Summary: Dietiker argues that although mathematics teaching has updated curriculum and standards, the way mathematics is delivered in the classroom largely remains the same as past tradition, where teachers explain, students practice, and meaning is expected to appear. She equates math class to reading an instruction manual: predictable, boring and focused on the end product. Dietiker suggests changing lesson delivery, so that it is more like a novel: an engaging experience that unfolds over time, with a sense of resolution at the end. By intentionally re-structuring lessons from organized lists of outcomes towards the aesthetic of literary art, students are drawn into the process, anticipating next steps, wanting to continue on in a way that resembles not being able to put down a good book. Here are some parallels that she uses between literature and mathematics:
Characters: mathematical objects - 3, 5
Actions: operations - 3 + 5 (these build the plot)
Plot: sequence of actions that withholds/reveals information building “pull” for the reader (p. 7)
Setting: space that characters are found - symbols on a page vs. tiles on a desk vs. points on a graph
When teachers arrange lessons so that patterns gradually become visible, learners can experience mathematical ideas before formalizing them. In this sense, understanding is not delivered through explanation but constructed through a carefully orchestrated experience. Here is a sample sequence that Dietiker lays out, involving the idea of a fair game:
- Read an outline of the presented game and discuss what might happen
- Play the game with peers and analyze results
- Play the game against teacher with a conjectured strategy
- Change the game to make it “fair”
- Share out the resolution with the class
Stop 1: “It is not an object’s attribute but the individual’s perception and interaction that is the locus of aesthetic.” (p. 2) This reminds me of the idea that beauty is in the eye of the beholder. It focuses on process rather than product. In previous courses, we have talked about stories for meaning-making in mathematics. The engagement factor was often a link to culture and personal reflection, while meaning formed from articulating ideas. While there is overlap, Dietiker seems to focus more on the story as an engagement tool by design, the way we would get sucked into the plot of a good book or a good show. The meaning-making coming from predicting ahead and wanting to figure out the story. I remember a while back seeing novel study work left in the photocopy room while I was in there with another science colleague (senior physics.) I asked - don’t you sometimes wish you just do a solid novel study? To which his reply was a resounding - nope! But, I did. I think this design strategy might work well for me.
Stop 2: “Imagining mathematics curriculum as a story opens up the possibility of reimagining the mathematical activities by changing the setting.” (p. 6) I am excited by the idea of setting. Some examples are: a linear function presented in a table vs a coordinate plane, on a calculator, with manipulatives, jumping on a number line. I am hopeful that by experiencing stories across varied settings, we might be able to increase transference of mathematical concepts to more contexts, something that I see students struggle with now.
Stop 3: “Stories that seem to have no point…or are easily predictable are quickly abandoned.” (p. 9) The idea of diversity in the design of the story is important as an engagement tool and something to keep in mind when developing lessons this way. They cannot be just cookie cutter, formulaic. Dietiker presents a solution to this as different genres of math stories. These can then target different interests or learning goals.
Questions:
- Our learners are so diverse, I wonder how this “novel” format can be universally designed so that it is accessible to all. My comfort zone of adaptations comes from the traditional methods that this paper is pulling away from. What happens when students do not make-meaning from process? How do we best support them? (Perhaps this is where collaboration comes in?)
Reference:
Dietiker, L. (2015). What mathematics education can learn from art: The importance of considering form and experience. Educational Studies in Mathematics, 89(1), 27–44. https://doi.org/10.1007/s10649-015-9592-3
Your summary got me hooked. I love the idea of writing math as if it were a story. Meaning-making happens more easily through narration (I too would love to do a novel study). I tell my students all the time that math and English class are closely related as both are languages, and ideas on the page can be explained in many ways that get to the same (or similar conclusion) that are equally valid. Taking this a step further and actually constructing our math lesson to resemble a story with a gripping plotline sounds like a fun exercise for an educator. I wonder how I can use these construction tools to develop future lessons: (1) Framework of natural pattern, exploration, conjecture, connection to terminology. (2) mathematical interactions as a narrative of various genres (ie. Mystery, thriller, action, romance, etc).
ReplyDeleteI usually think of two settings, hypothetically on paper, then in applications of real life. I am going to need time to wrap my head around these sub-divisions of setting that allow students to transfer concepts. How can I think about a table of values and a Cartesian coordinate system like different settings…?
I almost feel like they are doing the same thing, but with different tools. Would it be appropriate to say that one is what it would be as a connect-the-dot (table of values), and the other is what it would be as a painting (graphing). I think this might take more thought to process.
My concern for this process is for ELL students and those with very low attention spans. Perhaps I need more data on how much I speak in the class, but I feel like describing these things as a story will take more words, not less. I need to experiment with this in my classroom to see if engagement changes based on the words I use.
I think a major advantage behind concise notation is that it can describe the goings-on of mathematics with less. Perhaps we can use the analogy about a “picture is worth 1000 words” and apply it to an equation instead: “An equation is worth 1000 words”.
What is the mathematical equivalent to a car-chase sequence?
This will be a fun mental exercise to – at the very least – keep educators entertained.
Thanks for the thoughts! 😊
Yes! An equation is worth 1000 words, and I think as with pictures it is worth 1000 universal words - in any language.
ReplyDeleteI think that the idea of table and values and graphs telling the same story is correct. But being able to pull out the theme of the story regardless of the layout/setting might be a goal? Just giving different points of view? Like Romeo and Juliette and Westside story?
As i read through your reflection, I tried to think about how I as a student would feel to connect English class and Math class so closely through novel and storying. Honestly, I wouldn't like it. I was never strong in english class, at any level of my education, but I excelled at math. Math is my strongest language. Olly is right in saying that it is a language. I was not convinced while reading at the beginning, until you connected it to past courses where adding meaning to life and experiences enhanced learning, I hadn't made this connection to "novel". Then Olly's "an equation is worth 1000 words", now I'm interested in how this could play out!
ReplyDeleteGreat conversation and ideas here, reading group! I want to add the possibility of story-telling through other media too -- like drama, film/ animation, song and dance. Think too about bringing historical stories of mathematics into the picture . And I agree with the idea that concise notation is worth 1000 words. We so seldom talk explicitly about WHY mathematical notation is so beautiful and valuable, or that it comes after a good deal of experimentation, storying, exemplification, and 'boils down' those complicated insights in something beautiful in its simplicity, and easy to carry around in our minds.
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