Non-Curricular Routines
A few fun ways to get students thinking at the start of class
I love non-curricular routines. They've been a part of my teaching for a long time in different forms. In the last year I've rethought how I use these routines in my class and I want to share a bit about what they look like now.
A non-curricular routine has two goals for me. First, it gives students a chance to practice mathematical thinking. Each of the routines I will describe gets at different ideas: mental math skills, mathematical language, attention to detail, using structure, and more. Second, when they practice routines over time, students have the chance to get good at the routine. Lots of students feel dumb in math class because it feels like, right as they get the hang of something, the class moves on to something new. Routines give students a chance to get good at a specific type of thinking and recognize their own progress.
I've tried different models for routines, from a different routine each day of the week to using them regularly at different places in units. In the last year I've switched to a model where I use the same routine at the beginning of class every day for 3-5 weeks, then move on to a new one. One of my biggest goals is for all students to become familiar with the routine so they can worry less about what comes next, and focus more of their mental energy on thinking about the math. As students get used to a routine they get better and better at it, creating an environment where they can sink deeply into the math thinking and feel smart. Another benefit of doing a routine for a few weeks and then moving on is that I can use more routines over the course of a year than if I rotated days of the week or something like that. The final benefit is that using the same structure for a few weeks makes planing a lot easier for me. I plan them, copy the templates, and put slides together all at once.
My last argument for non-curricular routines is that they're fun. For every routine I've used there have been a few students who love it yet don't engage with lots of other stuff we do. And each routine appeals to different students. I enjoy teaching them because they bring out quirky and original thinking and start interesting discussions. That's something I want to experience in every math class! All in 5-10 minutes per day.
Final logistics: my school has a common structure of a Do Now when students come in. My dow now is a few quick problems and then a puzzle. I take attendance, collect the cards I use for random seating, and deal with anything else that's come up. After the Do Now we do the routine for the day. I have templates on the back of their Do Now for them to write their answers or explanations during the routine.
Here are the routines I have used this year:
Contemplate then Calculate
I use a shortened version of the routine described by the Fostering Math Practices folks here. The best problems for this routine are either addition/subtraction problems with a lot of terms like 9 - 8 + 7 - 6 + 5 - 4 + 3 - 2 +1, or number talk-style counting problems. I flash the image or problem on screen for a second or less, then ask students to write down anything they noticed about the problem that might be mathematically important. This is meant to be too fast for students to solve, but long enough to notice one or two things that might be helpful. We share out. Then I put the image back up, and students take a minute to solve silently. Students share their strategy with a partner, and I pick 2-3 strategies to share with the full class — connecting back to what students noticed before if I can. The goal is for students to practice thinking flexibly about problems and slowing down rather than rushing right to a solution.
Which One Doesn't Belong
Which is different from the others? The website curated by Mary Bourassa is amazing, and the problems are designed so that there is a possible justification for all four. Students take a minute to write down what they think, then share with a partner, then with the full class. My goal with the share-out is to hear a reason for all four, and to emphasize that there's no right answer. The discussions are often good ways to review geometry, divisibility rules, properties of numbers, and more. The whole exercise also encourages attention to detail and use of mathematical language. And it leads to some goofy fun!
Estimation 180
How many are there? I put up an estimation problem pulled from Andrew Stadel’s website and have students make guesses they think are too small and too big as well as just right. This way we can calibrate our guesses and create more opportunities for conversation than trying to guess the exact number. We start by sharing the too small and too big guesses, then the just right guesses, and then the reveal. The goal is to build some number sense, and also to enjoy the challenge of estimation.
Visual Patterns
My goals with Fawn Nguyen’s visual patterns are to scaffold an accessible skill and help students move a little bit from concrete to abstract thinking. First they draw the next step. Then they predict how many will be in the 6th step. Then, they try to predict how many will be in the 12th step. I scaffold by offering a table, but then try to highlight answers using the visual pattern itself as well so students have multiple tools to work with.
Decide and Defend
I've adapted this routine from the Fostering Math Practices website as well. The structure is simple: I give students a short problem and two possible answers, one from "Mai" and one from "Lin." Here are two examples.
Students take a minute to write down who they agree with and why, then turn and talk, and then we have a brief discussion. This routine is good for reviewing a mix of topics later in the year, and to get students talking and analyzing student thinking. I find it's also more accessible for many students to agree with someone else, rather than trying a problem themselves.
Answer, What's the Question?
Simple: I give an answer. They have to provide the question. I start with simple answers, like 6 or 88, but then go into stuff like 12 legs, -3, 0, 7/4, 0.3333333…, and more. Some students take the straightforward approach and give me answers like "7/4+0" and "0.3333333+0" but others are more creative, and the discussions are really fun.
Fraction Talks
First, the website from Nat Banting is a ton of fun to explore. My student struggle with fractions (fractions were kindof the sweet spot where the pandemic hit their learning the hardest) so I want to make it manageable. I use the pattern blocks category, and the prompt is to pick any color, tell me what fraction of the shape it takes up, and how they know. Then students share, and we dissect the figure in a few different ways and make connections between equivalent forms of fractions.
Number Talks
I use Fawn Nguyen's middle school number talks resources, and facilitate as she does with one difference. Every day we take our time and discuss one problem. Students have time to think, then share with a partner, then I pick a few strategies to share with the class and annotate. The twist is that I give students a similar problem to solve at the end once they have a some strategies to choose from. My goal is to give a quick round of practice with a strategy before we move on, especially for students who heard it for the first time. If the problem is 84 - 27, I ask 95 - 38 as a followup. If the problem is 15*12, I ask 15*14. The conversations are great, and the goal is to expand each student's repertoire of mental math strategies.
Quick Images
This idea is stolen from Jenna Laib. I flash a geometric figure for students for about two seconds then take it down. They write down a description of it in as much detail as they can. Then they try to describe it for me, and I play a game where I try to draw what they described, yet not what the image showed. If they say "a rectangle with a triangle missing" I draw a creative interpretation of what a rectangle might look like with a triangle missing. The goal is to practice using precise mathematical language, and students often see it as a goofy challenge. We do a few rounds of this, then talk a bit about some of the mathematical language at play. I try to pick shapes that aren't too easy but aren't too hard -- 3/4 of a circle, or an isosceles triangle, or an isosceles trapezoid, or a hexagon that is really just a square with a square cut out.
Slow Reveal Graphs
Another great one from Jenna Laib. I modify the routine to work with three slides. The first has very little information, and students write down and share something they notice. This is fun because they often notice things that seem obvious or silly — "it's blue" — that turn out to be important. The second slide has most though not all of the information, and students try to guess what the graph is saying. Then, we look at the third slide — the full graph — and discuss it as a group.
Between Two Numbers
I forget where I got this one but it's simple. I give students two numbers and an open number line. They need to label any number in between. Then we put a bunch of the students' numbers on my number line up front. I start with simple ones, then move to negative numbers, fractions, decimals, and mixed problems, and students try to think of weird ways to make a number hard to place. The version of this I saw originally asked for a number between 1.15 and 1.2, which is a great problem. There are lots of other fun ways to set it up, especially with fractions where students can choose to use a fraction or a decimal depending on what they feel confident with, leading to great discussions.
Those are the routines I used this year but I’m always looking for more. A few I’m considering:
A different Between Two Numbers from Fawn Nguyen about unit analysis/Fermi problems
Some sort of variation of Nat Banting’s Oops I Forgot
Some sort of “Number of the Day” where we do a few different arithmetic and geometric things with a different number each day
Connecting Representations via the Fostering Math Practices folks
I want to emphasize one more time that these are some of the most fun moments of class for me, and also moments that engage students who are slow to engage other times. Every student should feel smart in math class, and non-curricular routines are a great way to create a foundation of success for struggling students to build off of.