Reading Apprenticeship Goals in Mathematics
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Reading Apprenticeship Goals in Mathematics
Tim Jones teaches high school algebra and precalculus. He is acutely aware of how literacy interacts with mathematics, and he takes responsibility for helping his students see it as well:
As the person trying to get students to comprehend a math textbook better, I really have to be a specialist in coming up with strategies to help them address those issues, and then encouraging them to think about how they think about reading and get inside their own heads when they read a textbook. How do they have to switch if they’re looking at a history book or a science book? How does that change? And making them aware that when you go to a math textbook, there are some subtle differences and some very large differences in how they have to approach the way they’re reading.
Student goals for building knowledge about the discipline of mathematics identify what some of those dispositions look like to a mathematics apprentice reading and doing math (see Box 8.22).
For a perspective on the discipline of mathematics as practiced by some relatively young mathematicians, Dorothea Jordan and her grade 7 pre-algebra class illustrate a number of disciplinary goals, which are represented in Classroom Close-up 8.15. |
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The majority of Dorothea Jordan’s grade 7 transitional mainstream pre-algebra students scored at the “intervention” level on a grade-level math readiness test. Dorothea is firmly convinced that reading is central to students’ success in math: “There is a big initiative that all eighth graders will have algebra because algebra is the gatekeeper to success in college. And what I’m finding is that the gatekeeper to success in algebra is literacy.”
From classroom observations and interviews with Dorothea, the following sampling of student goals in mathematics can be anchored to students’ experience.
Mathematical Representation: Dorothea worked explicitly on academic literacy, helping her students navigate the many texts they encountered in the class. In addition to textbook word problems, students read lengthy supplemental multistep word problems, instructions, rubrics, and computer text, as well as graphs, charts, numerals, and algebraic symbols.
Talking to the Text, which Dorothy modified to include drawing and visual note taking, became an automatic classroom practice. However, knowing what to pay attention to in a word problem before reading it all the way through can be tricky (see the Problem Identification paragraph that follows).
Mathematical Language: In the first weeks of school, Dorothea frequently drew students’ attention to the language of math, guiding them in translating conversational English into academic math language and urging greater specificity and precision in students’ use of language: “Let’s put that in math language. The number of eggs is unknown.” When her student Chris offered, “Positive is always higher than negative,” Dorothea used the phrase “Positive always has higher value on the number line.” Chris later used parallel language to express a new idea, “Negative always has a lower value.”
Mathematical Reasoning: Many of Dorothea’s students brought with them a view developed from years of too many math worksheets that math is a meaningless, for-school activity. In challenging this view, Dorothea made it a point to bring the real world, and students’ experience of it, into the class-room to support understanding. For example, while working on a multistep word problem involving the diameters of a soccer ball and a Ping-Pong ball, she brought a real soccer ball and Ping-Pong ball into the classroom, musing to students, “Why would I want to have a real soccer ball? So it could be another tool, just looking at it.”
She elaborated on her reasoning during an interview: “Theoretically these students did not need the props to solve the problem. But one of the goals of doing math is mental math and predicting what the answer will be. If you can’t predict what that answer will be, you have no way of monitoring your answer.”
In promoting the reciprocity of concrete and abstract thinking, bringing in real balls conveyed the message that math should make sense. Internalizing this message, one of Dorothea’s students went to the store the night after the comparative diameters lesson. The next morning Sal brought in his research on the diameter of soccer balls. Using mathematical language, Dorothea thanked him for his independent confirmation of their solution.
Problem Identification: Dorothea’s students never read mathematics problems without Talking to the Text to identify the key math words and symbols they need to understand in solving the problem. She learned, however, to modify this routine to fit the structure of word problems, which often must be read all the way through before the actual problem begins to take shape. Rene, for example, had marked the phrase quite a while on a first read-through of a problem, expecting that it would be important, but later noted that it had not been necessary for solving the problem. After a number of similar incidents, Dorothea concluded that in mathematics reading, students need a complete first read-through of a problem before going back to reread, mark the text, and identify the particular problem to be solved.
Pattern Application: Dorothea noted a tendency for students to attribute success in math to luck or cleverness rather than to knowledge or skill. It was in the context of games that she began to chip away at the mystery of math, thinking aloud about her own decision making, and requiring students to explain the reasoning behind their moves: “Was that a good move? Why?” “Now why did you pick that one? How does that help you?” In addition, Dorothea challenged students to become metacognitive about the implicit theories behind their decisions: “See if you can find a general rule.” |
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In a mathematics classroom, students learn about the discipline of mathematics and themselves as readers and users of mathematics by way of the following discipline-specific goals.
CONCEPTUAL CATEGORIES*
I can identify the purpose for and use different areas of math knowledge such as number, algebra, functions, geometry, statistics and probability, and modeling.
MATHEMATICAL REASONING
I can think interchangeably about a math problem in abstract and quantitative terms.
I monitor the reasonableness of the relationship between my abstract and quantitative thinking.
MATHEMATICAL REPRESENTATION
I can read and represent mathematics with words, formulas, and mathematical symbols.
I can read and create diagrams, tables, graphs, and flowcharts for mathematical purposes.
MATHEMATICAL LANGUAGE
I understand the precise nature of mathematical language and use it to communicate exactly.
PROBLEM IDENTIFICATION
I can read and identify “the problem” in a math problem.
PROBLEM SOLVING
I make conjectures about and evaluate alternative approaches to a problem and then monitor the reasonableness of a solution approach as it proceeds.
ACCURACY
I understand that in mathematics there may be alternate approaches to a solution, but only one correct answer. I check that the final solution makes sense and all computation is correct.
PATTERN APPLICATION
I look for mathematical structures, approaches, and patterns that I can apply to the solution of new problems.
MATHEMATICAL IDENTITY
I am aware of my evolving identity as a reader and user of mathematics.
*These conceptual categories are drawn from the Common Core State Standards for Mathematical Practice. |
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From Reading for Understanding, pp 282-284 |
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