Takeaways from #NCTM14

Nothing is rejuvenating like spending a few days with like-minded people. I feel incredibly fortunate that I was able to speak with many innovative, passionate, and engaging math educators.

And… there’s more to it than that. Obviously, the people who are able to attend NCTM either have districts who are willing to foot part of the bill, received some kind of grant, or paid their own way. That is, the attendees are not representative of the larger math education population. But it seemed that I talked to two kinds of people: those who were content being there and those who were super hungry to connect.

I teach at a super small private school. We have about 220 kids in 6th – 12th grade. I don’t hear bullshit acronyms like “KWLM” – literally “kids we love the most” when talking about children who have not had success at school. I don’t hear kids being called “low” or “high.” I don’t hear these false classifications of children that have nothing to do with how they learn. Even a casual label like “lazy” is describing a behavior, not a person. I encourage you, if you hear a child being called by a label, call them on their shit. It can be awkward, but it can’t be a practice that continues or ignored.

End rant. Here’s a collection of takeaways I had.

The sessions by Christopher Danielson and Michael Serra respectively were the only two where I actually had to think mathematically. I think that it’s such a rich experience for math teachers to be able to be back in a learning environment with novel material. Students do it every day.

What I loved about Danielson’s presentation is that he didn’t know where it would lead: when students come up with their own classification system of hexagons, he doesn’t have a pre-defined notion of what we would create (unless he’s already come up with every possible way to classify his set of 14 or so hexagons.)

Michael Serra had created inquiry based materials which can be found here. We worked together in groups on some of his pre-created materials. I highly endorse the donut polygon problem set.

Recurring theme: Few people have a clear vision of what to do with technology in the classroom. I heard so many stories of computers, iPads, smart boards and other gadgets being stuck in storage because they were being used improperly or didn’t foster student learning (or both).

This was the first year I went to a talk on math and social justice. Rochelle Gutierrez’s sessions focused on the “unearned privilege” that those who can do math have in society. Math typically is heralded as the highest form of intelligence, and presents a high barrier to some. I really dug her point that “math needs people” as a converse to the traditional “everyone needs math.”

People I met put up with a lot of pressing questions. I don’t know how no one told me to shut the hell up all weekend. When asked about your craft, you people are much less defensive than I think I would be.

Steven Leinwand gave the sermon of the week. Among other items, he mentioned the “devastating impact of teacher isolation”; how standards and assessments are the bookends – necessary but not sufficient; and plugged the new Practice to Action text. I’m still confused why Practice to Action isn’t a free text, supported by NCTM, the Dept of Ed or some other professional organization. If the document is so important, why not put it in the hands of everyone? Maybe I’m naïve here.

“Don’t read off your slides. Don’t read off your slides. Stop it. Don’t read off your slides.” – me talking to myself and about half of the presenters.

Dan Meyer had a great presentation on lessons we can take from how video games are structured. He’s got a summary here. But what I really took away was a line at the end of his presentation. He pointed out that so many people are vying for kids’ attention: video game developers, shoe creators, wanna-be teen celebrities, musicians, etc. etc. etc. But what teachers are given (by law no less) is a captive audience. Students are “forced” to be with us for 180 days out of the year, and it’s a time that should be treasured and honored. It’s a time to innovate and connect kids with mathematical thinking instead of mechanical procedures.

I couldn’t have been more grateful for everyone who stopped by for my presentation. I was literally in the hallway grabbing anyone who looked at my sign in an act of shameless self-promotion. It was my first one, and I didn’t really feel nervous – it seemed like all who attended really wanted me to succeed.

I think that’s the vibe that should be supported by NCTM: we want you to be a badass and you can do that here at our conferences. Other than that, I’m still not sure why people come.

Thinking Like a Manager, an Economist, and a (sometimes) Mathematician

Every day I have a 20 minute commute, wherein I expect to be entertained! Dodging other commuters has an inherent challenge, but not the level of stimulation I crave.

I love Russ Robert’s weekly podcast series EconTalk. Through his series, I’ve learned how much wine my pregnant wife can drink, how potato chips fly 60 mph over a conveyer belt, and how incentives affect waste policy. His discussions with his guests are playful, inquisitive, and are a nice blend of education and entertainment. I have no training in economics, only Mathematics and education (sometimes at the same time).

I’m always jazzed to hear educators on the podcast… even though I usually have some strong reservations. Doug Lemov of Uncommon Schools and author of “Teach Like a Champion” was recently a guest. The direct link to their podcast is here. Their talk revolved around how to improve performance within schools and of teachers. Lemov framed his background by noticing a strong negative correlation between state-mandated test scores and poverty level (i.e. poverty went up and scores went down). However, there were some schools who systematically scored higher on the standardized tests. His work began by wanting to find the “industry secrets” of those schools.

They go on to discuss successful teacher techniques, systematic culture issues as well as the importance of education in a democratic community. I was incredibly inspired at some of the compassion, care and thought that Lemov had towards kids in the inner city. I don’t teach to that community, I teach at a private school in a fairly affluent community. And as an educator, I have some pretty strong beliefs with regards to mathematics education that I want to highlight no matter the demographics.

Making Teachers Great

Lemov begins by acknowledging that teaching is a performance profession. He mentioned that a good performance one afternoon does nothing to guarantee an equally good performance the next. Teachers are always on the spot. I agree that the discussion in teacher improvement must begin at that point. It gives light to how much preparation must go into every damn day. He offered up a few techniques for teachers. As I was listening, my cynical brain began lighting up. His book has 49 of these techniques, but he covered the following:

1. Call and response

Students are expected to call out certain concepts automatically. No need to invent the wheel here, Christopher Danielson already took care of this critique.

Without getting too philosophical, I still want to mention this: if schooling is a system of memorizing facts, it can easily turn into a tool of systematic oppression. But that’s another rabbit hole for another time.

2. At Bats

The idea of at-bats is simple – kids need to practice skills multiple times. They need to have opportunities to see them in slightly different permutations and practice with feedback, but not an overwhelming amount. This is probably an area where I am admittedly weak. At times I get stuck in the idea that we need to cover so many things that I don’t give students enough straight practice. As I reflect on my last semester, I realize that I selfishly emphasize co-creating concepts; this is a huge time commitment. Sometimes kids gotta practice.

The trap comes when we limit Mathematics to practicing skills. But we’ll come back to this in a moment.

3. Check for Understanding

This is a pretty straightforward technique. Teachers need to gain insight into what their students have learned and haven’t learned. Most educators would call this a formative assessment. It doesn’t have to be a big quiz or test. Lemov describes one way to check by putting up a multiple choice question on the board and having all the students raise their hands at once with their response. It’s feedback for the educator. I’m all for it.


Here’s where I get stuck. We get stuck in thinking of learning outcomes in a very narrow way. To most of those outside of the teaching profession, the only relevant teaching outcome is grades. If a girl is getting an A in Biology and Calculus, she is really doing well. If a girl is getting a C in Biology and Calculus, she is having a difficult time. The nuance of humanity is completely shuttled into a few digestible letters.

Lemov’s desirable outcomes seem to be tied to standardized tests. Forget for a moment that a recent study by Dr. Stroup found that about 10% of the variation in Texas test scores is accounted by previous scores. Forget that standardized tests are “insensitive to instruction.” Hell, forget that Lemov himself said that scores in all domains are highly correlated by students’ ability to read. My question is: what learning goals do we have for kids in math class?

My own most desirable learning outcome is that students think like a Mathematician. My short spiel on thinking like  a Mathematician is someone who can

  • Look at a problem in a novel context
  • Understand the structures of a problem
  • Think in extremes
  • Question assumptions
  • Experiment mentally (no getting our actual hands dirty!)
  • Make logical arguments to convince himself and others of a solution

My list doesn’t have anything to do with computations. I understand where leaders like Lemov come from – they are taking metrics like state-mandated test scores, and try to make his students successful in that way. The only problem is that we conflate doing well on these tests with thinking like a mathematician. In short, here’s an equation:


My heart breaks at one point when Lemov describes how often his kids practice math. He has his children practicing all the time, and he describes his kids at loving. When I think of loving math, my mind doesn’t think of things I can regurgitate facts like multiplication tables, trigonometric ratios or the power rule. I had a terrible experience when it seemed that college math was more memorization. I light up thinking of the time in 4th grade when I created a really strange algorithm for how to test divisibility. I think of my class in Combinatorics when I had to think about sets and functions without any assistance. Math is created, not memorized, and I sincerely hope Lemov’s kids aren’t crushed when they discover that fact.

The reification of those exams would be more interesting if we broke the SAT/ACT tests into: Reading fast, vocabulary, computational fluency and science trivia. So we get this idea that a kid knows how to do Calculus because he gets A’s in his math classes. He gets to higher levels of mathematics, and it’s a shit show. Because when someone is telling you how to find the nth derivative in multivariable Calculus, this beautiful creative process we call Mathematics turns into a chore. I don’t perceive mathematics as someone telling you how.

And I’m making huge assumptions here. Perhaps Lemov doesn’t have his kids “practice” Mathematics by asking them to calculate how to add or subtract repeatedly. Maybe he has them practicing estimation. Maybe he asks them questions where he genuinely doesn’t know the answer. But I’m guessing that most parents don’t view Mathematics in the same way Mathematicians do. They think of it as computations, or a set of rules to follow.

Improving Organizations

When Lemov discussed how he runs schools, I was incredibly impressed. He begins by creating powerful context: “The first obligation of an organization is that it makes its people better.” In all my discussions with administrators of any different organizations, this seems to be an undeniably underserved function. He said that a teacher should be observed every three weeks.


I can tell you that my admin team is pushed to the brink of responsibilities and time already, and I might be observed once or twice in a year. However, priorities are spelled T-I-M-E, and if managers aren’t looking to improve their team, they have to see what’s actually happening in classes. How to promote improvement in the teaching sector is not at all straightforward, but I’’ really appreciate his sentiment.

Listen to the rest of the podcast, there were some great other points when discussing schools as a system: the divide between admin and teachers, how incentives come into play, how autonomy and accountability have to live together.

So while I still have some qualms in Lemov’s approach to what math is, I loved hearing how he views schools as systems. And thanks for the invitation to reflect through writing.

Kick-Ass Parent Meetings and Pancakes

Here’s the situation. You’re in the middle of the school year. You’re checking your email at 10 pm, because at this point you have a terrible, terrible habit. If we lined up all bad habits from biting your fingernails to free basing cocaine, randomly checking your work email would be a solid 7.5. It doesn’t matter if it’s 3 weeks into the year and you’re checking to see if your boss approved the purchase of some rulers or if it’s the middle of October. Spontaneously checking email is a sickness.

So you’re delighting in your vice. You notice emails from your HR rep, a few questions about homework, updates to your favorite websites about kittens… when you notice an email from a parent. You glance at it inquisitively. You scan the email to see what it might be about. You see that your supervisor is cc’d. You see snippets like “crush her confidence”, “discuss this immediately”, and “It sounds like you are setting her up to fail”.

Your reptile brain immediately slips into a few traps. One of two things immediately take over your mindset – “How quickly can I quit this job. Do I still have that emergency resignation letter?” or “How dare they question me. Don’t they know I have a teaching certification??

Slow down. Look away for a moment. Take 5 good breaths. Count them.

I’m not here to solve your twitch-inducing email problem. But I am here to show you a quick how-to on handling your shit in a parent meeting. You might be saying, “Brandon, what do you know about running kick-ass parent meetings?”. Let me give you an analogy I call the General Mutant Pancake Theory (GMPT). Whenever you make pancakes, what happens to the first pancake? The heat on the pan is uneven, the oil is spread evenly, and the first few pancakes are terrible looking. After those first couple, you’re good to churn out some golden, syrup-sponging delights. I am great at parent meetings because I’ve burned so many pancakes and luckily wasn’t fired. My griddle’s hot and I’m here to help you out.

Brandon's mutant pancakes

My first pancakes I ever made

You’re quickly go through the 5 stages of grief.

Denial – this won’t last long. At first, you’ll think “There’s no way this is supposed to be sent to me. You’ll check who sent the email and it won’t take you long to conclude that it’s for you.

Anger – This is where first year teachers will stay before, during, and after the meeting. First year teachers work hard as hell because they basically have no idea what they’re doing. They have all the good intentions in the world. See GMPT.

Somehow, some way, you have to acknowledge you’re in this stage and move forward. There is no having a good meeting with parents if you are here. It is like an anchor that is made up of the opposite of pancakes weighing you down.

Bargaining – read: feeling helpless and vulnerable. No matter how awesome your friends/partners are, even if they are teachers, it’s hard for others how bad you feel when you are called out by a parent. For the lead up to the meeting, you will continually wonder what you could have done.

Depression – quick fix: happy hours.

Acceptance – friends, this is where you should be before the meeting starts. It’s not easy to get here, but it’s the only way you’re going to produce anything positive. You will char the life out of any pancakes you come across if you go into the meeting with anything but acceptance.

The first thing you must accept is that you and the parents have a common interest. Both parties deeply want to serve their child’s needs. There may be some disagreements on how to do so. However, if in your mind, parents are set up as adversaries rather than potential partners, you will fail to find any common ground.

Most of the time, parents want to be heard, deeply and sincerely. They want to share with you all kinds of details that may never surface again – what their kid’s previous education was like, what their social standing is, hell even what they’re allergic to. It is extremely important that you “Hear” what they are saying. I capitalize “Hear” because this must be  reverent act, not just word recognition. When you Hear what they are saying, you are not reacting, not judging (because you’re already past the anger stage, right?).

It is extremely important that you Hear them first, before recommending, prescribing, educating or responding. Hearing does not mean waiting your turn to speak. When it is your turn to speak, you will take what you Heard and feed it back to them. If they begin by talking about their kid getting their confidence challenged, make that something you talk about explicitly. If they talk about their kid not being challenged enough, acknowledge it explicitly. Bring a notebook if your mind has a hard time remembering all their points, don’t be afraid to look vulnerable, especially when you are desperately vulnerable. Make sure you are always framing your responses as a partnership-in-progress rather than defending what you do. Not being defensive is easier said than done. Again, see GMPT.

A few other general points

  • Call the parents by their first names, invite them to do the same for you
  • Sit in a circle
  • If they talk about your lack of experience (especially in front of admin) acknowledge it
  • Let go of your defensiveness
  • Finish with any agreed-upon, reasonable action items. (Note: do not tell parents you will call them every week, unless you are willing to call them every week. You will resent them for the rest of your year)

Look, first year folks. Your first pancake is going to look a little freaky. It can still taste delicious, it might be stackable, heck it might even be the right color. But it’s going to get better. You are a professional and you probably do a good job. Parents love their kids and are crazy and overprotective and over-questioning. It’s the job, but you are going to rock it eventually.

Brandon's good pancakes

Everyone was pumped, GMPT was born

One last bonus tip: within a few days, send a quick follow-up meeting and cc your supervisor. Make it short and sweet – a quick recap of the meeting and any action items. This is professional as hell, and everyone will think you are totally boss.

Antifragility, Efficiency, and Agency in Class

I am still finishing up Nassim Taleb’s Antifragile book. I wanted to jot down some of the ways we treat students if we classified them by fragile vs antifragile.

Primer: systems are fragile if they are harmed by low-level stressors (think of a light bulb degrading over time/use); systems are antifragile if they benefit from low-level stressors (think of your cardiovascular system after progressing through difficult runs).

If we view children (students) as fragile

 If we view children (students) as antifragile

Give them constant positive feedback Barbell approach to feedback (only short corrections or high praise)
Breaking curriculum into manageable “bite-sized” portions Give them the lowest, appropriate level of information transfer (Moore method is most extreme)
Use behavioral incentives to motivate students Let students’ natural curiosities drive motivations
Use grades as incentives to motivate students Let students’ natural curiosities drive motivations
Try to plan out every moment of class time Be flexible to natural questions and connections
Stick to the specific domain of class Use human brain as a natural conduit to connect across domains (non-linear)
Ask questions with goal of efficiency and convergence Ask questions with goals of multiple approaches and nuance

I’d like to make one note on the last point on the table. When we run classrooms where students have multiple approaches to solving, we run the risk of students developing a non-optimized solution/algorithm/heuristic.

Where we have to be careful is realizing that creation and invention should be primary goals of mathematics, not efficiency. Efficient thinking is desirable and has a purpose, but it’s too easy to give (or force) students the optimal solution.

Giving pre-determined, optimized solutions has a few effects. First, we remove that student’s agency. We no longer care what you created, only what’s fastest. The main fallacy we approach with the banking model of education is that there are already millions of computers that can come up with a solution to 99.9% of high school problems.

Second, mathematics becomes something done to students, rather than students creating. See Paul Lochkhart’s Lament for a more eloquent discussion on that end.

Third, that modus operandi is bleeping boring. I know it bores me as a teacher. All I have to do is watch all of the pupils in my students’ eyes roll back when I “prescribe” optimal solutions to know students aren’t thrilled either.

Extremely Simple Example

I have seen a few different versions of this. Background: we’ve talked about some of the common conventions for the names of pairs of angles (e.g. alternate exterior) as well as which are congruent.

If the measure of angle 8 is 40 degrees, find the measure of angle 2.

Instead of recalling (or as Freire says – making a withdrawl) that alternate exterior angles are congruent, perhaps a student knows that 6 and 8 are congruent, and that 2 and 6 are congruent. So it becomes two-stepped instead of one-stepped.

However, at the cost of a small efficiency (which they will develop later independent of the teacher), the student retains agency. Instead of a memorization, there’s a bit of playfulness involved.

Giving students a chance to play with a simple transversal on a plane gives them more agency to do problems like this


Look at this. Already your mind is trying to parse together how you can begin solving. It’s a delicious problem.

Bottom line: if you’ve been showing kids how to solve simpler ones, if you’ve been treating them as fragile creatures instead of the wonderful, resilient, antifragile creatures that they are, then their response to your outstanding problem is this:

“Can you show me how to get started?”

Socratic Seminar in Math class #NCTM13

Below is a recap that should be live on mathrecap.com soon. But I thought I should host it myself while I’m at it. Enjoy!

Socratic Seminar in Math: Development of Math Reasoning Collaboratively from Ryan M. Higgins

Ryan M. Higgins presented a markedly different methodology to scaffold discussions in a mathematics class. Although Socratic seminars are traditionally used in other domains, Higgins discussed how this method of engaging the class can develop “mathematical connections” and create “authentic assessments” as well as help students obtain procedural knowledge.

Structuring the room

Higgins emphasized the importance of the physical arrangement of the classroom. Since students are the primary speakers, they need to see each other. She advised seating students in circles, fishbowls (to enable students to still refer to the board), or concentric circles. If you look at the handouts, there is no seat designated for the teacher, further emphasizing the teacher as observer rather than participant.


A Socratic seminar can be very different from a traditional classroom environment. Because so much of her class is discussion-based, Higgins had to set very clear guidelines for students to have these conversations. Students were trained to agree or disagree with others’ statements, not with the person as a way of maintaining respect and the safety of the classroom. Once conversations began, students were limited to speaking twice, until everyone had spoken at least once. Students were also asked to structure their contributions in the form of statement –questions, such as “I understand that you are trying to find the area of a triangle, can you explain why you are adding the side lengths?”.

Typically, she had a student track who had and had not spoken on the board so everyone could see – keeping accountability of all, and furthering agency of the students as a whole.

Teacher’s Role

In typical Socratic fashion, Higgins explained that during class the teacher’s role is to primarily prepare and observe. Most of the work of the teacher is done beforehand – trying to come up with challenging problems/situations and developing potential probing questions. During class, she noted that she tried to avoid answering students’ questions. Higgins emphasized not asking low level questions of students in initial problems, as the conversation will be over too quickly.

Other takeaways

Higgins shared some resources such as the NAEP site and the UT Dana Center for interesting tasks. Socratic seminar can be very empowering and engaging for students, but talks like these always leave me slightly wanting. They usually serve as a source of introduction rather than advancement. Rarely do speakers like Higgins show concrete examples of what the class looks like or how interactions flow (no doubt due to IRB and privacy issues). While I understand the restrictions, they highlight the hypothetical rather than the practical. I didn’t leave her talk understanding what she meant by “authentic assessments”. But I definitely feel that the structure and intention behind her use of Socratic seminar is something I would like to take back to my class.

Attached below are her presentation materials.

181_Higgins_Socratic Seminar Handout

181_Higgins_Socratic Seminar

Balancing Assessment/Pedagogy and SBG

Over the past couple of weeks, I’ve been completely inundated by an inquiry into assessments. I’ve been trying to answer Michael Fenton’s call to create better assessments by analyzing some of the assessments I’ve given this year. Like I said before, our school is exploring standard-based grading techniques. I teach four different preps, and each is taught and assessed in different ways.

  • AP Stats
  • Pre-Algebra
  • 6th grade skills class – called Foundations
  • Geometry

My goal in this post is to reflect some on what type of pedagogy, assessment, and rigor are applied in class.

Note:When I say rigor, sometimes I think I mean “pacing” or maybe my own stress level?

Primarily Direct Instruction

Statistics is almost exclusively taught using a direct lecture. I basically talk and ask very directed questions. I tell them what buttons to push, and how to graph Gaussian distributions and write down definitions very explicitly. I assess using materials that were purchased alongside the text book. High rigor.

Pre-Algebra is predominantly question driven, followed up with working example problems and smaller lectures. The class is somewhat strangely put together. We’ll cover matrices and exponents in the same chapter (I’m not totally sure why we cover either). Aside from this class needing some pruning, it’s essentially directly taught as well. I assess with frequent small, fragmented quizzes.  Low rigor

Primarily Exploratory

In my Foundations class, I ask lots of questions, and we brainstorm how to solve them. We cover standards (like multiplying fractions) and the more light side of mathematics (like making hexaflexagons). Sites like estimation180.com have been a godsend for finding really simple, practical ways to frame questions – in that case for estimating. We take our time, and have lots of remediation when possible. Low rigor

In Geometry, I base the course loosely on the Modified Moore Method. If you’re unfamiliar, the MMM purposefully and carefully gives students definitions and axioms, and the student’s’ responsibility is to prove theorems, corollaries and such. I’ve modified the course in a way that students prove traditional theorems less, and are put into difficult environments in which they must navigate. Medium rigor.

For example:

“You are given a regular polygon with each exterior angle measured at 10°. (4 points each)

a. What is the measure of each interior angle of the polygon?

b. How many sides does the polygon have? (Hint: double check your answer)

c. What is the sum of the measure of all of the interior angles?

d. What is the sum of the measure of all of its exterior angles?”

This might look like a typical Geometry problem that you might see in any vanilla Geometry class. However, I feel like it’s not an assessment of how they use the interior angle/exterior angle sum formulas – because I never gave them one. The closest I came to giving them a formula was to ask them in class what the sums of various regular and irregular polygons is. They filled out and made some guesses and then I gave them this questions on a quiz.

When I hear from students afterwards about the time they spent on this problem – drawing out different scenarios, testing and re-testing hypotheses, (productively) struggling and learning from the quiz itself, it makes me feel good. AND I COULD HAVE TAUGHT THIS FORMULA IN ABOUT 38 SECONDS. So what exactly am I doing here?

Relating this to Standards Based Grading – what standard coincides with this assessment? Formula creating ability? Rigor? Tenacity? Badassery? When I try to separate my assessment with a basic pedagogical philosophy, I really run into trouble.

Closing thoughts

According to standards-based grading, the students either picked up that information or they didn’t. They essentially strip the time component as well as the character component. SBG is pedagogically agnostic. It leaves me really wondering what a best practice is for each of my classes. Sometimes I feel a little timid and afraid that I’ve picked the wrong pedagogy for the class I’m teaching. I still feel strongly attracted to standards-based grading, but there has to be room for giving students feedback on these other issues such as tenacity and grit. The lingering question for me is: how will they receive it?

Assessment in Math

Getting Ahead of the Curve

My school has recently begun an inquiry about how we assess. I have been struggling through the process for a number of reasons. I struggle most with trying to break away from assessing how I’ve been assessed. Mathematics lends itself to a certain type of grading – that where we think about the “skills” of the class.  I’m very good at this game we call school, but many of my students are incredibly bad at the game, yet great at mathematical thinking.

The main question I go back on forth on: What does it mean to earn an 80% in a Geometry class? Does it mean that you’ve mastered 80% of each of the objectives? Does it mean that you tried really hard, but did not understand all of the concepts? Or does it mean that you know everything, but were kind of lazy and missed deadlines? Where do expectations fit in?

Just like the Fed is supposed to control inflation and stimulate the economy with one lever (setting interest rates), so too does a teacher try to represent too much with its lever: setting grades.

I’ve spoken with some of my students about this, and we’ve come to two general conclusions. Historically grades represent ability (think skillsets and applications) and work ethic (meeting deadlines, trying one’s best).

One Assessment Model

I tried to graphically represent a model of this historical assessment:


This two-dimensional graph has a number of implications:

First, when we think of skills, we tend to think of “hard skills” such as multiplying fractions or using the chain rule correctly. We lose the “soft skills”, such as the highly touted 4 C’s: communication, collaboration, creativity and critical thinking. Each of those is difficult in and of itself to assess

Second, how do we weigh these two dimensions?

Should we base grades solely on the content that students mastered? This brings up a few thoughts for me: If you were really great at Calculus, you might be able to walk into Calculus 2, listen to the lectures, try a couple problems and understand all the content. If we weigh only the objectives of the class, how can we justify giving a 0 for a missed deadline? The grade will artificially be lower – the student understands the material but their grade does not reflect it.

And what of work ethic? If a student is really trying, improving, busting her ass on every assignment but still only masters 79% of the material, we give her a C. That somehow feels cold and unfair. But standards/objectives-based grading would imply that she earned exactly that grade. The counter would be: multiple studies indicate that humans with higher work ethics are more successful than humans with solid skill sets. So perhaps we should prioritize work ethic…

Third, we can think about the mathematics behind this model. On a 1-100 point grading scale, lacking a work ethic will destroy any representation of skills. Miss a couple of assignments, ace all your tests, your grade could be interpreted as skill-deficient. You can get by more with good work ethic/poor skills than poor work ethic/good skills. Either mindset lends itself to gaming grading.

Trying to Give Weight

I began playing around with what weighing these dimensions with grades might look like, and I came up with…


Yes, yes, I know the proportions are not exact. Feel free to create a better one.

And just for fun, I put some hypothetical students on the original scale. I started thinking how I approach those students, and the results were delightful and frightening.


I’m so drawn to people with high work ethic, but become too skilled, and I’m flustered on how to keep you engaged… One class wanted me to put a point where I thought each of them landed. I don’t advise you to try it at home.


I’m still at a loss for how I want to approach grading philosophically. I had a great professor at Baylor tell me that he did not want to give out any grades, but write two sentences about each student. I think that mindset is powerful, but rarely feasible. Maybe I’m approaching this the wrong way and equivocating grading with assessment. I would love to know some other math teachers’ (or just teachers) thoughts on this particular model.