Category: What Do My Students Really Know?

A Math Essay

In April, I asked my Accelerated Algebra 1 students to write an essay. The prompt was simple:

Compare linear and exponential functions. Explain the essential characteristics of each, and discuss what is the same and what is different about them.

I’d thought about having students write an essay in the past, and one of my colleagues had assessed her students using a similar prompt a few years back. Assessment wasn’t really my goal, though. With months of work with linear functions under their belts, my students could calculate slope, write equations for lines (in multiple forms!), determine where two lines intersect, and use linear functions creatively (Desmos art!) and practically (modeling-type problems). But when I asked the class one day about the key feature of a linear function, they didn’t have a great answer for me. Their answers were okay – “it’s a line!”, “y=mx+b“, “there’s a slope” – just not quite what I was looking for. The last response came closest, but my follow-up question of “What does it mean for a linear function to have a slope?” didn’t elicit the sort of responses I wanted.

A linear function has a constant rate of change.

Eventually, someone said that, and I probably shouted in joy. You might prefer to describe the essence of linear functions differently, but for my purposes, “constant rate of change” is the key phrase. And it’s one my students knew. They just didn’t think about it as much as they thought about all the procedural stuff. So I said to them:

More than anything else, I need you to finish this course knowing that a linear function has a constant rate of change.

Sometimes, I think, we get hung up on all of the skills and techniques and nitty-gritty details, so hung up that we forget about the essence of our content. It’s nice to be able to write an equation for a linear function, but in my eyes, it’s crucial to be able to know that a linear function can model a phenomenon with a constant rate of change. I had to make sure they owned this.

I decided an essay would help me achieve this goal for two reasons. First, an essay would force students to refine their thinking and work toward understanding and explaining key concepts. They couldn’t hide behind procedural fluency. Second, because a math essay seemed so strange, students would approach it differently than they approached other assignments. Writing involves skills and thought processes that we don’t often use in math class, and I felt like writing about math would help students gain a new perspective on the content.

I wish I could say that my students loved the idea of writing a math essay, but they didn’t. Students expressed a mixture of confusion and anxiety. To their credit, however, they complained very little. And for the most part, they completed their essays on time.

Here are some guidelines I offered:

  • Strive to make your writing feel like an essay. Don’t just give me a list!
  • One paragraph can be enough. You may write more, but you don’t need to.
  • Be sure to use mathematical vocabulary.
  • Use examples judiciously. I want to read about linear and exponential functions in general, not just about a specific linear or exponential function.
  • Feel free to supplement your writing with equations, tables, graphs, and images.

Just for good measure, I also provided an incredibly vague rubric.

A: Beautiful essay that demonstrates exceptionally deep understanding of linear and exponential functions and moves MC to tears.

B: Solid essay that demonstrates strong understanding of linear and exponential functions and makes MC feel pretty good about life.

C: Okay essay that demonstrates decent understanding of linear and exponential functions but that has some weaknesses that leave MC feeling unsatisfied.

D: Inadequate essay that demonstrates little understanding of linear and exponential functions and makes MC wonder how much you’ve really learned.

F: Complete lack of an essay leaves MC curled up in a ball in the corner of his classroom. Please don’t let this happen!

Then, I waited. A few students asked for advice, and I read through several essays so that students would know if their essays met my expectations. Interestingly, several students asked the English Language Arts teacher on my team to look over their essays. Although I was a little apprehensive about how the essays would turn out, I mostly felt good about the thought and the effort that I saw.

The deadline passed, and it was time to read the essays. Here are some excerpts – some great, some not great – that I found interesting. All spelling and grammar has been preserved from the originals.

When you deal with a linear function you add the same number each time, but an exponential function multiplies by the same number each time instead of adding.


In an exponential function the line is curved meaning that the rate is not constantly the same.


But from the beginning you can tell that they will be different from “line” and “exponent” in their names.


Another major difference is that exponential functions never ever, reach zero, this is called an asymptote.


An exponential function is a function that increases at a constant rate raised to a power. It is important to know that, in an exponential function the independent variable is the exponent and the base was consistent.


Instead of having a constant rate of change the exponential function changes by a common ratio.


In a linear function the line touches 0 but in a exponential function the line never touches 0, it’s an asymptote!


Because the rate of change is not constant exponential function are able to increase/decrease faster than linear functions do


Exponential will have a slight curve in it and will eventually get super steep.


linear functions unlike exponential functions have x intercepts, due to the fact that exponential functions are unable to ever reach zero because the amount will keep getting cut in half.


Another similarity is that both of the domains are all real numbers for both functions.


The main difference between these two types of lines is that exponential lines slope is increasing by a certain percentage each time whereas linear lines have constant slopes.


linear functions are arithmetic (adding the same number each time) and exponential functions are geometric (multiplying the same number each time).


Another similarity is that both of the functions intercept the y-axis and the x-axis. Likewise, Linear functions and Exponential functions both are functions which for each x-axis, there is exactly only one y-axis number.


exponential functions have an asymptote making it appear like there is an x-intercept but it does not meet it at a finite distance.


On the other hand, in a linear equation, there is never an exponent in the function.


A linear function is essentially adding your slope many times, and an exponential function is essentially multiplying your slope.


an Exponential function is more like a hill that gets steeper and steeper as it gets taller in length (so basically a curve).


In the long run exponential equations will always outpace linear growth.

Okay, I might have gone overboard in quoting from my students’ essays, but there’s a ton to think about here! I learned so much about what my students really understood. One particularly interesting misconception involved quadratic functions. We’d moved onto quadratics by the time the essay was due, and I noticed that several students confused exponential and quadratic functions. In one or two cases, everything non-linear was considered exponential. Far from being discouraged by these (and other) misconceptions, however, I found myself empowered to help students develop more robust understandings of the material. Maybe it’s easier to correct procedural mistakes, but it feels so much more meaningful to help students better differentiate between two related concepts and to refine their explanations of the similarities and differences.

So, I think the essay worked. I learned more about what my students knew. My students learned more about linear and exponential functions, and they got an opportunity to engage in mathematical discourse using a different medium. This was a powerful experience for me. I’m toying with the idea of asking my students to write one essay each quarter. Perhaps that’s a bit much, but if I can develop meaningful prompts, then why not do it? I really think writing in math class offers some great possibilities for enhancing learning, and if nothing else, it will allow for the creativity and humor seen in amazing quotes like these:

Have you ever heard of a linear or exponential functions? Well if you haven’t what are you doing with your life?!?!?!


I will preface this paragraph with a disclaimer: truthfully, there are not many similarities between linear and exponential functions, but regardless I will present those that I have knowledge of.


Functions are mathematical concepts that are a mitochondria for the algebra cell.

An Assessment on Desmos

An Assessment on Desmos

I gave my two Algebra classes an assessment on Desmos covering linear functions. The idea started as a joke – I told a student who had found Marbleslides challenging that I’d put some on the test. As I thought about it more, though, I decided that assessing with Desmos would be a great idea.

I used Activity Builder to create the assessment. The easy-to-use, intuitive interface made creating the assessment fairly easy, but I encountered a major unexpected challenge. Designing worthwhile questions proved much more difficult. It no longer made sense to ask students to simply graph or write the equation of a line. Instead, I focused on questions that ask students to describe how to graph a line, to explain why an equation’s graph would look a certain way, and to interpret a line’s equation in the context of a problem. This is actually the type of question I always want to emphasize but rarely do.

Desmos Assessment 3

Consider the question above. If I had asked students to simply graph the line given by the equation x=5, they might have been able to do so without truly understanding the equation for a vertical line, and I never would have known.

Desmos Assessment 10

Similarly, this word problem went beyond simply asking students to write an equation to making them connect the mathematics to the situation being modeled. Each of these five students wrote a correct equation, but their understandings of the problem clearly differ. So too does their ability to express their reasoning, something seen in the following example also.

Desmos Assessment 9

Do these students understand the relationship between the graph of a line and its equation? To an extent, they certainly do, but their explanations also reveal some gaps in their understanding. What I find most interesting is how students managed to express their thinking in so many ways. Some used mathematical vocabulary; others didn’t. Some provided precise explanations that anyone could follow; others used ambiguous language that might obscure their meaning. For as much as I think I emphasize communication in my classroom, my students’ responses make me want to spend even more time refining our ability to share our thinking in a clear, concise manner. Perhaps including more problems that call for explanations on each assessment will help me move in that direction.

And that’s probably my favorite part of using Desmos for an assessment. It’s so much easier for students to explain themselves on the computer than it is with pencil and paper. Consider the following responses.

Desmos Assessment 5Desmos Assessment 6Desmos Assessment 7

I know my students, and I can say with complete certainty that they would have written much less on a paper-and-pencil test. And I would have missed out on seeing and understanding their thinking. Between this problem and the one shown in the image at the top, I developed a clear picture of what my students know and don’t know about y-intercepts, something that may not have been possible the way I typically assess.

And, of course, Marbleslides. The incomparable joy of Marbleslides.

Desmos Assessment 4Desmos Assessment 11

I don’t see a lot of students absolutely beaming during tests, but I did this time. That student I mentioned earlier – the one who found Marbleslides so challenging – successfully collected all of the stars on this assessment, and she was so incredibly happy. Seeing her smile made the entire assessment worthwhile.

I suppose it’s worth discussing the nuts and bolts. Grading wasn’t really easier or harder than a pencil-and-paper assessment. It was just different. Take a look at the dashboard below.

Desmos Assessment 1

It’s easy enough to grade a question when a student gets a check, but everything else required me to take a closer look. Sometimes, as with the following question, that was pretty easy to do.

Desmos Assessment 8

I can quickly glance through student responses and get a sense of common misconceptions. But with questions that require an explanation (or an input that doesn’t get verified), I have to take the time to look through everyone’s individual work. And that’s totally fine. That’s what grading is usually like, and I think it’s important to see and assess each student individually. Desmos actually made it easier to do this.

As far as actually tallying scores and providing feedback, I had to improvise. I used Google Sheets to create a little rubric. I included a place for a numerical score and a place for brief comments on the individual problems. I also let my brain rest and made Sheets calculate the grades for me. Here are some examples.

Desmos Assessment 12Desmos Assessment 13Desmos Assessment 14Desmos Assessment 15

I printed these little rubrics and returned them to students. Then, I un-paused the activity and allowed students to look back at their work and correct it if they so desired.

Other miscellaneous thoughts:

  • The Ohio AIR test uses the Desmos graphing calculator, so this sort of assessment should help my students prepare. It’s also easy to create AIR-type questions using Desmos.
  • There isn’t really a way for students to “turn in” the assessment. I just told them to close the tab and shut down their Chromebook when they finished. This is totally fine; it’s just something I had to tell them about a hundred times.
  • It’s relatively challenging to monitor students to make sure they’re not just using Google to search “how to write a linear equation” or using Discord to ask each other questions. I emphasized honesty and integrity at the beginning, and that seemed to do the trick.

If you’re wondering if I’d give another assessment using Desmos, the answer is a resounding yes. I’m actually designing two more assessments (one for Algebra, one for Math 8) right now. And my colleagues have agreed to try using Desmos for one of their assessments!

Thank you to Desmos for being awesome! Thank you to Julie Reulbach and Jonathan Claydon for introducing me to the idea of Desmos assessments! Thank you to my students for making my job wonderful!

Update: Wow! This post received quite a response on Twitter! Here’s the link for anyone interested: https://teacher.desmos.com/activitybuilder/custom/5bc52d70744e4b427f3ce5a6

What Do My Students Really Know? Part 2

During the second semester, I only assigned homework using Delta Math. This worked well for several reasons, with the most important one being that it gave me better insight into what my students actually understood than pencil-and-paper homework ever did.

The Delta Math homework I assigned surprised me right off the bat. The image below displays one student’s results on the first assignment of the second semester. Many students had similar results.

HW 1 Linear 1

This assignment consisted entirely of review material. We had assessed linear functions at least once in December, and for the most part, students did fine. I knew that students would be a bit rusty after Winter Break. That’s why we completed this assignment. The surprise came from the misconceptions I saw in student responses.

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Although I could only see their answers, I found it relatively easy to determine their misconceptions. Unsurprisingly, I saw all of the classic slope mistakes.

  • Writing slope as an ordered pair instead of a ratio
  • Subtraction errors, especially with negative numbers
  • Putting the change in x in the numerator instead of the denominator
  • Mismatching x and y
  • Forgetting to simplify (not exactly a mistake, but Delta Math marks this wrong)
  • Misinterpreting \frac{4}{0} as either 4 or as 0

These aren’t new mistakes. I’ve seen them many times, and we’ve corrected them together many times. We said “vertical divided by horizontal” at least ten times each class period every day for a month. And the students got this! Like I said, students did fine when we assessed linear functions in December.

So why did so many students continue to display these misconceptions? I’m not naive. I realize that students master the material at different rates. But how could so many of them who had mastered the material make these mistakes? Why did they return to misconceptions that they had overcome a month earlier?

One explanation is that students managed to know the material well enough to pass an assessment, but they did not develop the robust understanding necessary to maintain their skills even a month later. That’s the explanation I originally subscribed to, but I think it lacks something important. Many of my students did understand slope. Throughout the fall, they worked hard to connect proportional relationships, steepness, and lines. They built a solid conceptual foundation. They thought mathematically. They solved problems. I think what they lacked – and this is on me – was a proper emphasis on procedural fluency. Sure, we had plenty of opportunities to practice and to develop that fluency, but rarely did they have the “Uh oh, I’m wrong!” moments that Delta Math gave them. My Delta Math assignments required students to get a certain number of problems correct to get credit. You just can’t fake a right answer. That’s what I like about Delta Math – it holds them accountable.

The other major advantage of Delta Math over pencil-and-paper homework? I can look through as many assignments as I want to in a relatively short time period without having to carry papers around. I have a bit of a problem with keeping papers organized, so moving homework online and avoiding paper altogether saves me some serious time and energy. The obvious drawback is that I cannot see the work that the students did. In the examples above, it’s relatively straightforward to identify misconceptions without seeing the student’s work, but that certainly won’t always be the case. I think, though, that simply being able to see that a student struggled with a problem type may be enough, especially given that some sort of intervention would need to take place anyway.

I’m not sure yet what homework will look like in my classes this year, but it seems like I’ll want to place more emphasis on procedural fluency. Perhaps such an emphasis earlier in the year will lead to better understanding all year long. I hope 2018-19 is the year I finally figure out how to make homework work for me and for my students!

What Do My Students Really Know? Part 1

What Do My Students Really Know? Part 1

My two Accelerated Math 1 classes have been working with sequences. They have completed some investigation problems to develop vocabulary and to grapple with the characteristics of different types of sequences. We have built definitions of explicit and recursive formulas as well as of arithmetic and geometric sequences, and as a class, we came up with explicit formulas for both of these types of sequences. Their work has impressed me, especially given that we’ve done all of this using function notation, which is totally new to everyone. Students don’t even seem particularly troubled by some of the more challenging problems (such as figuring out missing terms when given two specific terms or determining the term number for a far-off term) that have frustrated some of my past classes. I know that they need more practice, but overall, I’m pleased with their progress.

For practice, we completed a matching activity in which students had a card with either the formula for a sequence or the sequence itself. They had to find the student with the matching card. I designed the sequences to be fairly similar so that students really had to think about which sequence went with which formula. It wasn’t enough, for example, to notice that the first term of the sequence matched up with the formula because multiple sequences had the same first term.  My students handled this activity pretty easily.

I wanted to try something similar with Desmos, so I made this Polygraph activity. It features eight numerical sequences and eight explicit formulas. I expected students to ask questions like: “Is your sequence arithmetic?”; “Is the common difference positive?”; and “Is the first term negative?” Although I certainly saw questions like those, I noticed that many students used imprecise language and pursued a line of questioning that lacked a logical progression meant to eliminate possibilities. Their work surprised me and has made me wonder what they have really learned. Here is some of their work along with my commentary.

1

The questions about negative numbers make sense on the surface, but all decreasing arithmetic sequences will have negative numbers eventually. Is it important that this student was referring to the first three terms without saying so? Had I been the picker, I may well have answered “yes” to that question even if my sequence only displayed one negative number. Substantial mathematical background allows me to focus on this sort of nuance. How do I help students refine their language to remove the ambiguity that I see? Or am I seeing an issue where one really does not exist?

2

Does the formula for a sequence count as a sequence? Should that first question be worded as “a numerical sequence” to avoid confusion? Would n-1 count as having a negative? Does using “over” instead of “greater than” present a problem?

3

Although I’m happy to see “arithmetic” used correctly, the word “decimals” worries me. I’m not sure that “decimals” has a firm definition in my students’ minds. These students seem to be referring to numbers containing a decimal point, which makes sense, but I wonder what they think of 3.2\times 10^{3}. Does that count as a decimal? Perhaps a better question would ask about non-integers.

4

Sure, these questions get the job done with minimal ambiguity, but I’m not a huge fan of the final two questions. Without a need for vocabulary like “common ratio,” I suppose it makes sense to ask these questions. Maybe I need to force students into choosing between two geometric sequences with the same first term. But would students then simply ask about “the number in parentheses” or something similar?

5

Ah! Here’s a misconception! Seeing multiplication in the equation led the picker to call it a geometric sequence. This is why I love Desmos activities – concrete evidence of students’ understandings and misunderstandings that I can immediately work to address.

6

Another opportunity for improvement on my part! The two sequences remaining after the first three questions differed in two regards – their first terms and their common differences. A possible revision might be to change the highlighted formula to f\left ( n \right )=12+6\left ( n-1 \right ). This would even make a question about subtraction useless, so it should elicit a question about the common difference.

Lots going on here. Plenty of great vocabulary here (e.g. “explicit formula” and “common difference”) as well as some great questions about odd and even numbers. I’m particularly fond of the question “do you add to get to the next term.” Sure, I’d love to see “arithmetic sequence” show up in that student’s vocabulary, but it’s hard to fault a student for a description that just works.

12

Just a fascinating line of questioning here. Only one of the three visible sequences has exactly two negative numbers, but the students here only considered the visible terms. Why did the guesser asked about negative numbers anyway? Wouldn’t asking if the sequence increases (or decreases) be simpler? And to follow one interesting choice with an even more interesting question…wow! Why does this student bother including the word “arithmetic” in this question? Both of the sequences are arithmetic. And why ask if it’s adding 7 instead of simply asking if it’s increasing? This sort of specificity doesn’t really add anything other than the risk that one of the students miscalculates and leads the other astray. I wish I could have been inside this student’s head!

So much to say about these… One student was extremely curious about the starting number. What does that say about that student’s understanding of sequences? Several students asked about function notation rather than asking about formulas or equations. How about the student who asked about positive numbers, even numbers, double-digit numbers, AND the number 4? Yes, these questions got the job done, but what do they really tell me about my students’ understanding of sequences?

As you can see, my students used all sorts of vocabulary. They used lots of different math terms, but they didn’t use as much of the sequence vocabulary as I expected. On the one hand, I’m proud of their ability to use what they know and what they’re comfortable with to find success. But I wonder what this all says about our work with sequences. This Polygraph activity certainly differs significantly from the work we’ve been doing. Students need to do more than just perform calculations and interpret numbers. Polygraph forces students to wrestle with sequences in a unique manner and to draw distinctions between sequences represented numerically or as equations. Perhaps in the face of such a challenge my students retreated to the most natural language. Perhaps their understanding of sequences isn’t what I thought it was. Perhaps the numerical sequences and explicit formulas I chose did not lend themselves to the sort of questioning and language that I had hoped for.

So what does this all mean for my teaching and for my students’ learning? I actually feel like I’ve done a better job with sequences than I ever have before. So much more of our work this year has involved thinking through and solving problems. But maybe I haven’t gone far enough in including rich, meaningful tasks. Or maybe this was just a necessary step on the path toward mastery. As we continue working with sequences and eventually use our sequence knowledge to develop linear and exponential functions, I look forward to further probing my students’ understandings and challenging them to engage more deeply with the mathematics.