Let’s Build Strong Math Classrooms From Day 1

Hey math team!

I am really excited by the priorities and challenges that Scotty and Dave both talked about in their beginning-of-year blog posts. Dave’s growth mindset focus is so powerful for building mathematical agency, and Scotty’s student-centered approach reinforces the idea that knowledge is not something “held” by the teacher, but something that students themselves can create and own. I see in both of their visions for their classrooms a real emphasis on student leadership, and an asset-based approach to their students’ potential.

This got me thinking about the types of structures and routines that we can start the year off to help promote student leadership in our own classrooms. I don’t think there is any one answer to this, but there are lots of examples from other excellent math teachers who have used the first few weeks to build strong culture in their classrooms. 2014 teachers will be familiar with Grace Chen’s  Unit 0 Plan from Kick-Off, but here are some other ideas…

Bob Peterson

Bob Peterson, who has written a lot about teaching math for social justice (including Rethinking Mathematics, which has a lot of fantastic CRT examples), had this to say about how he starts his school year…

“I usually start the year with kids exploring, in small groups, how mathematics is used in their homes and communities. They scour newspapers for numbers, cut them out, put them on poster paper, and try to give sense to their meanings, which at times is difficult. They interview family members about how they use mathematics and write up their discoveries. As part of a beginning-of-the-year autobiography, they write an essay, “Numeric Me,” tying in all the numbers that connect to their lives, from height and weight, to the number of brothers and sisters they have, to addresses, phone numbers, and so forth. Over the years, I have pushed this assignment to go beyond the self, and I’ve encouraged the students early in the year to explore “math in the community” and “math in the world,” At times, this can be as simple as evaluating population figures, but it also has gotten into more engaging issues such as the amount of money spent on the Iraq War or the number of children without access to clean water. Some years I ask my students to write a “history” of their experiences in mathematics classes, what they think about mathematics, and why.

This process starts a yearlong conversation on what we mean by mathematics and why it is important in our lives. As the students increasingly become sensitive to the use of numbers and mathematics in news articles, literature, and everyday events, our discussions help them to realize that mathematics is more than just computation and definitions. It also includes a range of concepts topics – from geometry and measurements to ratios, percentages, and probability.”

Peterson goes on to discuss how he explicitly builds in issues of justice and equality after establishing trust with his students. Want to read more? You should join the Teaching Mathematics for Social Justice book study this quarter in Leland,  which is the book that I drew this excerpt from.

Justin Reich

I love the idea of explicitly incorporating math mistakes into student discussion. Reich talks about how to use Michael Pershan’s Math Mistakes blog (which is fantastic on its own!) to create a routine around identifying mistakes. Granted, this protocol is designed for adult learners, but I think it makes perfect sense as a student-centered math routine as well. The process goes like this…

1)   Look at three problems on the board. Predict all of the mistakes that students might make.

Michael gave me three problems to work with, and before showing any student work, I showed my pre-service teachers the original questions and problems. I asked my students to predict the different kinds of mistakes young mathematicians might make. I put students into three groups (of about 8 teachers each), and I ask them to consider all three problems.

2)   Look at what a student actually did. Make observations about their work, first. Then, start to ask questions about what you see. Then, start to make some predictions about what they may have been thinking.

For this section, I assigned each group to look at one problem. One issue that emerged here was that different Math Mistakes have different richness of student output. For instance, one problem just showed that a student wrote the number 0. Not much to observe there. Another problem showed several steps of work, including some non-standard notation that lends itself wonderfully to close parsing. So some groups raced through the step of making observations, whereas other groups needed more time.

3)   Enrich your conversations by bringing in voices of expert teachers from MathMistakes. What new ideas emerge here? What is the range of possibilities of what the student may have been thinking? What is the range of ways to respond?

When group conversation started to slow (pretty soon for the group whose student answered “0″), I gave them each a printed copy of the relevant comment thread from MathMistakes.org. I printed them in part for logistical reasons (the class didn’t need computers for anything else), and in part because I wanted them to be impressed by the heft of the discussion. The comments for the three problems I shared run 10 pages long, filled with insightful observations about student thinking, analogous mistakes, and instructional approaches. My sense was that students were quite impressed that a single mistake on one worksheet could generate so much thoughtful reflection from experienced educators.”

This routine seems similar to the 3-Read technique that we discussed at Kick-Off (and 2013 teachers can learn about in the Culturally Responsive Math – Evolving with our Regional Priorities session for Q1 (sign up through the course catalog!) What better way to build strong conceptual understanding of math than to directly call out common mistakes that are made with our math topics? I see a lot of potential in this sort of routine in building student comfort in making mistakes (everybody does it!) while allowing students to reflect deeply on mathematical thinking and work (where might a mistake happen? What is the mistake? What might this student have been thinking?) See Justin’s full blog post HERE.

Twitter Math Camp

Twitter Math Camp “is a conference run by teachers, for teachers. The participants, speakers and organizers communicate year round via twitter, blogs and other social media. This 4-day conference in July is a chance for everyone to get together for face to face interactions.” They JUST wrapped up for 2014, and there are some great ideas they discussed. One such idea is interactive notebooks. Are you worried about how students will take and interact with notes and content in your class? Perhaps this is a system that will work well for creating student ownership of their work (who doesn’t hate loose notebook pages?). Here you can find a collaborative discussion that the TMC participants had about interactive notebooks, culminating in the following list of examples that you can draw from for your own classroom:

Where can I learn more about the INB?
Ms Hester’s Classroom INB page

Sarah Hagan’s INB tag

Jonathan’s INB writings (his are the Low Maintenance variety)

CheesemonkeySF’s Why INBs are a tool for equity and social justice

Sarah Rubin’s Blog: http://everybodyisageniusblog.blogspot.com/p/interactive-notebooks.html

Megan Hayes-Golding’s Blog: http://kalamitykat.com/interactive-notebooks/

Algebrainiac1 Blog (not many): http://algebrainiac.wordpress.com/category/interactive-notebook/

A great resource of ideas WITH PICTURES: http://journalwizard.blogspot.com/

Foldables & Organizers: http://lisawilliamssocialstudiesclass.weebly.com/foldables–graphic-organizers.html

Lapbook Templates (GREAT): http://www.homeschoolshare.com/lapbook-templates.php

Another FANTASTIC routine that was discussed at TMC was “Talking Points.” I love love love this way of building student discussion in class, and better yet there are specific examples of how to incorporate this at the start of the year with non-math examples! The idea is pretty straight-forward:

First, Students are given “talking point” statements (i.e. “Talking is more important than listening” or “Some questions have multiple right answers” or “Longer numbers are always bigger than shorter ones”) and one student reads one statement without making any other comments.


ROUND 1 – Go around the group, with each person saying in turn whether they AGREE, DISAGREE, or are UNSURE about the statement AND WHY. Even if you are unsure, you must state a reason WHY you are unsure. You’ll be free to change your mind during your turn in the next round.

ROUND 2 – Go around the group, with each person saying whether they AGREE, DISAGREE, or are UNSURE about their own original statement OR about someone else’s statement they just heard AND SAY WHY. NO COMMENT. You are free to change your mind during your turn in the next round.

ROUND 3 – Take a tally of AGREE / DISAGREE / UNSURE and make notes on your sheet.

Groups should then move on to the next Talking Point.

At the end of ten minutes ring the bell and have groups finish that round. Don’t let the process go on — they need to stop and move on.


2 minutes

At the end of the ten minutes, give students exactly two minutes to fill out the Group Self-Assessment. They will use this during the whole-class debrief and will hand this in for a group grade.


5 minutes

Ask each group to report out about specifics, such as:

  • Who in your group asked a helpful question and what was it?
  • Who in your group changed their mind about a Talking Point? How did that occur?
  • Who in your group encouraged someone else? How did that benefit the conversation?
  • Who in your group provided an interesting additional idea and what what is?
  • Who did your group disagree about and why?

I generally choose two or three of these questions and then move on to the actual mathematical group work while the energy level is still high.

Wow. What a fantastic way to spur student discussion, and with a routine that you can introduce at the start of the year! HERE is a page with all of the resources for this routine, including the Activity Lesson Plan, a list of non-math talking points to start off the year with, the group self-assessment, and math-related talking points. I could easily see this evolving into a way of discussing specific mathematical concepts, using talking points like “A triangle can have 2 obtuse angles” or “a fraction must always equal a value less than one.” What do you all think? Who wants to try this out?

Grant Wiggins

Wiggins (who is probably most widely known as the co-author of Understanding by Design), gives some good advice for providing feedback, which should be top-of-mind as you dive into your first few weeks of teaching. A couple of great points he makes about feedback…

  1. Most so-called feedback is really advice or praise (as in the four examples above)
  2. The feedback is not clear and descriptive enough about what did and didn’t happen as a result of some action taken to achieve a purpose. (e.g. a total score of 72 out of 100 on a math quiz is the feedback; it’s meaning for action is unclear.)
  3. The purpose of the task is so unclear (or non-existent) to the performer that the feedback is either random or mysterious. (Without a specific teacher goal for the observed lesson, feedback and advice are pointless.)
  4. The learner has not been provided with any exemplars of excellence against which to compare their work and thus obtain feedback. (Rubrics are NOT specific enough for the performer; they are inherently general. Models plus rubrics provide the basis for useful feedback).
  5. The feedback is too late. (Thanks to a commenter for reminding me to highlight this crucial issue, as I have done in earlier posts. It is especially noteworthy on standardized tests and final exams: there is NO feedback.)

Some great things to keep in mind when you’re developing plans for how you will organize your feedback to students this year!

Jonathan Claydon

Claydon is a high school math teacher who “had a job in construction project management juggling millions of dollars and pounds of paperwork. Then the little voice said go forth and teach, and yea, verily, it came to pass.”

Want to see how he organizes feedback, note-taking, room arrangement, and grading in his classroom? Take a look at this page from his blog HERE!

So there you have it, a lot of different ways to be considering the atmosphere we are creating in our classrooms in the first few weeks and months of school. Now it’s your turn! I’d love to hear some comments below telling me how YOU are structuring your classroom for the start of the year. What questions do you have for our math team? What are you excited about? What are you nervous about? Which of the resources above really appeals to you? Fire off!

Until next time,



Ask, Play, Argue

Ask we’re all packing things up to head home for the holidays, I thought I’d give a quick, fun reduction of the Common Core Standards for Mathematical Practice, courtesy of Christopher Danielson:

1: Ask questions. Ask why. Ask how. Ask whether your answer is right. Ask whether it makes sense. Ask what assumptions you have made, and whether an alternate set of assumptions might be warranted. Ask what if. Ask what if not.

2: Play. See what happens if you carry out the computation you have in mind, even if you are not sure it’s the right one. See what happens if you do it the other way around. Try to think like someone else would think. Tweak and see what happens.

3: Argue. Say why you think you are right. Say why you might be wrong. Try to understand how someone else sees things, and say why you think their perspective may be valid. Do not accept what others say is so, but listen carefully to it so that you can decide whether it is.

So go ahead and do it–ask, play, and argue all throughout break. The more we practice being mathematicians, the better we can teach other mathematicians.

(Also, hopefully you’re having an easier and easier time figuring out what it looks like for students to ask, play, and argue in your own classrooms. But if you want one more example, check out the “estimation wall.”)

Math Games: Five Steps to Zero

I recently got a copy of Success from the Start: Your First Years Teaching Secondary Mathematics. It’s an expensive book, but as first- and second-year teachres, a very valuable one.

One of my favorite sections is actually an appendix in the back, filled with math games. They’re all great ways to fill small bits of time in your classes, simultaneously building in your students a sense of mathematical agency, a feeling of urgency, a deeper number sense, and the idea that math can be fun. Without stepping on any copyright laws, I want to share some of those games with you. If you want more details, I recommend you buy the book!

Five Steps to Zero, as the name indicates, is a game in which, with five steps or less, students take a given number and try to get it to zero. To make this fun, there are restrictions, of course:

  • Students can use the four basic operations (addition, subtraction, multiplication, division)
  • Students can use any whole number, 1 through 9. They MAY NOT use zero, numbers larger than 9, or negatives.

For example, given 213, students might:

  1. Add 2:                       213 + 2 = 215
  2. Divide by 5:             215 ÷ 5 = 43
  3. Add 2:                      43 + 2 = 45
  4. Divide by 9:             45 ÷ 9 = 5
  5. Subtract 5:               5 – 5 = 0

Once students have played for a while, they might realize there are some tricks to it. As you play, you might think of good variations, as well!

What is an “algebra”?

You learn something new every day.

Here’s how one teacher defines an “algebra”:

 An algebra is essentially a set of objects that can be both added and multiplied, with the two operations fitting together via the distributive property.

In other words, that means it’s a set of things where if you add (which really also includes subtraction) or multiply (which, ditto, includes division), the thing you get out is still in that set.  If you take a number, and add another number, you get out a number.

That property also holds true for: expressions with variables, including polynomials; matrices; functions; even transformations on the plane!

This allows me to see how some things we teach in algebra that sort of seem tangential–like matrices, for example, and also some of the geometry–make a whole lot more sense. We are figuring out what rules work on an algebraic system, starting with the system of real numbers that students have learned about all through elementary and middle school, and seeing how it also applies to other sets of mathematical objects.

Mr. Honner cues up a couple other “big ideas” in algebra. I’m rewriting those slightly, based on some feedback from Grant Wiggins, while trying to get somewhat student friendly:

  • Organizing sets into binary relationships allows us see when there is and is not equivalence (e.g., equality and inequality)
  • Some of those relationships can be matched perfectly to each other (e.g. functions, in a big simplification)
  • Those relationships can be mapped onto a simple coordinate plane (e.g. the Cartesian coordintes)

How many ways can you look at this problem?

I recently found the webpage for Harvard Calculus, and one thing I think is very cool is this “rule of nine” that shows nine different ways we might approach a math problem:

algebraically, analytically, geometrically, historically, graphically, numerically, conceptually, psychologically, as well as experimentally.

What the heck does that mean? This may require some calc knowledge, but for something like finding the derivative of sin(x) at x=pi/3, you could think in these ways:

algebraically we know sin’=cos and cos(pi/3)=1/2
analytically we know sin(x) = x-x^3/3! + x^5/5! – … and so sin'(x) = 1-x^2/2! + x^4/4!-… = cos(x) so that sin'(pi/3)=1/2.
geometrically sin(x) is the height of a right angle triangle with hypotenuse of length 1. The rate of change of this length in dependence of the angle can be seen geometrically.
historically the derivative can be derived from Euler’s formula exp(i x) = cos(x) + i sin(x) which has the derivative i exp(i x) = i cos(x) – sin(x). Comparing real and complex parts shows cos’=-sin, sin’=cos.
graphically draw the graph of sin(x) and determine the slope at x=pi/3
numerically take a small number 1/1000 and compute 1000 sin(pi/3+1/1000)-sin(pi/3)) which gives 0.499567.
conceptually since sin(x) increases for increasing for acute angles, the result is positive.
psychologically my teacher does not like to assign problems with irrational numbers as answers. The result should be a simple rational number. Because 0 and 1 are out of question, the next reasonable result is 1/2 ……
experimentally here is an esoteric experiment: why not use Fourier series and differentiate that series. To make it interesting, take f(x) = |sin(x)| since sin(x)=|sin(x)| around pi/3 …

One thing they note, though, is that if you try to approach a problem from too many ways at once, you’re going to overwhelm yourself before you even understand anything. So we should pick three, maybe four approaches, to make sure we really get something without going overboard.

Jo Boaler (yes, I know, I keep going on about her) has adapted this for algebra, and suggests that for algebra problem, we should have students try to see if they can express it in four of the five following ways:

in words, in pictures and diagrams, in tables, in graphs, and in symbols.

How many ways can you think through this problem?


Tired of that worksheet?

It’s late the night before school and you still don’t have a good practice activity for your kids… those Kuta software or Infinite Algebra worksheets are sure looking tempting right now, huh…

That kind of repetitive practice is important, once students have solidified their conceptual knowledge, but we also know in most classrooms they get used to often, and we can all (teacher and students) slip into a quicksand of boredom. And that kind of practice certainly doesn’t build good habits of mathematical practice.

No fear! Here’s a great example of how you can reframe a boring old worksheet and make it into an activity that gets students thinking at a higher level.

Conceptualize, then formalize

If there’s one simple trick that I think every teacher could master really quickly–and that would lead to more mathematical agency in classes, and students having deeper levels of understanding, it’s this:

Let students conceptualize something on their own before bringing in mathematical language and notation makes it more likely that the notation will aid in their understanding rather than provide another hurdle in learning.

I’m quoting Bowman Dickson there, and he’s got a whole post explaining how he did this in a calculus class. You should check that out. If it’s been a while since calculus (or if you never took calculus), that might be a bit overwhelming, but it shouldn’t be. It just means don’t teach vocabulary and symbols at the very beginning of class. I don’t know how many times I’ve seen that: a teacher up at the front, asking students to copy down definitions, which still mean nothing to students, and meanwhile they’re checking out one by one…

Just save them! Do the vocab at the end, rather than the beginning.

Nonsense can be useful

There are 125 sheep and 5 dogs in a flock. How old is the shepherd?

Is that even a math question? There’s certainly an argument that it’s not–but too many students would try to answer it like one. We teach students that math is basically just the act of smashing numbers together to get a new number out, so why wouldn’t students work up an answer to this?

In one of our professional development sessions last year, we talked about a study that showed students were more likely to come up with an answer to a question like this the older they got. That is, our math classes are teaching them to stop paying attention to reason and just try to spit out numbers.

That’s why I like the idea here that little kids be asked these sorts of questions. But I’d actually not limit this to second and third graders–I’d do it for middle schoolers, and probably even high schoolers, too.  I know a lot of you have built “problem solving” routines into your week somewhere. These sorts of “nonsense” problems would be a great thing to include to help students think more holistically about problem solving.  Make students articulate why these problems can’t be solve. Make them adapt the questions so they can be solved.

The first day!

It’s coming! A lot of y’all will be beginning the new school year on Monday. If you’re still searching for a way to fill the time, considering stealing some of the ideas from this great diagostic activity.

You might look at it and wonder if it’s really a diagnostic, since there’s only those three little math questions at the bottom. And that’s what I like so much about it: it’s only assesing the most essential mathematical ideas for the first unit. Meanwhile, it provides all kinds of great information about students’ habits and beliefs: how are they at analyzing a situation for potential mathematical content? at posing mathematical questions? at thinking flexibly?

This would have to be modified for your grade and your vision, but it’s a great start…