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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:

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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.


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:

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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?

The Common Core Shift in Algebra I

I’ve spent this year trying to teach a genuine Common Core Algebra I curriculum to high school freshman (my first time doing either of those) and I keep trying to find a way to write about with my experiences, but it’s hard not to get lost in all of the moving pieces that’ve happened this year. As the year wind downs, I guess the biggest thing I feel is: the Common Core shift is real and I feel it and I have a lot of uncertainty about what my students should leave my class with or that I’m preparing them in the best way for what’s coming. This post is about me really have to readjust what I thought the content of an Algebra I curriculum was, and trying to figure out what it needs to be now.

used to have a feeling for what students didn’t know when they walked into Algebra I – things I could assume I needed to teach from scratch and things I could assume they had seen before and had some familiarity with. I used to know what the non-negotiables were for when they left my class and moved to Geometry or, much down the road, Algebra II. I used to know the balance between procedural fluency and conceptual understanding; when to ground something in a purely mathematical process versus something necessitated by a real-world situation (example: are logarithms motivated by the need to take the inverse of an exponential? Or by the need to measure sound frequencies on a decibel scale?) I used to know where the overlap between Algebra, Geometry, and Algebra II was – where to draw limits in Algebra I that would get picked up in Algebra II. I’m much less confident now and I’m not sure what to fall back on. I feel things needing to be done differently but not knowing how or to what extent.

A lot of the uncertainty I’ve felt this year is also the product of some outside forces that I’ve been reacting to. Most of my district has adopted the Carnegie Curriculum, complete with textbooks, support materials, and once-a-week computer time with the the Cognitive Tutor program. Our middle schools are using this curriculum as well, which then ties directly to the curriculum we teach in high school. I teach in Arizona, which adopted the Common Core Standards and aligned itself with the PARCC assessment consortia a few years ago. In March of 2014, Arizona withdrew from PARCC. In November 2014, they adopted a new test dubbed AZMerit, which is similar to how Utah and Florida are implementing their state tests (colleague and fellow blogger Jason Dyer has a play-by-play of what those tests looks like). In February 2015  (this year), Arizona almost repealed the Common Core standards – instead, a group will review and revise them for next year. As a result: it’s hard to find solid instructional ground while also adjusting to a new curriculum, somewhat-new standards, and a new assessment.

Right now, thinking about next year, there is a tension with every Algebra I topic and how it:

  • aligns to the Common Core standards or my best prediction of what the Arizona standards might turn into
  • aligns to the AZMerit test, for which all I know is this blueprint.
  • aligns to my textbook & the common curriculum my district is trying to use, especially since our feeder schools are using this curriculum as well
  • aligns to skills or knowledge that my students need to know in order to be successful in their future classes or future careers.

So – reflecting on this year, starting to think about adustments for next year, and (I’m realizing now) as a place to process my thoughts, here are some things I’ll just call The Shifts and some thoughts on how to react to them

Shift #1: Self-Contained Units

I think the biggest thing I feel as a result of all of these moving pieces is the need for units that have a clearly defined ending and ‘wrap everything up’ quality. There were some topics where I used to be able to say “You need to know this so we can do ______ later”, but I’ve lost all intuition about when these statements are true anymore. Instead, my best units this last year were the ones that built towards a specific problem to solve or scenario to investigate or project that tied everything together – that was a natural culmination of the material we had been covering and didn’t rely on “trust me – you’ll need this later”. Trying to motivate material with “You need to know this for Geometry / the next unit / the state test” was a failure because, honestly, I no longer have any idea if what I’m teaching right now will truly be necessary as we move forward (I’m looking at you inequalities and solving absolute value equations).

Instead, thinking back to last year, I can remember a few units that had this nice self-contained feeling (start, middle, end). These were the ones I enjoyed teaching the most and the ones with the most in-depth questions, investigations, and sense of independence from the students. Thinking about next year, I want more of these – units with a natural progression towards some kind of self-contained question/scenario/project and with a clear beginning, middle, and end. I don’t want the motivation to purely be “they’ll need this later” because, really, everything is so fluid that I don’t know if they’ll really need it later. I want students to know something because they need it now, in this moment – otherwise it becomes just another random rule or procedure to memorize without any internal connection to it.

Shift #2: More Application, Less Procedural

One big shift I see in how my textbook presents new content is that it is almost always grounded in investigating a problem or scenario that is real and has consequences. Systems of Equations are introduced via break-even points, inequalities are introduced via ‘at least’ or ‘at most’ problems, exponentials are grounded in patterns or ‘doubling’/’tripling’ scenarios. I’m used to ‘word problems’ or ‘real-world scenarios’ being the last topic of a unit or even a unit all on its own – but in the age of Common Core, I see these showing up more and more as the entry point of problems and then again as the finishing point. I’ve become a big fan of this for lots of reasons: it has a lower floor for students to discuss a problem, it eliminates the ‘here’s what we’re learning next because (arbitrary reason)’ style of curriculum, it’s more relevant to what they’ll see outside of school, and its easier to engage students in something concrete instead of something abstract. I plan on trying to model every unit in this way – find something for my students to dig their teeth into before barging forward with the math content, then circling back and using the content to re-examine these problems/scenarios and see how useful all that math really was.

Shift #3: More Calculator

Holy crap – this thing is a game changer. Arizona’s old high-stakes test was no-calculator, so students were taught methods to answer questions by hand. But now? Calculators can be used on our exams, which means calculator fluency is a big deal in my class. And, frankly, I’m really glad calculators can play a bigger role in an Algebra I class – they’re a legitimate tool and, from a planning perspective, they let me ramp up the intensity for the types of problems we solve and scenarios we investigate. It used to be that I had to choose problems where the numbers ‘worked out nicely’ or the graphs ‘fit nicely in the standard window’. This was me creating artificial blocks for myself and my students that aren’t realistic and aren’t valid anymore. Now I have more freedom and more tools to show my students so they can solve problems that matter.

Or, at least, this is what it should be. This last year: it wasn’t. I hardly did any calculator-based lessons with my students. I skipped the sections in the textbook that explicitly used the calculator to answer problems. When I would try, the lessons would drag on and on as I tried to troubleshoot calculator problems and keep the class together. These lessons also would show up in the middle of a chapter on something else, so the switch to calculators usually seemed random and forced and took some time to get used to. I was never sure what the payoff was going to be and I was still so used to students needing to know how to do things by hand that I just defaulted to teaching familiar lessons that could be done ‘easily’ by hand.

Thinking about next year, my default ‘do it by hand, easy numbers’ mindset needs to shift dramatically. I need to spend time getting students familiar with calculators and seeing them as a valuable tool to solve problems. I want to plan an entire unit which is just on using the calculator, specifically the graphing functions (finding mins/max, finding roots, finding intersections, using the table, adjusting the window, etc). I want students to see the calculator as a valid option as a way to start investigating a scenario. I think this is a big deal, and I’m looking forward to explicitly planning lessons around using a calculator.

Shift #4: More Statistics

This may be a revelation that’s more for me than other teachers who’ve done Algebra I before, but I never thought of statistics as being a vital part of the course. Things like representating data, measures of center (mean, median, mode), and linear regressions were more like an afterthought or extension or digression rather than an integral part of the curriculum (or, at least, that’s always what it looked like to me). The Common Core seems to have shifted this a fair amount (with drastic shifts happening in Algebra II). The idea of ‘big data’ and analyzing (rather than simply representing) this data seems to have taken a much bigger role in a Common Core Algebra I class. And, seeing what’s to come in the Algebra II standards (standard deviations, z-scores, normal curve analysis), there’s a responsibility to prepare students for these topics in the next levels.

I didn’t do any of that this year. Even though statistics is 17% of our AZMerit test, my class didn’t do a whole lot with data and regressions and measures of central tendency – these standards fell to the wayside as I desperately tried to prioritize my time and guess what students would need moving forward. But, I don’t think I can let this happen next year – in the bigger picture of preparing students for their next classes (especially Algebra II) and the real world, there needs to be a place to prepare students to look at data and interpret it. I think the standards have made a statement that these items no longer lie solely in the realm of a statistics class and, if I’m to genuinely teach the Common Core, I can’t have these units be an afterthought that eventually falls to the bottom of my priorities.

Edit 6/20: This Tweet pointed me in the direction of these resources: Publications in Statistics Education, which looks to be a collection of resources and publications aimed precisely at this issue: the new influx of statistics required by the Common Core and the lack of resources I have to present these standards. So – that’s awesome, and the reputation of the person sending the tweet (she blogs at is enough for me to take the recommendation seriously.

Shift #5: Is Algebra I necessary for Geometry?

This question really challenged me as the year wound down and I thought about the impact this year would have on my students. This question really comes from the shift I’ve seen in Geometry: these standards are now grounded in proof or transformations or real-world applications. Gone are the days where Geometry was an excuse to solve an algebra problem. Fading are the days where a student could be unsuccessful in geometry purely because they had weak algebra skills. With the new emphasis on reason and explanation and coordinate planes, a lot of the algebraic foundation that is given in Algebra I may not be needed in a Geometry class. I don’t remember any student telling me that they had to solve an algebra problem on the Geometry portion of the AZMerit exam.

Personally, I like that Geometry wants to become more of its own discipline, free from the chains of algebraic problems being forced into a geometric context. But, it makes me wonder about the types of skills students need to have as they enter this class. Do I need to impart algebraic skills as students prepare to enter Geometry? Or do I need to impart algebraic strategies for their geometry experiences? Will they be more likely to graph an equation or solve for x, or more likely to analyze a scenario and find an entry-point into the problem? From what I’ve seen this year, it seems to be the latter. This, also, is even more of an argument for self-contained units in Algbera I.

This, again, makes prioritizing units and standards a challenge because it’s unclear when a particular algebraic skill may pop-up in Geometry next year. I used to try and pick ‘non-negotiable’ standards that students absolutely needed for their future, but I’ve lost most of my intuition about what these are as I’ve watched the Geometry curriculum change. It also makes me wonder, radically, if our course progressions need to stay in this same Algbera I – Geometry – Algebra II progression that they’ve been for as long as I’ve been in school. Before the AZMerit exam, this progression was mandated purely by the fact that our old high-stakes test included standards primarily from Algebra I and Geometry and was only given at the end of their sophomore year. But, with narrower end-of-course exams given at the end of each year, what’s to stop a school from switching up the order? It’s much easier for me to think about how Algebra I transitions to Algebra II rather than how Algebra I leads into Geometry. Without derailing this post into an argument for rearranging the course structure, I will say that I’ve started to think about ‘non-negotiable’ skills in terms of what they need for Algebra II instead of Geometry, even though they still take Geometry before Algebra II.

Shift #6: We Need Better Teacher-Given Assessments

This has been one of my biggest frustrations/regrets/source of anxiety this year – the gnawing feeling that I could never capture what we my students were learning and how they could communicate that learning in a pen-and-paper test. Writing tests that are purely procedural does a disservice to the complex scenarios we’ve discussed in class and what I’ve witnessed in my room as students talk to each other and present their ideas. Designing a test with large open-ended problems is tricky and takes practice to phrase the question exactly the way you’d like it so you can really parse if a student knows something or if they don’t. I haven’t figured it out yet, but almost every test I gave ended with the feeling that “these questions were not aligned with what we’ve been doing in class”.

A lot of this, especially early on, was (I think) my own unfamiliarity with how much less procedural Algebra I has become. Standards like solving equation and graphing lines and even solving systems have moved down to 8th grade, which means Algebra I is reserved for applying those skills to situations and interpreting the results. I found myself leading classes that focused on using algebra to analyze a scenario and spending most of our time discussing this analysis, but then giving a test with problems that were solely algebra and devoid of any context. Trying to find a way to write questions which are more than just application of skills has been tricky, and sometimes leads to open-ended questions where students aren’t sure what I’m asking or, in answering the question, they don’t demonstrate the skills I’d like to see as they solve the problem. Finding this balance has been tough.

Thinking about assessments next year, I was pointed towards this document from Achieve the Core on Publisher’s Criteria for High School Mathematics and was drawn to the pages that talk about rigor (although I probably could have found this same revelation from other common core documents). Rigor is discussed as being composed of equal parts: conceptual understanding, procedural skills & fluency, and applications. Thinking about my past assessments, I was usually too heavy on one of these areas while ignoring the others – maybe one test was entirely problems devoid of context, while another had mostly open-ended scenarios for students to analyze and solve. Looking into next year, I want to try and design each assessment so it has pieces of each of these: some questions emphasizing skills & fluency, some questions emphasizing conceptual understanding (maybe finding mistakes? or agree/disagree & explain why? or compare/contrast?), and some questions with an application focus (probably grounded in some sort of real-world scenario). I’m hoping this will give me some guidance so I can avoid those tests that lead to me thinking “this doesn’t match what we were doing in class”.

Classroom Interactions in a Common Core World

I’ve spent this year trying to teach a genuine Common Core Algebra I curriculum to high school freshman (my first time doing either of those) and I keep trying to find a way to write about with my experiences, but it’s hard not to get lost in all of the moving pieces that’ve happened this year. As the year wind downs, I guess the biggest thing I feel is: the Common Core shift is real and I feel it and the demands of the standards are affecting the types of interactions I have in my classroom. This post is about the challenges of encouraging genuine student-centered discourse in my classroom (and, specifically, the unexpected challenge of getting students to listen to each other), then sharing a strategy I tried this year that I think worked pretty well.

Looking at the wording of the Common Core standards and watching the architects talk about them, I’m convinced that genuinely teaching these standards requires a classroom where students are (1) talking to each other, (2) listening to each other, and (3) able to communicate their ideas in writing. I see this in the level of independence and self-startedness that the Common Core demands in their modeling standards and, if I want my students to be successful with these, then I need students to see themselves as bringing something valuable to a conversation instead of relying solely on the teacher as the one-and-only-knower of all information. I see this in how the standards are written – verbs like ‘Discover’ and ‘Understand’ are peppered throughout the standards, both of which encourage communication and debate and explanation rather than answering exercises. I see this in some of the released PARCC sample items (this question has always stood out in my mind) and, in Arizona on our AZMerit exam, in the emphasis on questions coming from levels 2 and 3 on the Depth of Knowledge chart.

This is tricky because none of these actions – talking, listening, writing – directly involve me, the teacher: my students need to be doing these and, more importantly, with each other rather than just with me. And yet, all of these indirectly involve me because I need to provide the opportunities for these to happen, which means I need to be very intentional about the types of questions I ask, how I elicit responses from my students, and how I provide feedback to encourage these behaviors. Intentionally trying to get students to talk to each other has become embedded in my practice – I’ve read strategies, seen resources from other teachers (the blogotwittersphere is especially strong in strategies for fostering communication), and I’ve seen this modeled enough times that I think I have a pretty good handle on trying to get students talking to each other and to me. The listening piece, however, has been a new and unique challenge I wasn’t expecting, but I think I’ve convinced myself that its the most important – listening, evaluating, and responding the what someone else says in a meaningful and productive way. Man that’s tough to teach freshman to do.

Example: I was at a conference this week where this pattern was put on the board: “3, 2, 5, 4, 7, 6, …” and the audience was asked what the next number was (it’s 9). We were then asked to share  how we got our answers, and these strategies were then shared with the group. Some people saw the sequence in pairs – (3, 2), (5, 4), (7, 6) – and therefore the next pair is (9, 8); some people saw it as two sequences smushed together (like a piecewise sequence) – 3, 5, 7, 9, … with 2, 4, 6, 8, …. smushed between the numbers; some people saw a numerical pattern – subtract 1, add 3; some people saw that each consecutive pair summed to a pattern of its own (3+2 = 5; 2+5=7; 5+4=9) and used that to generate the next number (6+?=15; ? = 9) [personal note: I found this strategy to be the most interesting and a new playground for generating sequences]. All of these are useful and insightful strategies. Most of us in the audience, being interested teachers and respectful adults, listened to the other strategies and thought “oh that’s clever” or “ehh – mine’s better”.

But – here’s how this experience might play out in my classroom: students take a moment to get their answers independently, they share with each other to see how their answers compare, and then I call on people. My students, having their answer, are confident that they can respond should I call on them – I imagine subconsciously they’re thinking “Okay – I have an answer if he calls on me, so My Work Is Done“. As I call around the room, they listen half-heartedly and evaluate how similar each response is to their own only because I end every conversation with “thumbs up if this answer was similar to the one you discussed with your partner”. At the end of this discussion, we’ve discussed about 3-4 different ways to approach this problem, but students are still latched onto their way to solve the problem in case I pull a surprise attack and still decide to call on them.

This leads to the challenge: my experience has been that students aren’t even able to repeat back these other responses if I asked them, much less understand what they mean and see their value. More so, my students don’t see the reason why they need to be intellectually responsible for responses different from their own. To them, they got an answer and could explain it, so their cognitive work is done – listening to different strategies and evaluating their reasonableness isn’t something they’re used to doing. This presents a challenge to me because maybe the goal of this activity was to introduce the idea of piecewise sequences and so the ‘two smushed sequences’ strategy is really important for us to talk about. And as I try to transition to the next part of the lesson, I’ve left behind everyone in the class who didn’t use that strategy and wasn’t listening when their peers presented it, but still feels satisfied that they did what I wanted them to do because they had their own answer and could explain it.

So, this has been my guiding question as I keep finding myself in these situations and trying to react: how do I encourage students to have a reason to really listen to their peers thinking and use that thinking as the lesson continues? How do I create a culture of ‘student-as-giver-of-new-information” instead of “teacher-is-only-person-who-can-give-new-information”?

Quick sidenote: it would be easy for me to add a teacher move saying “these are all great strategies, but the one we’re really going to talk about is … Let me explain it again to make sure everyone understands it before moving forward“, but there are some subtleties to this (especially the italicized part) that I have worries about. First, it reinforces that the teacher is still the most important person in terms of giving new information and, if I do this often enough, students will realize that they can always wait until the end of the discussion to listen to my little summary. Also, if I do this too often, it implies that there is a ‘correct answer’ that I’m looking for in this type of activity, which means my discussion of different strategies isn’t motivated by a genuine interest of how students solved the problem but by my search for the ‘correct strategy’ that we’re really going to talk about. I’m a big believer that once students have even the smallest suspicion of this deception, their motivation to take risks drops to 0 and their engagement bottoms out as they sit there wondering “when is he just going to tell us what he wants to talk about?”.

Other End Of the Spectrum: If I don’t do this teacher move – adding some teacher-guided explicitness to a specific strategy – then a lot of my students will struggle with the next activity/problem/task if they weren’t listening and evaluating different strategies (for example: if my next sequence was “1, 20, 4, 18, 7, 16, 10, 14”, the students who missed the ‘smushed sequence’ strategy would struggle much more than I want). This last year, I feel like I spent a lot of classroom time regrouping, re-eliciting the responses, then sending students back to work on their task. This takes time. I want this to take less time.

Anyway – this has been part of where my head has been – finding some good teacher moves to handle to situations above. I’ve been looking for resources and strategies to get students to listen to each other, and trying different things that I’ve invented in my own class. Here’s what I’ve tried so far (although I don’t think I’m anywhere near an expert yet):

Weekly Mathematics Discussion Notetaker

This was something I created to try and get students to listen to each other as we had class discussions. I expected it to be a purely extrinsic motivation (a grade), but it had some unintended consequences that led to some intrinsic motivations (more on that below). Anyway – students got it at the beginning of the week and turned it in at the end of the week. They were responsible for 5 quotes from other students where that student explained their thinking. This could be full-group or small-group. They couldn’t use themselves and they couldn’t use the teacher. Eventually, I tried (but failed (but will try again next year)) to add that they can’t use the same person twice during the week, hoping to encourage more people participating in discussions. This became a weekly grade and was one of the few things they couldn’t turn in late. I encouraged them to look for because statements and to think of this as recording notes that they could look at later, so the notes needed to be clear and complete. When I graded it, I didn’t look for correctness of the statements (so “The y-coordinate is 3 because the y-coordinate comes first” would still get full points) because that’s not the purpose of the assignment and, especially during some class discussions, incorrect things are said that are clarified later and I didn’t want to penalize students for not knowing the difference yet. I did, however, provide written feedback to clarify any incorrect quotes. This became a part of my students day-to-day operation in my class – they had their pencil, paper, and the discussion recorder. The back of the handout was a weekly planner because I thought that would be a good use of the space on the back, especially for my freshmen – it didn’t factor into their grade.

Some Observations: Overall, I’d call this a success with some really nice intended and unintended consequences. I noticed students paying more attention to class discussions which was the whole point of this. But, more than that, I noticed students being more conscious about phrasing their answers in complete sentences and using the word ‘because’ in their responses. In fact, this led to a new type of classroom interaction: I’d ask a student to answer a question and explain their answer; their answer is incomplete or mathematically imprecise (“what’s the better mathy word for that horizontal line on the coordinate plane?”); I keep pushing them until I’ve gotten a complete, precise response from the student but with lots of interruptions from me; AND THEN (here’s where the magic happens): I ask them to put it all in a complete sentence using the word ‘because’, which triggers the other students to get ready to write it down, and now I’ve got a student saying a complete, mathematically precise explanation and the rest of the class really paying attention to what they’re saying and how they’re saying it. But someone doesn’t hear it, so I make another student restate it, and now two people have said the complete, mathematically precise, student-generated explanation. Yes – this takes time, but I think it’s been worth it.

More consequences, most of them unintended: students got a sense of pride from being someone whose name gets written down on everyone’s paper as contributing a good response (next year, I think I want to make a wall of ‘good quotes’ to keep track of good responses through the year). My students who liked to answer questions and get called on started to get frustrated because: if they’re always answering questions, and they can’t write their own name down, then their discussion notetaker never gets filled out. So, these same very-active students started encouraging other people to interact in class so they could fill out their notetaker – it was a nice unintended student-driven reaction to the need for these to be filled out. Students also started holding each other accountable for more detailed responses – for example, let’s say I wanted a student to do a problem up at the board and talk through it. Maybe the discussion goes like this:

Student: “I did this. Then this. Then this. Then I got this and that’s the answer”
Me: “Did he/she say anything that you could write in your discussion notetaker?”
Class: “No”
Me: “Are there any questions you’d like to ask him/her so you can add something to your discussion notetaker?”
(Students starts asking why and can you explain questions to the student presenter)
(Or: no one asks anything and the opportunity is gone)

I’d be remiss if I didn’t at least mention: I had to adjust a few things after the first few weeks as I saw lots of people trying to copy responses on the Friday due-date. I had a conversation with the classes re-emphasizing the point (to reward those who share during class discussions and who are actively paying attention), I explained that I would make adjustments for absences so you didn’t feel pressured to complete the whole thing just because you were absent, and I said it was okay to copy a response if it just happened and you missed the tail end of it. Then I laid out some consequences and it wasn’t really an issue anymore.

So – in addition to this whole discussion notetaker thing, I’ve also been looking at…

Resource: 5 Practices of Orchestrating Productive Mathematics DiscussionsThis is a pretty stellar book and I like it as a starting point for how to orchestrate tasks and activities that really require students to generate different strategies. It’s also got some great advice for how to respond to students so the teacher still sits in this role of facilitator and guide rather than taking on the role of information-giver. I don’t really know what else to say – if generating productive discourse is going to be a focus in my classroom, this book is important.

Resource: MAP Project PD Modules on Questioning and Student CollaborationThe PDFs linked in bold have some really great questioning and interaction strategies, which I think is half the battle. Students can only have genuine conversations if I ask the right type of questions and encourage the discussion in just the right way. Students will see themselves as independent learners once I learn how to interact with them in a way where I’m out of the way yet still holding them responsible and pushing them forward. I haven’t looked at these in-depth, but I think I need to.

Reflection: I don’t want to write a ton on this because it would be easy to dovetail into a curriculum conversation, but I guess I wanted to mention: as I think about the topics/lessons/days that have generated lots of discussion versus the ones that didn’t, I tend to think that the truly rich discussion days happen when the topic lends itself to connections between multiple representations – specifically, connecting to some kind of real-world context. Teaching parabolas as connected to different scenarios & contexts led to explanations that weren’t arbitrary or rule-based but were instead interpretations tied to the scenario that we were investigating. The same thing happened when I introduced equations to represent parabolas – having graphs and equations and scenarios available as resources for students to share their answers led to deeper explanations and discussions than if we were examining each topic separately. On the opposite end of the spectrum were my lessons on factoring and properties of numbers (commutative, associative, etc). I didn’t connect factoring to anything else other than a procedure (admittedly a bad teacher move), so the explanations during those weeks were really weak and un-insightful. The properties of numbers were very rule-based and definition-based, so it was hard to get explanations that were anything other than a restatement of the rule or definition. These units need work; units that work in multiple representations seem to be an easier playground for meaningful explanations and discussions.

Conclusion: I don’t really know if I have one, but I’m excited to start the discussion notetaker at the beginning of next year, I’m hoping to connect as many topics as I can to a real-world context or multiple representations, and I’ll keep trying to shift the role of information-provider away from me and onto my students.

Reflections on: Trying to Teach a Common Core Curriculum

I’ve spent this year trying to teach a genuine Common Core Algebra I curriculum to high school freshman (my first time doing either of those) and I keep trying to find a way to write about with my experiences, but it’s hard not to get lost in all of the moving pieces that’ve happened this year. As the year winds downs (edit: did wind down. This post has been in the ‘draft’ pile for a few months and its already summer), I guess the biggest thing I feel is: the Common Core shift is real and I feel it and I have to rethink a lot of how I used to think about curriculum. This post is about me wrestling with what it means to try and genuinely implement a Common Core curriculum and trying to know where the wiggle room is.

In an effort to be proactive and give guidance to new (and old) teachers, my district aligned themselves with the Carnegie Textbook & Curriculum. This is the first time I’ve worked for a school that has aggressively pursued aligning lessons, materials, and curriculum to a textbook – in the past, I always viewed the textbook as a resource for problems and a baseline for what would get covered (and to what depth) in a class. Rarely would I look to the text for instructional guidance or activities or resources that I would use primarily in class with my students. I was used to creating (or stealing) lessons with an investigative aspect, or using a notetaker with problems embedded, or facilitating discussion and having students take notes. It was almost an unspoken rule that the textbook was a subpar mechanism for delivering instruction and it was better to create your own spin on the content. As such, I have lots of resources from my days supplementing a subpar textbook.

But this textbook isn’t like most textbooks I’ve seen. This text was developed from scratch in the last few years to address the specific shifts of the Common Core standards (just like EngageNY or the Math Vision Project) and, for a change, there are lots of things I like about it. The newness of the text means I don’t have any of these cut and paste issues from previous non-common core editions. I like that the textbook is consumable – students are encouraged to write in the book and take notes; that problems are embedded along with the notes. Non-traditional tasks & activities are embedded in chapters (organizers, sorting activities, matching activities) which are things I used to design independently. Questions encouraging discussion and debate are embedded into the section, which I can expand on if I want to. The textbook follows the ‘let’s do one problem in depth’ rather than the ‘here’s the vocab, here are some examples, here are lots of tiny problems’ philosophy, which I like. The textbook tries to ground content in a context or developed from prior skills rather than saying ‘here’s the new thing we need to learn for arbitrary reasons’. Each chapter is like a narrative – some starting problem leads to something new and everything wraps up by the end. Here’s a sample chapter if you’re curious what I’m talking about.

In practice, I’m having a lot of trouble embedding this textbook with my usual classroom procedures and remaining honest with the spirit of the curriculum. The biggest obstacle is: there’s a lot of reading and active learning and student-driven contributions and analyzing different methods which takes time and is especially challenging if my students aren’t on their game or the lesson has a high entry-point and my students begin to feel defeated. My lessons don’t fit nicely in hour-long class periods anymore – the bell rings in the middle of discussing a problem or a class debate, leaving all of us with a feeling that we spent the class period spinning our wheels without ever landing on something solid to take away from the day. I try to have routines grounded in bellwork and exit tickets and an occasional homework, but it’s been hard to plan when I can’t always predict where or when the conversation will end. I have trouble reacting when students aren’t contributing and analyzing where the struggle is. Is it an off day and I need to work on my questioning and engagement strategies? Or is the activity in the textbook too dense or the entry-point is too high or assumes too much from their prior knowledge? In the first semester, I spent one month on only half a chapter as I tried to parse all of this out, reacting to unfinished discussions and delayed assignments and unsure if the activity was flawed or if it was something with the class & how I was running it. By the time second semester rolled around, I didn’t open the textbook at all – in the spirit of ‘coverage’, my lessons needed to move faster, which meant isolating the most important parts of the curriculum and using that independently from the textbook. We didn’t do a single lesson on statistics, despite it being 20% of our Algebra I state test.

This all leads to this really delicate internal conflict for me: if this textbook/curriculum does a lot of things that I like in theory, but I’m struggling to deliver the material effectively in practice, which of the two needs adjusting? Do I need to find better strategies to implement the content and rigor of the textbook/curriculum effectively? Or, since the textbook & curriculum is only 2 years old, are there aspects of the text that are genuinely unsound and worth revamping in my own style? I’m already thinking about next year and I need to ask myself: where do I want to dedicate my time? To a more purposeful integration of this textbook and trusting that, after my students get used to it, things will be better? Or do I start supplementing aspects of the text with my own materials and routines that I’ve used in the past? In other words: do I want to dedicate time to trying to implement the textbook more effectively as intended, or to dialing back some of the expectations in favor of more tried and true methods/activities/assignments I’ve used in the past?

The pre-Common Core me would have said “Abandon the textbook – make your own stuff – use the blogotwittersphere for resources” in a heartbeat. The current me isn’t so sure. The current me is still getting used to the content and rigor shifts that are supposed to be occurring in our classrooms. The current me doesn’t know the Common Core standards as well as I knew the old ones. I’m worried it would be too easy for me to simply start using my old resources and slip into a curriculum that isn’t aligned to the level of rigor in the Common Core standards anymore. ‘Abandon the textbook’ could accidentally lead to ‘Abandon the Common Core’, and I don’t want that to happen. Very few of my old resources are connected to a context like the textbook is (and like the standards demand); very few of my old resources are as calculator-heavy as the textbook is (and the state assessment allows); my old resources aren’t aligned to the curricular shifts that have occurred (lots of my materials are suddenly more suited for 7th and 8th grade rather than 9th grade). To revert back to what is familiar and ‘worked’ in the past doesn’t necessarily mean I’m doing myself any favors as I try to implement a true Common Core curriculum.

Start SideNote: I started writing this post as a personal reflection on my experiences with a specific textbook and a specific curriculum, but after some reading (like here and here) and talking to some other math teachers at a conference here in Tucson, I think this post might resonate with anyone trying to genuinely implement a new ‘from scratch’ Common Core curriculum. End SideNote

I guess another important part of this shift is: it bangs its head against many of the ‘daily lesson’ frameworks and routines I’ve seen – things like the Essential Elements of Instruction (EEI) and the Madeline Hunter model of daily instruction where everything is self-contained and wraps up nicely at the end of the day. I’ve always found this to a be a useful way to make sure I don’t overreach in the course of a lesson, but I don’t think it works well with the way this curriculum is structured. I feel like an EEI self-contained lesson would present the mathematics so we can solve a problem, whereas I’m trying to investigate a problem for the purpose of unwrapping the mathematics inside. I’m used to bellwork every day and exit tickets at the end and the lesson is snuggled somewhere in between, but I haven’t been able to fall into that groove this year. It’s been really strange to read my own posts from 2 years ago describing my thoughts on bellwork and homework and exit tickets and how to establish those as routines, yet I’ve almost completely abandoned those this year as I try to make room for discussions and presentations and student-driven interactions. Is it okay that my lessons run over? Is it okay that I haven’t figured out how to assign homework in a common-core world yet? Maybe I need some new frameworks for what a ‘daily lesson’ should look like – something like Complex Instruction? Something that gives guidance for how to manage a discussion & an investigation over multiple days? Is this where I spend my time – on adjusting what my idea of a ‘daily lesson’ looks like?


MathyMcMatherson Note: So – these paragraphs (this one – right here – that you’re reading in this very moment) exists completely independently of the post above. Let me tell you a secret: I’m really good at starting blog posts, writing a fair amount, then getting distracted or not really knowing how to wrap them up or finishing my thoughts and then not caring enough to publish (much like how Hitchock got bored with his movies as he was filming them), and then they sit in an unpublished pile while life goes on. And I’ve decided I don’t like that. I’d like to try and have more of this stuff out in the world.  Is that egotistical? Maybe. Isn’t having a blog in the first place somewhat egotistical? Probably. Anyway.

So – I’m trying something radical: I’m going to start publishing posts even if they’re unfinished. Like this one – it’s unfinished – there’s no closure to anything that I wrote above. It’s like a mystery novel with the last 2 chapters torn off. But you know what? It’s better than having it sit for months unpublished (or is it?). And maybe it’s still useful. And maybe you, dear reader, will yell at me to finish it, which might be motivating. Who knows? This is an experiment. Expect more of these – deliberately unfinished posts – in the coming weeks and we’ll see what happens.

Curve Stitchings & Card Shufflings

Here’s something non-pedagogy related and rather brief and who knows. Here goes.

Fun Fact!

1) The solution to this problem (in short: how many perfect shuffles until a 52 card deck is back in its original order)


2) the creation of this picture:


Are the same thing:

Waning Moon

So… I’m just gonna let that sit out there in the open with very little explanation. Maybe you’re curious. Maybe not. I certainly am.

Related and very cool:


Two other sets of related questions:

1) Will an n card deck always take n perfect shuffles before it’s back in its original order?
2) If each of the n dots in the circle are really n nails in a piece of circular wood, can I create this design with a single thread of string?

Also, this post couldn’t have happened without Dan Anderson. All this math is his fault.

Idea: Intervention Table

The other day, MissCalcul8 asked me on twitter about ideas for setting up an intervention table. The exact text was: “Any ideas for setting up an intervention table? Mostly for students who don’t even know how to begin.” Twitter’s great for networking, but I’m far too verbose to fit my thoughts into 140 characters. So here are some thoughts on the idea of creating an ‘Intervention Table’.

When I’ve seen others try something like this – a designated intervention station in the classroom – I’ve seen two variations. One type of station is designed for students to use during the lesson as a signal to the teacher that they’re not understanding and need some help. Another variation is a designated station designed to be used during an assignment (such as bellwork or homework time or stations). There are some practical things to consider with these, but also lurking the background is creating a culture where advocating for yourself doesn’t have a negative stigma, and being careful to frame either of these interventions as an opportunity rather than a punishment.

Ideas for an Intervention Table to be used during a lesson

I once worked with a teacher who had a designated desk near the front of the room labeled the “Help Desk”. The idea was that if a student was struggling during a lesson – there was a step they didn’t understand or they couldn’t make it through a class problem – they could move to this designated desk with the promise that they’ll get help sometime soon from the teacher. This desk was right next to the teacher’s desk and had a direct line-of-sight to the board.

When presenting this idea to students, he framed it by giving a speech to his class that was something along the lines of: “If you’re paying attention to a lesson and you feel like you’re not sure what’s going on, feel free to come sit in the Help Desk. It’s near the front of the room so you can see easily, it lets me know that you need help so I’ll make sure to check in with you, and once you feel like you’ve got it then you can move back to your seat”. In this way, students were encouraged to advocate for themselves that they need help with the promise that the teacher will give them a little extra attention while they’re sitting there to help make sure they understand. It also lets the teacher know immediately that there’s a student who doesn’t understand in a way that doesn’t directly interrupt the lesson (related: Red, Yellow, and Green Cup Stoplights).

If I were to try this (which, in writing this, I don’t know why I’m not trying this), I’d want to add a ‘Tutor Desk’ next to the ‘Help Desk’. Both desks would be left empty during the start of the lesson and, If someone needs help, they’re free to move to the Help Desk to get some extra help. But also, students who feel like they know what’s going on can sit in the Tutor Desk as a signal that they’re available to help people too. I know that I have at least 2-3 students in each class who genuinely like helping others out, so offering them a subtle way to do so could be something that’s very appealing to them. I think I’d also find some subtle positive reinforcement to encourage students to act as tutors and to encourage students to ask for help (something along these lines).

Ideas for an Intervention Table to be used during an assignment

A Digression: Suppose you’re in a unit on Solving Systems of Equations. Suppose you’re on the topic of solving via substitution. As a scaffold to get students used to the concept of replacing variables with expressions, you’ve got a bunch of problems with the variable already isolated – for example: y = -2x + 8 and y = 5x – 22. Let’s say that, in working through these problems, you discover that a student understands that they’re supposed to set these two equations equal to each other (ie: -2x + 8 = 5x – 22), but then has no idea how to do the remaining algebraic steps to solve for x.

This digression also tends to happen during graphing lessons (can set up the expression but can’t plot the points using x and y), or during coordinate geometry (can plug into the formulas but can’t evaluate the integers) or during polynomial operations (knows they need to combine like terms but can’t evaluate the integers & get the right signs).

These situations are probably the motivation behind this idea of a designated ‘intervention table’ – a place for students to work on the underlying skills that they need before they can continue with the current class content. These situations were also the motivation for my Wall of Remediation, which I still use in my own classroom. Part of addressing these issues is deciding how you feel about the following statement: If a student can’t do the underlying steps to a problem (integers, algebra, graphing, etc), then there’s no point in having them keep trying these problems that are above their head. In other words, instead of a student learning integer arithmetic and algebra and systems (and, depending on your unit, graphing systems) all at the same time, why not just reduce it down to integers and then build back up to the other stuff?

Adopting this mentality is scary because, at times, it means throwing away your objective for the day. On the other hand, its relieving for the student since it means they can focus on the real roadblock to their learning and feel like they’re making tangible progress. It also means that you, the teacher, need to be really prepared for when these roadblocks manifest. You need to be ready right away with a worksheet or set of notes or something to be able to give to the student to say “Try these – just these – then we’ll build back up to what we’re working on today”.

So – with all of this in mind, if I were to have a designated intervention table, it would have:

—A collection of worksheets with isolated skills that are also curated and checked by me. Here are most of the ones that I use currently. I also get worksheets from (I like that their answer keys show step-by-step solutions) and rarely from Kuta. I would also have an Answer Binder for students to check their answers immediately. When possible, I would have an answer bank for students to check their answers even more immediately (something like this). I let students turn in any remedial assignments like these for points to make up past homework assignments, but I always pick the assignments they complete.

—Number lines. Grids for multiplying numbers or polynomials. Positive & Negative tiles. Graphing squares with the numbers written on the sides. Whatever other tangible representations that are usually used by elementary and middle school teachers to give concrete representations to the things that high school teachers have made abstract. These are things I can grab easily and quickly show a student how to use to solve problems, using it as a temporary scaffold that eventually gets removed. If I can find good procedural guides for how to solve problems, I’d have those too (I’m thinking of things like multiplying fractions or graphing coordinates) for students to look at and reference.

—Whatever notes or definitions or formulas they need for the lessons, even if they’re ‘supposed to know it’. For example, I recently stole Sam Shah’s folder system for organizing myself, but I’ve been added notes and definitions to the center of the folders for students to reference:

2015-04-04 09.26.09

This could also be included at an ‘intervention table’ – not necessarily printed versions of your lesson, but the bare minimum important diagrams/definitions/notes/etc that students can reference quickly. This goes a long way in giving students that entry into a problem, especially those with high absence rates.

Working at the Intervention Table

So – let’s say you’ve set up your intervention table and you’ve got a student sitting there with a basic assignment to work on. Let’s say that assignment is on basic 2-step equations and you’re working on the problem below:


In working with this student, you feel torn – you want to provide meaningful help that is worth their time, but you’ve also got a class full of students who are also working on an assignment and would like your feedback. The issue is: working meaningfully with this student could take at least 10 minutes and means addressing all of the misconceptions in this one problem, such as: which number do you start with? why that number and not the others? my teacher always told me to start with the number on the left. my teacher always told me to start with the positive number. my teacher always told me to start with the smallest number. why did you add the numbers instead of subtract? why is the answer negative and not positive? why are you dividing by -5 – why not 16? why is the answer positive? am I done? how do I start the next one?

The Tension: You’re trying to help this student in a meaningful way that they will remember for next time, but you also need to bounce around to everyone else in the room to help them with their assignments and for just basic classroom management sanity. If you spend a long time working with this intervention student and honestly addressing all of their questions, you’ll feel obligated to get up and walk around once you’ve done only one problem in depth – but the student at the table will probably need some reassurance before they can work independently and so, once left alone, will not have the confidence to do a problem completely. On the flip side, you could try to rush through a problem in order to get back to your classroom, but then the misconceptions and underlying questions are never really addressed, so the student can’t transfer this ‘band-aid’ fix to any of the other problems, and usually won’t remember the first step and can’t get started.

Here’s How I Navigate That Tension:

First, we’re never solving just one problem – we’re always solving at least 8. And, especially at the beginning, we’re not doing one problem to completion – we’re doing five or six problems one step at a time. The trick for me is to break this problem into pieces that are general enough to be applied to most problems and can be applied quickly to several problems. Once we do the step for the first problem, I make them repeat just that step for several more (allowing me to do a quick pass around the room). Along the way, I’m checking for understanding on just that step before moving on to the next one. Here’s a pretty much word-for-word account of what I would do with this struggling algebra student to navigate this tension:

1) “In an equation, the most important thing is the equal sign. Find the equal sign and put your finger on it. Now draw a line through the equal sign, splitting your problem into two pieces. Now do this for the next 6 problems” (Walk Around – come back – check work)

2) “Which side is your x on? If it’s on the left side, write an L. If it’s on the right side, write an R. Now do this for the next 6 problems” (Walk around – come back – check work)

3) “We need to get the x by itself, so I’m going to look at the other number with the x and that is still on the same side as the x. Find that number an underline it. Now do this for the next 6 problems” (Walk around – come back – check work)

4) “We need to find the opposite of this number so they can zero out. What number is the opposite? Write it underneath on both sides. Now do this for the next 6 problems” (Walk around – come back – check work)

5) “These terms zero out and we’re left with _____. On the other side, we need to do some math. What do you get when you combine these numbers?” (At this point, if they’re struggling with integers, we stop the algebra and start working just with integers). “Good – now do this for the next 6 problems” (Walk around – come back – check work)

6) “Now we need to get the x by itself. What does it mean when a number is next to a variable? And what’s the opposite operation? So what do you think we should divide by? Just on this side? What happened on the side with the variable? And on the other side? Good – now do this for the next 6 problems” (Walk around – come back – check work)

7) “Great! Now do these last 2 from start to finish” (This is important – that you always save a few problems to do completely on their own from start to finish).

Some things that are done intentionally: Each step has something tangible for the students to write/draw/circle/etc for me to check later. Each step is broken up so its manageable, but also lets me check for all of those tiny misconceptions that can crop up. And, by doing several problems at once, I can see a specific misconception that I may miss if I do only one problem at a time (for example: if the first 5 problems all have the variable on the left side, It’ll be a while before I uncover misconceptions a student may have about variables on the right side of the equal sign – but, if we’re solving several problems at once, I’m more likely to notice and ask questions about the one problem with a variable in a ‘weird’ spot). Last intentional thing: the decision of which number to work with first in solving that problem is not a trivial decision – I purposefully add scaffolds to help make that decision more concrete and logical rather than a series of special cases that feels closer to memorization than algorithmic problem solving.

A lot of times I come up with these strategies on the spot as I’m trying to navigate this tension between meaningful help that applies to several problems, while also managing my classroom. My guiding principles are: have them do something tangible, have them break decisions into smaller pieces, try to isolate the steps where I know most misconceptions can occur.

A Completely Valid Point: But the student still doesn’t really know what they’re doing or why they’re doing it! You’ve just given them a procedure to follow to get the answer!

Response: Yep. If we’re still talking about a single student in the middle of a class who needs a very targeted intervention, then yes – that’s exactly what I’ve done. It’s not perfect, but it’s how I’ve reconciled the cost-benefit game of these moments. There’s the benefit of trying to explain the conceptual underpinnings of algebra with balance scales or developing a real-world analogy, and then there’s the challenge of competing with their attention span, my resources at that moment, their motivation in that moment, and the time I have with them – all of these lead me to conclude: it’s not realistic that I can fix years of conceptual misunderstandings in a small moment that takes place in the middle of another classroom lesson. If I’m lucky, a student who suddenly gets the procedure will start asking “but why does this work?”, which can lead to that conceptual conversation, but it doesn’t always happen.

There is a place for these conceptual conversations though – its either a more in-depth tutoring session, or a dedicated intervention class (which is what I usually teach). This is when I try to build that conceptual framework and hold them accountable for it – but the middle of a lesson on a totally different subject is not the time or place for that.

So…. there are lots and lots of thoughts and ideas and opinions. Thanks for reading.

Resources for Intervention Classes

I gave a presentation last week on strategies that I use in my dedicated intervention classes for high school math students. The feedback was pretty positive, and I thank the local Tucson folk who attended. One of my favorite moments was letting teachers imitate their worst students, then trying to teach a lesson to them – cell phones out and shouting out answers and heads on desk and unprepared for class and all of the best behaviors of our worst students. That was fun. However, I still ran out of time – so much to talk about.

Anyway – for any other interested parties, I’ve compiled the resources from my talk all in one place: (the MEAD is from the Tucson Math Conference where I first gave the much-too-short version of this presentation). This collection of resources includes:

  • The presentation I gave
  • A collection of resources for how I teach basic Integers & Algebra
  • An article about Mindsets from Carol Dweck
  • Lots of resources for starting to implement SBG
  • Some resources & ideas for classroom engagement strategies
  • Some details of how my intervention classes are organized at my school from a placement & administrative perspective

Leading this workshop has made me want to do two things:

  • Have a video recording of the first week of my intervention classes when I do a lot of culture building and attitude reshaping and a little teaching.
  • Write a blog post detailing the process that me and my administration have gone through to implement our version of intervention. One of the things I could feel at this workshop is that a lot of teachers are being ‘thrown in’ to an ‘intervention’ class without clear details about what the needs of the students are or how to measure progress. This is a stark contrast to how our school has done intervention, so I’d like to share some of that – another time.

Anyway – for any other interested intervention folk, I hope those resources help. Hopefully I’ll find time to share more of my thoughts on intervention in the coming months.