So this year I’ve been teaching a brand new class called AP Computer Science Principles – and, it looks like next year my entire teaching load will be computer science classes. And that’s awesome and exciting and totally up my alley. Let’s talk about all of that.

Hey #MTBoS – long time, no see. I wanted to stop in and share something kinda cool I did with my AP Computer Science Principles class (wait – what?)

I’m teaching a brand new course this year, AP Computer Science Principles. I’ve mostly been following the curriculum provided by Code.org, which has been excellent – I dig their philosophy of providing open Creative Common licensed resources to benefit everyone, and I’m totally bought-in to their underlying principles of equity and ‘this is not just another coding class’. One of the big ideas of the course is **Big Data** – the idea that computer scientists manipulate and transform data into something presentable and look for actionable patterns or trends.

I had been looking around online for different ideas of how to address **Big Data** and, frankly, I wasn’t satisfied with what I was seeing. Most places suggested having students create a survey, have lots of people take it, then look at the data and perform some analysis on it to identify trends and patterns. I disliked this for two reasons, both of which come from my experiences as a math teacher and being acutely aware of *psuedocontext* – wrapping up a task in an inauthentic experience. Since the survey is a *means* to analyze the data rather than the true focus of the unit (as it might be in a statistics class), this almost necessitates that it be superficial and quick and and probably won’t lead to any truly meaningful insights – not great. I also didn’t like that a ‘large’ survey done this way would have maybe 100 data points, which isn’t anywhere near what a truly ‘large’ data set is in the computer science world.

If I was going to do this unit, I wanted students to look at *real* raw data on a scale where it is only feasible to use a computer to analyze it and whose analysis could provide *real* insights. So, I went around looking for raw data sources and found this Forbes article that pointed me to a lot of good places, but it wasn’t until I found FiveThityEight’s Elections page that I really got excited.

I don’t really know what else to title this post, but I’m about to share a few adventures I’ve had in improving my programming skills to make some problem-generators that’ll make my life easier. This is less of a reflective post and more of a “look at this!” post.

Our Algebra I curriculum starts the year by introducing students to a plethora of terminology to describe functions – linear, quadratic, increasing, maximum, continuous, discrete, etc. The curriculum then exposes students to a variety of graphs and asks them to classify them based on their properties. It then takes it a step further by developing stories to fit each type of graph – what type of story leads to a linear versus a exponential graph; what kinds of stories have maximums versus minimums; etc.

When I taught Algebra I, I found myself wanting to quiz students strictly on the vocabulary, which meant I needed to generate lots of graphs with lots of different properties and be able to categorize these graphs so I could discern correct vs incorrect answers.

This was my first attempt, made about a year ago: http://schneiderisawesome.com/desmos/oldGraphProperties/classifyinggraphs.html. It gets the job done, but its kinda hacked together. Drawing a continuous line by stringing together discrete points was fun. testing the checkboxes is done very literally. You can also click on the graph and a new window will pop up with the graph converted to an image, which let me copy and paste those into worksheets like this one.It worked okay.

Since then, I’ve increased my code-fu. I’ve learned Bootstrap, a bit of jQuery, and some Angular basics. Angular in particular had the annoying property of, initially, being more work and causing more confusion rather than saving time and effort. Only recently has it reached the point where I feel like I’m truly harnessing its power for time and efficiency.

But, the most exciting new thing I’ve started to play with is the Demos API. Having the power of the Desmos calculator at my fingertips was an incredible motivator to see what else I could come up with.

With Desmos as the catalyst, I updated my graph categorization program: http://schneiderisawesome.com/desmos/graphProperties/graphProperties.html. It’s way better and has more functionality, especially now that I don’t have to worry about the graphing part. I also really love how easy it is to add a ‘test’ button for people to test their predictions before submitting, since all I need to do from a programming standpoint is plug their expression into the Desmos calculator. I wish discrete graphs were easier to make and I wish there was a way to customize point size & line thickness – but other than that, things look gooood. So, now that’s out there for me to use with my students.

That was actually the second program I made with the Desmos calculator – the first one was focused solely on linear equations and was more of an experiment with Angular and Bootstrap rather than Desmos: http://schneiderisawesome.com/desmos/linearEqn/linearEqn.html. I wanted a way for students to test their abilities to write linear equations given two points, which is basically all that this program does.

So – now those programs are out in the open. They were fun to make and hopefully they’re somewhat useful.

Previously: ELL Math – 3 Weeks In

In between then and now:

- We’ve talked about fraction operations to decent success, decimals to mixed success, comparing numbers & comparing fractions to great success, and translating between English and Numerals (ie: fifty two thousand five hundred and six = 52,506) to decent success
- I have a group of students that I’m trying to prepare for Algebra I next year, who’ve been exposed to: order of operations, simplifying expressions by combining like terms and with the distributive property, solving 1-step equations, and writing linear equations from a context (ie: Joe has $5 and makes $2 per hour = 2x + 5).
- Less than half of the students I have now were with me at the start of the semester: many students have left, many students have joined. My class needs to be flexible enough to adapt to the changes.
- We’re still doing Khan Academy, which exposes students to other topics I haven’t taught as a full group: mixed numbers, simplifying fractions, etc.

Basically, at this point in the year, students have been exposed to a ton of different content and lots of different strategies to solve whatever math problems come their way. One of my favorite things is to see all the different strategies in action – one student may have latched onto a physical model for multiplication while another student does a repeated multiplication strategy; some students combine positives and negatives using the number line while another group may use a physical model by drawing open and closed circles. It’s been pretty fascinating watching all of these strategies play out and seeing which students have become comfortable with the procedural short-cuts that we try and move students towards (when comparing fractions, make the denominators the same and just compare the numerators) versus the students who still need that conceptual foundation to solve the problem (when comparing fractions, draw the two fractions and compare which one has the greater shaded area).

So, here’s a snapshot of a quiz I gave last week showing where we are and the strategies students are using to get here:

Some Comments:

- I’ve started splitting the class into A, B, and C groups so I can differentiate content and problems, which is why there are 3 different versions of the quiz. This helps me with a lot of logistical things I was trying to figure out in terms of only having one gradebook but several different levels in my room.
- Somewhere in there is a student who solved the comparing fractions problems by making a common denominator and comparing the numerators, but solved the ‘least to greatest’ problems by drawing the models. I wonder if that was intentional – if that student felt the models were easier for the least to greatest problems – or if that student didn’t realize that the former strategy can also be used for the latter problems
- Teaching students to create a common denominator has made me appreciate the difference between these two questions: “What is 18 divided by 2?” versus “2 times what equals 18?”. The latter question is the one that my students struggled with more than I had initially expected but, in retrospect, I see now that this latter question requires a different type of trial-and-error thinking than the former one, but you need this type of thinking if you want to solve the least to greatest problems in a procedural way. It made me re-think how I would teach multiplication next year – I would try to include more of these types of questions earlier to reinforce the relationship between multiplication & division.
- I’ve realized that the commutative property of multiplication is not as obvious as I’ve always believed it to be, especially when you create a model to represent the multiplication. I’ve had some good one-on-one discussions around this point.
- I’ve had a shift in opinion about the role of multiplication tables. I didn’t like that using tables removed some of the cognitive burden from all future multiplication problems or that, frankly, students would just use a neighbors table instead of making their own. Multiplication problems became an exercise in looking up data in a table rather than an application of a conceptual understanding of multiplication. I used to think that these tables were a crutch that students should be steered away from – instead, if they need to multiply something, draw a picture or make a model to solve the problem.But, when I would tell students to do this, I started to see is that students weren’t realizing that they can re-use their models for similar multiplication problems. For example: I would watch students multiply 7 x 6 by drawing 6 rows of 7 dots, then move on to multiply 7 x 7 by drawing 7 rows of 7 dots without realizing they could just add one more row to their previous drawing. And, since the commutative property of multiplication isn’t immediately obvious, 3 x 7 and 7 x 3 would result in two different models. This reliance on models and the time it took to draw them was starting to discourage students and distract from the other topics we were discussing (ie: adding & subtracting fractions with unlike denominators).
So, instead what I’ve started doing is: students can use premade tables during class, but they can’t use them on quizzes or tests. However, I will give them a blank table which they can fill in as part of the time it takes for them to complete their quiz. I like this system because they are still doing cognitive work and convincing me they know how to multiply when they make the table. This is also a totally valid strategy for any other test they take in any future class – if you need a multiplication table, take time to make your own.

- I discovered that the Marcy Math Pizzazz worksheets, while generally below my rigor expectations at the high school level, are a decent quick resource for additional problems at the middle & elementary school level. They are especially helpful because they allow for self-checking. Several students prepared for this quiz by working through problem sets from that book. It also makes my differentiation efforts a lot easier to manage.

So…. there are some updates on what I’ve been up to. Lots of fun.

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.

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:

**Fractions**

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:

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: 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.

**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.

**Artifacts**

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:

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:

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.

**The End**

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**. 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.