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?

In 2012 I saw David Pengelley give a wonderful talk in which he outlined a rather intricate system of assigning and grading calculus problems of varying difficulties over several sections of the textbook, all of which were meant to be done by the students before class and accompanied by an email to the professor reporting on the section of the textbook they had just read. This happened for every class meeting. The mental yoga of keeping track of so much material at once made my head spin, and I did not (and still do not) have the confidence to implement something on such a grand scale. (A summary of David’s method can be found here: https://www.ams.org/publications/journals/notices/201708/rnoti-p903.pdf)

But the ideas in the talk were inspiring. I left the talk telling myself, “I should create opportunities for my students work on math more regularly.” I started giving mini-assignments in all of my classes on a daily basis. The mini-assignments consist of about five relatively simple problems related to the material covered in a given class, which are due at the beginning of the next class.

I can mark the papers and record grades in about 15 minutes per assignment (I do this by hand – if you use WebWork it takes no time at all). It helps make sure the students are engaging with the class material throughout the week instead of just the night before a bigger homework set is due. It also lets me focus my grading efforts on problems with more substance that better measure what the students are learning. This works well for my teaching philosophy and helps make the students stay on top of the course material.

Our teaching experiences are shaped by a large number of variables, including student preparation, class size, resources, research expectations, and other demands on our time. Because of this, the things that my colleagues who work at ivy league schools are able to do in the classroom often are not feasible for me, just as the things that have worked well for me in a class of 20 students might not scale well to a class of 300 students. However, rather than finding all the ways that a technique cannot be applied to your classes, look for ways to transform that solution into something that will work for you.

When I dissect the “no, because…” statements that persistently come to my mind, I find two main themes. The first main issue is one of practicality – what works for person X may not work for person Y – and we have already discussed this. A second main theme, whether we like to admit it or not, can be boiled down to an issue of discomfort or fear.

Change is hard. For most of us, our educational lives consisted largely of lectures to be observed and internalized. This is comfortable because it is what we know best. It is a known quantity in which we hold significant control. Letting go of some of that control is scary. What if it doesn’t work? What if the students don’t like it? What if I don’t do a good job? These are all legitimate concerns, and we can work through those concerns once we are honest with ourselves about their root causes.

Change does not happen overnight. Maybe you are interested in having a more active classroom or having a flipped classroom where students do more work outside of class, but you don’t want to give up complete control to the potential chaos that might come from this. Maybe you don’t have the time or resources to completely re-invent your differential equations course because you’re on sixteen committees and teach three classes per semester. I get it.

You don’t need to do it all at once, and you don’t need to do it all by yourself. Ask colleagues to lend you worksheets or materials. Find activities that help you move towards this goal and try them out a couple times over the course of a term. After a few years of doing this, you will have developed more resources and gained more confidence in this approach, homing in on techniques that work for your personality and style while also better serving your students.

New teaching methods aren’t going to be perfect the first time. You’ll probably mis-judge the difficulty of some tasks. Students might get frustrated. You might get frustrated. But it also won’t be a disaster. Students will still learn. So will you. It will be better next time.

Be prepared for the eventuality that a lesson won’t go as planned. It is hard to know how long an activity will take, but you can bet that it will usually take longer than you think. Don’t be afraid to change an activity in the middle of class if it is taking longer than you expected or the students aren’t getting it. You can always change your course schedule to adapt to this change. Besides, if you try five new lessons in a semester and two of them don’t go as well as you had hoped, that still accounts for a relatively small proportion of the overall class.

At the end of a course where you’re trying new things, reflect honestly about the successes and struggles you faced. Were there common themes in the struggles you faced? How can you fix them next time? Make a list of three things you’d do differently next time. Don’t be too hard on yourself.

There’s a rule in improv comedy called the “yes, and…” principle. If you’re in the middle of an improv comedy skit, you can only react to the material you are given from your collaborators. You may have been ready to tell a very good joke about Care Bears, but now someone has decided that there’s a grizzly bear running around the stage and you need to act on that instead. You can’t stop the skit and ask for a retake, so instead you have to accept the reality of the grizzly bear and add your own brand of humor to it.

The same principle applies to teaching. We can all become better teachers by finding inspiration in others. This takes work, and it can be scary to take a risk and try something new in the classroom. In many cases, we fail to apply the lessons our peers have learned because we feel their experiences do not directly translate to our own. Next time you go to a talk about teaching, I challenge you to move beyond the naysaying gremlins in your head. View the reasons to say no as equations that bound the parameter space of your problem. Say yes to new ideas and apply them in your classroom in a way that works for you. Over time, these small changes can add up to more effective teaching.

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

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.

A very different experience comes from another course. At our university it is called “Foundations of mathematics”. Unlike other math classes it does not have a lot of content. A bit of logic, set theory, relations, maybe some number theory. It is the first class where students are learning to write proofs. This is a writing-intense class. There are essentially no calculations. I collect the homework every class period and grade the same way one would grade an essay. My first requirement is writing in grammatically correct meaningful English sentences. This is not an easy task for most. A lot of students by the time they start this class learned to perceive math as number and symbol manipulations. At the beginning of the course I often see in students’ work words that are strung together in rather random fashion. We go together over some of the responses asking questions like : “Is this an English sentence? What is the meaning? Are all the terms defined? How could it be misinterpreted?”

By the end of the semester I observe a turn-around: there is a palpable effort from even the weakest students to put their ideas into words. The change is most noticeable in weak students. The struggle for finding the right words and writing in grammatically correct sentences may be still there. While they did not suddenly became great at math, their mental activity and learning efforts are much more productive, since they are consciously directed towards comprehension and expressing their ideas verbally with a degree of precision.

I wondered if my students noticed this change themselves; that was until I was approached at the end of the semester by two of my “Foundations” students who emphatically told me how this course entirely changed the way they view and approach math. This is reflected in their grades in other math classes as well. For example, one of my Calculus II students was taking “Foundations” concurrently. Her grade in Calculus II changed from a D at the start of the semester to a B towards the middle. Most notably, she enthusiastically confirmed and told other students how much taking “Foundations” helps with Calculus II, despite having no content in common.

What is most interesting to me is the quantum character of this change and that it was especially noticeable in weaker students.

Young children come to school as a blank slate. Yet they have the innate ability for reasoning, they have curiosity, they are eager to play and explore. Over the years their teachers influence their perception of what math is about. Two of the possibilities are:

- math is a manipulation of numbers and symbols according to a predetermined set of rules;
- math is communicated through meaningful statements.

The students who do not do well in mathematics typically view math as a manipulation of symbols. The “making sense” switch changes this so the students begin to read and communicate mathematics as meaningful, logically connected statements.

To summarize, here is what I observed:

- Both exclusively formal processing of math tasks and making sense of math tasks are learned, eventually habitual, behaviors. Either one becomes a mental process which is practiced and reinforced in every math class.
- Effective learning of mathematics does not happen until mathematical communication is perceived as meaningful statements.
- Students who view math as a formal manipulation of numbers or symbols will habitually direct their effort and mental energy toward this in a math class, unless they are given problems which naturally invite reasoning and stay away from using formulae and rules. In a class with a computational component, such as pre-calculus or calculus, even if a teacher tries to teach reasoning and making sense, it has relatively little consequence: under stress, such as homework due the next day or a test, such students revert to their habits. Some of them spend a fair amount of time studying and reinforcing these habits, often getting frustrated because of the little return for their efforts.
- A dedicated computation-free and writing-intensive class which stays away from problems that may suggest formal manipulation can turn on the “making sense” switch. Students start to perceive mathematics as meaningful statements. They look for logical connections between the statements. Their verbal skills are productively challenged.

The important qualities for such class, assuming the main purpose is to turn on the “making sense” switch:

- The class should be writing intensive.
- The tasks are such that students can rely on their existing reasoning skills, common sense, intuition. They should not be too abstract. For example, it is easier to find appropriate problems in logic, set theory, elementary geometry or combinatorics than in abstract algebra. This allows students to scrutinize math statements using their own “sensometer” and keep working with them until they can make sense of them.
- The course should be light on content and big on thought, allowing sufficient time to think and write about problems.
- The class should not include tasks which could tempt students into formal manipulation.
- Feedback on writing is continuously provided by the teacher: students’ attention is brought to details of their writing, the meaning of what is written, and how the writing could be improved.
- The time required for the “switch” to turn on in such class is less than a semester. This is my experience with undergraduates.
- This works even if this writing intensive class is taken in parallel with other, computationally intensive math classes.
- Once the switch is turned on it stays on in other classes, even in those with computational components, as long as teachers pay attention to making sense and reasoning.
- A practical aside on grading: a class where non-routine and sometimes difficult problems are part of homework presents certain challenges for grading. I told my students that if they could not solve the problem, they should write down their attempts, for example, how they used problem-solving strategies discussed in class, such as looking at related simpler problems or generating examples and trying to find a pattern and showing why it did not work. Adequate effort and quality writing would earn almost full credit. Of course, it is important to also include easier problems which are within reach for nearly everyone. If I did not have sufficient time to grade the full homework, I selectively graded 2 or 3 problems.

Unfortunately, the “Foundations” class is a sophomore level for math and math education majors. In fact, no prerequisites are needed for it. So we started to encourage students to take this class as early as possible, when noticing that it helps them in other math classes. There is no reason why a class with similar characteristics and goals is not taught to seventh graders. It would improve their learning of mathematics for years to come. As an example, in Russia, the class that perfectly fits the bill is Geometry class. Systematic study of Geometry starts in 7th grade and continues through 11th grade. The Geometry class meets 2 or 3 times per week. All statements and theorems are proven based on what is already known. Thus, Geometry is presented as a unified theory and not a random collection of facts. The students are expected to state definitions and prove theorems and they solve problems involving proofs and geometric constructions. Of course, there are Russian students who struggle with writing proofs and deriving formulae. But they are used to the concept of intrinsic reasoning and they know that they are expected to articulate it. In the U.S., ask a class of either seventh graders or freshman Calculus students why a particular fact or formula is true and the answer invariably will be “Because our teacher told us so ” or “Because it says so in the book”.

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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 2*x*+3*y*=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.”

[1] 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.

]]>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.

Many readers will be familiar with the long struggle to find a rigorous underpinning for the calculus of Newton and Leibniz, leading to the modern definition of the limit in terms of and . In this section, I excerpt a few episodes, which will become important in the later discussion of student thought. Readers already familiar with the history of the subject are welcome to skim or skip this section.

While Newton and Leibniz published their foundational work on (what we now call) derivatives and integrals in the late 17th century, they based these ideas not on the modern limit, but on notions that look hand-wavy in retrospect.^{[8]} To Leibniz, the derivative, for example, was a ratio of “infinitesimal” quantities — smaller than finite quantities, but not zero. To Newton, it was an “ultimate ratio”, the ratio approached by a pair of quantities as they both disappear. Both authors would calculate the derivative of via a now-familiar manipulation: augment by a small amount ; correspondingly, augments to . The ratio of the change in to the change in is , or . At this point, they would differ in their explanation of why you can ignore all the terms involving in this last expression: for Leibniz, it is because they are infinitesimal, and for Newton, it is because they all vanish when the augmentation of is allowed to vanish.

A famous critique of both of these lines of reasoning was leveled in 1734 by the British philosopher and theologian Bishop George Berkeley, arguing that since to form the ratio of to in the first place, it was necessary to assume is nonzero, it is strictly inconsistent to then decide to ignore it.

Hitherto I have supposed that flows, that hath a real Increment, that is something. And I have proceeded all along on that Supposition, without which I should not have been able to have made so much as one single Step. From that Supposition it is that I get at the Increment of , that I am able to compare it with the Increment of , and that I find the Proportion between the two Increments. I now beg leave to make a new Supposition contrary to the first, i.e., I will suppose that there is no Increment of , or that is nothing; which second Supposition destroys my first, and is inconsistent with it, and therefore with every thing that supposeth it. I do nevertheless beg leave to retain , which is an Expression obtained in virtue of my first Supposition, which necessarily presupposeth such Supposition, and which could not be obtained without it: All which seems a most inconsistent way of arguing…

It was a long journey from the state of the art in the early 18th century, to which Berkeley was responding, to the modern reformulation of calculus on the basis of the – limit. The process took well over a century. I will summarize this story by quoting somewhat telegraphically from William Dunham’s book *The Calculus Gallery*,^{[9]} from which I first learned it.

Berkeley penned the now famous question:

… They are neither finite quantities nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

… Over the next decades a number of mathematicians tried to shore up the shaky underpinnings… pp. 71-72

… Cauchy’s “limit-avoidance” definition made no mention whatever of attaining the limit, just of getting and staying close to it. For him, there

wereno departed quantities, and Berkeley’s ghosts disappeared… p. 78… If his statement seems peculiar, his proof began with a now-familiar ring, for Cauchy introduced two “very small numbers” and … p. 83

… We recall that Cauchy built his calculus upon limits, which he defined in these words:

When the values successively attributed to a variable approach indefinitely to a fixed value, in a manner so as to end by differing from it by as little as one wishes, this last is called the limit of all the others.

To us, aspects of this statement, for instance, the motion implied by the term “approach,” seem less than satisfactory. Is something actually moving? If so, must we consider concepts of time and space before talking of limits? And what does it mean for the process to “end”? The whole business needed one last revision.

Contrast Cauchy’s words with the polished definition from the Weierstrassians:

if and only if, for every , there exists a so that, if , then .

Here nothing is in motion, and time is irrelevant. This is a static rather than dynamic definition and an arithmetic rather than a geometric one. At its core, it is nothing but a statement about inequalities. pp. 129-130

The Weierstrassian definition (i.e., the modern one!) allows the manipulation to which Berkeley objected to be carried to completion without ever asking to be zero.

The core of this blog post is a discussion of three learners’ encounters with the – limit, seeking to illuminate some of the subtle challenges that can arise. I begin with my own story.

I “did well” in my college real analysis class, by which I mean that my instructor (a well-known analyst at a major research university) concluded on the basis of my written output that I had mastered the material. However, I walked away from the course with a subtle but important misunderstanding of the – definition that was not visible in my written work and so went entirely undetected by my instructor, and, for many years, by myself as well.

From my previous experience with calculus, I had concluded that you can often identify the value toward which a function is headed, even if the function is undefined “when you get there.” To the extent I had a definition of *limit*, this was it: *the value toward which the function is headed*.

When I studied real analysis as an undergraduate, I found the class easy, including the – work. I mean, if is where is headed as , then sure, for any -neighborhood around , there is going to be a -neighborhood around that puts inside the -neighborhood. But I related to the notion that is headed toward as *conceptually prior to* this – game. The latter seemed like fancy window-dressing to me, possibly useful post-hoc if you need an error estimate. I did not understand that it was a definition — that it was supposed to be *constitutive* of the meaning of limit. So, I completely missed the point! But I want to stress that you would not have known this from my written work, and of course, I didn’t know it either.

I went on to become a high school math teacher. In the years that followed, I did detect certain limitations in my understanding of limits. For example, I noticed that I didn’t have adequate tools for thinking about if and when the order of two different limit processes could be safely interchanged. But it did not cross my mind that the place to start in thinking clearly about this was a tool I had already been given.

After a few years, I began teaching an AP Calculus course. About 3/4 of the way through my first time teaching it, my student Harry^{[10]} said to me after class, “You know this whole class is based on a paradox, right?” He proceeded to give me what I now recognize as essentially Bishop Berkeley’s critique. At the time, it did not occur to me to reach for epsilon and delta. Instead, I responded like an 18th century mathematician, trying to convince him that the terminus of an unending process is something it’s meaningful to talk about. I hadn’t really understood what the problem was. Of course, Harry left unsatisfied.

The pieces finally came together for me the next year, when I read Dunham’s *Calculus Gallery*, quoted above. I remember the shift in my understanding: *ooooohhhhhh.* The ‘s and ‘s are not an addendum to, or a gussying-up of, the idea of identifying where an unending process is headed. They are *replacing* this idea! It was a revelation to reread the definition from this new point of view. *Calculus does not need the infinitesimal!* I immediately wished I had a do-over with Harry, whose dissatisfaction I hadn’t comprehended enough to be able to speak to.

I concluded from this that a complete understanding of the – definition includes an understanding of what it’s *for*.

Having come to this conclusion, in my own teaching of real analysis I’ve made a great effort to make clear the problem that and are responding to. In one course, I began with a somewhat in-depth study of Berkeley’s critique of the 18th century understanding of the calculus, in order to then be able to offer and as an answer to that critique.

In doing this, I ran into a new challenge. To illustrate, I’ll focus on the experience of a student named Ty. Ty arrived in my course having already developed a fairly strong form of a more intuitive, 17th-18th century understanding of the limit; essentially the Newtonian one, much like the understanding that had carried me myself through all my calculus coursework. He quickly made sense of Berkeley’s objection, so he was able to see that this understanding was not mathematically satisfactory. I was selling the – definition as a more satisfactory substitute. However, Ty objected that important aspects of his understanding of the limit (what Tall and Vinner called his *concept image*^{[11]}) were not captured by this new definition. In particular, what had happened to the notion that the limit was something toward which the function was, or even *should have been*, headed? The – definition of studiously avoids the point “at which the limit is taken,” even speculatively. To Ty, it was the – definition that was, pun intended, missing the point.

Of course, this studious avoidance is precisely how the Weierstrassian definition gets around Berkeley’s objection. The Newtonian “ultimate ratio” and the Leibnizian “infinitesimal” both ask us to imagine something counterfactual, or at least pretty wonky. This is exactly what made them hard for Berkeley to swallow, and as I learned from Dunham’s book, the great virtue of and $\delta$ is that they give us a way to uniquely identify the limit that does not ask us to engage in such a trippy flight of fancy that may or may not look sane in the light of day.

But, at the same time, *something is lost*.^{[12]} What I learned from Ty is that this loss is pedagogically important to acknowledge.^{[13]}

Another subtle difficulty in working with the – definition is revealed when you use it to try to prove something. I think what I am about to describe is a general difficulty students encounter in learning the methods and conventions of proof-writing, but I speculate that it may be particularly acute with respect to the present topic. Consider this (utterly standard) proof that if are functions of $x$ such that and , and , then :

Let be given. Because , there exists such that implies . Similarly, because , there exists such that implies .

Take .

Then for values of satisfying , it follows from the triangle inequality and the definition of that

.

Since was arbitrary, we can conclude that .

Here is a surprisingly rich question: is the in the proof one number, or many numbers?

On one way of looking at it, of course it is only one number: is fixed at the outset of the proof. Indeed, if were allowed to be more than one thing, equations like $\epsilon / 2 + \epsilon / 2 = \epsilon$ would be meaningless. More subtly, we usually speak about as a single fixed quantity when we justify the existence of in terms of the definition of the limit: we know exists because by the definition of the limit, for any there is a , *so in particular*, there is a for , etc. Note the “in particular”: we produce from the definition by *specializing it*.

But on another way of looking at it, of course $\epsilon$ is many numbers. Indeed, it must represent every positive number, otherwise how can it be used to verify the definition *for all *? The singular, fixed with which we work in the proof is a sort of chimera: it actually represents all positive numbers at once. That we think of it as a single number is just a psychological device to allow us to focus in a productive way on what we are doing with all these numbers.

This dual nature of in the above was driven home for me by working with Ricky. Fast and accurate with calculations and algebraic manipulations, Ricky was thrown for a loop by real analysis, which was her first proof-based class, and in particular by the – proofs. After a lot of work, she had mastered the definition itself. But in trying to write the proofs, she found the lilting refrain *for all …* to be a kind of siren song, leading her astray. She was constantly re-initializing with this phrase, so that reading her work, there were 3 different meanings for by the end. “Look at how the proof works,” I would say, referring to the proof of above. “You don’t need to be less than any old . You need it to be less than *the particular* that you are using for .” “What do you mean *the particular* I am using?” she would ask. “I am trying to prove it works *for all !*”

Ricky’s difficulty has led me to a much greater appreciation of the subtle and profound abstraction involved any time an object is introduced into a proof with the words “fix an arbitrary…” In a sense, this is nothing more — nor less! — than the abstraction at the heart of a student’s first encounter with algebra: if we imagine an unspecified number , and proceed to have a conversation about it, our conversation applies simultaneously at once to all the values could have taken, even if we were imagining it the whole time as “only one thing.”^{[14]} But I don’t think I appreciated the great demand that “fix an arbitrary…” proofs in general, and – proofs in particular, place on this abstraction. The mastery of it that is needed here goes far beyond what is needed to get you through years of pushing around.

I offer the above anecdotes primarily as grist for reflection about learning, and especially about the nature of the particular landscape students tread as they encounter and .^{[15]} But I would like to articulate some lessons and reminders that I myself draw from them:

(1) A complete understanding of a concept might require to go beyond mastery of its internal structure and its downstream implications, to include an understanding of its purpose, i.e. the situation it was designed to respond to.

(2) Work that successfully responds to the standard set of prompts may still conceal important gaps in understanding, as mine did in my undergraduate real analysis class. More generally, do not assume because a student is “strong” that they have command of any particular thing.

(3) Conversely, take student thought seriously, even when it looks/sounds wrong. Ricky and Ty were producing unsuccessful work for very mathematically rich reasons; I learned something worthwhile by taking the time to understand what each of them was getting at. Harry’s issue, which I didn’t take seriously at the time, could have pushed my own understanding of calculus forward — in fact, it did, albeit belatedly.

Finally, I hope the combination of these anecdotes with the history above serves as a reminder both of the magnitude of the historical accomplishment crystallized in the Weierstrassian – definition of the limit, and of the corresponding profundity of the journey students take toward its mastery.

[1] There is an important contemporary argument that calculus’ pride of place in the curriculum should be ceded to statistics. (For example, see the TED talk by Arthur Benjamin.) That debate is beyond the scope of this blog post.

[2] The First Encounter with a Limit. *The Mathematical Gazette*, Vol. 19, No. 233 (1935), pp. 109-123. (link [jstor])

[3] In addition to the 1935 *Mathematical Gazette* article quoted above, see, e.g., E. J. Moulton, The Content of a Second Course in Calculus, *AMM* Vol. 25, No. 10 (1918), pp. 429-434 (link [jstor]); E. G. Phillips, On the Teaching of Analysis, *The Mathematical Gazette* Vol. 14, No. 204 (1929), pp. 571-573 (link [jstor]); N. R. C. Dockeray, The Teaching of Mathematical Analysis in Schools, *The Mathematical Gazette* Vol. 19, No. 236 (1935), pp. 321-340 (link [jstor]); H. Scheffe, At What Level of Rigor Should Advanced Calculus for Undergraduates be Taught?, *AMM* Vol. 47, No. 9 (1940), pp. 635-640 (link [jstor]). I thank Dave L. Renfro for all of these references.

[4] E.g., E. Dubinsky and O. Yiparaki, On student understanding of AE and EA quantification, *Research in Collegiate Mathematics Education* IV, 8 (2000), pp. 239-289 (link). In this and the next two notes, the literature cited only scratches the surface.

[5] E.g., D. Tall and S. Vinner, Concept image and concept definition in mathematics with particular reference to limits and continuity, *Educational Studies in Mathematics* Vol. 12 (1981), pp. 151-169 (link), S. R. Williams, Models of Limit Held by College Calculus Students, *Journal for Research in Mathematics Education* Vol. 22, No. 3 (1991), pp. 219-236 (link [jstor]), and C. Swinyard and E. Lockwood, Research on students’ reasoning about the formal definition of limit: An evolving conceptual analysis, *Proceedings of the 10th annual conference on research in undergraduate mathematics education, San Diego State University, San Diego, CA* (2007) (link).

Findings about students’ informal understandings of limits that generate friction with their study of and include that they are often dynamic/motion-based (like Newton), or infinitesimals-based (like Leibniz), and meanwhile, they are also often characterized by a “forward” orientation from to — “If you bring close to , it puts close to .” This is in contrast with the – definition’s “backward” orientation from to — “To make -close to , you have to find a to constrain .”

[6] E.g., J. Cottrill, E. Dubinsky, D. Nichols, K. Schwingendorf, K. Thomas, D. Vidakovic, Understanding the Limit Concept: Beginning with a Coordinated Process Scheme, *Journal of Mathematical Behavior* Vol. 15, pp. 167-192 (1996), Swinyard and Lockwood *op. cit.* (which responds to Cottrill *et. al.*), and C. Nagle, Transitioning from introductory calculus to formal limit conceptions, *For the Learning of Mathematics* Vol. 33, No. 2 (2013), pp. 2-10 (link).

[7] To avoid ambiguity, the learners referred to here are *myself* and the students I below call Ty and Ricky. The student I call Harry illustrates a difficulty one might have *without* the – limit.

[8] The brief account I am about to give represents an orthodox view of the history of calculus, see for example J. V. Grabiner, Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus, *The American Mathematical Monthly* Vol. 90, No. 3 (1984), pp. 185-194 (link). This orthodoxy is not without its detractors, e.g., B. Pourciau, Newton and the Notion of Limit, *Historia Mathematica* No. 28 (2001), pp. 18-30 (link) or H. Edwards, Euler’s Definition of the Derivative, *Bulletin of the AMS* Vol. 44, No. 4 (2007), pp. 575-580 (link).

Readers interested in more comprehensive accounts of the history of the – limit can consult Judith Grabiner’s monograph *The Origins of Cauchy’s Rigorous Calculus*, MIT Press, Cambridge, MA (1981) and William Dunham’s *The Calculus Gallery: Masterpieces from Newton to Lebesgue*, Princeton Univ. Press, Princeton, NJ (2005). A very interesting-looking new book on the topic is David Bressoud’s *Calculus Reordered: A History of the Big Ideas*, Princeton Univ. Press, Princeton, (2019) (link), which takes a broader view, looking at the development of integration, differentiation, series, and limits across multiple millennia and continents, and viewing the limit as a sort of culmination driven by the needs of research mathematicians in the 19th century. Bressoud’s book also considers questions of pedagogy in relation to this history.

[9] Dunham, *op. cit.* (see previous footnote).

[10] All names of students are pseudonyms.

[11] D. Tall and S. Vinner. Concept image and concept definition in mathematics with particular reference to limits and continuity. *Educational Studies in Mathematics* Vol. 12, No. 2 (1981), pp. 151-169. (link)

[12] Relatedly, recovering that which was lost from calculus when and superseded the Leibnizian infinitesimals is often given as the rationale behind Abraham Robinson’s development of nonstandard analysis.

[13] This observation is related to the body of research indicated in note [5]. I think it is subtly different though. As I understand that research, the theme is the difficulties students have with the – definition due to “interference” from their more informal understandings of limits and derivatives. In contrast, my focus here is on a difficulty Ty had not because of “interference,” but rather because he recognized (perhaps more clearly than I did) that this new definition is not actually doing the same thing, so if it was being sold it as a substitute, he was not buying.

[14] To help Ricky contextualize what she needed to do for the proof in terms of things she already understood, I asked her to consider this proof that every square number exceeds by one the product of the two integers consecutive with its square root:

Let be any integer. Then

,so any square number is one more than the product of and .

“I think of the in this proof as every number,” she said. “But you have to relate to it as a single number during the calculation itself,” I replied. “Otherwise, how do you know that ?”

[15] I first encountered the metaphor of a “landscape of learning” attendant to particular mathematical topics in the writings of Catherine Twomey Fosnot and Maarten Dolk.

]]>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.

Exercise 1: The Growing Cone

The images in Figure 1 (left) depict a growing cone. The graph in Figure 1 (right) represents the relationship between the outer surface area and height of the growing cone. Describe the rate of change of the surface area with respect to the height of the cone as it grows.

A typical solution to the Growing Cone exercise involves drawing a collection of tangent lines along the graph in Figure 1 and exploring the steepness of these lines as the height increases (see Figure 2). One could observe that the tangent lines become steeper, thus the rate of change *increases*.

So, what’s so bad about this strategy? This issue is that students often use this technique successfully without interacting with the quantities of surface area and height. Furthermore, to determine an increase in “steepness” they do not need to measure or determine slope at all, they only need to observe visual properties of the line without actually doing mathematics. Students begin to rely on and practice such techniques exclusively. As a result they are not able to reason in situations where these strategies break down, which we will see in Exercises 2 and 3 below.

What does a response for Exercise 1 with deeper understanding look like? At a beginning level, a student could describe that as the height of the cone increases, the surface area of the cone also increases. At a more sophisticated level, a student could describe that as the height increases in equal sized amounts, the surface area begins to increase by *larger* amounts. This means that the surface area increases at an* increasing rate *with respect to the height of the cone. As a result, the graph of the surface area and height would appear steeper for sections of the graph corresponding to points where the cone is taller.

Supporting students’ understanding of the quantitative relationships we have discussed above provides them with a deeper understanding of tangent lines and other tools in calculus. A student who is able to understand the quantitative relationship in Exercise 1, should be able to illustrate the changes in the surface area across equal changes in the height of the cone both in the picture of the cone (see Figure 3 (left)) and on the graph (see Figure 3 (right)). From here, students can more deeply connect slopes of secant lines on graphs with the changes in quantities of the growing cone as illustrated in Figure 3 (right) and revisit slopes of tangent lines as ratios of infinitesimal changes. A sophisticated understanding of tangent lines then should provide a student with the necessary tools to interpret a tangent line by comparing relative changes of the quantities the line represents.

The “sliding tangent line” strategy is exclusive to graphs in the Cartesian coordinate system with the independent variable on the horizontal axis and the dependent variable on the vertical axis. To help students develop understanding of the co-varying quantities represented in a graph, we can provide them exercises in which tricks like the “sliding tangent line” strategy don’t apply. For example, does this strategy work in Exercise 2?

Exercise 2: The Growing Cone Exercises Part II

Describe the rate of change of the height with respect to the surface area of the cone in the Growing Cone exercise.

A student who is able to quantitatively reason about the situation in Exercise 1 could respond to this question without needing to redraw the graph in Figure 1. The student could instead consider changes in the height of the cone that correspond to equal changes in the surface area of the cone. The cones and graph in Figure 4 illustrate the changes of height in the cone corresponding to an equal partition of the surface area. The changes in height *decrease* as the surface area increases in equal amounts. Thus, the student could conclude that the height of the cone increases at a *decreasing *rate with respect to surface area.

It could be argued that students could redraw the graph of height and surface area in the previous example so that the “sliding tangent” trick still applies. But what about in polar coordinates? Consider Exercise 3 which explores the graph of the cosine function in polar coordinates.

Exercise 3: Polar Coordinate Graph

Describe the rate of change of the radius with respect to the angle in the graph in Figure 5.

Standard tangent lines do not make sense in this problem. We could draw a line that is tangent to the graph as shown in Figure 6. The slope of this line represents the vertical change in relation to the horizontal change, not the change in radius with respect to change in angle.

We could, however, reason with the graph in Figure 5 in a similar way as we’ve previously seen by creating changes in angle and radius. In Figure 7, a collection of equally spaced angles has been marked with brown arcs and the corresponding changes in radius are drawn with orange segments. We can first observe that the radius decreases as the angle increases. We can then compare the changes in radius as the angle increases. The changes in radius get *larger* as the angle moves from 0^{o} to 30^{o }to 60^{o} to 90^{o}. A student could then conclude that as the angle increases from 0 to 90 degrees, the radius decreases at an *increasing* rate.

The focus in calculus should be the measurable attributes, or quantities, involved in rates of change, not on tangent lines drawn on a graph. The tricks and associations we teach students provide them quick ways to draw inferences but leave them unable to understand the implications of these inferences on the quantities. As a result, students come away from calculus knowing how to slide lines along graphs but not knowing how to make comparisons in changing quantities in their world. We can help students develop deeper understandings in calculus by asking questions that focus on the fundamental relationships between changing quantities and challenge students to think beyond memorized strategies.

Several of the ideas and examples here are inspired by the work of Advancing Reasoning, an NSF-funded research project whose mission is to support students’ and teachers’ mathematical thinking and learning by developing products that create transformative learning experiences. For more ideas on how to support students’ quantitative reasoning, see our project page at: https://sites.google.com/site/advancingreasoning/. All figures in this post were created in the GeoGebra application.

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By Benjamin Braun, University of Kentucky

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.

**Three Psychological Domains**

As I’ve written about previously on this blog, a useful oversimplification frames the human psyche as a three-stranded model:

The intellectual, or *cognitive*, domain regards knowledge and understanding of concepts. The behavioral, or *enactive*, domain regards the practices and actions with which we apply or develop that knowledge. The emotional, or *affective*, domain regards how we feel about our knowledge and our actions. All three of these domains play key roles in student learning, and when we talk about “mathematical maturity”, what we usually mean is that students have high-level functioning across all three of these areas.

As a first version of a better definition of mathematical maturity, we can specify that students who are mathematically mature have highly developed intellectual, behavioral, and emotional functioning with regard to their mathematical work. When we replace our colloquial phrase with this refined three-domain language, then we can clarify more precisely the distinction between students who have good technical skills but give up too easily (i.e. mature intellectually but developing in their behaviors), or who are persistent problem solvers yet are not confident about any of their results (mature behaviorally but developing emotionally), etc.

**The Five-Strand Model of Mathematical Proficiency**

Once we have become more familiar and fluent with using language that distinguishes between the intellectual, behavioral, and emotional domains, it is useful to further specify proficiency within those domains. One means of achieving this can be found in the 2001 National Research Council report *Adding It Up: Helping Children Learn Mathematics*, where a five-strand model of mathematical proficiency was introduced. While this model was motivated by research on student learning at the K-8 level, in my opinion it is an excellent model through at least the first two years of college, if not beyond. In this model, mathematical proficiency is defined through the following five attributes (see Chapter 4 of *Adding It Up* for details).

*conceptual understanding*— comprehension of mathematical concepts, operations, and relations*procedural fluency*— skill in carrying out procedures flexibly, accurately, efficiently, and appropriately*strategic competence*— ability to formulate, represent, and solve mathematical problems*adaptive reasoning*— capacity for logical thought, reflection, explanation, and justification*productive disposition*— habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

The five-strand model and the three psychological domains weave together well. In particular, one can view the first two strands as refinements of the intellectual domain, the third and fourth strands as refinements of the behavioral domain, and the fifth in alignment with the emotional domain.

In my experience teaching students in their first two years of college mathematics, most significant stumbling blocks for students fall clearly within one of these five strands. For example, when students are able to compute a derivative correctly, but are unable to use that information to find the equation of a tangent line, then this student is succeding in strand #2 but struggling with strand #1. As another example, suppose a student is able to do routine computations and is able to explain how formulas are derived, e.g. the quadratic formula from completing the square, but is challenged by multistep modeling problems such as a max/min problem that requires both introducing and solving an appropriate quadratic function. In this case, a reasonable argument exists that the student “knows the math”, i.e. is proficient with strands #1 and #2, but is struggling to develop mastery of the strategies to apply those skills, i.e. strand #3. As a third example, for students who have a negative view of mathematics and their mathematical capabilities, as related to strand #5, it is challenging to develop the persistence and self-efficacy required to do mathematics successfully.

Much like our mathematical conversations benefit from having clear definitions, our conversations about student learning benefit from having clear and agreed-upon language to describe key components of proficiency. The five-strand model provides an excellent starting point for more clear discussions on this topic.

**Mathematical Proficiency for Majors**

For students studying advanced mathematics, whether they be mathematics majors or math minors in math-intensive major programs, the five-strand model is not a sufficient foundation for articulately discussing mathematical proficiency. In this setting, I feel that one of our most useful resources is the 2015 MAA CUPM Curriculum Guide. Specifically, the following two recommendations copied directly from the Overview to the guide provide an articulate description of some advanced behaviors and intellectual knowledge that majors should attain.

*Cognitive Recommendation 1: Students should develop effective thinking and communication skills. *Major programs should include activities designed to promote students’ progress in learning to:

- state problems carefully, articulate assumptions, understand the importance of precise definition, and reason logically to conclusions;
- identify and model essential features of a complex situation, modify models as necessary for tractability, and draw useful conclusions;
- deduce general principles from particular instances;
- use and compare analytical, visual, and numerical perspectives in exploring mathematics;
- assess the correctness of solutions, create and explore examples, carry out mathematical experiments, and devise and test conjectures;
- recognize and make mathematically rigorous arguments;
- read mathematics with understanding;
- communicate mathematical ideas clearly and coherently both verbally and in writing to audiences of varying mathematical sophistication;
- approach mathematical problems with curiosity and creativity and persist in the face of difficulties;
- work creatively and self-sufficiently with mathematics.

* **Content Recommendation 6: Mathematical sciences major programs should present key ideas from complementary points of view: *

- continuous and discrete;
- algebraic and geometric;
- deterministic and stochastic;
- exact and approximate.

At the major level, the 10 items in the CUPM Cognitive Recommendation and the four items in the CUPM Content Recommendation provide a framework that further extends both the three domains and five strand model. The Cognitive Recommendations are primarily focused on the behavioral and emotional domains and on the third through fifth strands. The Content Recommendations further refine the idea of procedural and conceptual understanding in the first two strands by emphasizing that at an advanced level, students need to understand not only the techniques and concepts themselves, but how those techniques and concepts fit together within a broader vision of mathematics.

**Putting These Into Practice**

I will end this article with a few suggestions for how departments or faculty working groups can put these ideas into action.

- Have two or three faculty jointly present these frameworks/definitions of proficiency during a department seminar or colloquium.
- Gather a team of faculty to review the structure and content of a course for first-year students using the three domain and five strand model. Which of these domains/strands are targeted for development by assignments or activities in the course? Are there any that are being unintentionally omitted from the course curriculum or structure?
- Conduct a similar exercise for a major level course or sequence, this time using the language from the MAA Curriculum Guide. Which of these goals are students being explicitly trained toward? If any of these goals are not treated within that particular course, are there other required courses within the major where students are provided the opportunity to develop in that direction?
- Design a short activity/survey for students in a particular class based on this language. Have the activity introduce the language from one of these frameworks, and ask them to identify activities or experiences in their course that they felt helped them develop with regard to those domains/strands/goals. Discuss the results of this activity/survey with a team of faculty or at a department meeting.

It is important to keep in mind that the best way to be more effective in our considerations of student learning is to frame our discussions within clear and precise definitions of mathematical proficiency. For some courses or departments, the three domain model will be sufficient for this, and for others the five strand model or MAA Curriculum Guide goals will be needed. In any event, we need to move beyond overly-vague discussions of “mathematical maturity” and toward a more sophisticated language to discuss student learning.

]]>(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.

Nicholas’s journey, from the boy with special needs to an Oxford doctorate, is inspiring and important but his transformation is far from unique. The world is filled with people who were unsuccessful early learners and who received negative messages from schools but went on to become some of the most significant mathematicians, scientists, and other high achievers, in our society – including Albert Einstein. Some people dismiss the significance of these cases, thinking they are rare exceptions but the neuroscientific evidence that has emerged over recent years gives a different and more important explanation. The knowledge we now have about the working of the brain is so significant it should bring about a shift in the ways we teach, give messages to students, parent our children, and run schools and colleges. This article will summarize three of the most important areas of neuroscience that directly apply to the teaching and learning of mathematics. For more detail on these findings, and others, visit youcubed.org or read Boaler (2016).^{[2]}

The first important area of knowledge, which has been emerging over the last several decades, shows that our brains have enormous capacity to grow and change at any stage of life. Some of the most surprising evidence that highlighted this came from studies of black cab drivers in London. People in London are only allowed to own and drive these iconic cars if they successfully undergo extensive and complex spatial training, over many years, learning all of the roads within a 20-mile radius of Charing Cross, in central London, and every connection between them. At the end of their training they take a test called “The Knowledge” – the average number of times it takes people to pass The Knowledge is twelve. Neuroscientists decided to study the brains of the cab drivers and found that the spatial training caused areas of the hippocampus to significantly increase.^{[3]} They also found that when the drivers retired, and were not using the spatial pathways in their brains, the hippocampus shrank back down again.^{[4]} The black cab studies are significant for many reasons. First, they were conducted with adults of a range of ages and they all showed significant brain growth and change. Second, the area of the brain that grew – the hippocampus – is important for all forms of spatial, and mathematical thinking. The degree of plasticity found by the scientists shocked the scientific world. Brains were growing new connections and pathways as the adults studied and learned, and when the spatial pathways were no longer needed they faded away. Further evidence of significant brain growth, with people of all ages, often in an 8-week intervention, has continued to be produced over the last few decades, calling into question any practices of grouping and messaging to students that communicate that they cannot learn a particular level of mathematics.^{[5]} Nobody knows what any one student is capable of learning, and the schooling practices that place limits on students’ learning need to be radically rethought.

Prior to the emergence of the London data most people had believed either that brains were fixed from birth, or from adolescence. Now studies have even shown extensive brain change in retired adults.^{[6]} Because of the extent of fixed brain thinking that has pervaded our society for generations, particularly in relation to mathematics, there is a compelling need to change the messages we give to students – and their teachers – across the entire education system. The undergraduates I teach at Stanford are some of the highest achieving school students in the nation, but when they struggle in their first math class many decide they are just “not a math person” and give up. For the last several years I have been working to dispel these ideas with students by teaching a class called How to Learn Math, in which I share the evidence of brain growth and change, and other new ideas about learning. My experience of teaching this class has shown me the vulnerability of young people, who too readily come to believe they don’t belong in STEM subjects. Unfortunately, those most likely to believe they do not belong are women and people of color.^{[7]} It is not hard to understand why these groups are more vulnerable than white men. The stereotypes that pervade our society based on gender and color run deep and communicate that women and people of color are not suited to STEM subjects.

The second area of neuroscience that I find to be transformative concerns the positive impact of struggle. Scientists now know that the best times for brain growth and change are when people are working on challenging content, making mistakes, correcting them, moving on, making more mistakes, always working in areas of high challenge.^{[8, 9]} Teachers across the education system have been given the idea that their students should be correct all of the time, and when students struggle teachers often jump in and save them, breaking questions into smaller parts and reducing or removing the cognitive demand. Comparisons of teaching in Japan and the US have shown that students in Japan spend 44% of their time “inventing, thinking and struggling with underlying concepts” but students in the U.S. engage in this behavior only 1% of the time.^{[10]} We need to change our classroom approaches so that we give students more opportunity to struggle; but students will only be comfortable doing so if they have learned the importance and value of struggle, and if they and their teachers have rejected the idea that struggle is a sign of weakness. When classroom environments have been developed in which students feel safe being wrong, and when they have been valued for sharing even incorrect ideas, then students will start to embrace struggle, which will unlock their learning pathways.

The third important area of neuroscience is the new evidence showing that when we work on a mathematics problem, five different pathways in the brain are involved, including two that are visual.^{[11, 12]} When students can make connections between these brain regions, seeing, for example, a mathematical idea in numbers and in a picture, more productive and powerful brain connections develop. Researchers at the Marcus Institute of Integrative Healthhave studied the brains of people they regard to be “trailblazers” in their fields, and compared them to people who have not achieved huge distinction in their work. The difference they find in the brains of the two groups of people is important. The brains of the “trailblazers” show more connections between different brain areas, and more flexibility in their thinking.^{[13]} Working through closed questions, repeating procedures, as we commonly do in math classes, is not an approach that leads to enhanced connection making. In mathematics education we have done our students a disservice by making so much of our teaching one-dimensional. One of the most beautiful aspects of mathematics is the multi-dimensionality of the subject, as ideas can always be represented and encountered in many ways, such as with numbers, algorithms, visuals, tables, models, movement, and more.^{[14, 15]} When we invite people to gesture, draw, visualize, or build with numbers, for example, we create opportunities for important brain connections that are not made when they only encounter numbers in symbolic forms.

One of the implications of this important new science is we should all stop using fixed ability language and celebrating students by saying that they have a “gift” or a “math brain” or that they are “smart.” This is an important change for teachers, professors, parents, administrators – anyone who works with learners. When people hear such praise they feel good, at first, but when they later struggle with something they start to question their ability. If you believe you have a “gift” or a “math brain” or another indication of fixed intelligence, and then you struggle, that struggle is devastating. I was reminded of this while sharing the research on brain growth and the damage of fixed labels with my teacher students at Stanford last summer when Susannah raised her hand and said: “You are describing my life.” Susannah went on to recall her childhood when she was a top student in mathematics classes. She had attended a gifted program and she had been told frequently that she had a “math brain,” and a special talent. She enrolled as a mathematics major at UCLA but in the second year of the program she took a class that was challenging and that caused her to struggle. At that time, she decided she did not have a “math brain” after all, and she dropped out of her math major. What Susannah did not know is that struggle is really important for brain growth and that she could develop the pathways she needed to learn more mathematics. If she had known that, and not been given the fixed message that she had a “math brain,” Susannah would probably have persisted and graduated with a mathematics major. The idea that you have a “math brain” or not is at the root of the math anxiety that pervades the nation, and is often the reason that students give up on learning mathematics at the first experiences of struggle. Susannah was a high achieving student who suffered from the labeling she received; it is hard to estimate the numbers of students who were not as high achieving in school and were given the idea that they could never do well in math. Fixed brain messages have contributed to our nation’s fear and dislike of mathematics.^{[16]}

We are all learning all of the time and our lives are filled with opportunities to connect differently, with content and with people, and to enhance our brains. My aim in communicating neuroscience widely is to help teachers share the important knowledge of brain growth and connectivity, and to teach mathematics as a creative and multi-dimensional subject that engages all learners. For it is only when we combine positive growth messages with a multi-dimensional approach to teaching, learning, and thinking, that we will liberate our students from fixed ideas, and from math anxiety, and set them free to learn and enjoy mathematics.

*This blog contains extracts from Jo’s forthcoming book*: Limitless: Learn, Lead and Live without Barriers, *published by Harper Collins.*

[1] Letchford, L. (2018) *Reversed: A Memoir*. Acorn Publishing.

[2] Boaler, J (2016) *Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching*. Jossey-Bass/Wiley: Chappaqua, NY.

[3] Maguire, E. A., Gadian, D. G., Johnsrude, I. S., Good, C. D., Ashburner, J., Frackowiak, R. S., & Frith, C. D. (2000). Navigation-related structural change in the hippocampi of taxi drivers. *Proceedings of the National Academy of Sciences*, 97(8), 4398-4403.

[4] Woollett, K., & Maguire, E. A. (2011). Acquiring “The Knowledge” of London’s layout drives structural brain changes. *Current **b**iology**:CB*, 21(24), 2109–2114.

[5] Doidge, N. (2007). *The Brain That Changes Itself*. New York: Penguin Books,

[6] Park, D. C., Lodi-Smith, J., Drew, L., Haber, S., Hebrank, A., Bischof, G. N., & Aamodt, W. (2013). The impact of sustained engagement on cognitive function in older adults: the Synapse Project. *Psychological science*, 25(1), 103-12.

[7] Leslie, S.-J., Cimpian, A., Meyer, M., & Freeland, E. (2015). Expectations of brilliance underlie gender distributions across academic disciplines. *Science*, 347, 262-265.

[8] Coyle, D. (2009). *The Talent Code: Greatness Isn’t Born, It’s Grown, Here’s How*. New York: Bantam Books;

[9] Moser, J., Schroder, H. S., Heeter, C., Moran, T. P., & Lee, Y. H. (2011). Mind your errors: Evidence for a neural mechanism linking growth mindset to adaptive post error adjustments. *Psychological science*, 22, 1484–1489.

[10] Stigler, J., & Hiebert, J. (1999). *The teaching gap: Best ideas from the world’s teachers for improving education in the classroom*. New York: Free Press.

[11] Menon, V. (2015) Salience Network. In: Arthur W. Toga, editor. *Brain Mapping: An Encyclopedic Reference*, vol. 2, pp. 597-611. Academic Press: Elsevier;

[12] Boaler, J., Chen, L., Williams, C., & Cordero, M. (2016). Seeing as Understanding: The Importance of Visual Mathematics for our Brain and Learning*. Journal of Applied & Computational Mathematics*, 5(5), DOI: 10.4172/2168-9679.1000325

[13] Kalb, C. (2017). What makes a genius? *National Geographic*, 231(5), 30-55.

[14] https://www.youcubed.org/tasks/

[15] Boaler, J. (2016) *Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching*. Jossey-Bass/Wiley: Chappaqua, NY.

[16] Boaler, J. (2019). *Limitless: **Learn, Lead and Live without Barriers.*

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

We bid farewell to Art Duval, who has been on the editorial board for five years, and was one of the founding board members for this blog. His service and guidance have been indispensable to me in guiding the blog, and his voice will be missed.

We welcome two new editorial board members:

Yvonne Lai received her S.B. in Mathematics from MIT and Ph.D. from UC Davis, specializing in geometric group theory and hyperbolic geometry. Following a post-doctoral position at the University of Michigan in mathematics, she took a second post-doctoral position, this time at the University of Michigan School of Education. There, she began doing research in the area of mathematical knowledge for teaching, in the group led by Deborah Ball and Hyman Bass. Lai is now an associate professor in the Department of Mathematics at the University of Nebraska-Lincoln. She is founding chair of the MAA’s Special Interest Group on Mathematical Knowledge for Teaching, a member MAA Committee on the Mathematical Education of Teachers (COMET), and a member of the writing team for the NCTM publication Catalyzing Change.

Ben Blum-Smith received a B.A. in anthropology from Yale University in 2000, an M.A.T. in mathematics teaching from Tufts University in 2001, and a Ph.D. in mathematics from NYU in 2017, with a thesis in representation and invariant theory of finite groups. He worked as a middle and high school teacher in public schools in Cambridge, MA and New York City, and then as a mathematics professional development specialist for high schools and as a faculty member of Bard College’s teacher training program, before beginning his training as a research mathematician in 2011. He is currently a part-time faculty member of Eugene Lang College’s Department of Natural Sciences and Mathematics, and has also taught in the Bard Prison Initiative.

His research interests lie in invariant theory, algebraic combinatorics, their applications to data science, and connections between mathematics and democracy. He was a 2018 TED Resident, developing a TED talk about the relationship of mathematics and democracy, and is a founding organizer of the Mathematics and Democracy Seminar at the NYU Center for Data Science. He remains involved in teacher professional development through Math for America, an organization devoted to the career-long professional growth of teachers. He is also engaged in mathematical outreach. He has led math circles with students and teachers at the School of Mathematics, the New York Math Circle, the Westchester Area Math Circle, the LREI Summer Institute, the Center for Mathematical Talent at NYU, and the MathLeague International Mathematics Tournament, and is regularly a faculty member and faculty mentor at the Bridge to Enter Advanced Mathematics, an organization focused on creating a realistic pathway for underserved students to enter the mathematical sciences.

* * *

The theme of these two vignettes is how the teacher must value the student, must see as much value in his or her ideas—correct or incorrect—as in our own. Because sometimes they are correct.

I: Cecelia and the Grapes

This anecdote took place in a high school remedial class. For many years, I would take the 10 or so students who were not on any grade level at all and teach them together in an ungraded classroom. These were students who had struggled with mathematics, and had been failing, for many years. Many had learning disabilities. Some had significant troubles at home. All hated or feared mathematics. It was my job to un-teach them this fear.

Of course, the worst thing you can do in teaching remedial students is the same thing over again, even if you go slower and talk louder, even if make the definitions crystalline and the logic pristine. It won’t help. The students will tune out, will continue to hate and fear mathematics—and worst of all, will revert to the defensive learning habits that caused their failure in the first place.

Remember that definition of mental illness?

So in this class I used activities. Games. Manipulatives. Students measured and weighed to learn fractions. They walked the corridors to learn geometry. They went up and down stairs to add and subtract signed numbers. (Many of them would confuse left and right, but almost never up and down.) They analyzed card tricks—performed with different sized decks—to develop algebraic representation. My administrators, whose support of my work was vital and unflagging, were kept busy apologizing for my students being in the corridors or stairwells or shuffling playing cards.

I love this kind of teaching. It requires enormous flexibility, creativity, spontaneity. It means that you have to be right on top of the students’ cognition, reading their minds as best you can. And since their minds were full of ideas so different from my own, this was a challenge.

The other kind of teaching I like best is with gifted children—and for much the same reason. You cannot give them just more of the same, and faster. Sure, they will appear to succeed. But eventually they will hit a wall, will find some material that they cannot just read and understand. They need to experience early that understanding can be the result of struggle, or they will not have the means to surmount that wall. The teacher needs the same flexibility, creativity, spontaneity that the remedial classes require.

And for either group of outliers, you need the same ability to read minds. These students will come up with things you haven’t thought of yourself, which are correct or incorrect in the most amazing ways.

Cecilia was a fifteen year old girl in my remedial class. She was not angry, not stubborn, not resistant to school. Yet she had trouble with mathematics. I was trying to get her to pass a first-year algebra exam—without cramming her full of test tricks and meaningless technique. In this classroom, I had the time to do it.

I had given the class a challenge problem:

Marty ate ten grapes on September 1. Then he ate twelve grapes on September 2. Then he ate 14 grapes on September 3. He kept eating two more grapes each day than he ate on the previous day. How many grapes had he eaten at the end of September 10?

Remember: this problem, for these students, had nothing to do with an arithmetic progression. It was a simple arithmetic problem. We had been working on how multiplication for natural numbers is repeated addition, how the distributive law could shorten computation, how to recognize ‘complementary numbers’ that added up to 10 or 100 or 1000. In short, we were busy taking arithmetic off the paper.

But this problem stubbornly remained on the paper. Students tried this and that. They got wrong answers, thinking this was yet another multiplication problem. They tried to fit the problem into a pattern they had seen before. Nothing worked.

They worked in pairs, but I had an odd number of students that day. So Cecilia was part of a group of three. And she was not interested. She rearranged her books. She stared out the window. She powdered her nose.

My job was to keep the wheels turning. So I came over to her group and asked what they were doing. One student, Tim, had a good idea, although he didn’t know it. He drew lines to represent the numbers of grapes for each day:

xxxxxxxxxx

xxxxxxxxxxxx

xxxxxxxxxxxxxx

xxxxxxxxxxxxxxxx

. . .

. . .

. . .

xxxxxxxxxxxxxxxxxxxxxxxxxxxx

Tim could have observed that the first and last line, the second and next-to-last line, etc, complemented each other to form a rectangle. But he didn’t. I encouraged him and made a mental note to come back to this idea later on.

Other students, of course, were busy adding things up by hand. Students younger than these are perfectly content to solve a problem using such busywork. They don’t see it as compulsive or boring. But teenagers do—and that was lucky for me. They don’t learn much from tedious computation.

Cecilia saw me come over and wanted to look busy. So she picked up her calculator. I always let students use calculators, if they could tell me their plan for using it. If I sensed abuse, I would forbid them the calculator and discuss what they were going to do with it. Only after they had a coherent plan would I let them have the machine.

Cecilia’s calculator was pink and heart-shaped. Each of the keys was a different colored rhinestone, and each was also in the shape of a small heart. She began pressing the keys.

“What is your plan?” I asked. She had none, of course, and was busy looking busy.

I said: “I’ll talk to Chris. When I come back, I want to hear your ideas.” She looked at me as if the word ‘ideas’ was not in the English language. I fought hard the temptation to lecture her about paying attention to the task. That never works.

I went over to Chris, who was more or less randomly multiply numbers together—by hand, to impress me. I asked why he was doing this or that. He had the (correct) idea that somehow the small numbers made up for the big numbers. But couldn’t express this arithmetically. He had missed the essential circumstance that the numbers were evenly spaced—formed an arithmetic progression. He knew that 10×10 would be too small, and 10×28 would be too big. I let him work a bit more.

Then I came back to Cecelia. She was staring at her paper, but not vacantly. I could see on her face that something had happened while I was gone from her desk. On her paper was written

5 x 38 = 190,

and on her face was a look of relief—not quite a smile—but a look that told me that she knew what she was doing.

“How did you do it?” I asked.

“With my calculator,” she replied. This was not mere adolescent backtalk. She really thought I was asking about the arithmetic.

“The small numbers make up for the big ones, so you can shortcut multiply.” And she started to explain. As I write this, I cannot recall how she explained it. Her words made no sense to me, whose mind was full of formulas for general terms and partial sums. I tried to listen, but quickly got lost in her verbal explanation.

No matter. She clearly understood the problem. She had figured out that pairs of numbers added to 38. I was delighted, but she was merely relieved. I got her to explain her ideas to the others in her group, who also didn’t quite understand her words. But I gave them another similar problem, and they could do it—and certainly used Cecelia’s ideas. So something had been communicated.

I never recovered Cecelia’s words. To this day I don’t know how she thought of the solution, nor how she managed to communicate it to others in her group.

Years ago, I saw a film called *Defending Your Life*. It took place mostly in heaven(!), and some of the characters were angels. It was explained that angels are not really different from humans, that humans only really use 10% of their brain’s capabilities, but angels use 90%. Maybe. My point is that for 10 minutes I had an angel in my classroom.

Then she went back to powdering her nose.

II. Cold Weather: An Unfinished Story

Here is another incident that occurred in a remedial classroom. The students in this class were studying linear equations, starting with a story and generating mathematical models for the situations. They had worked on stories about cars traveling at constant velocity, about Mary working at the grocery and saving her money, about Bob spending the contents of his piggy bank at a constant rate, and so on. Then I gave them what I knew was a hard example for them, just to see what they would make of it:

At 50 degrees Fahrenheit, 30 people will complain about the temperature of a building. For every drop of 10 degrees in temperature, five more people will complain. How is the number of complaints received related to the temperature in the building?

I had expected a table of values something like this:

D |
50 | 40 | 30 | 20 | 10 |

C |
30 | 35 | 40 | 45 | 50 |

… and eventually the equation *C *= − (1/2) *D *+ 55.

This was not meant to be a realistic situation. I have found that such things do not trouble students. In this case, they had fun thinking of the occupants of the building shivering at their desks. It was in fact a cold January day.

The students understood the situation well enough to make a table of values. But they could not write an equation. At first, they could not decide which variable should be independent and which dependent. I described to them how an historian might look at the number of complaints to infer the temperature, but most of us would think the other way around. They had no trouble with this, once it was pointed out.

They were thrown by the fact that the table did not start at 0, although some of them had learned to extrapolate to get the value at 0. They were confused by the fact that the temperature went ‘down’, not ‘up’. And I had not yet talked about what to do when *x *jumps by more than 1 in a table.

As they worked, I observed. They were still not secure with the concept that the equation must be true for every pair of values they knew. They had somewhere learned to follow the ‘key words’ of the problem, so they had various ideas about how the words themselves generated the equation. And all the equations were wrong. This gave me the opportunity to show them that substituting values, rather than looking back at the words of the story, was what would tell them if their equation is correct.

Work was proceeding as I had expected, until Selma stopped me in my tracks. Selma was a vivacious 13-year old, the kind who seems to want to cling to her childhood. She must have weighed about 75 pounds sopping wet, all sinew and energy. And delighted with life.

Selma, among others, gave the equation C = 30 + 5D. Many students had realized that 30 and 5 play roles in the equation and were simply guessing about where to put them. One reason I selected this problem is that the slope is not an integer, and so it is less likely that they would get the correct answer by guessing. When asked, the class quickly saw that this equation was wrong.

But Selma persisted.

“Do I have to do the equation your way? Can’t I do it another way?”

There is only one answer that a teacher can give to this question, and I gave it. It turned out to be the best question I’d received all week.

“Well, what’s another way to do it?” I asked.

Selma came up to the board, and wrote the following table:

d |
0 | 1 | 2 | 3 | 4 | 5 |

C |
30 | 35 | 40 | 45 | 50 | 55 |

“See,” she said, “*C *= 30 + 5*d*.”

I was about to repeat my tiresome argument about plugging in values, but — just in time — I noticed the top line of her table.

“What is *d*?” I asked.

“Oh,” said Selma. “My *d *is different from yours. My ‘*d*’ stands for ‘drops’. One ‘*d*’ is one drop of 10 degrees. So when the temperature drops 10 degrees, we have 35 complaints: the 30 we had at first, plus five more. And for every drop, we add 5 complaints.”

I was speechless. But the class wasn’t. “That’s wrong!” “That’s right!” I had no trouble engaging them, but I myself didn’t quite know what to say.

So I played it safe. I told Selma that I understood her reasoning, and her representation. Could she use her ideas to get an equation in terms of the temperature Fahrenheit? She understood what I meant, and she also understood that she was right.

But what should I have done next? I might have exploited this idea of changing variables that Selma stumbled on. What are some fruitful directions? What Selma had discovered, without knowing it, was that a linear change of variables does not affect the degree of a polynomial function, so that a linear relationship remains linear. The trick of choosing your variables wisely is an old one. It lies behind much of the work of the Renaissance algebraists, and was made into an art form even earlier by Diophantos. I am still not sure what the best pedagogical strategy might have been on this occasion, but I feel that there is more here I could have done.

I often structure lessons so that a problem remains open at the end. This time, life structured the open problem for me.

*Some of this Blog post will appear in the forthcoming book, edited by Hector Rosario: *Mathematical Outreach: Explorations in Social Justice. Singapore, World Scientific Publishing, 2019. *Other portions have appeared in “Anecdotes and Assertions about Creativity in the Working Mathematics Classroom” (with Mark Applebaum), in Leikin, R., Berman, A., and Koichu, B.,* Creativity in Mathematics and the Education of Gifted Students. * Rotterdam: Sense Publications, 2009.*

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- 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.

So it was easy for me to meet James’ needs.

What was the ‘difficult situation’? I didn’t know, and it didn’t really matter. From the cryptic note, I assumed it was a divorce, and that the family was not anxious for everyone to know. But it could have just as well been a marriage: a new step-parent or step-sibling can take some getting used to. Or it could have been the birth of a new, much younger sibling. Any one of dozens of such circumstances looms large in an adolescent’s life. It didn’t matter to me–James needed to be cut some slack, so I cut it.

James was surly, a sure sign of instability in his life. He made little irritating comments, usually addressed to me–within the bounds of adolescent propriety, but on the rebellious side. Something’s going on, I thought. Of course, it wasn’t really me he was angry at. I was an authority figure and he needed a target. I was willing to play that role for him. I have found that the most constructive way to deal with this situation is to ignore the barbs and engage whenever the student shows a more positive side.

So I found ways to dodge his anger and get at the person it was hiding.

James had a habit of wearing bright orange gym shorts. I took to teasing him about them. “You’ve got to get rid of those shorts!” I would say, in various ways. James loved the attention and developed a variety of snappy comebacks. Eventually, he mellowed, promising me a pair of orange shorts as a gift at the end of the year.

James often asked to leave class during the lesson. I knew full well that he didn’t have to use the rest room and was hanging out with friends in the hall. I went a step further, asking myself why he was not engaged in school, and what he might need in his life in order to continue attending school. I knew there was an answer, thanks to the note from guidance. But I didn’t know what it was.

James was getting a straight C in the class. He wasn’t closing doors to mathematics, but couldn’t find the resources in his life to take the next step. I waited to see what I might do.

In April, his father called, angry. Why is James getting a C? He has to go to a good college. I went to Harvard business school. A C is simply not acceptable. Why wasn’t I notified earlier that James was doing so poorly?

In fact he was notified, quarterly, of James’ progress, but didn’t respond. It turned out that he had a lot on his plate. During the conversation I found out what the matter was. His wife, James’ mother, was dying. She had been in and out of the hospital for treatment, and was getting ready to leave her family, and this world, forever. Now I understood fully James’ anger, and his father’s anger.

We all feel helpless in these situations. There’s not much you can do or say. I told James’ father what a wonderful son he had, how James was doing a great job keeping his life together during this crisis, how he might not get the greatest grade, but that he would eventually be able to recoup his academic losses. In this most difficult yet of James’ years on earth, I said, it is amazing that he is able to keep up a C average in a difficult subject. These comments dulled the father’s hostility. I was part of the solution, not part of the problem.

After this call, things got better between me and James. He knew that I was aware of his problems, and sympathetic to them. He looked me in the eye when he asked to leave the room and made arrangements to retake quizzes he had done poorly on. Luckily, James learned mathematics easily, and a few sentences from me set him back on the right path when the work got tough.

I was moving from one classroom to another the next year, and part of my daily routine was searching the school building for useable cartons to pack away a 30-year accumulation of books and materials. One day, I found two empty cartons in the delivery room, and was carrying them down the hall when I passed James.

He was stretched out on a bench in the hall. “Do you need any help with those, Dr. Saul?”

I really didn’t, but he clearly wanted to engage me. I decided to accept the invitation. “I’d love some help, James. Would you like to help me pack some books?”

We went down to my classroom. James and I talked about the books we were packing, how heavy they were, who had written them, who had given them to me–and eventually how it feels to be living in a house with a dying parent.

Then he said, “My mom is going into the hospital again, for a week.”

I gave James what I could–mostly the same words I had for his father, and especially how a C average is not so bad, given what he had to handle outside of school. He couldn’t stay too much longer, because he had an appointment with the dean at 3:00. I told him I was glad for whatever help he could give me, and not to be late for his appointment.

Later I met him in the hall, where he told me that he his appointment was actually with a ‘cute girl’. I told him that I knew what that was like, and that we all shared certain feelings in life. And that we are not alone in handling them. He looked at me and smiled. He knew that I wasn’t just talking about cute girls.

The next day, I met James in the hall again. He had come to math class, but left in the middle, and hadn’t returned. We were working on completing the square, a challenge for this group, but a challenge they must meet if they are going to attach any meaning at all to the quadratic formula.

After an exchange of pleasantries, I asked James, “Do you know how to complete the square?”

“Yeah, it’s not that hard, except when you get fractions.” It was clear, from the way he answered, that he knew how to do it, and had practiced it enough to know where the difficulties lay.

“So come and take a quiz when you’re ready,” I said.

“Okay.” James looked down. Then he looked up at me. “Dr. Saul,” he said, “Give me a hug.” I quickly obliged.

Sometimes it is as important to complete a circle as to complete a square.

- The Chain Still Holding

“That building–that’s what scares me.” It was Ivan who said this, pointing to the Hancock tower in Boston, across the river. I was sitting with him on the MIT campus, talking about his problems adjusting to America. A brilliant mathematics student, he had come from Bulgaria for six weeks of study with equally gifted American high school students at a summer program I ran here. He was 17 years old, and the year was 1993.

I had noticed that Ivan was too much in his room, not playing enough Frisbee, not bonding with other students. My job was to draw him out. So we were sitting, late one night by the banks of the Charles River, and talking heart-to-heart.

“That building scares me,” he repeated. “America goes too fast, too far. The people move quickly. The cars goes fast. The food tastes like…nothing.”

We took our meals in a student cafeteria.

“But don’t you have trouble getting food back home?” I queried. The American experience was overwhelming him, and I hoped to turn him towards its positive aspects.

“Yes. It is difficult. Prices have risen and salaries have not kept up.” He paused before taking the bitter medicine: “I think, wherever Communism has come, it leaves garbage behind.” His face got hard, and he was silent for a bit.

Then, “I dream each night of having breakfast at home.” Home was Dobrich, a town in an agricultural area of Bulgaria which Ivan considered “not too small.”

His monologue was strangely compelling. There was some reason why I had to listen, had to respond, to the expression of bewilderment, of events running away with him, and of his own country and his own upbringing, betraying him.

For a while I didn’t understand my own reaction. And then, slowly and quietly, understanding came over me. I suddenly felt like a link in a chain, a stitch in a fabric.

There are moments when the meaning of life overwhelms you, forces itself into your consciousness, and thrusts still deeper, into your unconscious mind. Suddenly, Ivan’s confusion and awe, and even the tears he was clearly holding back, were mine as well. It wasn’t a tall building that scared him. It something greater.

In 1913, my grandfather Froim arrived in America from a small town in east Europe. Mother Russia had become more of a warden than a guardian to her Jewish children, who were leaving in droves to build America. Stories of his confusion have become family legends. When someone showed him the subway to the Bronx, he thought he was going down into a root-cellar. His cousin, who had arrived two years earlier, had to teach him how to drink liquids through a straw. Seeing the Woolworth building from afar, he tried to walk over for a closer look. But he didn’t realize how big it was, and how far away, and spent his whole lunch hour getting nowhere.

I grew up with these stories. But why did I react to them so strongly? Why did my soul vibrate, hearing them? Could it have been because I knew I would be having this conversation in Boston? Was it for this moment that my grandfather repeated them to me, these stories that took place in 1913, and were told me in 1963? Are these messages sent across the years, from one struggling immigrant to another, with myself as the medium?

It was more, though, than giving back to Ivan what my grandfather had given to me. More: I was Ivan, I was Froim, and I was at the same time a father and grandfather to both. They say this happens, that the roles reverse as one grows older, and you begin to care for your parents as once they cared for you. But I didn’t expect these feelings at forty-something, with both my parents in good health. I wasn’t ready.

“The child is father to the man,” says the poet. I hadn’t felt the feeling to these words so strongly before, and was overwhelmed.

And I wasn’t ready for the brush with eternity that these feelings would bring, of a contact with events that happen over and over again, and so occur outside of time. It’s not just the experience of the immigrant, or the foreigner. It’s the experience of the child and the parent. Somehow, in this conversation, with a youth I hardly knew on the banks of a dark river, I felt a contact with the future as well as the past. My own children, and their children, will feel these feelings, will go through these experiences. And my adventures will become their legends. The chain is still holding, the fabric unrent.

]]>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.

**How did we get started? **

In March, 2012, UTEP’s College of Education initiated a joint UTEP-EPCC meeting to discuss alignment of math courses for undergraduates in the teacher education program. Issues discussed at the meeting include aligning two lower-division math courses, selecting a textbook, creating a wiki to share resources/information, advising students, and improving communication. Although we were brought together by administrators in Dean’s and Provost’s offices, we the faculty quickly took ownership of the effort.

At the second meeting, we identified a need for professional development to help instructors structure their classes to increase student’s mathematical thinking. We discussed questions like: Why should instructors attend? What should they expect to get out of it? What topics are appropriate (e.g., lesson planning, problem solving, what mathematics thinking look like)? We chose “Helping Students Become Mathematical Thinkers” to be the topic for our first workshop, which was held in August, just a few months after we got started. We chose to hold the workshop less than two weeks before the start of the fall semester because we wanted to offer instructors some ideas and resources for their courses. We called our workshop Teachers Teaching Teachers (TTT).

The first workshop went well and we had another five:

- Summer 2012 –
*Helping Students become Mathematical Thinkers*(28 attended at EPCC) - Fall 2014 –
*Fostering Mathematical Thinking*(25 attended at UTEP) - Fall 2015 –
*Big Ideas in Statistics, Insights from Practicing Teachers, and Task Analysis*(26 attended at EPCC) - Fall 2016 –
*Active Learning*(20 attended at UTEP) - Fall 2017 –
*Reading Math Textbook*(24 attended at EPCC) - Spring 2019 –
*Proportional Reasoning*(27 attended at EPCC)

**What does a typical workshop consist of? **

We offer breakfast at 8:30 a.m. to encourage participants to come early and register. Our program starts at 9:00 a.m., with an ice-breaking activity. Most of our workshops consist of two main parts: a learning-and-sharing session before lunch and a working session after lunch. The learning-and-sharing session might have two or three activities in which attendees are participating as active learners or problem solvers. In some years, these activities were facilitated by the workshop organizers: math tasks that challenge participants to think (2012), hands-on-approach in understanding areas (2014), active learning (2016), and 10 essential understandings of ratios and proportions (2019). In other years, invited guests presented topics like big ideas in statistics (2015), perspectives of elementary or middle school teachers (2015), and three-levels of reading—stated, implied, and applied (2016). In the working sessions, attendees worked in small groups to create math tasks (2012, 2016), analyze textbook tasks (2015), generate questions to guide reading of math text (2017), and analyze and respond to student written work (2019). We typically close our workshop by encouraging the participants to take an online survey.

**What does the workshop-organizing process entail? **

Organizing the first TTT workshop took the most effort because we had to start from scratch. That first summer, a group of about 6-8 of us met almost every week to plan the content and the logistics. Subsequent workshops have involved less effort and coordination because we knew our duties (e.g. logistics, food ordering), and because there has been very little turnover among the organizing committee members. We now typically have 3-6 planning meetings in the summer to prepare for the workshop in the fall semester. Some years, it may take 2-3 meetings just to decide on a topic because we allow ideas to emerge and evolve. For example, the Spring 2019 workshop started with creating a model lesson on a topic that “all” instructors could adapt and implement in their courses. We then selected proportional reasoning to be the topic, then the topic evolved into identifying key concepts necessary for understanding proportional reasoning deeply and analyzing students’ misconceptions involving ratios and proportions.

Expenses are minimal, and consist mostly of breakfast and lunch for participants. This is covered by the hosting department (UTEP or EPCC) and/or a textbook publisher.

**How did participants respond to these workshops? **

One measure of our success is that many participants keep returning to the TTT workshop each year. The workshops seem to meet the needs of the participants. The participant comments revealed that they liked the topics, activities, and student-centered discussions. They enjoyed working with peers, sharing experiences, and learning from others in a friendly atmosphere. Participants from each institution were eager to interact with their colleagues from the other school. And they enjoyed the food.

The comments also revealed participants’ growing awareness of the importance of student thinking and engagement. One participant “learned about my own classroom practice, learned to stop and reflect between activities.” Another participant acknowledged that “it’s challenging to create activities that engage students.” Other comments included “thinking must be present in the class,” “you must create a need for students to be engaged in learning,” and “it’s important to work with others in solving problems.”

**What factors contributed to successful collaboration? **

We attribute the success of TTT workshops to the mutual respect and collegiality among UTEP-EPCC math faculty in the organizing committee. We are comfortable and enjoy each other’s company. Many of our meetings are held in the evening at a restaurant where we have a chance to dine, chat, and connect in addition to work and plan the activities for a workshop. We are open to ideas and willing to try new things. For example, we ran with the suggestion of having school teachers share their perspectives for the 2015 workshop because most of our workshop participants, who are college instructors, do not teach elementary and/or middle school students. We are reflective and adaptive. For example, our first workshop lasted 7 ½ hours and our second was 5 hours. We eventually found that 6 ½ hours is most appropriate.

Even well-meaning faculty (and staff) from a community college and a university working together face challenges arising from competing institutional demands and constraints. For instance, a community college has strong incentive for its students to complete the associate’s degree, while a university has a strong incentive for its students not to put off courses in the major for too long. We at UTEP and EPCC have a built-in advantage in that we are the only 4-year university and community college, respectively, in the area. This almost forces us to work together on issues such as articulation and enrollment. Many students from EPCC eventually transfer to UTEP and many students at UTEP started at EPCC. Some students are even taking courses at both campuses at the same time. To capture this reality, we repeated to each other, “Our students are your students are our students,” sometimes modifying it to “Our students are your students are **all of our **students.”

Working together year after year, faculty from both institutions have come to appreciate that we have more in common than we have differences. Both UTEP and EPCC faculty work towards a common purpose; that is, to increase the quality of our math courses for prospective teachers who would in turn improve the math courses they teach when they become teachers. We think an element of kindness among the organizing committee members underlies our success. That is, we care for each other, our fellow instructors, and our students.

** ****What have we learned? **

In our earlier workshops, each institution (UTEP and EPCC) was responsible for different sessions. We later realized that our collaboration would be stronger if each activity is co-facilitated by one faculty member from each institution. We learned that one or two ice-breaking activities help improve the workshop atmosphere. This year, we began by having workshop participants share how they implemented the ideas learned in the previous workshop and the outcomes. We encouraged them to implement the ideas and activities of this year’s workshop, on proportional reasoning, in their courses and then share their findings at the next workshop.

** ****What are some of our challenges or unfulfilled dreams? **

Our success in offering an annual workshop may be limited to creating awareness and may not have lasting impact on changing the way our instructors teach mathematics. We were very motivated at the end of our first workshop but we were not successful with follow-up efforts once the semester started. Workshop participants were not very responsive after the workshops and the organizing committee members were busy with their own routines. Effective professional development projects require follow-up activities and support throughout the academic year, and possibly over two to three years. If we could secure funding to support such an expansion to a full-blown project with more extensive follow-up activities, we would all benefit. On the other hand, the group dynamics and motivating forces would be different, and there is the risk that our current collaboration would dissolve at the end of the funded project.

When we started TTT, we thought one of the easiest things to accomplish would be to agree on a common textbook for both institutions. It is a surprise that we still have not done this. So students who transfer from EPCC to UTEP must buy a different textbook when they take the third (upper-division) course at UTEP. We have also had difficulty attracting adjunct faculty who teach only one or two courses to attend our workshop.

**Conclusion**

We cannot say for sure which parts of our story would work at other universities and community colleges. Some of what is going on here may be very specific to El Paso, or to the people who happen to be here. But one part that is perhaps universal is the value of just getting the faculty at the different institutions together to talk to each other, and seeing what they come up with.

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