## The Difference Between Sketching and Graphing

Here is what I mean. Ask a student to:

Give an algebraic function whose graph has one positive root, a negative y-intercept, and an asymptote at x = -5, if that’s possible. If it’s impossible, explain why you can’t.

Maybe the student can determine the function. At some point, an advanced algebra student should determine the function. But what do I learn from a student who can’t determine the function? What does a blank graph tell me?

The student might understand what roots, intercepts, and asymptotes are. She might understand every part of the task except how to form the function algebraically. I won’t know because I’m asking a very formal task.

This is why a lot of secondary math teachers ask a less formal question first. They ask for a sketch.

Sketch a function whose graph has one positive root, a negative y-intercept, and an asymptote at x = -5, if that’s possible. If it’s impossible, explain why you can’t.

Think of what I know about the student that I didn’t know before. Think of the feedback that’s available to me now that wasn’t before.

Desmos just added sketching into its Activity Builder. That was the result of months of collaboration between our design, engineering, and teaching teams. That was also the result of our conviction that informal mathematical understanding is underrepresented in math classes and massively underrepresented in computer-based mathematics classes. We want to help students express their mathematical ideas and get feedback on those ideas, especially the ones that are informal and under development. That’s why we built sketch before multiple choice, for example. I’m stating this commitment publicly, hoping that one or more of you will help us live up to it.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

1. #### Damien Raftery

May 24, 2016 - 6:22 am -

Thanks, sketching has just added another big reason why I should be using Activity Builder to get students thinking explicitly

2. #### Jennifer Poole

May 24, 2016 - 6:32 am -

Wow. This is great! I can see a lot of applications for this, from 5th to 12th grade level math. This would be a fantastic bellringer activity.

I’m thinking there are some games we could play here as well, similar to Telephone or Pictionary.

3. #### Michael Pershan

May 24, 2016 - 6:33 am -

Think of what I know about the student that I didn’t know before. Think of the feedback that’s available to me now that wasn’t before.

I love the ability to sketch. This is absolutely a wonderful feature.

I’d be very interested to know what your thoughts and experiments look like for feedback in the Desmos environment.

4. #### Travis

May 24, 2016 - 9:02 am -

I hope Sketching is not just limited to Activities.
It should be available when graphing. Think of it as a student’s rough draft. Then they can refine with an equation.
Sketching is fabulous.
Also, students can use it to *write* estimates, then go to the next level and graph or solve.
Math encompasses both a precise equation/graph and a quick sketch.

5. #### Abe Hughes

May 24, 2016 - 10:17 am -

This connects directly to what I am working on with students in my Algebra II class. I have it all backwards though. I am asking students for the algebraic representation (formal) first, then asking for the sketch (informal). Having the students either sketch a function that meets the requirements or explain why they can’t sketch one to meet the requirements lowers the “math dial” tremendously and will result in some really valuable conversations. The more formal component can follow, but to me, it seems more advantageous that students understand the sketching part rather than the organizing of factor parts in the rational expression.

6. #### Amy Hogan

May 24, 2016 - 12:43 pm -

Looks great. I agree with Travis. This would be fantastic as a regular feature.

7. #### Elizabeth (@cheesemonkeysf)

May 25, 2016 - 7:14 am -

This feature makes me want to stand up and yell, “SUPER-LIKE!!!!!”

However, I have to say that I don’t think of sketching as being informal at all. In fact, to me, the ability to sketch a graph is about having a precise, deep, and automatic understanding of the shape(s) and key features of a function graph.

This feels like the essence of precision, rather than an informal relationship with functions.

Too many students just blindly poke formulae into the calculator and take the graph as the Word of God without understanding WHY the graph has the shape and features it has.

Thank you for making @Desmos even more useful!!!!!

– Elizabeth (@cheesemonkeysf)

8. #### Dan Meyer

May 25, 2016 - 9:34 pm -

Cathy:

Nice. I think we need to hook you up with some colors, first, but we’re almost there.

Michael:

I’d be very interested to know what your thoughts and experiments look like for feedback in the Desmos environment.

“Teacher feedback” is currently the third most commonly requested feature, just behind “equation labels on the graph.” Fourth place isn’t close.

So we think about feedback a lot, though we have more questions than answers. Questions like:

• What kind of feedback are the different actors best at? (Meaning peers, teachers, teachers in another room, computers, etc.)
• When in the learning process should students receive that feedback?
• Let’s say we enable written feedback from teachers on student activity work in Desmos. Are we making life any easier for those teachers? If we can’t make good feedback easier to give on a computer than on paper, maybe we should back off.

My biggest feedback dilemma right now regards card sort. You know card sorts, right? Well we’re making an authoring tool for card sorts. The teacher can set up an answer key if one exists for that set of cards. Students do their best to match cards into groups.

Big question: the Desmos supercomputers know a lot about every student in class. They know which student matches are correct and incorrect, for example. They know the most common incorrectly matched set of cards. So there is an opportunity for feedback. What should Desmos do to give the student the right kind of feedback at the right time? Tricky, right? What do you think?

9. #### Elizabeth (@cheesemonkeysf)

May 25, 2016 - 10:08 pm -

Dan,

What, more specifically, do you mean when you talk about “card sorts”?

Malcolm Swan breaks them down a number of different ways: there are one-dimensional cart sorts (i.e., 2 or 3 different categories), two-dimensional card sorts (sort into 2 or 3 different categories along the “x-axis,” then sort these further along the “y-axis”), and card matching tasks of differing complexity for differing purposes.

I ask because I’ve been thinking about writing a blog post to further refine these categories of card sorts and how I use them, but I would like it to be generally useful (including to Desmos).

So any clarity you can shed on your current thinking here would help me to help you.

– Elizabeth (@cheesemonkeysf)
Lover of Taxonomies

10. #### Dan Meyer

May 25, 2016 - 10:29 pm -

Sure. Teachers create the cards one at a time. Then they can group them if they like. This takes more effort for the teacher than other hypothetical approaches, but it’s more flexible. This permits card sorts where students group in “always,” “sometimes,” “never” categories (one dimension) or where they match tables, graphs, equations, and situations ten times over (two dimensions). It also supports situations where the teacher wants students to group the cards however they want.

11. #### Michael Pershan

May 26, 2016 - 2:30 am -

What should Desmos do to give the student the right kind of feedback at the right time? Tricky, right? What do you think?

Here are my quick, running around getting ready for work thoughts.

I don’t want Desmos making any teaching decisions for me, and the decision to give or withhold feedback is definitely a teaching decision. I’m not interested in Desmos offering individual feedback to my kids.

Whole-group feedback, though. I would love to have an easier time sharing discussion-worthy images or artifacts with kids. If the Desmos computers are really smart, maybe they can pick a favorite “no” from the card sort put it on a slide ready for me to show to the class, if I choose. Or maybe it can recommend a specific student’s work that might be great to talk about, and have that ready on an AB-style page for me to share with the class. Maybe there’s a follow-up activity that the Desmos computers can generate to address issues that my kids had during the card sort.

OK have to run! Eager to hear what your thoughts are.

12. #### Eric

May 26, 2016 - 9:09 am -

Hey Michael (disclaimer – I work at Desmos).

> I don’t want Desmos making any teaching decisions for me, and the decision to give or withhold feedback is definitely a teaching decision.

Interesting idea, but I have some questions. I would argue that there already are (and I think have to be) a lot of teaching decisions embedded in the design of Desmos activities, both the platform itself, and the individual activities. The decision to provide or not provide exposition before asking a question, the decision to allow free movement through the content vs block on completing each part, the decision about whether to share your classmates’ responses after you answer a question, the decision to always allow you to go back and revise work, the decision to provide the same content to each student regardless of previous answers. Even if we tried to not make “the decision to give or withhold feedback”, I think practically that would have to mean deciding not to provide feedback.

Do you have a vision for what it would mean for Desmos activities to “not make any teaching decisions for you”? Or is there some other way to phrase what you’re aiming at (e.g. to have all the teaching decisions embedded in Desmos activities be observable / predictable as a teacher, instead of relying on an unobservable algorithm)?

13. #### Michael Pershan

May 26, 2016 - 11:57 am -

Eric:

Interesting idea, but I have some questions. I would argue that there already are (and I think have to be) a lot of teaching decisions embedded in the design of Desmos activities, both the platform itself, and the individual activities.

You are 100% right.

Is there some other way to phrase what you’re aiming at?

Let me try again. This will be rambly, because I find this all confusing.

The decision of when to give indications of success and when to give written comments should be made by the teacher. This sort of feedback can interact with a kid’s motivation and thinking in negative ways, and as long as Desmos is imagining an environment with a teacher, Desmos should be wary of making this decision for that teacher. Not because Desmos shouldn’t make decisions, but written feedback or judgments of correctness are too hot to handle.

Is feedback a concept that leads us astray? I often think it does be seducing us to think that (a) feedback is evaluates correctness and (b) feedback is always beneficial. What if we designed our lessons (or our card sorts) while trying to get all the benefits of feedback without using the term “feedback” or the concept it connotes? Would that helps things? Perhaps not, but I think we should try, because feedback is so slippery.

In designing an activity, Desmos wants to help students develop their thinking. To develop a student’s thinking, we usually need to know something about their thinking. One question we can ask is, should Desmos make a guess about what a student is thinking in its interactions with students?

If the answer is “no” then it might still be worthwhile for Desmos to make a guess in its interactions with teachers.

Anyway, if Desmos is going to make guesses, I would appreciate if Desmos focused those guesses into helping me plan a follow-up activity. The dashboard is cool, but it’s mostly cool after a lesson, because I’m usually quite busy during the lesson itself. I would appreciate Desmos’ help in creating rich artifacts from the Desmos experience that might be worthy of following-up on in future class sessions.

Ramble ramble ramble turns out this is very hard and confusing.

14. #### Dan Meyer

May 26, 2016 - 1:16 pm -

@Michael, if I understand you, it’s okay for Desmos and its technology to make some instructional decisions but feedback is an area where you think we should tread more cautiously.

We’re currently taking a do-no-harm-to-students approach to feedback, intent on minimizing false positives and false negatives at the expense of more work from the teacher. I suspect these decisions will resonate with you. I think other teachers will prefer more automated feedback at the expense of a few false positives and negatives. Desmos will have to figure that out.

Our current design plan for card sort feedback is to give the teacher accurate, timely data for conversations in class around the card sorts, data which she in now way could gather on her own on any reasonable timeline.

When a student has completed a card sort or asks for help, we will advise the teacher on the dashboard to either a) tell the student how many matches are incorrect, b) tell the student which matches are incorrect, or c) to go talk to another student, who we know has interesting differences with our first student.

We don’t trust ourselves to algorithmically pick between a, b, and c, or to algorithmically choose the timing of a, b, and c. But we can give the teacher data on a, b, c, data that is totally impossible for the teacher to quickly gather on her own (particularly c). We’ll see if that’s enough.

15. #### Michael Pershan

May 26, 2016 - 3:15 pm -

This all sounds smart and cautious. I’m sure it’ll be great.

16. #### Kevin Hall

May 27, 2016 - 4:57 am -

Dan, are you saying that the dashboard will prompt the teacher to choose feedback option a, b, or c, and then Desmos will automatically pop that feedback up on the student’s screen? So, for example, I don’t personally have to count how many matches the student got wrong, I just have to click option a, and a dialogue box appears on the student’s screen saying something like “Mr. Hall wants you to know that you got 3 matches incorrect”?

If so, this sounds really cool.

17. #### Dan Meyer

May 27, 2016 - 10:10 am -

@Kevin, in a class of fewer than fifty students, I’d prefer to give that feedback personally and verbally. What’s your interest in having a computer deliver, rather than just inform, that feedback?

18. #### Michael Pershan

May 27, 2016 - 10:19 am -

A random thought. As noted above, Desmos gives oodles of feedback to students in the form of the responsivity of graphs or diagrams or the ideas of classmates. Desmos doesn’t give feedback in the form of things that feel like the computer is the teacher.

Could it be that the issue is about the metaphors we use? The sort of feedback Desmos currently gives presents itself to the student using the metaphor of “Desmos as an environment.” The sort of feedback Desmos is reluctant to give the student would employ the metaphor “Desmos as a teacher.”

Can students juggle both metaphors at once?

19. #### Kevin Hall

May 28, 2016 - 10:43 am -

@Dan, lots of reasons:

(1) Feedback like, “You have 3 incorrect matches” is intentionally very limited so it doesn’t give too much away. It’ll prompt a bit of whining for more specific hints. So I’ll have to extricate myself from the conversation without caving to student requests for more help than I want to give. Rather than saying no, it’s simpler to engineer the situation so the request doesn’t arise. In addition, when feedback is so intentionally limited, I don’t see how delivering it personally or verbally will help. Personal, verbal feedback is only helpful for longer, more discursive teacher replies. I’m not ready yet to give that kind of help until the kids have had at least a few minutes to think about the little hint. What I’d really like is the option to let Desmos tell the group how many matches were wrong, and simultaneously to have Desmos start a little countdown clock next to the group names in my dashboard so that if they haven’t corrected their mistake in 5 minutes, I get prompted to go intervene in person.

(2) If I choose C, I’ll want to eavesdrop on the conversation between the groups so I know if one group just tells the other group the answers. To do that, I can’t be talking to any other groups at that time.

(3) I’m imagining a room with 28 kids. Once I launch your activity, I’ll want to sit with the English Language Learners and check their understanding of the instructions and hold their hands through the first part. I may also have 1-2 students who really need me to be their group partners because students with special needs sometimes struggle with group work. This can be the case for students with autism, students with emotional disabilities, or for students who have been absent for the last 3 weeks (e.g., grandma got sick back in Guatemala, so the whole family went down there to be supportive). Sometimes I have a co-teacher to help with needs like these, and sometimes I don’t. Even when I do, it’s really hard to support the students who are struggling while simultaneously giving good feedback to my top 50%. There are days when almost all of my energy is going to struggling students. On days like this, anything you can do to help me serve the top students better is a god-send.

20. #### Dan Meyer

May 28, 2016 - 5:52 pm -

Really interesting use case, Kevin. I’ll take this back to my design team for discussions.

21. #### Dina Kubba

May 31, 2016 - 9:55 pm -

This is a great idea – don’t we always tell our students to “draw a picture” as a way into problems. I have had students who could solve this algebraically (they have learned what an asymptote means in an equation) but then still didn’t understand how their equation related to the graph of the function. One of my main goals each year is to help my students see the connections between the “pictures” and the symbols.

22. #### Roberto Catanuto

June 3, 2016 - 4:59 am -

I absolutely support the idea of sketching as a necessary part of the learning path in Math.

My students have so many troubles while moving from the informal to the formal, especially because they’ve been erroneously taught to start with the formal.

23. #### Karim

June 4, 2016 - 11:22 am -

Any thoughts on having sketch and graph both be options on the same screen? To Travis’s point, it’d be nice to be able to draw a sketch and then try to create the corresponding equation/graph: refining towards precision. (If you want to preserve the role of each approach in a single screen, any thought on allowing copy-forward of a sketch into a graph?)

24. #### Dan Meyer

June 7, 2016 - 9:47 am -

Lots of happy thoughts about that idea. We have a concept of “copying forward” student work from one graph to the next. Copying sketches forward seems like a natural extension.