Backwards design: Antiderivatives (Individual project)

For our individual projects this quarter, we each re/designed a lesson plan using Backwards Design (BD) (Wiggins & McTighe, 2005). I have yet to teach a college math class, so unsurprisingly, I have never designed a lesson plan for a college math class before. Though I do frequently daydream about it… And for some reason I usually imagine teaching Calculus II. Possibly because the topic is so…integral to STEM fields? So for my lesson plan, I chose the first day (or two) of Calculus II.

I thought really hard about what I want students to experience on their first day of Calculus II and what class culture I want to establish. Since all I can do is simply imagine what I want to teach, I wasn’t going to limit myself to the topic that is typically taught on the first day. Instead, I wanted to allow myself to reimagine what should be taught on the first day so that they leave with a solid sense of how Calculus II is related to what they have done in Calculus I. After following a few different paths, I decided that the topic I wanted to talk about was antidifferentiation, which is a nifty section that simultaneously reviews Calculus I while setting the stage for Calculus II; antiderivatives introduce you to the whole idea of “undoing” the process of differentiation – i.e., integration. Perhaps unsurprisingly, this is indeed one topic that you might see on the first day of Calculus II (or maybe the last day of Calculus I).

For class culture, the elements of communication, collaboration, and conceptual understanding are very important to me. And if these elements are to be included in the culture, Mazur (1997) emphasizes the need to establish these class norms on day one (and consistently throughout the whole course). Therefore, alongside my Relationship to Calculus I BD, I wrote a Class Culture BD.

In support of the class culture goals, I have a Digital Citizenship BD that allows me to separately track the incorporation of ISTE Student Standard 2: Digital Citizenship – “students recognize the rights, responsibilities and opportunities of living, learning and working in an interconnected digital world, and they act and model in ways that are safe, legal and ethical.” I’ve thought about it a lot, and so far the only way I see to authentically tie Digital Citizenship standards into your typical college math course, is to weave a digital environment into the fabric of the course. Otherwise, it feels like a box to check – “Okay, we did an assignment with Digital Citizenship and talked about it for 10 minutes.” But I think a digital environment gives you two things simultaneously: a way to tie in Digital Citizenship and another set of pathways for students to collaborate and communication, and develop their class culture.

So for my imaginary class, I chose slack.com as our digital environment to use throughout the course (see my previous post on Slack for more about what it is and why I like it). Using Slack will give us a context in which to talk about Digital Citizenship Indicator 2b throughout the quarter – “students engage in positive, safe, legal and ethical behavior when using technology, including social interactions online or when using networked devices,” and Digital Citizenship Indicator 2d – “students manage their personal data to maintain digital privacy and security and are aware of data-collection technology used to track their navigation online.”

My Backwards Design(s)

Here is a link to my Backwards Designed lesson plan. I wanted to share my whole design, even though it’s a practice lesson plan, because I had a hard time finding Backwards Designed math lessons. Sharing my final product for this project (which I still consider a draft in the grand scheme of things) could be valuable to other math teachers trying to use BD to write a math lesson – even if that value is seeing that you disagree with something in my lesson.

Admittedly, there’s a lot going on in my BD since I have three components: Relationship to Calculus I, Class Culture, and Digital Citizenship. But I do think they make sense together, and I felt like it was really helpful for me to do these things in parallel if what I truly want (in my hypothetical class) is a specific class culture. In fact, I felt that writing these BDs in parallel helped me see how I could possibly accomplish this vision I have for my math classes, where students communicate and collaborate, and where conceptual understanding is a priority. That said, I will only talk about some of the elements of my BD in this blog post – feel free to send me questions about anything I do or don’t address here. I will go into the most detail about the Relationship to Calculus I BD elements, since that was the most challenging to wrap my head around.

Relationship to Calculus I Standards: While there are calculus Common Core standards since calculus is taught in high school, they did not resonate with me as “the standards that need to be met” for college classes. It could probably be argued that colleges should adopt something like Common Core standards, but presently, the standards used to compare college courses really are the textbook topics. To see if two courses are equivalent (like when transferring credits), colleges compare the syllabi of the courses – i.e., they compare the topics covered in the courses. So after some deliberation, I decided that the standards which made the most sense really were the theorems, definitions, etc. from a given chapter.

Relationship to Calculus I Essential Questions: My essential questions most closely resemble the third connotation of essential questions discussed by Wiggins et at. (2005): “we can consider a question essential if it helps the students effectively inquire and make sense of important but complicated ideas, knowledge, and know-how—a bridge to findings that experts may believe are settled but learners do not yet grasp or see as valuable” [emphasis in the original] (p. 109).

I could possibly be convinced that my essential questions “What function, when you take the derivative, gives you this function? What is this function the derivative of?” do not fit the definition of an essential question, but I included it because it is the fundamental question guiding the lesson. My other essential questions, “What is an example of two functions that have identical outputs only sometimes? What is a real life situation that could be represented by a piecewise function?,” fit the definition better. Both of these questions are getting at why the interval I is mentioned in the definition of antiderivative. Mathematicians have decided that it’s “settled” that this tidbit of information is required for the definition, but it’s a tidbit that students could have a hard time seeing the value in. So these questions will hopefully be entry points to making sense of “the I” in the definition of the antiderivative.

Relationship to Calculus I Academic Prompts: I would enjoy a discussion on what performance tasks look like in math. I tried to come up with performance tasks by looking at the examples Wiggins et al. (2005) give for geometry (p. 266) and their descriptions of the types of evidence (p. 153), but instead I think I came up with a type of academic prompt. I did write these questions myself and I think they are influenced by my background with University of Washington’s Physics Tutorials (McDermott & Shaffer, 2002).

Class Culture Assessment Evidence: For this, I kind of metaphorically “threw paint at the canvas” where the paint represents things that could indicate the development of the culture I am hoping for, and the canvas represents the BD Assessment Evidence box. It’s hard to assess class culture, and I imagine doing it informally. I wasn’t necessarily imagining that I would tell the students “I will be assessing our class culture and here is how,” but it would be a good next step to consider which assessment items I would want to share with the students and how to word those items or frame the discussion.

Digital Citizenship Assessment Evidence: I am not sure the best way to meaningfully assess, in the context of a math class, if they are thinking deeply about what information they share in their Slack profile (as per Digital Citizenship Indicator 2d). I think you can assume they put some level of thought into it if they update their Slack information, but you can’t really make any judgment what level of thought. And what if they thoughtfully decided that the default information shared on Slack is actually the information they want to share? That wouldn’t be visible by looking at their profile. I considered giving them a ranking question like “I thoughtfully considered what information I want available on Slack: 5 4 3 2 1” but I’m not convinced that giving them that assessment would add any value to the process. So for now, the only assessment I have is informally observing them participate in the class discussion or updating their information in Slack, which can only maybe indicate that they are thinking deeply, and cannot indicate that they are not thinking deeply.

Six Facets of Understanding

Wiggins et al.’s (2005) say that “to understand is to make sense of what one knows, to be able to know why it’s so, and to have the ability to use it in various situations and contexts” [emphasis added] (p. 353). The issue with “teaching for understanding” is that understanding is an ambiguous term (p. 35). There are many meanings to the word understanding, and Wiggins et al. claim that “complete and mature understanding ideally involves the full development of all six kinds of understanding” (p. 85) where the six facets of understanding are defined as:

  • “Can explain—via generalizations or principles, providing justified and systematic accounts of phenomena, facts, and data; make insightful connections and provide illuminating examples or illustrations.
  • Can interpret—tell meaningful stories; offer apt translations; provide a revealing historical or personal dimension to ideas and events; make the object of understanding personal or accessible through images, anecdotes, analogies, and models.
  • Can apply—effectively use and adapt what we know in diverse and real contexts—we can “do” the subject.
  • Have perspective—see and hear points of view through critical eyes and ears; see the big picture.
  • Can empathize—find value in what others might find odd, alien, or implausible; perceive sensitively on the basis of prior direct experience.
  • Have self-knowledge—show metacognitive awareness; perceive the personal style, prejudices, projections, and habits of mind that both shape and impede our own understanding; are aware of what we do not understand; reflect on the meaning of learning and experience” (p. 84).

On paper, I think my hypothetical lesson did a fairly decent job including these facets – the only one I don’t feel like I can see in my lesson is the facet “apply“. However, in practice I might feel entirely differently about how my lesson holds up to these facets. But for now, here’s how I see the facets showing up in my lesson:

The Relationship to Calculus I BD elements get at the facets of explaining and interpreting. Explaining will hopefully emerge when the students work to develop the idea of adding a “+ C” to the end of the most general antiderivative. Then the assignment where they start off by drawing their own graphs, but ultimately explain the features of their partner’s graph should also encourage explanation. Interpretation comes up when they work in groups to come up with a real life situation that could be diagramed by a piecewise function; they will need to understand what is graphed and be able to interpret how a graph matches their thought-up scenario.

The Class Culture BD element gets at the facets of empathy and perspective. I found these two elements somewhat difficult to disentangle, but there’s an image in my mind that helps me understand them separately. I’m imagining this in a math context, but really it’s suitable for many contexts. I’m imagining a scenario where, for some reason, you don’t feel like listening to someone’s idea. Maybe you feel rushed, you really want to talk to someone you know is always one step ahead, and you feel this is not that person. Empathy encourages you to listen to them anyway. Perspective is what you stand to gain by listening to their idea – a new perspective on the problem or idea, a new way to see the math. A class culture of taking other people’s ideas seriously encourages the habit of listening, which I think can help teach empathy and perspective.

Self-knowledge should come out through self-assessment by using the Clear and Unclear Windows (Ellis, 2001). This is a reflection technique where, at the very end of class, you have the students write down what they felt was clear during the class period on one half of the page, and what they felt was unclear on the other half of the page. Then they turn it in to you before leaving class. This helps them reflect and helps the teacher get a sense for where the students are at. I also hope that through learning to understand their peers’ ideas, they will find opportunities to reflect on their own habits of mind and thus develop their self-knowledge.

I didn’t feel like application truly made its way into my day-one lesson. This is unsurprising since my performance tasks turned out to be academic prompts. Recall the description of application – “effectively use and adapt what we know in diverse and real contexts—we can ‘do’ the subject” (Wiggins et al. 2005, p. 84). This description seems to link “real contexts” with “doing” the subject and I’m currently struggling with the pairing of those two things in math. While mathematical relationships can hold true in the real world, math is a construct in the mind. And what I think of as doing the subject may or may not involve what I think other people call real contexts. At this point in my life, what it feels like to do math, even in my real contexts, feels very similar to the academic context. So while I feel like my lesson does including “doing” the subject, I’m not sure it involves real contexts in the way that Wiggins et al. (2005) means.

Final Thoughts

I thought the BD process was really valuable, and it helped me break up my different goals into individual tracks that I could think through separately before bringing them together. But the ideas in BD are complex, and I think it’s easy to think you know what something means and then realize you’re off base. I don’t expect to turn out a whole, perfect lesson that any Calculus II teacher could pick up and use, but I hope that my lesson can contribute in some way to the discussion of how to create a BD math lesson for a college class.


 

References

Ellis, A. K. (2001). Teaching, learning, and assessment together: The reflective classroom. Larchmont, NY: Eye on Education.

ISTE: International Society for Technology in Education. (2016). ISTE standards for students. Retrieved from https://www.iste.org/standards/standards/for-students-2016

Mazur, E. (1997). Peer instruction: A user’s manual. New Jersey: Prentice Hall, Inc.

McDermott L. C., Shaffer P. S. (2002). Tutorials in Introductory Physics. Retrieved from https://depts.washington.edu/uwpeg/tutorial

Stewart, J. (2008). Chapter 4.9: Antiderivatives. In Calculus early transcendentals 6th edition (pp. 340-354). Belmont, CA: Thomson Higher Education.

Wiggins, G. P., & McTighe, J. (2005). Understanding by design. Alexandria, VA: Association for Supervision and Curriculum Development.