Morgan State University, Baltimore, MD 21251
Prisoners are provided with a college education so that when they are released, they will adjust easily to society and won’t return to prison. I was fascinated by the idea so much that I wanted to be a part of it. As a result, I have been teaching in prison for two consecutive semesters. In this essay, I will explain how the fact that I am an immigrant from Iran having a single-entry visa helped me to get along with students in a U.S. prison and also motivated them to rely on themselves, focus on their successes and do better in math. I will talk about the challenges my students and I have gone through and, at the end, I will come up with some suggestions that I believe might help any prisoner attending math class in prison.
Introduction.
My name is Zeinab Bandpey, a graduate of the Ph.D. program in Industrial and Computational Mathematics at Morgan State University. I have been an adjunct faculty member at Towson University and the University of Baltimore. The University of Baltimore was selected to participate in the U.S. Department of Education’s Second Chance Pell Grant Program, and I was privileged to be a part of this program as a math instructor, to teach fundamentals of mathematics and college algebra.
How it started
My Ph.D. advisor was always looking for ways to help me survive financially in the U.S. A professor at Towson University had informed him of this program and my advisor passed the information along to me. I was fascinated by the idea and I wanted to be a part of it: “How cool is that?” I thought. “Prisoners are offered college education. It’s called a second chance.” I was curious: how many of them are going to take this chance seriously, and what is my role to motivate them to get into science? I had all these thoughts in my head when I attended the faculty meeting;. I was the only international instructor! And it freaked me out, because it was just a few months since I had moved to the U.S. I could not speak English properly, but I passionately wanted to do the program and I knew from the bottom of my heart that I could do it.
After being introduced to the students by my supervisor in the program, I went to each one of them to have a small conversation, asking for their names and a little more information about themselves: What do they want to study? Do they like mathematics? Why have they decided to participate in this program?
Although they answered my questions, sounding more determined than students in college classrooms, they were more concerned about my perspective of them, and they asked me questions like “Why are you here?” “Do you see us as a bunch of criminals who will never change and who will never have a bright future?” “Have you ever been afraid of being among 30 inmates who are mostly African American?” These questions made me think that they are afraid of prejudgments and they need to know their professors believe in them and trust the fact that some, if not all, of them are going to change their lives using the opportunity they have been given. I noticed that my attitude might change everything and I must prove myself to them first, that I believe in them and I am sure they are going to pass this class and any other classes they have taken perfectly.
It was not hard for me to come up with a response: I am an Iranian woman studying math, I have been down this road before, where people judge me because of my nationality or my gender. So, I could easily see their point: they did not want me to judge them because of their race or because of where they were born and grew up. I started to know each one of them individually and it helped to persuade them to go through the exams and classes.
It is worth mentioning that throughout the very first conversation I had with them, I found three of them very interested in math and computer science and I promised them I’d help them with learning basic stuff in these majors, as the University of Baltimore does not offer STEM majors through this program, and all of them essentially have to study other majors for now. One of the students, with notes I gave him, was able to write code to calculate Catalan numbers. This had certainly been done before, but he did not know this and did it all by himself. I saw this as a huge success.
Challenges in the classroom
It was beyond my imagination how small problems would make teaching hard and how student support and help would make things better. When we started, we did not even have a proper classroom with a board. There was a big room called the library with huge fans to keep the room’s temperature normal as it gets hot even on the coldest day of winter. Those fans made a horrible noise which made it hard for people sitting at the end of the room to hear me. Also, with the lack of a normal-sized board, it was harder to explain concepts because I had to erase things as soon as I wrote them down. Surprisingly enough, no student complained in class. They used to turn off the fans and sit closer to the board. They asked their supervisor afterwards to provide us with a bigger board, and after half of the semester we actually got one.
Another issue that I found challenging was that the students’ ages ranged from 25 to 65. Some of them have been away from studying math for more than 20 years and some others were young and quick learners; it was hard to arrange the pace of the class. If you go fast, those in their 40s and more would be left behind, and if you go slow, it would get boring to the other group. A normal pace would be also too slow for fast learners and too fast for slow learners. But again, they were so passionate and they wanted it to work out, so when I asked them to make groups and distributed students in such a way that in each desk there would be mixtures of those two groups. Fast learners could help slow learners to understand better. They accepted the arrangement, and it worked nicely.
Exam anxiety was another important issue, and I believe the most important reason for that was lack of confidence. The other reason might have been they did not want to disappoint their instructors, and that would put more pressure on them. One of my students had a very hard time during the first 3 or 4 exams. He was a good student and he never left a homework assignment undone, but he had test anxiety so that he used to sweat a lot—so much that I was concerned something would happen to him. He would get upset and express that he hates math. One time I sat next to him and asked him to do his exam and talk to me whenever he does not feel all right. He did, and explained he knows the concepts very well and he does not know what is happening during the exam so that he could not answer the questions properly. What I thought would help was to distract him from thinking about his not being able to answer the test questions by asking for specific definitions or concepts which would refresh his memory. He liked math at the end, and he said he felt much better now. He added, “I do not hate it anymore but I still do not love it.” He passed both classes he took with me with a B.
The students rarely had access to the internet and computers. Some basic problems were that they could not use “MyMathLab” or tutorial videos. They did not have graphing calculators for college algebra, and they could not reach me whenever they had questions using emails or office hours.
How could I relate my experiences to theirs?
I was a student under a single-entry F1 visa, which had taken me more than a year to get. Having a single-entry visa and studying for my Ph.D. meant I could not travel back to my country to visit my family because it was too risky. I might have lost the chance to finish my Ph.D., so I was kind of a prisoner in a big country. Because I was following my dreams, I totally understood the feeling that you do not appreciate holidays as you cannot celebrate them with your beloved family. I felt it strongly when they said they could not focus on class as they were missing a lot by being in prison (like birthdays, weddings, funerals, …).
In his speech on orientation day, one of the students, with tears in his eyes, told us how hard it was for him to stay focused and keep trying when prison staff always saw him as a prisoner and discouraged him from what he was doing in school. This touched me personally: in the same way, many of the students (and staff) could not understand how a woman from Iran could be Muslim but not practicing, was not a terrorist or terrorist supports, and in fact did work in mathematical counter-terrorism.
In my personal life since I moved to the USA I have tried harder than anytime else in my life to achieve my goals and to make it worth being away from my family. It went through very well. So, I encouraged them to use the time they have in prison to create something nice, study hard and make themselves and their family proud. I’d like to believe it worked, as out of 30 students in my class, 28 of them passed the class successfully and with passion.
At the end, what I think made this experience a successful one was we all believed, no matter how hard things worked out, that we were in the same group and we helped each other out through the challenges. Teaching math is not just instructing them to deal with numbers: it is building confidence in them to understand the logic behind each problem. They believed in me and I believed in them, and we had a great experiences there, in prison.
From a Handwritten Letter
May 31, 2017
Dear Professor Bandpey
Thank you for your patience, hard work, and dedication to teaching me to believe in myself. Before I started this class I not only HATED math. I was also horrible at it and didn’t believe that I could do math. However, you’ve taught [me] to have confidence in myself and with that confidence I began to get better grades. I can now teach my son math in the future… Thank you for making me believe in myself. I pray that you are successful in all of your future engagements.
Respectfully,
XXXXX
Acknowledgment
I would like to thank Prof. Jonathan Farley for his support in writing this essay.
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Montana State University, Billings
Since the COVID-19 pandemic hit during the Spring of 2020, I’ve been nothing short of impressed and amazed at my colleagues’ resourcefulness and creativity in shifting their courses to an online modality. So when I was asked to teach an online Modern Geometry course this past Summer, I was eager to roll out an inquiry-based version of this course. But when planning this course I realized that I would face unique challenges that would make this difficult.
MSU-Billings is a comprehensive regional state university which serves central and eastern Montana and northern Wyoming, parts of the nation that are often very rural and distant from the physical campus. For this reason, the school has had a strong focus on its online course offerings even prior to the pandemic. In particular, Modern Geometry serves as the last course in a Math Teaching Minor, which certifies current teachers in the state to teach Mathematics in addition to their other certifications. Because this minor is intended for current working professionals from across the state, it is necessary for the courses in it to be online.
But this poses its own challenges. I know from previous experience that internet access in rural Montana can be spotty and unreliable, so I was unsure that a Zoom or WebEx based course would allow for equal opportunity amongst students to share and communicate with each other. Moreover, as working professionals, they would all be saddled with myriad other responsibilities, particular this year. Since our school does not designate a specified meeting time for online classes, I did not think it would be feasible to be able to designate a meeting time without excluding some students. I concluded that a synchronous model for Inquiry Based Learning (IBL) would not support the pillars of Equity and Collaboration.
I concluded that I would have to develop an asynchronous model for inquiry learning. I was highly skeptical about how this would go. IBL experts who generously lent me their time gave me advice in general on inquiry learning, but were likewise skeptical of the asynchronous approach. I didn’t see that I had any choice in the matter.
The Course Design.
I decided to use Charles Coppins Euclidean and NonEuclidean Geometries as the primary resource for the class. Given that we had an 8-week schedule, I thought that the layout of the course would match well with that of the text. We also scheduled a single synchronous meeting on the first day of the summer session to discuss the flow of the course. I encouraged them to think of themselves as a research team, uncovering the structure of neutral geometries together, and that the discussion board posts would serve as a replacement for the in person conversation and discussion they would have had, both in and out of the class.
The layout of the course was as follows:
My role in all of this, especially in the beginning, was to facilitate discussion and conversation. Each week I would grade each student’s participation based on a rubric, and give qualitative feedback for each student, including things they posted that were good, and suggestions for the coming weeks. The final grade was based on the final document, discussion posts, and Desmos Geometry interactives I asked them to create.
The Good.
Right away, I noticed that there were advantages to running a class in this format as opposed to synchronously or face to face. In face to face classes, conversations can be dominated by those who speak more quickly or more openly. Even in a class where everyone was willing to share, time constraints only allowed for so much conversation to be had. On the boards, I found that when everyone could take the time to craft their posts and responses, that there was far greater and deeper participation. During the ice breaker phase, I found students writing quite illuminating posts and responses to each other. On the other hand, I had the time to write detailed individual responses to each student, which other students could read and respond to. It was kind of a best of both worlds of one on one conversation and group discussion.
Hello,
My name is XXX, I live in XXX and teach 7^{th}, 8^{th} and Algebra 1 at XXX high school. I have three grown children and have lived in XXX 29 years – raised in XXX. I am returning to college to get an endorsement in mathematics. I taught 5-8^{th} grade at a colony school for the last 13 years and was asked if I would come into town to teach just math. I wasn’t sure I was up to the task, but working with XXX and creating a fantastic math program has been a lot of fun. My goal with mathematics is to reach tohe struggling students and encourage them to trust me and we will understand math together. Most of my students are terrified of math (As am I – just a bit) and hopefully when they leave our school they will feel mor secure in their math skills. I feel like I will learn to ask better questions and I like the fact that we are working together as a term to reach our final solutions. I think it will free us up to explore and take risks. Thank you for taking the time to read this and I look forward to working with everyone.
I don’t know if I’m more blown away with how geometry relates to all math or that I didn’t realize this years ago! These videos weren’t very long, but they started to open my mind about how geometry relates to algebra and calculus in ways that I’ve never thought of before. The math concepts in the videos were not new, but the way the information was presented is a different way of thinking.
Hi,
I know exactly where you are coming from…. Math does not come easy to me either and I truly feel it is to our advantage – we understand the “not understanding” and I do believe that we have great empathy! I am like you – for some reason I really like working my way through a problem – I never would have guessed that one day I would be a math teacher.
XXX,
I too was blown away with how all other math subjects seem to have stemmed from Geometry! I really didn’t know the history of math at all and never thought to explore it any. Like you said, the videos really opened my eyes to the connections between all the disciplines of math that I didn’t actually see before!
Likewise, there were unexpected benefits of the discussion board for collaborative proof writing. The asynchronous format allowed for wide flexibility in responses. Students could write long detailed posts, or short quick responses. They could also include pictures, videos, or links to interactives such as Desmos in their responses to help illustrate their ideas. Having a written record of the collective conversation also made it easier to reach back and pull from an earlier idea, or make minor edits to someone else’s argument.
Here are my initial thoughts on Problem 5, which reads:
“Problem 5. Prove or disprove the following: Each point belongs to a line.
I think you would agree that any geometry that contains two points not belonging to any line is not interesting. Therefore, we need the next axiom, Axiom 4. If A and B are different points, then there exists at least one line that contains both A and B.”
I felt this statement could be proven.
It makes sense to me that any point either has to be on line L (since A3 says at least one line exists) or
outside of line L, in which case A4 says that point has to make a line with one of the points on line L. Thoughts?
Problem 18: Suppose ABC and ACD.
Q1: Is it possible that two of A, B, C, and D are the same?
Initial Thoughts: By A5, A, B, and C are different points and A, C, and D are different points. It could be possible that B and D could be the same point. Suppose A, B, C, and D are all points on a circle, where B and D are the same point. ABC results from going clockwise around the circle and ACD (which equals ACB) results from going counter clockwise. This would prove that it is possible for B and D to be the same point.
Edited to add: I’m not sure this would satisfy A4.
Q2: Do they all belong to the same line or is it possible that A, B, and C belong to one line and A, C, and D belong to a second line?
My initial thoughts: if a line exists, do all of the points need to be notated. For example, if there are points W, X, Y, Z on a line, would I need to list all of them? Or could I say WYZ – X would still be on the line, just not listed?
So for this problem, if I had a line ABCD… would it be appropriate to say within ABCD, there is ABC and ACD? If that’s the case, then it is possible that A, B, C, and D belong to one line.
XXX: I like your circle for matching points for question 1. I was thinking of an analog clock where 12 am and 12 pm are the same point but different times.
This is what I imagined for question 2…
XXX:
Initially, I thought the same as XXX. Then after rereading AB(b), I think there exists a point between A and B (labeled “D” in AB).
I think AB could apply to intervals (it doesn’t specifically mention lines). Based on Abb, I think an interval must contain at least three points.
[reply]
XXX:
I approached this in terms of AB (b) as well. If two distinct points exists it makes sense that there have to be points in between them. Part of Definition 3 says that interval AB is “the set of all points between A and B inclusive.” That, combined with A8(b), which says there has to be at least one point between any two distinct points, makes me think that any interval has to contain at least 3 points.
[reply]
Here is another run at it from the top, considering A9’s role in forcing location of points. I’m still not sure that I am properly applying A9 though.
https://www.desmos.com/geometry/ohgnzbybvs
I found that the students were active and engaged, treated each other with respect, and worked diligently towards the goal of solving each problem. At the start of the course, I was very active on the boards, responding to each post right away, asking questions, prodding or challenging them. This was mostly to model the type of interaction I wanted to see in the course. As the class continued, I was able to step back as they took on these roles themselves, prodding and nudging in them occasionally. It was great to see them come into their own as mathematicians!
At the end of the course, they turned in a beautifully written document, including several proven conjectures, that they wrote together in Overleaf, coordinated by a student driven effort on the discussion board. Their feedback, both on the boards and in evaluations, were overwhelmingly positive, I could not be happier or more proud.
The Ugly.
I had vastly underestimated the number of students who would be in this class. I ended up with 16 students, and in some weeks there would only be a handful of problems. The result was that there would be sometimes redundant or disjointed conversations, which could be confusing to peruse. Even within a thread, discussions would grow quite long, and after a few layers of replies, my learning management system no longer kept track of who was replying to whom, making conversation difficult to track at times. In the future, I plan on assigning small groups, either per section or through the semester, so that conversations can be more streamlined and tractable.
I had not planned on having the final document be written collaboratively. I don’t regret acquiescing to the students, as their argument was cogent and I believe it gave them a greater sense of ownership of the course. I do wish I had built a collaborative writing component into the course from the ground up, following the ideas of Wikitextbooks by Brian Katz and Elizabeth Thoren. As a result, there was quite a bit of inequitable work distribution by the end of the course. With the classes permission, we resolved this by having some students write bookend introductions, summaries and conclusions in the document, which did improve the final document, but it was an ad-hoc solution, and could have gone badly.
The Future.
I found the Discussion Board format to be an extremely effective way of delivering an inquiry course, and in fact the only way I could have reasonably delivered one this Summer. I’ve incorporated the boards in other classes, incorporating the “Investigate!” sections of Oscar Levin’s Discrete Mathematics text into my Discrete course, and in lieu of face to face office hours due to the pandemic. This coming Spring, I plan on teaching a Linear Algebra using Drew Lewis and Steven Clontz’s Team Based Linear Algebra, adapted for discussion board conversation.
Some advice I would offer is:
In an uncertain and precarious time, most of us have already delivered courses in modalities we may never have considered. Many of these modalities may work very well for most students, but not for all. Depending on the needs and limitations of your students, know that it is possible to design a rich, engaging and meaningful inquiry experience for students in an asynchronous format.
]]>Nadia looked at me with big brown eyes and asked a question. My Spanish is minimal, so I called over a coworker, one of the caregivers at her shelter. She was working with tangrams (a geometric puzzle), and was asking whether she could turn a particular piece sideways to form a certain shape. This was not how the question was translated, and probably not how it was posed. But I understood it, despite the dual barriers of language and formality.
Nadia is a migrant child who has been separated from her parents and is under Federal custody with the Office of Refugee Resettlement (ORR). She may have come without authorization with a “coyote”, or been left with a relative and picked up in a raid, or just walked over the border herself. I do not know how she got here. But her bright eyes and her engagement with geometry tell me all I need to know. Her mind is alive, and I want to keep it that way. Like most of these children, she is resilient and resourceful. And like most of these children, highly motivated. These are immigrants, and immigration is a filter. Only the most energetic and future-minded are likely to pass through.
I am working today with three other facilitators at Catholic Charities of New York. Twice a week, from 11 AM to 2 PM, two or three of us meet with Dr. Usha Kotelawala, the director of this program (Math on the Border) for the Julia Robinson Mathematics Festival. They meet at the office of Catholic Charities, in lower Manhattan. The children are there to meet with a lawyer (typically for 20-25 minutes) to prepare for a court date that will determine their fate as immigrants. But they must wait around for hours before it is their turn. During that time, we engage them with mathematical puzzles, games, and activities.
The children love it. Their eyes light up. They intrigue each other. Language and social barriers tumble. And their minds are active. The work is similar to leaving food and water in the desert for thirsty immigrants. We are not offering them a complete diet or significant sustenance. But we are keeping their minds alive until their situation stabilizes.
We have been working with this population since November 2018, for six hours each week. To date we have had 216 hours of contact with more than 1000 of these children. We never know how many children will be in attendance. There can be as few as five, or as many as 25. The average size of a group is 15, and we have three facilitators, again on average, to work with them.
The teaching requires skill, but is not difficult. The children engage readily, and work with each other on the activities. If it is a game, they will challenge each other and arrange impromptu “tournaments”. If it is a puzzle, they will work together towards a solution. The children can be as young as 5 years old, and as old as 17. (At age 18 they “age out” of this program and are treated as adults.) In one case, a teenage girl brought her infant daughter to the session—and participated while attending to the baby.
Nadia, for example, has come with two younger siblings—or maybe cousins—and the three of them work on the tangrams puzzle. Nadia, as the oldest, takes the lead. Her two companions are excited to work with their older sibling on this “advanced” puzzle. A group across the table gets interested in the brightly colored plastic pieces and wants to know what the game is about. Soon they too are working with a set of tangrams.
I am part of a pool of about 20 facilitators. Since the children only see us once, it is a different group every week. Hence facilitators need not commit a large part of their schedule to the program. Most of them are retired teachers or STEM professionals. They typically know how to relate to the children, and understand the mathematics and its value. A minimum of on-the-job training is typically all that they require, and Dr. Kotelawala supervises that process. We usually achieve a ratio of 1 instructor to 6 or 7 students, which is perfect for this informal situation.
The backgrounds of the instructors reflect the demographics of the group of retirees from which they are drawn. Some of them have been mathematics specialists—we could not buy better expertise. Others have a particular interest in Latin America. One of them is starting a school in Nicaragua. Another grew up in Venezuela, the child of American engineers working there, and speaks colloquial Spanish as well as his native English.
Another valuable group of facilitators is college or graduate students. We get them whenever we can, and they are some of the most effective instructors. They typically speak Spanish: many of our connections are with Hispanic student groups. They have often had experiences similar to those of the children, and can offer themselves as role models, however brief the encounter. And they know the mathematics. Unfortunately, our program runs from 11 AM to 2 PM on weekdays, so these students are not regularly available, except during vacations or exam days.
The activities do not require any particular background of the children. They have been intensively field-tested. These are low-threshold, high-ceiling problems that can be worked by anyone with an interest. And these students are interested. They show the typical immigrant enthusiasm for learning. They engage readily and joyously in the activities. Their faces shine.
About two hours into our work, Nadia had to leave. It was her turn to have a legal consult. She took her two younger charges, said goodbye, and went off to see the lawyers. When she returned, she was not so happy. We don’t know quite what the lawyer said, but it doesn’t matter. Typically, the students are pensive and serious after their brush with legal reality. It takes them some time to re-integrate into the group and engage in mathematics. For them, time spent with us is a respite from concerns about the future. For us, it is a rewarding and uplifting experience.
EPILOGUE
In March, 2020 the COVID emergency precluded our meeting these children. Children are still being held without their parents by ORR, although fewer have been crossing the border. Perhaps by the time the COVID plague lifts, such children will be reunited with their families quickly, and we will not have need for a Math on the Border program.
(Math on the Border was partially supported by a generous grant from the Alfred P. Sloan Foundation.)
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Our conversation began with a conjecture about a line tangent to the graph of a function—specifically, about the interval between the point of tangency and the point at which that tangent line intersects the graph a second time—then moved to averages in finite and continuous contexts. These two types of averages are the “different things” we study here.
Poincaré’s “different things” are generally understood as objects such as vector spaces and groups, not constructs such as averages. Perhaps he would have considered our appropriation of his aphorism to be illegitimate. For the purposes of teaching and learning mathematics, however, “the same name for different things” fits finite and continuous averages. We (authors and perhaps readers) conceive of these averages as different instances of the same thing. It’s unrealistic to expect students to think this way. School mathematics remains procedural and formula-laden. Vis-à-vis finite and continuous averages, addition and integration are different computations, and the formulas look different. It’s likely that most students perceive finite and continuous averages as different things.
The mathematics of continuity and the infinite are barely present in school mathematics, yet many students wonder about, even puzzle over, continuity and the mathematical infinite. How might being unable to express their puzzlements and pose their questions affect these students’ mathematical learning? We take up this question later in this post.
Our work began with Dan’s conjecture to Whitney and Bill about a tangent to the graph of a function: “[o]n the interval between a point of tangency and a point of intersection farther down the line…I think the average value of the derivative function is equal to the derivative at the point of tangency.”
Bill thought of the Mean Value Theorem (MVT). Dan’s observation felt intriguingly strange to him, especially when he labeled the point of tangency as (a, f(a)) and the second intersection point as (b, f(b)), then expressed the conclusion as $f'(a)=\frac{f(b)-f(a)}{b-a}$. The MVT guarantees a c between a and b such that .
That is, it vouchsafes that somewhere between the two points of intersection, the tangent line is parallel to the secant line connecting these points. Dan appeared to have done the MVT one better by discovering that the point of tangency is always a “somewhere.” Closer attention to the MVT brought back the nuance that its conclusion insures a strictly between a and b, meaning that
doesn’t improve on the MVT in the case of a re-intersecting tangent.
Whitney approached Dan’s observation differently, beginning with a generic picture.
She drew lines parallel to the line through the point of tangency (a, f(a)) and the point of intersection (b, f(b)), whose slope is
(call this line LT), to see if it made sense that there could exist a line parallel to LT that would be tangent to the graph of f(x) at some point c in (a,b). Intuitively it made sense; however, it was not clear why LT’s slope is referred to as the average rate of change (ARC) of f from a to b. This name did not seem to fit. It made more sense that the ARC would be the average of all of the derivatives over (a, b). Since there was no easy way of computing this average—we can’t sum the derivative over all points in (a, b) and divide by how many derivatives there are—Whitney accepted that is a reasonable estimate to what she thinks of as the ARC.
Whitney’s thinking made a strong impression on Dan. The computation gives the value of the slope of the secant line, LT, connecting (a, f(a)) and (b, f(b)). Dan ventured that many students, having been taught that the derivative computes the instantaneous rate of change, might well wonder, “Why should this value be the way to compute the average of all the instantaneous rates of change on the interval,” if that’s what the “average rate of change” might mean.
Until Whitney brought it up, Bill hadn’t conceived of “the average value of the derivative function” as the mythical arithmetic average of the derivative’s values at all points in the interval. Average rate of change caused him to think of how “average” is used in other parts of analysis. For him, the phrase “average value of the … function” evoked only the Calculus II sense of defining the average value of any continuous (or just integrable) function f on [a, b] to be . The semantic logic behind Whitney’s conceptualization of the ARC enthralled him.
Whitney turned to a calculus textbook (Simon, 1992, p. 149) for a curricular depiction of the ARC. She found herself lingering on a passage from the MVT section.
The statement that “the instantaneous rate of change of f equals the average rate of change of f at least once in the interval (a, b)” unsettled her. In students’ prior experience, when they compute an average of a set of values, the result is not necessarily a member of the set. The average rate of change isn’t a familiar “average.” First, it is not the average of all of the pointwise rates of change of f on (a,b); now it is being equated to one of these instantaneous rates. Averages of a finite set of numbers do not behave in this way, and Whitney’s curiosity was piqued. When comparing these ways of taking an average, the salient difference seemed to be that one set is continuous and one is finite. What is it about continuity that would cause an average to be a member of the set of values that one is averaging? Although the MVT guarantees this outcome for rate-of-change functions — as does the MVT for Integrals for continuous functions in general — we’re unaware of a proof that explains it.
A second way in which finite and continuous averages are “different things” surfaced when Whitney revealed her dissatisfaction with the denominator of , the expression for the average value of any continuous function. Even though (b-a) is also the denominator of the average rate of change , how is it in any way analogous to the number of numbers that we use in the denominator of an average of a set of values? Absent a satisfactory explanation, it was difficult for Whitney to have complete confidence in the definition of a continuous average.
Out of the tangle of ideas spawned by Dan’s conjecture (most of which we spare the reader), the two questions italicized were most prominent in our consideration of finite and continuous averages as different things with the same name. The following dialogue is a pared-down version of our Zoom conversation about these questions and more.
Dan: My Algebra 1 students helped me with the (b – a) in the denominator of the average rate of change between two points. In thinking about how to compute the average fuel efficiency of a car, one student emphasized that the number of gallons is like the number of people when one computes an average bonus per person. It’s “how many ones you have,” to which you are equally distributing the total.
Whitney: My concern is about the average value of any continuous function, derivative or not. I don’t see how to use your student’s insight in the continuous case!
Bill: Perhaps I do. Almost 45 years since first teaching calculus, until our conversations, I remained uncomfortable with thinking of b – a as the continuous analogue of the number of data points in a finite set.
The discomfort was strongest with the average value of a continuous function, and starkest when I placed the expressions and side by side. Thinking of integration as the continuous analogue of addition made it make sense for the numerators to correspond. But the denominators? I could conceive of both n and b – a as measures of the size of the set over which numbers are being averaged. This association was too vague and weak to assuage my discomfort.
Dan, it took your Algebra 1 student’s wording, interpreting b – a as “how many ones you have,” to help me with b – a in the denominator of a continuous function’s average value. This interpretation set off a chain of realizations: The number of elements n in a finite set is calculated by adding 1s, one 1 per element. This is . The continuous analogue of this expression is . And the value of this integral is b – a. Finally, it made sense for b – a to correspond to n.
Finite average:
Continuous average:
Whitney: If we are going to use the word average in the phrase “average value of a function,” then we should compare how this “average” is the same and how it differs from the average of a finite set of numbers. In providing some intuitive thinking on the MVT, the excerpt from my calculus book states, “… the instantaneous rate of change of f equals the average rate of change of f at least once … .” This is not the case for finite averages. In many instances the average of a finite set of numbers is not also a member of the set of numbers. This is a fact that we emphasize to students when they compute averages. It creates a probable point of confusion for students. Why must a function’s ARC over an interval be assumed in the interval? In order to make sense of it I would look to the differences between conditions in the MVT and an average of a finite set. Specifically, the MVT presupposes continuity on the closed interval (and differentiability on the open interval). Changing a constant function’s value at an endpoint shows that a discontinuity can prevent the ARC from being assumed. So it seems that continuity should play a key role in the different ways we employ the term “average.”
Bill: When I ponder which differences account for this discrepancy between finite and continuous averages, the conjunction of continuous functions’ properties and the real numbers’ completeness satisfies me. In particular, the theorem that a continuous function on a closed interval has a minimum and a maximum, and takes on all values in between, carries much explanatory power. When you couple this property with completeness, there is no other value that exists to occupy a space in the function’s range between its minimum and maximum.
Whitney: Although I see how you are describing the two situations as parallel to each other, I still think this is an unsettling example of using the same name, average, for different things (the average of a finite set of numbers; the average value of a continuous function). I think this a particularly extreme case; it is a big move from the finite and the summation to the continuous and the integral. The art in using the same name for different things is being able to communicate what similarity warrants using the same name, while not overlooking differences, and, in the context of teaching, addressing the similarity and differences with students.
It was obvious to me as a student that there are counterintuitive and ambiguous ideas that professors never addressed but needed addressing. I always wondered why, and I still do. As a learner who is an African American woman, I often felt as though my instructors did not think it was worth their time or effort to explain things to me because to them, in some way, I was unworthy or unable to understand what they had to say.
Dan: It seems that you are using Poincaré’s statement to reflect on your experiences as a student; this reflection not only allows you to articulate your questions around averages, but also to address your experience as a Black female learner of mathematics. How does Poincaré’s statement help you understand your experience as a Black female mathematics learner?
Whitney: Yes, many might be surprised at how my experiences as a Black female have impacted my experiences as a learner of mathematics. I think I stuck with math for so long first, because I really like it, and second because I never wanted someone else to determine for me if I was able or willing to learn mathematics. As a Black female I have learned through my family teachings and through life experiences that it is not wise to take things at face value. It is never obvious what is trustworthy and reliable and when people may be leading you astray. I naturally brought this skepticism to mathematics classrooms and learning the subject. Thus my questions were essential to my understanding, but just as essential for me to be able to comfortably rely on what I was learning. If I could not find reasonable responses to my questions and justifications for the mathematical ideas I was being taught, then there was no reason that I had to accept them as true. Although I could still work with them in most cases, I was unable to totally embrace my learning or be completely comfortable with the ideas.
When my professors dismissed my questions, they were also dismissing me as a person. This made me skeptical about their commitment to my learning and it made me skeptical about the mathematical ideas I was learning.
Bill: This White male’s mathematical coming-of-age was quite different. Two high-school teachers took an interest in me. One gave me two SMSG books I’ve kept through floods and fire for 50 years. The “New Math” introduced me to axiomatics, and eagerly I bought in. I don’t doubt that feeling safe in my melanin-poor skin contributed to being able to keep faith with the axioms and resulting theorems. Whether it was New Math, old math, borrowed math, or blue math, I cannot recall doubting the truth of any true mathematical statement made by an author or teacher.
Whitney: If my calculus instructors had been willing, or maybe able, to let me know that the foundations of the real numbers are vital to what we were studying, it would have impacted how I oriented myself as a learner to the course material. We aren’t giving people—not only people like me—a full enough story about continuity, let alone completeness, to appreciate the character of the real numbers.
Bill: No question. We have no choice but to keep our own counsel on some of the full story’s characters and plot twists. None of us believes that calculus syllabi should include a unit on the real numbers as a complete metric space or equivalence classes of Cauchy sequences of rational numbers.
But neither should we stand aside while more generations of students endure a school mathematics inattentive to the real numbers’ intricacies. We consider factoring sums of cubes and arithmetic with radical expressions precalculus necessities, yet we ignore the properties of real numbers upon which limits, continuity, differentiation, and integration all rely as developed in elementary calculus. Doesn’t seem right.
Whitney: First, I agree with you to an extent. I would not advocate for teaching those ideas in their entirety in a calculus class. However, there are ways in which instructors can set a larger mathematical context for the ideas that they are teaching. Teachers could also address using the same name for different mathematical objects or processes.
Second, what you say connects to what I said earlier. It feels that continuity is a very deep concept that many people may not understand. Those who you may think understand it (i.e., professional mathematicians) apparently have much difficulty helping others to understand or appreciate it.
Dan: Whitney, you’re clearly placing learners at the forefront of your considerations. The earlier post confronted challenges students might have with shifts in the meaning of “tangent.” In addition to your point about learners who are highly skeptical of the mathematical authorities, you seem to be arguing that, at this time, mathematical differences between the finite and the continuous may not be well conveyed to learners.
Whitney: For me, many things have been uncovered in this discussion. I think we all want all of our students to see the wonder of mathematical ideas and to experience the joy of coming to understand difficult concepts. Also, we all knew going into this conversation that teaching mathematics is a complex art. Highly effective mathematics teachers are adept practitioners with a deeply woven and intricate knowledge of mathematics and various types of human experience. It is not sufficient to show students how to solve a variety of mathematics problems. We mathematics teacher educators should stress the importance of respecting students’ capacities for learning and making mathematics accessible by opening up the world of mathematics without watering the ideas down.
There is much room for what Poincaré has said about mathematics. Teachers can more accurately present mathematics by taking students’ questions seriously and helping them make connections to seemingly unrelated mathematics that students have previously studied.
Of particular importance is helping instructors to be mindful of the move from instances in finite contexts to those in infinite contexts. This transition is at the crux for much of mathematics and historically has been the site for much debate and the development of new ideas. Instructors should have some responses to questions like, “How have mathematicians in the past set out to mathematize the infinite? Which attempts were more successful than others and why? Why did the discipline set upon the path that it did and not on other paths?” These are questions that can give clarity to students—clarity in the sense that they can know more about the contexts for the ideas, which helps them understand the ideas themselves. Attending to these questions also helps students to see which questions are worth an investment of their time and energy and which are not (which decisions were made for the art of mathematics and which were made out of necessity). It may or may not help in understanding particular mathematical processes or solution methods. But we can remove mysticism and dogma and allow students to perceive mathematics as a human-made body of knowledge that they can make sense of if given a fair opportunity to do so.
And it’s further complicated by the fact that mathematics teachers are often teaching students who are different from themselves. Teachers are continually confronted with the ways in which their students make sense of the world—ways that may be alien or clash with their own. They can try to rope off the mathematics classroom or the learning of mathematics from the rest of the world, but this pedagogical strategy is unlikely to work.
Many students choose to study mathematics because, at the early stages of learning the content, everything appears to be unquestionable and reliable. However, as one continues to study mathematics, the complexity and vulnerability of the subject is revealed.
When students experience the world as always in flux, and always having to decide whether the people they are interacting with, the contextual clues, and the knowledge they are being given are sound and trustworthy they develop a healthy skepticism to navigate daily life. Chances are that these students will bring this level of skepticism to the learning of mathematics. Based on my own experience, students may have questions that challenge basic ideas, facts, axioms, theorems, etc., and they may need a reliable, sound response to their questions before they can believe or accept what they are learning.
By dismissing the integrity and core being of students, insufficient answers to these questions may inhibit their mastery of skills and understanding of mathematical ideas. A part of the art of teaching is understanding and respecting this skepticism in students and preparing to offer answers to students that honor who they are as human beings in a complex, confusing world. If the questions that I posed to my mathematics professors had been treated with respect and as valuable, my experiences in their classes and as a math major overall would have been profoundly different and better. I might even have a Ph.D. in mathematics instead of in mathematics education.
Reference
Simon, A. B. (1992). Calculus with analytic geometry. Glenview, IL: Scott, Foresman and Company.
]]>A child’s insight
“I know how to find out how many divisors a number has. You factor it into primes….” Alejandro was with a virtual group of four enthusiastic ten year olds, in the midst of exploring a problem. He gave the usual result, using his own somewhat makeshift words. But not too distant really from what I would have said: If $N$ factors as $p_1^{a_1}p_2^{a_2}p_3^{a_2} \dots$, then the number of divisors is $(a_1+1)(a_2+1)(a_3+1)…$. His description was less economical, but still accurate.
His virtual friend Xue said: “That’s great. Let’s look it up on Wikipedia.”
Then, “No. Let’s not look it up. Let’s pretend we don’t know it and see if we can prove it.” It is this insight into his own learning, not any mathematical breakthrough, that I remark on in the subtitle to this section.
Dear Reader: I swear to you, on Galois’ grave, that I am not making this up. Nor the rest of the vignette I will be recounting here.
The venue
This spring, in reaction to the COVID crisis, I was part of a team developing an online ‘webinar’. The team was from the Julia Robinson Mathematics Festival (JRMF) program. At the time, I was its Executive Director. In normal times, we run non-competitive after-school mathematics events (“Festivals”) in which students are offered interesting games, puzzles and problems, assisted by a facilitator. Since face-to-face work with students has lately not been possible, we have sought to continue the work virtually.
The program has met with success. The JRMF team works on the presentation of a problem each week, polishing it for a group of about 200 students who `tune in’ to the event internationally. The students are split into groups of fewer than ten, and put in breakout rooms to discuss the problem. An adult facilitator guides the discussion, not to achieve a particular goal, but as a moderator, letting the students’ insights emerge naturally. Facilitators meet for half an hour after webinar, to pool their experiences and offer ideas for refining the program.
Problems are “Low threshold, high ceiling”. That is, very young students can work on them, have fun, and achieve insights that will eventually take them farther. More advanced students can use them to engage in thorny issues or deep mathematical concepts.
For examples of such problems—and an open invitation to participate in these webinars—see www.jrmf.org. We post the problem on Mondays and discuss them on Thursdays so participants have time to explore the question. On Thursday, we ask students two questions: (1) How long have you worked on the problem? (2) What is your age? They are assigned to breakout rooms depending on their answers to the questions.
Attendance has grown steadily. We find that students who come to the webinar tend to return. Thus we have created a virtual community, all around the globe, of students who enjoy mathematics.
The Zoom Room
Last week, I was assigned a room with four energetic and highly motivated young students, ages about 10. The facilitators were familiar with these four. When we first started the program, we found them difficult to work with. They had often gone far into the problem: the amount of time they spent on it could not tell us that. They often had bits of mathematical background that other students lacked. And their youthful and overflowing exuberance made it hard to integrate them into a group. They were always a challenge to any facilitator.
So we decided to create a special breakout room for them, the “Zoom Room” where they could race ahead. The success of this effort varied with the mood of the children. At best, they urged each other forward and vied with each other for insight. At worst, they would try to show off to each other what they already knew, without contributing to either the group effort or their own knowledge.
This past week the group clicked. I was delighted to find that the four boys (they were all boys) worked beautifully together as a team, and got further than any one or two of them could have in the short time available. I led them with but a light touch of the reins.
They did not solve the given problem. They didn’t even work on it. They created their own, and the last thing I wanted to do was confine them to what I thought they should be learning.
Here was the problem we had set (briefly): Given a large square with integer sides, how can you tile it with smaller squares. also with integer sides? The problem was presented in a more structured way, to offer `on ramps’ to the mathematics. An interesting problem, combining elements of combinatorics and geometry. And, as is typical of JRMF problems, it can be worked on many different levels. I was eager to find out where the discussion would go with my four young students. It took a turn that I could not have predicted—or prepared for.
They looked at the first problem and immediately answered that for $1 \times 1$ squares, you can tile any $N \times N$ square. The important point here is that they saw this as a special case: it was a sophisticated insight for children that young. They then went on to consider the question of tilings with $2 \times 2$ squares. I asked if you could tile a $7 \times 7$ square with $2 \times 2$ squares. They again saw that they couldn’t, and articulated the reason: 2 does not divide 7.
So I asked, “If $a$ does not divide $B$, then clearly an $a \times a$ square cannot tile a $B \times B$ square. But is that enough?” My point was new to them. It was the difference between a necessary condition and a sufficient condition. Very generally, I find that the core difficulty in learning mathematics—for anyone, at any level—is the logical structure behind the assertions or computations. Even these very experienced students had to take a minute to understand what I was saying.
In fact the condition is sufficient as well as necessary. They seemed to understand this particular example, but I am not so sure that they will understand the distinction between a necessary and a sufficient condition in another context. No matter. They are ten years old.
To guide the discussion a bit, and to get what I could out of their intense engagement in it, I asked how many ways they can tile a $7 \times 7$ square with identical squares. Dan (I am not using the students’ real names here) immediately said, “Only with $1 \times 1$, because 7 is prime”.
“No,” countered Alejandro. “You can tile it with one big fat $7 \times 7$ square. Does that count?”
“Well,” said Titus, “A prime number has only two divisors: one and itself. So we can use the same idea to count these tilings, if we count $7 \times 7$ as a tiling.”
Titus may have wanted simply to show what he already knew. But this seemingly innocent and perhaps boastful remark turned out to be a fertile one. Dan generalized immediately: “For an $8 \times 8$ square, there are four tilings.” (He meant tilings with identical squares, and everyone knew it.) “That’s because 8 has four divisors: 1, 2, 4, and 8.”
And this is where we came in. Alejandro took up Xue’s challenge, and his ten-year old explanation was wonderfully simple. “Say there are two primes, $p$ and $q$. Say the number is $p^2q^3$ You just make a picture.” And he drew this on the shared screen:
In another group, Alejandro’s explanation would have been a mystery. But these four looked at it and understood.
“You need a 1 to count the 1,” said Dan, “and also the singles: $q, \ q^2, \ q^3$.”
“Right,” said Xue. “So if $p$ is squared, you have three columns, not two. That’s why we add one to the number on top.” He meant the exponent.
“But what if it’s like $p^2q^3r^4$?” asked Alejandro… and answered his own question. “Oh. It’s the same thing. You can just list the twelve divisors we have already down the side, and list $r, \ r^2, \ r^3, \ r^4$ on the top.” As facilitator, I squirmed a bit at the error. But in this virtual environment, no one saw it. And knowing these kids, I remained silent.
“No,” said Titus quickly. “You need five columns: $1, \ r, \ r^2, \ r^3, \ r^4$.”
“That’s right,” said Alejandro. My silence had paid off: the point was made better and faster than I could have. The interaction at once exploited the benefits of kids working together and increased the bond between them. Boastfulness and ego were quickly put on the back burner.
I didn’t want to rest there. They could recite the formula. They could prove it. I wanted to make sure they could use it. So I asked them a question that they were unlikely to have seen before: What two-digit number has the most divisors?
Their thought was swift, and collective. They quickly saw that they had to look at prime divisors and balance the number of divisors with the exponent in the formula. All this without writing anything down.
Titus led off: “It probably should have lots of 2’s and 3’s. Because we don’t want the number to get too big.”
Xue: “Well, it can’t have more than six 2’s, because $2^7$ is already 128. And $2^6$ is 64 and has seven divisors.” He had intuited the formula for the case of a single prime. I did not need to call his attention to this special case.
Titus again: “What if we put in a 3? Three times 32 is 96. It has. . . ” He thought a minute. “ It has $6 \times 2 \dots$ twelve divisors.” I didn’t have to ask him to explain.
Indeed, I didn’t have time. Alejandro jumped right in: “It depends on the exponents. The primes don’t matter. They just can’t be too big.”
Xue: “Can we have a 5 as a prime factor? Well, we can’t have two 5s. We can, but that will give us 25, 50, 75, and they don’t have enough divisors.” He was imagining what applying the formula would do, and his intuition told him (correctly) that these numbers would have fewer divisors than the 12 that they already saw for the number 96.
Dan: “And if we have one five, the rest of the number is 20 or less. We would need 6 or 7 divisors for that kind of number. Can we do it? ”
Silence.
Then Dan again: “Seven divisors can’t work. It’s prime. Six divisors? It’s $2 \times 3$, so we need $pq^2$. That’s $2 \times 3^2$ or $2^2 \times 3$. Eighteen or twelve. Five times these give 90 or 60. Each of these also has 12 divisors.”
Alejandro: “I don’t think we can beat 12. We just have to look at 2’s and 3’s. No. We can’t get 13 or 14 divisors. We would need too high a power.” (I didn’t stop him—everyone seemed to understand.) “Can we get 15 divisors? We’d need $2^2 \times 3^4$. That’s too big. Or $2^4 \times 3^2$. What is that? $16 \times 9$. No, still too big.”
Titus: “So only 12 divisors.”
I asked, “Which two-digit numbers have 12 divisors?” The list came tumbling out of them, and they all contributed.
Generalization
Unbidden, the group asked the next question: “What three-digit number has the most divisors?” They started working on this, and the ideas flowed. Ramsey Makan, my techical assistant, himself quite young, had been listening. The number 720 came up, and someone remarked that this was $6!$.
Ramsey asked them, “How many divisors does $6!$ have?” They worked it out. Then of course started thinking about factorials in general.
Titus was out of the discussion for a few minutes, then came back. “I wrote a Python program to list the divisors of $n!$.” They all wanted to see, so Titus ran it, for $n = 1$ through $6$.
“Can it do $10!?$” someone asked. Titus ran it for $n=10$. The screen went blank.
“The numbers are pretty big,” he finally said. “So it’s going slow.”
And indeed it was. The program was using brute force. I wanted to keep the momentum of the group up, so I said: “Can you figure it out yourselves? Maybe you can beat the machine.”
And they did. When the number finally popped up on the screen, it matched their result.
With time running out, I wanted to leave them with something to work on. So I said: “Suppose you know the number of divisors of 12!. Suppose some wizard told you how many there were. Would there be more divisors of 13!? Or fewer?”
The group responded easily: “More.” And then Dan said, “Twice as many. Because 13 is prime.” This was met by a chorus of “Oh, yeah.”
“But it wouldn’t work for 14! if you knew 13!,” said Xue. Then, a moment later, “What would work?”
They started thinking. Titus said: “Four times as many…”
Titus’s idea was not quite right. But the time was up. The breakout room was closing. I said goodbye and the webinar came to a close.
Conclusion
Teaching online can be tough. You lack certain means of communication: gestures, looks, posture. And if a student is silent, it’s hard to tell if he or she is engaged. My experience with these four students may not generalize easily. But it does give us a picture of what can happen when students encounter each other virtually.
And it gives us another picture. Sometimes it is argued that we must not do anything special for students who need more mathematics. It inflates their egos, makes them think that somehow they are ‘superior’. Well, it can, if done badly. And it can damage a child if we value the gift and not its bearer. The experience I chronicle here shows how important it is that such students meet other such students, that they come to see themselves as no different from a whole group of peers. My experience has shown is that once they are in such a group, they grind off each other’s rough edges. Being a ‘nerd’ or a ‘brain’ is a costly defense, like a suit of armor. And the defense, in such situations, is quickly and eagerly shed.
Acknowledgments
I would like to thank Ben Blum-Smith and Yvonne Lai for their invaluable input into the writing of this piece.
]]>For reasons that will not be considered here, I recently learned this dance:
Although I have no background in any style of dance, I can now do the whole thing, start to finish. I am very proud.
My purpose in attaining this objective was unrelated to mathematics or teaching. Nonetheless, the experience put an eloquent fine point on a certain basic dialectic in math education.
I spent a decade working in middle school and high school math classrooms before I trained as a research mathematician. Conversations regarding goals for students in elementary and secondary math education, and math education research, often distinguish between two types of knowledge: procedural and conceptual. These are fraught words, and you have your own ideas about the meanings.^{[1]} Nonetheless, for the sake of clarity (at least internal to this blog post), I will offer some definitions.
Conceptual knowledge: knowledge of what things really are, what they are all about, and how they are connected.
Procedural knowledge: knowledge of how to actually do things.
I hope with these definitions that I have not accidentally tripped any wires. If your background is anything like mine, the mere mention of this dichotomy may have already given you some unpleasant flashbacks. In one of my first teaching jobs, almost every department meeting eventually devolved, in a practically ritual way, into a bitter fight. And one of the perennial sticking points was which of these two knowledge types deserved priority. Those days were a high-intensity period in the Math Wars, and the “procedural vs. conceptual” dichotomy served, in my experience, as a kind of a “Math Wars bat signal”: once it came up in a conversation, powerful ideological fault lines showed up soon after, as though they had been summoned.
The terrain has shifted a bit since then. It eventually became fashionable, uncontroversial—indeed, obviously true—to assert that these two types of knowledge are both important, and are mutually reinforcing.^{[2]} Interest has grown in creative ways to serve both masters at once.^{[3]}
Nonetheless, educators still often have a propensity one way or the other at the level of educational values and aesthetics. For some, a calculus student who can differentiate elementary functions flawlessly, but doesn’t know what any of it means, ‘hasn’t actually learned any math.’ To others, ‘at least they can solve the problem!’ For some, it is distressing and concerning when a fourth grader can accurately identify a wide range of contexts modeled by subtraction, but can’t compute except by counting down on their fingers. Others feel this student has already learned the hard and important lesson, and believe that this will make learning a better computational method easy. These differences can persist even among educators who believe passionately in the joint value and mutual complementarity of the two types of knowledge.
For example, I fall on the conceptual side. Not intellectually: I believe strongly that mathematical knowledge comes in both types, that they’re both crucial, and that they’re mutually supportive. Every time I reflect on my own learning with this question in mind, it’s obvious how much my procedural knowledge has done for me. That said, I’m simply more passionate about teaching concepts than procedures. I am lit up by the challenge of getting students to perceive an unexpected connection or to understand the purpose of an important definition. I can also get excited about the how-to-do-this stuff when I know it will make my students feel powerful, or put them in a position to think about a particular interesting question or concept, but even in these cases it’s a means to an end. Meanwhile, my heart sinks a little when I read student work that evidences thoughtless application of a formula, even if the answer is correct.
These differences in taste can shape our curriculum design and our teaching choices even if we believe at the intellectual level in the importance of both types of knowledge. For example, my gut orientation toward conceptual knowledge means that when a student presents as stuck or lost and asks me what to do, my first instinct is always to pull their attention away from that question, down to the level of “what is this all about, and how is it connected to other things you know?”
I don’t think there’s anything wrong with these tendencies toward the conceptual or the procedural, and in any case, we have them whether we like them or not. But because they shape our teaching practice, I do think it’s useful to recognize them. Sometimes, the thing a student needs is conceptual; other times it’s procedural. I think both types of bias have their strengths, but each can also lead to teaching blunders caused by failing to recognize the needs of our students.
For example, my strong habit of assuming that the obstacle facing a student is conceptual, can make it hard for me to recognize when a student has a procedural need that’s not being met. I, and I think many conceptually-oriented educators, have a tendency to see the procedural knowledge—what to actually do—as a consequence or corollary of conceptual knowledge. So if a student presents with a difficulty doing something, I (we) take aim at the concept of which the desired action is (to us) a consequence.
This does actually work a lot of the time! And, there are plenty of times when it doesn’t, because it isn’t always reasonable or fair to assume that the student can get from “I know what’s going on” to “I know what to do” on their own.
Much to my surprise, learning a K-pop dance routine provided me with an incisive opportunity to reflect on both of these possibilities—from the student side.
When I set out to master the dance from BLACKPINK’s 2018 hit song “DDU-DU DDU-DU”, it was kind of like learning to walk. My lack of any kind of dance training, combined with my gender socialization, meant that half the stuff Jennie, Lisa, Jisoo and Rosé do in the dance practice video was missing entirely from my movement vocabulary. But I was up for a challenge.
I started with the chorus. I got as far as the first “Hit you with that DDU-DU DDU-DU,” but that little 4-beat bouncy lean thing that immediately follows it—
[The video is cued up at the exact point I’m talking about, but you lose the cueing once you play it. To rewatch, reload this webpage.]
I mean, I was lost. Right shoulder down, right hip up, lean back, left shoulder down, left hip up… while the hands are moving? How do you do the weight transfer smoothly while you’re bouncing? How do you bounce and lean at the same time? Where do I put my head this whole time?? Trying to assemble this strange little movement felt like trying to hold too many things in my hands at once: something was going to fall. If I got my hips in the right place, I’d forget about my shoulders. Get the shoulders? Mess up the bounce. The idea of doing all of it at once felt overwhelming. The idea of ever making it look cute felt way out of reach. I needed help.
My wife has an actual background in a highly relevant field, namely hip-hop dance. Also, as it turns out, she is a completely conceptually-oriented dance teacher. Her first move was to tell me to stop thinking about what to do with each body part. Instead, she said, focus on the attitude. She illustrated it with other, more familiar movements that differed in their details but shared the attitude. “It’s like, ‘Eyyyyyyyyyy!'” she said, demonstrating.
The parallel to how I respond to analogous situations as a math instructor was extremely apparent. There was a main idea here. My wife was pulling my attention away from the impossible-feeling task of assembling the whole out of a bunch of disconnected details, and toward a single main idea from which all those details would flow. She was elucidating that main idea through its connections to more familiar knowledge. The main idea was what was important. The details would work themselves out.
It worked! By focusing on the attitude, everything came together. The bounce was nothing more than feeling the music. The whole thing with the shoulders, the hips, and the lean, turned out to be nothing but a right-to-left weight shift shaped by the appropriate attitude. The hands were, like, I mean obviously, I just hit you with that DDU-DU DDU-DU—now I have to put the “guns” away, and where else would they go? The entire motion felt logical and coherent, and I could do it without even thinking too hard.
Score one for the conceptually-oriented lesson!
I kept going. Exactly 7 seconds deeper into the chorus, there is a second “Hit you with that DDU-DU DDU-DU,” and again the four beats that follow it threw me completely:
It’s just a turn. No fancy roll/lean/bounce stuff this time, just rotate 360 degrees over four beats, stepping on alternating feet, and end up in that same little shoulder-shimmy as before.
But I wasn’t getting it! I felt off-kilter, gangly, uncoordinated. I felt I had to keep lurching, yanking my weight in different directions—this did not feel cute at all. I kept being late to finish the turn and set up for the shoulder-shimmy. Furthermore, I didn’t understand how it was possible not to be late. I repeatedly watched my wife and all four members of BLACKPINK pull it off, but this seemed like magic.
Fresh off our previous success, my wife again took a conceptual approach. To her, the main idea of the turn is to feel the beat in the alternation of your steps. She had me practice those 4 beats without turning, just stepping right-left-right-left in place.
This was easy for me—but this time, it didn’t actually help. My problem wasn’t, as it turned out, a failure to feel the beat in my steps. I realized I had a more fundamental question: where should I put my feet?
When my wife responded with, “It doesn’t matter,” I had a little moment of acute empathy for every student I’ve ever driven up the wall by insisting they focus on an underlying concept when they want me to tell them what to do. In that moment, I was the student who needed some concrete steps to follow (pun intended), and I wasn’t getting them.
On the one hand, in saying “it doesn’t matter,” my wife was obviously telling me the truth. The four members of BLACKPINK are at that point in the song rotating their whole formation. They’re all turning, it’s all synchronized, but they’re not putting their feet in the same places at all. My wife’s own rendition involved turning in place, so that was different too. All five of them—Jennie, Jisoo, Lisa, Rosé, and my wife—were evidently successfully executing the same fundamental dance idea, while putting their feet in different places. It follows that this particular dance idea is not determined by the locations of the feet.
On the other hand, I understood the underlying concept, at least as my wife was presenting it to me, but this understanding was not clarifying for me how to actually do the turn. She saw the procedural knowledge as an immediate corollary of the conceptual knowledge, but to me it was apparent that she was using some additional, not-entirely-conscious prior knowledge to translate this underlying concept into actual steps to take, and this was knowledge that I didn’t have.
This elucidated a mistake I’ve made countless times in teaching. The student is stuck and asks me how to proceed. I assume it’s a conceptual problem and take aim at the underlying concept. The student seems to understand the concept and is frustrated I won’t just tell them what action to take. Because the appropriate action, AKA procedure, feels to me like an immediate corollary of the concept, I assume that there’s a subtler, undiscovered conceptual problem still lurking. Because, furthermore, I fear that I’ll short-circuit the student’s opportunity to address this underlying conceptual issue by revealing the appropriate action prematurely, I hesitate to answer the question about what to actually do.^{[4]}
But sometimes, that’s what the student needs! The piece the student is missing may not actually lie in the concept, but instead in the way the concept entails the appropriate action—this is a kind of knowledge often not even visible to me, as focused as I am on the concept. In this situation, the student may need direct information about what to do. Seeing a complete solution demystifies this missing link, providing an opportunity to coordinate the underlying concept with the appropriate action.
This is what was happening to me with the turn. My only way forward was to directly mimic a correct example. I played the video back several times, focusing on Jennie—she’s the one in front at the beginning of the turn. Right foot steps out; turn 180 degrees on the right foot while swinging the left foot around the front; shift the weight; turn the other 180 degrees on the left foot, this time with the right foot moving backward; shift weight again; left foot behind right; step out with the right. The body is moving in the same absolute direction the whole time. Lemme try that…
It worked! Directly mimicking Jennie’s footwork gave me a structure to follow that solved the problem of how to turn around in exactly 4 beats without awkward direction changes. The abstract concept of feeling the rhythm in my feet could now inhabit the concrete set of motions I was following.
Score one for the procedurally-oriented lesson!
All of this is to say—we are hopefully on our way out of the false dilemma of procedural vs. conceptual knowledge, and toward a consensus that they are both critical, and are mutually reinforcing. Nonetheless, this wisdom can function as a bit of a platitude—preached, without always being lived. So I think it’s a worthwhile exercise to look, both in the classroom and outside of it, for opportunities to go beyond knowing it, to feeling it. And—who knew?—but learning a K-pop dance routine gave me the opportunity to feel it in my bones. Literally.
[1] Indeed, the lack of consensus about the meanings even extends to the possibility that by calling them knowledge types, I’m not being entirely faithful to the full range of their uses. See J. R. Star and G. J. Stylianides, Procedural and Conceptual Knowledge: Exploring the Gap Between Knowledge Type and Knowledge Quality, Canadian Journal of Science, Mathematics, and Technology Education Vol. 13, No. 2 (2013), pp. 169–181 (link), which argues that while the terms refer to knowledge types among psychology researchers, they are better seen as referring to knowledge quality among math education researchers.
[2] An illustration: In 2015, in the Oxford Handbook of Numerical Cognition, Bethany Rittle-Johnson and Michael Schneider wrote, “Although there is some variability in how these constructs are defined and measured, there is general consensus that the relations between conceptual and procedural knowledge are often bi-directional and iterative.” B. Rittle-Johnson and M. Schneider, Developing conceptual and procedural knowledge of mathematics, Oxford Handbook of Numerical Cognition (2015), pp. 1118–1134 (link).
[3] An example: M. Schumacher, Developing Conceptual Understanding and Procedural Fluency, on the Illustrative Mathematics Blog (link).
[4] While this and the next paragraph are focused on the situation in which I am wrong to withhold the “what to do” information, I hasten to add that this is, in general, a reasonable fear. If a student is in fact missing a conceptual piece of the puzzle, premature information about what to do may allow them to walk away from instruction with the belief that they have fully learned the concept when they actually did not. The student who applies a procedure in inappropriate contexts probably mis-learned it in this way. Judgement is required to determine what the student needs.
]]>CUNY Brooklyn
The forced conversion to distance learning in Spring 2020 caught most of us off-guard. One of the biggest problems we face is the existence of free or freemium online calculators that show all steps required to produce a textbook perfect solution. A student can simply type in “Solve ” or “Find the derivative of ” or “Evaluate ” or “Solve ,” and the site will produce a step-by-step solution indistinguishable from one we’d show in class. With Fall 2020 rapidly approaching, and no sign that distance learning will be abandoned, we must confront a painful reality: Every question that can be answered by following a sequence of steps is now meaningless as a way to measure student learning.
So how can we evaluate student learning? Those of us fortunate enough to teach courses with small enrollments have a multitude of options: oral exams; semester-long projects; student interviews. But for the rest of us, our best option is to ask “internet resistant” questions. Here are three strategies for writing such questions:
● Require inefficiency.
● Limit the information.
● Move the lines
Require Inefficiency
One of the goals of mathematics education is developing adaptive expertise: the ability to identify which of the many possible algorithms is the best to use on a particular problem.
For example, consider a quadratic equation. We have at least two ways of solving quadratic equations: by factoring; or by the quadratic formula. Which do we use? Since the quadratic formula always works, there’s no obvious reason why we would ever want to use anything else. But sometimes using the quadratic formula is like using a chainsaw to cut a dinner roll: we wouldn’t use it on “Solve $(3x-7)(2x+5) = 0$ ,” and we probably wouldn’t use it on “Solve $x^2-9 = 0$,” though we’d almost certainly use it on “Solve $6x^2 – 19x – 36 = 0$.” The boundary between the problems we’d attempt to solve by factoring and the problems we’d solve using the quadratic formula can’t be taught: every student has to find it for themselves through firsthand experience.
It should be clear that requiring inefficiency is a possibility every time there is more than one way to solve a problem. This approach works even better when one method is clearly (to us) less efficient. Indeed, the least efficient method is one that doesn’t work, and in some ways, requiring inefficiency in such cases may give us more insight into student learning than their ability to solve a problem.
For example, consider the problem:
If possible, solve by factoring: $x^2 – 3x – 12 = 0$. If not possible, show why; then solve using the quadratic formula.
Since the quadratic expression is irreducible over the integers, no online calculator will produce a factorization. Thus, a student can’t simply look up the answer. More importantly, in order to provide an answer, they must check every possible pair of factors (and show that none of them work).
There’s an added bonus. On the same exam, we might ask students to factor various quadratic expressions. We argue that a student’s attempt to factor $x^2-3x-12$ will actually reveal more about whether a student understands factoring than the successful factorization of an expression like $6x^2 + 19x – 36$. Thus, we can omit straight factorization questions (which, in any case, can be “solved” by an online calculator).
Limit the Information
Another way to thwart the use of internet calculators is to provide incomplete data. For example, Wolfram Alpha can find the derivative of any function—provided you give it the function. Thus we might ask students to solve problems without giving them equations.
This might sound hard to do, but it’s actually pretty easy. Since the 1990s, state and national mathematics standards have called for increased use of graphical and tabular representations, so source material is plentiful. Even the most traditional texts include problems based on interpreting graphical and tabular data. For example:
Suppose you know $f(3) = 5$ and $f'(3) = -4$ . Let $h(x) = ln f(x)$ . Find $h'(3)$.
While this is an algorithmic question that can be easily answered by invoking the chain rule, doing so relies on correctly interpreting the written statements about the function and derivative values. As such, it is currently beyond the capability of online calculators.
We can also present data graphically:
The graphs of y = f(x) (solid) and y = g(x) (dashed) are shown:
Find the sign of $(fg)’)(0)$.
Again, this is an algorithmic question that can be answered by invoking the product rule. However, it relies on being able to extract information from a graph, then make a quantitative argument based on the signs of the functions and their derivatives.
Moving the Lines
Requiring inefficiency and limiting information should be viewed as stopgap measures at best. Thus, when calculators were first introduced, math teachers insisted on “exact answers,” since the student who returned the answer “1.4142135” instead of $\sqrt{2}$ was clearly using a calculator. But now, even a \$10 calculator can return “exact answers” like $\frac {3+\sqrt {5}} 2$ , so this distinction is no longer useful as a way of distinguishing between students who used a calculator and students who didn’t. Similarly, while I’m not aware of any app that allows for the user to select a solution method, or that can read graphical or tabular data, there’s no a priori reason why there couldn’t be one. This means we need a more powerful method of creating internet resistant questions that can adapt to advances in technology. This leads to a strategy I call “moving the lines.”
To begin with, it’s important to understand that the problem “Solve $x^2 – 3x – 12 = 0$” does not exist outside of a mathematics classroom. So we should ask two questions:
● Where did this problem come from? This moves the “starting line,” where the problem begins.
● Why do we want the solution? This moves the “finish line,” where the problem ends.
Our long-term goal as mathematics educators should be to shift the lines and turn a sprint into a marathon.
Let’s consider this problem. What leads to “Solve $x^2 – 3x – 12 = 0$?” For that, we might consider some of the basic steps in solving any quadratic equation. One of those steps is to get the equation into standard form. So our problem “Solve $x^2-3x-12 = 0$” might have come from “Solve $x^2 – 3x = 12$.” In fact, you’ve probably asked this question before, specifically to identify students who failed to understand the necessity of getting the equation into standard form.
Now where might we have gotten a problem like that? We might have gotten it from “Solve $x(x-3) = 12$.” In fact, you’ve probably asked this type of question as well, to identify the students who failed to understand the zero product property.
Note that we still have an equation that can be dropped into an online calculator, so the next step is important: What type of question leads to a product equal to a number? There are many times we multiply two numbers to get a quantity of interest; for example, the product of a rectangle’s length and width gives us the area. This takes us to the problem:
A rectangle has an area of twelve square feet, and its width is three feet less than its length. Find the length of the rectangle.
In order to answer this question, a student would have to translate the given information into a mathematical form. This is beyond the capability of online calculators (especially if, as in this case, the numbers are also spelled out). If you enter the question into Google, you’ll get examples of similar problems, but no solution, effectively reducing you to your class notes and textbook. If you’re clever enough to switch numbers for the words, you’ll get an answer—which is incorrect (4 feet).
We can further improve the problem by changing the finish line. Remember that once a student translates this problem into the equation $x(x-3) = 12$, an online calculator can produce the algebraic solution, showing all the steps. One way to further blunt the ability of the online calculator to answer all questions is to require another step beyond the mathematical solution. Thus we should ask why we’d want the answer.
Let’s consider: we obtain the length (and width, since we know it’s three feet less than the length). So why would we want the length and width of a rectangle? There are three obvious possibilities: to find the rectangle’s area; to find the rectangle’s perimeter; and to find the rectangle’s diagonal. Since we already know the area, we might want either the perimeter or the diagonal. So we could ask:
A rectangle has an area of twelve square feet, and its width is three feet less than its length. Find the perimeter of the rectangle.
Even better:
A homeowner wants to fence a garden in the shape of a rectangle. The garden must have an area of twelve square feet, where the width is three feet less than its length. The fence will cost two dollars per foot. How much will it cost to enclose the garden?
The best part about this approach is that as technology advances, we can shift the lines in response. Perhaps some day we’ll be able to enter the above problem into a search engine and get the correct answer. So the next step will be to shift the lines again: move the starting point further back by imagining where the problem might come from; and move the finish line further forward by considering why we’d want to know the cost.
The Road Ahead
Notice that we end with something that might be called a “real world” problem. But a homeowner rarely has to build a garden with a specific area and relationship between the sides: it would be a stretch to call the problem above a real world example of how to use mathematics.
What’s more important is that real world problems don’t come with instructions on how to solve them, so they must be solved inefficiently, by trying different approaches until we find one that works. Real world problems don’t come with formulas attached to them, so they must be solved without complete information. And real world problems often change, so we must expect that the starting and finishing lines will change on us.
What this means is that regardless of when or if we can resume traditional resource-restricted exams, we should consider requiring inefficiency, limiting information, and shifting the lines on all our assessments. Sooner or later, our students will leave our classroom. If what they learned can be replaced by someone using a free internet app, then they can be replaced by a free internet app. So it’s not just about making our questions internet resistant: it’s also about making our students internet resistant.
Addendum
We’re stronger together. Readers interested in sharing their “internet resistant” questions should email them to me at jsuzuki@brooklyn.cuny.edu, and I’ll put up a selection of these in a later post.
]]>In 2012, 100 years after Henri Poincare’s death, the magazine for the members of the Dutch Royal Mathematical Society published an “interview” with Poincare for which he “wrote” both the questions and the answers (Verhulst, 2012). When responding to a question about elegance in mathematics, Poincare makes the famous enigmatic remark attributed to him: “Mathematics is the art of giving the same names to different things” (p. 157).
In this blog post, we consider the perspectives of learners of mathematics by looking at how students may see two uses of the word tangent—with circles and in the context of derivative—as “giving the same name to different things,” but, as a negative, as impeding their understanding. We also consider the artfulness that Poincare points to and ask about artfulness in mathematics teaching; perhaps one aspect of artful teaching involves helping learners appreciate why mathematicians make the choices that they do.
Our efforts have been in the context of a technology that asks students to give examples of a mathematical object that has certain characteristics or to use examples they create to support or reject a claim about such objects.^{1} The teacher can then collect those multiple examples and use them to achieve their goals.
Kayla: Algebra 2 students often get a super minimalized and overbroad definition of an asymptote. Many leave Algebra 2 saying something like “a horizontal asymptote is a line the graph gets close to but doesn’t touch.” In calculus, they get a limit definition for asymptotes. As a teacher, I’m prepared for students to enter calculus with the Algebra 2 definition—it’s acceptable knowledge for Algebra 2—but if a student left calculus with the impression that a horizontal asymptote is a line we get close to but don’t touch, I would be mortified.
Willy: I think the purpose of learning about asymptotes changes too, right? In Algebra 2, students are getting an overview of a lot of functions and their general behavior. At that point, it seems fine to have such a loose definition. Calculus introduces limits to explain function behavior at various parts of the domain. That includes wrestling with infinity.
Kayla: Yes, yes, but what I hadn’t noticed until recently was that students’ understanding even of tangent in calculus might be influenced by what they retained from geometry.
Willy: Right! The terms shift meaning a bit. When I took calculus and geometry as a student, I don’t recall any emphasis or discussion of a shift in the definition of tangent. In geometry, the only use of tangent that I remember was with circles: the tangent is perpendicular to the radius. That’s not at all how we talk about tangents in calculus.
Dan: And that’s Poincaré’s “giving the same name to different things.” David Tall (2002) argues that evolutions in definitions of mathematical concepts are natural in a curriculum—he calls the phenomenon “curricular discontinuities”—because you can’t unfold the complete complexity of a concept all at once. In different contexts, you think about particular dimensions of concepts. So it’s natural that when we’re just talking about circles, tangent is a special case of a broader concept. It’s one that you meet first. Lines whose slopes describe the instantaneous rate of change in graphs of functions are mathematically different, but it can make sense to give them that same name in order to capture some way in which they’re the same. Kayla, it sounds like you hadn’t thought as much about how differently the word tangent was used in calculus and geometry. What in particular, now strikes you as different?
Kayla: I believe most calculus students learn the new definition—how to derive a tangent, what it looks like, what it tells us about a curve—but I worry they may leave calculus still expecting tangent to mean “touching only at one point” as it did in geometry. I also worry that the geometric idea that the tangent line must lie on just one side of the circle causes some students to trip up and struggle in calculus when they encounter a tangent line that crosses the graph either at a point of inflection, or just at some other point. I also have students who think it is not possible to have a vertical tangent; they conflate the derivative being undefined with the tangent line not existing.
Willy: I wonder if that could be a result of trying to make sense of the idea that there is no linear function of x that will give a vertical line.
Dan: Kayla, it sounds like you’re saying that, on the one hand, there are things that are called tangents in calculus that wouldn’t have been called tangents in geometry and also the reverse, that there were tangents in geometry that calculus students would not think are tangents.
Kayla: Yes.
Dan: That’s really helpful, because it identifies a challenge beyond the curricular discontinuity of changing definitions. When definitions change, people might recognize and remember the changes—a changed concept definition—but the things that come readily to their minds might not change, what Tall and Vinner (1981) call a “concept image.” So really, Kayla, what you were saying is that only some of the things that come to students’ minds as tangent lines from a geometry perspective remain useful when they’re thinking in a calculus sense. A tangent sharing more than one point with a curve is acceptable in calculus but didn’t make sense in geometry; a vertical tangent made sense in geometry but worries the calculus student. The tricky thing is that students might notice that while their concept definition has evolved, their concept images might not have.
Kayla: Yeah. A couple years ago, when we had students sketch a graph with a vertical tangent, a lot of what we got was graphs like x = abs(y), a 90° clockwise rotation of the absolute value graphs students have seen, which doesn’t define a function of x at all. And, they treated the y-axis as the “tangent.” I just wonder if, to students, the picture just seems really similar to a circle despite its shape.
Dan: Right. One point of contact with the vertex of the “v” curve, the curve all on one side of the “tangent,” just like the tangent to a circle. From a geometry perspective, a student could think, well, that’s a reasonable example of a tangent. But, from a calculus perspective, it’s not. In calculus, we want the derivative to be well-defined, determining one specific slope for the tangent at a point.
Willy: If there is an art to the way mathematics names different things with the same name, then students should be able to understand why mathematicians over time decided to use the same name. It seems like the teacher has to help students appreciate the benefit of having the derivative as a well-defined function, with either one unique tangent line or none at all.
Kayla: I agree, but I don’t feel like I have a great answer to a student who asks why it is important that there not be multiple tangents to a point on the graph of a function. I would probably say something like: “At the vertex of the graph of abs(x), the slope to the left of the vertex and the slope to the right of the vertex are really different (one positive and one negative) creating a drastic change in slope where the two lines meet. And unlike a parabola where the slopes change from positive to negative across, those slopes are both approaching zero—just one from the negative direction and one from the positive direction. So, when looking at the vertex of the graph of abs(x), when you go to draw the tangent line what slope would you choose? The two drastically different slopes is why the derivative does not exist at that point—the slope from the right and left are different and the derivative function cannot take on two values for one x.
Willy: This is one of the reasons that asking students to produce examples of concepts has been really thought provoking when I think about teaching. Asking students to sketch a function that has a vertical tangent has the possibility of having students stumble upon things that might challenge their conceptions of how mathematics operates across contexts.
Dan: Those sorts of tasks can also give teachers information about what definitions their students are using, and what kind of concept images they have. But then, Kayla, it seems you’ve also been saying that such tasks give you a way to influence students’ concept definitions and concept images. Is that true?
Kayla: Yes, tasks like these help surface students’ concept image for me to work on with them. With some tasks, students all basically submit the same thing, showing how limited their image is. And, this applies not just to tangents. I especially like asking students to submit multiple examples. When we were doing rational function tasks, we asked them to submit multiple functions that would have a seemingly identical graph to a linear function and students could not think of multiple ways to do so. And from these sorts of tasks, I can also learn about how students think about related concepts: Do students think that points of tangency are different from points of intersection or just special ones? Or, do students think that a horizontal asymptote is a tangent?
Dan: So, your comments are about not just the match between the concept image and the concept definition, but also the richness and variety of the concept image space and connections to nearby concepts. Having surfaced all of those examples from students, in what way do you feel that those are a resource for your teaching separate from their role in assessing students?
Kayla: For the past couple years, students’ submissions have ended up being used in future discussions. When you have this bank of submissions that students actually submitted, you can develop a whole lesson based on what a couple students have submitted. I think the ability to see all those submissions easily, pick ones that are interesting, and use those, is great. Sometimes just seeing someone else’s submission can shift your concept image or support the new definition you are learning in a way that you weren’t able to without that extra nudge. I think that part is key. It can be super powerful just for students to see each other’s work.
Willy: I agree! And in the context of teacher preparation I also think about how difficult and time consuming it is for teachers to make up a variety of examples. So using student generated work helps! The work is already done for you, and then you can select the most appropriate examples for your purpose and have more time for other things.
Kayla: And I think often we make fake student work to use as teachers, we are saying these are the common submissions we know to expect. But now that we’re presenting this task to students, it has been interesting to see examples year after year that I hadn’t expected the first time around.
Dan: What’s an example of that?
Kayla: Year after year, students seem to think that there is a horizontal tangent on an exponential function where the horizontal asymptote is; they think the same line is both an asymptote and a tangent.
Dan: And, they aren’t thinking about a point at infinity!
Kayla: This comes usually in response to a prompt like “Enter a symbolic expression for a function whose graph is a line parallel to the x-axis. Then write a function, or sketch its graph, such that the line is tangent to the graph of the function at two or more points.”
Willy: To help us learn how students think about a concept, we can design assessment tasks that reveal students’ concept images or the definitions they’re operating from. Students can produce examples that do not fulfill all or any of the requirements of the task but still reveal possible gaps in understanding or overly broad or narrow concept images. For example, the “sideways absolute value” graph is not a function and does not have a tangent at the vertex. We can also design tasks that push students in a particular direction to further their learning—to encounter a concept in a certain way so that there is no prescribed solution or method and responses will vary. Such tasks could be used to shift student thinking for the purpose of, say, evolving their definition of tangent lines from a geometry sort of definition to one more appropriate for calculus. Interestingly, when I spoke with calculus teachers from my old school, one of the teachers thought it was weird that we would care whether a tangent line intersected the graph somewhere else because the curriculum focuses on tangents locally, not more globally. I wonder how extending the tangent line in calculus is helpful.
Dan: I was asking myself that question with a focus on the mathematics. I don’t have anything conclusive, but I have an observation to offer. On the interval between a point of tangency and a point of intersection farther down the line, even if that point of intersection is not another point of tangency, I think the average value of the derivative function is equal to the derivative at the point of tangency or the slope of the tangent line. For example, consider Red(x) = (x-1)(x-2)(x-3), and Green(x) = 2(x-1). The point of tangency is (1,0) and Red'(1) = 2. The point of intersection is (4,6).
Think about the interval [1, 4]. This interval reminds me of Algebra One where we often work with average rates of change and linear functions, rather than more complex curves. As long as we know the values at two points, in order to interpolate or extrapolate, we imagine a hypothetical situation where the change is distributed evenly, rather than the messy reality of change that is not evenly distributed. This observation about the interval between the point of tangency and intersection seems like it might suggest a mathematical value for considering when the continuation of a tangent line intersects with a function.
Kayla: I see the mathematical promise in that direction but wonder how many teachers would see that as standard calculus material. I wonder what it might take to have my colleagues consider using these tasks. I know I am a bit of an outlier. At the beginning of the year, I generally move through content with my BC Calculus class at a slower pace than other teachers in my district. From what I’ve heard from other teachers, many either skip the limits unit (assuming students understand the content from precalculus) or simply do a quick review (a week or so of class time). Similarly, with tangent lines, the concept of tangent line is pretty much skimmed over (pun intended!). The introduction to derivatives usually begins with defining derivative and then a quick transition into derivative rules, the relationship between functions and their derivative graphs, and applications of derivatives (related rates, optimization, linearization, etc.). Our district’s curriculum materials frequently ask questions about calculating derivatives and writing the equation of tangent lines at specific point, but there’s little digging into what the definition of a tangent line is and how it might have changed from geometry. Personally, I think it’s important to spend time on the issues about tangents that we’ve been discussing, but I worry many teachers may find these tasks a distraction that would take time away from other topics and skills in the curriculum that they see as more important/relevant to the AP exam.
Willy: Does that influence what you are going to do next year?
Kayla: No, not really. Using these tasks over the last few years has surfaced important areas of student confusion, even beyond the ones we’ve talked about here. I want students to think hard about definition and how definitions change. These “give-an-example” tasks help. They engage students with something interesting and challenging, and help them to pay careful attention to mathematical definitions and to be precise in using them.
Endnote
1. For the last two years, we have been using the STEP platform developed by Shai Olsher and Michal Yerushalmy at the MERI Center at the University of Haifa (Olsher, Yerushalmy, & Chazan, 2016). The ideas represented in this conversation were spurred by use of this program with activities developed in Israel (Yerushalmy, Nagari-Haddif, & Olsher, 2017; Nagari-Haddif, Yerushalmy, 2018) and adapted for use in the US.References
Verhulst, F. (2012). Mathematics is the art of giving the same name to different things: An interview with Henri Poincaré. Nieuw Archief Voor Wiskunde. Serie 5, 13(3), 154–158.
Olsher, S., Yerushalmy, M., & Chazan, D. (2016). How might the use of technology in formative assessment support changes in mathematics teaching? For the Learning of Mathematics, 36(3), 11–18. https://www.jstor.org/stable/44382716
Yerushalmy, M., Nagari-Haddif, G., & Olsher, S. (2017). Design of tasks for online assessment that supports understanding of students’ conceptions. ZDM, 49(5), 701–716. https://doi.org/10.1007/s11858-017-0871-7
Nagari-Haddif, G., & Yerushalmy, M. (2018). Supporting Online E-Assessment of Problem Solving: Resources and Constraints. In D. R. Thompson, M. Burton, A. Cusi, & D. Wright (Eds.), Classroom Assessment in Mathematics: Perspectives from Around the Globe (pp. 93–105). Springer International Publishing. https://doi.org/10.1007/978-3-319-73748-5_7
Tall, D. (2002). Continuities and discontinuities in long-term learning schemas. In David Tall & M. Thomas (Eds.), Intelligence, learning and understanding—A tribute to Richard Skemp (pp. 151–177). PostPressed. http://homepages.warwick.ac.uk/staff/David.Tall/pdfs/dot2002c-long-term-learning.pdf
Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169. https://doi.org/10.1007/BF00305619
]]>by Jeff Suzuki
Unless you’ve been living under a rock for the past decade, you know that one of the buzzwords in education is active learning: Be the guide on the side, not the sage on the stage. One of the more common approaches to active learning is the so-called flipped or inverted classroom. In a flipped classroom, students watch lectures at home, then come to class to do problems. This is actually a 21st century implementation of a very traditional approach to pedagogy, namely reading the textbook before coming to class. Many of us embraced this idea, and shifted our approach to teaching.
Then came the era of social distancing and forced conversion to distance learning. It might seem that those who switched to the flipped classroom model had an advantage: Our lectures are already online. And that’s true. But the second part of the flipped classroom involves working problems in class. This is now impossible, and those of us who had embraced the flipped classroom model have spent the past few months in existential agony. The “sage on the stage” can still give lectures through Zoom, but the “guide on the side” can’t guide.
The New Normal?
And yet…it’s now more important than ever to be the guide on the side.
We don’t know how long the current phase of social distancing will last, but even after it ends, we can expect that distance learning will be the new normal: it’s a trend that began long before the pandemic. And this forces us to deal with a new problem: It is impossible to monitor student activity remotely.
We accept this when we assign homework, and expect students will do the work with their books open, their notes in front of them, and a half-dozen math help sites open in different browser tabs. Before the pandemic, we told ourselves it didn’t matter, since they’d have to do the exam without all these study aids. But in the post-COVID world, there is now no difference between the resources available to students on homework assignments and on exams.
Don’t believe the hype about lockdown browsers (which work fine for the students who don’t have smartphones). Live webcam monitoring can be defeated by taping cheat sheets on the wall behind the computer. And if a student turns in a textbook perfect answer, it’s possible they listened to us when we explained how the answers should be written.
Will students cheat on exams? We’ve found copies of our exam questions posted to Chegg (with answers). This shocked me: Who would pay for a Chegg subscription, when there are so many free sites that show all steps to solving a problem and, unlike Chegg, leave no evidence behind?
The bad news is that every exam question that can be answered by following an algorithm is now obsolete, because such questions can no longer distinguish between the student who understands the material and the student who knows how to use Google.
Here is where the flipped classroom can be our salvation. A key component of the flipped classroom is letting your students figure things out for themselves, and not giving them a step-by-step algorithm for solving a problem.
For example, let’s consider a basic problem in calculus: Finding the derivative of a function. In the internet era, any function that can be described algebraically can have its derivative found, with steps, by a free online problem solver. So we have to ask questions that can’t be resolved by typing the problem into www.findthederivativewithstepsfree.com (not, so far as I know, a real website, but a thirty-second Google search will give you a plethora of possibilities).
Transcending the Machine
The good news is that computers are good at exactly one type of problem: problems that have algorithmic solutions. If you can describe the exact sequence of steps needed to solve a problem, then a computer can implement those steps faster, more accurately, and more cheaply than any human being. The real moral of the story about John Henry is don’t compete with the machine in the machine’s areas of strength. Instead, find the things the machine is bad at. In this case, the easiest way to neutralize these problem solving sites is to make every problem a word problem.
Of course, “students can’t do word problems.” This is a meaningless objection: at the start of calculus, students can’t integrate, but we still ask them integration questions on the final exam! Our job is to teach these students how to do these things. Here’s where the flipped classroom becomes a key part of the solution. Don’t spend class time lecturing: students can view lectures on their own time. Instead, class time should be spent working problems, especially those that can’t be solved by following a sequence of steps.
It’s helpful in the discussion that follows to think of problems as falling into one of two categories:
Flipping Your Class, Social Distancing Edition
Here’s one possible structure for such a class (where “class” means any time you’re working with students in realtime). All of the following takes place before class:
How should you run class itself? Class time is the most valuable resource available to students; using it efficiently and effectively can be challenging. Here’s a few things that may help.
At the start of class (online or in person), take down a list of student questions. One risk is that the more outspoken students tend to dominate the discussion; taking down a list of all questions at the start of class is a way to make sure that every student has a chance of getting their question answered, and to ensure that a sufficient variety of problems are presented.
Establish from the start that the routine problems have lowest priority: these are problems that should be solvable by students who followed the assigned lecture. It is vitally important that you keep to this rule: The biggest challenge to running a flipped classroom is students who don’t watch the lectures beforehand. Depending on how you’ve set things up and the system you’re using, it might even be possible to determine whether a student has watched the assigned lecture (though trying to do this realtime requires a bit of practice); another option, which I use, is to assign simple 1-point problems that students answer after they’ve watched the lecture. Remember: Class time is the single most valuable resource available to the students; it should not be spent on things that can be done out of class time.
One way to efficiently use class time is to focus on the setup. For example, let’s consider the following problem, which probably appears in every calculus text ever written:
A 25-foot long ladder rests against a wall. The base of the ladder begins sliding away from the wall at 2 ft/second, while the top of the ladder maintains contact with the wall. How rapidly is the top of the ladder falling when the base is 10 feet away from the wall?
The “sage on the stage” would identify the relevant parameters and write down the mathematical problem to be solved. The “guide on the side” would lead students to the mathematical problem. For this, it’s important to ask leading questions and not give outright answers. For example:
In the end, we have the mathematical problem, “Find $\frac{dy}{dt}$ when $x^2 + y^2 = 25$ and $\frac{dx}{dt} = 2.$” At this point, it becomes a routine problem—and if you’ve established that minimal class time will be spent on routine problems, you can leave the problem at this point, perhaps with a directive of “Finish the problem after class.”
It’s worth noting that, at this point, the problem can be handed off to an online calculator, which can then solve the problem. You might even go so far as to point students to the online calculator, lest they develop a mistaken belief that you’re unaware of the existence of such things. This epitomizes the idea that humans should do what humans are good at, namely extracting the mathematical problem to be solved; while machines should do what machines are good at, namely applying an algorithm.
As the preceding example suggests, it’s possible to teach a flipped class with very little change in how you’re already teaching. The main difference is establishing the expectation that students watch lectures before class.
Let’s see how we might take a larger step, using a standard topic: finding the extreme values of a function. A traditional approach might be to have students find derivatives, then critical values, then apply some test to decide whether a critical value corresponds to a maximum or minimum.
In a flipped classroom, students wouldn’t be given this algorithm. Instead, they’d create their own approach, typically through some guided exploration of a question. Coming up with good questions is challenging; fortunately, thirty years of reform calculus have provided us with an abundance of material, and many of these questions have been incorporated into every standard calculus text, so you needn’t write your own.
For example, I like to give students the following question:
An accelerograph records the acceleration of a train (assumed to be moving in a straight line); some of the data values are shown below. Assuming the velocity of the train at t = 0 was 0 m/s, estimate when the train was moving the fastest; defend your conclusion.
t (seconds) | 0 | 1 | 2 | 3 | 4 |
a(t) (m/s^{2}) | 3 | 2 | 1 | -1 | -2 |
A sequence of leading questions can guide students to creating their own approach:
and so on, leading to an answer like: The train’s velocity appears to be increasing until at least t = 2, and is decreasing from t = 3, so there’s a maximum velocity between t = 2 and t = 3.
What’s worth noticing here is that none of these questions can be answered by appeal to a formula or an algorithm. Consequently, any attempt to use an online calculator on this type of question will result in, at best, a nonsense answer. The closest thing to an “algorithm” is recognizing that the change from increasing to decreasing is where the local maximum value will occur, but even then, since that change occurs “offscreen”, students must consider how they know that the change has occurred.
A Return to Normalcy
Suppose, against all predictions and the entire trend of human history, we go back to how things were at the beginning of 2020: traditional in-person classes, no social distancing, exams where we could control the resources used by students.
None of the preceding needs to change. In fact, all of the preceding alterations in our pedagogy and our assessment are worth doing regardless of how we will give exams. The hard truth is that sooner or later, our students will leave the classroom. If what they’ve learned from our classes can be done by a free internet application, then their education is worth a free internet application.
We owe it to our students to give them something more.
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Fostering an understanding and appreciation of the deep, beautiful threads that unite seemingly disparate areas of mathematics is among the most valuable outcomes of teaching. Two such areas that are ripe for bridge building—functions and geometric transformations—are the focus of our NSF project, Forging Connections Through the Geometry of Functions. In this post, we describe the pedagogical benefits of introducing students to functions through the lens of geometric transformations.
Geometric Transformations as Functions
The most common representations of functions are symbolic and numeric in nature. This emphasis on number limits students’ images of the variety of mathematical relationships that can be represented as functions. As such, it contributes to common student misconceptions. Students may conclude that:
Although students investigate reflections, translations, rotations, dilations, and glide reflections in a geometry course, they typically do not regard them as functions; the functions they encounter in algebra always have numbers as input and output. We can expand students’ horizons and deepen their concept of function by treating geometric transformations as functions that take a Euclidean point as input and produce another point as output. Coxford and Usiskin pioneered this approach a half century ago in their ground-breaking Geometry: A Transformation Approach, but very few of today’s geometry students encounter it.
In Figure 1, a student has used three Web Sketchpad tools to construct the independent variable x, the mirror m, and the reflected dependent variable r_{m}(x). The student then drags point x and observes the traces of both point x and r_{m}(x). This sensorimotor experience introduces students to four mathematical ideas.
Constructing a Dynagraph
Having explored reflection and other geometric transformations in two-dimensional Flatland, students then restrict the domain of these transformations into the Lineland (one-dimensional) environment of a number line (Abbott, 1886). They focus in particular on connecting the geometric behavior of dilation and translation to the observed numeric values of their variables on a number line.
In Figure 2, students use the Number Line, Point, and Dilate tools to create a point restricted to the number line and dilate it about the origin to obtain a point labeled D_{0,s}(x). While this notation may at first seem daunting, it actually may be less mysterious than the traditional f(x) language. Rather than write out or speak all the words “the Dilation (of x) about center point 0 by scale factor s,” we abbreviate the important parts with just single letters to encapsulate this long-winded phrase. The notation f(x) does that, to an extent, too, but leaves f a mystery that is unraveled somewhere else (on the right hand side of the = in the definition statement) but never seen again. Algebraic notation is an “abbreviation” of what can be said in English (and used to be), but much more clumsily.
Students measure the coordinates of x and D_{0,s}(x) and drag x to compare the values. When asked to describe how D_{0,s}(x) moves when x is dragged, a student might respond, “As I drag x, D_{0,s}(x) moves faster. It seems to move twice as fast, and I notice its value is always twice the value of x. I wonder if its speed is related to the scale factor s.” By experimenting with different scale factors, the student concludes that s represents the relative speed of D_{0,s}(x) with respect to x, and that the coordinates produced by this dilation satisfy D_{0,s}(x) = x·s. Students can then experiment with a translation restricted to the number line and conclude that this new function, translation by a vector parallel to the number line and of directed length v, causes the two variables to move at the same speed, and satisfies the equation T_{v}(x) = x + v. This understanding—that multiplication and addition of numbers are the equivalents of dilation and translation of points—represents a deep connection between algebra and geometry.
Students are now ready for a new task: What happens when you dilate x and then translate the dilated image; in other words, how does the composite function T_{v}(D_{0,s}(x)) behave? Students’ first attempts at this task becomes visually confusing with three variables and a vector stumbling over each other on the same number line. To alleviate the confusion, we introduce a Transfer tool that moves the dependent variable to a parallel number line, separate from but aligned with the first. In Figure 3, students use this tool to construct a second number line parallel to the original. This visual representation of a function, with the independent variable x on one number line and the dependent variable T_{v}(D_{0,s}(x)) on a parallel number line is known as a dynagraph (Goldenberg, Lewis, & O’Keefe, 1992).
Students who construct T_{v}(D_{0,s}(x)) = x·s + v have built a linear function. This may seem a lot of effort to get to what is more typically presented on a Cartesian graph with sliders that control the values of s and v (generally labeled mysteriously as m and b), but the geometric approach has advantages. Note that the terms slope and y-intercept reference the Cartesian-graph representation, without directly communicating the fundamental underlying concepts of relative rate of change and starting value. Lost in this representation is an understanding of the choreographed dance between independent variable x and dependent variable T_{v}(D_{0,s}(x)). Though this information is encoded in the Cartesian graph and its accompanying algebraic form, it’s difficult for many students to tease the deeper concepts out of these representations. A dynagraph makes this behavior more accessible by allowing students to drag point x and compare its speed and direction with the behavior of T_{v}(D_{0,s}(x)) on a parallel number line. To answer the questions below, all of which focus on motion, students experiment with the values of s and v to create linear functions that match the desired behaviors.
The Cartesian Connection
To conclude, students create the Cartesian graph of a linear function using geometric transformations. As Figure 4 illustrates, students start with the same initial tools that they used to create a dynagraph, but this activity’s Transfer tool rotates a variable by 90°, transferring it to a vertical number line perpendicular to the original, horizontal number line. After using this tool to rotate D_{0,s}(x) to a vertical axis and translating by vector v, students use the Perpendicular tool to construct lines that keep track of the horizontal location of x and the vertical location of T_{v}(D_{0,s}(x)). They then construct a traced point at the intersection of these horizontal and vertical lines and drag x to see how the traced point’s motion corresponds to the behavior of the two variables. After performing the construction, students try different values for the scale factor s and the translation vector v, and they observe not only how changing the scale factor affects the speed of T_{v}(D_{0,s}(x)) relative to x but also how the speed of the dependent variable determines the steepness of the traced line. After making this observation, it’s relatively easy for students to propose that steepness could be measured numerically based on how far T(D(x)) moves relative to the movement of x, leading to a mathematical definition of slope as Δy/Δx.
Conclusion
By using web-based dynamic mathematics software and tools tailored to carefully structured tasks, students can enact geometric transformations as functions, create them, manipulate them, and experiment with them. In the course of their explorations they can develop a solid understanding of geometric transformations, explore connections between geometry and algebra, and construct and shed light on linear functions by using a dynagraph representation.
By beginning with R^{2}→R^{2} functions (transformations in the Euclidean plane) and connecting them to R→R functions in algebra, these activities can help prepare students for later study of complex (C→C) functions, and functions with three-dimensional domains and ranges (R^{3}→R^{3}). A further benefit is the gentle visual introduction of the concept and notation of function composition.
Pedagogically, the constructive nature of activities such as these has the potential to engage students, to give them opportunities to assess their own work, to encourage mathematical discussions, and to help students bridge the gap between the concrete, physical world and the profound elegance of abstract mathematical insights.
Acknowledgments
References
Abbott, E. A. (1884). Flatland: A romance of many dimensions. San Antonio, TX: Eldritch Press.
Coxford, A. F., & Usiskin, Z. (1971). Geometry: A transformation approach. Laidlaw Brothers Publishers.
Goldenberg, P., Lewis, P., & O’Keefe, J. (1992). Dynamic representation and the development of a process understanding of function. In E. Dubinsky & G. Harel (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 235-260). Washington, DC: Mathematical Association of America.
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