After my day-to-day interactions with students, one of my favorite things about teaching is talking with other teachers. There is no shortage of amazing teachers who are working hard to make their classes better and improve student learning. Likewise, there are plenty of opportunities to find inspiration in our colleagues’ work, ranging from attending talks at conferences to simply getting coffee with coworkers to talk about how our classes are going.

A few years ago, I realized that the proportion of inspiring ideas that turned into measurable change in my classroom was essentially zero. As I thought more about this, I realized that I was the biggest hurdle to this change. There was a little voice in the back of my head with a constant and emphatic message: No. I can’t do that, and here are fifteen reasons why.

I know I’m not the only one who hears this voice. Of course, the reason we have these thoughts is that they are often true. No two people experience teaching in the same way. We have different personalities, different styles, and allow for organized chaos in different ways. As a community, it is easy for us to despair in the challenges we face in our teaching.

Joan Baez said, “Action is the antidote to despair.” At the end of the day we are all mathematicians and we have been trained in solving problems. To be apathetic in the face of the challenges put before us is antithetical to our training as problem solvers. And teaching, particularly teaching well, should be viewed as a problem that desperately needs to be solved. Like many real-world problems, the problem (“What does it mean to teach well?”) is not clearly defined. The data is messy. There is not one single correct answer.

In the rest of this post, I would like to discuss some methods for moving beyond the little voice that says “no” and changing your teaching without reinventing the proverbial wheel. And, as with many real-world problems, I will not answer the question at hand (“What does it mean to teach well?”) and instead I will address a different question – How do I teach better?

Elena Galaktionova sent us this article shortly before she passed away earlier this year.

Foreword by Cornelius Pillen

Elena Galaktionova received her first introduction to mathematics from her favorite middle school teacher in Minsk, Belarus, her hometown. After she had finished her education at the Belarusian State University she went on to receive a Ph.D. from the University of Massachusetts in Amherst. Her area of research was representation theory. She taught mathematics for many years at the University of South Alabama, after some earlier stints at the Alabama School of Math and Science and the School of Computing at USA. In Mobile, Alabama, she was one of the organizers and teachers of the Mobile Mathematics Circle. The Circle has been going strong for 20 years. Later she recruited local teachers and a middle school principal to participate as a team at an AIM workshop on Math Teacher Circles. Upon return to Mobile she founded the Mobile Math Teachers’ Circle. Twice she gave presentations at the Circle on the Road conferences. Her work with local middle schools and her interests in home schooling were motivated by her love for mathematics. She cared deeply about math education. Sadly, Elena passed away earlier this year after a long battle with cancer.

In all my classes I try to teach reasoning, writing and problem-solving skills. I noticed that if a class is heavy on computations and dense in content, such as Calculus, the result of this effort is barely noticeable if at all. I recall a memorable moment in a multi-variable calculus class. The topic was optimization. My students knew just fine how to use the Lagrange multiplier method given a function and a constraint, thank you much. But it turned out they were helpless in the face of even the simplest application problems. Some of these students were studying Calculus with me for almost three semesters and their grades were good and I tried so hard to teach them what matters in mathematics the most. I remember a chilling realization at the moment, that we — the students and I — wasted three semesters.

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[1] 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.”

The calculus has a very special place in the 20th century’s traditional course of mathematical study. It is a sort of fulcrum: both the summit toward which the whole secondary curriculum strives, and the fundamental prerequisite for a wide swath of collegiate and graduate work, both in mathematics itself and in its applications to the sciences, economics, engineering, etc.^{[1]} At its heart is the notion of the limit, described in 1935 by T. A. A. Broadbent as the critical turning point:

The first encounter with a limit marks the dividing line between the elementary and the advanced parts of a school course. Here we have not a new manipulation of old operations, but a new operation; not a new trick, but a new idea.^{[2]}

Humanity’s own collective understanding of this “new idea” was hard-earned. The great length of the historical journey toward the modern definition in terms of and mirrors the well-known difficulty students have with it. Although it is the foundation of calculus, it is common to push the difficulty of this definition off from a first calculus course onto real analysis. Indeed, mathematicians have been discussing the appropriate place for the full rigors of this definition in the calculus curriculum for over 100 years.^{[3]}

There is also a rich vein in the mathematics education research literature studying students’ engagement with the – definition. Researchers have examined student difficulties coming from its multiple nested quantifiers^{[4]} as well as its great distance from the less formal notions of limit with which students typically enter its study,^{[5]} and have also made an effort to chart the paths they take toward a full understanding.^{[6]}

This blog post is a contribution to this conversation, analyzing in detail three learners’ difficulties with and .^{[7]} If there is a take-home message, it is to respect the profound subtlety of this definition and the complexity of the landscape through which students need to move as they learn to work with it.

If you give calculus students graphs, they are going to draw tangent lines. As instructors we often encourage students to rely on tangent lines so heavily that discussions about rates of change become lessons about sliding lines along graphs, rather than about understanding the relationships that these graphs represent in the context of a given problem.

So how do we help students develop a deeper understanding of these relationships? Let’s consider an exercise that you might give your own students during a lesson on tangent lines and rates of change.

The phrase “mathematically mature” is frequently used by mathematics faculty to describe students who have achieved a certain combination of technical skills, habits of investigation, persistence, and conceptual understanding. This is often used both with a positive connotation (“she is very mathematically mature”) and to describe struggling students (“he just isn’t all that mathematically mature”). While the idea that students proceed through an intellectual and personal maturation over time is important, the phrase “mathematically mature” itself is both vague and imprecise. As a result, the depth of our conversations about student learning and habits is often not what it could be, and as mathematicians we can do better — we are experts at crafting careful definitions!

In this post, I argue that we should replace this colloquial phrase with more precise and careful definitions of mathematical maturity and proficiency. I also provide suggestions for how departments can use these to have more effective and productive discussions about student learning.

By Jo Boaler, Professor of Mathematics Education, Stanford University, and co-founder of youcubed.org

(This is the first of two of our most popular Blog posts that we repeat for the month of July. )

2018 was an important year for the Letchford family – for two related reasons. First it was the year that Lois Letchford published her book: Reversed: A Memoir.^{[1]} In the book she tells the story of her son, Nicholas, who grew up in Australia. In the first years of school Lois was told that Nicholas was learning disabled, that he had a very low IQ, and that he was the “worst child” teachers had met in 20 years. 2018 was also significant because it was the year that Nicholas graduated from Oxford University with a doctorate in applied mathematics.

For this month’s blog post, I offer two more vignettes from my classroom experience. My intention, as in the last column, is to communicate what I think of as the essence of teaching, which is the emotional—not just intellectual—bond between teacher and student.

But first, with the end of the school year 2018-19, we would like to announce several changes in this Blog’s editorial board:

As the Spring term ends, I thought I’d share with readers two vignettes from my teaching career. The intention is for us to remember how much of teaching is the emotional connection between student and teacher. For me, this is the reality of the experience, and is what makes possible the communication of mathematical ideas.

Completing the Square

This first story started when I got a terse note from the high school guidance office about James:

“James has a difficult situation at home. Any leeway you can grant him about deadlines, tests, or quizzes would be greatly appreciated.”

Well: my classroom was run with very few deadlines. Students could re-take quizzes and tests whenever they learned the material, except that I had to report to their parents quarterly about their progress, at which time they got a grade.

By Ivette Chuca, El Paso Community College; Art Duval, Contributing Editor, University of Texas at El Paso; and Kien Lim, University of Texas at El Paso

Every year, at the beginning of the school year, a group of about two dozen mathematics instructors gets together from the University of Texas at El Paso (UTEP) and El Paso Community College (EPCC). For most of a Saturday, we put on a workshop for ourselves about teaching courses for pre-service elementary and middle school teachers. We have no incentive other than a free breakfast and lunch. While we have enjoyed putting together and participating in the workshops, we did not think it was especially noteworthy. But then several outsiders pointed out to us that working across institutional lines like this, between a university and a community college, is not so common. But maybe it should be more common, because we have found our partnership to be valuable to our respective institutions and to our students. Continue reading →

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