Thinking about my unit on linear equations, there were lots of successes, but also lots of students who are still struggling. Unlike some other units, I see some consistent patterns of thinking in students who have struggled with linear equations.
I think I can summarize these patterns best in that students want problems to be simpler than they are. They want a problem to just be “multiplication” or “addition”; they want to look at it, decide quickly what tool to use, solve, and not look back.
This is antithetical to the Common Core’s “Attend to Precision”. These students consistently get questions wrong when, for instance, they see a graph like this:
and they rush through the problem too quickly to pay attention to the different axes. Or, they are given an equation for the height of a plant in inches, and are asked how long it will be until it is 3 feet tall. They are too busy scrambling to find an answer to notice that what they found doesn’t make any sense. Or they’re asked for the initial value of a table
And seize at y = 25, without bothering to think through what it means.
I’m reminded of two ideas I want to bring into my next unit. First, John Holt, in How Children Fail, quotes Scientific American, saying:
The creative scientist analyzes a problem slowly and carefully, then proceeds rapidly with a solution. The less creative man is apt to flounder in disorganized attempts to get a quick answer.
This is exactly the problem-solving mindset that my students need. I’m not sure I have the tools to teach it, but one idea came to mind — Dan Meyer shared this resource from Oakland Unified School District, outlining their best practices for math teaching. One of those practices is what they call “Three Reads”.
The entire document is worth reading, but to snip a summary of Three Reads:
3-Read is a mathematics and language comprehension strategy designed to delay the rush to an answer, deepen student understanding of both the situation and the mathematics, and help students make sense of a problem before setting out to solve it. The strategy consists of reading the stem of a problem (the problem without a question) three times aloud, in close proximity, while establishing a specific purpose for each read: 1) comprehending the text; 2) comprehending the mathematics; and 3) eliciting mathematical questions based on the information provided.
Too often, students disengage from math problems, and simply take the numbers and do something with them (add, subtract, multiply or divide). 3-Reads is designed to engage them in making sense of the problem first, and then drawing connections between the situation and the quantities presented. By asking students to come up with mathematical questions on their own, 3-Reads focuses their attention on the context and the mathematical structures, and helps to ensure that students understand both the explicit and the implicit information and quantities presented, setting them up for meaningful productive struggle with a math problem. It delays their need for an immediate answer, and helps students get to the mathematics of a lesson or a unit.
I’m excited to give this a try. It’s a valuable strategy both for my English Language Learners — and any other students who struggle with language in math problems. And the practice of slowing students down, and giving students opportunities to reason about mathematical relationships before solving. For me, this is what “Attend to Precision” means. It doesn’t mean write your units in your answer, or write the y-intercept as (0,4) instead of 4. It means analyzing a problem slowly and deliberately, taking note of all the information, and then proceeding with a solution.