I’ve spent this last year mostly out of the classroom doing quasi-administratory-things as a Math Interventionist. I’ve managed our online credit recovery program, gathered & analyzed data, and created personal relationships with our intervention students to the point where our school has a pretty robust math credit recovery program for our students who fall behind or transfer students whose credits never made it with them. I’ve also helped bring ST Math to our school (based in a large part on a recommendation from Christopher Danielson) which has been a huge boost to our ELL Math program we’ve developed. I’ve written grants for manipulatives, recruited tutors to participate in the class, and gotten to the point where basically a team of 3 teachers can support our 70 ELL students who are below grade level in both language and math, trying to get them ready for high school credits before they age out.

All of this has made engaging with this blog in the past year really difficult. I don’t have great lessons to post or student misconceptions to puzzle over. I haven’t been making worksheets that artfully scaffold basic understanding to procedural fluency to deep mathematical extensions that I can share (who am I kidding – I barely did that when I was blogging more regularly). The challenges I’ve faced throughout the year aren’t ones where writing about them can help give some clarity to solving them (which is one big way I’ve used this blog in the past) and the things worth celebrating are so steeped in context that it’s hard to share them without writing a dissertation about the demographics of the school I teach in.

But – I’ve also been teaching this AP Computer Science Principles class, which has been really awesome. I’m a big fan of the philosophy behind the class and teaching it has been a lot of fun. I’ve also been promoting the course and pushing for more Computer Science classes at my school – and, apparently I did a pretty good job, because I’m teaching all CS classes next year: Computer Science Discoveries, AP Computer Science Principles, AP Computer Science A, and an Electronics course where I think we just get to play with microcontrollers and robots for a year. I also became a PD facilitator for Code.org (I used most of their curriculum while teaching Principles this last year), so if anyone will be at TeacherCon in Phoenix: I’ll see you there. This is all pretty exciting and, since I’ll be back in the classroom more often next year teaching something pretty new, I suspect I’ll have more to write about.

Anyway, I decided to create a new blog exclusively for my Computer Science adventures: Codey McCoderson. I’m hoping the ‘blank slate’ setting will help re-acclimate myself to the process of writing about my teaching, and I like the idea of preserving this blog as it stands – a time-capsule of sort from my first 5 years of teaching. So, if you’re interested in reading about me trying to teach Computer Science at all sorts of different levels, maybe you’ll enjoy Codey McCoderson.

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**What’s Up With AP Computer Science Principles?**

AP Computer Science Principles (AP CSP) is a brand new AP course intended as an introduction to different areas of computer science, most notably: the internet, programming, big data, and social issues in privacy and security. There are two things that make this course exciting for me: (1) the explicit goal of the course is to attract students to computer science, which means there are no pre-requisites or minimum-requirements to take the course – anyone with an interest and the student habits to support an AP class can take the course; and (2) unlike the AP Computer Science A exam which mandates that students learn Java, AP CSP lets the instructor choose a programming language, which means I can teach Python (I have some strong opinions about Python being an ideal introductory programming language). Here’s more info about the course:

- College Board Course Overview Page
- College Board Teacher Pages: Course Overview & Exam Info
- Programs that offer AP CSP-aligned curricula, resources, and PD: College Board and more info from apcsp.org

**How Did You Choose A Curriculum?**

Before I started teaching the course, I looked at programs that offered College Board supported resources – at the time, these were Code.org, uteachcs through the University of Texas, and Project Lead the Way (PLTW) (there are now two more programs that are now supported: Mobile CSP and the Joy and Beauty of Computing). My biggest concern was that I didn’t want to get locked into delivering lessons in a prescribed order with a prescribed programming language / tool / textbook without room to improvise or substitute some of my own resources. I also wanted to parse out the philosophy behind the curriculum – why are units presented in the order that they are? What are the guiding questions behind each section? Is this curriculum really an overview of computer science and aligned with the curriculum framework, or is it just another programming class masquerading as a CSP class without much depth in the internet or big data or the social issues inherent in the course. My last concern, which came from me doing this legwork myself, was I didn’t want programs that cost money – trying to find a grant to send me off to PD where I wasn’t exactly sure what I would get was not an appealing thought.

In the end, I decided to get involved with Code.org‘s program mostly because, at the time, their curriculum was completely available online for free under a Creative Commons license (it still is, and I’m a big fan of the sharing-resources, rising-tide-raises-all-ships philosophy) and, when I contacted one of the curriculum writers, I was told improvising with their curriculum was encouraged – that there was no mandate to use their resources exactly as given (which was not always true of the other programs when I inquired). As of writing this, the Joy and Beauty of Computing is also freely available under a Creative Commons license and parts of the MobileCSP curriculum are available too. It also helped that Code.org’s professional development options were free, including paid airfare and hotel stay, at a 5-day event called TeacherCon along with continued PD and support throughout the school year.

After the summer PD, I put together a syllabus and some recruitment resources and started the year.

- Here’s my Syllabus (this is the one I gave students – not the one I submitted to the college board)
- Here is the big AP CSP Framework (which includes the ‘standards’)
- Here’s Code.org’s curriculum

**Who’s In The Class?**

I started the year with 12 students (which is my fault – I didn’t recruit) – got up to 21 students – and I’m starting the second semester with 17 students. We restricted the class to sophomores and above who earned at least at least a C in Algebra I – next year, I’d like to push for students who earned at least a C in English (writing is a much bigger part of this course than I initially realized). The class is pretty evenly split between sophomore, juniors, and seniors. It’s also pretty well split between male and female, which is a good thing. I had several seniors join my class a few weeks into the semester because they discovered this was an AP-weighted class offered later in the day (most AP classes are earlier in the day at my school) and they thought it would look good on their college applications / boost their GPA – which is great, because I’m getting students enrolled in my class who otherwise may have never taken a computer science course.

**How’s It Been Going?**

This class is awesome. Seriously. Awesome. Here’s some cool stuff we’ve done. Not listed on there is the Election Data Analysis we did too.

I’ve mostly been following the Code.org curriculum, especially for the units on The Internet and Data – these are areas where I didn’t have a lot of material prepared and was looking to lean pretty heavily on other resources. I was skeptical at first as to why they wait so long to introduce programming, but I learned more about their philosophy at their summer PD – a big focus is on equity. If there’s any aspect of this course that students will have seen and explored before, it’s the programming – which is why, as soon as you introduce it into the class, this ‘gap’ appears: the students who ‘get it’ and are ‘fast’, and the students who ‘miss it’ and are ‘slow’. It’s pretty much the exact problem math teachers face at the start of every school year – if you decide to review the past year’s material, that gap appears and it just reinforces the positive or negative self-images that students have about themselves. So, waiting a while to introduce programming concepts is a conscious equity decision – letting students become familiar with things they’ve probably never studied before (like the TCP protocol stack) and gaining some confidence before letting the programming gap appear.

In the first semester, we talked about the Internet, How Data is Encoded, Organizing Big Data (although I skipped most of this chapter and condensed it to just the Excel manipulation stuff), and some Basic Graphics Programming. The Code.org widgets are pretty awesome and students responded well to them – very hands-on and engaging and, at times, hard to put down (especially the text compression and pixelation widgets). We also spent a few days with the Lightbot Hour of Code because there are surprisingly similar problems on the AP Exam. My school also has these half-days every other week, which I used to do rapid research on a particular topic – over the course of the semester, we looked at Virtual Reality, Internet Technologies, and Artificial Intelligence & Driverless Cars (have you seen this?!?!!). This gave students a chance to research and cite articles from reputable sources, then summarize and present arguments from them in preparation for the AP Explore task. This particular type of exercise culminated in the final exam for my class, which was based on the show Shark Tank: they had to research a technological innovation, write about why it’s important and the impact it’ll have, then present to a series of judges. It was pretty fun.

My biggest struggle has been with absences. There are lots of activities we’ll start in class in partners, but then the next day a student will be absent and their partner is left to fend for themselves. These aren’t activities where partnering up serves to reduce the work burden – these are activities that can **only be done in pairs** (like sending binary messages using IP Addresses) and are explicitly built into the course framework (the verb ‘collaborate’ shows up a lot). I’m also conscious of the fact that, even though this is a course about technology, I can’t assume students have access to technology at home – so, for longer projects, I tend to give in-class time to work on assignments and programs. But, if a student is absent, this basically guarantees they won’t be able to turn in the assignment because they missed the allotted time to work on it. And lastly, there’s just the content that students miss when they’re gone. There’s no textbook for my class – content is gained through collaborative experiences or demonstrations, which are hard to replicate independently. Thankfully, Code.org has some nice videos that usually summarize their lessons and I could show these to absent students – but, now that I’m starting to use my own materials, I’ve also started to record my own lessons (in a makeshift way – no judging) just to accommodate the inevitable absences I’ll have.

Another struggle has been finding structures to help guide student writing, which is a completely new area for me. A lot of the AP Performance Tasks are writing based, so I find myself needing to train students to read an article or analyze a piece of code, then write complete and descriptive responses to these artifacts. If I had time, I think I’d talk to the AP US History teachers and ask them how they prepare students to write about Document-Based Questions (DBQ’s). I also need to spend more time scaffolding the prompts I give students or being more intentional about the articles we read and how they should structure their responses – these are all things that I just don’t have in my teacher toolbox yet. We start this next semester with a unit on Data Security & Privacy, which will have lots of article reading and responding, so I’m hoping to get better at this as we move through this unit.

Some positives have been: students are really into the content – it’s relevant to them and also unlike anything else they’re learning about my school. We have really great discussions about how the mechanisms we’re learning about inform the devices they use in their daily lives (ie: how the internet works affecting how their data gets transmitted and recorded). A top-5 teaching moment was watching students figure out that the relationship between keys and their SHIFT counterparts (ie: 1 and !, 7 and &) has to do with the ASCII numerical representation. It’s also been great diving into multiple ways to solve problems – the curriculum encourages a lot of pair work, which leads to many solutions to a particular problem and the discussions that come out of those are really worth-while.

**On Feedback w/Google Classroom & Repl.it**

I’ve been using two websites pretty consistently to help collect assignments and give feedback to students: Google Classroom and the Repl.it Classroom. I think it’s completely changed the role of feedback in how I interact with students in a really positive way. I realize I only have 17 students, but I actually *like* ‘grading’ homework these days because it’s a chance to start a feedback-based conversation with students. Here’s what I mean:

I’ll have an assignment created as a template document in Google Classroom. I can tell google to make a copy of this assignment for each student, which they can then write in and submit. I can also include resources that go along with the assignment – handouts, links, etc. Some students have told me they like having everything on Google Classroom because they don’t have a ton of papers to manage or eventually lose. I make my assignments due a week from when they’re assigned – but, I tell students they can submit early and *if they do*, I will respond with feedback on their assignment and give them a chance to re-submit as many times as they want before the due date. With this system, about half of my students will submit at least once ahead of time, giving me a chance to see misconceptions and address them individually with each student. It also lets me give question-based feedback to prod students in the right direction of how to think about their answers (I think this post by Michael Pershan, or a similar one, has been bouncing in my head all year – the idea of feedback as reinforcing where they’re on the right track, then nudging them in the direction of where they need to be going). It’s been awesome – sometimes I’ll have actual conversation with students through this turn-in process: I make a comment, they comment on my comment asking a clarifying question, I answer the question, they adjust their answer and have learned something in the process. It’s one of the first times I’ve felt that the written feedback I give is being looked at and used productively.

Repl.it’s classroom has a similar model: I can assign programs, students can turn them in, and I can return them to students with feedback. The advantage here is: all of this is done via an online programming environment, which solves a lot of my equity issues: students don’t need to download and install any programs on their home computers – they just need to go somewhere with an internet connection and they have access to all of their files (and, unlike Cloud9, there’s no complicated development environment to setup). I also like that Repl.it seems to be actively investing in this program and is constantly updating it: I made a request that their teacher dashboard page have an ‘export to csv’ option so I could transfer grades to my gradebook, and two days later it was there.

Both Related and Unrelated: If I was still doing traditional math classes, I would jump all over the new Quizster app, which seems to be this same sort of feedback-cycle but for traditional math assignments.

**What’s Next**

We’ll see how the AP exam goes at the end of the year, but I’m confident my students will do well. Next year I’ll be teaching Computer Science Discoveries, another Code.org curriculum designed for middle-school, but we’ll be doing it at the freshman level. There are also mini-twitter discussions that happen every once in a while between me, Kaitie O’Brien (her blog name is great), and @reilly1041, so that’s pretty cool.

Thinking about the future, the only other thing I’m not sure about is how all of these new Computer Science courses relate to existing department progressions, specifically Math & CTE. Some states have said AP CSP should count as a math credit, but for others it takes more convincing (I’ve been looking into the process in Arizona and it takes some hoops to jump through). There’s also the place it has with existing Career and Technical Education (CTE) programs – my school already has a web design track through our CTE program, with the goal being to gain the skills needed to get a job straight out of high school doing web design. Many of the skills in that course overlap with mine, but AP CSP doesn’t exist as a stepping-stone to get an entry level job post-high school; this class exists as part of the bigger computer literacy movement and as a pathway to college. And yet, it’s tempting to make this course part of a CTE progression just because some of the skills seem to overlap with other CTE courses. I’m really curious to see where new schools thinking about offering this course decide to place it – is it a math class? a CTE class? or a regular elective class?

So… that’s been my year so far. In a nutshell.

]]>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.

FiveThirtyEight has a reference to every poll it uses in its model. One of those polls is a Google Consumer Survey, which opens automatically in Google’s DataStudio (brief pause: HAVE YOU SEEN GOOGLE’S DATASTUDIO?!?! Let’s talk about this more at the end of this post because oh man oh man does it look cool). The survey has 4 questions: how likely are you to vote, who would you vote for, what’s your gender, and what’s your age range. At the bottom of the page is a link to the data used to generate that model – but, even better, is a link to all the historical data going all the way back to August 2016. Each .csv file is a 20,000+ entry file that I could open in Excel and start playing around with.

And this is when I got really really excited because I had, at my fingertips, *real, raw, meaningful* data about an event that was *real, raw, *and *meaningful* for my students. **I had a vehicle for students to use a computational tool to become more critical citizens and have a meaningful interaction with data that wasn’t superficially imposed by me** – analyzing whether more girls or boys like dogs versus cats sounds pathetic when placed next to predicting who will run our country for the next four years. And suddenly this Big Data unit became one of the most exciting things I was doing with my students.

Jumping ahead a bit, here’s what they came up with (flaws and all). The assignment was to pick two consecutive data sets and find a story to tell about them. We did this towards the end of September, so most picked dates around then (which happened to be before and after the 1st presidential debate). I showed them how to create pivot tables and charts (more on that below), then had them work in pairs and choose a state to focus on. They put their analysis in a shared Google Slide so everyone could see each others work. Even though this was the finished product, this isn’t where I started – there were a few days that led up to this:

On day 1, I wanted to build up the idea that visualizing big data is important as a way to understand and communicate the data, and no one is better at making that argument than Hans Rosling. I showed the first half of his 2006 Ted Talk where he looks at 3rd world vs 1st world using Gapminder, then I showed a minute or so from the end where he gives a mini call-t0-action regarding connecting big data to visualizations so it can be communicated to the world. We looked at some really terrible data visualizations – I got mine from Code.org, but you can find them anywhere – then we visited Gapminder World and I had them pick one of the example graphs to explore and pick their own. One really cool thing about Gapminder is all of their data is available for download, so we grabbed the Population Data with Projections, imported it into Excel, cleaned it and filtered it, then picked two countries to compare the populations. Pretty full day – here’s what they came up with.

On day 2, I explained the Netflix Prize as a way to introduce the next data set we’d look at: movie ratings with demographic info. Code.org has a lesson based around analyzing a subset of this rating data, but they also have entire large data sets available for download which is really awesome – again, I’m a big fan of their open sharing philosophy. We looked at the smaller subset first and I showed them how to make pivot tables in Excel so they could average ratings by gender or age, or look at the distribution of ratings for a particular movie, etc etc. Lots of stuff to play with. We ended by looking at the full data set – they had to pick at least 5 movies and analyze them in some way. Here’s what they came up with.

From there, we dived into the Election Data. I gave them the files, told them they’d need to use the skills we developed in the last few days, then set them loose. The only real requirements were to work together and try to find a pair of graphs that you could use to support a statement about a change in the voting pattern for a particular demographic. We spent a few days working in-class, and still some groups weren’t able to finish. The freedom to play with the data and choose something that interested them led to some great conversations and lots of engagement – I was happy with how seriously they took the assignment. I also really liked everyone having a common Google Slide for students to post to – its something I’ve started doing with other aspects of this class and its worked out really well so far.

In retrospect, I wish I had narrowed the states students could choose – one group picked Vermont and, for some reason, barely anyone in Vermont was polled (I think it’s less than 30), so it was difficult to make a bigger statement because the sample size was so small. I also wish I defined my expectations a little better – ultimately I added the checklist at the front of the presentation, but that wasn’t there to start with. I would keep the overall vagueness of the prompt – ‘make the data tell a story’ – but I would have some exemplars or examples to show, similar to how Gapminder has a few examples students can see before playing around with the data themselves. Sentence frames would also help students form the type of written response I was looking for, but even that can subtly restrict what students look for or find important.

So – all of that is the gist of the project. Here are a few more random thoughts:

- Exploring this project also led me to discover Google’s DataStudio, which looks really awesome for creating visual analytics for data. I went through their tutorial of linking data to their dashboard widgets and its pretty powerful. If I had more time, this is definitely where I would push students to develop their skills.
- Code.org has a link to this nifty style guide on data visualizations, which was really great to show students on the second day of doing the election slides.
- There’s still 10 days left in the election, so you’ve got some time to try this yourself if you’d like. I highly recommend it if you’re teaching Computer Science Principles, and maybe even if you’re not for the reasons below.

**One Final Note:** I just got back from NCTM Phoenix where I was lucky enough to attend Karim Kai‘s session on the value of application problems in a math curriculum. Part of his argument [paraphrased by me] is that a true application task starts with mathematics as its premise, and uses that to understand the world (rather than using the world as its premise and using that to understand mathematics). We went through this Mathalicious lesson which was really awesome, then he answered some questions about ways to implement these types of lessons on a teacher or school level.

One comment that’s really stuck with me was his appeal that **doing more true application problems has the potential to create a better informed student who is an active, rational, thoughtful participant in our society**. I’ve been thinking about this as I’ve been reflecting on this task: the goals were a bit vague, we spent less than a week on it, and we’re going to move on to something completely different afterwards. These are all the things that usually swing the pendulum towards “better to skip it”, which I almost did (remember that? 3rd paragraph in this post?). But, I’m glad I didn’t: in the moments where my students were analyzing their data and trying to find the story within, it *felt* like something bigger was happening. That I was engaged in conversations that were bigger than the little universe that lives inside the four walls of our classroom, that the energy level and thirst to know more was amped up a little more than usual, and that these conversations were continuing beyond our walls with people whom I’ve never met. And this made the lesson valuable in ways I didn’t expect, probably because I could see Karim’s point being acted out right in front of me.

So, I guess I’m saying: if this fits into your class in any way – sorting and filtering data, making visualizations in Excel, making Pivot tables – I’d highly recommend giving it a try as a true application task.

So – there’s that. Thanks for reading.

]]>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.

]]>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.

]]>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.

]]>**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.

]]>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’.

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

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