Learning to Be Less Helpful

FREDERICK PECK

University of Montana, Missoula

 

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.

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Sense-Making and Making Sense

WILLIAM MCCALLUM

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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A Physicist’s Lament

By Judah L. Schwartz, Harvard University

From whence this blog

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. Continue reading

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What is \(0^0\), and who decides, and why does it matter? Definitions in mathematics.

By Art Duval, Contributing Editor, University of Texas at El Paso

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.

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Illuminating skills learned from teaching

By: Mary Beisiegel, Oregon State University

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.

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#thestruggleisREAL: Reflection in a Real Analysis Class

By Katharine Ott, Department of Mathematics, Bates College

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. Continue reading

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Comparing Educational Philosophies

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:

  • Which blend of these pedagogical approaches have you found congenial for specific audiences?
  • Are there other approaches that are essentially different that I did not include in this list?
  • What steps do you find helpful when transitioning students who are used to one pedagogical approach to another?
  • There is a tendency for students from high poverty schools to be exposed primarily to Traditional Math approaches. The Common Core State Standards represent an effort to slightly improve on this prevailing norm, by pushing teachers in the direction of Conceptual Math instruction. What do you think people who love math and teaching should do to improve access to high quality math education for these students?

I look forward to hearing your ideas!


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Ideas under construction: children saying what they know

Alli entered kindergarten quite skilled at mental addition and proud of her skill. Subtraction followed quickly. Near the end of her kindergarten year, Alli bounced into class and said that her father had taught her about negative numbers. To assure that I knew about them, she explained, “If you subtract 20 from 10, you get negative 10.” I asked, “And what if you subtract ten from seven?” She thought a second and chirped “Negative three.” Then she explained how to write a negative number—“Just put a minus in front”—and added “There are negative numbers and positive numbers.” And that was it. As with many conversations with 5-year-olds, this one ended as abruptly as it began. Continue reading

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Mathematical Practices

My work and that of my colleagues at Education Development Center has always put mathematical thinking—the habits of mind that are indigenous to our discipline—at the core of our work with teachers. What we’ve learned from expert teachers has led me to think more carefully about what it means to “work like a mathematician.” The attached essay details some of the things I’ve learned.

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Beyond Grades: Feedback to Stimulate Rethinking and Intellectual Growth

By Cody L. Patterson and Priya V. Prasad, Department of Mathematics, University of Texas at San Antonio

We frequently use writing assignments to encourage students to examine topics in greater depth than what we cover in class, and we emphasize to our students that writing assignments constitute one of the most important dimensions on which students’ thinking will be assessed. Yet in our early implementation of these assignments, we frequently received work that did not reflect students’ full potential for understanding the topic explored in the assignment. In these cases, because we were using a roughly linear scale to assign each submission a numerical grade, which would then become part of a student’s overall grade in the course, we faced a difficult decision.

  • We could assign low grades to work that did not meet our expectations; this would have the advantage of sending students a clear message about whether their work meets the standards of the course, but it might demotivate students or limit the potential of an otherwise competent student to earn a good grade in the course.
  • We could assign moderate-to-high grades to such work; this would lower the stakes of failure for students, but it would also require us to endorse work that does not meet a high standard.

Neither option, however, seemed to address our greatest concern: that some of our students had not explored and communicated about the topic of the assignment with the depth desired. Moreover, numerical grades allowed many students to decide that they had gained enough, grade-wise, out of the assignment, and did not need to take advantage of opportunities to revise their work. In this article, we’ll talk about our journey toward crafting and implementing a grading scheme for writing assignments that provides greater opportunity for student learning and growth. While we use writing assignments specifically in the context of content courses for preservice teachers, we believe much of our advice is adaptable to other mathematics courses. Continue reading

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