For a long time I have tried to make a mental note of the real math I do all the time and find ways to package it into lessons for random points in the school year. Again and again I find that other than managing finances, I am constantly needing to plan events, or buy stuff at classroom scale. Like you, I’ve bought classroom quantities of supplies before, whether its notecards or glue or snacks. Each year we have a laser tag party for our AP math kids, an event for 70 people that requires a decent amount of food. Turns out juggling unit prices and headcounts is just as important as talking about polynomials, if not more.

Finally, this desire came to a head earlier in the week when I was extremely tired, hungry, and not in the mood for planning a 90 minute College Algebra lesson. We’d just finished up some topics and taken an assessment so I figured, you know what, let’s take a little break.

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Yesterday and today, students were required to plan a party for 50 people. They could work alone or with a partner. They had to plan two scenarios: sourcing all the supplies from a grocery store, and sourcing the food from a restaurant or other vendor. In both cases they had $500 to play with, which makes for a pretty decent party.

I wrote up this outline about an hour before the kids came in:

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Viewable Copy You Can Duplicate

These are all seniors and have surely attended and planned many family events where alcohol was part of the preparation, so yes, there’s an item in there where they could consider beer costs. Let’s not ignore that kids are already in the real world, shall we?

Once we went over the directions, it was just…magic. Flipping through grocery ads, planning menus, discussing appropriate quantities, it was awesome. There was a very quiet buzz as they went through everything. I have a couple kids who are always hesitant to start work and they jumped on this. They spent their 90 minutes period figuring out their plan and doing research, and they spent their 50 minute period today wrapping it all up and submitting.

One kid did tell me straight up I couldn’t come to their party. 😭

The best problems can be brilliant in their simplicity. I could see throwing this at the AP kids after the test and getting an equally amazing result.

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AuthorJonathan Claydon
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Every so often the stars align and we have a real casual experience all day long in my room. Yesterday was the end of a marking period and every kid was in “finish this stuff” mode.

In Calculus we were wrapping up a function analysis project:

BC Calculus was up to something similar, and we talked about a benchmark they took recently.

Over in College Algebra kids had two tasks to accomplish, wrap up a project on polynomials:

…and take a test. After submitting their project, they grabbed their weekly (ish) assessment and completed it on their own time. College Algebra is a very casual environment. Whole group instruction isn’t really a thing in there. In any given week kids have a to do list (new lesson, assignment, assessment) and have to complete it by the end of the week. There are 23 kids in this class and they work at 23 different paces. Often that means a handful are done early. Most of the time they’ll get a bonus activity (via Desmos or something else), or sometimes I’ll bust out the puzzles:

So, in yesterday’s 50 minute period, lots of kids were wrapping up their assessment, a few handed in their projects and started/finished the assessment, and others were finished, all with some Weezer in the background. 23 kids, all hanging out, taking care of whatever they needed to get done.

And it played out that way all day long. Hanging out, doing some math, listening to music. It is my favorite learning environment. To spend the whole day in it is simply sublime.

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AuthorJonathan Claydon

Not long ago I had this simple idea for introducing the idea of an antiderivative. Turns out it wasn’t long before the idea would pay off again. In College Algebra, we were looking at radical equations. Solving stuff with square roots wasn’t new, but solving stuff with cube roots and fourth roots was. I place a huge emphasis on connecting algebra to graphs in College Algebra. If we’re solving a cube root equation, I want them to be able to graph it.

I cheat a little bit and tell them Desmos “can’t” accept a cube root as written with a radical symbol. It very much can, if you go digging around in the function buttons. However, I use it as an opportunity to introduce root and fractional exponent equivalence. It was a bit of a struggle to get kids to remember, so I thought we’d play a game with some of the extra index cards I had lying around.

As before, groups of students were given a random pile of cards. Each card had 1 match in the deck. I did not tell them how they were supposed to make matches.

Thrown into the mix were some rational expressions and their equivalent form with negative exponents. Students had no exposure to this other than from previous classes. I made the coefficients unique enough to where they were able to put things together using that as a context clue. As groups concluded their pairs, I went around and had a discussion with each group about what rules they had developed to make their matches.

Within 10 minutes, we had decoded fractional exponents and negative exponents without any boring lecture about exponent laws.

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AuthorJonathan Claydon

College Algebra is an interesting course to teach because for the kids involved, the topics aren’t really new, but there are certainly new things they can discover within them, or get better insight hearing about something a second time. This last week we were starting an introduction to transformations. That prompted this bit of lesson planning:

Opening Acts

I used a set of three pre-built Desmos Activities with the group. I intended to use these last year but just never forced myself to do so.

Opener: Transformation Golf

Middle Innings: Translations with Coordinates

Closer: Practice with Symbols

For the ability level of the kids involved, these went really quickly. They are simple, straightforward activities but do present some interesting challenges. The kids really enjoyed transformation golf in particular.

It prompted a lot of good discussion and offered just enough challenge for everyone. We completed that activity in one 50 minute class period (about 40 minutes or actual working time). The other two (coordinates and symbols) were done in a single 50 minute class period. The combination of these three activities were just to job some memories and reacquaint with transformation vocabulary.

Proving Activity

A longer version came later, but using the polygon() tool in Desmos, we did a short proving behavior. We built polygons using a table, and applied some coordinate rules to those polygons. Students had to modify their polygon in 4 ways, writing down what they did. Then they submitted a link to their graph (my subtle way of teaching them how to sign-in with Desmos and save things). This took about 25-30 minutes of real class time:

I really liked the progression. Kids got to take a familiar skill and learn something new about the calculator. A few days later they did a more involved polygon transformation and applied what they learned to transformations of various parent functions (quadratic, absolute value, radical, natural log). The best part? These three days worth of lessons only took 15 minutes to map out thanks to the great resources in the Desmos Activity Library and the incredibly slick polygon command (launched only a few months ago).

Really happy with how all this came together.

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AuthorJonathan Claydon

A few nights ago I was working on an assessment and I spat this in my twitter machine:

I do this often as a way of talking to myself while working on things late in the evening. Often it's just to be funny, sometimes it's a little more serious, but I always I figure it's late and not a lot of people are reading. Not so much this time. It happens. There were a few reactions I want to address though, as my use of "points" there was misinterpreted in a couple ways.

Don't use them!

Ideally, yes. I feel you here. Unfortunately I have a gradebook I have to maintain with a minimum number of assignments per grading period, so I've got to do something.

Let the kids decide!

I tried this once before. In 2015 I implemented A/B/Not Yet grading in Calculus. We'd take an assessment, kids would look at the solution, and then rate themselves. Generally, kids were not adept at rating themselves. I had no good system for dealing with students who rated Not Yet, I was too busy with athletics to have any kind of viable after school system. Collecting the ratings was very time consuming and I was poor at communicating how to determine what should be what. A rubric you say? At that point this work saving system has now become more work than another system would be, so no thanks.

It was interesting experiment, but one I chose not to continue. Your experience may be different.

SBG!

I never never never assume someone is familiar with my teaching journey. These responses were expected (and welcome!) and I chose not to reply to them, because it'd be too easy to come across as that guy who is all "well I wrote the book on SBG blah blah blah..." because that's not a good look. But to those who suggested SBG, yes, I love it as a system and it works super great in a lot of contexts. I have used in Algebra 2, Pre-Cal, and College Algebra with great success. If you are interested in my history with the systems, I believe I have tagged the posts appropriately.

What I do these days...

In general, most classes work great for SBG. I have an SBG system in place with College Algebra and the kids like it. It's extremely similar to the system I came up with a long time ago. However, AP Calc has really never been SBG friendly in my opinion. Implementing a built-in retry system is really the problem. And with the speed you have to move with AP Calc, eventually in class assessments just become a burden. Last spring, AB Calc shifted entirely to free response based assessment because that's what we needed to do. It didn't work, but I still liked it and have some thoughts for this year. In general, with Calculus I will break stuff into a topic, assign some general value to the category, and give a handful of questions about that standard. The points vary, the kinds of questions vary, and there is no built-in retry. It's not really SBG. It's also not a test worth 100 points.

Here's the assessment I was working on when I tweeted:

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This particular assignment was for my BC group. The complaint was about how to weight the various sections based on the time it would take to complete them and the complexity. When I grade something like this I take an overall picture of the work. I check for correctness, offer comments, and give students a chance to discuss their work with others. Each one of these sections is an entry in the gradebook. But 1 point ≠ 1 correct problem, I take the whole body of work into account based on any trends in error I may see.

Maybe that clears things up, but maybe it doesn't. Non-traditional graders of the world I'm very with you.

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AuthorJonathan Claydon

Over the years I've tried to start incorporating financial literacy lessons into what I do. Seniors in particular get that "wasn't I supposed to learn this?" feeling about this kind of stuff and I aim to help a little bit. Especially to offer some perspective on the rent vs own debate. A majority of my students rent their living space and have heard all about how it's allegedly a waste of money. Even more are eager to save up for a car of their own and need some insight on the process.

Last year I formalized that into Let's Buy a House. Last year I had about 10 days with this lesson. I had students do a lot of comparison shopping, visit a make shift bank, and then organize their findings. I had waaaaaaay less time with this in Calculus after the AP test this year. More or less two class periods and then school was over. A condensed version was necessary.

Objective

Fast forward to your early 30s and assume you have the cash saved up for a car and a house. Find a new or used car for under $30,000. Find a house in certain zip codes for under $300,000 (not unreasonable for the area around school). Approximate the cost of property taxes for the house. Calculate the monthly payments and see what monthly income would be necessary to afford both. Answer some questions related to what you observed while researching.

All of students have Google Apps accounts and used my class set of Chromebooks to complete the task.

That's the short version. Here's what students were presented with:

Because of the limited time available, they only needed to run calculations and present their findings on 1 house and 1 car. 3.5 hours of class time over 3 days was allotted for this and most students finished in about 2.5 hours. This was their absolute last assignment of the year, finals started the day after this was due.

CAR PAYMENT CALCULATOR

HOUSE PAYMENT CALCULATOR

You'll have to make a copy of these files to use them.

Results

Despite the rapid end of school approaching, students did a great job with this activity. They took their time and asked a lot of good questions along the way. Last year we spent a couple days building the payment calculator together. I didn't have the time this year so it was just given to them. Thanks to the suggestion of someone at TMC 17, I presented three credit rating scenarios for the car payments. That prompted a LOT of questions of what it takes to be considered in the Bad, OK, and Good camps. Students who had taken some of our finance electives were able to assist those that didn't understand it as well.

Many many students made good observations about how people in worse credit situations are often offered lower monthly payments not seeing the big disparity in money going towards interest. I think I successfully scared most of them off 30 year mortgages too.

A sample:

Extension

Earlier in the year I did the same exercise with College Algebra. We were nowhere near as constrained with deadlines so they had a much longer version of this project. They had to research 3 cars, run the calculations, and present their findings. Separately they had to find 5 houses, run the calculations, and present their findings. For their 5th house they were given a $4 million budget, just to give them an idea what the payments would be like for something like that. Collectively they were a little less organized, so in addition to giving me two presentations, they had to gather and submit their house findings on a worksheet:

They were a little dazzled about the kind of numbers that resulted from their "dream house:"

College Algebra spent about a week on each half of the project. It was part of a series of tasks they had to complete in the final grading period. As it was more of a self-paced environment, students worked on other things as they did this.

Conclusion

College Algebra found it useful, if a bit tedious. I may shorten their requirements for next year. Calculus did a great job despite the constrained schedule. Calc BC didn't do this activity at all because schedule quirks left us with even less time (1 class day really) and I had other things I wanted them to do. A majority of those students attended summer camp and did the exercise there anyway, so it wasn't a huge loss.

More than one student made me laugh out loud as I scanned through their presentations:

Many of them incorporated a "I don't wanna grow up" vibe that was adorable.

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AuthorJonathan Claydon
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Seniors graduated a few days ago and I'm making an attempt to be a little more proactive about some things this summer. Mainly addressing some to do items that have been languishing for a few years. Before it starts in earnest, a few final thoughts about the year.

Ten years ago when I informed an advisor from college I was switching to education, his only comment was "well, will be interesting to see how you like teaching Algebra 2 forever." Interestingly enough, there was only a small run where I taught the same thing multiple years in a row. This year's biggest challenge was all the creation that was necessary. At the end, it all turned out pretty well.

College Algebra

I took a giant gamble in January where I decided to stop whole group instruction. Somehow, we made it through the entirety of the second semester and it really wasn't a problem. It required a lot of effort on my part to properly script moves. Accounting for the time kids would take on things was constantly in flux and I was always over or underestimating. Though it felt like there was a lot of wasted time, it could be argued it all evened out because even if it took forever to get kids started, they were doing something quite a lot of the time.

Essentially, I had 4 hours of class time each week and kids had about 3 hours of work to do (exploration, discussion with me, classwork). Early in the week a lot of time was burned with setup as I had to float around and give some introductory information and outline the requirements of the week. Each little pod took a different amount of time to buy what I was selling. In any given week maybe 8-10 kids out of 32 would finish their week's work early and not have a lot to do Friday. This was not ideal, but a reasonable trade off to allow the ones who needed longer to take longer. A mistake on my part is letting a particular group of students set the pacing for everyone some weeks. Often they were taking longer not because they needed to, but because they actively chose to. It was very hard to decide whether I should penalize them for taking forever (and thus make grades about compliance) or surrender the time now to avoid delays in the future (inevitability there'd be an assessment they had 0 clue how to do because I forced them to stop working on a topic). The long term benefit of having everyone in the same ballpark was more valuable than getting into a protracted skirmish with 4 kids (it was highly likely that weaponizing grades would've caused behavior problems from 1 of them).

I am teaching College Algebra again and I think starting with the small group model from the beginning will be interesting. There are a few procedures I can tighten up as well. In the end, teaching this class was a good experience and the students as a whole were very good. There is a lot of joy in helping seniors rediscover an interest in math when a lot of people have told them they aren't good at it.

Calculus AB

The eternal struggle. There's sort of three things going on. Vocab, concepts, and deep mechanical fluency. And you only have time to pick two. I have always chosen to focus on some core fundamentals to the detriment of smaller ones to improve the whole group proficiency. I just don't like leaving kids behind. As a group, there was a lot to like. Many many kids put in a good effort and showed promise during our AP reviews. What that will translate to in July remains to be seen. Last year saw sizable increases and the sense I had was more kids were prepared and at a better level of preparation than last year. Fingers crossed.

There are always new efficiencies to find and I think I've got some we can work on next year. Wrapping my head around presenting good, concise, mathematical arguments was a late game discovery this year, something that will be helpful if we can put it into practice for longer.

Post Exam reactions were fairly positive. The ones who I thought would do well didn't seem too frazzled and the free response questions were incredibly restrained and well within stuff my kids would've known how to do. As I said to them several times (with only a little snark), with 62 people taking the test, I would think more than 2 could pass.

My big goal this summer is to finalize my classwork. I change it so much each year that I think it's finally time to decide what I should be doing and stick with it, as a sanity saver throughout the next school year.

Calculus BC

Probably the biggest questions here. This was our first group of students taking it as a separate class. The sheer scope of the material caught up to a few of them at the end of the year. But they were all incredibly capable students. It would be almost impossible to pick a better group to start a class like this with. They really embraced the task at hand, validating the recommendations they were given to take it.

There was some mild complaining about free response with this group, but after seeing the question they were talking about (#6), I agree with their assessment. We didn't dive into series quite thoroughly enough, so there was a lot of surprise that could've happened in free response scenarios. The big relief was that as with AB, there were minimal comments otherwise. All the students felt like the material was accessible. We also had some very calming conversations about what it takes to show proficiency on this thing and I think that helped. I really hope some of them do well and that there are some universal good results for the whole group.

In their exit comments, they did mention that they'd like assessments to be a little more intense. Throughout the year all of their assessments were collaborative (sometimes with notes, sometimes without). In my mind it made the most sense for such a small group, with only 15 students the grades shouldn't be important, the focus should be on collective understanding. In effect, their request was for me to force them to be stronger individuals, as a few noted that while they understood what was going on, they found themselves becoming dependent on others. Interesting to seem them recognize this with no connection at all to the "I need to know I have a better grade than other people" mindset.

New Frontiers

As the old ones leave, it's time to start thinking about the ones that will come to take their place. I had a meeting with the BC students of 2018-19 and they all seem very excited. Especially when I said not only would they be getting their own personal calculator (not for keeps, but for use throughout the year), but that they could give it a goofy name. Varsity Math is proving a successful recruiting tool, with Statistics numbers finally headed to the right direction (30 next year, up from 9 this year) and kids pumped to be involved in all of our AP offerings. Summer Camp enters year three, and it continues to be a fun way to onboard kids into the Varsity Math universe.

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AuthorJonathan Claydon

Every spring, during a brief period of time when it's very pleasant outside before the ravages of summer, a few hundred students venture outside and make over a half mile of public art. We call it Sidewalk Chalk Day.

Sidewalk Chalk Adventures
Return of Sidewalk Chalk
Sidewalk Chalk Three
Sidewalk Chalk, the Fourth One
Sidewalk Chalk Five
Sidewalk Chalk Six

Now etched into school tradition, Sidewalk Chalk Day features students from all kinds of math classes displaying graphs of whatever it is they have been learning recently. As one faculty member who was strolling outside as we worked put it, "I always know it's spring when the chalk goes down." The first iteration involved two sections of Pre-Cal. This year 11 classes (AB Calculus, BC Calculus, Pre-Calculus PreAP, Pre-Calculus, and College Algebra) went outside throughout the school day.

Setup

Myself and a colleague pick the day a couple weeks in advance. It's trickier to pick a day than you might think. Because of our bell schedule, only Tuesdays and Fridays work, and, as the subject that started the movement, it has to coincide with Pre-Calculus classes wrapping up Polar Equations. Go to early and the material isn't covered thoroughly enough. Wait too long and suddenly it's testing season and nothing works.

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Day in mind, we incorporate a unit that will culminate with students graphing something. Pre-Calculus students create a pair of polar equations (rose curves/cardioids or something in between); AB Calc creates regions between curves with stated integrals for area and volume using that region; BC Calc creates polar equations with integrals of area for them; and College Algebra showed off logarithms and their corresponding inverses. Each student created their own set of equations/graphs that fit the requirements. They were given two panels (or one double panel) of sidewalk to show off their work. If complete, extra panels could be decorated however they like.

While students worked I flew around a drone and took pictures. Last year there I made a video compilation. The weather didn't cooperate this year, so I had to settle for pictures. Enjoy!

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AuthorJonathan Claydon

This school year has been a bit weird. I scaled back my duties yet I still have a lot of work to do outside of school. What gives? Well, each of my preps has presented some interesting challenges.

College Algebra

By far the class I underestimated the most. Not from a material point of view, but how the students would consume it. Turns out, whole group instruction doesn't really work here, I could blab all I want and some percentage is still going to need an individual explanation. In January, I started catering to that need. The challenge has been creating material that offers students a script they can follow and leaves me opportunities to have a conceptual discussion with them. I do not want it to devolve into a "do this, ok great, moving on" kind of thing. I want to retain the discussion aspect and that is requiring a lot of energy, as I have the same conversation 4 times in a class period. Pacing is still weird, many finishing assignments quickly, others taking their time (distracting themselves has started to play a part in this).

But on the plus side, the group in general is understanding new concepts, I'm probably saving time by cutting out the note taking, and it keeps the atmosphere really casual. Everyone still gets tons of time to work and enough face time with me than they can stand.

Calculus AB

Though we are slowly turning a corner, there's a lot of work to do here. I'm using yet another assessment system, and employ Desmos for benchmarks. Until we reach some future performance nirvana, I'm going to keep tweaking my approach in this course. Despite missing all that time at the beginning of the year and then another two days for ice, I think we're in a good place. It's that magical time of year when all the concepts finally start to make sense and you can discuss free response questions without major headaches. In a couple weeks I'll see if I was able to grow my exam participation (last year was 50%).

Calculus BC

We've entered uncharted water. We cruised through the AB material with little problem and now we have to tackle the material that's unique for them. It feels like we have eons of time to get ready. Since the class is small and they're all doing great, there isn't same benchmark mechanic necessary here, they're all taking the exam. As much as it pained me, I decided to bite the bullet and jump into sequences and series now before it starts to feel like we're running out of time. It is my weakest content area, so there's been a lot of studying and restudying and restudying to try and make sure I have it all straight. I knew it was going to be a rough period. But slowly, the light at the end approaches.

Common Theme

What's the problem really then? I have to make TONS of stuff. College Algebra is loosely based on my last attempts at Algebra 2 but the "no talking" structure of the class has really made me rethink a lot of things. Pacing still throws me. Calculus BC is all new territory as I relearn the material myself AND come up with ways to teach it. Calc AB is the one comfort zone, I know what I need to do, but there's still a lot of retooling required.

One day I will take a summer and write some kind of definitive practice book I can use throughout the school year, but that day has yet to come.

 

Posted
AuthorJonathan Claydon

Though I don't write about it much, I've been teaching College Algebra this year. The kids aren't earning college credit for it, it's more of an alternative to Pre Cal we're offering for students that need another year of algebra. The Pre Cal we teach isn't like some places where you're doing Algebra 2 all over again, so our version of College Algebra makes a nice option.

Interestingly enough, I have two very small sections, 18 and 16 kids in each. It's been a long time since I've had classes this small. We had a bit of a population boom since I started and some quirks happened to bring class sizes down this year despite a very high overall population. The biggest lesson I learned from those early years with small classes is that they were totally wasted on early teaching me. I always wanted a chance to do a better job with that kind of environment.

The first challenge was their work pace. These kids are great, but they're all over the place. I developed a technique for dealing with that last semester.

The new challenge became holding their attention. For whatever reason, they aren't interesting in listening to me speak in front of the group in a traditional sense. I noticed this back in 2013 when I reframed Algebra II around lots and lots and lots of classwork. The kids got so used to knowing math class would require them to do something, that listening became problematic. They just wanted the classwork and it was crazy. Same thing here, this College Algebra crew eats up their classwork. But lecturing? Forget it.

Strategy

How did I fix it? Let's look at a multi-part assignment I gave for inverses:

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On the surface, this looks like bad textbook worksheet moves. Steps are labeled. Kids follow the steps. Repeat.

The Mini Lesson

As mentioned, these groups work at varying paces. As the semester has gone on, they've gravitated to sitting with kids who move at their same pace. Generally, I know which table is going to finish first and which takes some time. This frees me to wander, stop, and initiate a discussion.

This multi-part assignment came with three discussion phases. As a group finished a part, we discussed what they learned. In Part 1, we made some observations about the data points from the table. We ironed out the relationship I asked them to observe, many noticing right away that it looked like some kind of reflection. I floated around and had the Part 1 discussion several times, with audiences of 3-6 kids each time.

For Part 2, we expanded the idea. Can entire equations be reflections of one another? Do the observations about points still hold true? Are the points of one equation just reflections of another? Could we come up with a counter example?

Part 3 is a culmination of the first two discussions. Now that we seem to have a definition of inverses in general, can you determine them on your own and check your work with a graph.

Observations

My classroom is uniquely designed for this kind of set up. Near each table is a screen replicating my teaching computer. I can carry a keyboard and trackpad with me to manipulate stuff from wherever. I can also make use of their work as we talk since it's right there with us at their table.

I think it went really well. It was easy to get the kids to focus because there's no hiding when it's a group of three, or even six. The kids were eager to share their observations and made some good ones, all catching on quickly. The pacing allowed kids to work independently and have a piece of my attention. In this classwork heavy setting, I have been very free to offer more face time than these kids could ever want.

From an outside perspective it probably looks chaotic. From a breakthrough perspective it's not that special. It's just some scripted tasks leading kids towards an observation. Kids do some work, practice some things, get tested over those things. The key part I think is that the teaching element hasn't gone away. I'm still their main source of information, rather than videos or whatever. I'm still facilitating discussion and offering new ideas even if the tasks aren't particularly exciting, such is the nitty gritty algebra sometimes. And the kids are more than willing to give everything a try, mistakes are no big deal when it's just a few people sitting around a table.

I like where this is going.

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AuthorJonathan Claydon