The language of mathematics is considered “scientific”— free of opinion, emotion, and personal connection; a language of truth and facts. But the relationship between math and language is far more complicated.
A student’s ability to use “academic” words can be misheard as proof of mathematical knowledge, and the absence of these same words can be misheard as a lack of knowledge. Thus a child raised in the dominant culture is often heard as “smart,” “articulate,” and “mathematical” even though his ability to use the conventional words may mask confusion and uncertainty. A child whose way of speaking differs from that of the “academic” world, meanwhile, is at risk of being unheard. Her mathematical ideas are likely to go unrecognized.
In my classroom I have shaped my practice with the goal of trying to truly hear all of my students’ own voices. As a white teacher, I have to be aware that society has trained me to recognize white voices most easily.
Because I grew up as a working-class female and have developed feminist and class consciousness, I also have experiences that help me to recognize voices outside of the dominant culture. I strain to become aware of language and the powerful role it plays in school success and failure. I try to see and understand the complex relationship between knowledge and language.
What follows are three examples of students’ mathematical discussions from my classroom of first and second graders in a public school in Cambridge, MA. This year, about two-thirds are students of color, mostly African American, from working-class or low-income families. About one-third of the students are from white middle-class families.
In math class, I routinely ask students to develop their own problem-solving strategies and to record their ideas in their own ways. Students explain their thinking and questions as they discuss problems and construct mathematical understanding together. During these discussions, I point out connections between students’ ideas, maintain high expectations for listening to each other, and model how to ask for clarity. In this way, the students learn that their own ideas are always the starting places for problem solving.
On one particular day we had been discussing fractions, and we had worked on a number of problems from real life situations that involved the fractions one-half, one-fourth, and one-third. I then gave the students some problems to work on their own. Jahmai asked me for help on the following problem: “Fabienne had $6.30. She spent one-third of it on a gift for her baby brother. How much money does she have left?”
“What do you know about this problem?” I asked.
“I know Fabienne has $6.30 and she spent one-third of it for a present,” Jahmai said.
“What would one-third of six dollars be?” I asked.
“I don’t get it,” Jahmai said.
“What does one-third mean, one-third of something?” I asked.
“What does one-half mean, one-half of something?”
“It’s when you split it in half so each person gets the same.”
“So, what does one-third mean?” I pressed.
“Oh yeah! It means when there are three people and you split it in half. Like we each get the same.”
“So if you have to figure out one-third of 6 dollars … ?”
“Yeah I know,” said Jahmai. “We would each get two because it would be split up so we each got the same.”
Jahmai used the words “split it in half” to express his understanding of the mathematical idea of dividing something up into any number of equal parts; what was important was not the number of parts but that they are consistently equal. His understanding became clear when he further explained what it would be to “split up” six dollars into thirds.
However, if I had stopped Jahmai when he originally explained his understanding of one-third as “it means when there are three people and you split it in half,” I might have thought that he had no concept of one-third and that he only understood halves. I probably would have corrected him that one-third does not mean “split it in half.”
By continuing to listen to his ideas, though, I discovered that Jahmai knew exactly what he was talking about, and that he did understand what one-third meant. I decided that I did not need to go back to what he had said in his explanation and say, “So you didn’t mean ‘split it in half’, you meant ‘split it equally.’” I decided to allow Jahmai’s meaning to exist in his own words. I knew that in subsequent discussions about fractions, other students would help him to clarify the term.
As the teacher, I had to keep listening and asking questions to try to get at Jahmai’s understanding. An easy mistake is to retell students what we thought they really meant in our own words. I try to always be aware that language can both mask and reveal knowledge.
Another example comes from a discussion between two students over a math problem I had assigned. We had been taking the temperature and recording it on a graph every day, and comparing the different temperatures. The problem was: “The temperature was 28 on Monday, 37 on Tuesday, and 27 on Wednesday. What is the difference between the temperature on Monday and the temperature on Tuesday?”
Joshua: “I don’t really get it.”
Fabienne: “I can help. OK, you know, like how are they different?”
Joshua: “I don’t get it. I mean, I know they’re different. But I don’t get how are they different.”
Fabienne: “OK. Tuesday’s warmer, right?”
Fabienne: “So how much is it. Like how far from 28 to 37.”
Joshua: “Oh you mean like what’s in between?”
Fabienne: “Yeah like to get to 37.”
Joshua: “Oh I get it. Like you mean you go 28 to 37 and how much is it.”
Fabienne: “Yeah, like they’re different, that’s how much they’re different.”
Not only is Fabienne’s explanation not the expected mathematical language of “take 37 and subtract 28 and that’s the answer,” but it shows that her thinking about the problem is much more complex in terms of the real relationships between the temperatures 28 and 37. She is able to express a sophisticated understanding of number relationships in terms of quantity and direction, a more algebraic understanding of subtraction rather than simply the arithmetical idea of “takeaway.”
Joshua, meanwhile, is able to engage with her immediately because they are peers and because there is the expectation in the classroom of listening for understanding. Students know they should take all the time they need to try to say just what they mean in their own way. Thus Joshua and Fabienne expect that their language will be identifiable to each other.
Because of the way that they are talking to each other, Joshua is also able to express his idea of “what’s in between,” a remarkable understanding of the wholeness and complexity of number relationships. He becomes more solid in his own understanding by articulating his ideas and having them recognized by Fabienne. Consequently, they both further their understanding of the problem and the mathematical relationships involved and they expand their ability to express their ideas mathematically.
A final example has to do with teaching my students a word that is recognized as a standard mathematical term and the ways in which they worked on making the word their own.
We had been working on geometry for about six weeks. During these weeks I had asked the students to describe in their own words different shapes they noticed in their physical surroundings, to draw different shapes and to make them out of various materials. After many discussions and sessions where students worked together as a class to create mutual understandings of geometric descriptions, I had taught them some of the standard mathematical names for specific shapes such as “pentagon,” “hexagon,” and “trapezoid.”
I asked them to create and compare different kinds of shapes within the same categories. The day before the one described here, I had asked the students to work in partners to make four different hexagons on a geoboard and then copy and cut each one out on dot paper. During this discussion, the idea of “sameness” of the shapes came up again, as it had in previous discussions. I taught them the word “congruent,” explaining that it is a word that mathematicians use to describe the idea of “sameness” that they had been describing. The students tried out the word in different ways.
The next morning as we were all sitting down for classroom meeting, Manny said, “What me and Eli had for dinner last night is congruent because we both had spaghetti.” Many students responded with excited agreement that that situation of “sameness” — spaghetti dinners — could be a possibility of congruency.
During math class later that morning I asked the pairs of students who had worked together the day before to choose two of the four different hexagons that they had made to share with the class. We gathered in a circle on the rug and I called on pairs of students to put their two hexagons in the middle of the circle and share about them — how they made them, what had been difficult, and what they had noticed in the process. They then took questions and comments from the group.
Manny put his shape out, which was exactly the same as Christopher’s, who had shared before him. From the group came excitement, whispers, “That’s the same.” “That’s like Chris’.” Many hands went up. Manny called on Jelena who had her hand as high up as she could make it go.
Jelena: “That’s … it’s … it’s … ca – ca – gru – int with Christopher’s! Because, because it’s the same … because it’s ca – gru – int … because … they got the same sides and it’s ca – gru – int ’cause … it’s the same.”
Everyone was listening to Jelena. No one interrupted the pauses.
Manny (listening, looking at Jelena with respect): “Are you done?”
Joyce: “I agree with Jelena because it is ca-gru-int. Because it has the same sides and same corners and it’s the same, like the same size. It is ca-gruent.”
Jelena was using the word “congruent” for the first time. She was making her understanding of the word and of the relationship between the two hexagons as she spoke it.
She used the word to get her mathematical idea out, at the same time that she used her mathematical idea to try out and make the word hers. Her pronunciation, her emphasis, her body, the pauses in what she said, made it clear that this was happening. And the attentiveness of the other students made it clear, too. They knew it was important — mathematically and linguistically — the moment she said it. They were all in it with her, together.
In order to participate in discourse — to understand and to be understood — there must be common ground between participants. When teachers, particularly white teachers, fail to understand what students of color, girls and low-income students know, it is not just because there is a lack of common ground. Rather, it is often because the teacher fails to recognize that the ground is not common. This unrecognizing, though usually unconscious, is purposeful because it comes from the ideas that society has ingrained in us about intellectual inferiority based on class, gender, and race. Teachers are taught to teach from the place of these dominant beliefs about “intelligence.”
Too often, teachers do not acknowledge our students’ ground as different and, in schools, unwelcome. We simply expect them to step over onto the school ground. Because they don’t match the place that we have learned to teach from, we refuse to recognize their ground. And then we cannot hear their words or recognize their intellect.
To make a change, our job as teachers is twofold. First, we must expand the idea of what constitutes mathematical language. We must push out the constricting and exclusive walls that enclose the authoritative words as the only real mathematical language. We must learn to recognize the diverse language that our students use to express their mathematical understandings as specifically mathematical language.
Second, we must ensure that students of color, girls, and low-income students — all those to whom the “authoritative” language has not traditionally belonged — have access to the traditional, standard, academic mathematical language. All students need to learn, use, and come to own the dominant academic language. Students must have the experience of learning and using these words in order to see that they came from human ideas and processes of expression. Only then can students demystify the academic language and learn how it came to be known as official and standard in the mathematical world.