Some Background: I’m using Khan Academy with my ELL Math classes because their exercises are pretty good for remediation with language (especially the Early Math content) and it is self-paced, which gives my students some freedom to progress or practice as they need. But, since my students have severe language barriers, I need to curate the curriculum that I give them so it emphasizes visual and de-emphasizes language. I’ve created my own progression through Khan Academy, which I have them keep track of via the checklists in This Post.
It’s one thing to look at the checklists and verify that a student completed an exercise, but I also want to be able to double-check using Khan Academy’s robust coaching system. This leads to A Problem: All of Khan Academy’s reports are synced to their Missions and their exercise progression, not to my custom exercise progression that I want my students to move through. This also means that when I download the reports from Khan Academy, they include every exercise that a student has completed, including the ones that I don’t want them to do. The data is a bit overwhelming and it is time-consuming to parse out only the exercises that I want. I need a way to filter out the unwanted exercises and organize by my own exercise progression. I especially need this for grading and accountability.
The Solution: I wrote a Python program that generates custom Khan Academy reports that are aligned to my own custom curriculum. The reports look like this. And, if you’re someone who is doing something similar, you can probably use this too.
How I Did It:
First, I needed to download the data on my students, which I could do from the Student Progress section of the Coaching dashboard. I needed to make sure I had ‘All Students’ selected and I needed to make sure I set the Activity window to ‘All Time’.
This file automatically downloads as an excel file with 5 worksheet tabs at the bottom. I deleted all of the worksheets except the Exercise worksheet, then saved the file as a Comma Separated Value (.csv) file. When I was done, it looked like this. In order for the program to work, this file needs to be a csv file that is only the exercise section of the Khan Academy report.
Second: I created a text file that had all of the exercises that I cared about in it. It looks like this. Each line is a title of a Khan Academy exercise, which I took directly from the Khan Academy report. The lines with hashtags in front of them are the different categories that I give my students (see this blog post for more info). When the report is generated, these hashtagged lines will be the ‘headers’ of the table that is created.
Third: I run the Python File, which can be found here. The program works by first asking you for your student data (the .csv file), then asking for the list of exercises (I called it the MasterRecord). From there, the program looks through your students and compares it to the master exercises, then creates an HTML document that has it all organized for you. Each student is saved as a separate document in the same location as the Python file (so if you save it on your desktop, you’ll end up with a ton of new HTML files on your desktop – be careful!). Again, the report ultimately ends up looking like this.
From here, I can use this for grading purposes or intervention purposes or to share with students.
If You Wanted to Do This Too: You would need to follow the instructions in the First step to download your students’ data and save it as a .csv, then you would need to create your own MasterRecord.txt file (you could probably download mine and edit it, but make sure there aren’t any typos), then you would need to download my Python File and run it. If all goes well, you’ll have your own custom reports.
All the Files in one Place:
A sample report: http://schneiderisawesome.com/blog/KAReport_amphitest11.html
A sample Khan Academy .csv file: http://schneiderisawesome.com/blog/KhanReportCSVBlog.csvMy MasterRecord.txt document: http://schneiderisawesome.com/blog/MasterRecord.txt
The Python Program: http://schneiderisawesome.com/blog/KhanReport.py
I’m still figuring lots of stuff out with my ELL mostly-refugee students and I’m loving that I get to process it all through this blog. Here’s what’s on my mind:
We started going fractions, which has been going really well. My general classroom structure is to pick a topic for the week, go over it every day, then quiz on it on Friday. Last week was just representing fractions – things like writing a fraction given shaded pieces or given a point on a divided number line. I made these Geogebra programs to help me with the practice problems and to keep the visual theme going:
Writing Fractions from Visuals: http://tube.geogebra.org/material/show/id/1754581
Writing Fractions from Number Line: http://tube.geogebra.org/material/show/id/1754587
Once we had the concept of identifying fractions, we moved onto combining fractions with common denominators. Here are some artifacts from how that went:
- If I had it to do again, I’d add a third box to the worksheets for students to draw their answers too.
- I love that I was able to circle back to positives and negatives, but now in the context of fractions – so, I get to hit the skill again, but not in a way that seems repetitive and like we’re spinning our wheels moving nowhere. It was also a really easy transition to represent fractions with +’s and -‘s instead of just shading them in, so our visual language of ‘zeroing out’ was re-used with these fractions.
- Most of problems were along the lines of: given a symbolic fraction problem, draw a picture to help answer it. However, my favorite types of problems were actually the reverse: given two pictures of fractions, write the symbolic representation and then answer it. These really helped cement the visual language I wanted them to use.
- For the most part, the students who drew pictures got correct answers, whereas the students that didn’t tended to get incorrect answers (not pictured). Even some of my more advanced students reverted to drawing pictures to check their work. Seeing that was one of the most tangible manifestation of one of my biggest overall teaching philosophies: teach a representation and rules on that representation, then let students recreate that representation to solve problems. This is basically my philosophy behind everything in this class – the symbols have a visual representation and rules on how they interact which gives you the answer. The students who took the time to create the representation tended to get correct answers – the students who moved too fast got incorrect answers.
- Some students still struggle to identify the sign of numbers without anything in front of it. For example, in the expression “3 – 5 + 4”, students are confident that the 5 is negative and the 4 is positive, but are unsure of the 3. This is curious to me and I don’t really know how to fix it other than “if there’s nothing there, it’s always positive!”, which is an arbitrary rule and is hard to communicate in the absence of a common language. For some students, I wonder if its because their native language reads right-to-left whereas they are suddenly learning a language that reads left-to-right.
The plan after this is to go into fraction multiplication, then into combining fractions with different denominators. For a while I was struggling with how to teach this visually, but this demo lesson from ST Math was invaluable in informing how I’ve been teaching fractions: http://www.mindresearch.org/play/. I use it with every student now, even my non-refugee students.
Also – I wish I could erase the part of my brain that wants to draw a circle as the default way to represent fractions. From a pedagogy standpoint, everything is much easier to teach if I default to drawing an array of rectangles of a number line (but especially an array of rectangles). A rectangle divided into fractions segues segues to fraction multiplication easier, it segues to decimals easier, and its easier to draw and manipulate if I make a mistake while drawing.
When I’m not teaching full-group lessons, my students work self-paced on Khan Academy (more info here). The self-paced aspect is working great as is the alignment between what I’m doing full-group and what they work on individually. For the earlier exercises, KA also has lots of different types of exercises for students to work on with multiple representations, so students get lots of practice on the same thing even though they’re progressing through the curriculum.
In general, I think the ‘Missions’ are pretty useless with my students – there isn’t a lot of logic to how the problems are generated and it all seems chaotic. My students get frustrated and want to give up in the face of being unsuccessful and not seeing how the previous exercises connect to the next exercises. However, if students have been working through individual exercises, then they can use the ‘Mastery Challenges’ to revisit exercises and gain ‘mastery’, which I like. I had my students do this for a week and had them skip the ones they had never seen before. It was especially interesting to see them work on the Early Math problems – many of them are explicitly language based (like these ones on ‘shorter’ and ‘longer’ and ‘bigger’ and ‘smaller’), so many of my students learned how to use Google Translate to answer these questions, which I thought was a valuable teaching moment even if it wasn’t necessarily a ‘mathematical’ teaching moment.
Khan is also really good at adding new exercises – almost as if they read my mind, they added a Multiplication Using Array’s exercise this week, which is pretty much exactly how I taught multiplication to students. But – I wish there was a place where I could see when new exercises are added or updated (I asked them on twitter, but no response). I just happened to ‘discover’ these ones – it’d be great to receive email updates or an rss feed or something when new exercises are added.
Christopher Danielson recommended this program called ST Math for this class since their philosophy is almost entirely aligned to my goals in this class: start with the visual, then add in the symbolic later. Running through the Demo lesson linked above, it seems like a pretty awesome program for this very specific demographic. I’ve even convinced my school to look into purchasing it, but we’re having a lot of trouble getting a hold of someone from the company who we can talk to about buying the program. So, if anyone from ST Math happens to read this, I’d love to get in touch to look into using this software in my classes.
We started Multiplication this week. I needed a quick way to determine if students knew their multiplication tables or not that segued quickly into me working with the students who had no concept of multiplication. I decided to give 3-minute Multiplication Fact quizzes all week – the students who knew their facts completed the quizzes quickly and bought in from the challenge of improving their scores and the competition of comparing their numbers with their peers. The students who needed me to teach them multiplication were clueless on the quiz, but since it only lasted 3 minutes, I could quickly see how they were doing and start remediating immediately. By the end of the week, here’s what their work looked like:
- I defaulted to showing the area model because it’s the most visual of the models. Most kids gravitated to that. By the end of the week, some students got tired of drawing all the circles and wanted a faster method, which let me show them how keeping track of each column let you build a multiplication chart so you didn’t have to draw circles for every problem.
- Some students used the ‘count the tallies to create a chart’ method (similar to skip counting), but since language is an issue with my students, counting was also an issue and these students tended to make smaller counting mistakes (pointed out about with red arrows). The students who used the visual models tended to fair better than the students who tried to rely purely on the symbolic/procedural models.
- In one of the pictures above is the Lattice Method, but its crossed out. I showed this to several students, but no one latched onto it, which I thought was curious because it’s also very visual. In thinking about next week (division), I’m actually kinda glad no one latched onto it – it’s pretty straightforward to show how to divide using the same area model and the tally model that you use for multiplication, but the lattice method is a little less straight-forward to reverse-engineer to get division.
- Check these out:
This is from my student who had absolutely no concept of multiplication – she thought the ‘x’ symbol still meant ‘add’. I showed her the area model, but she has trouble counting past 15, so I also told her to cross off circles as she counted them – every 10th circle, write ’10’ on the side, then start over. Using this strategy, she was basically grouping circles by 10s, then adding all the groups together to get her answer. This was the basis for my comment on twitter: I’ve never appreciated place value and grouping-by-10 than when my students can’t reliably count past 15.
- I thought this was curious: several students used this strategy
In other words: students doubled one multiplier while halving the other multiplier, then added the former number as many times as the latter number. So 7 x 8 is the same as 14 x 4 which is 14 + 14 + 14 + 14. I was surprised several students did this by default without needing to be shown this ‘trick’ – very clever.
- I give a quiz every Friday over whatever weekly skill I decide to cover. This week was a 20 question quiz that was all multiplication problems. Every student passed, even the ones who spent the whole period creating their models and only learned multiplication on the Tuesday (we had Monday off from Labor Day). I feel like this is the only way this class can work – every skill needs a model that students can create on their own and use to solve problems, even if it means it takes much longer than might be considered ‘reasonable’.
Artifacts From This Week:
I think my favorite thing this year will be collecting the work that these students do as they solve problems. So far, it’s completely fascinating. This week was addition & subtraction – here are some artifacts from the week:
This last picture is fascinating to me:
The red and green work are subtraction problems without borrowing, which this student got incorrect. The blue and yellow problems are subtraction problems with borrowing, which the student got correct. So somehow, in trying to create a visual intuition about subtraction in order to motivate the concept of borrowing (which looks like it was a success), I un-taught this student their original intuition for subtraction without borrowing. I guess we’ll work on this next week.
Next week is multiplication. Some students know their multiplication facts already; some students have no idea where to start. It’ll be a curious week.
Some Khan Academy Things
I’ve got a Wall of Champions – Khan Academy Version going in my classroom:
Some notes & clarifications:
- I only use Khan Academy for the exercises – we don’t watch the videos, and I don’t really want my students to either. However – the ‘hints’ that are provided for each question are the most useful for my students in terms of feedback and learning a process on their own. Some students, when they see something they don’t understand, display all of the hints and then follow through the problem to see how it was solved. I have two students who have started displaying the entire hint text, then copy and pasting the text into Google Translate, then reading the explanation in their native language. Another neat feature of Google Translate is you can highlight particular words or phrases and it’ll show you the corresponding word or phrase in the other language. So if a student sees two unfamiliar words in English (like numerator and denominator), then sees unfamiliar words in their language – they don’t have to guess which word is which: google translate will highlight the text and they can know for certain.
- That last bullet points is too long. A shorter way to have said that is: we don’t use Khan Academy videos for instructional purposes ever. Some students watch them because they want to learn the English, not because I want them to learn the math. However, the hints in the exercises have much more instructional value, especially when paired with Google Translate.
- My kids are really enjoying the structure of the Khan exercises – the isolated skills and repeated problems, that they have positive reinforcement and some gamification elements, that they can compare their progress with others, that they have tangible goals and intangible rewards, and that they can work at their own pace despite the clear language barriers.
- I am also really enjoying the structure of the Khan exercises because it takes a lot of the management of a differentiated classroom out of the picture, letting me focus more on helping the students who need the most help and need things taught at a fundamental level, while letting the students with a strong mathematical background progress at their own pace and help each other out.
- I wish Khan Academy wouldn’t consider a skill ‘practiced’ if they get the very first problem correct, even though I understand the intent of allowing students to move quickly through content they already know. I wish it was ‘first 2 correct’.
Something that’s been fun for me is: the goal is to eventually prepare these students to enter an Algebra I class once their language and fundamental skills catch up, which means I’m basically teaching a condensed 3rd-8th grade curriculum to a group of motivated, intelligent students – and, since I know how things in 4th grade (like multiplying multi-digit numbers) connect to things in high school (like multiplying binomials), I can be very purposeful with how I present certain topics and how the ground work is laid for future conversations (like using the box method, which can be used for both skills). Most traditional students have to wait almost a decade before this connection is made. My ELL students will have to wait a year at the most before this connection is made – a much shorter amount of time – and I’m curious how the ‘quickness’ of this connection will effect how well they internalize the concepts.
Paging Christopher Danielson
I just want to put out there that, of all the potential people who read this series and react to it and have feedback or pushback, Christopher Danielson is someone whom I am most interested in hearing from (he’s already provided some neat insights that I’m looking into). I’m positive that many of the problems and solutions and strategies that I will end seeing this year will overlap almost entirely with the same problems and solutions and strategies that one would see when presenting mathematics to a child for the first time, which is one of several niches that Chris is a part of (Have you seen Talking Math with your Kids?). The thing I’m curious about is: if/how these strategies break down as I adopt them for my ELL demographic. Next week, as I teach an older ELL student (with broad experiences in the world) how to represent multiplication for the first time, can I do it the same way I would for an English-speaking child who is learning multiplication for the first time? The nuances of this situation, if they exist, are curious to me.
Thanks for reading
If you’re wondering what the first 3 weeks of a math class for primarily refugee ELL students who don’t speak any English and several possible languages (arabic, spanish, kinyarwanda, somali, swahili, kirundi, etc) – it looks like this:
Here are some thoughts and explanations and etc:
- In the absence of being able to communicate in a common spoken language, I’ve been working on developing a common visual language to describe mathematics. Two places where I was already familiar with this were: positive and negative numbers, and place value. Which is what you’re looking at.
- I decided to teach integers using physical tokens (closed circles are positive, open circles are negative) rather than a number line approach – I think in my mind I briefly rationalized that it would take less words to describe what’s happening than if I used the number line. My approach is very similar to Kate Nowak’s from this video.
- Positives & Negatives segued very nicely to place value – the inconvenience of drawing 50 dots leads to the desire to represent numbers in groups of 10s.
- These students can explain their answers to each other with only the words “open. negative. closed. positive”, which is awesome. Actually, This is an interesting pedagogical problem: imagine you are teaching a new topic to students who don’t know any of the words you’re about to use. What is the minimum number of words you need in order to communicate the idea (you are allowed infinite body gestures and pictures) and what are the minimum number of words needed so students can explain their answers to each other.
- Now that we know place value, I can check in on how well they understand multi-digit addition & subtraction. Carrying and Borrowing mean even less when students don’t even know the words – visually regrouping is a better way to communicate.
When I’m not teaching full-group lessons, students need something with a low language threshold, based in a visual language, and differentiated so students with a strong mathematical background from their own country can advance while the students with a weak background can get feedback and work on the problems they need.
There’s actually a pretty stellar solution to this problem that hopefully doesn’t cause too many ideological waves: it’s Khan Academy. Here’s what we do:
You can also access these on my class website: http://www.schneiderisawesome.com, then clicking on Math Lab 6th and 7th Period.
I don’t know if Khan Academy realizes it or if they do this intentionally, but they’ve got some pretty stellar exercises for students with a low language threshold that need to learn both the language and the math at the same time. I’m thinking specifically of their Early Math exercises, focusing almost exclusively on the connection between symbol and language. And their videos, while not always great in content or pedagogy, usually have several options for translation, which helps students make connections between the words in their languages and the equivalent word in English.
They’ve also got some pretty clever exercises in the early math grades that emphasize connections between pictures and mathematical symbols, especially with some of their fraction exercises. I’ve been going through and vetting exercises specifically to avoid wordy exercises and to try and hit as many visual exercises as I can so I can use the visual language later to help them understand something.
So – if you’ve got a group of students who don’t speak English and are at varying degrees of mathematical ability, strategic use of Khan Academy is a pretty good idea.
More updates to come. This class is tons of fun.
This is for all of you out there who are teaching a class of English Language Learners, primarily refugee students who have been in the country for less than a year with a limited knowledge of both math and English and need something to do on the first day. If this describes you (anyone?), then boy is this a neat thing to do and we should talk some more about math strategies for this totally awesome and unique demographic. And if this doesn’t describe you, then maybe that first sentence intrigues you enough to keep reading.
I knew I wanted to have a word wall for this class. I knew I wanted to have some kind of language assessment on the first day. I knew I wanted to have some kind of math assessment on the first day. I knew I wanted to begin this year by validating that part of learning math is also learning the language that describes math, and translating between languages is a valuable skill.
I also did not know how fluent my students would be, what their previous math experiences were, or even if they had even been in school before. There is a lot of uncertainty on the first day with these classes.
So, I made this document:
(Basically, it asks everyone to write the word for numerical digits, mathematical operations, variables, and a few others in both their language and English)
We went through the top two sentences together so I could know what languages were in my room, then went through how to complete the first few lines of 0 and 1, then had them continue to get as far as they could. I answered any spelling questions on the board (ie: parenthesis) and helped them fill in the English side, and let them fill in the side for their own language.
As they finished, I grouped students by common language and had them compare, then gave them post-its (a different color for each language) and had them write their words on the post-its, then put them on the wall I had already created. The result looks something like this:
Each post-it is a different language. They are: kinyarwanda, somali, swahili, french, and spanish.
If a student didn’t know a word in their own language, I got them on a computer (oh – there are computers in this class – most of the work they’ll do will be paced on a computer. That’s another story) and had them find an online translator, then look up how to translate from the English word into their own language. This was an unintended consequence of this activity, but a good one – I knew eventually I wanted students to be comfortable accessing translators, but I hadn’t intended it to be something that happened on the first day in this activity. So, getting them on a translating website on the first day was a nice added benefit to this.
When I first had the wall setup, I didn’t have the English words written out yet – I had planned to write those myself with the students during the activity. But, what ended up happening was I had one student who didn’t know any of the words in either English or her own language, so I had her write the English words on the notecards (which explains why some of the words are slightly misspelled in the pictures above) and match them to the correct symbol, which was a good use of her time in starting to learn the words for each symbol.
Other Things That Happened:
- I learned ‘Zero’ is basically the same in every language
- The letters on keyboards are capital letters – which means if I ask a student to type something but I write it in lower case, they can’t find the keys to type it in.
- I was expecting most students to know these symbols in their own language but maybe not in English. The opposite was true – more students than I expected didn’t know these words in their language but did know them in English (but couldn’t spell them). This tells me a lot about the students in my room and what to expect, and validates this activity as a really excellent pre-assessment.
- A cool thing that happened: There were two girls who spoke the same language – girl A was very timid and didn’t understand a lot of English, girl B was more involved and interactive and had clearly been in the country for longer than girl A. During this activity, girl B knew all the English words, but not the words in her own language – but, girl A knew the words in her language but not in English. So, a neat peer-teaching moment arose as they worked together to teach each other the words in the different languages, and I’m hoping this inspired some confidence in girl A to engage more with her peers and with the class.
- The word wall has already come in handy – students could refer to it when we did some translation exercises the next day (ie: what is ‘five plus three equals eight’ written in math?), especially for the new students who came into my class the next day.
One of the ways that I teach solving linear equations (things like 2x + 4 = 3x – 5) is by using balance puzzles:
squares are x’s, circles are constants. The puzzle above is the same as 2x + 2 = 8. The solution, at the bottom, is the same as x = 3. Credit: the puzzle above, as well as this whole idea, came from James Tanton’s book Math Without Words.
This ‘puzzle’ way of introducing equations is great for my role as an intervention teacher because it ‘tricks’ students into solving algebra problems without them realizing it.
But, when I went looking around the internet for more puzzles like these, I couldn’t find very many, which made me very sad.
So, I made a website that generates these puzzles for me. And, even better, I can use these generators in class with students as we solve puzzles together. Here, see for yourself:
How I Use These:
These puzzles have a very simple, concrete set of rules: equal terms on opposite sides weigh the same and can be ‘canceled’ out; equal terms can be added to both sides of the balance since they weigh the same; positive and negative terms ‘zero’ out when they’re on the same side of a balance; the puzzle is ‘solved’ when the circles and squares are on separate sides of the balance.
I find my students come to me with a very procedural understanding of algebra – it’s a series of arbitrary rules that don’t make sense and somehow get an answer that the teacher cares about but doesn’t have any personal meaning to me. I use these puzzles as a way to bypass this very negative mentality, and I use the puzzles to make the algebra concrete for the student again. X’s and numbers stop being arbitrary symbols and start being squares and circles (which explains why you can’t combine them). The equal sign is no longer this random symbol in an equation, but the divider between one side of a balance to the other side. This ‘negative’ perspective of algebra gradually gets overwritten with the positive memories of solving puzzles and explaining their reasoning.
I usually spend a day or two using these generators at the front of the classroom and doing problems with students. These days have been some of the most successful lessons I’ve ever done – students can verbalize how to solve the puzzle while I record their words in symbols on a whiteboard; soon their description isn’t in terms of circles and squares but in x’s and numbers; soon there’s no puzzle at all but an equation instead, but I can still go back to having students think of the puzzle if necessary (which is a big deal in terms of not stepping too far up the ladder of abstraction all at once).
Lastly, knowing the rules to the puzzle provides a self-checking mechanism for the rules of solving an algebra problem. If a student is unsure if they’re allowed to do something, I can relate it back to the puzzle and ask if they can make the same move in the puzzle. Students are usually more confident with how they would solve the puzzle rather than the equation, but this confidence slowly starts to transfer to the actual equation and soon they can speak with confidence about the rules of algebra that let them solve an equation.
My Lessons (.pdf) (multiple days)
Update: @Borschtwithanna shared this related and cool-looking resource with me: Mobile Puzzles. These, in turn, reminded me that another source of inspiration for this whole activity was Paul Salomon’s Imbalance Puzzles.
Disclaimer: I made these myself and they work for my Windows computer when I run Google Chrome. They also work on my Android phone. They also work on my SMART Board. I’m not a software designer who cares about checking these on every platform in every situation, so I sincerely hope they work for you too – but, if they don’t, I probably won’t spend a ton of time to fix it. You (yes you – reading this) are welcome to make your own edits if you’d like – I’d love to see these get better.
Well, I remember Do You Know Blue? because I still use it to teach a unit on number systems in a computer science class that I teach every summer. And I have been terribly upset because www.doyouknowblue.com is DEAD! It’s disappeared into the ether leaving me without an amazing amazing resource.
So, like any good programmer, I made a new one and you can find it, in pieces, with instructions, here.
To be clear: this is a big deal to me as a computer science teacher because one of the fundamental problem-solving strategies for a programmer is “how do trick this computer into doing something that I do naturally?”. This is at the heart of almost any programming endeavor and is a huge roadbloack to students. This is an amazingly difficult, fundamental, painstaking problem and simultaneously the source of every aspect of joy that comes from programming something correctly – tricking the computer to doing what I want it to do is the cause of 100% of the times I’ve hit my head on a lamp as I’ve leapt from my seat in celebration. But, getting students to appreciate how big a deal this is – that it takes hidden acrobatics to do even the simplest things – isn’t always an easy sell.
Which is what makes Do You Know Blue? so amazing – it effortlessly prompts students to consider how many hoops we need to jump through just to do something that we, as humans, do effortlessly. It emphasizes how easy it is for us to take for granted something that computers have absolutely no way of understanding (until we trick them). And further, the solution to this problem is completely disconnected from the concept of ‘color’ – we’re just manipulating numbers in a strange way that, in a happy accident, does what we want it to do (related: simulating dice rolls, simulating computer choice, anything having to do with computer graphics). These are big ideas.
But here’s the sad part: my version of Do You Know Blue is so unbelievably inferior to the original website that it breaks my heart. The original website was almost a precursor to the peer-interaction, scaffolded, seamless lessons that Desmos is producing like a boss. My websites work for me and the lesson I need to use them for, but that’s it. I don’t even know if they work on other internet browsers – I just use Google Chrome for everything. But, there they are, the 5 pieces of Do You Know Blue? that I’ll use again in a few weeks when I teach this class.
But, if you know any hungry techy developer folks who may want to take this and make it better, more interactive, more seamless, less clunky and boxy, then that would be a wonderful byproduct of having this out in the world. And I hope someone does make recreate the old Do You Know Blue? progression, because it was awesome and why should awesome things disappear from the world?
I’ve spent this year trying to teach a genuine Common Core Algebra I curriculum to high school freshman (my first time doing either of those) and I keep trying to find a way to write about with my experiences, but it’s hard not to get lost in all of the moving pieces that’ve happened this year. As the year wind downs, I guess the biggest thing I feel is: the Common Core shift is real and I feel it and I have a lot of uncertainty about what my students should leave my class with or that I’m preparing them in the best way for what’s coming. This post is about me really have to readjust what I thought the content of an Algebra I curriculum was, and trying to figure out what it needs to be now.
I used to have a feeling for what students didn’t know when they walked into Algebra I – things I could assume I needed to teach from scratch and things I could assume they had seen before and had some familiarity with. I used to know what the non-negotiables were for when they left my class and moved to Geometry or, much down the road, Algebra II. I used to know the balance between procedural fluency and conceptual understanding; when to ground something in a purely mathematical process versus something necessitated by a real-world situation (example: are logarithms motivated by the need to take the inverse of an exponential? Or by the need to measure sound frequencies on a decibel scale?) I used to know where the overlap between Algebra, Geometry, and Algebra II was – where to draw limits in Algebra I that would get picked up in Algebra II. I’m much less confident now and I’m not sure what to fall back on. I feel things needing to be done differently but not knowing how or to what extent.
A lot of the uncertainty I’ve felt this year is also the product of some outside forces that I’ve been reacting to. Most of my district has adopted the Carnegie Curriculum, complete with textbooks, support materials, and once-a-week computer time with the the Cognitive Tutor program. Our middle schools are using this curriculum as well, which then ties directly to the curriculum we teach in high school. I teach in Arizona, which adopted the Common Core Standards and aligned itself with the PARCC assessment consortia a few years ago. In March of 2014, Arizona withdrew from PARCC. In November 2014, they adopted a new test dubbed AZMerit, which is similar to how Utah and Florida are implementing their state tests (colleague and fellow blogger Jason Dyer has a play-by-play of what those tests looks like). In February 2015 (this year), Arizona almost repealed the Common Core standards – instead, a group will review and revise them for next year. As a result: it’s hard to find solid instructional ground while also adjusting to a new curriculum, somewhat-new standards, and a new assessment.
Right now, thinking about next year, there is a tension with every Algebra I topic and how it:
- aligns to the Common Core standards or my best prediction of what the Arizona standards might turn into
- aligns to the AZMerit test, for which all I know is this blueprint.
- aligns to my textbook & the common curriculum my district is trying to use, especially since our feeder schools are using this curriculum as well
- aligns to skills or knowledge that my students need to know in order to be successful in their future classes or future careers.
So – reflecting on this year, starting to think about adustments for next year, and (I’m realizing now) as a place to process my thoughts, here are some things I’ll just call The Shifts and some thoughts on how to react to them
Shift #1: Self-Contained Units
I think the biggest thing I feel as a result of all of these moving pieces is the need for units that have a clearly defined ending and ‘wrap everything up’ quality. There were some topics where I used to be able to say “You need to know this so we can do ______ later”, but I’ve lost all intuition about when these statements are true anymore. Instead, my best units this last year were the ones that built towards a specific problem to solve or scenario to investigate or project that tied everything together – that was a natural culmination of the material we had been covering and didn’t rely on “trust me – you’ll need this later”. Trying to motivate material with “You need to know this for Geometry / the next unit / the state test” was a failure because, honestly, I no longer have any idea if what I’m teaching right now will truly be necessary as we move forward (I’m looking at you inequalities and solving absolute value equations).
Instead, thinking back to last year, I can remember a few units that had this nice self-contained feeling (start, middle, end). These were the ones I enjoyed teaching the most and the ones with the most in-depth questions, investigations, and sense of independence from the students. Thinking about next year, I want more of these – units with a natural progression towards some kind of self-contained question/scenario/project and with a clear beginning, middle, and end. I don’t want the motivation to purely be “they’ll need this later” because, really, everything is so fluid that I don’t know if they’ll really need it later. I want students to know something because they need it now, in this moment – otherwise it becomes just another random rule or procedure to memorize without any internal connection to it.
Shift #2: More Application, Less Procedural
One big shift I see in how my textbook presents new content is that it is almost always grounded in investigating a problem or scenario that is real and has consequences. Systems of Equations are introduced via break-even points, inequalities are introduced via ‘at least’ or ‘at most’ problems, exponentials are grounded in patterns or ‘doubling’/’tripling’ scenarios. I’m used to ‘word problems’ or ‘real-world scenarios’ being the last topic of a unit or even a unit all on its own – but in the age of Common Core, I see these showing up more and more as the entry point of problems and then again as the finishing point. I’ve become a big fan of this for lots of reasons: it has a lower floor for students to discuss a problem, it eliminates the ‘here’s what we’re learning next because (arbitrary reason)’ style of curriculum, it’s more relevant to what they’ll see outside of school, and its easier to engage students in something concrete instead of something abstract. I plan on trying to model every unit in this way – find something for my students to dig their teeth into before barging forward with the math content, then circling back and using the content to re-examine these problems/scenarios and see how useful all that math really was.
Shift #3: More Calculator
Holy crap – this thing is a game changer. Arizona’s old high-stakes test was no-calculator, so students were taught methods to answer questions by hand. But now? Calculators can be used on our exams, which means calculator fluency is a big deal in my class. And, frankly, I’m really glad calculators can play a bigger role in an Algebra I class – they’re a legitimate tool and, from a planning perspective, they let me ramp up the intensity for the types of problems we solve and scenarios we investigate. It used to be that I had to choose problems where the numbers ‘worked out nicely’ or the graphs ‘fit nicely in the standard window’. This was me creating artificial blocks for myself and my students that aren’t realistic and aren’t valid anymore. Now I have more freedom and more tools to show my students so they can solve problems that matter.
Or, at least, this is what it should be. This last year: it wasn’t. I hardly did any calculator-based lessons with my students. I skipped the sections in the textbook that explicitly used the calculator to answer problems. When I would try, the lessons would drag on and on as I tried to troubleshoot calculator problems and keep the class together. These lessons also would show up in the middle of a chapter on something else, so the switch to calculators usually seemed random and forced and took some time to get used to. I was never sure what the payoff was going to be and I was still so used to students needing to know how to do things by hand that I just defaulted to teaching familiar lessons that could be done ‘easily’ by hand.
Thinking about next year, my default ‘do it by hand, easy numbers’ mindset needs to shift dramatically. I need to spend time getting students familiar with calculators and seeing them as a valuable tool to solve problems. I want to plan an entire unit which is just on using the calculator, specifically the graphing functions (finding mins/max, finding roots, finding intersections, using the table, adjusting the window, etc). I want students to see the calculator as a valid option as a way to start investigating a scenario. I think this is a big deal, and I’m looking forward to explicitly planning lessons around using a calculator.
Shift #4: More Statistics
This may be a revelation that’s more for me than other teachers who’ve done Algebra I before, but I never thought of statistics as being a vital part of the course. Things like representating data, measures of center (mean, median, mode), and linear regressions were more like an afterthought or extension or digression rather than an integral part of the curriculum (or, at least, that’s always what it looked like to me). The Common Core seems to have shifted this a fair amount (with drastic shifts happening in Algebra II). The idea of ‘big data’ and analyzing (rather than simply representing) this data seems to have taken a much bigger role in a Common Core Algebra I class. And, seeing what’s to come in the Algebra II standards (standard deviations, z-scores, normal curve analysis), there’s a responsibility to prepare students for these topics in the next levels.
I didn’t do any of that this year. Even though statistics is 17% of our AZMerit test, my class didn’t do a whole lot with data and regressions and measures of central tendency – these standards fell to the wayside as I desperately tried to prioritize my time and guess what students would need moving forward. But, I don’t think I can let this happen next year – in the bigger picture of preparing students for their next classes (especially Algebra II) and the real world, there needs to be a place to prepare students to look at data and interpret it. I think the standards have made a statement that these items no longer lie solely in the realm of a statistics class and, if I’m to genuinely teach the Common Core, I can’t have these units be an afterthought that eventually falls to the bottom of my priorities.
Edit 6/20: This Tweet pointed me in the direction of these resources: Publications in Statistics Education, which looks to be a collection of resources and publications aimed precisely at this issue: the new influx of statistics required by the Common Core and the lack of resources I have to present these standards. So – that’s awesome, and the reputation of the person sending the tweet (she blogs at statteacher.blogspot.com) is enough for me to take the recommendation seriously.
Shift #5: Is Algebra I necessary for Geometry?
This question really challenged me as the year wound down and I thought about the impact this year would have on my students. This question really comes from the shift I’ve seen in Geometry: these standards are now grounded in proof or transformations or real-world applications. Gone are the days where Geometry was an excuse to solve an algebra problem. Fading are the days where a student could be unsuccessful in geometry purely because they had weak algebra skills. With the new emphasis on reason and explanation and coordinate planes, a lot of the algebraic foundation that is given in Algebra I may not be needed in a Geometry class. I don’t remember any student telling me that they had to solve an algebra problem on the Geometry portion of the AZMerit exam.
Personally, I like that Geometry wants to become more of its own discipline, free from the chains of algebraic problems being forced into a geometric context. But, it makes me wonder about the types of skills students need to have as they enter this class. Do I need to impart algebraic skills as students prepare to enter Geometry? Or do I need to impart algebraic strategies for their geometry experiences? Will they be more likely to graph an equation or solve for x, or more likely to analyze a scenario and find an entry-point into the problem? From what I’ve seen this year, it seems to be the latter. This, also, is even more of an argument for self-contained units in Algbera I.
This, again, makes prioritizing units and standards a challenge because it’s unclear when a particular algebraic skill may pop-up in Geometry next year. I used to try and pick ‘non-negotiable’ standards that students absolutely needed for their future, but I’ve lost most of my intuition about what these are as I’ve watched the Geometry curriculum change. It also makes me wonder, radically, if our course progressions need to stay in this same Algbera I – Geometry – Algebra II progression that they’ve been for as long as I’ve been in school. Before the AZMerit exam, this progression was mandated purely by the fact that our old high-stakes test included standards primarily from Algebra I and Geometry and was only given at the end of their sophomore year. But, with narrower end-of-course exams given at the end of each year, what’s to stop a school from switching up the order? It’s much easier for me to think about how Algebra I transitions to Algebra II rather than how Algebra I leads into Geometry. Without derailing this post into an argument for rearranging the course structure, I will say that I’ve started to think about ‘non-negotiable’ skills in terms of what they need for Algebra II instead of Geometry, even though they still take Geometry before Algebra II.
Shift #6: We Need Better Teacher-Given Assessments
This has been one of my biggest frustrations/regrets/source of anxiety this year – the gnawing feeling that I could never capture what we my students were learning and how they could communicate that learning in a pen-and-paper test. Writing tests that are purely procedural does a disservice to the complex scenarios we’ve discussed in class and what I’ve witnessed in my room as students talk to each other and present their ideas. Designing a test with large open-ended problems is tricky and takes practice to phrase the question exactly the way you’d like it so you can really parse if a student knows something or if they don’t. I haven’t figured it out yet, but almost every test I gave ended with the feeling that “these questions were not aligned with what we’ve been doing in class”.
A lot of this, especially early on, was (I think) my own unfamiliarity with how much less procedural Algebra I has become. Standards like solving equation and graphing lines and even solving systems have moved down to 8th grade, which means Algebra I is reserved for applying those skills to situations and interpreting the results. I found myself leading classes that focused on using algebra to analyze a scenario and spending most of our time discussing this analysis, but then giving a test with problems that were solely algebra and devoid of any context. Trying to find a way to write questions which are more than just application of skills has been tricky, and sometimes leads to open-ended questions where students aren’t sure what I’m asking or, in answering the question, they don’t demonstrate the skills I’d like to see as they solve the problem. Finding this balance has been tough.
Thinking about assessments next year, I was pointed towards this document from Achieve the Core on Publisher’s Criteria for High School Mathematics and was drawn to the pages that talk about rigor (although I probably could have found this same revelation from other common core documents). Rigor is discussed as being composed of equal parts: conceptual understanding, procedural skills & fluency, and applications. Thinking about my past assessments, I was usually too heavy on one of these areas while ignoring the others – maybe one test was entirely problems devoid of context, while another had mostly open-ended scenarios for students to analyze and solve. Looking into next year, I want to try and design each assessment so it has pieces of each of these: some questions emphasizing skills & fluency, some questions emphasizing conceptual understanding (maybe finding mistakes? or agree/disagree & explain why? or compare/contrast?), and some questions with an application focus (probably grounded in some sort of real-world scenario). I’m hoping this will give me some guidance so I can avoid those tests that lead to me thinking “this doesn’t match what we were doing in class”.
I’ve spent this year trying to teach a genuine Common Core Algebra I curriculum to high school freshman (my first time doing either of those) and I keep trying to find a way to write about with my experiences, but it’s hard not to get lost in all of the moving pieces that’ve happened this year. As the year wind downs, I guess the biggest thing I feel is: the Common Core shift is real and I feel it and the demands of the standards are affecting the types of interactions I have in my classroom. This post is about the challenges of encouraging genuine student-centered discourse in my classroom (and, specifically, the unexpected challenge of getting students to listen to each other), then sharing a strategy I tried this year that I think worked pretty well.
Looking at the wording of the Common Core standards and watching the architects talk about them, I’m convinced that genuinely teaching these standards requires a classroom where students are (1) talking to each other, (2) listening to each other, and (3) able to communicate their ideas in writing. I see this in the level of independence and self-startedness that the Common Core demands in their modeling standards and, if I want my students to be successful with these, then I need students to see themselves as bringing something valuable to a conversation instead of relying solely on the teacher as the one-and-only-knower of all information. I see this in how the standards are written – verbs like ‘Discover’ and ‘Understand’ are peppered throughout the standards, both of which encourage communication and debate and explanation rather than answering exercises. I see this in some of the released PARCC sample items (this question has always stood out in my mind) and, in Arizona on our AZMerit exam, in the emphasis on questions coming from levels 2 and 3 on the Depth of Knowledge chart.
This is tricky because none of these actions – talking, listening, writing – directly involve me, the teacher: my students need to be doing these and, more importantly, with each other rather than just with me. And yet, all of these indirectly involve me because I need to provide the opportunities for these to happen, which means I need to be very intentional about the types of questions I ask, how I elicit responses from my students, and how I provide feedback to encourage these behaviors. Intentionally trying to get students to talk to each other has become embedded in my practice – I’ve read strategies, seen resources from other teachers (the blogotwittersphere is especially strong in strategies for fostering communication), and I’ve seen this modeled enough times that I think I have a pretty good handle on trying to get students talking to each other and to me. The listening piece, however, has been a new and unique challenge I wasn’t expecting, but I think I’ve convinced myself that its the most important – listening, evaluating, and responding the what someone else says in a meaningful and productive way. Man that’s tough to teach freshman to do.
Example: I was at a conference this week where this pattern was put on the board: “3, 2, 5, 4, 7, 6, …” and the audience was asked what the next number was (it’s 9). We were then asked to share how we got our answers, and these strategies were then shared with the group. Some people saw the sequence in pairs – (3, 2), (5, 4), (7, 6) – and therefore the next pair is (9, 8); some people saw it as two sequences smushed together (like a piecewise sequence) – 3, 5, 7, 9, … with 2, 4, 6, 8, …. smushed between the numbers; some people saw a numerical pattern – subtract 1, add 3; some people saw that each consecutive pair summed to a pattern of its own (3+2 = 5; 2+5=7; 5+4=9) and used that to generate the next number (6+?=15; ? = 9) [personal note: I found this strategy to be the most interesting and a new playground for generating sequences]. All of these are useful and insightful strategies. Most of us in the audience, being interested teachers and respectful adults, listened to the other strategies and thought “oh that’s clever” or “ehh – mine’s better”.
But – here’s how this experience might play out in my classroom: students take a moment to get their answers independently, they share with each other to see how their answers compare, and then I call on people. My students, having their answer, are confident that they can respond should I call on them – I imagine subconsciously they’re thinking “Okay – I have an answer if he calls on me, so My Work Is Done“. As I call around the room, they listen half-heartedly and evaluate how similar each response is to their own only because I end every conversation with “thumbs up if this answer was similar to the one you discussed with your partner”. At the end of this discussion, we’ve discussed about 3-4 different ways to approach this problem, but students are still latched onto their way to solve the problem in case I pull a surprise attack and still decide to call on them.
This leads to the challenge: my experience has been that students aren’t even able to repeat back these other responses if I asked them, much less understand what they mean and see their value. More so, my students don’t see the reason why they need to be intellectually responsible for responses different from their own. To them, they got an answer and could explain it, so their cognitive work is done – listening to different strategies and evaluating their reasonableness isn’t something they’re used to doing. This presents a challenge to me because maybe the goal of this activity was to introduce the idea of piecewise sequences and so the ‘two smushed sequences’ strategy is really important for us to talk about. And as I try to transition to the next part of the lesson, I’ve left behind everyone in the class who didn’t use that strategy and wasn’t listening when their peers presented it, but still feels satisfied that they did what I wanted them to do because they had their own answer and could explain it.
So, this has been my guiding question as I keep finding myself in these situations and trying to react: how do I encourage students to have a reason to really listen to their peers thinking and use that thinking as the lesson continues? How do I create a culture of ‘student-as-giver-of-new-information” instead of “teacher-is-only-person-who-can-give-new-information”?
Quick sidenote: it would be easy for me to add a teacher move saying “these are all great strategies, but the one we’re really going to talk about is … Let me explain it again to make sure everyone understands it before moving forward“, but there are some subtleties to this (especially the italicized part) that I have worries about. First, it reinforces that the teacher is still the most important person in terms of giving new information and, if I do this often enough, students will realize that they can always wait until the end of the discussion to listen to my little summary. Also, if I do this too often, it implies that there is a ‘correct answer’ that I’m looking for in this type of activity, which means my discussion of different strategies isn’t motivated by a genuine interest of how students solved the problem but by my search for the ‘correct strategy’ that we’re really going to talk about. I’m a big believer that once students have even the smallest suspicion of this deception, their motivation to take risks drops to 0 and their engagement bottoms out as they sit there wondering “when is he just going to tell us what he wants to talk about?”.
Other End Of the Spectrum: If I don’t do this teacher move – adding some teacher-guided explicitness to a specific strategy – then a lot of my students will struggle with the next activity/problem/task if they weren’t listening and evaluating different strategies (for example: if my next sequence was “1, 20, 4, 18, 7, 16, 10, 14”, the students who missed the ‘smushed sequence’ strategy would struggle much more than I want). This last year, I feel like I spent a lot of classroom time regrouping, re-eliciting the responses, then sending students back to work on their task. This takes time. I want this to take less time.
Anyway – this has been part of where my head has been – finding some good teacher moves to handle to situations above. I’ve been looking for resources and strategies to get students to listen to each other, and trying different things that I’ve invented in my own class. Here’s what I’ve tried so far (although I don’t think I’m anywhere near an expert yet):
Weekly Mathematics Discussion Notetaker
This was something I created to try and get students to listen to each other as we had class discussions. I expected it to be a purely extrinsic motivation (a grade), but it had some unintended consequences that led to some intrinsic motivations (more on that below). Anyway – students got it at the beginning of the week and turned it in at the end of the week. They were responsible for 5 quotes from other students where that student explained their thinking. This could be full-group or small-group. They couldn’t use themselves and they couldn’t use the teacher. Eventually, I tried (but failed (but will try again next year)) to add that they can’t use the same person twice during the week, hoping to encourage more people participating in discussions. This became a weekly grade and was one of the few things they couldn’t turn in late. I encouraged them to look for because statements and to think of this as recording notes that they could look at later, so the notes needed to be clear and complete. When I graded it, I didn’t look for correctness of the statements (so “The y-coordinate is 3 because the y-coordinate comes first” would still get full points) because that’s not the purpose of the assignment and, especially during some class discussions, incorrect things are said that are clarified later and I didn’t want to penalize students for not knowing the difference yet. I did, however, provide written feedback to clarify any incorrect quotes. This became a part of my students day-to-day operation in my class – they had their pencil, paper, and the discussion recorder. The back of the handout was a weekly planner because I thought that would be a good use of the space on the back, especially for my freshmen – it didn’t factor into their grade.
Some Observations: Overall, I’d call this a success with some really nice intended and unintended consequences. I noticed students paying more attention to class discussions which was the whole point of this. But, more than that, I noticed students being more conscious about phrasing their answers in complete sentences and using the word ‘because’ in their responses. In fact, this led to a new type of classroom interaction: I’d ask a student to answer a question and explain their answer; their answer is incomplete or mathematically imprecise (“what’s the better mathy word for that horizontal line on the coordinate plane?”); I keep pushing them until I’ve gotten a complete, precise response from the student but with lots of interruptions from me; AND THEN (here’s where the magic happens): I ask them to put it all in a complete sentence using the word ‘because’, which triggers the other students to get ready to write it down, and now I’ve got a student saying a complete, mathematically precise explanation and the rest of the class really paying attention to what they’re saying and how they’re saying it. But someone doesn’t hear it, so I make another student restate it, and now two people have said the complete, mathematically precise, student-generated explanation. Yes – this takes time, but I think it’s been worth it.
More consequences, most of them unintended: students got a sense of pride from being someone whose name gets written down on everyone’s paper as contributing a good response (next year, I think I want to make a wall of ‘good quotes’ to keep track of good responses through the year). My students who liked to answer questions and get called on started to get frustrated because: if they’re always answering questions, and they can’t write their own name down, then their discussion notetaker never gets filled out. So, these same very-active students started encouraging other people to interact in class so they could fill out their notetaker – it was a nice unintended student-driven reaction to the need for these to be filled out. Students also started holding each other accountable for more detailed responses – for example, let’s say I wanted a student to do a problem up at the board and talk through it. Maybe the discussion goes like this:
Student: “I did this. Then this. Then this. Then I got this and that’s the answer”
Me: “Did he/she say anything that you could write in your discussion notetaker?”
Me: “Are there any questions you’d like to ask him/her so you can add something to your discussion notetaker?”
(Students starts asking why and can you explain questions to the student presenter)
(Or: no one asks anything and the opportunity is gone)
I’d be remiss if I didn’t at least mention: I had to adjust a few things after the first few weeks as I saw lots of people trying to copy responses on the Friday due-date. I had a conversation with the classes re-emphasizing the point (to reward those who share during class discussions and who are actively paying attention), I explained that I would make adjustments for absences so you didn’t feel pressured to complete the whole thing just because you were absent, and I said it was okay to copy a response if it just happened and you missed the tail end of it. Then I laid out some consequences and it wasn’t really an issue anymore.
So – in addition to this whole discussion notetaker thing, I’ve also been looking at…
Resource: 5 Practices of Orchestrating Productive Mathematics Discussions. This is a pretty stellar book and I like it as a starting point for how to orchestrate tasks and activities that really require students to generate different strategies. It’s also got some great advice for how to respond to students so the teacher still sits in this role of facilitator and guide rather than taking on the role of information-giver. I don’t really know what else to say – if generating productive discourse is going to be a focus in my classroom, this book is important.
Resource: MAP Project PD Modules on Questioning and Student Collaboration. The PDFs linked in bold have some really great questioning and interaction strategies, which I think is half the battle. Students can only have genuine conversations if I ask the right type of questions and encourage the discussion in just the right way. Students will see themselves as independent learners once I learn how to interact with them in a way where I’m out of the way yet still holding them responsible and pushing them forward. I haven’t looked at these in-depth, but I think I need to.
Reflection: I don’t want to write a ton on this because it would be easy to dovetail into a curriculum conversation, but I guess I wanted to mention: as I think about the topics/lessons/days that have generated lots of discussion versus the ones that didn’t, I tend to think that the truly rich discussion days happen when the topic lends itself to connections between multiple representations – specifically, connecting to some kind of real-world context. Teaching parabolas as connected to different scenarios & contexts led to explanations that weren’t arbitrary or rule-based but were instead interpretations tied to the scenario that we were investigating. The same thing happened when I introduced equations to represent parabolas – having graphs and equations and scenarios available as resources for students to share their answers led to deeper explanations and discussions than if we were examining each topic separately. On the opposite end of the spectrum were my lessons on factoring and properties of numbers (commutative, associative, etc). I didn’t connect factoring to anything else other than a procedure (admittedly a bad teacher move), so the explanations during those weeks were really weak and un-insightful. The properties of numbers were very rule-based and definition-based, so it was hard to get explanations that were anything other than a restatement of the rule or definition. These units need work; units that work in multiple representations seem to be an easier playground for meaningful explanations and discussions.
Conclusion: I don’t really know if I have one, but I’m excited to start the discussion notetaker at the beginning of next year, I’m hoping to connect as many topics as I can to a real-world context or multiple representations, and I’ll keep trying to shift the role of information-provider away from me and onto my students.