Monthly Archives: June 2015

The Master Master Plan for Precalculus: (4) Understanding Concepts & Going Beyond

I’m changing things big-time next year. This is the fourth post. Click one of the links to go to the other posts.

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

One of Michael Pershan’s complaints of SBG has always stuck with me: by breaking math into standards and grading it as such, we compartmentalize math and students see it as a clustering of individual skills. This 20% of their grade is intended to encourage them to make connections. Here’s their view of the gradebook for reference:

See the different stars? I hope students get as excited about it as I am!

See the different stars? I hope students get as excited about it as I am!

There are two “levels” (stars!) that students can earn for each concept standard. The first is for giving an acceptable answer on the quizzes to the “understanding” question, which is intended to group the skills together. Because those questions change every time, students can lose this star if they show that they understand one part of it but not others. This will work similarly to the skills: I’ll either take the mode or latest grade: whichever is higher. When the options are “1 or 0”, then mode makes a lot of sense to me. I intend to make a list of possible “understanding question” and have this available to students so that they can prepare for these, though I won’t promise to cover ever understanding question which may show up on the quiz.

Not every concept is broken down into 3 skills, but I kept the number within the range of 1 to 3 for each concept. Each “concept” is a section of the textbook which should help me as a teacher find good “understanding” questions.

Going Beyond (to Infinity and…)

The second star cannot be obtained just by taking quizzes. Students must do something outside of class to demonstrate a connection between two or more concepts. This could be a project, paper, explanation (through Doceri or similar software), presentation to the class, or anything that is “above and beyond” what I would expect of students. Something like the Google Maps project or the Stop-Motion Parametric Videos project. This makes these projects optional, except for students who want to earn an A. I hope that many students will go for these projects as (1) getting the second star automatically gets you the first star (a funny thing about how the gradebook is setup) and (2) you cannot lose the second star (“Sorry, you know that project you did and took all that time on that I made you redo? Well, your recent quiz renders all of that void and useless.”)


I’m considering making “getting all coins within a concept standard” as a prerequisite for getting the star for that standard. After all, can students understand what they’re doing if they don’t know what to do in the first place? I’m also considering making “getting one star” a prerequisite for getting the second “Going Beyond” star. However, this might discourage students from going for the bigger projects and as long as I’m strict about the projects clearly demonstrating understanding, then I can feel comfortable giving them both stars at once.

Growth Levels

I want the growth levels to be given to students who become better at mathematics while I’m teaching them. Students who “get things the first time around” aren’t growing, so won’t get anything for skills if they master it the first time around. Because of this, I want something for them to “reach for”, so each 2nd star (“Going Beyond”) that they get, they earn a growth level. That way the growth level rewards weaker students who really have to struggle to get skills down as well as stronger students who push themselves beyond the bounds of the course. I hope that everyone can always find some way to be growing in my class, and that they are motivated to that end.

Oh, and as the growth levels go up through different video game characters. I won’t reveal the full list in case a student finds this blog, but I will say that it includes the likes of Mario, Yoshi, Link, Zelda, and Sonic. I’m pumped.


I’m very excited about the new direction that my classroom is going. I hope that students get excited about it as well. Please give me feedback and let me know if you’ve done something similar in your class!

Gotten better at Math? Bwhahaha!

Gotten better at Math? Bwhahaha!


Filed under Teaching

The Master Plan for Precalculus: (3) Skills

I’m changing things big-time next year. This is the third post. Click one of the links to go to the other posts.

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

Although math classrooms over-focus on procedural skills, they should not be entirely cut from the curriculum. There’s a reason they’ve become so important in the math classroom. By making skills 40% of the grade, I can say “a majority of your grade does not come from skills” and yet “you must master a majority of these in order to pass this class” (70% is passing at our school).

Here’s a quiz that’s I’ve made for the near the beginning of the year:

Some things you may or not be able to notice:

Skill Standards are separated into pairs of questions

I don’t expect both questions to cover the entire skill, which is why I intend to give each standard multiple times. See the “Form C” in the top right? That means it’s the third iteration of this quiz. Their grade is then a 0, 1, or 2, depending on how many they show me they know how to do [1]. Their grade will then be either the mode or the latest grade, whichever is higher [2].

Each question is taken from or modeled by a question from the HW in the book

The more similar the questions are to students’ HW, the more likely it is that they’ll do their HW. Many of these are taken straight from their textbook and I’ll point this out frequently throughout the course so students start making the connections.

Each Skill Standard has 3 boxes for grades

One goal of SBG is for students to get better at self-assessment. The first box is for them to predict how well they’ll do, either before they’ve seen the questions or after they’ve tried them. The second box is for them to give themselves a grade as they grade themselves against an answer key [3]. Students are surprisingly bad at giving themselves a grade even though they have the answer key! The third box is the grade that I give them after checking after them. This should help them to get better at assessing themselves.

The Skills Standards are clearly grouped together in one larger box

I’ll explain more about the “understanding” standards, but this reinforces the idea that the skills are closely linked and students should be thinking about how these relate to one another. The final question is my attempt to tie these together in an over-arching “understanding” question.

Here’s how I plan on reporting this to students:

Gradebook: Student View

An older version, where you can see the “Math Practice” bar chart filled out on the right.

Student's View of their Grade

The newer version has a link with more practice material to the right of their grade (stars).

The coins are my gamification of their grade. I tried to group each set of coins (skills) into a star (understanding). One of the things that contributed to the growth level (at the top) is how much they “improve” from the first time they take a skill. This rewards students who don’t do well the first time around, but study and do better later times. I hope this little incentive encourages those who don’t always “get it” the first time around, as they’re the ones who often need encouragement in math class. Their growth level has no bearing on their grade.

In the next post I’ll explain more what the stars mean.

[1] NOT how many they “get right”. If they show their work, which students are more apt to do in a SBG system like this, they’ll get credit but lose a point in “Attend to Precision” from their math practices grade (see previous post).

[2] I wanted to do just mode, so if they take a quiz 3 times, get a 2, 2, then 1, I’ll reward them for their sustained ability the first two times. However, I didn’t want students to dig holes so deep that they couldn’t get out, so 0, 0, 0, 0 isn’t automatically a failing grade, hence the “latest grade” opportunity.

[3] An idea I’ve gotten from other blogs: orange pens in the back of the room along with an answer key. When students finish their quiz, they leave their own writing utensil at their desk and get instant feedback. They’re not just to mark “right or wrong” but they’re supposed to fix mistakes in work and write how to do it correctly in the orange pen.


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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.


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The Master Plan for Precalculus: (1) An Overview

I’m changing things big-time next year. This is the first in a series of posts, giving an overview of what I’m changing. Other follow-up posts will cover specific parts of the plan. Here are the other posts:

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

For this overview, I just want to show what motivated me to do such an overhaul of Precalculus. It was motivated by multiple people on the MTBoS, but a few places in particular:

Some goals and inspirational thoughts I got from these articles:

  • I need to be grading more than just procedural skills. I can say that “thinking mathematically” and “asking good mathematical questions” are good things, but if I don’t back it up by assessing it, it won’t stick and radically change students the way I want it to.
  • Standards Based Grading, as designed, compartmentalizes skills too much. I need to make connections and encourage students to make connections between skills, showing that math is more than a set of individualized skills.  Oh, and I need to do this through assessments (see bullet point immediately above).
  • I can’t totally abandon procedural skills. I need to find a way to encourage students to practice these outside of class.

My Grading Breakdown

  • 40% Skills
  • 20% Understanding
  • 40% Math Practices

I’ll explain more what each of these mean, and how I plan to assess and teach each of these, but for now here are some comments that I believe are true about my grading system and why I like this breakdown at the moment. Oh, and all 3 of these categories will be graded using SBG.

  • A majority of a student’s grade is not be based on procedural skills.
  • A student cannot pass my class without focusing on “math practices”–habits of mind which are used by mathematicians.
  • In order to get an A in my class, you cannot just show up and take the assessments–you actually have to do something outside of class which goes “above and beyond”.
  • In order to get a B in my class, you have to understand the math that you are doing.
  • Students can target procedural skills very specifically for both practice and reassessment.

In my next post, I’ll explain more what I mean by “math practices”.


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A Preview

Reading MTBoS blogs are great because not only do I get motivated to try new things, but I also get ideas (and steal shamelessly).

The following idea was motivated primarily by reading (and hearing on Global Math) Dale Ehlert.

I’m working on a new way to communicate grades to students. I’ve made a rough-draft (read: non-working draft) of what I want students to see.  I’ll explain it in future posts, but I’m excited enough about it to post a snapshot of what I want it to look like here.

Screenshot from 2015-06-09 23:27:16

More sources of inspiration will be given in future posts, hopefully along with links to a working copy. Have I gamified it too much??

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