Posted in Assessment

The case against “The Case for Not Allowing Test Retakes”

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Last night on Twitter I came across this tweet: 

Go ahead and read the article… I’ll wait 🙂

The article really struck a chord with me, and not in a good way. Actually it sent me on a 10-tweet rant, and when I woke up this morning, I still felt compelled to write about it.

Anyone who has worked with me knows my passion for Standards-Based Grading. I started this in my first year of teaching (before I even knew that SBG was a thing), and I am continually striving to improve our grading system so that it meets the needs of both students and teachers. We have modified every year to address different needs/issues that arise. It’s never been perfect. It’s still not. But the pros so outweigh the cons that I would never go back. [As a side note, here’s a pervasive problem in education – if something doesn’t work perfectly the first time, we throw it out instead of working to solve the problems we have. But I digress…]

The author bases this article on his “experience with policies around allowing test retakes, dropping late work penalties, and prohibiting zeros.” I am not clear on what these experiences are – what this teacher has tried, what issues arose, what he did to try to fix them – but I’ll build my arguments here on my own experience.

The traditional policies—giving each assessment only once, penalizing late work, and giving zeros in some situations—help most students maximize their learning and improve their time management skills, preparing them for success in college and career.

I find this statement problematic for many reasons. First, the idea that these policies help “most” students. How many is most? 99%? 51%? There’s a big difference. Yes, we have a lot of students who have figured out how to play the game of school, but what about the other 1-49% of students who are not helped by these policies? I see grading reform as looking for ways to move closer to “all.” I started using SBG to help students help themselves. I wanted all students to be able to look at an assessment and know where they excelled and where they needed to improve. I found that students did not understand how to “maximize their learning” with “traditional” grading policies, so I looked for ways to make the learning process more coherent for all students. Am I reaching every single student? Not yet… but now I’m better able to strategically intervene with students based on specific areas that they haven’t yet grasped.

But this author doesn’t focus on content learning as much as improving “time management skills, preparing them for success in college and career.” So let’s talk more about these interpersonal skills that are taught through “traditional” grading practices.

Deadlines motivate force me to finish something. I am a procrastinator/perfectionist: I am so sure that the longer I keep working on a project, the better I can make it, so I tend to wait until the last minute to submit. (This is why it takes me so long to blog. EDIT: this is why I’m still working on this blog post 2 days later…) But if I’m past a deadline, do I keep going? If I don’t get “credit” for my late work, what’s the point? Oh well, maybe next time. Except what have I learned? I’ve learned that if I don’t submit something on time, I never have to learn that content; in fact, my teacher won’t even let me turn in that project to show what I’ve learned. I have NOT learned how to “improve [my] time management skills” so that this won’t happen again. I am NOT prepared for college because I still don’t know how to “self-assess and then self-advocate to get the help that [I] need.” I need a teacher to help me learn how to do this.

The author purports that these new policies are designed for students who are intrinsically motivated, and that many students need extrinsic motivation. I’m not sure what evidence there is to support the first point, but I would argue that traditional grading practices decrease motivation. On a traditional 100 point scale for grading, a zero is detrimental. Let’s say there are four major assignments in my class; if I earn a zero on the first assignment, even if I earn 100% on the next three, the maximum score I can earn is 75%. If the best grade I can earn is a C, even if I try my very hardest, why bother? Using SBG, on the other hand, I’ve seen an increase in motivation and buy-in from students because they know that I am continually assessing their progress. They know that if they don’t get it now, they can keep working at it and keep learning. Students learn at different rates and we must honor that in both our words and our actions, which is why the following quote hurts my heart the most:

In math classes, where concepts constantly build on one another, traditional policies hold students to schedules that keep them learning with the class. This makes kids practice and check their practice on a timeline determined by the teacher that will have them ready to test before the test.

As much as teachers plan lessons and try to keep a certain pace, it is student learning that truly dictates the timeline of the class. Please don’t misunderstand me: I don’t stop moving forward until every single student gets a concept – that’s not good for anyone. But it’s also not good to stick to a timeline for the sake of a timeline, at the expense of student learning. I find that SBG allows me to continue with the course at a reasonable pace, while allowing for students to continue to improve their understanding at their own pace.

But isn’t math sequential? The author points this out in his paragraph on mental health:

My math class builds sequentially: Mastery in early units helps students be successful in the following units. In my experience, traditional policies motivate students to maximize their learning in the first unit, which helps them on every later unit.

The great thing about math being sequential is that we’re constantly practicing the skills that we’ve learned! For example, in my Algebra class, we would learn the distributive property early in the course. Students would make mistakes (distributing a negative!) that would lead to lower scores on assessments. However, we would use it all the time in every unit that came after, and guess what… they got pretty good at the distributive property! Of course, this happens in every class, but with SBG, their grade now reflects their current understanding, instead of their quiz score from September.

Since the author brought it up, let’s talk about mental health for a second. I’m going to ignore the HUGE leap from soft deadlines to Snapchat and instead focus on this statement in particular: “Under retake policies, parents at my school have expressed concerns about how overwhelmed their children become due to being caught in a vicious cycle of retakes.” Although my entire school hasn’t enacted these policies (yet), I have seen the opposite happen in our math classes. Instead of stressing out over an assessment, students feel more relaxed because they know that one bomb won’t tank their grade, that we’ll keep working at it until they get it. I also employ highlight grading to refocus assessments on learning rather than grades. I got several sweet notes this year from students who specifically named these practices as helping them “relax and focus on learning rather than stressing about grades.”

The author’s third point is on teacher effectiveness. I’m not going to lie; the first year of reforming grading policies is tough stuff. But I find that to be true with everything! The first time you try something new, it feels clunky and time-consuming because you’re having to make conscious decisions about everything. Think about your first year teaching: those lesson plans probably took a long time. Now they are second nature, so it doesn’t take you nearly as long. Same with implementing SBG. Once you have a system in place, it becomes second nature. The author points out, “every minute writing and grading retakes or grading long-overdue work is a minute that I’m not planning effective and creative instruction,” but I have come to find that I can’t plan a lesson without knowing where my students are. It is the assessment that feeds the instruction, and with SBG, I know exactly where my students are and SO DO THEY. Ultimately they are better equipped to make decisions as to how to improve, and they start doing the heavy lifting, not me. Presto changeo, teacher effectiveness.

Finally, and most importantly IMHO, I’d like to address specifically the fundamental mindset shift that must occur before making any changes in grading policies, the mindset that seems to undergird this article and many arguments like it. Many teachers think that the goal of changing grading practices is giving students better grades, which is reflected in the author’s focus on options (“test retakes, dropping late work penalties, and prohibiting zeros”) rather than interests (increased student learning). Although I’m thrilled students are earning better grades, I’m more concerned with aligning student grades with their current understanding.

Shoutout to @mrdardy for sharing this article and getting me all fired up! 🙂

Posted in Assessment, Calculus, First day

AP Calculus BC: reflection and ideas for the future

I loved teaching AP Calculus BC last year! I was a little intimidated at first… I hadn’t taught or even used Calculus in 15 ish years. It was a great experience and I can’t wait to do it again! But alas… not enough sections this upcoming year. I do want to reflect on what I did and what I would like to do in the future so I don’t forget.

My favorite lesson was actually the first day of school. I showed three linear graphs and asked students to write down everything they knew about them.

First day graphs

After talking about key features – intercepts, end behavior, domain and range – and linear vs. constant functions and a line that’s not even a function, I surprised them: these aren’t lines!

First day graphs 2

Students worked in teams to match each graph with an equation and come up with a convincing reason why. Then teams presented their solutions via gallery walk.    

First day student 2   img_1357

We touched on domain and range, even and odd functions, and other key features of functions; we also began exploring the ideas of limits and local linearity. Students really latched onto the idea of “zooming in” on the graph, which ultimately paid huge dividends throughout the course. I found myself saying weekly, if not daily, “remember the first day of school when we zoomed in on those three graphs?”

Another win last year was practicing derivatives and (some) antiderivatives at the same time. This is one of my favorite teaching strategies in general; if you can do something one way, you should be able to think “backwards.” I only did basics, like the table below, but it really got students thinking both directions.

I like to give a little peek into what’s coming, and since we already knew that the derivative of a constant is zero, it wasn’t too challenging to sneak in conversation around this “+C” business early.

Here’s another one, this time after trig derivatives and chain rule:

In the future I would like to do this even more intentionally and often.

Around the same time, we were sketching derivatives given a graph, so I threw in some antiderivatives in the same way: if this is the derivative graph, what could the original graph look like? Here they could really see the “+C” as teams came up with different vertical translations of the same graph.

As much as I loved exploring and wondering about these different ideas, the biggest struggle I had was the order in which to introduce concepts. Everything is so connected, and I didn’t want to muddle their understanding by doing too much at once. I ended up sticking pretty close to our textbook; especially in my first year, especially since my PLC teammate would also be following the textbook, especially since the instructor of my APSI used the same textbook at his school and provided me with a bunch of already aligned supplemental resources (!!!), it felt important to stay consistent.

Now that I have a little more “big picture” vision, I would do things a little differently.

First, standards-based grading! I started this in second semester last year, when we began studying series. It was a natural break to begin something new, but the content turned out to be perfect for the kind of self-reflection and improvement that I needed from students. (I’ve been implementing various versions of SBG every year I’ve been teaching; this might be my favorite version so far, but still far from perfect.)

Learning Objective 4.1A – Determine whether a series converges or diverges.

This learning objective encompasses A LOT of different tests for convergence, and what I found is that our textbook gave very bare answers to homework problems, so students thought they were doing great when they were actually missing big chunks of explanation that would communicate sound mathematical reasoning (and let’s be honest, would cost them points on the AP exam). By changing my grading strategy, I was able to better communicate my expectations and give feedback for improving.

After our first quiz on the topic, I gave students their highlighted quizzes back (no points marked, my typical M.O.) along with a Google doc for reflection. This included a rubric for them to score their own progress on this learning objective.

Rubric series

I showed an example of a 4 to the whole class, emphasizing three parts to communicating whether a series converges: conditions, evidence, conclusion. Many students were providing the evidence, but not first checking that the conditions for using the convergence test have been met or providing an appropriate conclusion; we looked at two examples of 3s that illustrated these omissions. Then students made corrections to each problem on their quiz and gave an overall score for their understanding with evidence to support their score. Ultimately they were very accurate and I only had a couple of conversations with students whose evaluation of their learning was different from mine.

The best part – every student earned a 4 on their next assessment of the same learning target. After a whirlwind first semester, I finally felt like I was doing something right!

I always give cumulative quizzes/exams, so this quiz also included derivatives from first semester, along with a rubric for self-evaluation similar to the one above. I got some very honest, “I thought I was a 4, but this evidence says that I’m currently at a 2” evaluations. They realized for themselves that they still had some work to do on the “easy” stuff! I really hope to start this from the very beginning of the year when/if I get to teach this course again… I think the potential for continuous learning here is huge.

The other thing that I would really want to do is address the sequence of content; I still don’t have a good feel for this yet. In my other courses, I try to space out related content so that students have time to practice content before trying to build on it. For example, in EM3, we teach unit circle and trig functions (all no calculator) in unit 2, practice it throughout homework in units 3 and 4, then unit 5 brings in inverse trig functions after they have built up fluency. I don’t know yet what this looks like in Calculus, but I would like to attempt it. Thoughts so far:

  • Do some work with series earlier in the course. Chapter 10 in our textbook is all about series and it is HEFTY. It could easily be two units with all of the convergence tests taught earlier and practiced before needing them to test endpoints for intervals of convergence later.
  • Teach derivatives and integrals simultaneously??? Maybe start with graphical and tabular interpretations of both, then move to analytic approaches? Can this be done??? I have more questions than answers.
  • I felt like I needed more time for teaching and practicing integration techniques, so find a way to do this earlier. My students sucked at u-substitution, honestly because I think I sucked at teaching it. Maybe by teaching chain rule and “backwards chain rule” together, students will become more fluent. Maybe by separating it from other integration techniques and letting them get good at one technique at a time, they can feel more equipped when I ask them to compare and contrast strategies. Maybe Coaching Questions would be a good routine to use practice using integration techniques.
  • Polar area – I feel like I can sneak this in earlier with other area integration problems. Still not sure.
  • The construct of teaching all AB content first, then moving to BC content, seems forced. Is there a reason for doing this, other than my textbook said so? Yes, the AB content is foundational for BC content, but can/should it be better integrated?

And there we have it folks – a lot of partially formed ideas about AP Calculus BC! I really do hope to teach this class again. Shoutout to my department chair – if you’re reading this, in case I haven’t said this enough, I would LOOOOOOOOVE to teach AP Calc again in the future. 😉 😉 😉

Posted in Assessment, Calculus, Instructional Strategies, Intervention

Exit Slips and Intervention

I’ve been trying to figure out the “best” way to write this blog post… finally I decided to just write it in whatever imperfect way it comes out because I think it’s one of the most powerful things that we can do as teachers and I want to spread the gospel far and wide!


But it’s not just the exit slip, it’s what you do with it. You’re getting some information about your students’ understanding, and then following up in a very strategic and targeted manner.

Here’s what I did recently with an exit slip on limits and continuity.

Exit slip 1

Intervention #1: Pulled small groups

I gave this exit slip at the end of a class period on Tuesday. On Tuesday afternoon, I quickly sorted exit slips into “yes” and “no” piles based on their answers to #1(a)-(c). Then I sorted the “no” pile into subcategories based on the mistake that was made. I ended up with these categories:

  • On #1(a) – students who answered 2, which is the value of f(-2) not the limit as x approaches -2;
  • On #1(b) – students who answered 3, which is the limit from the right not the limit from the left;
  • On #1(c) – students who answered 2, which is the limit from the left and the value of f(3);
  • On #1(c) – students who answered 1, which is the limit from the right.

I made new student teams based on like mistakes. On Wednesday, I projected a PowerPoint slide with their new teams; as they worked on some practice problems together, I visited each group to address the misconceptions in their exit slips and answer any remaining questions. (In the past I have pulled small groups to work with me at a side table, but this year I’ve decided to regroup teams temporarily and visit each team.)

Intervention #2: Targeted whole class intervention

When I looked at students’ responses to #1(d), I noticed that all students had said that the function was not continuous at x=1, but most did not reference the definition of continuity in their explanations. This told me that I needed some whole group instruction around the definition of continuity, specifically in how to justify their reasoning. So after I addressed each team’s individual needs on #1(a)-(c), I asked students to discuss the prompt below in their teams.


We had a discussion of the three conditions required for a function to be continuous at a point, and students were able to specifically identify the reasons for which each function was not continuous at x=0.

Intervention #3: Peer teaching

Finally, on problem #2, the mistakes were all algebraic/computational and had nothing to do with limits: students either forgot how to factor a difference of two cubes or made a mistake in dividing. In every team, there was at least one student who had correctly factored, simplified, and evaluated the limit, so those students supported their teammates in finding their mistakes and making corrections.

So how much time did this take?

Well… it has taken me longer to write this blog post! 🙂  It took me about 5 minutes to sort the exit slips into piles for intervention #1, maybe another 10 minutes to read the responses for #1(d) and create intervention #2, and about another 5 minutes to look at the work for #2 and make sure that each team had at least one student who had completed the problem correctly.

It took me about 25 minutes of class time to check in with every small group (and would probably have been less if I remembered my timer… oops!). And this is happening as the rest of the class is working on another task, so I’m not losing any time; I’m actually gaining time because I’m not reteaching to the whole class.

Is it worth it?

Absolutely! On their quiz last week, every student correctly identified a two-sided limit, a one-sided limit, and a limit that did not exist; nearly every student was able to correctly show that a function was continuous at a point using all three parts of the definition.

In all honesty, I think this is one of the simplest changes a teacher can make to improve student learning. And although this example is from my Calculus class, I have used this strategy in every class, at every level, with every student population. IT WORKS!

Shoutout to Cassandra Erkens of Solution Tree, who has been a consultant in my district for many years now, and who has helped me to refine my assessment and intervention strategies every year!