{"id":2372,"date":"2018-12-17T13:47:24","date_gmt":"2018-12-17T18:47:24","guid":{"rendered":"http:\/\/blogs.ams.org\/matheducation\/?p=2372"},"modified":"2018-12-17T13:47:24","modified_gmt":"2018-12-17T18:47:24","slug":"learning-to-be-less-helpful","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/matheducation\/2018\/12\/17\/learning-to-be-less-helpful\/","title":{"rendered":"Learning to Be Less Helpful"},"content":{"rendered":"

FREDERICK PECK<\/p>\n

University of Montana, Missoula<\/span><\/p>\n

 <\/p>\n

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<\/a><\/u>, 101 Q\u2019s<\/a><\/u>, Desmos<\/a><\/u>, NCTM\u2019s ShadowCon<\/a><\/u>, to name just a few. But he initially rose to prominence on the basis of a TEDTalk<\/a><\/u> 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.<\/p>\n

Be less helpful.<\/p>\n

Teachers often understand their job as involving \u201chelp,\u201d in some form or another. Most of us would like to think that we help students learn. Dan\u2019s 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<\/a><\/u> for teachers at all levels.<\/p>\n

What does it mean to be less helpful, and why should that be a goal?<\/p>\n

In his TED Talk, Dan shows how interesting mathematical questions are often surrounded by scaffolding:<\/p>\n

\"\"<\/a><\/p>\n

\"\"<\/a><\/p>\n

Image credit: https:\/\/www.ted.com\/talks\/dan_meyer_math_curriculum_makeover<\/a><\/u><\/p>\n

This scaffolding, in the form of mathematical structure and sequences of steps, works to obscure the interesting question (\u201cwhich section is the steepest?\u201d, buried in question 4).\u00a0Students 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.<\/p>\n

More perniciously, the scaffolding takes away much of the mathematics. Hans Freudenthal believed that structuring was at the heart of mathematics.\u00a0In this problem, the scaffolding has already structured the problem. There is no structuring\u2014and therefore no mathematics\u2014left for the student to do. By presenting students with a ready-made structure for getting an answer, \u201cmathematics\u201d is reduced to answer-getting.\u00a0 To be less helpful means to remove the scaffolding, and to let students do mathematics.\u00a0 That is hard work for a teacher. In this post, I’ll give an example from my own practice, and discuss how I\u2019m learning to be less helpful.<\/p>\n

<\/p>\n

Background<\/h1>\n

I have always considered myself a \u201creform-oriented\u201d teacher\u2014at least in content courses, my classroom practice adheres closely to the \u201cconceptual math\u201d described by Amanda Serenevy in an earlier post on this blog<\/a><\/u>. In calculus, I have followed many of the recommendations from the calculus reform movement of the 1990s, including the \u201crule of four\u201d: 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:<\/p>\n

\"\"<\/a><\/p>\n

Following the rule of four (actually, I also use a lot of models, so let\u2019s 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: \"\"<\/a>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 \"\"<\/a>, and thus we can use the fundamental theorem to find the derivative, \"\"<\/a>.<\/p>\n

Finally, we come to a point in the course where it\u2019s 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:<\/p>\n

    \n
  1. Using the rule of four, students would explore the derivative of y=2x<\/sup> from graphical, numeric, algebraic, and verbal perspectives.<\/li>\n<\/ol>\n
      \n
    1. Putting these perspectives together, students would conjecture that the derivative of \u00a0y=2x<\/sup> is itself an exponential function\u2014in fact, it is a vertical compression of the original function, with a scale factor of about 0.69. That is:<\/li>\n<\/ol>\n

      \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Conjecture<\/b>:\u00a0 \"\"<\/a>, where a\u22480.69<\/p>\n

        \n
      1. Do a similar analysis for \u00a0y=5x<\/sup>. Here we come to a similar conjecture, only now the vertical scaling is a stretch, rather than a compression:<\/li>\n<\/ol>\n

        \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Conjecture<\/b>: \"\"<\/a>, where a\u22481.6<\/p>\n

          \n
        1. Putting these two conjectures together leads to three interesting observations.
          \nObservation:<\/b> The derivative of an exponential function seems to be a vertically-scaled exponential function with the same base.Observation:<\/b> Some scale factors are <1, and they compress <\/i>the derivative, relative to the initial function (e.g., the scale factor for y=2x<\/sup>). Other scale factors are >1, and they stretch<\/i> the initial function (e.g., the scale factor for y=2x<\/sup>).Observation: <\/b>Bigger bases seem to have bigger scale factors.<\/li>\n
        2. These observations, in turn, lead to a very interesting question. If the derivative of 2x<\/sup>\u00a0 is a compression of the initial function (a<1), and the derivative of 5x<\/sup> is a stretch of the initial function (a>1), and if the scale factor generally grows as the base grows, then reasoning by continuity tells us that there must be a base between 2 and 5 where the scale factor is exactly 1.\u00a0 What is this magical base?<\/li>\n
        3. Using technology, explore this question. Can we find such a base?<\/li>\n<\/ol>\n

          This plan hinges on students developing a good sense of the behavior of the derivative of\u00a02x<\/sup>, that is, steps 1 and 2. Thus, for the rest of this post, I\u2019ll limit my discussion to these steps.<\/p>\n

          My first attempt<\/h1>\n

          In my first attempt at steps 1 and 2, students competed this worksheet<\/a> in groups.\u00a0 The worksheet guides students to consider the derivative of 2x<\/sup> from a variety of perspectives.<\/p>\n

          Graphical:<\/h2>\n

          \"\"<\/a><\/p>\n

          Numerical:<\/h2>\n

          \"\"<\/a><\/p>\n

          Algebraic:<\/h2>\n

          \"\"<\/a><\/p>\n

          Verbal:<\/h2>\n

          Throughout, students are asked to reflect on or explain their findings.\u00a0 The worksheet culminates with a prompt to make a conjecture about the derivative of\u00a02x<\/sup>:<\/p>\n

          \"\"<\/a><\/p>\n

           <\/p>\n

           <\/p>\n

           <\/p>\n

          Can you guess what happened at this point?<\/p>\n

          Well, let me tell you what didn\u2019t <\/i>happen. Students didn\u2019t take a holistic perspective on the work that they had done. They didn\u2019t say things like, \u201cfrom step 1, I see that the graph of the derivative is a vertical compression of\u00a02x<\/sup>. 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 \u2032(x) is some constant times 2x<\/sup>, and I see that the constant is 0.69. Taken together, I conjecture that the \"\"<\/a>.\u201d<\/p>\n

          Instead, students treated question 8 as if it were independent from the rest of the worksheet. They generally conjectured that the derivative of 2x<\/sup> was x\u22c52x-1<\/sup> (i.e., they used the power rule). Even though students had just completed a graphical, numeric, analytic, and verbal analysis of the derivative of 2x<\/sup>, they did not draw on this analysis in answering the big question.<\/p>\n

          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\u2019t. This is because I <\/i>was the one who was doing the structuring. I created a cookbook-style \u201crecipe for mathematics,\u201d 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.<\/p>\n

          Deciding to be less helpful<\/h1>\n

          After a few of years of this worksheet, always with the same good intentions, always with the same disappointing results, I finally took Dan\u2019s advice to heart. I decided to be less helpful.<\/p>\n

          Dan suggests that we start<\/i> with the big question rather than burying it under a pile of scaffolding. My big question was, what is the derivative of 2x<\/sup>?<\/p>\n

          To be less helpful, I thought, \u201cwhy not just ask it?\u201d<\/p>\n

          So that\u2019s 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.<\/p>\n

          Now, can you guess what happened?<\/p>\n

          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 \u201cbig question.\u201d Rather, my question seemed to them to be a straightforward application of a known rule.<\/p>\n

          I learned that I needed to set up the question better.<\/p>\n

          In math education, we call the setup to a problem the \u201claunch.\u201d 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\u2019t provide enough grounding, limiting students\u2019 ability to engage in the problem.<\/p>\n

          A good launch should do three things:<\/p>\n

            \n
          1. Help make sure that all students are familiar with the context and content of the problem<\/li>\n
          2. Produce a perplexing question<\/li>\n
          3. Remind students about relevant prior knowledge without \u201cgiving away\u201d the solution or solution strategy<\/li>\n<\/ol>\n

            Here is how my launch works now:<\/p>\n

              \n
            1. Just ask it. Ask students to find the derivative of y=2x<\/sup>.<\/li>\n
            2. When the majority of the class produces x\u22c52x-1<\/sup> as the derivative, have them check this by graphing their conjecture against the numerical derivative produced by their calculators.<\/li>\n
            3. When the graphs do not match, this produces a perplexing question: What\u2019s going on here, and what is the actual<\/i> derivative of y=2x <\/sup>?<\/li>\n
            4. Before asking students to explore this question, I give a brief presentation that summarizes our previous strategies for finding derivatives of unknown functions: numerical analysis, graphical analysis, algebraic analysis using first principles, and algebraic analysis by rewriting the unknown function as a combination of known functions.<\/li>\n
            5. Just ask it, redux. What is the derivative of 2x<\/sup>?<\/li>\n<\/ol>\n

              In my experience, this sequence accomplishes the objectives of a good launch. Students now see the question about the derivative of\u00a02x<\/sup> as a \u201cbig question,\u201d worthy of exploration, and they have tools that they can use to explore that question. At the same time, I haven\u2019t 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.<\/p>\n

              Groups go at the problem in different ways.<\/p>\n