Teaching Mathematics Through Immersion

By Priscilla Bremser, Contributing Editor, Middlebury College.

Chapter 1 of Make It Stick:  The Science of Successful Learning [2] is called “Learning is Misunderstood.”  That is an understatement, as demonstrated by the remaining chapters.  The 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.

Since the early 1990’s, I have been using a form of what we now call Inquiry-Based Learning (IBL) in my Abstract Algebra course; more recently I’ve been doing so in Number Theory as well (using [6]).  This all started when Professor Bill Barker of Bowdoin College described an Algebra course built around small-group work, and I was hooked.  Surrounded here at Middlebury College by excellent immersion language programs, I realized that Bill was describing a mathematics immersion program.  I 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.  That first post-conversion semester, there were numerous classes that went overtime before any of us noticed, so engaged were the students.  

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.  The results were less satisfying; those sessions felt like an add-on rather than an integral part of the course.   I assumed that I couldn’t abandon lectures completely because of the list of topics I felt compelled to cover.   Last 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.  My 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.

Make It Stick confirmed my preference for an active learning model as soon as page 3:  “Learning is more durable when it’s effortful.  Learning that’s easy is like writing in sand, here today and gone tomorrow.”  It’s easier for students to copy my problem solution from the blackboard and then imitate it in a bunch of similar homework exercises, but it’s no wonder that they don’t seem to retain much in that setting.  “When you’re asked to struggle with solving a problem before being shown how to solve it, the subsequent solution is better learned and more durably remembered.” [2, p. 88]  What I’m reconsidering is the way in which I choose problems; I want the particular struggle to be productive in ways that the authors describe.

Naturally some students resist a shift from passivity to activity.  The student evaluations for my first IBL-ish course were quite positive, except for one that said “You’re the expert; you should tell us what to do.  I learn better in a lecture,” an assertion that I continue to hear from a few students.  According to Make It Stick, those students may well be misunderstanding their own learning:  “We are poor judges of when we are learning well and when we’re not.  When the going is harder and slower and it doesn’t feel productive, we are drawn to strategies that feel more fruitful, unaware that the gains from these strategies are often temporary.” [2, p. 3]

To confront such resistance, I put some effort into what I thought of as a sales job:  “This way I can help you speak mathematics in real time, and it gives you practice collaborating for later in life, and aren’t we lucky to have small classes at Middlebury,” and so on.  These days I think of such effort in the context of metacognition, which I first encountered in [7].  In being explicit about why I structure my courses the way that I do, I’m also encouraging my students to think more critically about their own learning, which is in itself an asset to that learning.  This semester I’ve put a page on the course website with information about the science of learning.

The work of the social psychologist Carol Dweck ([1], [4]) comes up in Make it Stick. Perhaps I’m biased, but surely mathematics learners are particularly prone to the curse of the “fixed mindset” rather than having a “growth mindset.”  This semester, my first assignment in Calculus was to read “Bad at Math is a Lie” [9] and then have a class discussion.  First my students shared their “bad at math” moments in groups of three or four, and then we heard some in the full group.  I know that one event won’t move everyone into a growth mindset, but it’s a start.

For some reason – the relentless “coverage” drumbeat? – a while back I stopped my practice of taking mini-surveys on Fridays in Calculus classes.   These had three questions:  (1) What were the important themes this week?  (2) What concept(s) intrigued you?  (3) What concept(s) are still muddy to you?  They helped me know what students were thinking, and communicated to the students that I wanted to know what they were thinking.  I’m reintroducing the surveys, not just for those purposes, but also because they ask students to reflect on their learning.  “Reflection can involve several activities … that lead to stronger learning.” [2, p. 89]

On the other hand, I’ve always resisted quizzes because of the added stress.  According to Make it Stick, however, frequent low-stakes assessments that require students to retrieve new knowledge can assist in the learning process.  So this term I’ve scheduled weekly quizzes in which anything from the semester so far will be fair game.

The authors of Make It Stick suggest that instructors “be transparent.”  [2, p. 228]  One way in which I convey my intentions to my students is by including this quote at the end of my syllabi:  “Trying 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.” [2,  p. 101]  I am still trying to come up with the best ways to provide corrective feedback; that effort might be the subject of a future post.  In the meantime, I am grateful to Bill Barker and many others who have been transparent about their pedagogy as I refine my own.

References

[1] Braun, Benjamin. Persistent Learning, Critical Teaching: Intelligence Beliefs and Active Learning in Mathematics Courses.  Notices of the American Mathematical Society, 61 (January 2014), 72-74.

[2] Brown, Peter C., Roediger, Henry L., and McDaniel, Mark A. Make It Stick:  The Science of Successful Learning. Belknap Press, 2014.

[3] Christie, Hazel, in The Times Higher Education, April 3, 2014.

[4] Dweck, Carol S. Mindset: The New Psychology of Success. Ballantine Books, 2007.

[5] Lang, James N. Making It Stick, in The Chronicle of Higher Education, April 23, 2014.

[6] Marshall, David C., Odell, Edward, and Starbird, Michael.   Number Theory Through Inquiry. Mathematical Association of America, 2007.

[7] National Research Council. How People Learn: Brain, Mind, Experience, and School: Expanded Edition. Washington, DC: The National Academies Press, 2000.

[8] Stover, Catherine.“For the most part, we are going about learning in the wrong ways.A Fine Line blog, April 10, 2014.

[9] Waite, Matt.  Bad at Math is a Lie. Math Horizons.  September 2014, p. 34.

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3 Responses to Teaching Mathematics Through Immersion

  1. Emily Triffault says:

    this is a fantastic article to motivate maths teachers to let their pupils struggle and maybe set up a problem or a series of questions first, and in fact getting them to create their own annotations/lesson notes from their learning, which must have first been effort full
    Such a good word to describe the mathematical struggle we all face on different levels.
    In being transparent to your class, can I ask how old they are? As the language seems quite complex to get through to high school pupils.
    Thank you,
    Emily

    • Priscilla Bremser says:

      Thank you, Emily. I’m glad you found this post to be useful.

      I teach at the university level. At my institution, this means most students are 18 – 22 years old. It would be an interesting exercise to translate the message to a lower grade level. Here’s a quick attempt: Scientists who study learning say that if you have to think hard to solve a problem, you’ll remember the solution better, even if you make mistakes, as long as you get help figuring out where you went wrong.

    • Ben Braun says:

      There are nice examples of this at the high school level described in Jo Boaler’s book “What’s Math Got to Do With It?” (the title in the UK is “The Elephant in the Classroom”). See Chapter 3 in particular. Boaler’s website youcubed.org is also a good resource for K-12 teachers interested in this line of thought.

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