What surprises you mathematically in your classes? When do you witness students’ creative moments? How often does this happen?
When instructors develop an environment where students are willing to put themselves “out there” and take a risk, interesting moments often happen. Those risks can only build one’s creativity, which is the most sought-after skill in industry according to a 2010 IBM Global Study.
How do we get students to be creative? And how does that balance with the content we are required to cover? Below, past and present members of the Creativity Research Group present reasons on why and how we each teach for creativity. We all have different but synergistic teaching practices we engage in to foster creativity in our students. Gulden focuses on having students making connections, while Milos has students take risks through questioning and sharing wrong answers. Emily focuses on tasks that have multiple solutions/approaches; Gail emphasizes the freedom she gives students in exploring these tasks. Mohamed provides time for students to incubate their ideas. Houssein and Paul reflect on their teaching practices and how teaching for creativity has been integrated into theses practices. There is also the thread of opportunities for student self-reflection woven throughout these stories.
One common aspect is that we try our best to saturate our courses with chances for students to be creative from beginning to end. These stories are our attempts at being creative about fostering creativity. Enjoy!
Gail Tang
I recently watched an Ugly Delicious episode where world-renowned chefs talked about their definitions of pizza and what it meant to them to make pizza. These chefs fell into two types: those who stuck to traditional ways to make pizza and those who departed from these ways. Those who ended up leaving the traditional ways behind did so because they felt stifled to operate under the strict rules of making pizza; they felt their identities were being compromised. With the courage of their convictions, they left to forge their own pizza paths. These chefs rejoiced in their freedom; finally they made pizza in a way that paid tribute to their identities. Chef Christian Puglisi said “If you only look at how it used to be done, or how it’s supposed to be done, you don’t allow yourself to move it forward.” This episode really resonated with me; replace “pizza” with “math” and “make pizza” with “teach math,” – you get the same story of suppressing innovation in the name of tradition. The idea of teaching others in the same way I was taught suffocated me. I was not interested in producing generations of students who could mimic my every mathematical move.
I started with baby steps in Calculus. I found exercises with more than one solution path to the same answer and assigned these without any direction. Students wrote different solutions on the board. Students were not used to seeing each others’ creations let alone creating their own solution paths. The energy in the room was thrilling. Unfortunately, I did not collect this data at the time, but fortunately Houssein El Turkey (see his narrative below) has an example of three students’ work on computing the limit below.
Letting students try problems on their own with little direction has the opportunity to have a profound impact on their mathematical identities. For example, one student started my Calculus 1 as a biology major and ended Calculus 1 as a math major! She wrote in her Calculus 2 weekly reflection:
I think having math ‘done to me’ rather than getting to explore it and have fun with it in high school is the reason I didn’t enjoy math in high school. I love how in your classes we get to try problems our own way and don’t have to use your method. I also think it’s super cool that you encourage using different methods. I would never have considered myself a creative person until I started working with numbers. I’m not anywhere near as creative as I should be, but I feel like math is helping me become more creative. The other night I did a problem with a method that I knew was going to be wrong, but I just wanted to see what happened. It actually helped me understand why that method doesn’t work.
Emily Cilli-Turner
A turning point in my thinking about students’ potential for creativity and how to foster it in the classroom happened while I was teaching a Linear Algebra course using the inquiry-oriented linear algebra materials. One task asked for the solution to a system of three equations in two variables; if there was no solution, find the “best” approximate solution. This task purposefully did not define the word “best” so the students would be forced to think about what qualifications the best approximate solution would have. Every group graphed the linear equations (which bounded a triangular region) and most presented “best” solutions as averages of $x$ and $y$ values of intersection points. However, one student was able to find the exact least squares solution by using optimization of functions of two variables techniques from calculus. Once this student presented his solution to the class, the other students were intrigued and could see the drawbacks of their own solutions. This was very unexpected for me. It drove home the point that students can and will be creative when we give them the tools and the freedom to hone their creativity.
In my mind, teaching for creativity has two main components: task design and collaboration. A large part of teaching for creativity is providing students with tasks that involve multiple approaches/solutions. The above episode would have turned out very differently if I had given the definition of the least squares solution and then several problems finding the least squares solutions of a system. The student would have never had a chance to find his original solution because all of the mystery would have been taken away with a provided definition. Yet, if the students had not been working on the task in groups, bouncing ideas off of each other, and using the group whiteboard to do scratch work, I think this vignette would not have happened. Collective creativity can be greater than that of the individual, and the students’ discussions helped that individual student refine his ideas and come up with the idea of using optimization to solve the problem.
Milos Savic
I believe teaching for creativity addresses a lot of issues in mathematics education. When a student is trying to be creative, there are many side effects, including more saturation with the content, greater mathematical confidence, and the ability to manipulate mathematics in different or new ways. Also, I believe mathematical creativity allows a student to be more of themselves instead of more like me. Finally, solving problems in STEM fields requires creativity, and I strive to create authentic experiences so that students can engage in being creative. These beliefs either are expressed explicitly (“Class, I want you to play with this idea”) or implicitly through tasks, quizzes, tests, and other requirements.
To generate curiosity, I have students ask two questions for every homework assignment; one is intended to be about the concepts, and the other is about their mathematical processes. I also give many routine problems with little twists. For example, I pose problems that require a student to go backwards instead of forwards (e.g. what non-linear function, when integrated from 1 to 2, is 17?). I also ask them to provide their own definitions or theorems using what they know. For example, using what they knew about groups and semigroups, a student created an anti-identity (the additive inverse of the multiplicative identity) and anti-inverses. My actions support these tasks; I celebrate many wrong answers in the class in order to show the process of mathematics and the creative moments within it.
Houssein El Turkey
It is interesting how my teaching has evolved to focus on more than just covering the basic learning outcomes in the classes I teach. I have become aware that teaching mathematics can and should include discussions with students on the novelty and flexibility in problem solving (or proving). Now I seek different approaches from my students to show them that there are often multiple ways to solve a problem. I also point out to my students when we build on something we discussed a while back to show that making connections is crucial. Another action I have taken is to explicitly show how certain processes generalize to a bigger picture.
Seeing an AHA moment from a student is one of the best highlights from a class. For example, when I asked my students to factor $x-1$ many of them were baffled but with hints and guidance, some of them came up with: $x-1 = (\sqrt{x}-1)(\sqrt{x}+1)$. The majority were taken by the simplicity and/or originality of the solution and the look on their faces was priceless. These AHA moments have been occurring more than before and they are constant reminders that they don’t have to be ground-breaking in order to have significant impacts on students. To a first-year college student, these simple tricks generating AHA moments can be crucial to show the originality aspect of doing mathematics even though instructors might find these tricks standard.
I also noticed that teaching to foster creativity has lifted my expectations of my students. I now see more potential in them and I work harder to get the best from them as I challenge them with tasks that require incubation and effort. An example of such a task that I used this semester was finding the limit $\lim_{x \rightarrow 1} \frac{\sqrt{x}-1}{x-1}$ in three (or more) different ways. Students struggled in their first attempt in class to see it as a slope of a tangent line but I left it for them to think about it at home and we picked it up the next class. I was happily surprised that some students had made that connection. Whether it was due to incubation or someone suggested it to them remains unknown to me.
Assigning challenging tasks was accompanied by emotional support through my continuous encouragement and emphasizing that it is OK to fail or not complete a task and it is OK to struggle because that shapes the learning process.
Mohamed Omar
Many students come to mathematics with an inherent curiosity, but traditional teaching practices tend to suppress this. It is essential that we tap into the curiosity that students have as a way to make mathematics come alive for them. Infusing creativity in the classroom fosters this in a way unlike any other.
To facilitate creativity, I ask open-ended or open questions on assignments. For example, a central theme in one course was counting the number of regions that a set of hyperplanes partition an $n$-dimensional space into, and determining what data about the set of hyperplanes are sufficient to answer that question. The related open-ended question asked how these results change if we used circles or other geometric objects instead of hyperplanes. I gave students plenty of time to play with these open-ended problems, and supported them along the way. The key to unlocking creativity was to create a structure where students were rewarded for their efforts and diversity of approaches, rather than their final output. To facilitate this, I required a three-page reflection using the Creativity-in-Progress Rubric (To read more about the rubric, please see our short article in MAA Focus Feb/Mar 2016). Students had to submit all their scratch work and all partial results, and subsequently use the rubric to reflect on their problem solving process. For instance, if a student used definitions and theorems from the course in conjunction definitions and theorems from outside resources, they could provide direct evidence for this in their work and comment on how the conjunction occurred. This allowed students reflect on their thinking processes, facilitating the pathway to creativity.
Gulden Karakok
Through active learning teaching practices, I plan learning situations that provide opportunities for discovery and making connections. Making connections has been an important component of my teaching, as I believe this process facilitates learning of the new topics and also allows students to transfer learning to other situations. Making connections comes in many forms in my courses — connections between definitions, theorems, various solution approaches, examples, and representations. I often ask my students if they have seen a similar topic, definition or example before. My goal is to discourage compartmentalization of ideas and topics. Unfortunately, our education system seems to train students to see ideas in disparate categories. To address this concern and foster creativity, I have been pushing for the process of making connections. I think asking students to find similarities and differences between ideas from their perspectives and background not only helps students to “own” these ideas but also develop sense of “usefulness” of them. With this ownership, students will be more equipped to be creative.
One example of how making connections promotes creativity comes from my preservice elementary math content course. During a class discussion on definitions of even and odd numbers, one student raised her hand and asked how we can determine “quickly” if a number in base 5 is even or odd (e.g., Is 123 base 5 even or odd?). This particular student was making connections to different bases discussed during the first week. Students worked on this problem and came up with several generalizations. We then discussed connections between those generalizations.
Paul Regier
Far too many students are afraid of math. I believe the antidote to their fears concerning math is experiencing mathematics by their own creativity. I suspect that in removing creative exploration from teaching mathematics, we run the risk of damaging our students. Just as the processing of modern food removes most of its nutritional value, removing creativity from math robs students of the most significant benefits they can gain from studying mathematics. Although a few students may appreciate the sugar rush of an already processed solution presented to them, it does not nourish them. What they gain does not last. It does not stick with them.
I teach for creativity by thinking creatively myself! When I lesson plan, after I have some kind of basic connection to the material and to past experience, I try to incubate for at least a day before I think about how I’m going to structure the class. Then I ask myself, “How little time can I spend presenting an idea, so that students are motivated and ready to start thinking about it themselves? What do I subconsciously withhold from students’ experiences (the joy of discovering and creating mathematics for themselves) that I can give conscious attention to?” In my experience, it is much easier to facilitate this kind of awareness by having students work together in groups with limited (but carefully planned) guidance and encouragement from me. It’s may be easy to give students quick answers, but this often takes away the student’s creative drive. Thus, I am learning to acknowledge students’ own thinking to be able to better provide opportunities for their own creative discovery.
Conclusion
As this blog ends, we hope that our stories serve as the beginning to your adventures in teaching for creativity! The Creativity Research Group has recently been awarded an NSF IUSE Grant (#1836369/1836371), “Reshaping Mathematical Identity by Valuing Creativity in Calculus”. To learn more about how to participate, or to communicate any of your ideas about fostering creativity, email us at creativityresearchgroup@gmail.com.
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In December 2017, the MAA released the Instructional Practices Guide (IP Guide), for which I served on the Steering Committee as a lead writer. The IP Guide is a substantial resource focused on the following five topics:
The IP Guide was designed with the intention of having independent sections be relatively accessible, so reading it from start to finish is not necessarily the best way to use it — I do recommend that everyone begin by reading the Manifesto and Introduction in the Front Matter of the IP Guide. My goal in this article is to provide three suggested starting points for faculty who are interested in using the IP Guide to inform their teaching, since it can be a bit daunting to identify where to start with this document. I want to emphasize that these suggestions are meant to be inspiration rather than prescription. My hope is that this article might be useful as a roadmap for department leaders incorporating the IP Guide for seminars, workshops, or other professional development activities with their faculty.
My belief is that faculty can be effective teachers using many different teaching techniques — there is no single “best way” to teach. Thus, our goal for faculty should be to gradually expand the teaching techniques they are familiar with, in order to create a “teaching toolbox” full of methods, ideas, and activities. With this in mind, I will frame my suggested starting points for the IP Guide based on the level of previous experience a reader has had with different teaching techniques, assessment structures, and course design frameworks.
For Faculty With Experience Using Mainly “Traditional” Teaching Techniques, e.g. Instructor Lectures, Problem Sets for Homework, In-Class Exams, etc.
Many readers of the IP Guide will have had their primary teaching experience be with what I refer to as “traditional” methods — by this I mean that the majority of class time is spent in a lecture format (even if it is somewhat interactive), the students are evaluated using problem sets for homework given once or twice per week and using 3-4 in-class exams, and the goals for student learning listed in the course syllabus are a list of content topics to be covered. For faculty who feel that this generally describes their previous teaching experience, I recommend starting with the following sections.
The selections from the Classroom Practices chapter all focus on short and simple teaching techniques that can be used in any class. For example, CP.1.1. discusses how to handle the first day of class in a student-centered manner, and then the following CP sections build on this by providing examples of techniques such as think-pair-share or paired board work that use 3-10 minutes of class time to engage students. These are natural first steps for instructors who are most familiar with teaching exclusively via lecture.
The selections from the assessment chapter focus on two goals. The first goal is for instructors to be able to increase the quality of exams that are given to students, which is the focus of AP.3. on Summative Assessment. Second, the example assignments in Section AP.4. provide an opportunity for faculty to have students reflect on their experiences in their courses, both from a mathematical and personal perspective.
The final two sections regarding course design and equity are intended to spark reflection on the part of faculty members. For example, section DP.1.1. contains a list of important questions for faculty to ask themselves prior to the start of a course, and section XE.4. provides three concrete examples of ways in which faculty can and should include equity considerations in their teaching.
For Faculty With Experience Using Some Non-Traditional Teaching Methods, e.g. Think-Pair-Share, In-Class Group Work, Reflective Essays, Lab Reports, Take-Home Exams, etc.
For those faculty who have some experience with non-traditional teaching methods, I recommend starting with the following sections.
For the selections from the Classroom Practices chapter, these sections focus on more “ambitious” teaching techniques that are more approachable once an instructor feels comfortable with smaller-scale methods such as think-pair-share. The three subsections from CP.2. are not discussions of techniques per se, but rather general frameworks through which to consider questions about task design for student activities.
The three sections from the assessment chapter indicated here provide three perspectives on student learning. The first is through the lens of formative assessment, and AP.2. provides both research-based frameworks for defining formative assessment and examples of how to implement formative techniques. Complementing this, AP.3.3. provides a taxonomy for instructors to use when writing exams (summative assessments) to evaluate student learning. Finally, as mentioned in the previous set of starting points, the example assignments in Section AP.4. provide an opportunity for faculty to have students reflect on their experiences in their courses, both from a mathematical and personal perspective.
For instructors with more experience using non-traditional teaching methods, the idea of framing their activities within a more coherent overall course design should be a natural progression. Because it is most natural to begin to engage with deeper consideration of course design from the perspective of activities and discussions, DP.1. an DP.2.2. are the recommended sections to begin with. Supporting an increased awareness of the role of course design, section XE.2. discusses important aspects of equity that should play a prominent role in every course.
For Faculty With Experience Using Multiple Pedagogical Techniques and Strategies, e.g. Inquiry-Based Learning, Online Courses, Service Learning, Explicitly Designed Formative and Summative Student Assessment, Course Projects, Universal Design for Learning, etc.
For faculty who have extensive experience with a range of teaching methods, the sections of the IP Guide that will likely be of the most interest are those that dive into some of the theoretical frameworks for teaching and learning, such as the following.
The extended Classroom Practices section on selecting appropriate mathematical tasks provides a broad discussion of theoretical frameworks such as Vygotsky’s ZPD theory and cognitive load theory, along with illustrative examples. Further, the discussion of Error Analysis in subsection CP.2.7. provides a refined perspective on how to handle student slips, errors, and misconceptions. The Assessment Practices section on formative assessment provides research-based frameworks for defining formative assessment and examples of how to implement formative techniques. The Assessment section on conceptual understanding is a detailed explanation of concept inventories, including examples and a discussion of how to incorporate items from concept inventories into assessment schemes.
Finally, the discussion of theories of instructional design in DP.4. introduce the reader to the frameworks of backward design, realistic mathematics education, and universal design for learning. Since these three topics are far too extensive to cover within a single section, references are given for further reading. The entire section on equity should be read by every instructor, as the ideas and language from studies of equity in mathematics education are not always widely-known; however, they are especially critical for faculty who have significant experience using ambitious classroom practices but who have not had much experience with explicit equity considerations in their classes.
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Dan Meyer is as close as we can get to a rock star in the world of mathematics education. These days, Dan is known for many things: 3-act tasks, 101 Q’s, Desmos, NCTM’s ShadowCon, to name just a few. But he initially rose to prominence on the basis of a TEDTalk that he gave as a high school teacher in 2010 (I recommend it highly). In his talk, he speaks directly to math teachers, and gives one simple piece of advice.
Be less helpful.
Teachers often understand their job as involving “help,” in some form or another. Most of us would like to think that we help students learn. Dan’s advice can therefore be understood as a challenge to a core aspect of our identity. And yet, the advice has become a mantra of sorts for teachers at all levels.
What does it mean to be less helpful, and why should that be a goal?
In his TED Talk, Dan shows how interesting mathematical questions are often surrounded by scaffolding:
Image credit: https://www.ted.com/talks/dan_meyer_math_curriculum_makeover
This scaffolding, in the form of mathematical structure and sequences of steps, works to obscure the interesting question (“which section is the steepest?”, buried in question 4). Students are asked to apply a mathematical structure (a coordinate plane) and accomplish sub goals (e.g., find vertical and horizontal distances) before they even know the goal (find the steepest section). Collectively, the scaffolding works to make math seem like an exercise in rule-following, rather than an opportunity to explore and make sense of interesting questions.
More perniciously, the scaffolding takes away much of the mathematics. Hans Freudenthal believed that structuring was at the heart of mathematics. In this problem, the scaffolding has already structured the problem. There is no structuring—and therefore no mathematics—left for the student to do. By presenting students with a ready-made structure for getting an answer, “mathematics” is reduced to answer-getting. To be less helpful means to remove the scaffolding, and to let students do mathematics. That is hard work for a teacher. In this post, I’ll give an example from my own practice, and discuss how I’m learning to be less helpful.
I have always considered myself a “reform-oriented” teacher—at least in content courses, my classroom practice adheres closely to the “conceptual math” described by Amanda Serenevy in an earlier post on this blog. In calculus, I have followed many of the recommendations from the calculus reform movement of the 1990s, including the “rule of four”: concepts and procedures should be explored from graphical, numerical, algebraic, and verbal perspectives. For example, early in the year, before any formal procedures are introduced, students explore derivatives graphically and verbally:
Following the rule of four (actually, I also use a lot of models, so let’s say, the rule of five) we use many strategies to develop differentiation rules and make connections between them. We use numerical analysis (looking at successive differences) to make conjectures about derivatives of polynomials. We couple this with verbal and graphical analyses (such as the examples above), which we also used to make conjectures about the derivatives of sin(x) and cos(x). We use first principles (algebraic) to prove these conjectures about elementary functions. We use models (area and see-saw, respectively) to develop the product rule and chain rule. We use algebraic strategies to write unknown situations in terms of known situations. This allows us to develop new rules, e.g., the quotient rule by rewriting a quotient as a product: and to find derivatives of new functions (e.g., developing the derivative of tan x by writing it as sin x / cos x . Later in the course we develop the logarithm as an accumulator function , and thus we can use the fundamental theorem to find the derivative, .
Finally, we come to a point in the course where it’s time to find the derivative of exponential functions. This is one of the crown jewels in a first-semester calculus course and I want to ensure that students understand it conceptually. As a first-year teacher, my plan was as follows:
This plan hinges on students developing a good sense of the behavior of the derivative of 2^{x}, that is, steps 1 and 2. Thus, for the rest of this post, I’ll limit my discussion to these steps.
In my first attempt at steps 1 and 2, students competed this worksheet in groups. The worksheet guides students to consider the derivative of 2^{x} from a variety of perspectives.
Throughout, students are asked to reflect on or explain their findings. The worksheet culminates with a prompt to make a conjecture about the derivative of 2^{x}:
Can you guess what happened at this point?
Well, let me tell you what didn’t happen. Students didn’t take a holistic perspective on the work that they had done. They didn’t say things like, “from step 1, I see that the graph of the derivative is a vertical compression of 2^{x}. From step 2, I see that the ratio of the derivative to the initial function is about 0.69. From step 3, I see that f ′(x) is some constant times 2^{x}, and I see that the constant is 0.69. Taken together, I conjecture that the .”
Instead, students treated question 8 as if it were independent from the rest of the worksheet. They generally conjectured that the derivative of 2^{x} was x⋅2^{x-1} (i.e., they used the power rule). Even though students had just completed a graphical, numeric, analytic, and verbal analysis of the derivative of 2^{x}, they did not draw on this analysis in answering the big question.
I tried this worksheet for a few years, each time with the same result. What I finally understood was that I needed to be less helpful. Even though I saw the mathematical structure in the worksheet, students didn’t. This is because I was the one who was doing the structuring. I created a cookbook-style “recipe for mathematics,” and in doing so, I had done all of the mathematics. The only thing that was left for students to do was to follow steps and get answers.
After a few of years of this worksheet, always with the same good intentions, always with the same disappointing results, I finally took Dan’s advice to heart. I decided to be less helpful.
Dan suggests that we start with the big question rather than burying it under a pile of scaffolding. My big question was, what is the derivative of 2^{x}?
To be less helpful, I thought, “why not just ask it?”
So that’s what I did. I put the question on the board, and asked students to explore it. I was really nervous, because I had never been this unstructured before. Structure is comforting to a teacher. Removing it is risky.
Now, can you guess what happened?
Actually, the same thing as when students got to question 8 on the worksheet. They applied the power rule and thought they were finished. Even though we had previously used graphical, numeric, and analytic approaches for finding derivatives of unknown functions, students did not use them here. In fact, they did not even recognize that I had asked a “big question.” Rather, my question seemed to them to be a straightforward application of a known rule.
I learned that I needed to set up the question better.
In math education, we call the setup to a problem the “launch.” Recently researchers have identified the launch as a key phase in a successful problem-solving lesson. When done well, the launch prepares students to engage in a complex problem. When done poorly, the launch either gives away too much, decreasing the cognitive demand of the problem, or doesn’t provide enough grounding, limiting students’ ability to engage in the problem.
A good launch should do three things:
Here is how my launch works now:
In my experience, this sequence accomplishes the objectives of a good launch. Students now see the question about the derivative of 2^{x} as a “big question,” worthy of exploration, and they have tools that they can use to explore that question. At the same time, I haven’t told them what to do or what the answer is, and so there is still plenty of mathematics to be done by students. The evidence of a successful launch is deep student engagement in the problem, and that has been my experience with this launch.
Groups go at the problem in different ways.
I don’t get every approach every year, but each year I get at least a few of these approaches. I have students share their approaches, and we make connections between them as a class. By the end of our class discussion, students are ready to conjecture that , and I can continue with the lesson. All together, this part of the lesson takes about 50 minutes: 10 minutes for the launch, 20 minutes for group exploration, 10 minutes for group presentations, and 10 minutes for me to summarize and connect approaches.
I’m learning that when I’m less helpful, students learn more. This is because rather than going through steps that are initially disconnected from a larger goal, students are consciously engaged in exploring a big question right from the beginning.
I’m learning that a lot of mathematics materials are cookbook-style “recipes for mathematics,” much like my initial worksheet was^{1}. I’m learning to use these materials to identify the key question, and then be less helpful by removing much of the scaffolding.
I’m learning that being less helpful is risky. Being less helpful means giving some of the control of the class over to students. Not all students will use every approach in their groups (unlike in my worksheet, where every student did the same thing). Instead, groups have discretion about the strategies that they use. And because the class discussion is dependent on the group work that came before it, the particular strategies that one class sees may differ from those that another class sees. There is less structure and more heterogeneity.
I’m learning that heterogeneity can be a resource. I depend on different groups taking different approaches. This allows me to structure our follow-up discussion so that students can see the connections between graphical, numeric, and algebraic approaches.
I’m learning that it actually takes a lot of work to be less helpful:
I’m learning that being less helpful doesn’t mean that I can’t help students. During group work, I often prompt students to develop their strategy further or connect it to a different strategy. Sometimes I’m pretty explicit in this prompting. For example, I’ll suggest that students try to make a table. Sometimes, if no group produces a particular strategy, I’ll introduce it in the class discussion.
I’m learning that being less helpful isn’t a magic bullet—not all of my students are engaged all of the time or learn all I’d want them to learn. But when I have a good question and a good launch, more students are more engaged in the activity and learn more of the mathematics than when I do the structuring for them.
I’m learning that one can learn to be less helpful. These days, it doesn’t feel so risky to ask the big question about 2^{x}, because I have a very good sense of what will happen. And it doesn’t feel quite so risky to ask new and different big questions. I know that I’ll fail sometimes (as in when I just ask the big question with no launch), but I also know that this is part of the process.
1 This includes many so-called “reform-oriented” materials. It’s likely that many reformers share my goal of engaging students in activity, and are trying desperately to shed the urge to explain—the traditional lay-it-all-out-and-then-let-the-student-practice approach, in which none of the puzzling-through is left for the student. However, it seems that often reform-oriented materials replace traditional explanations with a series of leading questions. Much like in my initial approach, these questions do all of the structuring for students and leave little mathematics for students to do.
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In the early part of this millennium, when the math wars were raging, I gave some testimony to the National Academies panel that was working on the report Adding it Up. Somewhat flippantly I said that which side of the math wars you were on was determined by which you were paying lip service to, the mathematics or the students. I was recently invited to give a plenary address at the ICMI Study 24 in Japan on school mathematics curriculum where I decided to expand on this remark, because I think it is worth going beyond the flippancy to map out an important duality of perspectives in mathematics education. What follows is an edited summary of what I said in that address.
In my address I talked about two different stances towards mathematics education: the sense-making stance and the making-sense stance. The first manifests itself in concerns about mathematical processes and practices such as pattern seeking, problem-solving, reasoning, and communication. It is an important stance, but it carries risks. If mathematics is about sense-making, the stuff being made sense of can be viewed as some sort of inert material lying around in the mathematical universe. Even when it is structured into “big ideas” between which connections are made, the whole thing can have the skeleton of a jellyfish.
I propose a complementary stance, the making-sense stance, which carries its own benefits and risks. Where the sense-making stance sees a process of people making sense of mathematics (or not), the making-sense stance sees mathematics making sense to people (or not). These are not mutually exclusive stances; rather they are dual stances jointly observing the same thing. The making-sense stance views content as something to be actively structured in such a way that it makes sense.
That structuring is constrained by the logic of mathematics. But the logic by itself does not tell you how to make mathematics make sense, for various reasons. First, because time is one-dimensional, and sense-making happens over time, structuring mathematics to make sense involves arranging mathematical ideas into a coherent mathematical progression, and that can usually be done in more than one way. Second, there are genuine disagreements about the definition of key ideas in school mathematics (ratios, for example), and so there are different choices of internally consistent systems of definition. Third, attending to logical structure alone can lead to overly formal and elaborate structuring of mathematical ideas. Just as it is a risk of the sense-making stance that the mathematics gets ignored, it is a risk of the making-sense stance that the sense-maker gets ignored.
Student struggle is the nexus of debate between the two stances. It is possible for those who take the sense-making stance to confuse productive struggle with struggle resulting from an underlying illogical or contradictory presentation of ideas, the consequence of inattention to the making-sense stance. And it possible for those who take the making-sense stance to think that struggle can be avoided by ever clearer and ever more elaborate presentations of ideas.
A particularly knotty area in mathematics curriculum is the progression from fractions to ratios to proportional relationships. Part of the problem is the result of a confusion in everyday usage, at least in the English language. In common language, the fraction a , the quotient a ÷ b, and the ratio a : b, seem to be different manifestations of a single fused notion. Here, for example are the mathematical definitions of fraction, quotient, and ratio from Merriam-Webster online:
Fraction: A numerical representation (such as 3/4, 5/8, or 3.234) indicating the quotient of two numbers.
Quotient: (1) the number resulting from the division of one number by another
(2) the numerical ratio usually multiplied by 100 between a test score and a standard value.
Ratio: (1) the indicated quotient of two mathematical expression
(2) the relationship in quantity, amount, or size between two or more things.
The first one says that a fraction is a quotient; the second says that a quotient is a ratio; the third one says that a ratio is a quotient. These definitions are not wrong as descriptions of how people use the words. For example, people say things like “mix the flour and the water in a ratio of 3 .”
From the point of view of the sense-making stance, this fusion of language is out there in the mathematical world, and we must help students make sense of it. From the point of view of the making-sense stance, we might make some choices about separating and defining terms and ordering them in a coherent progression. In writing the Common Core State Standards in Mathematics we made the following choices:
(1) A fraction a as the number on the number line that you get to by dividing the interval from 0 to 1 into b equal parts and putting a of those parts together end-to-end. It is a single number, even though you need a pair of numbers to locate it.
(2) It can be shown using the definition that a/b is the quotient a ÷ b, the number that gives a when multiplied by b. (This is what Sybilla Beckman and Andrew Isz´ak call the Fundamental Theorem of Fractions.)
(3) A ratio is a pair of quantities; equivalent ratios are obtained by multiplying
each quantity by the same scale factor.
(4) A proportional relationship is a set of equivalent ratios. One quantity y is proportional to another quantity x if there is a constant of proportionality k such that y = kx.
Note that there is a clear distinction between fractions (single numbers) and ratios (pairs of numbers). This is not the only way of developing a coherent progression of ideas in this domain. Zalman Usiskin has told me that he prefers to start with (2) and define a/b as the quotient a ÷ b, which assumed to exist. One could then use the Fundamental Theorem of Fractions to show (1). There is no a priori mathematical way of deciding between these approaches. Each depends on certain assumptions and primitive notions. But each approach is an example of the structuring and pruning required to make the mathematical ideas make sense; an example of the making-sense stance. One might take the point of view that the distinction between the sense-making stance and the making-sense stance is artificial or unnecessary. A complete view of mathematics and learning takes both stances at the same time, with a sort of binocular vision that sees the full dimensionality of the domain. However, this coordination of the two stances does not always happen. Rather than provide examples, I invite the reader to think of their own examples where one stance or the other has become dominant. This has been particularly a danger in my own work in the policy domain. I hope that spelling out the two stances will contribute to productive dialog in mathematics education, allowing for conscious recognition of the stance one or one’s interlocutor is taking and for acknowledgement of the value of adding the dual stance.
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Nearly twenty years ago Paul Lockhart wrote a brilliant essay, A Mathematician’s Lament^{[1]}, on the parlous state of mathematics education. In it, Lockhart laments that mathematics education does not celebrate mathematics as an art and as an important part of human culture. I write this essay in the same spirit, lamenting that mathematics education does not do well in preparing our students to use their mathematical skills to model the world they encounter in the practical, economic, policy and social aspects of their lives.
I have spent many years trying to understand why so many people seem to have difficulty with mathematics. Many people have a distaste for the subject and will go a long way to avoid engaging any use of their mathematical knowledge.
Elementary and secondary schools, the social institution to which we entrust the education of our young, present the subject of mathematics as a “right answer” subject. What other inference is to be drawn from questions like “what is the sum of 34 and 28?” and then “what is the sum of 41 and 24?” Seldom do we see problems like “Make up an addition problem with two whole numbers the answer to which lies between the sum of 34 and 28 and the sum of 41 and 24.” One could follow that problem with “How many such problems are there and how do you know?” And follow that with “Suppose you could use integers rather than whole numbers, how many such problems are there and how do you know?” It is no surprise then that the public at large thinks of mathematics as a body of knowledge in which any question has a unique correct answer and an uncountable number of incorrect ones. On the other hand, I claim that there are important cognitive benefits to be derived from posing problems with multiple demonstrably correct answers and multiple demonstrably incorrect answers.
I have never thought of mathematics as a “right answer” subject. I have always believed that mathematics provides a way of approaching the world and making sense of what one sees, hears and feels there. Indeed this is precisely the reason most often given for the inclusion of mathematics in the curriculum of our K-12 schools. Mathematics offers the promise of allowing people to make reasoned decisions about their daily lives.
Mathematics provides a set of tools that can help to confront the extraordinary complexity of phenomena that surround us. Although the things that interest us and the relationships among those things are inherently complex, mathematics can clarify essential elements in the surround. Mathematical models can often provide great insight into the way our world works. That having been said, our ability to deal with the complexity of our world by using our mathematical models is complicated by the constant need for –
Judgment and Imprecision
Counting and measuring are essential to all modeling. Even with these simplest acts of modeling, normally considered as an integral part of the K-12 mathematics curriculum, uncertainties about consistency and validity and the need for judgment are present.
Judgment is a serious matter in the act of counting. Does the head count in a ball park include only paying customers? People with complimentary tickets? Attendants, vendors, players, coaches? Clearly, purpose needs to be taken into account.
Imprecision and judgment are serious matters in the act of measuring. Consider the act of making simple measurements, even of such quantities as length or time. No measuring instrument can resolve the attribute it is being used to measure with infinite precision and as a consequence the best we can do is to assign a range of rational values as a magnitude. Additionally, judgment is required in deciding the extent of the attribute to be measured.^{[2]} Needless to say, no person has ever measured a quantity that resulted in an irrational number for a magnitude.^{[3]}
Further, judgment is required is assigning magnitude to the distance from Boston to San Francisco. An appropriate choice of unit might be the kilometer. It would hardly make sense to report the distance in millimeters, leaving aside all issues of from where in Boston to where in San Francisco. Reporting that distance in light-years would display a similar lack of judgment^{[4]}.
The need for judgment extends well beyond counting and measuring.
Suppose for example one needs to construct a cube of whose volume is as close as possible to 8 cm^{3}. Deferring the issue of precision for the moment, our mathematical model of a cube tells us that a side length of 2 cm will work. But it also tells us that side lengths of –1+3^{1/2}i and –1–3^{1/2}i would do as well. The decision to discard the extraneous roots of the equation x^{1/3} = 1 is a judgment that is made based on the context. The craftsperson fashioning the cube from a block of steel will certainly be able to make the required physical choice and not either of the permissible mathematical choices.
Similarly, our mathematical model of how a ball thrown on a flat Earth moves tells us that there are two times that the ball is on the ground. Newton’s laws of motion have two correct mathematical solutions – the time in the future when the ball will hit the ground and the time in the past when the ball was launched [neglecting the fact that the ball was almost certainly not launched from ground level]. We are likely to be interested in only one of these times.
Some postulates about essences
At least for the sake of this discussion please allow me some postulates about essences –
The essence of
…learning is posing for oneself a provocative next question
…teaching is posing for students a provocative, engaging next question at the proper moment
…education is learning to ask of the world “what is this a case of?” and “what if not?”
…mathematics is making, exploring, proving and disproving conjectures^{[5]}
The reader will note that all these essences center on the posing of questions and the making of conjectures. This is particularly true of mathematics. None other than Georg Cantor once said, “In mathematics the art of proposing a question must be held in higher esteem than solving it.” By posing problems, David Hilbert, at the International Congress of Mathematicians in Paris in 1900, set the course of research for much of the mathematics of the twentieth century.
A physicist’s view of posing problems
Because as physicists we are engaged in modeling the world, albeit with mathematical tools, we do not always have the luxury of asking questions that have unique correct answers. Our problems, which we pose both to our students and to ourselves, are about models, their predictions and their consequences.
Models are never “correct” – at best they are “adequate.” Unlike an unsolved problem in mathematics, models in physics cannot ever be proven to be “correct.” This observation suggests, at least to this physicist, two guidelines about the posing of problems in mathematics. They are –
The reason underlying the first guideline is that a unique correct answer can only address the internal consistency of a model but not its validity or utility.^{[6]}
The reason underlying the second guideline speaks to the adequacy of a model—no model can take into account the full complexity of the phenomena being modeled. Judgment must be applied when deciding whether an answer to a problem is “good enough.” Perhaps the most widely known examples of the use of this guideline are Fermi estimation problems.^{[7]}
How might we think about mathematics when posing problems that follow these guidelines?
Objects and Actions – Measures and Models
The idea of mathematical objects and mathematical actions that can be performed on them or by them is central to the nature of mathematics. Simple mathematical objects and associated mathematical actions are often combined to form more complex objects which in turn have actions associated with them.
The power of this idea derives from the fact that essentially all of the languages spoken by people have utterances that are composed of a noun phrase and a verb phrase. A noun phrase tells us about objects and their properties. An associated verb phrase tells us about the actions that an object carries out or that is carried out on the object.
Measures are quantifications of attributes of nouns. Examples include the weight of a person or a gallon of milk, the circumference of a person’s waist or the height of a bookcase, the surface area of the body or that of a mountain lake. Other such measures are the volume of a human body or that of a bottle of wine, the length of a person’s lifetime or that of a movie, the number of a person’s red blood cells or the number of people in the US on any given day.
The measures cited here^{[8]} are basic ones. We are equipped by nature with sensory tools to assign some sort of magnitude to these measures or, at least, to order a collection of objects having that attribute.^{[9]}
These fundamental measures can be composed with one another to make composite measures. Composing a distance traveled with a time interval yields velocity as a composite measure. Composing velocity with a time interval yields acceleration as a further composite measure. Composing the number of people in a city with its surface area yields a measure we usually call population density.
Models are assertions of relationships among measures. Newton asserted a relationship between the measure of acceleration and the magnitude of the push or pull causing that acceleration. Mendel asserted a relationship between the color of the seeds of one generation of pea plants and the color of the seeds of cross-bred members of the next and subsequent generations.
The reason schools include mathematics in the K-12 curriculum^{[10]} is that the judicious use of mathematical models can help our students make sense of what they see and hear in the world around them. Such use offers the promise of allowing people to make reasoned decisions about their daily lives.
If, indeed, the end goals of our teaching mathematics and our students learning mathematics is to help them use mathematical models in their lives, we must put greater emphasis on modeling in our teaching—and not at the expense of reliably and correctly executed mathematical manipulations.
In a future blog, I hope to discuss one approach to doing just that.
Endnotes
^{[1]} Reprinted in MAA Online, March 2008, Devlin’s angle
^{[2]} Does one determine a person’s weight before or after a haircut? Does the surface area of a mountain lake include [all, part, none] of the runoff stream?
^{[3]} For example, the π that appears in C = πd is defined as the ratio of a circumference to a diameter, but the magnitude of the number, itself, is not computed from measurements. A measurement of the circumference yield a value, C ± ΔC, and a measurement of the diameter yields a value, d ± Δd. The ratio of these two values (C ± ΔC)/( d ± Δd) is a bounded range of rational numbers. The definition C = πd expresses a model of the relationship between circumference and diameter, perhaps inferred initially from many measurements of circumferences and associated diameters but then determined by logical reasoning.
^{[4]} I recall an incident in which a sixth grade student who fully understood the issue, displayed this understanding in an extraordinary way. I had asked him to formulate an estimation problem. With a huge grin on his face he said, “What is the average size of a postage stamp in square miles?”
^{[5]} Proving and disproving each have a different meaning in mathematics than they do, for example, in law.
^{[6]} A train is normally thought of as a rigid body. Thus, in this model, the caboose begins to moves the instant the locomotive does. Rigid bodies, inextensible strings, point masses are all instances of models that are valid enough and useful enough for many purposes. In the idealized world of such models, problems with single, correct answers can be, and often are, posed.
^{[7]} For example, how long does it take a person to eat his/her own weight in food?
^{[8]} Length, area, weight, volume, time, number.
^{[9]} Measures can be defined within mathematics itself. Consider a collection of rectangles. People of all ages are willing to grant that a 5 x 6 rectangle is ‘squarer’ than a 2 x 15 rectangle. Thus one can pose the problem of defining a measure of ‘square-ness.’ Having done this with both elementary school and college students, I can attest to the fact that many reasonable answers are possible.
^{[10]} I have written elsewhere on the topic of the reasons for the inclusion of mathematics in the school curriculum. See Can Technology Help Us Make the Mathematics Curriculum Intellectually Stimulating and Socially Responsible?, International Journal of Computers for Mathematics Learning, 1999, Vol. 4, Nos. 2-3, pp. 99-119.
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How is \(0^0\) defined? On one hand, we say \(x^0 = 1\) for all positive \(x\); on the other hand, we say \(0^y = 0\) for all positive \(y\). The French language has the Académie française to decide its arcane details. There is no equivalent for mathematics, so there is no one deciding once and for all what \(0^0\) equals, or if it even equals anything at all. But that doesn’t matter. While some definitions are so well-established (e.g., “polynomial”, “circle”, “prime number”, etc.) that altering them only causes confusion, in many situations we can define terms as we please, as long as we are clear and consistent.
Don’t get me wrong; the notion of mathematics as proceeding in a never-ending sequence of “definition-theorem-proof” is essential to our understanding of it, and to its rigorous foundations. My mathematical experience has trained me to ask, “What are the definitions?” before answering questions in (and sometimes out of) mathematics. Yet, while we tell students that the definition needs to come before the proof of the theorem, what students apparently hear is that the definition needs to come before the idea, as opposed to the definition coming from the idea.
Why definitions?
What is a definition anyway? Or rather, what gets defined? We could make a special name for the function that maps \(x\) to \(5x^{17} – 29x^2 + 42\), but we don’t. On the other hand, we give the name “sine function” to \(\sin(x)\), the ratio of the length of the side opposite an angle with measure x to the length of the hypotenuse of a right triangle. We give a name to the sine function, even though it takes much longer to describe than \(5x^{17} – 29x^2 + 42\); in fact, we give it a name in part precisely because it takes longer to describe. If we need to refer to \(5x^{17} – 29x^2 + 42\), it’s not that hard, but we do not want to have to write down that definition of sine every time we use it in a statement or problem. We give definitions to ideas for two related reasons:
Brevity: It’s clearly easier to write “\(\sin(x)\)” instead of the huge sentence above. Further, packing this idea into a single word helps make it easier to chunk ideas in an even longer statement, such as a trigonometric identity.
Repetition: If we have to use the same idea more than once, then giving it a compact name increases the efficiency described above that much more. Sometimes an idea repeats just locally, within a single argument or discussion, and then we might temporarily give it a name; for instance when finding the maximum value \(x e^{-x}\), we would write \(f(x)=x e^{-x}\), so we could then write \(0 =f'(x)\), but we are only using \(f\) this way in this one problem. On the other hand, the ideas that show up over and over again, in many different contexts, such as \(\sin(x)\) or “vector space”, get names that stick.
This begs the question, “Why do certain ideas, or combinations of conditions, repeat?” Consider “vector space”. The idea of \(R^n\) is clear enough, but of all its properties, why focus on the simple rules satisfied by vector addition and scalar multiplication?
First, because several additional examples have been found that satisfy these rules, such as the vector space of continuous functions, the vector space of polynomials, and the vector space of polynomials of degree at most 5. Second, because once the key properties that make up the definition are identified, we may find that the proofs only depend on those key properties: The Fundamental Theorem of Linear Algebra, for instance, is true for arbitrary finite-dimensional vector spaces, so we don’t need a separate proof for \(R^n\), for polynomials of degree at most 5, etc. (Purists may argue that all finite-dimensional vector spaces of the same dimension are isomorphic, but this isomorphism is defined in terms of vector addition and scalar multiplication, just reinforcing the significance of those operations.)
Choices
But there are often still choices to be made. Must a vector space include the zero vector, or could it be empty? (Is the empty set a vector space)? For that matter, since vectors are often described as being determined by “a direction and a magnitude” and the zero vector has no direction, is the zero vector even a vector? The answers to these questions are no and yes, respectively, but why? The zero vector is a vector, because it is so helpful for a vector space to be a group under addition, which requires an identity element. (I know — this only takes us back to why are groups defined the way they are. Let’s just take this as a piece of evidence for why groups are an important definition.)
As for the empty vector space, there’s nothing inherently wrong with it, except perhaps for the need for a zero vector as discussed above. (This also takes us back to why groups are not allowed to be empty. Let’s stick to vector spaces for now.) But how would we define the dimension of an empty vector space? How would we define the sum of the empty vector space with another vector space? And then, even if we do make those definitions, how do we reconcile them with this identity?:
\[
\dim (A+B) =\ \dim A\ +\ \dim B\ -\ \dim (A \cap B)
\]
This example shows that, even though we cannot write the proof of a theorem until all the relevant definitions are stated, we do often look ahead at the theorem before settling on the fine points of the definition. At research-level mathematics, we might even modify our definitions substantially to make our theorems stronger, or to deal with potential counterexamples. (For more details on this, read Imre Lakatos’ classic Proofs and Refutations [1].) I will stick to smaller cases where we adjust definitions mostly just to make the theorems easier to state.
More examples
Why is 1 considered to be neither prime nor composite? When you first learn this, it may seem silly. The definition of prime is so simple and elegant — an integer \(n\) is prime if its only factors are 1 and \(n\) — and 1 seems to fit that definition just fine. Why make an exception? The answer lies in the Fundamental Theorem of Arithmetic, that every integer has a unique factorization. Well, except of course that we could change the order of the factors around; for instance, it makes sense to consider \(17 \times 23\) to be the same factorization as \(23 \times 17\). And also we need to leave out any factors of 1, otherwise we might consider \(17 \times 23, 1 \times 17 \times 23, 1 \times 1 \times 17 \times 23\), … to all be different factorizations. If we take a little extra effort at the definition, and rule out 1 as a prime number, then the theorem becomes more elegant to state.
Is a square also a rectangle? In other words, should we define rectangle to include the possibility that the rectangle is a square, or exclude that possibility? When children first learn about shapes, it’s easier to simply categorize shapes, so a shape could be either a rectangle or a square, but not both. But when writing a careful definition of rectangle, it takes more work to exclude the case of a square than to simply allow it. Similarly, theorems about rectangles are easier to state if we don’t have to exclude the special cases where the rectangle happens to be a square: “Two different diameters of a circle are the diagonals of a rectangle” is more elegant than “Two different diameters of a circle are the diagonals of a rectangle, unless the diameters are perpendicular, in which case they are the diagonals of a square.”
Is 0 is a natural number? It doesn’t really matter; just pick an answer, be consistent, and move on. It’s even better if we can use non-ambiguous language instead, such as “positive integers” or “non-negative integers.” To be sure, mathematics is picky, but let’s not be picky about the wrong things.
Finally, what about \(0^0\)? If you just look at limits, you’d be ready to declare that this expression is undefined (the limit of \(x^y\) as \(x\) and \(y\) approach 0 is not defined, even just considering \(x \geq 0\) and \(y \geq 0\)). And that’s fine. But in combinatorics, where I work, setting \(0^0 =1\) makes the binomial theorem (\((x+y)^n = \sum \binom{n}{k} x^k y^{n-k}\)) work in more cases (for instance when \(y=0\)). And so we simply declare \(0^0=1\), at least in combinatorics, even though it might remain undefined in other settings.
(See here for a list of other “ambiguities” in mathematics definitions.)
In each of these examples, there is a human choice about how to exactly state the definition. This is a great freedom. But, to alter a popular phrase, with great freedom comes great responsibility. If you declare \(0^0\) is a value other than 1, now you are limiting, not expanding, the applicability of the binomial theorem. And if you want to declare that \(\frac{1}{0}\) has any numerical value, you will have to sacrifice at least some of the field axioms in your new number system.
In the classroom
The issues that arise with developing precise mathematical definitions is well-known to mathematicians, but we generally don’t share it with our students enough. If we stop hiding this story from our students, then they will see that mathematics is a human endeavor, and that mathematical subjects are not handed down to us from on high. This can be one factor in convincing students that mathematics, even advanced mathematics, is something they can do, that it is not just reserved for other people. And even students who already “get it” will not be turned off — we should not abandon definition-theorem-proof, we can just pay more attention to sharing why each of our definitions is written the way it is. If students know where a definition comes from, what motivated it, and why we made the choices we did, they may have a better chance of making sense of the idea instead of memorizing the string of words or symbols. (See also my earlier blog post, A Call for More Context.)
An anecdote that Keith Devlin tells, near the end of a blog post about mathematical thinking, illustrates the power of crafting the right definition. To summarize much too briefly, his task was to “look at ways that reasoning and decision making are influenced by the context in which the data arises” in a national security setting. His first step was to “write down as precise a mathematical definition as possible of what a context is.” When he presented his work to government bigwigs, they never got past his first slide, with that definition, because the entire room spent the whole time discussing that one definition; later he was told “That one slide justified having you on the project.”
We might not have the luxury of spending an entire hour discussing a single definition, but we can still let students in on the secret that the definitions are up to us, and that writing them well can make all the difference.
References
[1] Lakatos, Imre. Proofs and refutations. The logic of mathematical discovery. Edited by John Worrall and Elie Zahar. Cambridge University Press, Cambridge-New York-Melbourne, 1976.
]]>This past spring, I received an email from a graduate student who was concerned about applying for jobs in industry. The student wrote: “I’m having a difficult time trying to market my teaching experience. I’ve been teaching for three years now and I want to leverage that in my applications. I’m just not sure what to say beyond ‘improving communication skills’.”
Whether their interests are in academic positions or not, many graduate teaching assistants (GTAs) are concerned about the jobs they will find and whether they are prepared for those jobs. I have led the graduate teaching assistant training in the Department of Mathematics at Oregon State University since 2013. In that time, I have come to realize it is critical to help GTAs understand the professional skills they develop during their graduate careers, particularly as they learn to teach. My goal in this note is to unpack and describe some of the processes of teaching to help the GTAs appreciate the skills they learn through teaching, and see that these skills can be applied to a variety of jobs beyond academia.
I searched the internet for recent articles that describe the skills employers are looking for, now and in the future. In the list that follows, I highlight some of skills that were common across these articles and discuss how GTAs develop these skills through their teaching. This list is not meant to be exhaustive.
In providing this list, I want GTAs to see that teaching is much more than writing mathematics on a board, and that there is much to be learned through the processes of teaching. Illuminating the skills learned through the processes of teaching will help our GTAs reflect on their practices, help them to reflect on what they are doing as teachers, and inspire further exploration. This reflection in turn helps GTAs better describe their relevant experience in cover letters, on CVs, in their teaching statements, and in conversations about their work as teachers. I believe that explicit attention to these skills can contribute greatly to the professional development of GTAs.
Communication is a critical skill recognized on multiple websites, and teaching is all about communication. As teachers, we learn to communicate complex ideas in multiple ways, and we communicate much more than the mathematical concepts we write on the board. We use many modes of communication (speaking, writing, body language, facial expressions, written assignments, handouts, and online materials). We communicate many messages (encouragement, positivity, enthusiasm), and of course we communicate mathematical content (mathematical ideas, problem solving strategies, multiple representations of mathematical concepts).
Most of what we communicate to our students comes through public speaking in classrooms – standing in front of small or large groups of people, who look at us all at the same time, waiting for us to speak to them and to get the class going. Through the processes of teaching, we learn to build skills of responsiveness and to adapt our instruction to the different ways students solve problems. Teaching naturally provides GTAs with opportunities to cultivate the ability to speak publicly and give presentations. Beyond this, teaching helps GTAs learn to convey abstract ideas effectively, in ways that people with varying backgrounds and learning styles can understand. Effective public speaking is a skill that applies in many different situations: job interviews, conference presentations, or presentations in the work place.
What other communication skills might we learn and hone through the processes of teaching?
Several websites noted that facilitation is an important skill for people interested in leadership positions. In fact, one author [1] referred to facilitation as the “key to the future of work.” What work is involved in facilitating student learning? Leading a group of 10, 20, 30, or more students in productive group learning and problem-solving activities is rich, complex work. Facilitating students’ group work on mathematical tasks and conversations about their work requires giving clear instructions and setting expectations. Effective facilitation requires attention to inclusivity and equity to ensure that all students’ voices are heard and supported, and that every student’s work is recognized as valuable and contributing to the course. Facilitation also means actively listening, responding in ways that lead to productive conversations, helping students learn to how to communicate their mathematical thinking, and getting them to work as a team and support each other.
Developing interpersonal skills is also essential for any kind of work with people. Some websites noted that people with strong interpersonal skills are more successful in both their personal and professional lives ([2], [3]). Another website noted that interpersonal skills are considered “employability skills” [4] because hiring managers do not want to hire people without them. A few websites noted that applicants should highlight interpersonal skills in cover letters and/or resumes. So, what are interpersonal skills? They are also called ‘people skills’ – the behaviors and characteristics we use when we communicate and collaborate with others, such as active listening, empathy, collaboration, problem-solving, adaptability, and leadership.
Interpersonal skills can also be learned outside the classroom, during office hours and in tutoring sessions when teachers work with individual or small groups of students. Through these interactions with students, GTAs can learn what mathematical explanations and representations might be best for helping students understand a mathematical idea. In addition, GTAs can reflect on their interactions to learn how their communication with students can be positive and encouraging. Office hours can also include tough situations and conversations, which may require honest or critical feedback that is also supportive and encouraging. GTAs can learn a great deal from their work with individual students and translate that learning into deeper interpersonal skills that can be applied to any number of future work places.
Much of what we do as teachers is lesson planning. Planning for and teaching a course is a form of project management. Before I became a teacher, I was a project manager at an educational software company. Multiple websites offer anywhere from five to ten steps for successful project management ([5], [6]). These steps include determining the objectives of the project (learning outcomes), initiating the project (writing the syllabus, planning the term, finding course materials), executing the project (doing the work of teaching and working with students), managing the project (monitoring progress, re-calibrating, staying connected to project outcomes), and completing the project (getting to the end of the term having covered the material of the course). Indeed, the work of planning a course, sequencing concepts, problems and tasks, connecting past ideas to current ideas, foreshadowing what students will encounter later in the class or in future classes helps our TAs learn about project management.
Once a course is mapped out for the term, there is the day-to-day implementation of the project through specific lessons. GTAs can learn to plan specific features of the lesson, such as lecturing, group work, and student presentations, when and how those features will occur, and how they might create alternatives should a lesson not go as planned. Planning for specific lessons gives GTAs the opportunity to think deeply about what mathematical concepts they will present to students and how they might have students engage in mathematical activities around those concepts. Some questions GTAs might ask when planning for a lesson include: What mathematical concepts do we want to communicate? What representations might we use? At what points in a lesson might we pause and let students do some mathematical work – and why would we pause for those particular tasks? These questions can be applied to multiple work places in terms of how work happens and why, the sequencing of tasks, and revisiting and revising work to improve outcomes. How might homework sets reinforce classroom learning and prepare students for the next class, to keep the mathematical momentum going?
An important skill listed on several websites is analysis of data. Often, the work of grading is simply described as marking students’ work as correct, incorrect, or somewhere in between. I would argue that ‘grading’ does not adequately describe the various ways that teachers assess students’ learning through homework, quizzes, exams, formative and summative assessment. Assessing students’ work is much more complex and requires much more thought. Teachers regularly analyze data as they administer assessments, analyze student work, and make conclusions about how they might modify their teaching to improve student outcomes. Assessing student learning requires the teacher to meaningfully interpret student work and try to understand what students were thinking when they solved a problem. By assessing student learning, GTAs will learn how to analyze data, understand it, and respond to it. They will learn how to keep electronic records, compute statistics, and make decisions on how to proceed.
GTAs learn to use software programs during their graduate course work (e.g., Matlab, Python, SAS, SPSS, and so on), and they also learn about various learning technologies used in the classes and recitations they lead. These technologies include the use of Clickers, Canvas, Blackboard, MyMathLab, Geogebra, Desmos, Learning Catalytics, and open resource materials, among many others. Not only do GTAs learn those programs, but they also learn how to help students to use those technologies. In these ways, GTAs are developing many skills that can be applied to multiple work places.
I believe the list above is a good first step in helping GTAs to appreciate the skills they learn through teaching. However, it is essential that we go beyond informing GTAs that they are learning these skills because, eventually, they will have to present themselves as people with these skills. It is critically important that GTAs become conversant in these skills, to develop their own voice and an understanding of how they authentically present themselves as someone with these skills. Consequently, I ask the GTAs to reflect on their experiences in classrooms with these skills in mind. They write about how they see themselves learning these skills through different teaching situations. Some of these situations might include how they communicate a difficult idea to students, how they sequence a lesson and why they thought that sequence was effective (or not), or how and why they might improve on their assessments of student learning. In writing about these situations, GTAs reflect on how they are growing as teachers and what skills they have learned through the processes of teaching, which helps them see the way their teaching experience can be applied across multiple professions.
Through these types of activities, I think we can help GTAs appreciate what they learn from their teaching experiences and help them to translate that learning into a concrete, explicit set of skills they can apply to multiple professions. And, for students like the one cited above, we can provide them with ways to present themselves as learned professionals.
[1] Klein, B. (2017). What “facilitation” really means and why it’s the key to the future of work. Retrieved August 28, 2018 from https://www.fastcompany.com/40467377/what-facilitation-really-means-and-why-its-key-to-the-future-of-work.
[2] Terrell, S. (2018). What are interpersonal skills and why are they so important? Retrieved August 28, 2018 from https://blog.mindvalley.com/what-are-interpersonal-skills/?utm_source=google_blog
[3] The Skills You Need. (n.d.). Interpersonal skills. Retrieved August 28, 2018 from https://www.skillsyouneed.com/interpersonal-skills.html
[4] Doyle, A. (2018). Interpersonal skills list and examples. Retrieved August 28, 2018 from https://www.thebalancecareers.com/interpersonal-skills-list-2063724
[5] Sundwall, H. (1996). Seven steps to success for project managers. PM Network, 10(4), 31–32. Retrieved August 30, 2018 from https://www.pmi.org/learning/library/seven-steps-success-project-managers-3313
[6] Lucidchart Content Team. (2017). 5 essential project management steps. Retrieved August 30, 2018 from https://www.lucidchart.com/blog/5-essential-project-management-steps.
]]>Real Analysis is a rite of passage for undergraduate math majors. It is one of my favorite courses to teach, but I recognize that the course is challenging for students, and, for many, downright intimidating. In Fall 2017 I was scheduled to teach Real Analysis for the third time in my career. Prior to the semester starting, I knew that I wanted to alter the grading scheme of the course to de-emphasize exams in favor of effort. Ultimately, I wanted to promote a growth mindset and to help students identify their strengths and weaknesses independent of exam performance. During our annual summer visit, my good friend and graduate school classmate Matthew Pons described to me his new project with Allison Henrich, Emille Lawrence, and David Taylor called The Struggle is Real: Stories of Struggle and Resilience on the Path to Becoming a Mathematician. (For more information on their project, check out https://math.roanoke.edu/tsir/.) I loved their idea of gathering and sharing personalized stories around this topic and immediately thought of adapting the exercise for my students. Since I was teaching Real Analysis, I decided to include reflective homework problems and activities under the label #thestruggleisREAL. I was worried that the hashtag was too gimmicky, but decided that with the right sales pitch students would embrace the pun. In this post I describe how this well-trodden hashtag injected a great deal of reflection, and a bit of levity, into my students’ experience in Real Analysis.
I’ve spent a great deal of time privately considering my own struggles in mathematics. I’ve exchanged stories with classmates from graduate school, with my spouse and other confidantes, and one-on-one with students during office hours. But as a student I never had these conversations as a part of a class, and as a teacher I have not offered a place for sharing such stories beyond one-off conversations. Thus I decided, without figuring out any further details, to incorporate into the class writing prompts where students could share their own stories of struggle and resilience. The two things I did decide upon from the beginning were that the writing assignments would be regular throughout the semester and graded (but low stakes). The pertinent language on the syllabus read:
#thestruggleisREAL: Throughout the semester there will be homework problems and activities under this hashtag. At the end of the semester, students may elect to complete a reflective project elaborating on this work. For students who complete this option, the project will count towards 10% of the final grade, and the exams will each count 15%. [As opposed to the higher of two exam scores counting 25% and the lower exam score counting 15%].
I previously used different grading tracks in Calculus classes when including a Community Engagement component in the course, and I felt that this would be another appropriate time to give students agency in how their work was evaluated. It was important to me that they participate in reflection throughout the semester and that everyone was rewarded equally for this effort. At the end of the semester, students could then choose what type of work was the best measure of their performance in their class. As is to be expected there were questions regarding the mechanics of the grade distributions, but there were no complaints about separate grading tracks. Indeed, several students with high exam averages completed the final reflective project, and several with low exam scores opted out.
The Assignments
As in previous semesters, I assigned weekly homework in the class consisting mainly of proofs. This time around, I added to each assignment one problem labeled #thestruggleisREAL. In addition to the reflection that these problems prompted, I think that by putting narrative writing side-by-side with formal proof writing I was able to strengthen the case that mathematical writing is “writing”. In most cases, students received full credit for completing the problem. I relied on the fact that all Real Analysis students have taken a writing intensive first-year seminar. This means that the quality of exposition was generally acceptable, and besides that I used the formal proofs in the homework to hold them accountable for writing mechanics.
Below is a selection of the writing prompts that I used during the semester.
First assignment: I teach at a small liberal arts college where course reputations carry a great deal of weight. With this in mind, the first reflection asked students to write down their preconceived notions of the class. Students did not turn in their assignments, but shared them in pairs during our second class meeting. I wrote my own reflection and shared it on the class webpage.
Graded assignments: The writing prompts asked students to reflect upon a particular homework problem or proof completed in class. I generally asked for a paragraph response and it counted as a regular homework problem. Some examples are:
Optional (fun) assignments:
Final homework assignment: Should Real Analysis be a requirement for the math major? Why or why not?
Final project: Finally, as stated in the syllabus, students were given the option to complete a more formal writing piece reflecting on their semester in Real Analysis. I directed students who were interested in the project to first read an article from 2010 in Math Horizons, The View from Here: Confronting Analysis, by Tina Rapke. (If I had found the article sooner, I would have assigned it as reading in Week 1!) The written assignment required students to choose 3-5 of their own proofs from the semester as representative examples, and then to write 1-2 pages narrating their experience in Real Analysis using these proofs to illustrate their experiences. Roughly half of the class turned in a narrative and most exceeded the requested two pages in length. I was extremely pleased with the quality of reflection they demonstrated in the final projects. Yes, most students who did the final reflection did so in order to boost their course grade, but many of their writings achieved what I had hoped for at the beginning of the semester: they catalogued moments of growth, success, and failure while taking a look back on their experience in Real Analysis.
Overall Impressions
The most telling evaluation of #thestruggleisREAL is that word got back to me about the hashtag from colleagues in my department. Students were talking about the assignments outside of class, which gives me hope that they were sharing stories of their struggles with each other in addition to completing the writing. The class had a great sense of community throughout the semester and whether by chance or due to the invitation to voice their stories of struggle, they were very supportive of each other.
I use a well-known text when teaching Real Analysis – Understanding Analysis by Stephen Abbott – so the Internet is always an issue when it comes to homework problems. This semester I noticed a decline in blatant misuse of online sources when grading homework. I would like to think that by being asked to document their struggle, and being incentivized to acknowledge the difficulty of the work, students were less inclined to simply copy an answer from an outside source.
I will certainly incorporate a regular reflective component into future Real Analysis classes, using #thestruggleisREAL as long as it has some relevance to students. The components that worked the best were the regularity of the reflective writing prompts, the periodic inclusion of “light” activities (such as the meme contest) to allow for creativity and humor, and the optional nature of the final project. In the future, I have three concrete ideas for improving upon #thestruggleisREAL:
I have recently heard several mathematicians claim that the educational philosophies of Math Circles and the Inquiry Learning Community are essentially the same. I disagree. I will contrast the differences between these two approaches, along with two other common educational philosophies in the United States. All four approaches to math education differ significantly both in terms of the overall instructional goals and in terms of the primary methods used to achieve these goals.
In my experience, even the originators and staunchest advocates for specific philosophies incorporate the other approaches when putting their favored one into practice. Instructors should think carefully about the goals they have for a given set of students, and then choose a combination of approaches they believe most likely to meet those goals.
I would like to invite you to comment on any thoughts that you have about these lists. A few questions that I have for readers include:
I look forward to hearing your ideas!
The primary goal of the Traditional Math approach is to teach students to solve problems of a specified type as easily and efficiently as possible. This approach arose out of a need to broaden the pool of people able to accurately perform specific computations.
Because of these goals, Traditional Math instruction has the following characteristics:
The primary goal of the Conceptual Math approach is to guide students to a deep enough understanding of common math topics that they can devise multiple approaches to solve those kinds of problems, and make sense out of their answers. In our current economy, employees need to know when to apply common computational approaches more than they need to know how to fluently perform multi-digit computations. Most employees are asked to devise ways to solve a range of problems rather than simply following a procedure laid out by someone else.
On the other hand, the Conceptual Math approach still aims to be accessible to all students and to all teachers. This means that course and lesson designs must be simple enough that teachers with many students and busy schedules can implement them easily. As a consequence, the Conceptual Math approach puts more emphasis on deep understanding of traditional math topics rather than developing the ability of students to research and tackle realistic practical problems or to create and tackle their own mathematical questions.
Because of these goals, Conceptual Math instruction has the following characteristics:
The primary goal of the Inquiry approach is to teach students to create and investigate their own questions. This approach to instruction originated with those interested in preparing students to be scientists, engineers, programmers, or entrepreneurs.
The instructor often guides student inquiry by posing the initial question, which usually does not provide all of the needed information, and is deliberately badly defined. Problems often involve messy, realistic numbers. Students pose sub-questions and have substantial control over the direction their investigation will go. Students not only re-contextualize their results, but often present their results to outside audiences in a variety of written and verbal formats (including videos and web pages). During concluding discussions, the group creates anchor charts to codify strategies and facts they have discovered.
Communication and collaboration are explicit goals of the Inquiry approach. Students share their thinking verbally and in writing and give one another meaningful feedback. There is significant emphasis on teaching students about ways they can contribute positively to a team effort.
Because of these goals, the Inquiry approach to instruction has the following characteristics:
The primary goal of the Math Circle approach to instruction is to teach students learn how to work creatively in the discipline of mathematics. They create new mathematical playgrounds, brainstorm new questions for existing mathematical playgrounds, make original approaches to questions posed, generate data for given approaches, design ways to organize information obtained, propose conjectures about patterns they see, seek proofs of conjectures, find ways to define terms that make it easier to explain results, and express their results using diagrams, mathematical notation, and terms the way a mathematician would. Students learn to seek connections between seemingly different situations.
One of the goals of a Math Circle is to enculturate students as mathematicians. Students cannot develop this culture on their own working in small groups, so a Math Circle instructor frequently models the norms of mathematical discourse. Most of the ideas for solving problems come from the students (though the instructor may ask leading questions when needed). However, the instructor frequently intrudes while students are presenting their ideas to impose the cultural norms of math as a discipline.
Students learn about mathematics as a discipline. They learn to value (and collect) failed attempts as an aid to eventually solving a problem. They practice common proof techniques, and learn to use terms and notation so that other mathematicians will understand what they say and write. Students are exposed to the history of the mathematical ideas they encounter. They also learn what makes a question mathematically interesting, and how to deal with being stuck (emotionally and mathematically). Students learn to interact appropriately with fellow researchers, including being able to communicate effectively in verbal and written form, balancing personal emotional needs against those of a group, building a collegial atmosphere capable of producing interesting mathematical insights, and enjoying the process of mathematical discovery.
Because of these goals, the Math Circle approach to instruction has the following characteristics:
Later, the teacher showed the kids a mathematical tug-of-war game. Each pair of children would have a single die, a small plastic bear, and a number line laid out like this.
The bear starts on the 10 and children take turns rolling the die, one child moving the bear that many steps toward 20 and the other child moving the bear toward 0. Each child also each had a sheet to record the bear’s moves, one sheet with addition templates the other with subtraction Using this format, the children were to record where the bear had started when their turn began, the size of their move, and where the bear landed.
They all understood the mechanics—roll the die and move the bear that many spaces toward their side. I was surprised that several didn’t seem to understand that they were playing one game, together, rather than taking turns re-starting the bear at 10 and rolling their die to see how far it went this time. It was no surprise, though, that only a few recorded their jumps. Frankly, that made sense. The recording step may (or may not!) serve learning but, to the children, it was simply an arbitrary rule with no logical role in the game. Nothing about the game was enhanced by recording it.
We played, cleaned up, and then it was snack time.
During snack time, Alli asked me “how do I write positive three?” I thought, of course, of her early morning announcement about negative numbers. Her question was so clear and specific that I didn’t think (as I always should) to say (as I often do) “I’m not sure I understand. Tell me more.” I too quickly assumed that I knew what she meant.
“Well, we usually just write three, just the way you always write it.”
“But I mean positive three.”
I should have realized right then that I’d mistaken what she had in mind, but I plowed on.
“Just 3—we could put a plus sign in front, but we don’t usually.”
“No but I was on 17 and I rolled 6. How do I write positive 3?”
“Well, Alli, what is seventeen plus six?”
“Twenty-three. But how do I write positive three?”
Now I understood.
Communication with kindergarteners can feel like a string of non-sequiturs when we don’t see the connective tissue, the theory in their mind that they assume we know and that they therefore don’t bother communicating.
It turns out that what Alli meant tells us a lot about the theory she had constructed when her father told her about negative numbers. Prior to hearing about them, Alli had never heard of positive numbers, either. There were just numbers. Now she knew there are kinds of numbers. I don’t know what her father did or didn’t say, but it’s easy to believe that he, like I, would have assumed that nothing further needed to be said about positive numbers; after all, Alli was already quite adept with them. But for Alli, it wasn’t yet clear that the familiar numbers were just getting a new name, positive. For all she knew, the designation positive might well be reserved only for some special use.
And that does explain her question. She learned that going below zero called for negative numbers, and that they contrasted with positive numbers somehow. Perhaps she first thought that positive numbers were all the numbers she had already known (or, less likely, that 0 was yet a third category), but in the context of the number line tug of war game, she built a competing theory. The line contained the numbers from 0 to 20—just plain numbers. She knows that there are other numbers, not shown. Now she knows that below 0 were negative numbers. Perhaps the designation positive also refers to numbers not shown, but above 20. In other words, the categories she created were not “above and below zero,” but “above and below the range we’re attending to.” With astonishing ease for a kindergarten child, she mentally computed 17 + 6 = 23, but now she assumed that “positive three” was the way to express that excess above 20 and she wanted to know how to write it.
The point of relating this story is not to show how impressively smart kindergarteners can be. And it’s certainly not to note a “misconception.” It’s to illustrate what I think is a subtle aspect of teaching mathematics. As teachers, we can’t fully control what ideas our students build, even if we believe we are being are quite clear and precise. What people (children and adults) put in their minds is what they construct, not what someone else says or even shows, and it combines what they already know with their interpretation of what they are currently seeing and hearing. Because that construction combines current experience with past, our “clear and precise” communication will reach different people differently: each makes something of it, but not necessarily what someone else would make, and not necessarily what we expected would be made. We say/write what’s in our mind; what gets in the mind of the listener/reader isn’t conveyed there but built there. Communication is not high-fidelity.
Alli was working out a piece of mathematics. That’s where her dad was no doubt focused when he mentioned negative numbers and that’s where I focused as I tried (and at first failed) to answer Alli’s question. But Alli was also working out a piece of English, a definition. In many contexts, we do report how far some value is above or below a range. Although she’s unlikely to have examples like blood-pressure or cholesterol levels, any kindergartener does already know that some categories name whole ranges of numbers above and below another range of numbers. For example, with no particular precision about which numbers demarcate the categories, they know that babies are below a certain age and adults are above a certain other age and in between are children. Alli has no information yet from which to conclude that this isn’t how the words negative and positive are used when referring to numbers. But it could be, whence Alli’s interest in knowing how to (or whether we should) treat 23 as “positive three.”
In this story, the uncertainty about the meaning of a word is of no real consequence. Though someone might wonder why knowing about “negative” was insufficient to clarify for her what “positive” meant, there’s no risk that Alli’s confusion would lead anyone to conclude that she’s “bad at math.” And, aside from her own interest, there’s no rush for her to know: she is, after all, still in kindergarten and will surely sort this all out in time.
But there are times when the vagaries of communication cause mischief. In US elementary schools, it’s common (probably close to universal) practice for teachers to instruct children to pronounce numbers like 3.12 as “three and twelve hundredths,” not as “three point one two,” what I call a spelling pronunciation. (In my opinion, the insistence on a fraction pronunciation in school is not helpful—for one thing, just think how you’d be expected to pronounce 3.14159—but I’ll save my many reasons for a later blog post.) In one fourth grade classroom that I was supporting, the teacher asked the students to read 3.12, and then wanted to check their understanding of the place value names, so she asked “how many ones?”
The class chorused “Three!”
“How many tenths?”
“One!”
“And how many hundredths?”
Dead silence.
Then a timid “two?” and a more timid “twelve?”
The context “how many ones, how many tenths” seemed to call for the answer two, which is what we know the teacher wanted to hear, despite the loose wording of her question. But children don’t yet have a way to be sure. They’d just read the number as “three and twelve hundredths,” so twelve was a sensible answer. Nobody, of course, answered “three hundred twelve,” which would have been a delightful response showing deep understanding, just as nobody answered the earlier questions with “3.12 ones” and “31.2 tenths.” All of these answers are mathematically correct but they’re “wise guy” answers because they violate norms for communication. They are correct, but clearly not what the teacher meant by the question. In the case of “how many hundredths,” however, students might well be unsure which the teacher meant.
Because the teacher didn’t recognize the source of the confusion—just as I had not at first understood the source of Alli’s confusion—she heard the hesitation and mixed answers as evidence that the class didn’t really understand the mathematics. I had the luxury of being the observer, hearing and following up individual children’s queries rather than having the full responsibility of the teacher addressing and trying to manage the entire class. What I heard and saw made it clear that virtually all of the children did understand the mathematics; the confusion was only about which of two very reasonable interpretations of the teacher’s question was the one she intended.
Unlike the story of Alli, this miscommunication did have consequences. One consequence was a review that was unnecessary, and therefore a turn-off, and that still didn’t clarify the question (the English) and so left several children feeling like they “don’t get it,” despite being able to respond correctly to unambiguous questions on the same content. The worst consequence, in my opinion, is that the lesson some children are getting is not about decimals but that they “just don’t get math.”
So what can we do to reduce negative consequences of missed communications?
At times, I read laments about teachers’ imprecision in language; these are decent examples and I’ll say a bit more about the issue, but later.
In my view (and in all kinds of circumstances), we give students a valuable message when we try to figure out what is sensible about their responses and explicitly state it: “Ah, you were thinking about the twelve hundredths we had just read, and [to the other student] you were thinking about just the number shown in that hundredths place.” In a case like this, it’s valuable even to acknowledge that can now see why they hesitated to answer and that we didn’t at first understand: “Oops, I wasn’t clear about which of those I meant.” Such responses from us teach several things. Possibly the most important is that students know that their thinking is valued even if it takes us a while to catch on. Another is that students see that our focus is on the logic, the sense they were trying to make even if it did not match our intent, and that we are assuming that’s their focus, too. That sets logic, not an answer to a particular question, at the center of the mathematical game. It values clarity, and it shows that we, too, struggle to communicate clearly. It detoxifies errors without fanfare and without “celebrating mistakes,” which students recognize as school propaganda. (Nobody ever says “Woohoo! I made a mistake!”) It models asking questions when we get lost in communicating an idea. (After all, if the teacher does that, it must be a useful and respectable tool.) And it acknowledges that trying to express mathematical ideas in words is clumsy and difficult—the problem is often not the thinking, but the communication—and that’s why mathematics has special vocabulary, notation and conventions. It’s not because mathematicians like fancy words and symbols.
And when we can’t understand students’ logic, we can admit that, legitimizing “I don’t understand what you mean” by showing that that happens to us, too. Kids’ explanations, even when they are totally correct, are often elliptical or garbled, so there’s plenty of opportunity for us to say, “Wait, I don’t get it. Could you explain again?,” giving you a chance to understand and giving them a chance to clarify and perhaps even rethink.
Finally, what about that issue of teachers’ imprecision in language? Being routinely more precise takes a lot of thought, a lot of knowledge, and a kind of self-consciousness and control that is hard to achieve, but building good “mathematical hygiene” (I attribute that lovely term to Roger Howe) with appropriate use of mathematical vocabulary and correct use of notation is a certainly a thing for teachers to think about. On the other hand we must also recognize that there will remain times when conveying a rough idea of what we mean is the best we can do, times when communication, especially with a child, can’t achieve understandability and precision at the same time. Teaching must walk a fine line.
Mathematics is so much easier than English.
(Just as I was finishing writing this blog post, I saw a brief article “Linguistic Ambiguity” by Ben Hookes in issue 103 of the Primary and Early Years Magazine on the NCTEM website, https://www.ncetm.org.uk/resources/52245, which gives other examples in which kids’ sensible interpretations of language leads to answers we might, but shouldn’t, consider wrong.)
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