ATTENDING TO PRECISION AND REVISITING DEFINITIONS

Samuel Otten and Christopher Engledowl

The Common Core State Standards for Mathematics [CCSSM] (National Governors Association & Council of Chief State School Officers, 2010) explicitly included definitions twice in their paragraph about attending to precision, the sixth Standard for Mathematical Practice [MP6]. They stated that mathematically proficient students should “try to use clear definitions in discussion with others and in their own reasoning” and by high school they should “have learned to examine claims and make explicit use of definitions” (p. 7). Koestler and colleagues (2013) expanded upon the CCSSM paragraph on MP6, writing about how definitions in mathematics are especially precise because they create the objects being defined rather than merely describe them after the fact. This precision of mathematical definitions makes them powerful tools with regard to justifying properties of mathematical objects or reasoning about relationships between different classes of objects. The power of definition is not realized, however, when we simply provide students with a list of terms and polished definitions because this approach can hinder students’ acquisition of mathematical language (HerbelEisenmann, 2002) and circumvent the potential benefits for conceptual development that come from participation in the various “aspects of definitional practice” (Kobiela & Lehrer, 2015). Instead of being given definitions,

students should be actively involved in the process of defining because it allows them to simultaneously explore the characteristics (and non-characteristics) of a mathematical object and gain insight into the nature of definitions themselves. With respect to MP6, Koestler and colleagues (2013) argued that, not only are definitions precise, but defining is an important part of MP6 because it involves moving from a somewhat vague notion of a mathematical concept to a more refined phrasing that captures the necessary and sufficient conditions without any extraneous ones. In this article, we present examples from a high school mathematics classroom where previously defined terms were opened up for conversation. In this way, the students were afforded new opportunities to engage in the practice of defining in ways they perhaps did not have when they first learned the terms. We point out how this involved MP6 and how allowing room for such interactions can provide students with opportunities to participate and develop more refined mathematical understandings. BACKGROUND Mr. Forrest is a high school mathematics teacher at a rural, Midwestern middle school, who at the time had been teaching mathematics for 10 years. He was involved in de-

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partmental leadership and in professional organizations. His approach to teaching involves students being “the drivers of the mathematics” because they “retain more when they ‘discover’ concepts on their own versus having [the teacher] deliver it to them.” He also strives to incorporate the Standards for Mathematical Practice into his teaching on a regular basis. This article focuses on his blockschedule advanced algebra class consisting of eighteen students in their junior or senior year. Mr. Forrest’s classroom contained individual student desks arranged in rows facing forward but he had the students rearrange themselves to work in groups on a daily basis. Students frequently used TI-84 calculators and were accustomed to coming up to the SMART Board at the front of the room to share their thinking. The excerpts below are from a lesson on the links between the graphical behavior of polynomials and the features of the corresponding polynomial equations. We focus particularly on defining and MP6 during the first segment of the lesson in which Mr. Forrest was reviewing the term polynomial. MAKING A DEFINITION MORE PRECISE To introduce this lesson on polynomial properties and graphical behavior, Mr. Forrest began by asking students about the definition of

polynomial. In the subsequent interaction, Mr. Forrest pressed them to articulate the precise definition. We use pseudonyms for identified students and MS to indicate an unidentified male student and FS an unidentified female student. Mr. Forrest: Polynomial functions. Do you remember what a polynomial is? Because we have a pretty specific definition.

Shannon: An equation with multiple terms? Mr. Forrest: Okay. It has multiple terms. That’s true. There’s one more thing that’s pretty important. It can’t just be anything with multiple terms. MS: They have exponents. Mr. Forrest: Okay. They have exponents. But even more important about the exponents. You’re barking up the right tree here. Amy: [reading from her notes] A single term or set of terms containing variables with whole number equations. Mr. Forrest: That’s not quite it. I don’t think it’s whole number equations.

Amy: I don’t know. That’s what I wrote. I wrote something wrong. Shannon: Coefficients? Mr. Forrest: Nope. Doesn’t have to have whole number coefficients. Dustin: Solutions?

Mr. Forrest: You are really close now.

Clark: Exponents? Mr. Forrest: What? Clark: Whole number exponents? Mr. Forrest: Exponents! Polynomial functions have only whole number exponents on the variables. Clark: You’re welcome. This interaction involved MP6 as they moved from a definition that is too broad (“an equation with multiple terms”) to phrasing that is more precise (“a single term or set of terms containing variables”) but includes a nonsensical condition (“with whole number equations”), then finally to a correct condition and a summary statement from Mr. Forrest (“Polynomial functions have only whole number exponents on the variables”). Throughout we can see Mr. Forrest spurring the increases in precision. For example, he acknowledged that the initial attempt at a definition is not false but is too broad (“It can’t just be anything with multiple terms”) and he indicated to the students when they were getting closer to the target (“You’re barking up the right tree here.” “That’s not quite it.” “You are really close now.”). Although this interaction involved the refinement of a definition as the class worked to recall what they had previously learned, the nature of the MP6 was essentially a progression toward the precise definition with Mr. Forrest serving as the indicator that they were on the right track or had arrived at the destination. Clark’s final playful comment (“You’re welcome”) also highlighted that they were all collectively trying to articulate the precise definition and were happy (relieved?) when they had

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done so. This progression is distinct from what happened next, which we characterize as an exploration of imprecise definitions on the way to a precise definition. ATTENDING TO THE PRECISION OF A DEFINITION Having stated a definition for polynomial function, Mr. Forrest’s very next move was to ask about the definition of whole numbers—also a concept from the students’ past. Mr. Forrest: What are whole numbers? Jesse: Like, number one would be a whole number. Mr. Forrest: Number one is a whole number. That’s true. Specifically, what is the set of whole numbers? Because we learned this multiple years ago. (pause) Because when we say whole numbers, it’s actually a very specific mathematical set, right? Zack: One, two, three, four. Mr. Forrest: OK. Ben: Like zero to infinity, positive. Mr. Forrest: Zero to positive infinity? Ben: Mm hmm. Jesse: Wait, is there other kinds of infinity? Henry: It could be fractions and decimals, and those aren’t [whole numbers].

Mr. Forrest: Alright, so if we say zero to positive infinity, we’re including too much? Is that what you’re saying, Henry? (Henry nods) OK. So be pacific [sic], as the small children would say. (laughs) Dustin: Starting at one and counting up by ones. Mr. Forrest: Starting at one and counting up by ones. Is that the whole numbers?

Amy: It doesn’t have zero? MS: No, zero isn’t [a whole number]. Mr. Forrest: I don’t really like pestering anyone, but I see lots of college algebra books right in front of people. You could find out, right? We see that Mr. Forrest repeatedly asked students to be more precise in their attempts to define whole numbers (“Specifically, what is the set of whole numbers?”, “Is that what you’re saying?”, “be pacific [sic]”). The students moved from a single example (“one”) to a pattern of examples (“one, two, three, four”) to a working definition (“zero to infinity, positive”). These attempts at defining whole numbers are similar to the progression in the previous section on polynomials, but then things become different. In the previous section, the class did not explicitly consider what it would look like to have objects that fit the imprecise definitions—for example, they did not generate expressions with whole number coefficients rather than whole number exponents. Here, on the other hand, the class con-

sidered the implications of the working definition “zero to infinity, positive.” Henry identified some numbers that would fit under this working definition (“fractions and decimals”) but that should not be included in the set of whole numbers. Mr. Forrest revoiced Henry’s ideas and clarified that it meant the working definition is too broad. This then provided an intellectual motivation for the next move toward precision (“be pacific [sic]”). Dustin provided a new working definition (“Starting at one and counting up by ones”) and Amy continued the exploratory efforts by asking about whether zero should be included in the set of whole numbers, because it is not included in Dustin’s working definition. Not all ideas were explored to their full conclusion. Jesse’s wondering about multiple types of infinity and the question about zero (which is a valid and sometimes contentious mathematical debate to have) were not taken up, but such limitations are an unavoidable reality of teaching and learning because of time constraints and necessary prioritization. Mr. Forrest’s move to consult the textbook’s definition of whole number was reasonable given the fact that he wanted to proceed to the heart of the lesson on polynomial graphical behavior. With regard to MP6, however, even in the brief time they did spend on recalling the definition of whole number, it was interesting to see how they explored some implications of the imprecise definitions rather than simply progressing directly through them.

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CONCLUSION

By involving students in the process of defining, they can gain better understanding about the concepts being defined (Kobiela & Lehrer, 2015) and also gain experience engaging in MP6 and seeing the value of mathematical definitions as tools for precise communication and reasoning (Koestler et al., 2013). The excerpts above showed two different ways that defining might occur in a high school classroom. In the first example, the teacher encouraged the students to be more precise until they had generated a correct definition for polynomial functions. In the second example, the teacher and students took brief moments to contemplate imprecise definitions and motivate their next effort toward precision. Both involve MP6 in subtly different ways. The examples above focused on the concepts being defined without explicitly “stepping out” to consider the process of defining or to talk about the characteristics of a good definition. This focus made sense given Mr. Forrest’s goals for the lesson, but in other situations, it might be worthwhile to have these reflective discussions. Additionally, teachers can look for opportunities to engage in MP6 by developing the definition of a term that has not been previously learned by the students, building up the definition through careful consideration of examples and non-examples. In the excerpts above, the students had already been exposed to definitions for polynomial functions and whole numbers, so the process was geared

toward recreating or recalling the definitions. We wonder whether the differences in the two excerpts may be related to the different levels of familiarity the students had with the two concepts. For polynomials, they seemed to have a somewhat vague sense of the concept but not yet a full handle on it. For this reason, perhaps some of the students were relying on recollection of phrases without deep meaning. For whole numbers, on the other hand, we presume that most students have an intuitive sense of what those numbers are but do not know a precise definition of the set. In this case, the process can be focused on putting those intuitive meanings into precise words rather than having to involve developing the meanings and putting them into words simultaneously. In either case, defining can be a rich learning opportunity that goes far beyond simply presenting students with a list of terms and definitions. Defining can also involve recognizing patterns across examples or non-examples of the mathematical object being defined, thus gleaning the defining characteristics, and can lead to the definition being a powerful tool in reasoning-and-proving. In these ways, engaging in the defining practice can lead MP6 to be generative—spurring other mathematical practices (Engledowl & Otten, this issue).

ACKNOWLEDGMENTS

This work was supported by the University of Missouri System Research Board and the University of Missouri Research Council. We thank the teachers and students for allowing us to work with them, and we thank Vickie Spain for her work on the project.

REFERENCES Engledowl, C., & Otten, S. (2015). Attending to precision: A gateway to other practices. Mathematics in Michigan 48(2), 18-22. Herbel-Eisenmann, B. A. (2002). Using student contributions and multiple representations to develop mathematical language. Mathematics Teaching in the Middle School, 8, 100-105. Kobiela, M., & Lehrer, R. (2015). The codevelopment of mathematical concepts and the practice of defining. Journal for Research in Mathematics Education, 46, 423–454. Koestler, C., Felton, M. D., Bieda, K. N., & Otten, S. (2013). Connecting the NCTM Process Standards and the CCSSM Practices. Reston, VA: National Council of Teachers of Mathematics. National Governors Association Center for Best Practices, & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Author.

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Samuel Otten is from the U.P. and earned his B.S. from Grand Valley State University and his Ph. D. from Michigan State. He currently works at the University of Missouri researching the Standards for Mathematical Practice and students’ mathematical discourse. [email protected]

Christopher Engledowl is a graduate student in mathematics education at the University of Missouri. He is interested in students’ reasoning and is currently teaching mathematics methods courses. [email protected]