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?

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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:

image

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…

image

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.

image

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.

Conclusions

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.