It’s HOW students solve the problem that counts in Math
By Alex on February 13th, 2012
Is it the journey that counts, even in math class?
We recently caught up with Marilyn Burns, an esteemed math educator whose team at Math Solutions has created a new free assessment tool that tests not whether students get the right answer — but HOW they get the answer. The unique tool, called the Math Reasoning Inventory (MRI), was created to help teachers evaluate a key skill outlined in the Common Core State Standards. With MRI, teachers can dig deep into whether or not students understand the processes of solving problems. See what Marilyn has to say about her new assessment tool MRI, student reasoning skills and their true understanding of mathematics principles.
Why are understanding and reasoning skills so important in math?
By asking a student to explain their reasoning it makes them think about the process of solving the problem, and shows whether or not they understand the logic behind the answer. Reasoning is also used in everyday life from tipping a waiter or waitress at a restaurant to putting the Thanksgiving turkey in the oven for a certain amount of time—being able to reason and understand mathematical problems in everyday context is critical to life. Reasoning also plays a role in a child’s success in algebra and high school math, which is needed for practically every career choice
Asking questions and calling on students to give answers has always been a regular part of classroom teaching. In my experience, especially as a beginning teacher, when students answered questions correctly, I usually accepted their responses with a nod or comment of approval, rarely prodding them to explain their reasoning. When students were incorrect, however, I was more likely to probe further by asking, “Are you sure about that?” or “Why do you think that’s right?” Follow-up prompts like these then became signals to the students that their response was not correct or acceptable.
How do you know if a student truly understands math concepts?
What we learned from our work developing Math Reasoning Inventory (MRI) is that asking students to explain their reasoning, even when they solve problems correctly (or, perhaps even especially when they are right), provides insights into their thinking and understanding in a way that answers alone can not provide. Also, the interview experience itself models for students what’s important about learning mathematics, that while answers are important, reflecting on their thinking and communicating how they reason are integral to their math learning. It’s all about getting the student to apply their knowledge to solve problems now in the classroom and problems they will encounter in the future throughout their lives.
Tell me about MRI, what is it designed to do?
MRI has been designed to help teachers make their classroom instruction more effective. Formative assessments are not meant to replace high stakes summative assessments, like our state tests. They fill a very different role. They help shift the focus from “covering” the curriculum, to a focus on “uncovering” the curriculum for the students.
Teachers rely on homework assignments, quizzes, listening to students in classroom discussions — a variety of ways to judge how students are doing in class. While these are all useful tools, none of them are as reliable or informative as listening to student’s answer questions specifically chosen to reveal their strategies and understandings. And more than answers, MRI also probes how students reason, which gives teachers the kind of deep insights that are needed for effective instruction. MRI makes it easy to find out about what students really understand about math.
What are some common problems you find during “interviews” with students?
We ask students to solve most MRI problems mentally, without using paper and pencil, and we’ve chosen problems that should be accessible to them. Students who aren’t accustomed to explaining how they think, who primarily having been doing math with paper and pencil, often have more difficulty when we probe their thinking. These students may be able to compute successfully with paper and pencil, but seem to rely on these procedures without having access to other ways to reason.
A few years ago, when we were focused on assessing younger students, I had an experience with a third grader, Ellen, that then influenced our work on MRI. When assessing what Ellen understood about subtraction, I asked her, “How much is 100 minus 3?” Ellen thought for a moment and answered correctly, “It’s 97.” Then I asked, “How much is 100 minus 98?” She thought and then said, “I can’t count back that far. Can I use pencil and paper?” I agreed and Ellen set up the subtraction problem, writing 98 under 100, then crossing out to borrow and figure the answer.
What understanding did Ellen lack? She didn’t understand the inverse relationship between addition and subtraction, a key number property that is essential for students, not only with problems like these, but also when they learn to operate with positive and negative integers, and when they learn to solve algebraic equations. It’s a foundational necessity for algebra.
But Ellen was a third grader, and since MRI was focused on assessing incoming sixth graders, we changed the problem to 1000 – 998. We initially included 1000 minus 3 as well, but we eliminated that question since all students we tested answered it easily. When developing MRI, we tested many, many questions. We eliminated questions that all students easily answered correctly, and we eliminated questions that hardly any students answered correctly. We looked for questions that gave us a good statistical spread and that produced a variety of reasoning strategies from students. So this question, 1000 – 998, is now the first question in the Whole Number interview.
How does MRI connect with the CCSS?
MRI asks questions that the Common Core expects all middle school students to answer successfully. MRI responds to the CCSS recommendation for a “balanced combination of procedures and understanding” and cautions that “students who lack understanding of a topic may rely on procedures too heavily.” We’ve all know students who borrow, carry, invert and multiply, and more, yet are unaware when answers are reasonable, have difficulty making estimates, and lack numerical intuition. The focus of MRI is on uncovering students’ reasoning strategies and understandings, and also their misconceptions and gaps — information that teachers need to make their classroom instruction effective.
The Standards for Mathematical Practice describe the expertise that we’re looking to develop in students. They define eight practices that rest on “processes and proficiencies with longstanding importance in mathematics education.” The MRI Interview focus on reasoning directly addresses this section of the Common Core. In a way, reasoning is the core within the core of the Standards for Mathematical Practice.
The Standards for Mathematical Content include “a balanced combination of procedures and understanding.” The questions in MRI draw from three Domains—Numbers & Operations in Base Ten, Numbers & Operations—Fractions, and Operations & Algebraic Thinking. The Common Core cautions that “students who lack understanding of a topic may rely on procedures too heavily.” This caution challenges us as teachers not to judge students’ numerical proficiency solely or primarily on their ability to perform procedures. We’ve all known students who borrow, carry, invert and multiply, and more, yet are unaware when their answers are unreasonable. These students typically have difficulty making estimates, they typically lack numerical intuition, and they often don’t see the sense in mathematics.
Educators — we’d love your thoughts in the comments!