Message 45 by Cathy L. Seeley from “Faster Isn’t Smarter: Messages About Math, Teaching, and Learning in the 21st Century, 2nd Edition“, (2015)
A fifth-grade student, Marisa, is working one-on-one with a teacher. Marisa looks at a problem that asks how many buses it will take to hold all of the 295 students in school if a bus holds 25 students. She sees the word all and decides it must be an addition problem, so she adds 295 and 25 to get an answer of 320 buses. She pauses for just a second as she’s telling the teacher her answer, likely thinking that the answer doesn’t make sense. But rather than reconsider, she decides that she has done what she was supposed to, so the answer must be right. (The video of Marisa is located on the “Mathematics Reasoning Inventory” website.)
Expecting mathematics to make sense is a core habit among proficient users of the discipline. When something doesn’t make sense in mathematics, it should be disturbing and cause one to dig deeper.
—Al Cuoco, mathematics teacher and author
Unfortunately, Marisa’s situation is not unique. Often students don’t think about the context as they march toward a solution to a mathematics problem. It’s not uncommon to see students other than Marisa work out a “story problem” and arrive at the conclusion that it will take two people twice as long to do a chore as it takes one person. Or in other situations, with or without a context, students may generate an answer on a calculator that’s off by a factor of 10 or 100 or 1000 and not even question the result. Teachers and nonteachers alike have lamented the state of mathematics learning today when they see such examples of students accepting answers that just don’t make sense.
Some would have us believe that a willingness to accept answers that are unreasonable—even ridiculous—comes from an over-reliance on calculators. But in many cases, technology isn’t involved at all. The reality is that students (and the adults they become) have for years blindly accepted mathematical answers they generated with a pencil and paper, even if clearly unreasonable, as long as they believed they had applied the correct procedure. The instruction to “check your work” often leads people simply to do over again what they first did, rather than to verify that the answer they determined makes sense with respect to the original problem. When Marisa came up with an answer of 320 buses, wouldn’t it have been wonderful if red lights and sirens had gone off in her head alerting her to a problem? If she had said to herself, “Wait a minute! That doesn’t make sense. And math is supposed to make sense!”
The notion of making sense of mathematics has always been considered important. In recent years, sense making has become a well-identified mathematical habit of mind, with students always expected to verify that answers are reasonable. But there’s a deeper, more subtle level of sense making I’d like to advocate—that is, students need to develop a habit of expecting, insisting, and demanding that mathematics make sense, whether it’s an answer to a problem they’re solving, an explanation someone else is presenting about their thinking, or a new mathematical idea they’re learning. When something doesn’t make sense in mathematics—when what they just did or found or heard or read isn’t reasonable—they need to pay attention. Maybe they need to ask a question for clarification. Or maybe they need to reconsider the approach they just tried. Regardless, their cognitive dissonance—the realization that something doesn’t make sense—should drive them to dig deeper until whatever it is does make sense.
The student errors described at the beginning of this message demonstrate the importance of making sense of answers to problems, both those posed in a nonmathematical context and those that may be purely mathematical. Whether strictly computation or a straightforward contextual problem or a more complex situation or modeling problem, students need to constantly ask, “Is what I’m doing making sense Does the answer I just generated make sense?” If they develop a habit of expecting answers to make sense, when they arrive at one that doesn’t, their inner mathematical voice should shout, “Wait a minute. That doesn’t make sense. And math is supposed to make sense!” That alarm should drive them back to adjust what they did until they find an answer that does make sense.
Likewise, in a classroom based on student discussion of mathematical ideas and problems (my favorite kind of classroom), as students learn to listen to and interact constructively with each other, they should also be expecting the mathematics they hear to make sense. When a student listens to an explanation or a comment from a peer, the student should be able to make sense of what the other is saying. If something doesn’t make sense, then the student should think: “Wait a minute. That doesn’t make sense. And math is supposed to make sense!” We should help all students learn to communicate in constructive ways what doesn’t make sense to them and be able to explain why, perhaps emphasizing that a particular thought doesn’t make sense to them. As they learn to interact with others in productive discussions about mathematical ideas and problems, they can help others clarify their thinking, as well as themselves. In the process, both the individuals within a group, and the group itself, grow as mathematical thinkers and communicators.
When students are learning a new concept or procedure, whether that learning takes place within the context of a group activity, individual work, or the direct presentation or explanation by a teacher, students should expect the new idea to build on their existing knowledge and to make sense to them. What they’re learning should follow logically from what they already know about mathematics. If it doesn’t, we want students to say, “Wait a minute. I don’t get it yet. That doesn’t make sense to me. And math is supposed to make sense!” That realization should push students to dig a bit deeper and should give the teacher a signal that the student needs another or a different experience until something does make sense. Sometimes that might be as simple as asking another student to explain the new concept or skill in his or her own words. And sometimes it might call for the teacher to try a different approach other than the one used the first time. We all know that if one student isn’t yet making sense of something, it’s quite possible that others aren’t, either. Investing in students making sense of what they’re learning is always a good use of time.
Students making sense of what they are learning and doing should be a foundational expectation in every mathematics class at every grade level. “Does that make sense?” should not only be the first question the teacher asks a student or a class when work is being shared; it also should be the first question every student asks himself or herself about every answer generated. And it should be the primary question students ask about the work of other students as part of their classroom discourse. Asking this question internally or out loud should become a deeply entrenched habit, arguably the most important mathematical habit of mind our students develop. Classroom norms should reinforce the importance of all students raising this question as a routine part of their learning.
Mathematics does make sense, and when students internalize this understanding, it can help with both their attitudes about and their success in learning mathematics. In their quest for expecting, insisting, even demanding that mathematics make sense, students will find opportunity after opportunity to develop other powerful mathematical habits of mind—persevering to solve all kinds of problems, developing logical arguments, communicating with and about mathematics, representing mathematical ideas in many ways, making generalizations from patterns they notice, and so on.
We have to remember that math and science naturally make sense!
—Zak Champagne, teacher
Developing the expectation that mathematics will make sense does not happen in any single year in school. This expectation needs to be nurtured and reinforced year after year, making teaching for this habit an important priority for articulation and discussion across grade levels and from elementary to middle to high school. Developing the habit of expecting and insisting that mathematics make sense should be part of the work of professional learning communities and should be a thread in all professional development activities in mathematics.
It’s the responsibility of teachers and those who support teachers and students to help students come to the fundamental belief that math really does make sense. By teaching in ways that engage students in deeply thinking about mathematical ideas and problems, by structuring classrooms where students constantly question themselves and each other, our overarching goal should be to help every student demand to have what they are doing, learning, or finding make sense. One of the best models to develop the habit of sense making might be upside-down teaching. In upside-down teaching, rather than present students with procedures and then give them problems where they use those procedures, teachers offer students problems they may not already know how to solve.
Sense making—insisting that mathematics make sense—should be the driving force in everything students do in mathematics. Whether students simply don’t yet grasp the connection between what they are seeing or hearing and what they already know, or whether an answer they just determined simply isn’t reasonable, alarms should go off in their heads telling them, “Wait a minute! That doesn’t make sense. And math is supposed to make sense!” And that realization should drive students to recalculate, reconsider, reevaluate, or rethink, until what they find does make sense.
Message 45 by Cathy L. Seeley from “Faster Isn’t Smarter: Messages About Math, Teaching, and Learning in the 21st Century, 2nd Edition“, (2015)