Tag Archives: Standards Based Grading

[2016 Blogging Initiative] Week Three: Understanding Questions


After making the transition to Standards Based Grading a few years ago (which is awesome!), over time I realized that my class had become too skill-oriented. To fix this I first tried to create standards that were “understanding standards”, but this overwhelmed my students with too many grades.

It took me an entire semester, but I realized that what I should be doing is asking “understanding questions” on assessments instead of only skill-oriented questions.

I try to limit my assessments to three questions (sometimes a question might have multiple parts), but I now always try to include an “understanding question” as one of those three questions. I never grew up answering these on math assessments, and they’re harder to grade because there’s usually not “one right answer”, but it has helped me get a better grasp of what my students understand (or don’t).

What do these look like? Here are some examples.


Exponents: Explain why b^x \cdot b^y = b^{x+y} is true.[1]

Polynomials: What does multiplying polynomials have to do with the distributive property?

Polynomials: Why can you combine some terms of a polynomial but not others? (3x^2 + 4x^2 can be added but 3x^2 + 4x^3 cannot)

Rational Expressions: Before factoring was the opposite of simplifying. What has changed and why do we factor first to simplify rational expressions?

Functions: Give an example of a function and a non-function outside of math class.

Transformations: Why does (x+3) move a graph left and (x-3) move a graph right? Isn’t that that the opposite of what you would expect?

Logarithms: Explain why \log_b{M} + \log_b{N} = \log_b{(M \cdot N)} is true.[1]

Reflection/My Own Questions

(1) Are these “understanding questions” enough to check for understanding? Probably not by themselves. So I need to get better at assessing repeatedly over time to check for retention of understanding.

(2) Should I give students the questions beforehand? Right now I do because if they want to figure out the answers on their own, great! As long as I have enough possible questions so they’re not simply memorizing and spitting back what I say, but really understanding it. (Or should they be able to get these questions even without me providing them ahead of time?)

(3) Is there a place to get these types of questions? I primarily look in the textbook or come up with my own questions, but surely there’s a bank of these somewhere online that I haven’t found yet.


These questions are ones that get at understanding, though harder to grade (at least they take longer), are worth it. When I started these questions, I was sorely disappointed how little my Precalculus students understood (even though they have seen some things, like exponents, in Algebra II and probably even in Algebra I!). I’m really curious what people think about the three reflection questions above.

Thanks for reading!


[1] I’ve flip-flopped between using the vocabulary “prove” and “explain”. The former suggests there’s one right answer to students whereas the latter allows for various explanations. “Explain” also is harder to grade, but I’m very excited if I see students start to write out examples in their explanations. No, it’s not as rigorous as professional mathematicians, but it shows me that they’re starting to understand.



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Reflections on a Grading System

This semester in Precalculus I used my own grading system that I created through Google Spreadsheets. The program worked well enough, but I am going back to ActiveGrade for a few reasons.

Screenshot from 2015-12-19 15-22-43

Left: Student View. Right: Teacher View.

Why I’m Leaving the Google Spreadsheet Grading

The students were confused with the overwhelming number of grades they had to wade through. The simplicity of ActiveGrade is also one of its strengths.

Occasionally the students couldn’t see their grades because the function ImportRange sometimes didn’t work.

The teacher view is every student’s assignment, so I can’t easily see where students stand on a given topic.

I initially liked separating skills and understanding concepts, but there are a few problems from it. For one, the skills and concepts are on different tabs, so it’s more cumbersome to input grades.

I separated the standards on each quiz, but through some teachers on Twitter I see the perspective of not putting standards on quizzes and having students figure out which standard is which. Breaking the standards into tiny chunks was good for encouraging students to “retake” standards.

One more advantage of ActiveGrade is the ability to send an e-mail to every student and parent with their grades. Currently our school sends a weekly grade e-mail to parents, but since I don’t keep my grades there, parents don’t see student’s grades.

Math Practices

I tried to keep track of certain math practices: (1) Good reasoning & explanations, (2) Checking work, (3) Math Modeling, (4) Asking good questions, and (5) Reasoning when doing estimations. I wasn’t good at “catching” (1) when students did it in class (that was supposed to be the primary way I got that one), students didn’t do well enough on quizzes to get to the point of (2) checking their work, and the system I had designed for (4) asking good questions didn’t allow for enough students to demonstrate their ability. The only ones that worked were (3) and (5). The (3) Math Modeling worked since they had to turn in the sheet to show that they worked during the 3-act lesson. The (5) estimations worked because it was a warm-up [1], once a week, and the students submitted responses into a Google form (bit.ly/rcsguess), which I could then plug in later. However, the warm-up took longer than I wanted (~15 minutes), so I think I’m going to drop it for 2nd semester.

Because of all this, the Math Modeling is the only Math Practice I’m going to keep grading. I’ll probably plug this into the school’s grading system and set it as %30 of a student’s grade. This should be a gimme for students that are present and turn in the work that they do in class, once a week, on block days.

The New Plan

I still want to emphasize understanding. I plan to do this on a case-by-case basis. I will no longer break one standard into sub-standards for grading purposes (e.g. 1.7 Combinations & Composition of Functions had the sub-standards 1.7A Combinations of Functions, 1.7B Function Composition, and 1.7C Function Decomposition) because I want students to know what a question is asking by reading the question, not the standard heading. However, it is often still helpful to introduce new ideas by breaking it down, so I’ll hold on to those names somewhere. Quizzes will be broken into 3 “sections”, and students receive a grade: 1, 2, or 3, for each quiz. To earn a 3 (Mastery), students will have to demonstrate understanding at some point[2]. I am still going to make “going beyond” available (trying to decide between making it a 3.5 and a 4) for students who want to reach and get an A. I’ve really appreciated the Nrich math problems to that end.


Quizzes (or discussions) to demonstrate previous standards will take longer: to create, take and grade.

It will take more work for students to see which skill or understanding they need to focus on when going back over previous standards, since the standards aren’t broken into tiny sections anymore.

Students won’t see that “growth level” (and the video game characters to go with it!0 which I was so excited about. I think it accurately captured which students worked the hardest to improve in math in my class and I hope to find ways to encourage that kind of growth again.


Those are the only drawbacks? Grading should be faster to plug in, parents will know where students stand more frequently, and students will be required to recognize when a skill or technique applies rather than just memorizing a single skill and applying it. Why didn’t I make this change sooner? I’ve still got a ways to go, but hopefully I can start focusing on content more.


[1] Also, the Estimation 180 website is already created, and super cool!

[2] Before a 2 “felt like a 2” and a 3 “felt like a 3”. Now there 3 questions and it’s easier for a student to predict how well they’ll do based on their own grading of themselves after a quiz. From Chemistry I have appreciated making it clearer what a student will get based on how well they perform on each question.


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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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[SBG] Re-examining Classroom Assessment

The past two years I’ve used standards based grading (SBG). Last year I did a drastic overhaul. At the end of this year (when I had time again!) I started re-reading many of the blogs that originally got me onto SBG, and I forgot how incredibly deep and helpful they are! There are so many things I liked but forgot to implement, so I’m going to post here to help me remember everything I forgot last year.

Quick overview of how I do grades

100% standards, 0% HW.  Standards are topics not skills right now. I assign HW but don’t check it: I encouraged students to keep track of that themselves and a small fraction of the class continued to do that throughout the year.

What I didn’t like

A majority of the class did not do their HW and quizzes were sub-par. I have been struggling with this idea: “Students must turn in HW (or some equivalently difficult assignment for those that felt my usual HW to be “busy work”) before they may assess.” I’ve decided against this, thanks to several bloggers. This is from Jason Buell:

Your students must trust you. The number one question I and others get is wondering if students will still do homework or other classwork if it’s not worth points. I can answer with 100% certainty the answer is yes. Yes they’ll do whatever you ask them to do, but only if your students trust you. They’re trusting that what you’re giving them will help them reach their goal. It’s not busy work. It’s not assigned out of habit. It’s meaningful and will help them get from A to B. They will do it because they believe it will help them learn. They must trust that you are helping them get there.

You must trust your students. Allow them to surprise you. Give them freedom. Allow them to fail but allow them to learn from those failures. If you don’t trust your students, they will fail. If you believe they won’t do it if you don’t make it worth points, then they won’t do it. Trust your students.

You must trust yourself. Deep in your heart, you’ve got to trust that what you’re giving them will help them learn. Everything you do is to help them learn. If you don’t believe that, they’re not going to believe it either.

So I realized that making HW required was a cop-out for me. The alternative is going back through what I give as HW and making sure every assignment will help them on the assessments I give them. I also need to go back over my assessments and make sure I’m assessing them on what I want them to learn & know (I’m doing better at this than my first several years, but there’s always room for improvement here!). To that end I’m trying “Understanding by Design” as Sam & Bowman explain it in this post.

Change the formula of the classroom

This post from Shawn Cornally explains exactly how I feel I’ve been teaching, but he is clearly much more focused and intentional in how his classroom has changed. I want that. I think it goes hand-in-hand pretty well with the Understanding by Design.

I also noticed that a big part of SBG should be allowing students to become better self-assessors. I’ve done this sporadically, but something simple I can do to greatly improve this is put something like this at the top of every quiz:

  • Before the quiz, grade yourself on how well you think you’ll do:
  • Now that you’ve taken the quiz, grade yourself on how well you think you’ll do:
  • Grade yourself using the answer key and give yourself a grade [1]:
  • Your actual grade from Mr. Newman:

Think about what your scores mean

What does a 1 mean in my class? How about a 2 or a 3 on a specific standard? Yes, I have the general vocab (“Not yet”, Proficient, and Mastery), but what does that mean for 6.1 Balancing Chemical Equations?  Jason hammers this home (and explains how much time it will take!). I wanted to start this last summer but kinda burned out [2].

One thing I’m still hesitant on his is idea of what a 4.0 means, and why he has a grade of 4. I want to encourage students to make connections, but I think that should be separate standard (something I don’t do well in math right now!).  I also like his idea of “a step above what’s expected” but I suppose that’s what my “3” is.

Summary/List of Tasks to complete over the summer

Here’s what I need to do this summer:

  • Leave Chemistry (mostly) alone.
  • Start with enduring understanding and re-vamp the Precalculus curriculum (again).
  • Embrace the fact that textbooks are not wholly un-usuable resources. I know they get a bad rap on the MTBoS, but last year I think I didn’t hand out the Precalculus textbook for a whole month or two. I need to recognize that they can be useful in skills practice, especially for students without internet access. I’m not going to start teaching straight from the book, but I won’t shun it either like it’s the anti-grail of mathematics education (Khan isn’t that either, btw).
  • Create the “real-world application starting points” for every topic [4]. Weekly, if possible. Like what Shawn Cornally did for his calculus classes back when he was in a normal school.
  • Re-examine my assessments and decide (and write out) what explicitly make “proficient/passing” and what makes “mastery”. This is what I wanted to tackle last summer but failed. Now I’ve added on 3 other steps and think I can still do this?!? Perhaps. At least now I have last year’s assessments to work with rather than starting from relative scratch.


[1] I’ll include this if I have my act together and can put out answer keys to all of the assessments I give. Here’s what I’m doing.

[2] “Burned out” is the wrong expression because in implies that I was working really hard all summer.

[3] It’s the class I always tackle first (nearly 50% of my students are in chemistry so it makes sense…) and I should really give Precalculus & Physics a fighting chance in my planning time. Okay, I should really give Precal a fighting chance. If things stay as they are right now, I’ll only have 10 physics students next year. I guess I sure do a good job of scaring them away in Chemistry…

[4] I actually picked my textbook because in each section the author begins with explaining a real-world application of the topic. Some of them are cheap cop-outs (I’m still looking for a real-world application of trig identities & proofs), but most are something for me to start with. I’ll also dig through yummy math, Dan Meyer’s 3 acts, Robert Kaplinsky’s lessons, and many, many more.


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[SBG] Exam Time

Like in previous years, I’m going to allow students to select what is on their exam. This year I’m doing things a little differently in that I’m allowing them to select their entire exam. They may choose between 3 and 4 standards from throughout the year to take. If students need more to pass, they may speak with me, but I require them to get a parent signature ensuring (a) that I’m covered & the parents know they’re not in a good position to pass and, more importantly, (b) that the student is studying a lot each night and doing their best to ensure that they pass this exam.

I use Google Forms for students to sign up for exams and, for the first time, I thought to look at the summary of the responses. Pretty cool stuff here.

Standards to Retake

If you notice, the distribution of what students selected for their 3 (or 4) standards was really evenly distributed, which I think is cool!  It means that students need/want to improve a huge variety of standards, and that everyone didn’t bomb the same one or two quizzes. It also means my job is harder, both for review and creating the exams, but still worth it.

In the graph, we see more recent (within the last month) essential (required to pass) standards are more common, I think, due to students not having as much time to improve those standards. [1] We also see more 2.1 and 2.3 which are two essential standards that I re-assessed the entire class on later in the year. In general, students probably chose later standards because they feel more comfortable re-visiting those, although the standards early in the year were not totally ignored.

I’m struggling with how to do a final in my other two classes (Physics and Precalculus) because of my new format for standards (some are essential and students MUST pass these in order to pass the class). Dilemma: if I give students even one essential standard on the final and a student doesn’t pass it, then they’ve failed the class. This harsh system works during the year because students can retake standards, but as a final format? I’m not a fan.

How do you do your exams if you’re into SBG? Does it replace all of your grades in your grade-book? It is, *gasp*, averaged?!? Or do you have another cool system? Please help!

[1] Or it could be that there wasn’t a progress report sent home since then, so parents didn’t get on students whose only motivation comes from their parents. *Sigh*


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