You know how teachers are supposed to join in when they do “Drop Everything and Read”? We should have the same attitude in math.
My third recommendation to help students reconcieve of math as a discipline that goes beyond calculation is to do math with students. Most teaching time is spent with the teacher doing math at students. The teacher demonstrates a procedure and the students must mimic that procedure. Math is something that happens to students, not something in which students are involved in its creation or development. The teacher is the ultimate authority and the true doer of math.
What kinds of things can we do for foster a more flattened classroom hierarchy where we are all exploring math together? This post will share a few specific strategies, but essentially the mantra is the lesson: do math with your students.
One thing we must do is offer tasks in which students are generating ideas moreso than the teacher. Number Talks come to mind.
Number talks have evolved considerably. Whereas they were once posed solely as simple computational problems (and still work excellently doing just that!) we also now have versions that incorporate patterns, fractions, equations, images, etc. The prompt can differ but it typically follows the template of “What do you see and how do you see it?”
Students think silently for a time to allow for everyone to develop an idea and then we’re off! We ask students for their ideas and that’s where we start doing math with the students. It’s crucial that you express authentic joy and delight at student ideas and engage with those ideas. Consider the following two scenarios.
Scenario A
Student: “… and that’s how I solved the problem 18 x 5.”Teacher: “Very nice! Does anyone else have another strategy?”
Scenario B:
Student: “… and that’s how I solved the problem 18 x 5.”Teacher: “Wow! Let me think about that for a second! What a brilliant and new strategy, Student! I’m so in love with that strategy I want to see if it works for 21 x 6! I’m going to try that for myself right now!”
I’m pleased by the multiple strategy approach from the teacher in Scenario A, but I’m very excited by the enthusiasm and willingness to explore the student ideas in Scenario B. This is also a nice opportunity to ascribe academic status on a student by naming the method after him or her. You can refer to the strategy as “Beth’s rule” or “the Jalen Method.”
Have students come up with the task – or at least a portion of the task.
We know and love “Which One Doesn’t Belong?”. One of the cool things that the good folks at WODB.ca provide is a template for student-created or student-assisted WODB tasks.
Currently, there are only a handful of incomplete sets, but the idea is brilliant: have students generate the artifacts that may not belong for a reason (or two or three). By doing this, then you, the teacher, can authentically participate in the fun.
Consider having students create a WODB entirely from scratch. Give each student or each student group a topic or concept and have them look through their notes to craft a quality WODB task with which they may challenge the whole class.
Got It!
During one of Howie Hua’s talks, he introduced us to the game of “Got it!” in which students are challenged to find a number by following a path along a game board. For example, can you find a pathway (starting anywhere and moving horizontally and/or vertically) to obtain the number 23?
You can have students challenging one another (and you, the teacher) to obtain a number.
Bring in authentically new math-y things.
Teachers of literacy or English are encouraged (and often do) discuss the books that they are reading to showcase and encourage a lifelong love of reading. Why don’t we do that in math?
One weekend I discovered (or rather I got twitter-algorithm’d) something called a “golygon”. A golygon is (apparently) a shape in which each adjoining side is one unit larger than the previous. So an octagon in which the side lengths are 1, 2, 3, 4, 5, 6, 7, and 8. On monday I came in to class and was like “Hey I found this cool/weird thing! Let’s all draw some golygon!” And it was then that I drew my first golyglon.
Another Monday, I told my students about an egregious mathematical error I made: ordering too many bags of volcanic rock for the pathway I was building. I also shared how I neglected to pay attention to the units when ordering serrano peppers from a grocery store. You see, I thought I was buying one (1) serrano pepper but actually was buying one pound of serrano peppers. That’s a lot of serrano peppers if you are unfamiliar.
Also when I pressed all these apples and only got this much cider.
OK now is the time when I talk about mathematical portfolios. I talk about portfolios a lot and at the risk of being the hammer that sees everything as a nail, constructing a mathematical portfolio alongside students is a great way of showcasing the satisfaction that math brings you, the teacher. You may even wish to refer to it as a math journal.
Here are a few tasks/places that encourage doing math with students.
Measurement by Paul Lockhart
Clothesline Math
Open Middle
When have you done math with students? Toss your stories/ideas/activities in the comments and I’ll add them to the blog post!
And there you have it. This Mini mini-series offered three strategies to help students think more broadly about the discipline of math: do creative math, do useful math, and do math with students. Check out the first two posts if you haven’t done so yet:
Part 1: Do Creative Math
Part 2: Do Useful Math
These strategies are recycled from Necessary Conditions, but presented with fresh, updated, and expanded resources and ideas. You can also view these and dozens of other strategies in my Necessary Conditions card sets.
While you’re here, here are some other projects of mine that might interest you, including my previous, larger mini-series.
A Rubric Masterclass
Your Math Syllabus Bootcamp
Routines, Lessons, Problems, and Projects
Algebra Warm Ups for Geometry
Problem Based Curriculum Maps (Grades 3-11)
Geoff’s Professional Development Services and Stuff
emergent math 
