We don’t see things as they are. We see things as we are. —Anaïs Nin Imagine your beginning algebra class. Together you and your students engage in making sense of notation, representations, and terms. Your students watch you—and one another—covertly to see what makes an acceptable question, what strategies are valued, what pictures and symbols mean and how they are used, how mathematics is written and discussed, and how to justify a solution. These “hidden regularities . . . become the taken-for-granted ways of interacting” that constitute the culture of doing mathematics in your class (Wood 1998, p. 170). Classroom culture is established through both conversations and practices. Traditionally in mathematics class, we focus primarily on the latter; that is, we show our students what “doing mathematics” looks like and then ask that they try it themselves. In this article, I suggest three mathematical conversations that help bring covert—and often ineffectual—meanings into the light. The process I describe, sometimes called interrogating meaning, allows students to make explicit their assumptions about how, when, and for what purpose a mathematical notation, representation, or term is used (Rosebery et al. 2005). In other words, these conversations can help students recognize the strengths and weaknesses of their own interpretations and give them agency in changing them. Further, they allow us as teachers a window into our students’ thinking. Each mathematical conversation begins with a question for your class to ponder and discuss. I will describe some typical student thinking about each question and suggest some ways to build on students’ ideas. Each topic has been addressed in the classroom with students at a variety of levels and makes a powerful point about mathematical culture. CONVERSATION 1: THE EQUALS SIGN The equals sign, =, signifies that two quantities are the same. It does not mean “write the answer.”During the first week of my algebra classes, I write the problem shown in figure 1 on the board and ask that everyone decide, without talking to one another, what numbers go in the blanks. Then we discuss their ideas. More than ninety percent of upper elementary school students will interpret = as “write the answer to the preceding computation” and will fill in the blanks with 19 and 25 (Falkner, Levi, and Carpenter 1999). At the university level, a majority of students in introductory mathematics courses, such as college algebra, mathematics for elementary school teachers, and liberal arts mathematics, will also respond with 19 and 25 (Szydlik, Kuennen, and Seaman 2009), so it is a good bet that your beginning algebra students harbor this alternative conception about the meaning of the equals sign. This interpretation is not consistent with that of the mathematical community, but it is not objectively incorrect. We could have decided as a mathematical culture that = means “write the answer.” However, we did not, and we had important reasons for defining equality the way that we did. Equality is a fundamental mathematical relationship between quantities signifying that these quantities are exactly the same. Students need to be told all this explicitly and helped to understand that meaning. Carpenter, Franke, and Levi (2003) recommend that elementary school teachers pose number sentences in ways that better reveal limitations of the “write the answer” interpretation. For example, a teacher might write equations in a variety of forms— ___ = 4 + 7 3 + ___ = 12 8 + 4 = 5 + ___ —and ask students what values make the equations true. Those of us who teach algebra might take a similar tact before formally solving an equation. For example, after making the point that equality is a relationship between two numbers, I will write something like 2x = x2 – 3 and ask, “What values of x make this equation true? Can you find any without writing anything down?” Then, when we discuss algebraic moves that will help find solutions, I can reinforce for my students that the point of algebraic manipulation is to give us exactly those moves that allow us to preserve equality. I find a conversation about equality particularly valuable when my students use the equals symbol to mean something like “and then I did this . . .” during the course of performing a series of algebraic steps. For example, a student solving for a semicircular area in which she needs to first square the circle’s radius, then multiply by π and then divide by 2 might write something like this: A = 42 = 16 = 16 = 16/2= 8. In a class in which we have had an equality conversation, the student and I might have this exchange: Teacher: What did you mean here when you wrote 16 = 16? Audrey: I was showing all my steps. First, I squared the 4, and then I multiplied by . Teacher: So in this calculation, = means “and then I did this. . . .” Audrey: [laughs] And then I did that and that. Teacher: I get it. But looking at this, remember that I would think you meant that 16 is exactly the same quantity as 16. And that might even lead me to solve for and think that = 1. Audrey: How would you think that? Oh, I see. You’d divide both sides of 16 = 16 by 16. Teacher: [nods] Algebra is about all the things you can do that keep two sides of an equation balanced, and mathematicians have decided to use the equals sign to show that balance. It is important to write mathematics in a way that is consistent with that meaning. It gives you power in organizing your work, and it allows you to communicate with others who are learning the language of algebra. How could you change what you wrote so that it made sense to me? The focus here is not on the student being wrong. Rather, I acknowledge that the student did have a meaning for the symbol, that her meaning made some sense, but that she will not be able to effectively communicate her thinking in the language of algebra unless she adopts the mathematical culture. CONVERSATION 2: REPRESENTATIONS Representations do not carry meaning. People bring the meaning. I ask each student to take a minute to make up a mathematical meaning for the representation shown in figure 2. Then I solicit a variety of ideas. What follows is a typical start of a discussion. Teacher: What do you see in this picture? [Fifteen seconds or more of wait time] Sara: I see a side view of a three-dimensional house. Teacher: Do you mean the white part? Sara: Uh-huh. You might get the view from the different sides and have to imagine the whole house. Richard: I’ve seen those types of problems before. Iiona: I see that. But I was picturing it [as] just the fraction 5/12. Teacher: Okay. How did you see that? Iiona: Five white squares out of the total of twelve squares. Teacher: Who else saw that fraction? [Several hands are raised; many students see fraction representations by what is not shaded rather than by what is shaded.] Did anyone see the fraction 7/12? [Hands are raised.] Either one is reasonable, right? So I guess when we make pictures of fractions, we should say whether we are looking at the shaded part or the unshaded part. Other ideas? Charlotte: Could it be the fraction 5/7? Iiona: No. That would mean 5 out of 7, and it is 5 out of 12. Charlotte: I mean 5 white and 7 shaded. Teacher: So you are thinking of the ratio 5 white to 7 shaded [writing on board: 5 : 7]. Maybe we could write it like this? Does this seem okay to you, Iiona? [Iiona nods.] Teacher: The big point here might be that this picture has lots of reasonable mathematical interpretations. There is not just one correct way to see it. When we make a representation, we need to talk about what it means for that particular problem or situation. Can anybody think of another possible meaning for this picture? Diego: It could be an area model for a probability problem. Teacher: Ah. Can anyone come up with a problem for which that picture would be a model? I have heard all these responses (depending in part on the current content of the class in which the question was posed) and lots of others too. For example, students have said that the picture shows 12 – 5 = 3 + 4. That it is the number 17 (on a digital clock). That it suggests the expression (3 • 4) – 5. That it is showing that 3/12 + 4/12 = 7/12. That it shows an impossible net for an open box. The idea that representations do not carry meaning is not new. In the 1980s, researchers published empirical studies showing that even “standard” mathematical representations have many viable interpretations (Schipper 1982; Feller 1983; Radatz 1986). For example, when Schipper asked 109 first-grade children to interpret pictorial representations (like that shown in fig. 2) from standard first-grade mathematics textbooks, he found that about a third of the children gave alternative meanings to pictures that had (or were similar to those that had) been used in their classes; another third declined to attempt a representation. Thompson (1994) suggested that without awareness of alternative meanings, teachers may assume that students see what we intend for them to see, and he warns that communication can break down when students see something other than what we intend. Conversation 2 gives me the opportunity to see and validate many student conceptions and to acknowledgethat representations require clarification. It also lets me explicitly tell students that if they do not understand what a picture or symbol represents in a particular context, or if they are seeing something different in a representation, they need to bring this to the attention of the class. We talk about the fact that pictures, diagrams, and symbols can have many reasonable meanings and that it is our job as a class to make sure that we discuss and agree on what representations mean. Having an alternative conception does not imply that the student is wrong or bad at mathematics. Pictures do not carry one correct meaning. Bauersfeld (1995) argued that these types of conversations about alternative conceptions can help students build groundwork for future mathematics. “As soon as we narrow the students’ interpretations of pictures and situations toward an unequivocal ascription of mathematical meaning,” he warns, “we throw away the opportunity for an early and powerful preparation for later problem solving” (p. 146). So not only does this conversation give students opportunities to describe their thinking, but it also allows them to argue why a certain interpretation may be valuable in one context, whereas another interpretation may be valuable in another, and to create scenarios (such as the probability model problem) that may be useful in future mathematical contexts. CONVERSATION 3: MATHEMATICAL LANGUAGE Mathematicians have agreed on precise meanings for words. Pay careful attention to their language. Many students do not appreciate the subtleties of mathematical talk. As part of a research study (Szydlik, Kuennen, and Seaman 2009), we posed the item shown in figure 3 to a large sample of college students in their beginning mathematics class (such as college algebra, mathematics for elementary teachers, or liberal arts mathematics). Try it with your students as a way to begin a conversation about how mathematicians use language. Only approximately half the students in our study responded in a way that is consistent with the interpretation of the mathematical community—that neither (a) nor (b) need be true. Students who chose (b), the most common alternative response, argued that if you are talking about people in a pool, there must be someone in the pool. This led us to question the precision with which students attend to language and, specifically, the meanings that students may be giving to statements containing quantifiers (e.g., for all, there exists), quantifying language (e.g., at least, at most, exactly), and words such as unique, distinct, and, and or. Precise statements are a hallmark of algebra. Algebraic identities are those statements that are true in all cases (e.g., for all x and for all y, (x + y)2 = x2 + 2xy + y2 ). When we solve an equation, we are implicitly thinking, “If there exists a solution, x, then I could do all these moves to find it.” We might tell students that two distinct lines in the plane can intersect at most once or that a cubic equation has at least one real root. When talking about solving an inequality, we may explain that x > 7 or x < –7. But what do students make of this language? First, we need to ask them. Second, we need to share explicitly the mathematical culture regarding the precision of our language. Some researchers have found that revoicing (repeating or rephrasing) student talk can help students clarify ideas, learn mathematical vocabulary, or attend to specific words and their meanings (O’Connor and Michaels 1993). In my algebra classes, I look for opportunities to amplify words by emphasizing their importance when revoicing student talk and by emphasizing them in my own speaking. Revoicing might sound something like this: Aidos: I got x is bigger than 7 and x is less than negative 7. Teacher: [to the class] Aidos says x is bigger than 7 and x is less than negative 7. [short pause] Hmm. Give me a number that x could be. Violet: 10. Teacher: Okay, then 10 has to satisfy Aidos’s statement. Let’s read it with 10 in there. Ten is bigger than 7 and 10 is less than negative 7. [Five seconds of silence] Aidos: I meant that it just has to be one or the other. Violet: So it should be x is bigger than 7 or less than negative 7? Teacher: Yes. Then you are saying that all values of x that are bigger than, greater than, 7 along with all values of x that are less than negative seven make the inequality true. I also share with students, through stories, mathematical culture regarding language. For example, I explain that if a mathematician has six children and you ask whether she has three children (in the context of mathematics—and probably outside it too), she will answer in the affirmative, because if she has six children, then she also has three children. I tell them that my father (a mathematician too) will respond to and-or questions with either yes or no. I learned quite young that if I asked him if he wanted peas or beans at dinner, he would simply say yes. (I am delighted when students adopt this language and start to answer my questions in that manner. Teacher: “True or false? (x + y)2 = x2 + y2?” Class: “Yes.”) Stories like these give us opportunities to share meanings that mathematicians give to language. OPENING GAMBITS The types of conversations described here pay high mathematical dividends for the class time invested. They allow us to hear student thinking about mathematical symbols, representations, and language and share meanings given to these objects by the mathematical community. They provide teachers and students opportunities to lay groundwork for future problem solving and to discuss larger mathematical values and practices. In addition, they specifically address the Common Core Standards for Mathematical Practice regarding attention to precision: “Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equals sign consistently and appropriately” (CCSSI 2010, pp. 7). Further, the conversations encourage contributions from students who may have been reticent to engage because they allow us to validate their different ways of seeing mathematical objects. In other words, these types of discussion can enhance the culture of participation in our classrooms because they allow us to shift the notion that mathematical rules are based on the teacher’s authority to a more inclusive and empowering view in which mathematical understanding is developed by a community of learners. This is not to say that three conversations are sufficient; changing classroom norms is an ongoing project. The practice of interrogating meaning of symbols, representations, and terms must be ongoing if this type of participation is to become the norm. These conversations are meant to serve as openings to begin that transformation. REFERENCES Bauersfeld, Heinrich. 1995. “The Structuring of Structures: Development and Function of Mathematizing as a Social Practice.” In Constructivism in Education, edited by Leslie Steffe and Jerry Gale, pp. 137–58. Hillsdale, NJ: Lawrence Erlbaum Associates. Carpenter, Thomas P., Megan Loef Franke, and Linda Levi. 2003. Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Heinemann: Portsmouth, NH. Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf. Falkner, Karen, Linda Levi, and Thomas Carpenter. 1999. “Children’s Understanding of Equality: A Foundation for Algebra.” Teaching Children Mathematics 6 (4): 232–36. Feller, Gisela. 1983. Diagnosis and Analysis of Mathematics Achievement in Elementary School. Frankfurt am Main, Germany: Peter Lang Verlag. O’Connor, Mary Catherine, and Sarah Michaels. 1993. “Aligning Academic Task and Participation Status through Revoicing: Analysis of a Classroom Discourse Strategy.” Anthropology and Education Quarterly 24 (4): 318–35. Radatz, Henrik. 1986. “Graphical Representations and Understanding of Subject Matter in Teaching Mathematics at the Elementary Level.” In Beitrage zum Mathematikunterricht, pp. 239–42. Hildesheim, Germany: Franzbecker. Rosebery, Ann, Beth Warren, Cynthia Ballenger, and Mark Ogonowsk. 2005. “The Generative Potential of Students’ Everyday Knowledge in Learning Science.” In Understanding Mathematics and Science Matters, edited byThomas Romberg, Thomas Carpenter, and Fae Dremock, pp. 55–80. Mahwah, NJ: Lawrence Erlbaum Associates. Schipper, Wilhelm. 1982. “Selection and Order of Mathematical Content in the Early Grades.” Journal fur Mathematik-Didaktik 2: 91–120. Szydlik, Jennifer E., Eric Kuennen, and Carol E. Seaman. 2009. “Development of an Instrument to Measure Mathematical Sophistication.” In Proceedings for the Twelfth Conference of the MAA’s Special Interest Group on Research in Undergraduate Mathematics Education (SIGMAA on RUME). http://www.rume.org/crume2009/Szydlik_LONG.pdfThompson, Patrick W. 1994. “Concrete Materials and Teaching for Mathematical Understanding.” Arithmetic Teacher 41 (9):556–58. Wood, Terry. 1998. “Alternative Patterns of Communication in Mathematics Classes: Funneling or Focusing?” In Language and Communication in the Mathematics Classroom, edited by Heinz Steinbring, Maria Bartolini Bussi, and Anna Sierpinska, pp. 167–78. Reston, VA: National Council of Teachers of Mathematics.