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by Annie and John Selden

The way teachers explain mathematics depends, to a large extent, on the conceptual grasp they acquire in their college classes. In addition, they often teach as they were taught, modeling themselves after their college mathematics teachers as much as their K-12 teachers. Mathematics departments share in the preparation of all preservice teachers, and in particular, secondary education mathematics majors take many of the same courses as regular mathematics majors.

What views of mathematics and its teaching do preservice teachers get while in college? A lot of research effort has gone into probing their understandings of both mathematics and how it is taught. Somewhat curiously perhaps, getting at students' understandings of mathematics seems to be a very different enterprise from assessing students' work for the purposes of assigning grades. Students can often carry out algorithms well (i.e., have sufficient procedural knowledge to pass a course), yet are unable to explain why the algorithms work (i.e., have little underlying conceptual knowledge). Thus, it can be hard to gauge college students', including preservice teachers', conceptual grasp through normal testing alone -- hence, the need for in-depth studies of their knowledge of mathematics and how to teach it.

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SUBJECT MATTER KNOWLEDGE VS. PEDAGOGICAL CONTENT KNOWLEDGE

Whatever one may think of the *NCTM Standards* (currently undergoing revision) and the *Professional Standards for Teachings*, they are an attempt to bring more genuine understanding of mathematics into K-12 classrooms, while continuing to teach basic skills. It was always intended that both conceptual and procedural knowledge should be taught. What kinds of (mathematical and other) knowledge should a teacher have in order to promote both of these? Whether it's mathematics or another discipline one is attempting to teach, Lee Schulman of Stanford, and his colleges have distinguished two relevant knowledge domains: * subject matter knowledge*, which includes key facts, concepts, principles, and explanatory frameworks of a discipline, as well as the rules of evidence used to guide inquiry in the field, and * pedagogical content knowledge*, which consists of an understanding of how to present specific topics in ways appropriate to the students one is teaching. (There is also just plain *pedagogical knowledge* which concerns teaching and learning generally.)

For K-12 teachers of mathematics, along with knowing basic formulas and rules, subject matter knowledge would include a rich conceptual and connected, rather than compartmentalized, knowledge of the mathematics they are to teach. It would also involve knowing how to do and talk about mathematics, as well as having appropriate beliefs about the nature of mathematics. Pedagogical content knowledge, on the other hand, would comprise knowing several representations of each mathematical concept, having a number of powerful analogies, examples, and explanations for each topic, as well as knowing which ones are likely to be difficult for students and what conceptions, misconceptions, and preconceptions they might typically bring with them.

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THE STUDY: LEARNING TO TEACH MATHEMATICS

In this multi-year project, eight seniors in a K-8 education program, who had chosen mathematics as an area of concentration (approximately 20 semester hours of mathematics, statistics, and computer science) and had average-to-above-average academic performance, were followed through their senior year and into their first year of teaching. The study's goal was to assess these students' procedural and conceptual knowledge of mathematics, as well as their ideas and practices regarding the teaching of mathematics. The researchers wanted to describe their knowledge, beliefs, thinking, and actions related to the teaching of mathematics over the entire period. Data was collected using semi-structured interviews, questionnaires, written documents, and observations. While a number of articles came out of the project, the case of a single individual (Ms. Daniels) stood out and was selected for in-depth analysis.

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The Case Of Ms. Daniels

Ms. Daniels' first three years of college had been spent as a mathematics major. She had maintained a C average through two years of calculus, an introductory course in mathematical proof, a first course in modern algebra, and four computer science courses. Like a number of other mathematics majors, Ms. Daniels hit a wall upon getting to the second modern algebra course and an advanced calculus course, earning very low grades. She wanted to become a teacher, but was turned down by the secondary mathematics teacher education program because of her grades in mathematics. So she decided to major in elementary education with a mathematics concentration. Because of her background in mathematics, she was exempt from the first two of a three-course mathematics sequence designed especially for elementary education majors and missed the opportunity to explore elementary number concepts and operations.

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Ms. Daniels' Beliefs About Mathematics Teaching And Learning

Ms. Daniels believed that good mathematics teaching included making mathematics relevant and meaningful for students, that teachers should incorporate applications from students' everyday lives, as well as applications students might believe were useful to someone someday. She also thought it important to plan activities students might enjoy. She felt that, to make mathematics meaningful, a teacher should provide good explanations "on the students' level." However, she seemed unable to come up with specific examples, and as the year with its four student classroom practice assignments progressed, she started saying how difficult it was to relate mathematics to students' lives. In fact, she seemed incapable of providing the applications she wished to have, and the methods' course, which she was taking concurrently, did not seem (to her) to provide them.

Ms. Daniels felt both procedural and conceptual knowledge were necessary for understanding mathematics, yet she was clearer about how to teach the former. This was done, she felt, by carefully demonstrating algorithms, explaining each step in detail, and providing opportunities for students to practice until the algorithms were "engraved in their brains." However, her ideas of how to teach concepts and reasoning were vague. She thought students "can discover things on their own, relate things in their own mind without being told mathematically," but had little idea how a teacher might help students do so. In her classroom placements, she attempted, to some extent, to teach for conceptual knowledge, however, limitations in her own knowledge sometimes forced her to focus almost entirely on procedures. She compensated by providing students with mnemonics and memory aides. For example, one day she told her students, " . . . a circle is like a pie . . . [and pi] is my friend that we always have around when we are working with circles." Later (to the researcher) she confided, "I don't just like saying 'Well, this is pi. Remember it,' . . . [but] where does pi come from? Well, I don't know."

Ms. Daniels' beliefs in the importance of learning mathematics with understanding, of considering applications, and of providing visualizations were undercut by her inability to come up with specific examples. She believed her knowledge of mathematics was adequate (a belief supported by testing out of the Concepts of Mathematics course for elementary education majors). During her college years, she had never been forced to rethink her understanding of learning to teach, her own knowledge of mathematics (or lack thereof) for teaching middle school mathematics, or her uncritical belief in the efficacy of (drill and) practice.

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Ms. Daniels Comes Up Short: Division Of Fractions

In interviews at the beginning, middle, and end of her senior year, Ms. Daniels was asked to describe how she would teach division of fractions to a sixth-grade class. In particular, how would she respond to a pupil who says, "I know that when I'm supposed to divide two fractions, I have to turn one of the numbers upside down and multiply, but I don't know why all of a sudden it gets changed to multiplication . . ."? Ms. Daniels replied, "A good way to remember this problem would be the number you are dividing by is the number you want to invert," which gave little evidence of conceptual understanding. Later, she admitted, "I don't know why you invert and multiply, I just know that's the rule," but she added , " . . . someone could very well ask me that question and I couldn't tell them why. I should know that."

Ms. Daniels knew multiplication and division were inverse operations. When asked about a problem such as 1/2 divided by 1/4, she would frequently restate it as "How many 1/4's are there in 1/2?" She stated that students should be able to draw a diagram to represent their understanding of division, but when asked to provide one, she began to describe a diagram based on a Cartesian product interpretation of multiplication, then hesitated. When the interviewer suggested she diagram the simpler situation of 2 divided by 1/2, she said, "I don't know . . . I mean, we didn't really do that [in our methods class]."

Ms. Daniels' knowledge of fractions was put to the test one morning in April whilst reviewing the division-of-fractions algorithm with some sixth-grade students in preparation for a standardized test. After carefully explaining the algorithm with 3/4 divided by 1/2, one of the girls asked, "I was just wondering why, up there when you go and divide it and down there you multiply it, why do you change over?" Recognizing a conceptual explanation was called for, Ms. Daniels attempted to provide a concrete example using a wall with 3/4 left to paint, but having only enough paint for 1/2 of that. She divided a rectangle into fourths and shaded one-fourth, then she divided the unshaded part in half, and said, "There is 1/4 on each side, plus half of a fourth." She then divided each of the fourths in half, and started to indicate how many eighths they had paint for, when she said, "I did something wrong here."

She stared at the board for about two minutes and said, "Well, I am just trying to show you so you can visualize what happens . . . but it is kind of hard to see . . . let me see if I can think of a different way of explaining it to you . . . But for right now, just invert the second number and multiply." While the children worked practice problems, she stared at the board, looked at the teacher's manual, and said to the researcher (who was observing the lesson), "I just did multiplication." However, she did not tell the students the example illustrated multiplication nor did she attempt a correct presentation the next day. For the rest of the lesson, she continued practicing algorithms. When questioned later, Ms. Daniels indicated that she was basically pleased with how the lesson had gone, just concerned that she'd spent too much time.

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Some Implications Of The Learning To Teach Mathematics Project

(1) Prospective teachers should be given opportunities in their university course work to strengthen the subject matter knowledge of the topics they are going to teach. However, the simplistic solution of just increasing the number of mathematics courses required for certification, as has been suggested by the Carnegie Forum ("A Nation Prepared: Teachers for the 21st Century," New York, 1986) or the Holmes Group ("Tomorrow's Teachers," East Lansing, MI, 1986), is unlikely to work. Current offerings in mathematics departments are *not* providing what's needed -- for example, courses that focus on the conceptual development of rational number concepts.

Indeed, research going back to the 1970's provides little evidence to support a direct relationship between teachers' knowledge of mathematics, as indicated by the number of university-level mathematics courses successfully completed, and student learning. [Cf. School Mathematics Study Group, "Correlates of mathematics achievement: Teacher background and opinion variables," in J. W. Wilson and E. A. Begle (eds.), NLSMA Reports (No. 23, Part A), Palo Alton, CA, 1972 and Eisenberg, T. A., "Begle revisited: Teacher knowledge and student achievement in algebra," *JRME* 8, 1979.]

In line with these ideas, MAA's COMET says that preservice teachers need opportunities to " . . . explore, analyze, construct models, collect and represent data, present arguments, and solve problems . . . " (*A Call for Change*, 1991). However, an "introduction to mathematical proof" course, as taken by Ms. Daniels, did not provide her with practice in the kind of arguments/explanations needed for presenting fractions to sixth-graders. Sixth-graders ask different questions from college mathematics students. They also need different explanations (based on different starting points) -- they are not learning to prove theorems from axioms and definitions, but nevertheless understand reasoned arguments.

(2) University course work should provide prospective teachers with an opportunity to strengthen their pedagogical content knowledge, in particular, help with relating various representations (models) and applications to algorithms and procedures. University methods courses and student teaching experiences should provide these opportunities.

[For more information on the Learning to Teach Mathematics project, see Eisenhart, et al, "Conceptual knowledge falls through the cracks: Complexities of learning to teach mathematics for understanding," *JRME* 24(1) 1993 and Borko, et al, "Learning to teach hard mathematics: Do novice teachers and their instructors give up too easily?", *JRME* 23(3) 1992.]

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OTHER STUDIES SHOWING PRESERVICE TEACHERS' SUBJECT MATTER KNOWLEDGE OFTEN COMES UP SHORT

Because there is a plethora of studies on preservice teachers' conceptions and misconceptions, we will indicate only a couple of recent research results here. [An annotated bibliography is provided for those wanting more information.] Recently, Zazkis and Gunn investigated preservice elementary teachers' understandings of set concepts, after the students had completed their study of the topic. In answering the question, Why is the empty set a subset of every set?, these students did not resort to the definition of subset, nor had they earlier used the definition in looking for nonempty subsets of a set. This disinclination of beginning students to use mathematical definitions, in favor of other things they know about a concept (their concept image), has been noticed before. Of course, mathematics majors eventually need to learn to use mathematical definitions, but what of preservice teachers (and others) who only take a couple of college mathematics courses? For Zazkis and Gunn, what seemed most effective and psychologically satisfying for those preservice teachers who had elected to do an "ISETL project" was to use an action-process approach, saying that they could construct subsets of a set *A* by taking "any number of elements" of *A*. At first, "any number" did not include the possibility of zero elements being chosen, but eventually the students came to accept this. [Cf. "Sets, subsets, and the empty set: Students' constructions and mathematical conventions," *JCMST* 16(1) 1997.]

Preservice teachers can often give a general explanation of the meaning of The Fundamental Theorem of Arithmetic, yet also give responses that appear to deny uniqueness of the prime decomposition. When asked by Zazkis and Campbell to determine (and explain) whether *M* = 3^{3} x 5^{2} x 7 was divisible by 7, 5, 2, 9, 63, 11, or 15, twenty-nine of fifty-four preservice teachers in a Foundations of Mathematics for Teachers course stated that 3, 5, 7 were divisors since those were among the factors in the prime decomposition. However sixteen were unable to apply similar reasoning to 2 and 11, noting instead that *M* is an odd number so "2 can't go into it" and reaching for their calculators to divide by 11. Zazkis and Campbell conjecture that the concept of divisibility is usually constructed prior to the concept of indivisibility. They observe that "7 is one of *M*'s prime factors, therefore *M* is divisible by 7" and "11 [is prime but it] is not one of *M*'s prime factors, therefore *M* is not divisible by 11," while sharing a similar linguistic structure, are neither logically nor conceptually equivalent. The latter requires an understanding of the uniqueness of prime decomposition, whereas the former does not. [Cf. "Prime decomposition: Understanding uniqueness," *JMB* 15(2) 1996.]

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WHAT'S TO BE DONE?

While there are no "quick fixes," there are some things that might be done by those within the university. Students, like Ms. Daniels, should not be exempt from courses that would (potentially) help them teach elementary mathematics. Perhaps not surprisingly, neither Ms. Daniel's content courses like calculus, nor her introductory course in mathematical proof, seemed to prepare her to give explanations for the division-of-fractions algorithm, nor did they seem to provide her with the necessary curiosity, perseverance, or self-confidence to attempt to figure such things out for herself. There are many mathematically sound, child-understandable illustrations, explanations, reasoned arguments (i.e., examples of pedagogical content knowledge) that are non-trivial to discover, even given an understanding of the relevant mathematics.

If we sometimes feel about the students in our "math for elementary ed" courses, as a colleague did, that we don't want some of them teaching our (grand)children, then perhaps we should review what's being taught there. Are we challenging preservice teachers to come up with conceptual explanations, models, and applications which are relevant to the grade level they are to teach? What good will it do them if they really understand integral domains and are able to construct satisfactory (abstract) proofs, if they have nothing to offer at the level their students can grasp?