{"id":386,"date":"2014-09-20T00:01:15","date_gmt":"2014-09-20T04:01:15","guid":{"rendered":"http:\/\/blogs.ams.org\/matheducation\/?p=386"},"modified":"2014-09-17T22:27:53","modified_gmt":"2014-09-18T02:27:53","slug":"teaching-mathematics-through-immersion","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/matheducation\/2014\/09\/20\/teaching-mathematics-through-immersion\/","title":{"rendered":"Teaching Mathematics Through Immersion"},"content":{"rendered":"

By Priscilla Bremser, Contributing Editor<\/a><\/i>, Middlebury College.<\/i><\/p>\n

Chapter 1 of Make It Stick: \u00a0The Science of Successful Learning<\/i> [2] is called \u201cLearning is Misunderstood.\u201d \u00a0That is an understatement, as demonstrated by the remaining chapters. \u00a0The book has received several strong reviews ([3], [5], [8]), so rather than providing a critique, my aim here is to explore the ways in which its account of cognitive science research has validated some decisions I have made about my teaching and gotten me to reconsider others.<\/p>\n

Since the early 1990\u2019s, I have been using a form of what we now call Inquiry-Based Learning (IBL) in my Abstract Algebra course; more recently I\u2019ve been doing so in Number Theory as well (using [6]). \u00a0This all started when Professor Bill Barker of Bowdoin College described an Algebra course built around small-group work, and I was hooked. \u00a0Surrounded here at Middlebury College by excellent immersion language programs, I realized that Bill was describing a mathematics immersion program. \u00a0I modeled my course on his so that my students would learn mathematics by speaking mathematics with each other, while I roamed the room as consultant. \u00a0That first post-conversion semester, there were numerous classes that went overtime before any of us noticed, so engaged were the students. \u00a0<\/p>\n

Meanwhile I began making less drastic changes in my Calculus courses, devoting at most one class session (out of four) each week to small group work. \u00a0The results were less satisfying; those sessions felt like an add-on rather than an integral part of the course. \u00a0\u00a0I assumed that I couldn\u2019t abandon lectures completely because of the list of topics I felt compelled to cover. \u00a0\u00a0Last spring, however, considering data showing that few of our Calculus I students go on to Calculus II, I decided to ditch the massive textbook in favor of fewer topics and an interactive format. \u00a0My goal had shifted from getting them through a fixed set of material to having them engage the ideas deeply enough that they thought differently about measuring change, whether in their economics and biology classes or when reading the news ten years from now.<\/p>\n

Make It Stick<\/i> confirmed my preference for an active learning model as soon as page 3: \u00a0\u201cLearning is more durable when it\u2019s effortful.<\/i> \u00a0Learning that\u2019s easy is like writing in sand, here today and gone tomorrow.\u201d \u00a0It\u2019s easier for students to copy my problem solution from the blackboard and then imitate it in a bunch of similar homework exercises, but it\u2019s no wonder that they don\u2019t seem to retain much in that setting. \u00a0\u201cWhen you\u2019re asked to struggle with solving a problem before being shown how to solve it, the subsequent solution is better learned and more durably remembered.\u201d [2, p. 88] \u00a0What I\u2019m reconsidering is the way in which I choose problems; I want the particular struggle to be productive in ways that the authors describe.<\/p>\n

Naturally some students resist a shift from passivity to activity. \u00a0The student evaluations for my first IBL-ish course were quite positive, except for one that said \u201cYou\u2019re the expert; you should tell us what to do. \u00a0I learn better in a lecture,\u201d an assertion that I continue to hear from a few students. \u00a0According to Make It Stick<\/i>, those students may well be misunderstanding their own learning: \u00a0\u201cWe are poor judges<\/i> of when we are learning well and when we\u2019re not. \u00a0When the going is harder and slower and it doesn\u2019t feel productive, we are drawn to strategies that feel more fruitful, unaware that the gains from these strategies are often temporary.\u201d [2, p. 3]<\/p>\n

To confront such resistance, I put some effort into what I thought of as a sales job: \u00a0\u201cThis way I can help you speak mathematics in real time, and it gives you practice collaborating for later in life, and aren\u2019t we lucky to have small classes at Middlebury,\u201d and so on. \u00a0These days I think of such effort in the context of metacognition, which I first encountered in [7]. \u00a0In being explicit about why I structure my courses the way that I do, I\u2019m also encouraging my students to think more critically about their own learning, which is in itself an asset to that learning. \u00a0This semester I\u2019ve put a page on the course website with information about the science of learning.<\/p>\n

The work of the social psychologist Carol Dweck ([1], [4]) comes up in Make it Stick. <\/i>Perhaps I\u2019m biased, but surely mathematics learners are particularly prone to the curse of the \u201cfixed mindset\u201d rather than having a \u201cgrowth mindset.\u201d \u00a0This semester, my first assignment in Calculus was to read \u201cBad at Math is a Lie\u201d [9] and then have a class discussion. \u00a0First my students shared their \u201cbad at math\u201d moments in groups of three or four, and then we heard some in the full group. \u00a0I know that one event won\u2019t move everyone into a growth mindset, but it\u2019s a start.<\/p>\n

For some reason \u2013 the relentless \u201ccoverage\u201d drumbeat? \u2013 a while back I stopped my practice of taking mini-surveys on Fridays in Calculus classes. \u00a0\u00a0These had three questions: \u00a0(1) What were the important themes this week? \u00a0(2) What concept(s) intrigued you? \u00a0(3) What concept(s) are still muddy to you? \u00a0They helped me know what students were thinking, and communicated to the students that I wanted to know what they were thinking. \u00a0I\u2019m reintroducing the surveys, not just for those purposes, but also because they ask students to reflect on their learning. \u00a0\u201cReflection can involve several activities \u2026 that lead to stronger learning.\u201d [2, p. 89]<\/p>\n

On the other hand, I\u2019ve always resisted quizzes because of the added stress. \u00a0According to Make it Stick<\/i>, however, frequent low-stakes assessments that require students to retrieve new knowledge can assist in the learning process. \u00a0So this term I\u2019ve scheduled weekly quizzes in which anything from the semester so far will be fair game.<\/p>\n

The authors of Make It Stick<\/i> suggest that instructors \u201cbe transparent.\u201d \u00a0[2, p. 228] \u00a0One way in which I convey my intentions to my students is by including this quote at the end of my syllabi: \u00a0\u201cTrying to come up with an answer rather than having it presented to you, or trying to solve a problem before being shown the solution, leads to better learning and longer retention of the correct answer or solution, even when your attempted response is wrong, so long as corrective feedback is provided.\u201d [2, \u00a0p. 101] \u00a0I am still trying to come up with the best ways to provide corrective feedback; that effort might be the subject of a future post. \u00a0In the meantime, I am grateful to Bill Barker and many others who have been transparent about their pedagogy as I refine my own.<\/p>\n

References<\/strong><\/p>\n

[1] Braun, Benjamin. Persistent Learning, Critical Teaching: Intelligence Beliefs and Active Learning in Mathematics Courses. \u00a0Notices of the American Mathematical Society, <\/i>61 <\/b>(January 2014), 72-74.<\/p>\n

[2] Brown, Peter C., Roediger, Henry L., and McDaniel, Mark A. Make It Stick: \u00a0The Science of Successful Learning.<\/i> Belknap Press, 2014.<\/p>\n

[3] Christie, Hazel, in The Times Higher Education<\/i>, April 3, 2014<\/a>.<\/p>\n

[4] Dweck, Carol S. Mindset: The New Psychology of Success.<\/i> Ballantine Books, 2007.<\/p>\n

[5] Lang, James N. Making It Stick<\/a>, in The Chronicle of Higher Education<\/i>, April 23, 2014.<\/p>\n

[6] Marshall, David C., Odell, Edward, and Starbird, Michael. \u00a0\u00a0Number Theory Through Inquiry. <\/i>Mathematical Association of America, 2007.<\/p>\n

[7] National Research Council. How People Learn: Brain, Mind, Experience, and School: Expanded Edition<\/i>. Washington, DC: The National Academies Press, 2000.<\/p>\n

[8] Stover, Catherine.\u201cFor the most part, we are going about learning in the wrong ways.<\/a>\u201d A Fine Line<\/i> blog, April 10, 2014.<\/p>\n

[9] Waite, Matt. \u00a0Bad at Math is a Lie. Math Horizons. <\/i>\u00a0September 2014, p. 34.<\/p>\n

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By Priscilla Bremser, Contributing Editor, Middlebury College. Chapter 1 of Make It Stick: \u00a0The Science of Successful Learning [2] is called \u201cLearning is Misunderstood.\u201d \u00a0That is an understatement, as demonstrated by the remaining chapters. \u00a0The book has received several strong … Continue reading →<\/span><\/a><\/p>\n

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