtraction with decimals demonstrated problems not only

with students understanding of place value, but also with their number sense. Students were asked to set up0.572 0.86. When rewriting the problem vertically, a few lined up the 2 and the 6, disregarding dissimilar

place value. Nearly all wrote the first (smaller) number above the second and didnt foresee that a problem

would arise when used to calculate. Though not prompted to, some students did attempt to solve. They werestymied when it came to subtracting 0.8 from 0.5. One student turned 0.5 into 1.5, so that there would besomething from which to borrow. She didnt see a problem with her resolution.

We saw problems also in students conceptions of fractions. When we asked them to compare a/5 to

a/8, we saw that their rationale infrequently included the mention of the number of pieces of a whole. Althoughsome compared the two values by assessing their distance from 1, far more relied on the application of a oft-

practiced procedure: creating common denominators. Though not incorrect, it is further evidence that students

apply known procedures rather than using reasoning, even when reasoning is more efficient.We later provided students with multiple fractions with different denominators (i.e., 4/5, 5/8, -3/4, 5/4)

and asked students to place them on a number line. It isnt surprising that students would choose to use division

to convert to decimals. Doing so would ease the task of segmenting the number line for use with multiplefractions. What is surprising is that several students used division to convert a fraction into another fraction.

They did so by dividing the denominator by the numerator and thus the fraction they created was different fromthe one with which theyd started. With this method they lost the potential advantage of a decimal conversionand placed a fraction on the number line that was not equivalent to the one theyd been given. Moreover, theyfailed to be bothered by either problem.

On a more general level, we saw a basic lack of number sense among interviewees. They wereuntroubled when their answers to mental multiplication problems should have suggested to them that theydmade an error, as when the product of 31 x 13 was less than the product of 30 x 13. They also seemed not to

notice when they placed 4/5 to the right of 5/8 on the number line immediately after they had clamed that 5/8was the larger of two numbers. Finally, when selecting from a several expressions those that represented half

of n, they were unconcerned when they selected n, n , and n 2. Any pair of options should

together have triggered a thought about incompatibility, but they rarely did.

Can we get students to reason about mathematics?One series of questions we posed to students was intended to tap (1) their ability to reason about values, and (2)their ability to transfer that reasoning to mathematical notation. We first asked, What would happen if you hada number and added 1/3 to it? Would it be more than what you started with, less than what you started with, the

same as what you started with, or can you not tell? Most students answered readily and correctly. We followedup immediately with the question, Ifa + 1/3 = x, is x more than a, less than a, the same as a, or can you nottell? The latter question was never perceived to be as straightforward as the first, nor were students answersto it always correct. Perhaps most intriguing was the large number of students who thought the second questionwas unanswerable unless either a orx (or both) was provided, in spite of the fact that they just answered thesame question. Of course, their difficulties may have been due to a failure to understand the mathematics

notation. When we asked students to substitute the value of 6 fora, they could generally answer correctly forthat case. Multiple substitutions often led them to conclude that the equation would be true for positive wholenumbers, but they were often unwilling to generalize to fractions or negative numbers. We then posed a similar

pair of questions about multiplying a number by 1/3 and comparing x to a in the equation a * 1/3 = x. Asexpected, this pair of questions was more difficult than the first pair. Generally though, students were able toreason about the initial part, even when unable to apply the same reasoning to the mathematical notation.

In another series of questions, we presented students with a measurement representation ofa + b = cand asked them to evaluate the truth of other equations that used a, b, and c. They frequently reasonedeffectively, though it was common forc a + b = 0 and c a b = 0 to pose problems. Students were troubled

by the inclusion of 0 in the equation. Two students stated explicitly that that wasnt allowed.The final mathematics problems of the interview involved the equation x y = 0. Students very often

said that the statement could be true only ifx andy were the same value. However, when asked to re-write the

equation in terms ofx, they were often unable to say thatx =y.The commonality across these examples seems to be that students can reason about math when

presented a mathematical scenario, but not necessarily when the same scenario is represented in mathematical

notation. At some level, students understand important mathematical concepts, but being required to work withmathematical notation seems to undermine it.


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Case Study: Roberto2When interviewed, Roberto was in his first semester in community college, having graduated from high schoolthe prior spring. He recently turned 18, and plans to become a history teacher. To reach that goal, he must

eventually transfer to a four-year university, which in turn requires that he successfully pass through a sequenceof three developmental math courses at his community college (i.e., Arithmetic, Pre-Algebra, and ElementaryAlgebra). Roberto took Algebra 1 both as an 8

thgrader and as a freshman and sophomore in high school. In high

school, his class was a one-year course spread over four semesters. Despite this extensive experience in Algebraclasses, he is still two semester-long courses away from the developmental course that is equivalent to a high-school level Algebra 1 course.

For Roberto there seems to be a difference between doing in-school mathematics and out-of-schoolmathematics. Over the course of the interview, he asked repeatedly if he was allowed to solve a problem oranswer a question a certain way. For example, when asked what number would fill in the blank to make the

equation 7 + 5 = ___+ 4 true, Roberto explained that both sides would need to have the same value because ofthe equal sign. He asked if he could do it just by looking at it, or by finding a way to do it. When theinterviewer said that the choice was his, Roberto chose to do it just by looking at it. He said the left sidetotaled 12, so the right side must also be 12. For this reason, 8 should fill in the blank. However, when askedimmediately after about the value ofx in the equation 7 + 5 = x + 4, Roberto remarked Well, if were talkingalgebra, you would subtract 4 and move it to this side. He initially saidx would be 12, but caught himself and

said I didnt do it right. He corrected his mistake, mentioning that he forgot to subtract 4 from 12, and agreed

that no number other than 8 could be a solution for either equation. It is striking that he was very concernedabout equality in an equation with a blank space. But when the blank was replaced with x, he deferred to

common algebraic procedures and by carrying out the procedure incorrectly he ruined the balance he said wasso important. When asked if he thought it would be okay to think about the second equation the same way hethought about the first, Roberto shared that he thinks you should be allowed to if you can, but that none of his

teachers allowed him to. He said they thought he was cheating. If stated that if he were a teacher, he wouldallow students to solve equations the way it makes sense to them.

Robertos understanding of the meaning of the equal sign in an equation was quite robust. Whenpresented with other equations, such as 7 + 5 = x, = 2/4, and 2 = 2, Roberto maintained his conception of theequal sign as showing a balanced relationship. Roberto understood the equal sign beyond what many studentsunderstand. Research shows that it is not uncommon for students to see the equal sign as a cue to do

something like find an answer (Knuth, 2006). Not so with Roberto. He understood the equal sign as relational,and could use this relationship to solve equations. However, it is not clear if he saw algebraic manipulations

such as subtract 4 from each side as a mechanism for maintaining the relationship.Could Roberto reason if provided the opportunity? When he was asked which was larger, a/5 ora/8

(given that a is a positive whole number), he replied that fifths are larger than eighths. He did not initiallyrecognize that the numerator must be the same across the two terms for that rationale to be sufficient. Once this

was cleared up, he explained that a/5 would always be larger because some number of fifths would be largerthan that same number of eighths.

As with a great many of our participants, Robertos responses indicated a disconnect between his

mathematical understanding and his performance on mathematics tasks involving algebraic notation. Whenasked to comparex to a in the equation a + 1/3 =x, Roberto responded that he would need to first find out either

a orx. He saw the equation as something to solve rather than something he could use to reason about the

relationships among the quantities in it. Roberto then responded that the sum would be smaller. This response isflawed for two reasons. First, Roberto explained that in order to add a whole number and a fraction, one needsto rewrite the whole number as a fraction. He said that ifa = 1, he would need to rename it as 1 over 0. Then

he added the numerators (1 and 1) and the denominators (0 and 3) to get a sum of 2 over 3 or two-thirds. Nexthe compared the sum to 1/3 rather than to the original number, and changed his answer to larger. When askedto try and add 2 + 1/3 without changing 2 to a fraction, Roberto said that he was probably taught to do that, but

didnt remember how. He said, I was taught so many things.Roberto was then asked to think about multiplying a number by 1/3 and whether the result would be

smaller than the original number, larger than the original number, equal to the original number, or if it was

impossible to tell. The equation a * 1/3 = x was written for him. Roberto choose a number for a, and thenmultiplied the number by 1 and by 3 to get a fraction equivalent to 1/3. He then simplified this fraction andconcluded that the result would be the same, again comparing the result to 1/3 rather than to the original

number. In the middle of this process, Roberto tried to recall what he had been taught to do and commented thathe has been taught by like seven million teachers how to do this. He used the same process with a = 3, withthe same result. The interviewer then asked him to think about of a number without writing anything on the

paper. Roberto talked about how finding half of a number is dividing the number by 2, and successfully found

2Roberto is a pseudonym. To listen to his interview, go to http://vimeo.com/7045271.


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of a few numbers. He correctly stated that of a number would be smaller than the number you start with, and

added that it depends on if the number is negative or positive. The interviewer asked if he could use that sameprocess to find 1/3 of a number. Roberto replied that taking 1/3 of a number is like dividing the number intothree equal parts. When the interviewer reminded Roberto that he earlier said that multiplying a number by 1/3

gives a number equal to 1/3, he said that he didnt remember what hed done with that problem. Clearly, there isa disconnect between his ability to reason about 1/3 of a number and thinking about this as multiplying by a

fraction.The implications of this disconnect and his difficulty with notation became clear when Roberto was

asked to choose from among several expressions which could be used to find of a number. Roberto chose the

expression with the exact wording 1/2 ofn, but also chose two incorrect expressions. He chose n and n

, which are consistent with how he described his process for finding half of a number. In his mind, the result issmaller, and he had talked about both subtraction and division. Furthermore, must be part of the expression(even though he talked about dividing by 2). The expressions he chose are the only two expressions that fit these

criteria. Roberto was able to find half of positive even numbers, but he was unable to choose an algebraicexpression that he could use to find half of a number.

Throughout his interview, Roberto often attempted to explain his thinking process by referring to a

number line, doing so even when the problem was not about a number line. Each time he did this he wascorrect. Also throughout the interview, Roberto mentioned trying to recall what he was supposed to do based onwhat he had been taught by seven million teachers. He claimed that he is good at math because he has a good

memory, though his responses render that judgment suspect. In fact, besides basic whole number calculations,he was incorrect with almost every calculation procedure he tried. Does this mean he does not understandmathematics or cannot reason about mathematics? Not necessarily.

Roberto invoked inverse operations, cleverly used number lines, and made generalizations. Whenasked to describe the equationx y = 0, he used an analogy of prison, saying that 0 keeps x and y from beingwhatever it wants to be. When the interviewer was able to give Roberto no other choice but to reason, he could

do it. One example is Robertos interpretation of the relationship ofx andy in the equationx y = 1, asx alwaysbeing one more thany. He also knew there would be infinitely manyx,y pairs.

Roberto reported that mathematics is interesting and fun when he is being challenged to think.However, it is not clear what Roberto has been asked to think about in his many mathematics classes. He clearlyhas some understanding and the ability to reason, but one wonders if his shallow knowledge of procedures has

been his downfall. What may at first appear to be gaps in understanding eventually reveal themselves to be gaps

in procedural knowledge and notation, exacerbated by the disconnect between his often correct reasoning and

his often incorrect procedures. Sadly, Roberto was not able to recognize this disconnect and too often deferredto his memory. We should not expect otherwise, as he professed at the beginning of the interview that a good

memory is the determining factor in being good at mathematics.

ConclusionsSadly, our community college participants are similar to Benny in many ways. We see in them more emphasison rules (correct or not) than we would care to see. We see also a lack of number sense that allows forinconsistencies across solutions to go unnoticed. For Benny and the college students alike, misunderstandings

have either gone unrecognized by their teachers or have been unsuccessfully addressed by them.The picture we paint is disturbing, and shows the long-term consequences of an almost exclusive focus

on teaching mathematics as a large number of procedures that must be remembered, step-by-step, over time. As

the number of procedures to be remembered growsas it does through the K-12 curriculumit becomes harderand harder for most students to remember them. Perhaps most disturbing is that the students in communitycollege developmental mathematics courses did, for the most part, pass high school algebra. They were able, at

one point, to remember enough to pass the tests they were given in high school. But as they moved intocommunity college, many of the procedures were forgotten, or partly forgotten, and the fragile nature of theirknowledge is revealed. Because the procedures were never connected with conceptual understanding of

fundamental mathematics concepts, they have little to fall back on when the procedures fade.It is clear from the interviews that students conceive of mathematics as a bunch of procedures, and one

often gets the sense that they might even believe it is inappropriate to use reason when memory of procedures

fails. Roberto, in our case study, asked at one point: 'Am I supposed to do it the math way, or just do whatmakes sense (paraphrased)?' He appears to think that the two are mutually exclusive. Roberto, remember, hadtaken elementary algebra three times in K-12: once in eighth grade, and then again for two years in ninth and

tenth grades. He showed signs of being able to reason, but didn't bring reason to bear when his procedures werenot working, nor was he able to notice that his answers resulting from procedures did not necessarily match hisanswers resulting from reasoning. He, like most of the students in this study, looked at each problem, tried to

remember some procedure that could be applied to the problem (e.g., cross-multiply), and then tried to execute


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the procedure. Unfortunately, much of the time, either the procedure was not the correct one, or it was executed

incorrectly, which led to the high incidence of mathematical errors.What is clear from our data is that the reason for students procedural difficulties can be tied to a

condition we are calling conceptual atrophy: students enter school with basic intuitive ideas about mathematics.

They know, for example, that when you combine two quantities you get a larger quantity, that when you takehalf of something you get a smaller quantity. But because our educational practices have not tried to connect

these intuitive ideas to mathematical notation and mathematical procedures, the willingness and ability to bringreason to bear on mathematical problems lies dormant. The fact that the community college students have somuch difficulty with mathematical notation is significant, for mathematical notation plays a major part in

mathematical reasoning. Because these students have not been asked to reason, they also have not needed therigors of mathematical notation, and so have not learned it.

But there also is some good news. In nearly every interview, we have found that it is possible to coaxthe students into reasoning, first, by giving them permission to reason (instead of doing it the way they were

taught), and second, by asking them questions that could be answered by reasoning. Furthermore, the studentswe are interviewing uniformly find the interview interesting, even after spending well over an hour with theinterviewer thinking hard about fundamental mathematics concepts. This gives us further cause to believe that

developmental math students might respond well to a reason-focused mathematics class in which they are givenopportunities to reason, and tools to support their reasoning.

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