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ELL Math: 25123 x 81956

I regret having fallen out of the habit of posting student work from my ELL Math classes. This blog has missed the fun work we did with fractions and decimals. But, starting the semester, we did multi-digit multiplication. We did 2 digits x 2 digits, then 2 digits x 3 digits, then 3 digits x 3 digits. Today, without any explanation, students were presented with 25123 x 81956. It took most students the entire period. Here are a few choice student samples:

2016-01-14 11.36.21.jpg2016-01-14 12.10.46.jpg

Super fun. Super engaged. Perseverance all the way.

Spend A Semester In My Head

I have a pretty good relationship with the universities near my school – they tend to send pre-service teachers to check out my classes as part of their observation hours. This semester was no exception – I had a pre-service teacher come twice a week to watch my freshmen intervention classes.

Whenever I have observers, I give them a notebook during class to write down any questions they have about something they see in class but don’t really have an opportunity to ask about. Why did I answer this student’s question a certain way? Why did I handle this interaction a certain way? Why is the classroom arranged like this? Why did you teach this lesson in this way? What was going through your head when this happened? I imagine what it would be like if my room was being videotaped and we were watching the replay – the notebook has all of the questions they would want to ask during the replay, but can’t because we’re not actually recording my class.

The result is usually an interesting relic of things that have happened in my class – moments that I reacted to and hints at the lesson I was teaching. It’s also a reflection of the types of issues and questions that the observer has as they watch – are they looking with an eye to classroom management? to instructional delivery? to classroom arrangement? to curriculum choices? Are the questions big and philosophical and reflective and ideological? Or are they detail-oriented and logistical and fine-tuned to specific aspects of my classroom?

The pre-service teacher in my room this year had some pretty stellar questions (she’ll be an awesome teacher one day), which led to about 40 handwritten pages of me reflecting on lots of things in my classroom this semester. On re-reading the entire notebook, I realized it captures a lot of thoughts and beliefs and tangible things I do in my classroom as a 5th year teacher working mostly with intervention students. So,  I decided I wanted to immortalize it here in this post since I think it’s pretty fascinating and it’s another artifact in my curious evolution as a teacher, which has pretty much been entirely documented on this blog.

So – if you’re interested, here’s the entire question-and-answer notebook in a single document from Scribd:

And here is each week separated into separate PDF files:








Hope its worth the read. Sorry/not sorry for my handwriting.

Custom Khan Academy Reports

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:
A sample Khan Academy .csv file: MasterRecord.txt document:
The Python Program:

ELL Math – Fractions

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:

Writing Fractions from Number Line:

Once we had the concept of identifying fractions, we moved onto combining fractions with common denominators. Here are some artifacts from how that went:

2015-10-04 18.05.37 2015-10-04 18.05.50 2015-10-04 18.06.13 2015-10-04 18.06.34 2015-10-04 18.06.51 2015-10-04 18.06.59 2015-10-04 18.07.08 2015-10-04 18.07.25 2015-10-04 18.07.34

Some Notes:

  • 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: 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.

Khan Academy

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.

ST Math

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.

ELL Week 4 – Multiplication


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:

2015-09-08 17.05.42  2015-09-10 13.48.31 2015-09-10 13.48.57 2015-09-10 13.49.14    2015-09-10 13.50.30 2015-09-10 13.50.40

Some notes:

  • 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:



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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

2015-09-10 13.49.38

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’.

More ELL Math – Student Work, Khan Academy, Christopher Danielson

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:

AddSubWork1 AddSubWork2 AddSubWork6 AddSubWork3 AddSubWork4 AddSubWork5

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:

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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.

The End

Thanks for reading

ELL Math – 3 Weeks In

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:

2015-08-29 12.13.49  2015-08-30 18.40.54    2015-08-30 18.39.042015-08-29 12.15.062015-08-29 12.13.242015-08-29 12.12.202015-08-29 12.15.39

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:, 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.

Math Word Walls for ELL Students

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.

The Activity

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:

2015-08-07 16.08.52 2015-08-07 16.08.55 2015-08-07 16.09.03

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.

Balance Equation Puzzles

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:

Balance Puzzles – Positive Terms Only
Video Showing how to use the Puzzles

Balance Puzzles – Positive X’s, Negative Constants
Balance Puzzles – Positive & Negative Terms
Video Showing how to use the Puzzles

Create your Own Puzzles

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.

More Resources:

My Lessons (.pdf) (multiple days)

All Other Equation Resources (worksheets, lessons, etc) (.zip)

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.

Do You Know Blue?

Does anyone else remember Do You Know Blue? The lesson/activity/website put together by Dan Meyer, Dave Major, and inspired by Evan Weinberg? Do these posts ring any bells?

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 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?


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