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 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 DB 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.



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

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

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.

Doing math online (Module 5, Creative communicator, Global collaborator)

Module 5 is about investigating ISTE Student Standard’s 6 and/or 7:

ISTE Student Standard 6: Creative Communicator – students communicate clearly and express themselves creatively for a variety of purposes using the platforms, tools, styles, formats and digital media appropriate to their goals.”

ISTE Student Standard 7: Global Collaborator – “students use digital tools to broaden their perspectives and enrich their learning by collaborating with others and working effectively in teams locally and globally.”

When I read these descriptions and their corresponding indicators, the thing that stood out to me was that students are going to need a good way to talk math in a digital setting. Whether students are connecting with learners from a variety of backgrounds and cultures (indicator 7a), working with peers, experts or community members, to examine issues from multiple viewpoints (indicator 7b), contributing to project teams (indicator 7c), or collaboratively investigating solutions to local and global issues (indicator 7d), students will need to communicate complex math ideas clearly (indicator 6c) and choose appropriate platforms to do so (indicator 6a). This led me right to my investigation question:

What platforms can students use to talk math and do math with each other in an online setting? More specifically, what platforms would be better than typing in chat windows or video chatting and sending pictures of your handwritten work?

The short answer is: I didn’t find an interactive, online tool that I think is obviously better than chat windows/video and pictures, or even a clear winner among the tools I did find. They all have their pros and cons (or glitches). I’ll discuss a few of the tools that I want to keep playing with, but it will hardly even scratch the surface of what’s available. For 60+ online whiteboard-like resources, I will refer you to these two links:

The platforms I’d like to keep playing with are:

  • Ziteboard – an online whiteboard I tried it out after reading Kar Romkodo’s (2016) mini rave review in the comments here.
  • iDroo – a whiteboard with extra things like a function editor and chat window.
What I Wanted

It’s hard to even articulate what I wanted in my online, interactive space. I wanted a smooth feel. I wanted to be able to draw and type. I wanted a function editor so that x^2 + y^2 could look like x^2 + y^2. I wanted it to feel fluid as I transitioned between these functionalities.

As I played with some whiteboards, I decided that chat boxes are still beneficial, and as always, I prefer having sharing and permissions settings. With further play, I found that I want the eraser to erase whole brushstrokes (as opposed to erasing only the pixels that the eraser is covering), and yes, I want to be able to export my whiteboard image… (that’s not always standard?!)

What that Meant – a Stylus

I immediately realized that in order to get what I want, I will need a stylus. Without that, drawing math on an online whiteboard is, to me, an absolute no-go. Trying to draw an integral symbol (\int) or even a simple x with my mouse or touch pad just filled me with anxiety. I would never choose that over sending a picture of written work, which makes it hard to see myself confidently suggesting that a student use a whiteboard that way.

Unfortunately, adding a stylus removes a level of accessibility for these tools. Some college students already have devices with a stylus, but many don’t and it’s not something I’ve ever owned. I decided it was worth purchasing though, because I’ll be starting a math master’s program in the fall, and I will be doing the whole program as a virtual, out-of-state student. However, I don’t want a new computer that supports a stylus. Instead, for about 30-dollars, I found a highly recommended USB device that works on the laptop I have: Huion’s H420 Graphic Tablet. The stylus interacts with a little pad that plugs into a USB port and acts as a fully functioning mouse and then some, with the capability of converting your handwriting into text.

As I hoped, the stylus made a world of difference as I tried to draw math on various whiteboards, but the experience is still not as smooth as I wanted…

What I Found, Generally

Unsurprisingly, on all of the online whiteboards, drawing is not nearly as smooth as it is on a program like MS Paint. In particular, the online whiteboards can’t keep up with fast writing. As long as you are making a continual brushstroke, fast is fine. But the moment you start making multiple brushstrokes, you have to be careful; I can’t write at a normal speed on these sites. In comparison, when I’m on a computer program that supports drawing, like MS Paint, my normal-speed writing is just fine. Here’s an image showing slightly careless, normal-speed writing on both Paint and Ziteboard.

Of course, I could use a little practice to make my writing look nicer, but the difference is obvious. Online, my equal signs regularly look like the L shape you see in the first Ziteboard column, and the second Ziteboard column got really out of hand. This issue makes me want to scrap online whiteboards and instead do a screenshare so I can use a program like Paint, but that takes away the interactive piece. That might be okay, depending on the circumstance, but and interactive board was part of the point.

What I Found, Specifically


In spite of the need to draw slow, there are a lot of great things about Ziteboard.


  • Clean, intuitive interface.
  • Mirror view option: helps everyone on the board see the same section of whiteboard.
  • Laser pointer option: allows others to see where your cursor is.
  • Lock all: locks the current objects on the board, allowing you to erase and clear all without deleting those objects.
  • Add images to your board.
  • There appears to be a Slack integration if you’re using Chrome (details here), which I should look into considering what I loved about! (See this post for my thoughts on Slack.)
  • Text editor.*
  • Others can edit what you have typed. So if you type an equation in a text box and your collaborator wants to copy+paste that equation, they can.*
  • They have a really nice FAQ page here, which I think is worth highlighting.*

The main con is that it’s been glitchy for me on Internet Explorer – but everything is glitchy on IE, so I don’t blame them. (Which, who uses IE?, I know. I use three browsers and IE is among them; it’s the browser I use for anything related to my Digital Education Leadership (DEL) MA program. When an interactive website isn’t functioning properly during our online class meetings, the first questions is always “what browser are you using?” and the answer is nearly always “IE.”) That said…

CONS in Internet Explorer:

  • Navigation button doesn’t work properly.
  • Side panel items work intermittently.
  • “Export whiteboard” won’t export.
  • Once you add text, the editing options don’t function as intended, but you are still able to edit the text.*

None of these things have been issues on Firefox, so I recommend using a different browser. But until I switch all my DEL stuff to a new browser, it’s something I’ll have to work around in IE.

Regarding what I said I wanted in a platform, there’s no typing and* no function editor and I don’t see a chat window, but what it does do, it does well (except in IE) and I think Ziteboard is worth recommending. If I don’t need to use a function editor, this is probably the site I will use for my future online whiteboard needs.

*Edit: Ziteboard saw this article and emailed me a very nice email! They let me know that there is typing, and honestly I should have known because there’s a text-size selector in the same menu as the brushstroke-size selector. To access the typing (on devices with a keyboard) you click to draw a dot, then simply start typing.


iDroo, on the other hand, more of what I was looking for, so in a sense I feel like I can’t complain, but there are a few things I wish it did better.


  • Function editor.
  • Text editor.
  • Chat window.
  • Import images.
  • You can move all items you create (including drawings) and edit any text/function.


  • I can’t see a way to export your whiteboard, which is a big con, I think.
  • Again, IE is glitchy and I can’t copy+paste in iDroo when using IE. Being able to copy+paste is super important to me because after painstakingly entering the math in the function editor, I definitely want to be able to copy+paste it. Thankfully, this doesn’t appear to be an issue in Firefox.
  • Zooming in and out is very hard to control, in both IE and Firefox.
  • The header bar never goes away so the board-space is a tad limited.
  • You can’t edit text or functions created by other board users, which I think is a bummer. I don’t know if this would actually be an issue in practice, but it seems like something I would want to be able to do.  Edit: I’m wrong, you can edit a text or function box that someone else creates. But you can’t edit a text/function box that someone else is currently editing. (I use two accounts to play with these tools and I must have been coincidentally clicking only on text/function boxes that I was already clicked into on my other browser.) Initially I said I didn’t know why I wanted to be able to edit what others had typed, but I realized, as I said earlier, it’s nice because then you can copy+paste the math that others have entered.

iDroo doesn’t feel as smooth as I want, but I’m having a hard time identifying why. It could just be that there’s a learning curve, and with practice, transitioning between functionalities would feel more fluid. Nevertheless, it does actually have a lot of what I was looking for, and I think it’d be worth giving iDroo a shot for online math-collaboration.

Final Thoughts

The Huion Tablet seems like a great product so far. It is working exactly as I had hoped (minus wishing online sites kept up with fast writing) and I’m excited to try it out in a real collaboration. I am expecting that the tablet and interactive whiteboards will be beneficial to me in my math program, which leads me to believe that the stylus + whiteboard combo is worth exploring as a way to meet ISTE Student Standards 6 and 7, Creative Communicator and Global Collaborator. And actually, with the text and/or function editors, Ziteboard and iDroo may not even need a stylus to be good platforms, depending on your needs. Hooray for free tools!

Hmm…I wonder if there exists a keyboard/device that has only math related keys…? That’d be something.


Grech, Matt. (2016). 10 Best Online Whiteboards with Realtime Collaboration [Blog post]. Retrieved from

iDroo. (2017). Retrieved from

ISTE: International Society for Technology in Education. (2016). ISTE standards for students. Retrieved from

Romkodo, K. (2016, May 15). Re: 5 Free Online Whiteboard Tools for Classroom Use [Blog comment]. Retrieved from

Smith, C. (2017). Interactive Whiteboards. Retrieved from

Ziteboard. (2017). Retrieved from

Bunco and MATLAB (Module 4, Computational thinker)

“One,” they say in approximate unison.
“One. Two. Three. …Eight! Nine!”
“One. Two. BUNCO!!!” they scream, with a curious comradery considering they’re playing against each other.
“Nooo!” the other tables tease.
“What was your score? Sweet, I’m the loser!”

Once a month, my extended family gets together to play a game called Bunco. See here for some details; rules vary, and indeed our rules are a little different than what I linked. It’s a dice game. It might sound complicated, but I swear it’s incredibly easy. You take turns rolling and (in our rules) you want to be the first person to reach 21 points; there are specific point values associated with rolling certain things. At most Bunco parties we’ll play through the game six times. That’s a lot of dice rolling. It takes us about three hours. So one might eventually wonder, as I did, how many times, on average, does an individual need to roll to reach 21 points?

There is surely a mathematical solution to my question. You could also brute-force the answer by counting the number of times you had to roll to reach 21 points, over and over and over, and then averaging the results. That would take ages, but if you know how to play the game, you could do it. Or, with a little coding, you could have the computer brute-force the answer in no time at all! (Well, in 2 min and 27 sec, which actually felt like forever.)

This week’s module, Module 4, is about investigating ISTE Student Standard 5: Computational Thinker – “students develop and employ strategies for understanding and solving problems in ways that leverage the power of technological methods to develop and test solutions.” In response to Computational Thinker Indicator 5b, “students understand how automation works and use algorithmic thinking to develop a sequence of steps to create and test automated solutions,” I asked the investigation question:

What resources or programming tools are there that would be appropriate for students who have not previously done any programming? Maybe some good beginners tutorials for MATLAB, or something to teach the ideas used in programming (like vectors, for loops, if/else statements).

After poking around the internet and thinking about the setting of a college math class, I decided that I really did want to take this opportunity to look for a good beginner’s resource for MATLAB. MATLAB doesn’t have to be used as an advanced tool, and if you think of it as simply being a different (but epic) calculator, then I see no reason why beginning programmers shouldn’t be introduced to MATLAB. And furthermore, if you pursue math you’ll surely be introduced to it eventually.

MATLAB Resources

My two favorite resources that I found are:

Mikhelson’s video tutorials have the main things that I was looking for:

  • Zooming in on computer screen so you don’t have to squint.
  • Short-ish videos. I’m not really looking for entire hour-long, lecture-style lessons, just quick videos that give enough to know some basics.
  • Few enough videos, and covering the topics that I would want my students to know. Again, not looking for an entire course, I just want some good basics. The main topics I had in mind are: variable declaration, vectors, matrices, for loops, while loops, and if/else statements. But I like the other topics he included.
  • A slow enough speaking tempo.
  • Easy to follow visually.

I mostly just skimmed Xenophontos’ PDF, and I liked what I saw. It had a nice tone and layout. I found it easy to look at. Lots of good basics; more than the video tutorials, but not an exhaustive MATLAB manual. I think it could pair well with video tutorials as a reference.

Connecting MATLAB to ISTE Student Standard: Computational Thinker

Learning and using MATLAB easily touches on Computational Thinker Indicator 5b. For loops and while loops are two basic ways of programming the computer do repetitive tasks for so many iterations, or while some condition has yet to be met. When you write a script (a.k.a: code, program), you are writing a sequence of ordered steps. Coding in MATLAB or any other program is a manifestation of computational thinking.

But What Does This Have to do With Bunco?

Recall my game question: how many times, on average, does an individual need to roll to reach 21 points? This is a perfect question for MATLAB and I think a great example of using programming to answer a real-life question. The answer, by the way, is approximately 32.67 rolls. In order to program MATLAB to “roll the dice,” count how many times it took to reach 21, and then average the results, all I needed was: variable declaration, a vector, a for loop, a while loop, a few if/else statements, and two other commands – one to generate a random integer between 1 and 6, and one two average the number of rolls. Aside from the two other commands, all of these things are covered by Mikhelson’s tutorials in about 35 min.

It’s very freeing and rewarding to be able to answer your own questions, and a program like MATLAB opens you up to a new set of questions you can answer. It gives students a tool they can leverage while being a computational thinker. In addition to answering real questions, another reason a program like MATLAB is great tool to have at your disposal is because of what processes it can automate for you. For example, one time I was creating tons of bar graphs, which required calculating dozens and dozens of percentages. The tediousness and repetition of the task was making me cry on the inside. In comes MATLAB to save the day! Write a little code; copy and paste some tables; change a few numbers every now and then. Bam! Tables complete. I wanted to cry tears of joy over how much time MATLAB had saved me.

When I read Computational Thinker Indicator 5b, “students understand how automation works and use algorithmic thinking to develop a sequence of steps to create and test automated solutions,” the first thing I think of is MATLAB. It is a tool that enables you to embody this indicator. And as much as MATLAB can do, knowing even just a few basics can add such a powerful tool to your technology-toolbox. It’s a tool I want people, and my future students, to have access to.

(By access I meant the knowledge to use it, but speaking of access, Octave is the free “equivalent” to MATLAB; nearly all of its commands are identical. And typically, colleges will give students a discount on MATLAB and/or have MATLAB available for use on the school computers.)


ISTE: International Society for Technology in Education. (2016). ISTE standards for students. Retrieved from

GNU Octave. (2017). Retrieved from

MathWorks. (2017). MATLAB. Retrieved from

Mikhelson, I. (2014, March). MATLAB tutorials. Retrieved from

Xenophontos, C. A beginner’s guide to MATLAB. Department of Mathematical Science, Loyola College. Retrieved from – Communication management (Module 3, Innovative designer)

Module 3 is about investigating ISTE Student Standard 4: Innovative Designer – “students use a variety of technologies within a design process to identify and solve problems by creating new, useful or imaginative solutions.” In response to Innovative Designer Indicator 4b, “students select and use digital tools to plan and manage a design process that considers design constraints and calculated risks,” where digital tools are defined as “brainstorming tools, flow charts, drawing or mark-up tools, 2D or 3D design software, note-taking tools, project-management tools,” I asked the investigation question:

What project management tools are there?

My investigation question is fairly narrow, and my answer is even narrower…but I am so excited about it! At the suggestion of program director, Dr. Wicks, I looked into which is a communication management tool. It’s basically a place to create discussion channels (i.e., chat rooms) for group/project/team members. And these channels are only available through invite – the world doesn’t have access to them.

When it comes to group communication, I’m a huge fan of FB, and until Slack, nothing else has been able to compete. But as a student, I have observed that students tend to keep their FB activity fairly separate from their school activities, and some students don’t want to be on FB. In my experience, communication between peers often happens through texts, emails, comments within Google Docs, and LMSs like Blackboard and Canvas. Rather than go through what I think the disadvantages of these platforms are for group communication, I’ll jump right into…

What I love about Slack for online, group communication (from 5 days of playing with it):
  • Notifications and tagging – two communication features which I think are an absolute must if you want to foster a sense of community or “teamness” during online interactions. The notifications are super customizable, and I’ll just note that you can even tell Slack to notify you when a specific word or phrase is said! What?! Love it.
  • The apps – I’ll mention this right now because notifications are most effective, I think, when they come from an app. There are desktop and phone apps; I have both.
  • Threaded comments – if notifications and tagging are a first-tier must, threaded comments is a second-tier must. It’s just super necessary for discussion organization.
  • Edit your comments – in a school setting, when I find a typo in my post/comment and can’t edit it, it drives me bonkers.
  • /remind – you can set a reminder for yourself…OR someone else! Even more conveniently, with the click of a button you can also have Slack remind you about a specific message.
  • “Apps and integrations” – Not to be confused with the Slack apps themselves, there are tons of things that you can add to your Slack group, like polls (I recommend Polly), RSS feeds (I recommend RSS), dice rolling, and calendars.
  • Other convenient features: star messages, see all starred messages, see all things you were tagged in, private messages, multiple channels, and search discussions.

Slack is super easy learn and very intuitive – getting started is a breeze. There is a lot to explore and Slack just keeps surprising me with cool things!

Regarding any downsides or limitations (that I’ve seen within these 5 days):
  • I wish the apps and integrations had user ratings. I sent them /feedback about that from within Slack (super cool feature) and they quickly got back to me – they’re working on how to do that well.
  • Slack uses a non-trivial amount of computer memory in order to run (~375,000K for desktop app; ~475,000K in Chrome), but what can you do? So does running FB (~501,000K in Firefox).
  • There are free and paid versions of Slack. Limitations on the free version are: search limited to the 10,000 most recent messages, storage capacity limited to 5GB, only 10 apps or integrations, and voice and video calls limited to two people. For a list of differences based on version look here.

The limitations don’t deter me – they are what they are and something to keep in mind. If Slack were integrated into a 10 week college course, a class of 30 students could go past 10,000 messages if they’re very active. They’d have to average over 33 messages per student, per week. Regarding storage, 5gigs is quite a bit of space. That’s about 16 hours of some .mp4 video files I have, twice as much space as I have on my free Dropbox account, and way more space than the 100MB of storage that Google Sites gives you! 10 apps/integrations is quite a bit, and there are other ways to do free group voice/video calls.

Tying Together Slack and ISTE Student Standard 4d

Aside from using Slack to manage project communication, the apps and integrations (which I will now just refer to as “apps”) within Slack seem to help corral other design and project management tools by keeping extra tools linked in one place. However, it can take some time to find apps that are actually free, so if I were suggesting Slack to my future students, I think it would be nice to give them a head start with a variety of suggested apps.

quickly browsed the app store, looking for apps that could support collaboration and the goals of ISTE Student Standard 4d. Below are some functionalities I saw, with apps I feel confident suggesting:

  • Create diagrams (It has some limitations, but I checked out Lucidchart enough to know it is free. You can make flow charts, Venn diagrams, etc.)
  • Project management (Trello and Trello Alerts, which is free)
  • To-do lists (To-do bot)
  • File management (Dropbox, Google Drive)
  • Calendars (Google Calendar)

Between apps like these and Slack itself, I can see ways for students to brainstorm, make flow charts, take notes, and track project tasks, timelines, expenditures, materials, and potential obstacles. I think this makes for a pretty nice set of tools. Or at least it does in theory. I have yet to try it all out with a real project, so I can’t speak from experience, but I’d love to give it a go!

Using/Moving Over to Slack

Getting a group of people to use a new communication product can be a little challenging. It’s another thing to log into and check, another program to download and run, another app to put on your phone. Another thing to learn. But to me it seems worth it. Among other groups that could benefit from Slack, I think it could be a great tool to help students manage project communication as they work on their innovative designs.


ISTE: International Society for Technology in Education. (2016). ISTE standards for students. Retrieved from

Lucidchart. (2017). Help center: Account types. Retrieved from

Slack. (2017). App directory. Retrieved from

Slack. (2017). Pricing guide. Retrieved from

Forums, curation, and mathematics (Module 2, Knowledge constructor)

Module 2 is about investigating ISTE Student Standard 3: Knowledge Constructor – “students critically curate a variety of resources using digital tools to construct knowledge, produce creative artifacts and make meaningful learning experiences for themselves and others.” Looking at the indicators generated a lot of questions for me about how I can foster these things in a college math class. I want to document all of my questions here for future reference, but I really investigated just two of them (in red text).

In response to indicator 3a, “students plan and employ effective research strategies to locate information and other resources for their intellectual or creative pursuits,” I asked:

In physics education research, there is a need to study how students use their textbooks (Docktor & Mestre, 2014, p. 22) and while there are some research-based textbooks, most courses do not use them (p. 21). So I’m curious if there is research on how math students use their textbooks. Are there any research-based math textbooks? What resources do students use when they have math questions – what do they do when they are stuck – what strategies do they use to get unstuck?

In response to indicator 3c, “students curate [i.e., to gather, select and categorize resources into themes in ways that are coherent and shareable] information from digital resources using a variety of tools and methods to create collections of artifacts that demonstrate meaningful connections or conclusions,” I asked:

Can I find a place where students are sharing resources in a coherent way? (Places to look: Reddit – YouTube – FB groups.) Can my website host a forum for students to share their resources? Can I use Facebook groups as part of the course? Can I require college students to participate in a FB group? (Should I?)

In response to indicator 3d, “students build knowledge by actively exploring real-world issues and problems, developing ideas and theories and pursuing answers and solutions,” I asked

I would like to see some examples of students exploring real-world issues in math. Can I find some real-world-related final projects? Or can I find some examples of inquiry-based math?

I decided to look into creating a forum on my website and using Facebook groups in college courses because I wanted to make sure I could offer my students a space for sharing class-related things with each other (and I’m not a huge fan of the LMSs that I’ve used – as a student – when it comes to sharing resources and communicating).

Question: Can my website host a forum for students to share their resources?


WordPress has a variety of plugins for this. I installed “Forum – wpForo” (more information about the plugin can be found on their website, here). I am fairly happy with the forum. It has the main thing I want, which is threaded comments. Implementing the threaded comments theme was a little confusing (directions below), but otherwise installation was very easy. There are a few things I don’t love about the layout/display of the forums. For example, if you choose to “Answer” a post, you will add a normal comment, if you choose to “Add comment” you will reply in a threaded fashion – I wish they were called “comment” and “reply.” But overall, I’m pretty happy with the plugin.

(Version 1.1.1) To implement the threaded comments theme go to: Dashboard side panel > Forums > Forums > (click edit on the blue category) > (in the upper, right-hand box choose the “QA” category layout).

Question: Can I use Facebook groups as part of a college course? Can I require students to participate in a FB group? (But should I?)

Yes. Yes. And…no?

I’m sure it depends on the college, but from the looks of it, generally you can use FB groups in college courses, and it looks like some instructors do require FB participation. I found a great blog by Nisha Malhotra, PhD where she reflects on implementing FB in her course. The comments on the blog are also very insightful and show differing opinions on whether or not you should require FB participation.

Considering this blog was posted four years ago, I would like to find a similar resource but more current. A lot has changed in four years and I have a feeling students’ feelings about FB have changed. Indeed, it was just this last year that I heard for the first time, from a high school student, that FB is for old people! Who knew?! I would bet there are more people consciously abstaining from FB today than there were four years ago. (Not just because it’s “for old people,” but probably because of that too.)

While I really like the idea of a FB group for a class, I don’t think I could bring myself to require FB participation in a course. Based what I think FB can mean in our culture today, I think it is important to respect a student’s choice to not use FB. This is one reason I really wanted to look into putting a forum on my website. Then I could offer both as an option for online participation.

But is this really curation, or is it just collection?

By the end of this module, I decided that what I have really done is found resources that aid in sharing curations, rather than resources for curating. A classmate of mine found a wonderful blog about curating by Saga Briggs. I think Briggs paints a clear picture of what curation looks like and I now imagine curation as being able to say, “Here are some resources that I think are valuable, and here’s why I think they’re valuable together.” A forum or a FB group could be used in that way, but I think it would require prompting if the goal was to have every student curate resources. Additionally, Briggs includes a list of 20 resources for curating.

Possible Curation Assignment for Math

A while back I wrote a possible prompt that is more in line with collection. It needs to be adjusted to align with curation.

Initial prompt: Find a resource that helps you with something related to the course. Maybe identify something you struggle with and find a resource for that. Or maybe find a resource that helped you understand a topic better or helped you with a homework problem. Write a summary explaining what the resource is with the idea that you are helping someone decide if the resource would be valuable to them. Be sure to reflect on why it was helpful to you.

To turn this into a curation project, they could either share multiple resources that helped them with something and include in the summary why the resources are helpful together; or they could find additional resources after the fact to go with their “personally helpful resource.” The goal would be to create a “resource bundle” to help someone else with the same thing/topic/problem they needed help with. They could share this bundle to my forum or in a FB group.

I anticipate needing to help students learn how to find resources, but I also hope that they can learn from each other, and that this assignment could help them do that. O’Connor and Sharkey (2013) and Kingsley and Tancock (2014) both discuss how students struggle to find information when it requires digging, and during much of my undergrad that was definitely true for me. Somewhere in the beginning to middle of undergrad, I realized how unskilled I was at searching for information and using my textbooks. I realized this because I saw how my close friends/peers used their resources. They didn’t actively teach me how to do the same, but I began learning how to use my resources by watching them. I know what it’s like to not know how to search for information, and I know how valuable the skill is when you can.

Moving forward, I would like to check out the resources listed in Briggs’ blog and practice using them to get a better feel for the process of curating (as opposed to collecting).


Briggs, S. (2016). Teaching content curation and 20 resources to help you do it [blog]. Retrieved from

Docktor, J. L., & Mestre, J. P. (2014). Synthesis of discipline-based education research in physics. Physical Review Special Topics-Physics Education Research, 10(2), 020119, 1-58.

ISTE: International Society for Technology in Education. (2016). ISTE standards for students. Retrieved from

Kingsley, T., & Tancock, S. (2014). Internet inquiry. The Reading Teacher, 67(5), 389-399.

Malhotra, N. (2013). Experimenting with Facebook in the college classroom. Retrieved from

O’Connor, L., & Sharkey, J. (2013). Establishing twenty-first-century information fluency. Reference & User Services Quarterly, 53(1), 33–39.

wpForo. (2016). WordPress forum plugin. Retrieved from

Sanity checking in mathematics (EDTC 6102 Module 1, Empowered learner)

For this module we are investigating ISTE Student Standard 1: Empowered Learner – “students leverage technology to take an active role in choosing, achieving and demonstrating competency in their learning goals, informed by the learning sciences.” In response to Empowered Learner Indicator 1c, “students use technology to seek feedback that informs and improves their practice and to demonstrate their learning in a variety of ways,” I asked the investigation questions:

What are some methods for sanity checking in mathematics, examples of a time when it was needed, and how do you teach this skill? How can students demonstrate sanity checking?

My own working definition of sanity checking is: the act of using tools, techniques, and information to answer the question “Does this even make sense?” This definition is very general and not math specific, though I will focus on its application in math. Sanity checking can be used while solving problems, or problem solving, and to check a final answer.

I felt sanity checking was related to ISTE 1: Empowered Learner because when a student spontaneously sanity checks their own work, they are taking an active role in their learning and they are seeking some type of feedback to inform what they are doing. Tools such as calculators, mathematics software, and Wolfram Alpha can be creatively utilized to get feedback/aid in sanity checking, and of course there’s always Google. I figure that while they may have not articulated their own learning goals, they are implicitly demonstrating an achievement-related goal, whether it’s to achieve sense-making or achieve the right answer.

In my experience, sanity checking was not something that was ever explicitly taught, in spite of its value as a skill and way of thinking. This led me to wonder what kinds of educational tools and research exist related to teaching sanity checking. To my surprise, I found exactly zero published journal articles related to the combined key words “sanity checking” and “mathematics education” (and related searches, like “sanity testing”). This makes me wonder if sanity checking goes by different name in research. Turning to Google, I found a few websites and books with some relevant information.

Q: What are some methods for sanity checking in mathematics?

Finding a list of methods was more challenging than I imagined. But from a few resources, I have put together the start of a list:

  • Estimation: For example, you can use estimation to check that an answer is reasonable (Petrilli, 2014).
  • Plug in numbers: If two things should be equal, are they in fact equal when you plug in a random number (Wood, 2015)?
  • Comparing against external information: Suppose you know that a penny weighs about 3 grams, and you calculate that an adult weighs 6 grams (i.e., two pennies). Compared to the external information, this answer does not seems reasonable. Wood (2015) and Yaqoob (2011, p. 33) mention related things.
  • Do the units make sense? (Wood, 2015) or Dimensional analysis: If you’re calculating a velocity (meters/second), but end up with kilograms*meters/second, something went wrong. More generally, you can often use the “fundamental dimension” like length, time, and mass instead of meters, seconds, and kilograms to sanity check.
  • Definition/rule/fact based: For example, checking that the hypotenuse is the longest side of a right triangle, since it always is. Fenner (2013) gives many examples of this kind of sanity checking.
Q: What are some examples of a time when it was needed?

We have to show students that sanity checking can be a meaningful activity but I did not find any information on how to do this. So I would love to know some real examples of when sanity checking was needed or valuable. More specifically, I would love to know about the experiences students have had where they found value in sanity checking. Perhaps this could be an interesting assignment – have them turn in a reflection at some point throughout the quarter explaining a time when sanity checking helped them while working on the coursework.

Qs: How do you teach this skill? How can students demonstrate sanity checking?

I did not find any information on these questions either. Of course, demonstrating real-time sanity checking while working with students can help teach the skill by exposure. Additionally, let students see your “backstage performance” (Olitsky, 2007) to show them the struggles you have and how sanity checking is helping you.

One of the difficulties with demonstrating sanity checking is that sometimes it leads you along a messy path. You might erase little things here and there. You might scrap the whole page and start over. I’m not sure that turning in a homework set including all those changes would be easy or very valuable (maybe it would), but this is one reason I like the idea of a reflection assignment that has them explain one meaningful sanity checking experience from the quarter. I will also keep thinking about ways that students could utilize technology to share their sanity checking stories or resources with each other.

Moving Forward

Sanity checking is a valuable skill in mathematics, and teaching sanity checking can help students become self-directed learners, which Kivunja (2014) considers to be an important 21st century skill. Considering the value of sanity checking, I would like to find some more resources on the topic, for example best practices to teach it or how students utilize sanity checking. I will keep my eyes out for synonyms that might lead me to the vein of research on this topic, and if none exists, then perhaps this is something I would like to research in the future.


Fenner, S. A. (2013). Basic mathematics for engineers (8th Ed.). Lulu Press, Inc. (link)

ISTE: International Society for Technology in Education. (2016). 1: Empowered learner. ISTE standards for students. Retrieved from

Kivunja, C. (2014). Teaching students to learn and to work well with 21st century skills: Unpacking the career and life skills domain of the new learning paradigm. International Journal of Higher Education, 4(1), p1. Retrieved from

Olitsky, S. (2007). Facilitating identity formation, group membership, and learning in science classrooms: What can be learned from out‐of‐field teaching in an urban school? Science Education, 91(2), 201-221.

Petrilli, M. J. (2014). The Common Core sanity check of the day: Estimation is not a fuzzy math skill. Retrieved from

Wood, B. (2015). Sanity checking. Retrieved from

Yaqoob, T. (2011). What can I do to help my child with math when I don’t know any myself? Baltimore, MD: New Earth Labs. (link)