We—Emily McMillon and George Nasr—are graduate students at the University of Nebraska-Lincoln. We implemented mastery based testing for two sections of a course on geometry for pre-service elementary teachers during the Spring 2020 semester, and found that our students
In this post, we will discuss what led us to try mastery based testing for this student population, how we implemented mastery based testing in our courses, and some student survey responses.
We first heard about Mastery Grading late in 2019 when Austin Mohr gave a talk at UNL on the topic. At the time, we were both teaching mathematics courses for pre-service elementary teachers. Hearing about Mastery Grading, we both, independently, thought this type of grading would be excellent for pre-service teachers. Hence, with permission from our department, we decided to implement mastery grading in two sections of the same course in the Spring 2020 semester.
Before describing what exactly Mastery Grading is, we would like to discuss some general learning goals we find valuable in a course for future elementary school teachers. Our first goal is to guarantee that our students fully understand most of the course concepts upon leaving the class. We feel that it is particularly crucial that students in an education program fully understand concepts, given that they are responsible for being able to articulate similar concepts to their future students.
A second goal is to encourage students to revisit and reflect on their previous work and mistakes. It is particularly imperative that future teachers understand that mathematical ability can be improved upon, as studies have shown that elementary teachers pass on their views of mathematics to their students.
Our third goal is to broaden the scope of students’ understanding of the purpose of assessments beyond a numeric score. As future teachers, it is important that they are at the very least aware of different styles of assessment, and, ideally, critically assess different styles of assessment to determine which is ideal for their own future students.
Overall, we believe it is important that elementary education mathematics classes are designed in a way that encourages future teachers to continue working on concepts until they have demonstrated understanding. We want our students leaving these classes feeling confident that they have truly mastered the concepts that they may one day teach for themselves. We also want assessments to be seen as low-stakes opportunities for students to show us the progress they have made, while also incentivizing them to look back at their mistakes and try to understand what it is they have yet to learn. We believe this can be accomplished with Mastery Grading.
Mastery Grading is a grading scheme by which students are expected to show complete understanding of course objectives. This is done by offering multiple opportunities throughout the semester to reattempt course objectives for all or nothing credit. There is no penalty for students taking longer to master a course objective. The goal is for students to eventually show that they understand the material, not for students to necessarily demonstrate complete understanding of material the first time it is assessed.
There are many variations on mastery based grading; our implementation as described below is but one example. Many additional resources are available online. We found the following blog very helpful and so pass it along to the interested reader: https://mbtmath.wordpress.com/.
We believe that Mastery Grading helps achieve the three goals we mentioned in the preceding section on our motivation. Mastery is designed to encourage students to revisit concepts to receive full credit for learning them. In a point-based class, students can earn partial credit for partially learning something and then may never have to revisit that concept again. In this way, students will ideally leave this course with a robust understanding of the course content.
Another feature of mastery is that it only rewards students points for a problem once they have shown full understanding of the underlying concept. This incentivizes learning from mistakes and has the potential to help students cultivate a growth mindset toward mathematics. We also feel that mastery provides students with another perspective on how to run a class and assign grades.
Geometry Matters is a required course for most elementary education majors at UNL. The course covers geometry and measurement and follows chapters 10-14 of Sybilla Beckmann’s textbook Mathematics for Elementary Teachers. This course is part of a three-course sequence that covers chapters 1-14 of the aforementioned textbook. The first course in the sequence is Math Matters, which must be taken prior to Geometry Matters, covers chapters 1-7 in the textbook. The other course in the sequence, Math Modeling, covers chapters 8-10 and can be taken at any point.
The course is taught by faculty, lecturers, and advanced graduate teaching assistants, depending upon instructor availability in any given semester. The course grade is usually determined by some combination of assessment scores, homework scores, and written project scores (so-called “Habits of Mind” problems). Students tend to do well in the course — in the last six years, pass rates have ranged from 79% to 100%, with most semesters having over 90% of students pass the course. Hence, grades and pass rates were not a reason we decided to implement Mastery Grading.
We divided the course content into 18 Learning Outcomes. Student grades were based 60% on mastery of these outcomes, with homework and project problems making up the remaining 40%. Grading for individual Learning Outcomes was for all or nothing credit, and homework and project problems were graded with a traditional points-based system. Our original plan was to test outcomes 1–7 on the first assessment, 1–13 on the second assessment, and 1–18 on the third assessment. The final would not cover new material but would be a final opportunity to master previously not mastered outcomes. In addition, we planned to offer occasional opportunities to take one to two outcomes as “quizzes” in class as the opportunities arose.
As these courses were taught during the Spring 2020 semester, we were forced to move the courses online in March of 2020. We chose to make some modifications to the course assessment structure to better work in the online, asynchronous format required by our university.
Before the move to online, we had given the first assessment as well as two mastery quizzes. The second assessment had to be taken online. We decided to eliminate the third assessment and instead replace it with weekly mastery quizzes that would each test a single new concept and offer an opportunity for students to reattempt up to two learning outcomes they had not yet mastered. Recall that quizzes were made up of exam-level problems—the only difference between these and exams was the quantity of problems. The final exam remained as previously scheduled, albeit online.
The following is a description of one of our 18 Learning Outcomes assessing areas of polygons other than rectangles, which spans sections 12.3 and 12.4 of our textbook.
That is, to earn points for this learning outcome, students would have to show mastery of both parts A and B.
Here is a sample two-part problem assessing this learning outcome, along with work from a student that did not master the concept on their first try. Students knew they may use the formulas for the areas of standard shapes such as rectangles, triangles, and parallelograms. Students also knew they were expected to express reasoning for their conclusions, and to substantiate their reasoning with ideas such as principles of area.
On part (a), the student was very close and would have earned most points for this part, but we would have liked the student to say that you can form a rectangle out of two triangles of equal area, and hence, half of the area of the rectangle is the area of either triangle. One can infer from the dashed lines the student drew on the triangle provided that they are thinking about this as two triangles forming a rectangle, but being explicit in their explanation was critical for us to ensure their understanding.
However, the work on part (b) is what really led us to feel it was critical to have the student spend more time reviewing this outcome. The point of this part was for the student to recognize the shaded region could be decomposed into a triangle and parallelogram, and that adding the area of these shapes would yield the area of the original region. The student’s attempt still showed some understanding of how one can decompose and move regions in an effort to figure out the area, which showed a desire to use principles of area. However, if one carefully checks, it is not possible to fit both triangles the student creates in the specified regions, and this is critical for the student’s method to work. (Though coincidently, they do get the right answer for the area.)
Recalling that each learning outcome was worth 15 points. We would say the student would have earned around 8 points on this problem had this been graded with points. However, due to the mastery grading system, this student had a second chance to demonstrate their understanding. Below, we show a second version of a problem for this learning outcome and the student’s response.
We recognize that part (a) has some unconventional notational choices, but we feel it is clear the student showed comprehension of the underlying concept being assessed. On part (b), we see clear improvement from the first attempt in the student’s ability to find the area of a large shape by decomposing it into smaller, familiar regions. This student earned certification of mastery on this attempt.
We would like to give a brief overview of how our students did. By the end of the semester, 37 of our 42 students had mastered at least 17 of our 18 learning outcomes, and no student mastered fewer than 14 learning outcomes. Many concepts were mastered by students on their first attempt, and the majority of students needed at most two attempts to master a concept.
At the end of the semester, we surveyed our students on their experiences in the course. There was no concrete incentive to complete the survey, but 41 out of our 42 students completed the form.
This survey consisted of two parts — a series of Likert questions, and a series of open-response questions.
We asked students to respond to the following three statements on a scale of 1 to 5, with 1 meaning “strongly disagree” and 5 meaning “strongly agree”.
Below are the results.
We noticed two students who wrote overwhelmingly positive things in the next part of the survey but responded with “strongly disagree” to these questions, so we infer that these responses to the Likert survey were the opposite of their intended responses.
We also wanted to give students a chance to share, in their own words, how their experiences with mastery grading impacted their experiences. We asked our students a few questions regarding their experiences with mastery grading. We also asked them to compare these assessments with points-based assessments they’d had in Math 300 (a prerequisite course also about mathematics for future elementary teachers) and to reflect on how these experiences would impact their future teaching.
Working through the responses, we found several themes that were shared among many students, which we now discuss, categorized into expected and unexpected results.
Content Understanding: As instructors, we noticed that the work being turned in by our students was of exceptionally high quality as compared to previous semesters of mathematics courses we had taught for future teachers. A few students remarked on their personal feelings that they had taken more away from the course than they might have under a points-based system.
Learning from Mistakes: We found that mastery grading encouraged our students to look back at their mistakes on their exams. Of our survey respondents, 16 mentioned learning from mistakes as a positive takeaway of the course in their open survey responses. Many commented that they would have never looked back at mistakes they made on exams in other classes. The following two quotes are representative of the types of responses in this category. Some students mentioned specific learning outcomes they learned best, while others gave more general responses indicating that looking back at their mistakes benefited their learning.
Math Confidence and Growth Mindset: As one may expect from our third Likert question, several students indicated feeling more confident in mathematics. Students mentioned how they were able to learn content they did not think they would have been able to learn at the start of the semester. We also found some encouraging comments about students’ development of their growth mindset. In total, 8 respondents explicitly mentioned math confidence or an increased growth mindset in their responses. A representative comment is:
One of the interesting results was that some students even commented on growth mindset oriented toward their future students, as in the comment that follows.
Stress and Anxiety: 15 students indicated in their responses that exams felt a lot less stressful since they could redo their mistakes. Several of our students admitted to struggling with testing anxiety and said that this grading scheme gave them some relief to that. It should be noted that several students commented that at the beginning of the semester, the “all or nothing” nature of these exams seemed daunting. However, all these students said that things improved once they became more familiar with the grading scheme and started passing outcomes.
Exams as Opportunities: Mastery grading also affected how at least 6 respondents felt about exams. In particular, they felt that exams were an opportunity to show their knowledge and understanding as opposed to a hurdle to be overcome. The following quotes represent these responses.
In our experience, students sometimes view mathematics exams as being antagonistic, unfair, or that our goal as instructors is to trick them or make them get a lower grade. To us, these responses show that students saw this grading scheme as being friendlier and more conducive to allowing them to demonstrate their understanding.
Student Learning: Our future teachers also demonstrated an immense capacity to think about their future students. It appears mastery grading encouraged some to think more carefully about what their concrete goals are as instructors, such as with the following student.
They also showed an ability to preemptively empathize with their potential students. In particular, many students who admitted to struggling with testing and/or math anxiety commented on wanting to try mastery based grading with their own students as a way to alleviate their students’ testing anxiety.
Other students perhaps did not struggle with testing anxiety, but still saw the importance of giving students multiple opportunities to demonstrate their knowledge.
Challenges: It is important for us to acknowledge that not all feedback we received was positive. There were two common themes among those that found issues with Mastery Grading. First, students did not enjoy having to redo outcomes when they thought they misunderstood only a small portion of the outcome or made only a small error. Second, students still wished they could get some partial credit for the ideas for which they did demonstrate a good understanding. A small number of students commented that having multiple chances led them to care less about any individual assessment, and so they studied less. We also noted a trend that students who had taken more “traditional” math courses, i.e., calculus sequences courses, seemed more frustrated by having to retake outcomes when they made fairly small errors.
We believe we accomplished two of our three main goals. Students seemed to be successful in understanding to course content. In addition, students appeared encouraged to learn the content and felt motivated to understand their mistakes. We even saw that students felt more confident with mathematics and demonstrated a growth mindset. However, we are less confident that we broadened the scope of students’ understanding of the purpose of assessments beyond a numeric score, although based on some student comments, it appears that our students at least started thinking about this.
There were a few less expected results that we were delighted to see. Students generally reported feeling less anxious about exams since they knew they would have multiple opportunities to show what they know. Students felt assessments gave them the chance to accurately show their knowledge. They also reflected on their experiences with mastery and how it might inform their future teaching.
We feel that future teachers were the most amenable to this style of grading as they themselves tend to value the opportunity to grow and learn.
More to this point, we have already seen evidence of how mastery has affected their future experiences. We highlight one piece of evidence here. During Fall 2020, the semester after we implemented mastery grading, some of our students took another math for future teachers class with our colleague, Kelsey Quigley. At one point during the semester, Kelsey offered an opportunity for her students to earn points back on their first exam. As she discussed logistical considerations with them, a student suggested that the way they should earn points back would be to redo the problems they individually did not do well on, as opposed to the problems the class did not do well on overall. They said, “It’s like you’re mastering the concepts you missed versus going back and doing the ones that you understood.” Kelsey had the impression that this student gained this perspective through their experiences with mastery based testing in our course.
The challenges mentioned in the previous section are important to address. While many of the challenges presented by the students are inherent to mastery grading, we feel that there are a few things that instructors can do to address the issues.
In conclusion, we found mastery grading to be a rewarding experience both for us as instructors as well as for our students. This testing style felt like a perfect fit for pre-service teachers, and we would encourage any instructors of pre-service elementary teachers to consider giving mastery based grading a try in their courses.
Acknowledgements
We wish to thank Allan Donsig and Michelle Homp for backing our desire to teach this class using mastery-based testing, Wendy Smith for her help in designing our study and methods of data collection, and Yvonne Lai for her helpful feedback and guidance in writing this article. Finally, we would like to thank Austin Mohr for introducing us to this testing method and inspiring us to try it ourselves.
]]>Due to the global health crisis, a huge amount of instruction that was happening in person a year ago is now happening online. One theme highlighted by this change is the question of control. When students are in buildings with us, we^{[1]} have a high degree of control over the environment in which instruction takes place and the materials the students have access to. We even have a significant level of power over students’ movements and choices, at least while they’re in front of us. This is most obvious in primary and secondary school, where there is usually a whole “disciplinary” administrative apparatus designed to support instructors’ ability to dictate the movements and choices of students. But even at college and university, where for example there is often no explicit rule against a student getting up and leaving the classroom or building at any time, physical and social aspects of the classroom setting serve as a mechanism of influence. Continuing the example, to leave a classroom in the middle of class you have to physically stand up and collect your stuff, which means everybody knows you’re not coming back, and then face everyone as you walk past them on the way out. The instructor will certainly notice, will probably be hurt, and won’t necessarily respond kindly. It’s very rare for students to do this—in fairness, this is probably (hopefully) mostly because they don’t want to—but it’s very rare even when they do.
A fundamental aspect of the switch to distance learning is its disruption of all the usual structures and processes by which this control is exercised. In our running example, you can leave a Zoom class just by clicking “Leave”, with no need to awkwardly face anyone and a reasonable likelihood, depending on the size of the class, that the instructor won’t even notice. To cover your bases, you can instead leave without leaving—just mute yourself, turn off video, and go about your business while remaining formally in the meeting.
For a different and much-discussed example, while we are used to being able to design students’ environments rather meticulously during exam proctoring to head off both distraction and temptation, there is no analogous form of control over the exam environment built into distance learning.
How are we collectively responding to the challenges this change presents?
One major approach has been to use surveillance technology to try to claw back the lost control. For example, “remote proctoring” of exams has exploded, with colleges and universities spending millions of dollars on software that monitors students via webcam. When colleges first moved to distance learning last spring, 54% of institutions surveyed by the higher education IT association Educause said they were using remote proctoring software, while another 23% said they were planning on or considering it.
This is not going over especially smoothly. Objections have been raised to the disturbing privacy implications of video-monitoring students in their homes—
Finished (hopefully) my last ProctorU exam and never thought I'd be saying I'm happy I don't have to be subjected to Black Mirror-like testing anymore where they track your eye movement, see all 4 walls of your room, and tell you to take off your "hair cap" in your own home :))))
— Jonté (@noturaveragept2) December 17, 2020
4. These packages allow an instructor to watch a student from up close, over a webcam, for the entire duration of an exam without the student's knowledge. The potential creep factor on that goes right off the scale.
— Carl T. Bergstrom (@CT_Bergstrom) October 31, 2020
—the inevitable glitchiness of the technology—
The fact that I have to put in my contacts to take proctorio is dumb. But what's extra dumb is that even with contacts, it still can fail to detect my eyes
— Shalyssa (@CallMeAlyssaD) December 16, 2020
I was doing a 120 min exam and 12minutes in the @ProctorU proctor wasn’t able to see my video. She put me through tech support for over 60min, got a new proctor and guess what!? I had 41 min when my exam resumed. Not 110 minutes. #fail literally
— suzsuzbar (@suzsuzbar) December 20, 2020
—its inequitable glitchiness—
The coordinated course I'm teaching has decided to stop requiring the use of Honorlock. It's not only an incredibly invasive surveillance system, but also discriminates against Black students by failing to recognize their faces & repeatedly locking them out of assessments. https://t.co/9CXLtTZxhE
— Dr. Marissa Kawehi (@MarissaKawehi) October 13, 2020
ExamSoft, Proctorio, & ProctorU must address the alarmingly long list of equity, accessibility, & privacy issues students are facing on their exam platforms. Students of color & those with disabilities must not be locked out of tests or wrongly accused of cheating. pic.twitter.com/Kp5Yb7sMRx
— Richard Blumenthal (@SenBlumenthal) December 4, 2020
—the stress and anxiety of having involuntary body movements scrutinized—
Yikes — schools are using 'learning integrity' software that tracks students' head and eye movements in an effort to deter cheating. Instructors are evidently dinging students for moving their heads and eyes too much. https://t.co/fUXnQ6yx93
— Christopher Ingraham (@_cingraham) September 11, 2020
—and needing to jump through extra hoops in the already high-pressure situation of an exam—
Here are the requirements for my math midterm next week. You tell me if these instructions are easy to follow and don't make you nervous. pic.twitter.com/WnW0YSXmhr
— Spotted At Laurier (@SpottedLaurier) October 22, 2020
—and the list goes on. This article in the Washington Post (also linked above) does some more comprehensive reporting, including revealing that law students who sat for New York’s first online bar exam in October urinated in their seats to avoid violating the online proctoring rules.
The pushback goes beyond individuals on Twitter. There is organized resistance from students and parents. The press has joined the fray. Nonprofit organizations have too. In some cases the decision to implement remote proctoring is being blocked or reversed by faculty. Lawmakers have gotten involved.
Nonetheless, usage of online proctoring services has continued to grow.
Why is this turning into such a fight?
From one angle, this question is entirely rhetorical. In preparing this blog post, I read a large number of tweets from students about their experiences with remote proctoring software. The main thought/feeling I was left with was horror that academia has embraced this Faustian deal. Surveilling students in their homes and subjecting them to suspicion based on automated interpretation of their involuntary body movements is transparently creepy, unfair, unreliable, and harmful to students. They are stressed out, and it’s making them do worse. Concerns about cheating are real, but they look petty and tiny in the face of these harms.
In view of this, I think students (and their allies in the professoriate, the press, etc.) are fighting back because remote proctoring is a travesty. What, are they going to take this lying down? They should be fighting back! It’s natural that they’re fighting back. I hope their fight grows. I’ve decided to join it by writing this. I hope you will too.
That said, I would like to proffer an additional explanation that I believe illuminates the situation from an angle that is useful to educators. Remote proctoring is meeting resistance because it is going against the grain of the situation.
The global health crisis has forced us online. The online format presents the institution of education with a new challenge—the disruption of its usual mechanisms of control. The wide-scale adoption of remote proctoring software during the pandemic is, in my view, an attempt to wish that challenge away rather than confronting it. Online proctoring companies are selling educational institutions the fantasy that it is possible to recreate the important elements of in-person testing online. It’s a transparent falsehood, but we want to believe, because the alternative—a deep and serious reconsideration of testing in view of everything that has happened—feels like too much. The changes in our own and our students’ lives already forced by the pandemic are hard enough to wrap our minds around! And we have to rethink testing too?
From this point of view, the pushback was to be expected because institutions implementing remote proctoring are straining against reality. Reality always strains back.
The reason why I think this angle might be useful to educators is that it seems—well, to me anyway—to point in a freeing, and expansive, direction. There’s some relief available for us in admitting that remote teaching cannot recreate a reasonable in-person exam environment. When we admit this, what can we start imagining instead? How can we work with the situation, rather than against it?
More broadly, I think that surrendering to the reality that we don’t have as much control over our students when they’re far away encourages us to ask productive questions about the functions our control was serving in the first place, and how else these functions might be served.
In Spring 2020, when we first went online, circumstances conspired to compel me to relinquish some forms of control I didn’t even fully realize I was holding onto, leading to an experiment in letting go whose results surprised me. I hope to describe this experiment in a future blog post. In the meantime, here are some other folks, from both math and other disciplines, investigating the sorts of questions I have in mind:
If you know of other writing addressing these types of questions (or have done some yourself), please comment with links!
[1] By “we” I mean the collective consisting of all educators, including instructors at all levels, administrators, etc., and the institution of education writ large. This is an intentionally broad and encompassing meaning. Some of below pertains to decisions individual instructors make in their classes, while some of it pertains to decisions made by people acting as part of institutional bureaucracies, but I hope it is useful to see either kind of decision (and everything in between) as informed by a conversation we are all a part of.
]]>Mathematics and mathematicians rarely make press. So it was a bit sweet, but mostly bitter, to read in the New Yorker of the deaths of John Conway, Ronald Graham, and Freeman Dyson, three great losses to our profession. (Yes, Virginia, Dyson published in ‘pure’ mathematics as well as in physics.)
And of course as soon as this article appeared, friends and colleagues wrote about others we have lost who were not mentioned in the press. It is likely that each of us has suffered some loss, some grief. I write here of my own, and what we can learn from it about our work.
My old and dear friend David Dolinko passed away last week, a final stab-in-the-back from the year 2020. His career can tell us something about our field. Mathematics is the heavy industry of the sciences, but also of other intellectual endeavors. The tools of thought that we develop are not apparent when a vaccine is tested, when an election is contested–or when a legal precedent is set. Mathematics is largely unseen by the public, and even sometimes by the people who are using it.
David started his intellectual life as a mathematics major, but became interested in logic, and earned his Ph.D. in philosophy. From there his interest ‘drifted’ (or progressed) to a law degree. He spent his career teaching the philosophy of law at UCLA until his recent retirement. He will be mourned by his students as well as friends and family.
David had a roving intellect, from the music of Mahler to the poetry of Eliot, from the history of Fascism to molecular biology. He became a prominent writer about punishment and the death penalty. In all of this, I think, his early training in mathematics betrayed itself. He had an uncanny ability to make intuitively clear ideas that were too often cloaked in formality.
Indeed, it was he who was responsible for my own initial epiphany in mathematics, a moment to which I can trace my love affair with the field. We were thirteen years old, in ninth grade. Our enlightened math teacher, Ms. Funke, spent her lunch period (and ours) meeting with interested students to form a math team. (At the time, competition was the only extra-curricular activity available for students interested in mathematics.)
Ms. Funke defined an arithmetic progression, providing us with the usual formulas for the nth term and the sum of n terms. Then she gave us a problem, straight from Hall and Knight, something like: “Insert three arithmetic means between 11 and 23.” I loved formulas. I could do this. David was sitting next to me, drawing some organic molecule he had been reading about.
“C’mon! Let’s do this! This is interesting!” I urged him. Anything that is fun to do is more fun to do with others.
But David wasn’t particularly interested: “Oh, I did that. It’s 11, 14, 17, 20, 23.” And he went back to drawing his molecule. It was at that moment, from my friend David’s answer, that I realized that algebraic formulas were invented to capture intuitions—here, that the numbers we seek are evenly spaced. It is unlikely that David had used the formula. Rather, he knew what he was looking for, the numbers were easy, and he found them without bothering with formulas. This astonished me.
For the purists among my readers: yes, intuitive methods are not general. David would probably not have inserted four arithmetic means as easily. But intuition drives formalization, seeds discovery, lends meaning to what we have discovered. And intellectually—putting aside the personal and emotional—it was through his lightning intuition that David taught me the most.
And not just me. His death has occasioned an outpouring of sympathy and recollection by lawyers and public figures who had been his students in law school. The same value of intuition, of making ‘obvious’ the meaning behind formalities, seems to have marked his teaching of the law as well.
And it is for students—David’s, mine, and yours, Dear Reader, for whom I write these thoughts. We all know that the year 2020 was an annus horribilis, a year of loss for all of us. As we plan our first ever Virtual, and last ever Joint, Mathematics Meeting, my wish is that 2021 be a year of renewal. For education is about renewal: e-ducare, to lead out. Out of darkness and grief, towards hope for the future. To renewal, to a passing of the torch to our students.
Somehow, sometime, whatever the cause, we all come to the Same End. As researchers, we advance knowledge in the present. As teachers, we build the knowledge of the future.
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It is 2020. You are taking a high school mathematics teacher licensure exam. Suppose you see these questions. What do you do? What do you think? (Warning: Your head may spin. These are not licensure exam problems from 2020. Further commentary to come.)
I own a horse and a farm. One fourth the value of the farm is four times the value of the horse. Both taken together are worth $1,700. Find the value of each. Write out a complete analysis.
A merchant gets 500 barrels of flour insured for 75% of its cost, at 2 1/2%, paying $80.85 premium. For how much per barrel must he sell the flour to make 20% upon cost price?
Perhaps you are thinking about proportional reasoning and percentages. You might also be thinking: How quaint. These numbers are unnecessarily contrived; and owning horses, farms, and flour barrels is unrealistic to most students and teachers these days.
It is 1895. You are taking a high school mathematics teacher licensure exam and you see those same questions. What do you do? What do you think?
You might still think the numbers are contrived, but the context may seem more realistic.
Context and mathematics have never had an easy relationship. For one thing, it’s virtually impossible to hit the trifecta of precision, accessibility, and truth to reality. Bring in specific standards to be addressed, and that is a perfecta harder to achieve than predicting what would have happened next in Game of Thrones. (Example: Try writing a set of realistic problems for standard CC.5.MD.2; what are data that make sense to interpret as fractions to the nearest 1/8 that also make sense to add, subtract, multiply, and divide?) And we haven’t even brought in whether the context is actually appealing.
In this post, I present a case that determining whether context can support learning and teaching, through humanizing mathematics, is neither a simple yes-or-no calculation, nor is it value neutral. It is a pedagogical and ethical consideration. Context can motivate mathematics; and it can also extinguish interest. Context can be part of the mathematics; and it can distract. Context can humanize mathematics learning; but humanizing mathematics is not exclusively about context.
In a course for prospective high school teachers, I often teach properties of exponents, with a focus on why definitional conventions make sense and on ways to lead future high school students to discover these conventions (such as defining b^0 = 1 for nonzero b). Year after year, I would reach some teachers but not others. It was hard for some teachers to invest in building up identities that they had been using for years. Then, two years ago, I launched the unit on exponential properties with Dan Meyer’s Three-Act Math Task, Double Sunglasses, which begins with a whimsical clip of Meyer donning one pair of sunglasses, then another pair, then both. In keeping with the Three-Act Math Task structure, I asked teachers after viewing the video, “What questions comes to mind?”
The teachers pounced. “What does he see?” “What’s the tint?” “What does tint mean?” “Would it be 100%?” “No, it’s 0%!” “Or 25%?”
I explained:
With double sunglasses, it is like Dan is wearing 75% tint sunglasses. With double sunglasses, there was suddenly an intellectual need to find the tint, and an intellectual sense for why 100%, 25%, and 0% cannot be right. Teachers in that class, and subsequent classes where I have used this video, speak with gusto to propose, critique, and defend their views. Teachers also almost all come to the conclusion that 75% is correct, even if they originally thought 100%, 25%, or 0% was correct, and they identify various exponential properties along the way. For example, a 0th power corresponds either to wearing 0 sunglasses or wearing sunglasses with 0% tint. In both cases, one sees 100% of the light headed toward your eyes.
Double sunglasses are not realistic. They are not practical. They are ridiculous. And in its concrete absurdity, teachers found a way into the mathematics more deeply than any naked math problem, or any more realistic scenario featuring compound interest.
Context can add to mathematics learning, even when the context is not perfectly realistic, because the context gives just enough of a flavor of reality to hold onto lived experience. When Freudenthal (1971) proposed that one should “[teach] mathematics related to reality” (p. 420), he did not mean that the context had to be practical, realistically messy, or even realistic enough to constitute applied mathematics. He meant that students should be able to draw on real experiences to explore ideas and make inferences.
Finding a right context for particular mathematics is hard. Deborah Ball (1988) and Liping Ma (2001) famously posed:
Write a story problem for 1 3/4 ÷ 1/2.
This task is hard for a variety of reasons. Crafting story problems, and evaluating their correctness, can help prospective teachers understand ideas more deeply as well as appreciate student difficulties. This problem was devised by Ball for her dissertation and then used by Ma in her cross-national study.
Sometimes, even when there is a mathematically accurate context, it still may be distractingly unappealing (Nabb, Murawska, Doty, et al., 2020):
(The Condo Problem) In a certain condominium community, 2/3 of all the men are married to 3/5 of all the women. What fraction of the entire community is married?
This Condo Problem was first proposed in Lester’s (2002) Making Sense of Fractions, Ratios, and Proportions. It can be a beautiful mathematics problem that can help students make sense of operating with ratios. In the wake of Obergefell v. Hodges, though, the context is gauche at best, and offensive at worst.
In the changing acceptability of the condo context, Keith Nabb and Jaclyn Murawska, and their students found an opportunity. They asked their students:
How can you rewrite the Condo problem to be more inclusive? (Nabb et al., 2020, p. 696)
Nabb and Murawska report that over time, students in their courses had become increasingly uncomfortable with the problem. And so, their students appreciated the prompt to rewrite, and learned from this experience not only ways to rewrite it (with guinea pigs, penguins, college education rates, and jeans and shirts), but also the important lesson that they have the power to rewrite problems. I would also hazard that writing isomorphic problems also gave students more contexts with which to examine and understand the ratios. Mathematics gives us the power to describe the underlying structure of seemingly disparate situations. Recognizing this power, and wielding it, is part of mathematics learning and teaching. Imagining isomorphic contexts allowed these prospective teachers to confront discomfort about the original context, with mathematical integrity.
Coming up with contexts and then solving the problems with those contexts can be the mathematics. Explaining why contexts do represent a particular mathematical idea is itself mathematical work, and work that poses mathematical challenges beyond the underlying bare mathematics (Ball, Thames, & Phelps, 2008).
Contexts can also inspire mathematical inquiry because of their relevance. I have been struck by a story from Ricardo Martinez about teaching high school students about slope. They saw slope as a bare formula that was not consequential to them.
But when Ricardo presented data of their own school’s population over time, broken into racial and ethnic categories, these students, many of whom identified as Black, Hispanic, or Latinx, wanted to know more. Suddenly, slope was no longer italicized letters on a page but a concept with vivid and personal explanatory power. They asked Ricardo whether they could look up more data and compute more differences. They wanted to use the math to make sense of their life.
The Gerrymandering and Geometry materials, developed by Ari Nieh, use unit squares to model pieces of districts. Across the times I have used these materials, the results are similar: the teachers start with unit square geometry and end with fascination about area metrics more generally and questions about the political implications of any metric. These gerrymandering problems, despite the unrealism of exactly square land, have sparked more conversations about alternative metrics and their consequences than any lesson I have ever taught on hyperbolic or spherical geometry. The context of gerrymandering and its sobering consequences motivate earnest work with a simple model, and exploring the simple model inspires mathematical curiosity.
Turning back the opening problems, context can distract from the mathematics. If I were to in fact pose these problems today to students, I would predict that the second problem would be confusing because the language used is (no longer) commonly known jargon. (Footnote: These problems were actual teacher licensure exam problems in 1895 (Hill, Sleep, Lewis, & Ball, 2007). Perhaps the jargon was common then.) As for the first problem, it suffers from multiple problems, including a cantaloupe problem. Unrealistic context has the potential to teach a person to ignore previous knowledge, such as how much a house might cost, or how many cantaloupes an individual can reasonably purchase. The result can be mathematics as an idiosyncratic silo rather than as a discipline of beauty and descriptive power.
As Matt Felton-Koestler has written, “real-ish” problems, such as the Double Sunglasses task, or Gerrymandering tasks, have a place as stepping stones, but not as the only kind of context.
Exclusive use of real-ish problems, no matter how good they are, can teach students that in math, problems should always be well-posed: students should never have to seek out information, and the answer is always the operation-of-the-day with the numbers featured.
Moreover, adding real world context can hurt a mathematics task. Consider the problem
How many different three-digit numbers can you make using the digits 1, 2, and 3, and using each digit only once?
Show all the three-digit numbers that you found.
How do you know that you found them all? (Ball & Bass, 2014, p. 302)
Perhaps there could be a reason that someone needed to arrange the digits 1, 2, and 3; or to arrange a particular three other numbers or objects. But this task needs no real world context to make it work. Ball has used this task for multiple years with rising 5th graders in a summer “turn around” program for students who have not been successful in mathematics at school (Ball & Bass, 2014). As her teaching demonstrates, so long as students understand the conditions of the problem (using each digit exactly once), it is a task that is highly accessible and engaging.
There is another way to think of this problem’s apparent lack of context and its appeal to students nonetheless. Sometimes, mathematics itself can be a context for other mathematical ideas, and adding more context than that would take away from the task. Curious phenomena themselves can be context for discovery and explanation.
One of my favorite classes in grad school was the day we learned about complexifying vector spaces. With apologies to that professor, I will use complexification as a metaphor here. When we complexify a real vector space V, we extend the vector space to have twice as many dimensions and we construct a complex vector space. It’s not just about tossing in more basis vectors; it is also about endowing the new space with a new structure that allows for working with complex number scalars in a way that is compatible with the original vector space structure in V.
When we add context to mathematics presented to students, we are not just adding a dimension to the mathematics. We are also adding potential interactions between the context, the mathematics, and the class community. These potential interactions can enrich learning while also complicating our work as mathematics instructors.
Suppose that you use a context with statistics about race, such as income (Casey, Ross, Maddox, & Wilson, 2018) or honors class enrollment (Berry, Conway, Lawler, & Staley, 2020). Discussing race can be uncomfortable, because it is so charged. But increasingly, Black scholars are calling for explicit discussion of race in the classroom (e.g., Berry et al., 2020; Love, 2019; Milner, 2017; Tatum, 2016). As Casey, Ross, Maddox, & Wilson (2018) wrote, “Race … is a reality in our cultural moment, and it too important to be ignored when discussing issues of equity in education.” (p. 84)
It is often said that one power of mathematics is to understand the world around us. There are powerful and important statistics and mathematics to be done with real world data. Avoiding charged contexts can also mean losing opportunities for our students and ourselves to understand the world around us.
At a local Math Teachers Circle a few years ago, teachers worked on the Midge Problem, which asked them to find ways to differentiate two kinds of insects based on antenna and wing length:
[Given existing data] How would you go about classifying specifies Af or Apf?
Suppose that species Af is a valuable pollinator and species Apf is a carrier of a debilitating disease. Would you modify your classification scheme and if so, how?
This context gave teachers a concrete entry into determining metrics, and spurred a lively discussion. When it came to defending points of view, teachers drew on data, and made precise connections to others’ ideas. There was also a sense, though, that this kind of sorting was all in fun and an academic’s hobby – until it was mentioned that cutoffs are something that happens with standardized exams. Context can change how mathematics is taken up. In retrospect, the midge context may have allowed for more play, which was important for exploring the math. But the standardized exam context was more convincing, to this audience of teachers at least, of real life consequences of mathematical decisions.
With issues of race, students may inevitably begin hypothesizing potential reasons for disparity, which raises the danger of accidentally reinforcing stereotypes. Whether you disagree or agree with these hypotheses, and whether other students do or not, it is in the spirit of mathematics and statistics, or any other disciplined inquiry, to consider counterfactuals. This opening of explanations is part of what makes teaching problems with charged context so difficult: How do you approach the explanations with integrity and sensitivity?
How to work with complex social contexts is an open question for educators, and a hard and important one at that. For the issue of race, Casey et al. (2018) propose one way that has worked for instructors using their materials. Students begin by discussing explanations for disparity in small groups. Then, students are asked to consider causes and separate them into causes from within the group and causes from outside the group. Then from readings, videos, and data investigations done throughout the semester, students are asked to constantly reflect on their lists of causes, adding and removing items from the list based on what they’ve learned, with encouragement to add notes to themself with respect to why they are making an edit to the list when they make it. The materials ask students to examine data sets related to various within-group causes, including ones that debunk likely potential explanations from within the group.
Organizing proposed explanations seems outside of mathematics, yet upon reflection is central to both the discipline and its teaching. As mathematicians, when we come up with potential explanations or theoretical counterarguments, we don’t stop at the proposal – we see the proposal through with proof and evidence. But doing this also takes time and energy that we may not have before our 9am class. I am optimistic that in coming years, there will be more teaching practices and materials that will make teaching with charged contexts more possible.
There have been calls to humanize mathematics. Humanizing mathematics has come to encompass many meanings, including finding joy in mathematics; finding utility in mathematics to explain and understand real world phenomena; seeing the role of mathematics in making the world a better place; and using mathematics as a resource for social justice. Context in mathematics problems is then integral to some definitions of humanizing mathematics, especially those with social agendas.
Context is one variable in humanizing mathematics because of pedagogical and ethical considerations. Pedagogically, context may or may not enhance a task. Ill-chosen context can turn people off from the mathematics. At the same time, carefully considered context can give students a foothold to discovery and joy, though context is not always necessary for this purpose.
Federico Ardila-Mantilla, in a Notices article, urged mathematicians to consider the question of what it means to “do math ethically” (Ardila-Mantilla, 2020, p. 986). Context is one variable in addressing this question. Context can challenge narratives about what “is” mathematics, who does or needs mathematics, and the consequences of using mathematics. As Ardila-Mantilla observed, mathematics can be used to improve lives, and mathematics may be used for harmful purposes. At the same time, there is a place for mathematics for mathematics’ sake, as emotional responses to the work of teachers like Jo Boaler or Francis Su suggest. Mathematics–like song, poetry, and art–can and should be a place to play, find beauty, and experience joy.
The calculation for whether, when, or what context should be incorporated is not simple. I do not believe that we should prescribe context as a cure for all educational disease, and I do not believe that for all instances of mathematics learning, there should be a fitting contexts. Improving mathematics teaching and learning, through humanizing mathematics, can have to do with context, but is not exclusively about context.
Acknowledgments. I am grateful for encouragement from Erin Baldinger and Younhee Lee to pursue this topic, and for critical feedback on this essay from Stephanie Casey, Andrew Ross, Paul Goldenberg, and Mark Saul.
Errata. Hyman Bass has pointed out that it was Deborah Ball who devised the “1 3/4 ÷ 1/2” problem in her dissertation, and Liping Ma who then used this problem in her cross-national study. The post has been corrected with this information.
References
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.
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