The Master Plan for Precalculus: (2) Math Practices

I’m changing things big-time next year. This is the second in a series of posts. Here are the others (links will be added as they’re written).

  1. An Overview
  2. Math Practices [this one]
  3. Skills
  4. Understanding & Going Beyond

Jo Boaler’s piece on complex instruction convinced me that my teaching (and assessment) should be “multidimensional”. That means that I teach more than just skills. I’ve tried to communicate this, but now it’s going to be a whopping 40% of the students’ grade, so I’m going to have to clearly define what it means to earn a grade here, and what assessment looks like. I’ve turned to the common core math practice standards to help me with this and I’ve settled on these 5 math practices for a few reasons. The main reasons are (1) I think they’re important math skills, (2) they are not assessed (easily) in a standard quiz, and (3) I think they are (more or less) easily assessed through an alternative method. Here are the five math practices that I’m assessing:

  1. “Explain Why”: Construct Viable Arguments and Critique the Reasoning of Others
  2. “Model”: Model with Mathematics
  3. “Check Work”: Attend to Precision
  4. “Good Questions”: Ask Good Mathematical Questions
  5. “Estimations”: Make Accurate Estimations

I’ll go into more detail for each of these below, but for assessment, students can earn a point in each of these practices in different ways. They can earn a total of 40 points, but can only receive a maximum of 10 points in any category. I mean to explain to them that “you should become really good at 3 of these (get 10 points), but nobody is awesome at everything, so I expect you to simply improve in the other two areas (say, 5 points each). If you can do this, you’ll get the maximum possible 40 points in this area by the end of the semester.” On their grade chart (Google Sheet), there is a bar chart for them to see how they’re doing in each of the five areas. Here’s how students earn points:

Explain Why

When working, students should constantly be demanding AND giving explanations and justification for their math. If I hear students either giving a good explanation why or if they are being persistent and asking why some bit of math works, from either a peer or myself, then they earn a point in this category. I’ll use something like Class Dojo to keep track of this during class and tally the points later. I already foresee students complaints: “I asked why but you didn’t hear me do it!” or “I gave a really good explanation but you didn’t see it!” I’ll be up-front about this aspect with them: “Explaining and asking why should become second-nature to you.  You ought to be doing it every day in class, so if you do it 90 times (once a day) and I see only 1 out of every 9 times you do it, you’ll reach ’10’ and make your quota. Don’t do it 10 times throughout the semester and expect me to see every time you do it. Make it become second-nature, like breathing, and I’ll catch you more than enough times! I only expect to get at best a quarter (1/4) of the times you do this. Do this so often that I can’t ignore you and you won’t have a problem.” I’ll also give them opportunities to come in outside of class and explain “why” on topics, or ask questions, and that should cover any problems of me missing some students entirely (“You never hear meeee!”).


I’ll explain this more, but I plan on starting every unit possible with a 3-Acts lesson and working into the math after we’ve already go a situation. Students will have opportunities to model with the mathematics by doing multiple representations, both for projects and classwork. They will get a point for each good model they do (I’ll let them fix what’s wrong with projects to earn a point for the model if they wish) and turn in, and should easily have more than 10 opportunities throughout a semester.

Check Work

I always teach students how to check their work, but never assess them on it. This gives me an opportunity to do so without directly tying it to whether they can do the procedural skill or not. Every quiz where they have checked their work for every problem, they get a point in this area. I’m tossing around the idea of them losing a point for a “careless” quiz where they miss too many problems on a quiz due to careless errors (and not checking their work), making this the only math practice that they can lose points on. With at least 18 quizzes in a semester, there’s plenty of time to improve and get 10 points in this area.

Good Questions

Starting each “unit”/week out with a 3-acts lesson (roughly 18) should give students plenty of time to hone their math-question asking ability. We’ll start with Alex Overwijk‘s cool “What makes a good math question?” lesson, where students discuss & work out what it means to ask a good math question (not exactly this post, but something like this post). I’ve always typed out their questions before, now it’s just a matter of me doing that somewhere I can save it (Evernote) and putting their names next to questions (probably a good idea even if I don’t use it for a grade!). Each good question gets one point, so students will be clamoring to figure out what makes a question a good math question.


Inspired by estimation180 and various teachers (Dan Meyer) talking about students getting “buy in” to 3 Act lessons by guessing has let me to realize that estimating a quantity is a mathematical skill that so many math students sorely lack. Especially when you look at “pick something way too high and way too low and then your best guess”, very often some students’ “way too high” is lower than other’s “way too low”, and visa versa. So I’m going to award points for good estimations (top 3 or within 10% is my current model–that’ll have them doing a bit more math!). This should increase their buy-in for the 3 Act lessons and have them reminding me to do estimation180’s at least every Monday (perhaps even a few each Monday, so that they can all have a chance to win and get points). I’ve seen students get excited without attaching a grade to it–should I not attach a grade so that it just remains fun?

That could go for all of these: should I even attach a grade to these things? In my (current) opinion I’m making the goal so low (only 10 a semester) that they can still have fun and see themselves as improving. I want them to see that I value when they do these things, not just when they can factor a quadratic. So I think it’s essential for me to give them credit for this, even if it’s super-easy to pass this part of their grade (and I hope that it is!).

In the next post, I’ll talk more about the procedural skills.



Filed under Teaching

6 responses to “The Master Plan for Precalculus: (2) Math Practices

  1. Pingback: The Master Plan for Precalculus: An Overview | Hilbert's Hotel

  2. Pingback: The Master Plan for Precalculus: (3) Skills | Hilbert's Hotel

  3. Pingback: The Master Master Plan for Precalculus: (4) Understanding Concepts & Going Beyond | Hilbert's Hotel

  4. great post! I’ve been thinking about how to help students generate good questions as well. for inspiration I’ve been reading Warren Berger’s book, “A More Beautiful Question.” One idea I’ve been kicking around for assessing my students ability to ask questions is to give a scenario/photo/graph related to the topic we’re studying on the assessment, and have students generate a meaningful or insightful question, then provide a possible approach to answer it. A thought for your process: have students rank the top three questions and give reasons why these are powerful/beautiful questions.

    • Yes, that sounds like exactly what I need, thanks for the suggestion! By putting it in the gradebook, students will now keep me accountable for getting me to make them ask more questions! I hope that it turns into a really positive thing. Thanks!!

      • Ah but the hard part is figuring out a rubric to determine what makes a “good” question. Still ruminating on that. can’t wait to see what you come up with over the summer on this very topic. thanks for all of your great posts!

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