by Paul Goldenberg and Al Cuoco
Content is essential; so are strategies and craft for teaching; but there’s more. It’s often said that “many elementary teachers don’t really know the content; the content they ‘know’ they don’t really understand; often they don’t realize that there is anything to understand.”
However true that characterization of elementary teachers is, we think it’s a distraction. There is no kindergarten teacher anywhere who doesn’t know how to count and add and subtract, which is most of what her children will encounter during the year. And if the teacher isn’t sure of the name of some particular geometric shape, that’s way down in the noise of what will matter for teaching. Lack of mathematical information—even a lack of understanding of why particular algorithms work—is not the biggest roadblock in the earliest grades. The remedy might involve more courses in mathematics, especially mathematics they will teach, but we think that the key issue is not more but different, even for secondary teachers.
As we see it, what hurts elementary mathematics teaching most—and hurts secondary teaching as well—is some of the ways in which teachers know “too much” math without a tempering sense of what the mathematical enterprise is and what not to teach. We will give two examples, one from fourth grade and one from first year algebra, to illustrate what we mean.
Fourth grade. Over a span of years and in several schools, we’ve watched many fourth-grade teachers as they present , which their curriculum uses as a two-minute warm-up puzzle in advance of a unit that introduces the distributive property that students will apply to multi-digit multiplication. Every time we’ve watched, many children blurted out that had to be 8 as soon as they saw the puzzle, pleased to show that they knew the multiplication fact 7 8. The purpose of the puzzle, at this point, was merely to have kids recognize that the “8” could be the sum of two numbers and to have them come up with several possibilities for and . At this introductory moment, even checking that if is 8, then + also gave 56, would be overkill because that’s exactly the work students would next do on their own.
And yet, all of the teachers felt a need not only to point that out right away, before the unit started, but also to dive into vocabulary and other formalities. Referring to the and , several teachers asked questions like “What do you think the word ‘variable’ means?” even though the term isn’t used anywhere in the lesson. Children generally had no idea. Some teachers then defined variable as “a letter, like x, that stands for a number,” even though and aren’t letters! Why?! The teachers recognized something they knew and felt compelled to teach the children the “right way.” One teacher wrote out as a justification for the 8 that the kids had already shouted out. One teacher put up a table to show how values for and “should” be recorded. And after kids had offered a few possibilities for and , that same teacher took extra time to say that 4 + 4 — which no kid had suggested — “would not be right because then the equation would have to be written with two squares or two circles.”
The last statement is wrong, of course, but the big problem, in our view, is not the teacher’s error, but the teacher’s apparent feeling, common to all of the observations we made on this puzzle (and a vast number of other observations of other teaching) that everything the teacher knows about the situation is relevant now.
Teachers need to know what (and when) not to teach.
Of course, good teaching practice does involve looking for learning opportunities, sometimes milking a problem for more than what appears on the surface, but part of the mathematical preparation of teachers at all levels must include ways to decide when not to do that—what is not relevant at a particular time, or is essentially a diversion that will, at best, dilute the focus on an important idea and, at worst, mean nothing at all to the students. “A letter, like x, that stands for a number” was certainly one of those; it wasn’t needed and didn’t clarify anything. The purpose of the puzzle was to set the stage for a key mathematical idea—a property of multiplication that the nine-year-olds would first explore, then apply, and only after that formalize to extend their ability to multiply. The distraction subverts (perverts!) the mathematics. Worse, because it became nonsense, it can convince students that understanding is neither necessary nor, perhaps, even possible for them. By contrast, when students are given time to solidify an idea first, naming the idea becomes useful, helping them talk about it and even indicating that the idea is important enough to warrant a name.
Starting mathematics lessons with vocabulary and notation seems nearly universal, even among teachers who know from their language arts instructional methods that vocabulary is best learned in context. And elementary school (and often secondary) mathematics teachers seem not to distinguish conventions and vocabulary from what can be reasoned out or understood. We’ve seen that showing and asking “What can you say?,” encourages lively thought and participation even though it’s a piece of notation, because its form allows kids to make sensible guesses about it. On the other hand, asking “What do you think variable means?” shuts down logical thought. There’s no context. Kids who know vary might come up with a reasonable thought, relevant or not, but to these fourth graders variable could mean anything. Vocabulary and conventions are needed for clarity and precision of communication, but the mathematics is something else: logic and the inclination to puzzle through a problem and figure it out rather than the disposition to treat each problem as something for which one must first be taught a rule or method. When learning vocabulary and formulas becomes the focus of mathematics education children move away from the skills they need to be mathematicians and they don’t develop confidence in their own mathematical abilities. That is because people can puzzle through mathematics, but what things are called or how they are notated is convention and can’t be “figured out.” Children who proclaim themselves to be “bad at mathematics” are likely not to have seen mathematics as an exercise in logic and reasoning, and have likely not had enough opportunity to see how good they can be at that. Readers of an AMS blog know that memorizing vocabulary and formulas, while it can be useful, has little to do with mathematical aptitude, but many teachers have been prepared to think otherwise and thus emphasize those at the beginning of every lesson, giving students the false and often destructive idea that those are the math.
In our view, the fix for this particular problem with elementary teaching is not for teachers to learn more mathematical content, but to change teachers’ perception of what mathematics is—their sense of how the discipline works—staying mostly within content they already know or once knew.
For example, how many fourth-grade teachers have students do age-appropriate research to find patterns in multiplication facts? Here is a particularly striking pattern that most teachers have never even seen. Presenting to students can be entirely silent—no “explaining.” On a number line, choose a single number like 4, draw two arrows up from it and write 16, then two arrows from its neighbors 3 and 5, and the product 15. Then start the process from one other number (e.g., 3), writing the square 9, draw the neighbor arrows and let students call out their product 8. If students need another example to “get” what you’re doing, give the 8 and start a new pair (e.g., at 6) leaving the numbers to the students. Keep going until kids are bouncing up and down dying to describe the pattern they see.
Then suggest some new research projects for the children to try on their own. For example, what if the outside pair of arrows are drawn from neighbors that are two spaces away from the original (squared) number? Or, what if the inside pair of arrows does not come from a single number (squaring it) but comes from adjacent numbers (e.g., 3 and 4) and the outer pair comes from their nearest outer neighbors (e.g., 2 and 5)? Do the patterns hold with negative numbers? What patterns do we see if the line is numbered with consecutive odds? Consecutive squares? Consecutive Fibonacci numbers? Students get plenty of “fact drill” doing research projects like this, and have opportunities to describe what they see.
For teachers in pre-service preparation, this is one example of what it means to do mathematics within a territory they already know. There are many others. For teachers, this does offer opportunities to develop new mathematical ideas, terms and notation, but if the preservice goal is treated as “more math to know,” rather than how to do mathematics (research, problem posing, puzzling through to find results), it stamps in the very problems we see so often in classrooms. Teacher preparation cannot ignore content, but it cannot be about content; it must be about mathematical ways of thinking, using content as the opportunity to do that thinking. Students come to view mathematics the ways their teachers view it. That, in turn, is influenced by the mathematical experience teachers have in their preparation. Though classroom curricula also influence students’ image of mathematics, teachers are key.
High school. We observed a class on graphing linear equations using the “slope-intercept” method. For readers outside the culture of middle and high school, this means that you transform whatever equation you have into the form y = mx+b, and, from this, you produce the graph. There is, of course, a sensible and simple method for graphing an equation like 2x+3y=9 but on this day, the teacher’s goal was the slope-intercept method.
So, students transformed the equation into y = (–2/3)x + 3. Then a 3-step procedure is used: (a) go up 3 units on the -axis and put a point; (b) from here, go to the right 3 units and down 2, and put a point; (c) connect the two points.
Most kids followed the procedure and produced the correct graph. Almost as an afterthought, the observer asked one student if the point (1, 2.5) was on the graph. The kid looked baffled, plotted it, and said that it looked as if (1, 2.5) was on the graph. When asked if (300, –595) was on the graph, the kid had no idea how to tell—it was off the paper.
We’ve seen this phenomenon in most classes. For many students, y = (–2/3)x + 3 is a kind of code; from it, one obtains three numbers (–2, 3, and 3) and uses them to produce a picture. Completely missing was the idea of determining if a point is on a graph by testing to see if its coordinates satisfy the graph’s equation. Assessments didn’t detect this deficit because, given an equation, students could transform it to slope-intercept form and produce a correct graph. The goal was about procedure, so the gaping hole in students’ understanding remained hidden.
This example might seem just plain weird to many readers, but this kind of thing happens often in secondary classrooms. There’s the “box method” for setting up equations to model word problems, a different box method for factoring quadratics, the “switch x and y and solve for y” method for inverting functions, and a host of other special purpose methods and terminology that have no existence or purpose outside of school.
Note the parallels to the fourth-grade example. In both, the teaching emphasis was on form, one particular way of writing and doing the problem, not on what the problem meant, which the fourth-graders naturally and instantly gravitated to and which the high-schoolers could have, too.
What can we learn from this? Part of teachers’ mathematical preparation must include an understanding of fundamental results and methods—content specified in state standards. Missing, though, is the aspect of mathematics that involves research, play, experimentation, sense making, and reasoning. Mathematics is not about how much you know but about how much you can figure out with what you know.
The problem these stories illustrate is not just what’s missing, but what’s there––a view of mathematics that most mathematics professionals would not recognize. Wu has written about “textbook school mathematics” as a dialect of the discipline that lives in precollege curricula. Wu’s main criticism is lack of precision, sloppy (or missing) definitions, absence of logical sequencing, and missing distinctions between assumptions (again, ill-formulated) and results that follow from those assumptions. But how best to mend those flaws? In many classrooms that attempt to remediate these deficiencies, the current practice is to put instruction in vocabulary and memorizing forms and formulas first, to teach without first (or perhaps ever) allowing students to build the mathematical sense of the underlying logic. This practice has failed. Wu calls for reducing “teachers’ content knowledge deficit,” remaining “consistent with the fundamental principles of mathematics (FPM).” We would concur, but his FPM seems easy to misread, allowing undergraduate instructors to conclude that it supports what they’ve always done. Wu’s FPM starts with “every concept has a definition,” which is not a claim that teaching must start that way. But it is easy to interpret as such.
For us, the classroom stories above illustrate something deeper and more fundamental than the “flatness” that raises convention to the same level of importance as matters of mathematical substance. And they are only partly about deficits in content knowledge. What they illustrate is a lacks of the perspective that learning mathematics means developing a collection of practices that help you figure out what to do when you don’t know what to do—developing the habits of mind that underlie flexible proficiency in the discipline. These classroom examples treat mathematics as a collection of special-purpose methods that allow one to perform specific tasks that are the calisthenics––the finger exercises––of mathematics. Practice is valuable for mastery in any field, but exercise as an end in itself produces muscle-bound results that can impede performance. Knowing how to transform an equation to some canonical form is an important skill, developed best through orchestrated exercise. But knowing when to use a particular form is much closer to what mathematics is about. More generally, it’s the doing of mathematics that gives people an understanding of the discipline. Learning mathematical facts and methods is absolutely essential but, by itself, builds a view of the subject that emphasizes getting to a particular form, like , rather than understanding the connection between an equation and its graph. By itself, it elevates what you know over what you can figure out.
Is it a stretch to trace the roots of such stories back to teacher preparation? We don’t think so. Yes, other forces are at play––curricula, pressures from high-stakes exams, oppressive working conditions, school lore. But a mathematical preparation that focuses on the doing as well as the learning of mathematics would give prospective teachers some tools to overcome the schoolish nonsense common in commercial curricula, to prepare students for state tests while immersing them in real mathematics, and to downplay the clutter in district syllabi so that there’s time to concentrate on what’s really important. That doesn’t mean ignoring the district syllabi—often a teacher can’t. Instead, one might seek a mathematical context, topic or activity of genuine intellectual worth as a venue for presenting the lightweight clutter. Teacher educators could look seriously at school curricula, think hard about how to prepare teachers to find the mathematics within or behind the school-only terms or methods such as “the box method” for whatever, or idiosyncratic or curriculum-specific terms like “number buddies,” or terms like “friendly numbers” that have a mathematical definition but appear in school with completely unrelated meanings. Children and teachers will hear these in school, and they may sometimes even be useful in school, but they are school-only, and will never be used outside of school. Educators could help teachers learn how to craft age-appropriate research activities that respect time constraints and content requirements but help kids experience the doing of mathematics. One way is by giving prospective teachers such experiences of doing mathematics.
Teachers know that what they value is communicated to their students. When teachers come to understand and value the heart of mathematics, they communicate this focus to students even when a particular day’s lesson must be about “what you have to know for the test.”
 E.g., Wu, Hung-Hsi. 2015. Textbook School Mathematics and the preparation of mathematics teachers. https://math.berkeley.edu/~wu/Stony_Brook_2014.pdf Retrieved September 15, 2019.
I find the first example of fourth-grade teaching nice and convincing. Learning what to leave out is important in teaching, but it strongly relies on the level of the learner. For accomplished learners, throwing in additional albeit peripheral terms or concepts (correctly defined!) might be fun and broaden the horizon. But don’t try to do it when the the topic is barely introduced.
I found the number line example and discussion, however, meaningless and lacking purpose. There are many things one can do on a number line, and all of them will give some kind of patterns. Are those patterns meaningful for instruction? Yes, if they can be analyzed and “explained.” Not really, if they serve just to entertain. This example could be the former for students in late middle school to illustrate formulae such as (a+b)(a-b), but is the latter for fourth grade students in here.
The high school example of graphing, if taught the way described, is indeed deficient. Yet is graphing linear equations commonly taught in such mechanical and mindless way? I doubt it. After all, asking whether a point lies on a line is just another sub-category of teaching linear equations similar to graphing a line using a slope-intercept formula, or deriving the formula of a line crossing two given points. Any half-decent textbook addresses all those sub-categories, and testing for all of them–even in a simple multiple-choice format– is pretty easy and routine. Seems to me the authors here are generalizing based on observing a badly taught class.
Finally, I do agree that strong focus on terminology and formalism, particularly in early grades, if counter-productive. I also agree that all too often teachers incorrectly feel they are not sufficiently “mathematical” if they do not enforce them. Using common–but precise–jargon-free language in early grades would probably go a long way to curing children from disliking mathematics, and it would be wonderful if schools of education paid attention to it. Unfortunately, it seems few ed-school professors seem actually know, let alone teach, the use of precise but jargon-free mathematical language.
My apologies to all, and thanks to the several people who commented that some items—equations and numbers, all items that had originally been expressed using a mathematical formatter—were missing. Thanks also to AMS that helped me re-edit when the draft I managed to get my hands on was even worse than what was live. I’ve re-checked and I now think this is complete.