This year, I have tried to engage my students in a more thoughtful homework process. I don’t think any math teacher, ever, has been satisfied with the way homework works in their class, and I would certainly put myself in that boat. My frustrations in the past have been that students sometimes would do something wrong and then continue to cement that wrong thing by repetition, I would get 30 homework assignments that look basically the same and spend tons of time giving useless feedback that they didn’t really even look at, and students were focused on completion over learning. I attribute this to the structure of the homework over students being their nutter butter selves. Here are the changes I made this year:
1. Every homework assignment comes with a full solution (not just answer) guide. It’s more work for me, but also makes me assign a reasonable amount of homework.
2. Students go through the assignment and do whatever they can without the solution guide.
3. Then they check the solution guide to check what they did and finish what they couldn’t. Anything they write after this point (or using the solution guide) is in a different color – which is a crucial point. They check their answers, fill in the rest of incomplete solutions and give themselves feedback on what they did well and what they did poorly.
It takes a little longer for the students, so I try to assign a little less. And some students haven’t bought totally into it yet (slash never will). But as a teacher grading it, I can see so much more. Like…
- Where students struggled and what they still don’t understand well, which is so obvious with the colored pen. What they did in pencil is their work and what they did in pen is their work with the solution guide.
- Evidence of learning – instead of doing something wrong over and over, they correct it and do it better the second time around, or at least know that what they did is wrong and need to get help from me.
- Where to give them feedback on the specific things that they are struggling on.
- Who is engaging with the homework and trying to learn from it, vs. who is just tryna get-r-done.
I also spend less time grading homework while still giving better quality feedback. I think they spend about the same amount of time doing homework but get more out of it.
The training process for this has been an investment, but worth it. I share with the class examples of things they can do to do this better, like this:
Feedback from students has been that they almost either really like it, or are fine doing it. They almost all indicate that it’s better for learning, which is what I care about.
How do you feel about the method of doing homework where you check your own answers?
It is very helpful XXXXXXXXXX
- It allows you to learn the right way of doing it while it’s still fresh in your mind.
- I like understanding what I did wrong right after I did it so that I can grasp what I did wrong.
- Being able to look at the answer and find what I did wrong at my own pace helps me understand the problem and how I should do it next time.
- Writing my own feedback is more helpful than skimming any you would give on homework.
- Self check is a way to see what you did wrong right after you did the work instead of a couple of days later,
It’s fine XXXXXXX
- I feel as though that making corrections and not totally understanding my mistakes is perhaps the biggest downfall.
- maybe if i came back after a longer period of time it would be more helpful to me in particular.
- I understand that it’s good to correct ourselves but I think I get more out of simply going up to you to clarify he things I’m struggling with.
- I only feel like feedback is necessary for some problems if I really don’t get it
- Well it is helpful some of the time but it does take a really long time to do this.
- I think that it’s helpful like 85% of the time, and then other times it confuses me
Meh, I don’t really do it. XX
Still experimenting! Would love some thoughts.
I have been quieter here than I normally would be during summer planning because I have been blogging at Klingspace, a blog run by the graduate school program from which I graduated in May. Below are the posts I have published there with a brief summary, if you are interested in reading about topics that may not be as math education focused as I usually am.
I am excited to rejoin a math classroom in the fall and hope to re-engage in the math education discussion on this blog that I am used to!
- 5/22 – Structure Is Not the Opposite of Autonomy – We shy away from procedures, structures and limitations in the name of creativity, but that structure can actually promote creativity.
- 5/28 – Keeping the Change: How > What - Success naturally breeds resistance to change, which means we must be sensitive to the fact that our change-filled futures are challenges to our success-filled pasts. Give people time to process change.
- 6/4 – Teacher Observation: Informing Practice, Not Judgment – The way most schools structure observations and evaluations make us see them as moments of judgment instead of opportunities to improve our practice.
- 6/11 – Have a GSA? Great! But It’s Probably Not Enough - There are queer students at our schools who aren’t served by simply having a GSA. More generally, we should not assume that because we have programming for X type of students that it serves every student who identifies as X.
- 6/28 – Using “Creative Tension” To Communicate Change – If leaders effectively show faculty the gap between their vision and the current program, faculty will be more likely to feel the need to move toward the vision
- 7/11 – Cultivating a Growth Soulset – Just as we can always learn more with a growth mindset, we need to tend to our emotional intelligence with the attitude that we can always become more emotionally adept.
- 7/21 – The Case Against a Linearly Sequenced Curriculum – Research about distributed practice suggests that studying something with space between is always more effective than studying it for the same amount of time uninterrupted. How can we incorporate this finding into our curriculum design?
- 7/29 – TBA
- 8/15 – TBA
(here are some excerpts from a paper I wrote for grad school about structuring math homework for better learning – the full paper is below)
The traditional structure of math homework (e.g. 1-67 odd) forces students to work hard, but not effectively, as all students blindly do the same assignment consisting of a similar number of each type of problem regardless of each student’s personal weaknesses. In “Practice Perfect,” Doug Lemov’s book on how to practice more effectively, the authors compare this type of practice to shampooing your hair, something we repeat daily but probably never improve on.
Math students ought to practice math the way that experts in other fields practice. When a musician learns a piece of music, they do not just play whole piece over and over. Instead, they workshop specific parts that they need to work on. When I was learning to play piano, my teacher would have me play difficult parts repeatedly – first each hand separately, then both hands together slowly, and finally at full speed. Students do the exact opposite on math homework – they do the problems with which they are comfortable and then leave blank those they do not know how to do. Thus, they are only practicing the very material that they do not need to.
If teachers would like them to engage with it differently, they need to make intentional changes in the structure. To make math homework more like expert practice, teachers should:
Force students to differentiate their homework experience.
Though the other suggestions below would be a helpful addition to traditional homework assignments, this first one would require a more radical shift. Instead of a linear assignment that encourages students to spend an equal amount of time on each part of the course, a math assignment should consist of minimal core problems for each learning objective that each student must complete, and a bank of other problems that the student could use to remedy any misconceptions. The set of core problems should be small enough that students can complete all and still have time to tackle their weak areas. Students should be instructed that a wrong answer should be a sign to reflect for a moment about what went wrong, perhaps even formally, and then immediately try more problems until they understand. To give space for this thoughtful type of work, a teacher might have to assign less work, but the quality of the work that is completed has the potential to be much higher.
(some thoughts on grading and accountability in the full paper)
Make homework objectives transparent.
With a differentiated homework assignment that required metacognition about weaknesses, students would need the tools to pick out their own weaknesses. When picking out problems from a math textbook to assign, I have an objective in mind for each group of problems. In retrospect, it seems obvious that I should simply share these objectives with students, paralleling Standards Based Grading for the wider structure of the class. Simply grouping math problems into learning objectives would help students focus their effort more effectively by allowing them to isolate skills and measure their success.
Ensure students can get immediate and actionable feedback
But even with clear learning objectives, students can’t make progress without feedback. Too often, students will power through their entire homework doing something wrong the entire time, encoding something in their brains the wrong way. Worried that students will copy answers out of the back of the book, teachers will assign problems that do not have attached solutions. We have to get over this fear – if students cannot check their work, or are not in the habit of doing so even when they are confident they got a problem correct, they risk not knowing that they are doing something wrong. Doing one problem wrong, fixing a misunderstanding and then doing a few more correctly will lead to far better results than doing five wrong and having to unlearn something incorrect a week later. For complex problems that only have a simple answer in the back of the book, teachers could post a solution guide that details not only the answer but the process that it takes.
No idea if this will work! I’m interested to try it out when I get back to the classroom. Here is the full paper:
One of my grad school professors taught me how to read.
Okay, so I knew HOW to read (hold your snarky math teacher comments, English folk), but I didn’t realize I had no idea how to read for an academic context. For one of our very first assignments, our professor set up a very sneaky experiment that taught me I wasn’t reading very well. He told the whole class that for one of our more dense and academic readings for the next week*, one person would be randomly selected to lead a class-wide discussion. This was early on in a program with a group of 22 all-star, experienced educators. Very scary.
I was terrified into competency.
Instead of just reading it straight through and perhaps highlighting, I wrote questions in the margins, connected various parts of the text, made a list of the main ideas, pulled out quotes that could generate discussion, and generally actively thought about the content of the article.
It turned out he was bluffing, which he revealed in class the next week. Phew, *changes underwear*. But with this exercise, he made the point to us that the way we read that article was totally different from the way we probably read most of the other stuff. And more effective. I was thankful for this because for the rest of grad school, I read much more effectively. Even if I didn’t have time to read an entire article, I would spend a bit of time diagramming, writing questions in the margins, and actively engaging with the content. Instead of expending more effort, I used my effort more effectively.
How did I make it through so many years of education without knowing how to read? How much more could I have gained from both my high school and college education? How does this apply to our math students? How many of them are trying to do better by working MORE instead of by working MORE EFFECTIVELY? What can we do to show the how to do homework?
What’s a good meta-assignment that can show students how to do math homework effectively (without making them sh*t some bricks to learn the lesson)?*Alfred North Whitehead, The Aims of Education. NB: I don’t really remember what it was all about 9 months later, but hey, I guess good teaching techniques have their limits?
Okay. It’s time I hopped back on the blogwagon™. I have been in graduate school this past year earning a Master’s degree in Private School Leadership (yeah, that exists) but I haven’t felt like I had anything to contribute to the discussion around math TEACHING up in my ivory tower. I want to end the year by reflecting on some of the things that I have been thinking about, ranging from little random thoughts to unanswerable questions, all of which I am excited to test out next year in my triumphant return to a math classroom!
So that’s what the next series of posts will be. But first… I have wondered if it was a good/necessary move for me to take a full year away from the classroom to reflect. One the one hand, it’s great to have the space and time to really delve into issues from human cognition to the use of data to improve student learning. On the other hand, I haven’t been able to workshop any of the ideas running around in my brain. Regardless, it has driven home to me the importance of reflecting, so I thought I would share the small change last year I made in my lesson planning that helped me become a more reflective, and thus more effective teacher.
You don’t need to take a year off or find more hours in the day to journal. You just need to add a column to your lesson plans.
This is an actual screenshot from my Evernote planning notebook. I chunk my class into activities, so that’s what you see on the way left. The next column describes any other necessary details or files that the activity requires. The third column is initially blank, and this is where I reflected. Every day, before planning the next lesson, I would go back to the one before and jot down a bullet point or two about each activity. Sometimes I would have a lot to say and would write some notes for my future self, but I would, at the very least, note how much time the learning activity took. Before long, this became a habit of my lesson planning that took no more than a few minutes.
I initially conceptualized this as a way to keep notes for myself in the future, should I teach the same class again, but found that I reaped the benefits far more quickly. Just sitting down for even 5 minutes to think about what happened that day started a recursive process where my reflections allowed me to make decisions in a different way in the future.
So I hope that some of my reflections from grad school might be helpful (for both readers and myself) — but I am also looking forward to reflecting next year in the classroom in a way that has a more immediate effect on my teaching and student learning.
(hey, I’m a math teacher, not an artist).
Has this situation ever happened to you? One thing that my colleague and I talked about over and over was how we engage in Socratic dialogue portion of our classes. What kind of questions are we asking? What kind of answers are we getting? What do these answers actually tell us? One thing that we both noticed was that occasionally you can get in a flow where you keep a learning conversation going by taking one word student answers and fleshing out their thoughts fully, or just finishing their
sandwiches sentences for them. It feels so good because it feels like THE WHOLE CLASS is on this EFFICIENT AND WONDERFUL thought train going at just about the same pace as someone who is basically an expert in the subject. And as “duh” how counterproductive this is to learning, it’s something that I totally didn’t notice until I had someone in my classroom to point it out to me.
What’s the whole point of Socratic dialogue in the first place? Well, for you, it is a way for you to check for understanding. Are students listening and understanding what is going on? And for them, it is a way to get them to think. Are they just taking in what you are saying or are they turning it over in their heads and taking it to new places? If we finish their thoughts for them, we are not only robbing them of the opportunity to think and learn, but we are deceiving ourselves about what they actually know and understand.
Here are some strategies that we discussed to avoid this:
HOLD THEM ACCOUNTABLE: Turn that into a full sentence please. Try that again with better mathematical terminology.
FOLLOW UP QUESTIONS: How do you know that? What do you think of _____’s answer? (Crucial to do this for both incorrect AND correct answers).
ASK BETTER QUESTIONS: Explain…. Why… How… If the question can be answered in one word, it’s probably not a great question.
STRUCTURED RESPONSE: Think-Pair-Share. Quick written reflection with cold calling.
BE EXPLICIT: Be direct about what you value in student answers.
I grew more and more aware of this as the year went on and I think I got much better at not only asking better questions but eliciting better responses from my students. I think this heightened the level of mathematical discourse in my classroom, and also gave me a much better idea of where they actually were in the learning process. It didn’t mean that students wouldn’t try the age old SHOUT ANY MATH WORD YOU CAN THINK OF to answer a complicated math question, but at least they understood why I pushed them further.
This past year, I had the distinct pleasure of serving as a mentor to an excellent new math teacher who was an absolute pleasure to work with (and is definitely going to be a superstar). All of the teaching fellows at my school took a seminar with an administrator, and then each one had a mentor who observed them and generally gave them advice. Though we did some lesson planning together, and we worked a little more closely at the beginning of the year, our main form of interaction was weekly observations.
For the whole year, she came to my class once a week, I went to her class once a week and then we met once a week to debrief. Sure, she tells me she learned a lot, but this was also the best professional development possible for me too. I learned so much over the course of the year and engaged in so many excellent conversations about teaching – I grew so much from a commitment of a little over an hour a week.
It makes me realize that I should have been doing this all along with a colleague, apart from the whole school appraisal process. Though it seems so easy, and I of course exchanged the common “I’d love to come visit your class” with so many colleagues, it never happened before. I think the thing that made it work with us was a structured commitment, and the formation of a habit in our schedules (it didn’t feel like something ON TOP of everything else, it was part of what I did every week). Any observation program, even if an informal agreement between colleagues, needs to be structured and scheduled so that we don’t push it away for the million other things we can do with our time. It can’t be something like “Go and visit someone’s class in the department at least once this term.” In my experience, that just does not work.
As I pore over my notes from the past year, I am going to dedicate the next few blog posts to my major takeaways from a year of observations. But first…
How We Observed Each Other
There are a million different ways to do observations, but here is how we did it:
- All visits were mutually scheduled beforehand, at the beginning of the week. We got so comfortable with each other that knowing someone was in the room was REALLY not a big deal, and wouldn’t change how we would plan our lesson.
- We did this so we could touch base beforehand to see if there was anything that the observer should look for. This was helpful when we were workshopping a technique, trying out a new type of activity or just wondering about something that is happening in the classroom.
- The observer would always take notes in the following format, which is something we developed based on other formats, and which we found simple and effective:
The time column really helped us pay attention to something that is often the last thing you pay attention to when directing a classroom. The middle column helped us talk about what happened and helped us remember everything that happened in a class. The column on the right was an acknowledgment that we aren’t just robots sitting there, but have helpful opinions and suggestions, but it helped us focus our subjective comments by basing them on what was actually happening in the class.
- Then, we would find a time to exchange notes and just talk about everything that happened in a comfortable and frank manner. We would talk about the objective things that happened in the class and share ideas and observations based on those. It wasn’t hard to talk for 45 minutes about teaching, but we could share notes in 15-20 minutes if pressed for time.
What worked for you for observing? Any other way of taking notes or organizing a program?
What can you do with this?
Today, we did what I thought was one of the coolest explorations I have seen in a while. It is called the “One Cut” or “Fold and Cut” problem, with inspiration from Patrick Honner (@MrHonner, this part of his website). The premise is simple: you have a shape drawn on a piece of paper. How can you fold the paper so that you can make one straight snip with the scissors and cut the shape out?
Weirdly enough, there is a theorem that says that no matter what shape, this is possible (concave, convex, numerous closed figures, as long as the sides are all straight) . Which totally blows my mind. Because it’s really friggin’ hard in practice (though, the star above is pretty easy). Check out some patterns for some other shapes here.
With one group today, we started doing this problem as a way to see what vocabulary they knew with polygons and to talk a bit about symmetry. I printed out little shapes on 1/4 sheets of paper and the levels went as follows, getting a bit harder as the levels go up:
Equilateral Triangle –> Square –> Isosceles Triangle –> Rectangle –> Regular Pentagon –> Regular 5 pointed Star –> Scalene Triangle –> Arbitrary Quadrilateral
They were TOTALLY hooked. Every single kid was working on their own and kept either having the “YAYY, TEACHER LOOK!” reaction or laughing hilariously at the silly shapes that they made by accident. The kid in the picture with his hand over his face kept yelling out “AGHHH, TRICKY TRIANGLE, TRICKY TRIANGLE” because he couldn’t do the scalene triangle. But he wouldn’t accept a hint from me because he wanted to find out on his own!
45 minutes later, the classroom was a total mess, and I was wondering where all the time had gone…
Now, working with the regular polygons you might get duped into thinking it’s pretty easy. But try a scalene triangle. Or a non-special quadrilateral (i.e. most quadrilaterals). It’s actually VERY difficult. And I think the solution is pretty fascinating, because my solution to it (which I got after about 10 triangles and an hour) heavily involved the triangle’s incenter (the intersection of the three angle bisectors). Which made me think that this would be a super cool thing to do in a geometry class when talking about angle bisectors. Are there multiple “incenters” on shapes with more sides, even if you have to define the idea in a different way?
I won’t share the solution but just post a picture to show you that YEAHHH it’s possible.
(If you haven’t heard me bemoaning how much energy it takes to do math for 2.5 hours with 11 and 12 year olds at a summer camp, well then FYI: I am currently teaching at a summer camp for 6th and 7th graders from China who are here doing an academic program that is half ESL, half math. It’s getting more and more fun, but certainly has been an adjustment from teaching seniors in high school for only 45 minutes at a time).
Both years I have taught AP Calculus AB, I have kind of dreaded the couple weeks of review. They are hard to plan for and probably really boring for students. On top of that, last year, I felt like I squandered the review time. I mostly gave them free time in class to do whatever they needed to do, and I am not sure how effective this was. 45 minutes straight of studying really dragged and I felt like students didn’t really even know what they needed to work on. In addition, the lack of structure I think prompted some students just to look at answer keys instead of struggling through problems themselves.
This year, I was dreading review again, but it really went much better and I think was much more engaging and effective. A few things I changed this year:
1. We started reviewing in class earlier, even before we finished all the material.
2. Review was more structured by me at first and slowly led to more independence, with opportunities for students to see which topics they needed most work on.
3. I spent literally 1% of the time explaining at the front of the classroom and 99% of the time having them to do the work.
The learning structures I used for review:
- 5 MINUTE SKILL DRILL
- TIMED FREE RESPONSE QUESTION
- MULTIPLE CHOICE JIGSAW
- MOODLE MULTIPLE CHOICE
- MOCK EXAM
- and then…. FREE TIME WITH PAST QUESTIONS
5 MINUTE SKILL DRILL
For about 3 weeks before we started review (i.e. while we were still doing DiffEQs, volume etc), we started class with 5 to 6 quick skill based questions. I tried to put a range of topics, from evaluating limits to writing a tangent line to finding the average value of a function. Students pulled out their notebooks and worked on the questions silently (or as silently as I could get my 85%-chatty-bro class to work). They did the ones they could do and tried the rest. After the timer ran out, I would scroll down and show the answers and show what Standard that the question corresponded with. Then, after explaining anything that needed explaining, we would vote as a class on whether to retire a topic if they felt confident or keep it on for the next day. This took about 10 minutes at the beginning of class.
I loved this because even though it ate up class time during the end of the year and forced the actual material to take longer, by the time we were ready to review, students had already brushed up on the skills and could focus on big ideas.
Next time, I will try to be more organized about it and have a booklet printed, or sheets for them to glue – I was improvising with this and I felt like it took students too much time to copy things down from the projector.
(this is sort of what it looked like below, but this is for integrals earlier in the year – I’m between computers right now and don’t have all my old files!)
TIMED FREE RESPONSE QUESTION
At the end of many units towards the end of the year, we would do a 12 minute timed Free Response Question, and this is something that we did almost daily during our review time. I would hand out a free response question on a little slip of paper, they would glue it into their notebooks and work on it for 12 minutes silently. If they didn’t know how to do it, they would just try as hard as they could, struggle through it and write down what they know. Then, after 12 minutes, I would hand out the answer key and they would grade themselves, AP style.
I loved that this forced them to struggle through a question and see what they actually know, and I loved that this got them used to AP grading (I had a much lower incidence of unit-forgetting and less-than-3-decimal precision). The trick for both of these benefits is in really holding out on the answer keys until the end of the time!
Next time, I will try to coordinate the 5 minute skill drill with this so that students can recall the topic before a tricky free response question, as I had some students who were so stuck that they didn’t really write anything down and got nothing out of the exercise.
MULTIPLE CHOICE JIGSAW
I find multiple choice harder to integrate into class than free response, but one learning structure I liked for multiple choice was Jigsaw. For those that don’t know this (I assume it is fairly common), there would be a set at 12 questions and groups of 3-4 would all work on a third of the questions together (1-4, 5-8, 9-12). Once every group got through theirs, I would rearrange the classroom so that each new group had one person who had worked on each of the sections. Then, they would either work on the rest of the questions individually and then check with each other when they got stuck, or they would just take turns and teach the other members of the group their questions. Some students reported to me that the process of explaining a question out loud really helped them understand what was going on.
I loved the interactions that this activity prompted and I loved how efficient it was for getting through many multiple choice questions (students could do this much faster than working on them on their own).
Next time, I will try to deal with the awkwardness of groups finishing at separate times and weak students incapable of explaining questions to their classmates, though I am not sure how.
MOODLE MULTIPLE CHOICE
I didn’t trust my students to do free response questions at home. They would just look up the answers and get NOTHING out of the process! But we did do a lot of multiple choice questions at home, through Moodle. It is super easy to set up quizzes, so I would just upload images of the questions from a multiple choice collection I had and set the correct answer. I would do 15 questions in a quiz, and it would take my students about 40 minutes to do. We started this about a month and a half before the exam, and then all the homework during the review time become these online multiple choice questions. Before the test, every single student did about 130 multiple choice questions, which amounts to about 3 full tests, and then many did more questions on their own outside of that.
I loved that the work was immediately self checked and automatically graded, as I think this did a lot for their learning from these questions.
Next time, I don’t think I would do so many of these as I think they got a bored with them. Also, I felt like some students were just clicking through the questions, so I would try to think of ways to get them to take these learning opportunities a bit more seriously.
This is, of course, nothing original, but if you have the luxury of stealing a few hours from your students on a weekend for a Mock Exam, do it! Correct it for them, but don’t put a grade on it so that it can be a truly diagnostic tool. This was the most helpful thing for my students in prepping for the exam, because, on top of everything else, the Mock really helped them figure out their weaknesses so that they could really be productive when finally I gave them…
FREE TIME WITH PAST QUESTIONS
By the time I was giving them large chunks of time to work in class on their own, most students knew what their weaknesses were (from the Mock, timed Free Response, Moodle Multiple Choice etc). Whether they needed to improve their multiple choice or their free response, or they needed to work on specific topics (and could with a packet I gave them with AP Free Response questions split up by type), I felt like most students REALLY used the time well, to the point where a lot of students didn’t even bother studying the night before the exam. All the structure and diagnosing we did at the beginning, and all the work that THEY were doing instead of me talking helped them become far more independent and effective in the review process. I hope it worked – I will find out in a few weeks!
Any review structures you used that worked well?
One of our PD sessions in the past was about how to support students with learning differences. One of the points that the presenter made was that most pedagogical tools that you would use the better serve these students are great tools to reach all learners. This struck me especially because I teach almost entirely students for whom English is their second language, and sometimes when I do something specifically to help students with the language of mathematics I come to larger and more general pedagogical understandings.
For example, this past week, I introduced solids of known cross-section in AP Calculus in a way that I thought would ease my students understanding of the tricky language involved in the problems, but what I ended up doing was really effectively let them develop their own conception of how these solids are formed and THEN interpret the AP problem language and integral notation in those terms. Conceptualize and then add mathematical formality to their own conceptual framework.
Here’s how it worked. I put 4 of these solids out around the room:
- First, I gave them 1-2 minutes to SILENTLY write down in bullet points how they would describe to someone else how the solid was formed.
- Then I gave them 2 minutes to share ideas in groups.
- Then I cold called on 7 or 8 students via a deck small cards with their names on them (which is by far my new favorite teaching tool). After I called on some students, I called for volunteers with any other ideas.
- LAST, I asked them to flip to the back of the paper and read the actual description.
During the “share” part, students said some of the craziest, random stuff, but most of the important parts of the description were said by various students. When it came time for them to read the description, at first they were like “whoa” because the language is still a bit daunting. But after a minute or so of close reading, they connected everything in that description with things that they themselves had said. So when it was time to do the actual integral, the intermediate notation I use made 100% sense:
So general pedagogical moral of the story? Letting students conceptualize something on their own before bringing in mathematical language and notation makes it more likely that the notation will aid in their understanding rather than provide another hurdle in learning.
A teaching fellow (a first year teacher) was observing my class (and has been observing periodically throughout the year). Afterwards, she remarked that she felt this was one of the most effective 10 minutes of the year, and I agree! And I think 10 minutes on this (instead of just 1 minute reading the question) will save lots of time in the future. Next week, I hope to try the same strategy with solids of revolution!!