I’m late to the party—the Virtual Conference on Mathematical Flavors—but I’ve been pondering for awhile the questions asked in the prompt: “How does your teaching move the needle on what your kids think about the doing of math, … what math feels like, or who can do math?”
I work one-on-one with students and while preparing for the start of the academic year I’ve reflected on what has gone well and what changes I’d like to make going forward.
In my sessions with students, my goal is to help them master skills and concepts taught in the classroom setting. I’m remediating, coaching, explaining, test-prepping, and only a little bit extending beyond what their main teacher has taught. I seem to get good results: students do well on their in-class assessments, grades improve or are steady, and students and parents are happy. However, I worry that while my students are learning some mathematical skills in the short or medium term, they still view themselves as needing the support of a tutor to succeed; that they are “not good at math”. I’m concerned that my teaching practice doesn’t push them that much farther into confidence, agency, or deep understanding as I would like, that I haven’t “moved the needle” enough.
Therefore, I have three goals for the coming year, in order to meet the needs of my students AND to better equip them to experience the mathematical flavors they encounter elsewhere in their academic careers.
1. [I will] Talk and write less.
In the past, I routinely guided students through the concepts of a math unit, making summary notes for them to keep, and doing much of the talking. I generally avoided the “do you get it?” self-report trap that can occur in a classroom, because the individual setting necessitates that my students demonstrate their understanding by doing examples, showing me their steps, and fixing their own mistakes.
But I’ve learned from my reading in cognitive science* that if I make students write things themselves, generate their own questions and examples, reflect on their results, and practice self-explanation, their learning will be deeper and more durable. That my students need to engage in their own processing time, in order to solidify and build their framework of math concepts and procedures. This takes more time, but the idea is that it makes the teaching episode much more productive; instead of an hour racing through lots of topics and examples, we can study in a way that ensures each math component we discuss is synthesized with prior knowledge and able to be retrieved for future use.
In the past I’ve shared some cognitive science research with my students, particularly about spaced retrieval practice and the limits of working memory (e.g. why you should “show your work” as you do your math). For the coming year, I will make this a key part of our curriculum, so that students learn how to build their own effective study practice. And I’m planning to talk and write less, taking the time to have my students talk and write more, because it is their learning that is the focus of our work.
2. [They will] Stock the mathematical toolbox.
Students often say to me, “I don’t know anything about” the math topic they are facing. I typically have jumped right in, explaining the topic from the beginning, providing scaffolding and support for the learning.
But in truth, the students probably do know something about the topic, and if I insist that we begin there, I will enhance their learning in two ways. First, by meeting them where they are in their conceptual development, I can build on what they know, expanding and strengthening their math expertise, rather than starting from nothing at all or repeating techniques they have already mastered. Second, by having students do the cognitive work of retrieving concepts, connecting to other knowledge, and applying prior skills to new problems, they are engaging in productive struggle that will ultimately make the learning deeper and stronger. That work might be hard work, but their active participation in doing the math is a critically important ingredient.
So I plan to explicitly help my students stock their mathematical toolbox—fill it with strategies, vocabulary, big ideas, and “things to try” when faced with a math exercise. Things like these:
- Draw a diagram & label what you know.
- Use inverse operations to do and undo.
- Try a different representation to get more information (graph, table of values, algebraic expression, etc.).
- Look at structure to figure out how to solve or how to graph (What does this equation look like—variables, powers, coefficients, fractions, etc.).
- Explain it in your own words (before using mathematical terms).
- Try a simpler problem first.
- Test out with numbers in place of the variables.
- Think about the computation before grabbing the calculator.
- And more…
Of course, the mathematical toolbox includes resources for help when you are really stuck: search Google, Khan Academy, or YouTube for help, or find a friend or classmate who is willing to show you how to do it, and gives you enough explanation so that you can do it yourself afterwards.
The final component of the toolbox is for students to actively engage in their main math class: taking notes, working examples, making sketches, and thinking about why and what makes sense. When something seems confusing, mark it to ask about later. Come to the tutoring session prepared with some questions to ask or trouble spots to work on, so we can target our efforts together.
3. [We will] Build self-confidence and a positive outlook.
This one is a perpetual challenge, with students who haven’t consistently succeeded in their math classes, or who have only achieved with supports. I have always begun the year asking students how they feel about math, or when did it first become difficult, or what part of it feels straightforward. As the year goes on, we discuss what topics are “hard for everyone” (so don’t worry if it feels hard for you) or what is “easy to do” (so once I explain it, you will get it too).
But there are dangers in labeling topics as “easy” or “hard” because each individual student experiences it in a unique and nuanced way.** That saying “this is easy” can backfire in the event that the student still feels lost, and then feel anxious because they should have been able to understand, even though my message is intended to be “you’ve got this”. And saying that something is really tough can make it feel like an insurmountable mountain.
So I’m planning to explicitly work on promoting a growth mindset*** with my students: that anyone can be good at math, that mistakes and challenges help their learning, and their efforts and practice will strengthen their understanding. That if they say “I’m not good at math” they need to add on “yet” to the sentence. And that everyone experiences difficulty some of the time, they are not alone, they have to believe they can do it (and I believe they can).
Finally, we need to approach our work with a positive attitude, tackling tough math with our full efforts. We will find accessible entry points and break the material into manageable pieces. We will build your self-confidence. We want to move the needle on how you view tutoring: moving towards the view that it is a helpful opportunity and an occasional safety net, and away from the view that it is something that you can’t do without, or (worse) something that releases you from responsibility to do your work. We’re in this together, but we’re working towards you being able to go it alone. I believe in you!
The Virtual Conference on Mathematical Flavors is a wonderful set of blog posts well worth your time to read and reflect upon. Thanks to Sam Shah for hosting, compiling, and cheerleading.
*Two books discussing cognitive science research and its implications for teaching and learning are Make It Stick: The Science of Successful Learning by Peter C. Brown, Henry L. Roediger & Mark A. McDaniel, (2014) and How I Wish I Taught Maths: Lessons Learned from Research, Conversations with Experts and 12 Years of Mistakes by Craig Barton (2018). The accompanying websites are http://makeitstick.net and www.mrbartonmaths.com/teachers/ (check out Craig’s podcast page and recommended research papers list). Two other helpful websites are www.retrievalpractice.org/ and www.learningscientists.org.
**Tracy Zager discusses the trouble with “this is easy” in this post.
***Lots about a growth mindset and how to foster it among your students from Jo Boaler and her team at www.youcubed.org/resource/growth-mindset/ and in the book Mathematical Mindsets (2016).