Category: math

A classroom with quality, complex problems as its cornerstones can support English Language Learners. First let’s check out a few “ground rules” about supporting English Language Learners. The following ground rules are not exhaustive, but are pulled strategically from English Language Learners and the New Standards by Margaret Heritage, Aída Walqui, and Robert Linquanti.

• Use authentic and meaningful tasks to build student agency and sense of purpose, as well as background knowledge and schema
• Learning needs to be social. Scaffolding should be interactive and occur through discourse.
• Learners need to use language themselves in meaningful and purposeful ways.

Problem-based learning can go hand in hand with these principles, provided you establish strategies to promote them. The use of authentic, quality tasks paired with discourse can yield significant gains in language acquisition. However, you must be conscious and strategic. Here are five strategies to help that process.

Use word walls. Post challenging vocabulary on your classroom walls to offer visual reminders of content terms. Be sure to provide not only the word itself, but pictures and examples when possible.

Use Sentence stems. Provide students sentence frames and sentence stems. For example,

Be judicious about sentence stems as time goes along. You don’t want to stifle student creativity and their ability to flex their English going forward. Still, sentence stems can be an effective way to bridge that gap.

Double Entry Journals. Double entry journals are a strategy of writing to learn. Student divide their notes with a vertical line. On the left is the source material. It may be a problem, a vocabulary work, or a strategy. On the right they can write their interpretation or process or even additional questions.

Use Graphic Organizers. Thankfully in math we have a wealth of graphic organizers at our disposal. Graphic organizers can include concept maps, Venn diagrams, bar charts, etc. Be sure to take your time with them. If you’re placing items in a particular part of a Venn diagram, encourage a discussion and come to a consensus as a class.

Amplify, don’t simplify. Don’t shy away from using sophisticated vocabulary. English Language Learners need multiple and varied exposures to rich, and even technical vocabulary. Offer the challenging vocabulary word and support it by providing synonyms, definitions, and examples.

This is only a small bank of strategies to support pupils. What are some of your most effective strategies to support students for whom English is not their primary language?

(Note: While I use the term English Language Learners (ELLs) in this blog post, I recognize that there are other – probably better – terms for students for whom English is not their first language. By putting “English” first in the term and implying English acquisition, the term promotes anglo-supremacy. It is the accepted term from the U.S. Department of Education. Other similar but distinct terms include Emergent Bilinguals (EBs) and First Language Not English (FLNE) students.)

There is no shortage of criticism of Khan Academy around these parts. In fact, Khan Academy criticism was among the first unifying themes of the math blogosphere. Since then, however, KA has made their platform more robust and useful. And those of us who swore it off might want to take another look.

I’m taking another look because my son is using it – not because his school is assigning it, but because he wants to learn the material. This in and of itself was perplexing to me, so I decided to figure out why he was so invested that he’d want to use KA as his limited screen time for the weekend.

Part of the hook is that it comes through a screen. For whatever reason, he’s more apt to do work when it’s on a computer. He’ll struggle to write an essay on physical paper, but struggle to keep his word count low in Google Docs. He doesn’t love to draw with a pencil, but will spend hours creating an interactive image in Scratch. Or even in the KA computer science section.

So that’s one hook. There are also probably lessons to take away around customization and visualization. For example, here is my son’s “bio.”

But there are two crucial things that make KA engaging and even pedagogically sound (or at least, can push us to be more pedagogically sound).

1. It lowers the pain threshold for incorrect answers. By allowing instant retakes on problem sets, there’s less of an admonishment for incorrect answers. Students who are troubled by a “7/10” on the top of a paper may be more prone to give the assignment another shot if they know that score will essentially disappear.

1. It offers immediate and actionable feedback. There’s no feedback more timely than immediate. Even the hardest working teachers return tests and quizzes the following day, if not several days later. And with the immediate feedback comes suggested instructional how-to videos.

Zaretta Hammond in Culturally Responsive Teaching and the Brain offers a few characteristics of quality feedback:

[Quality feedback] is specific and in the right dose… It is timely… It is delivered in a low stress, supportive environment.

Hammond (2015) p. 103

Hammond also cites research suggesting that students of color in particular often do not receive timely, quality feedback from teachers. I’d recommend checking out the book and especially this section about the types of feedback students receive and what type of feedback they should receive. While KA is not the solution to problems around timely feedback, it is instructive.

There remain countless valid criticisms of Khan Academy. The problems are rote and allow for little in the way of problem solving. It’s anti-social. Left in the hands of a less discerning teacher, it can reinforce negative attitudes about math and about one’s self as a mathematician. Even the timely feedback mechanism has pitfalls: a student can retake tests easily and quickly, which allows them to bypass reflection about the content and why the answers are incorrect.  If KA is a cornerstone of a classroom, I’d have concerns.

But there are elements that we can learn from to make our own instruction better, particularly in the realm of supportive, timely, actionable and low-stress feedback. And for pet drawings

When we got a Trader Joe’s in our humble little burg of Fort Collins there was much rejoicing. Now we have a place to get all sorts of goodies, which I’ll describe in more detail in a moment. Sadly, due to some byzantine Colorado laws they cannot carry Two-Buck Chuck. Nevertheless, I hit up TJ’s once or twice a week.

Trader Joes are pretty funky. There are a few distinguishing characteristics. They are small grocery stores with seemingly bizarre, diagonal aisles. It’s actually quite an inefficient layout when compared to a big box grocery (which, as we’ll learn, is part of the success). Most of the products in a Trader Joes are branded as a Trader Joes product.

I came across this article about what makes Trader Joe’s so successful. And boy is it successful (it sells twice as much as Whole Foods on a per square-foot basis). Naturally, my mind turns to professional development and teacher training. Naturally.

What are some lessons we can take from the success of Trader Joes to make our professional development more effective

1. Simplify the supply chain, get rid of decision paralysis.

Whereas in a typical grocery store you’ll find over a hundred pasta sauces, Trader Joes has less than ten. In the past when I’ve wanted to show how much awesome math material is out there, I just blasted out dozens of dozens of websites. Attendees are left not knowing exactly where to go because there are too many places to go. It would have been more effective if I’d just given participants three to explore.

2. Open it up, explore leisurely without that blast of cold air.

The frozen section in most grocery stores consist of a few aisles of glass doors, keeping the food frigid. At TJs, the frozen food is right there out in the open in super cold bins. A shopper will walk by and pick up and hold the food and wind up buying it. By letting customers interact with the food without barriers, they’re more likely to buy it. Professional development is so much more effective when we can get our hands on it immediately. Let participants play around with the concepts quickly (for instance, via an engrossing math task).

3. Create an element of discovery.

The reason most TJ branded products are hand-drawn is because it gives customers a folksy sense of discovery. Like, they’re the one who discovered this bag of russet potatoes.

How can we incorporate more discovery into professional development teacher training? Spend some time ideating with one another. Ensure participants are quick to share tasks and strategies that have worked well for them in the past. And for goodness sake, let attendees explore those (three) websites. Don’t just throw them a link and expect them to check it later.

And my personal favorite reason to shop at Trader Joes:

4. Offer a mix of pre-prepared items and fresh produce

If I want to bring the groceries home and have something ready in five minutes, I can grab a bag of fried rice and go with that. Or I could snag some produce and meat if I wanted to cook something. Most of the time, I do a bit of both: something quick for tonight and something that requires a bit more effort for tomorrow. While most grocery stores have both, TJs’ pre-prepared foods are cheap and good.

When I conclude professional development, I want teachers to have something quick-n-easy to implement tomorrow while also knowing where to turn for more intentional concepts. In a math PD, I may introduce number talks (interacting with them) so they’ll be able to turn around and facilitate one the next day. I’ll also often showcase a Problem-Based task or Project Based Learning unit. These require more planning than our PD often allows, so it’s something I can follow up with afterwards.

Similarly, I like to offer a task to facilitate as well as a pathway to build additional tasks. It’s a nod to a framework I wrote about earlier: Find, Adapt Create.

To conclude this blog post, as a Trader Joes sycophant, I’d like to pivot entirely away from professional development and share Geoff’s Trader Joe’s Essentials shopping list. These are the things I purchase every time I go there, because it’s always good to have these in stock:

• Potatoes: russet, yukon gold, and sweet
• Coffee (the $4.99 whole bean tins)
• Fried rice (one chicken, one veggie)
• Orange chicken
• Chicken Schwarma Thighs ($4.99/lb)
• Dried sweet mangoes
• Yogurt
• Bananas
• Either the pollo asada or beef bulgogi, whichever I’m feeling for later in the week
• Brie cheese

Recently I had a conversation with a special education coordinator. He was struggling to keep his kids in their classes. They kept getting sent out for disruptive behavior, being off task, or not playing well with others. He talked about parents who would leave IEPs in tears. These meetings – and other informal meetings – sometimes devolved into a laundry list from teachers of how the student is misbehaving or why they’re not up to the challenge of complex work.

The special education coordinator and I talked for a good long while. He would describe it as “therapy” afterwards, and the feeling was mutual. I shared my experience as a parent of a child who has special needs and gets kicked out of class routinely for disruptive behavior. I shared how every day I’m afraid of leaving my phone in a drawer for five minutes because I’ll miss a call from his school saying I need to come pick him up because he’s thrown a clipboard or flipped off an adult. Every time my phone rings during school hours I’m certain I’ll have to rush off to rescue him from the school, and the school from him.

And this has been going on for years.

Supporting students with special needs is a daunting challenge. Perhaps the biggest impediment is that label: “students with special needs.” That’s such a catch-all term to the point where it’s meaningless. When a teacher asks “how can I support students with ‘special needs’?” it’s like asking how they can support students who are wearing blue t-shirts. Students with “special needs” have very different needs.

Sometimes the special needs are even contradictory. A student may need one type of accommodation during one part of the day and a different for another. Teachers can be understandably frustrated and feel understandably powerless. As a parent, and speaking for my special educator friend, there may be an accommodation that will work, but A) we may have not found it and B) there might not be an accommodation that works in the moment.

There are certainly well-researched best practices. For starters I’d recommend Count Me In by Judith Storeygard and following Andrew Gael on (on twitter @bkdidact and his blog). There are effective strategies that may work for certain students at certain times.

But they might not.

That’s why my advice to teachers is to first understand. Why is a kid acting out? It might not be because he’s defiant, but because he is exhibiting his cognitve disability. If a student is yelling at other students, it may be because the student has a social communication disorder. These are very real diagnoses, just as real as a student with a physical disability. Any teacher in the world would make every accommodation for a student with a physical disability. We should be equally accommodating for students whose disabilities are invisible.

Does this mean every teacher needs to become an expert in the latest DSM? No, but it does suggest that if you have students with special needs you try to understand. Understand what makes him or her tick. What are his or her antecedents? What are his or her triggers? What are the warning signs? What makes him or her happy? Who are his or her friends (the few they do have)? What can you do to break through when he or she isn’t in an elevated state? And yes, eventually, what are some strategies we can try out?

Well meaning teachers want to know the strategy, the “magic words,” the trick that’ll make it better. I’ve had a teacher ask and even beg for that: “Just tell me the accommodation and I’ll do it! I want the student to succeed!”And I’m here to tell you that there isn’t a single, magical catch-all strategy or accommodation.

The most successful teachers in my son’s life have approached him from the standpoint of “how can I understand this guy?” It’s a subtle, but crucial, mindset switch from “how can I get through to him?” I’d recommend temporarily tabling questions around strategies and teaching moves. Those are important questions, but first, approach from a point of trying to understand the student. Then we can proceed to step two.

I’ve given the book talk (by other names) a few times now, and I’m noticing some patterns of what’s really resonating. One small, but significant piece that’s fostering conversation is a section around Active Caring vs. Passive Caring. I’ve blogged a bit about this in the past, so feel free to check out those posts. There appears to be an appetite for this conversation to occur in schools. Feel free to use this chart as a starter set of active caring action moves.

One question that comes up is, “How do I find time to display active caring to each and every student?” A secondary teacher may have well over a hundred students a day, segmented into blocks of time possibly as low as 45 minutes. How is a teacher supposed to show active caring to every student every day?

The short answer is: you probably can’t. Let’s be real honest. If you have a tight schedule and a large student load it’s challenging, bordering on impossible, to take time out for every student every day. It’s a simple math problem: if a teacher has, say, 120 students and five classes of 50 minutes (250 minutes total), you can spend about two minutes per kid before even getting into the day’s lesson.

Rather than throwing up our hands and saying we can’t do it, I’d propose the opposite: we need to be structured, methodical, and intentional with our actions around active caring. Here are three suggestions for tackling this math problem.

1. Make a list. 

Print out a class roster and with days of the week and record when you’ve had an interaction you’d classify as one of active caring. If you have a good memory, you could even do this at the end of the day or after a hald-a-day. Try to get around a quarter of your students every class period. That way, by the end of Thursday, you can see which students you have yet to have an active caring interaction with and you can make sure to be intentional on Friday. Keep yourself accountable to showing each student active caring no fewer than once a week.

2. Build in structured personal check-in time.

As students are working, build in, say, five to ten minutes where you are not answering questions about the assignment, but are rather floating and checking in with students. Be disciplined about it. Set a timer if you need to. Depending on the length of your class period and the way your day’s lesson is structured, consider whether you want to block off this time toward the beginning, middle, or end of the class period (or possibly bookending the class period).

3. Work as a staff or grade level team to identify personal connections 

I’ve seen a few staff, department or grade level teams do this.

Print out the name of every student and place them on the wall around the room. Teachers place a sticky dot by every student they have a personal connection with. Look for patterns and anything (or anyone) that stands out. This can help a school know which students might not be receiving the level of care that we’d hope. It can prevent students from falling through the “care gap.”

What are some of your strategies for ensuring you are demonstrating active care for all students?

Chris and Melissa gave a great talk on the importance of mathematical play at NCTM-Seattle last week. You can see their Math-on-a-Stick work on their website. There you can see pictures and examples and of children enjoying and playing with math in interesting and delightful ways. One of my many takeaways from their keynote was that play is math and math is play. In their talk, they referenced research that lays out seven attributes of play. Play (a) is purposeless, (b) is voluntary, (c) is inherently attracting, (d) involves freedom from time, (e) involves a diminished consciousness of self, (f) has possibility for improvisation, and (g) produces the desire to continue.

When I saw the mics set up I A) assumed there were going to be questions and B) just knew one of the questions was going to be “but what about older kids?” Sure enough, there was a question about how adolescents might play with math. The premise – which I kind of (but not entirely) disagree with – was that older kids wouldn’t be engaged by things like pattern machines, tiling turtles, and Truchet tiles. Chris and Melissa gave good answers about the age band of the kids of math-on-a-stick and spoke to the non-zero amount of older kids, but I’d like to offer a few examples of older kids playing with math. Unfortunately, I didn’t take as much time playing with math as I should have as a teacher. So I’ll share a few ways I and my kids play with math as regular ol’ humans.

Me: Baseball Prospectus and Sabermetrics

I was trying to think of the first time I played with math post-pubescence. I was such a baseball fan in high school, partially because the Cleveland Baseball Team was quite excellent at the time (despite not having any rings to show for it), but also because of stats. I began reading the great baseball writer and sabermetrician, Rob Neyer. I began organizing various baseball reference spreadsheets. I felt like I was finding out secrets of baseball that most managers (and fewer commentators) knew. Things like “on-base percentage is more important than batting average” and “home runs yielded are a better predictor of future pitching success than other categories.” This secret information yielded by mathematics helped me understand the game while also helping me win my fantasy league to boot. (Note: I’ve written a bit about this before.)

My daughter (age 13): Animation

My daughter is a phenomenal artist. She draws all day, every day. If there is a Gladwell-ian 10,000 hours rule, she eclipsed that at least year ago. She likes to create animations, frame-by-frame. Moreover, she likes to animate to music. She plays with math by timing out the different scenes in a potential song and crafts them into a music video.

My son (age 11): Scorigami

Scorigami is a concept created by Jon Bois, a content creator for SB Nation. A scorigami is a final score of an NFL game that has never occurred before. For instance, the score of seven to eight has never occurred before. Were two teams to end up with that final score, that would be a scorigami. Because of the interesting numbers and combinations of numbers that occur in a football game, many scores have not been achieved in an NFL game. Scores in football come in 6 (touchdowns), 3 (field goals), 2 (safety or two-point conversion after a touchdown), or 1 (extra point, but that has to come after a touchdown, 6).

For instance, there has never been an 18 to 9 final score. There has been an 18-10 final score, but never 18-9.

Every Sunday we watch football and keep an eye out for potential scorigamis. Once it gets to the fourth quarter and we’re looking at, say, a team with 11 points, we’re in scorigami red alert mode. My son plays with math by keeping an eye on the scorigami grid, including the density map, to identify how scores could occur throughout the Sunday games.

Here are a few more rapid fire examples of mathematical play I’ve seen or experienced from adolescents:

• Google Sketch up
• Messing around with pascal’s triangle
• Fantasy sports
• Games of chance
• Des-man

What about you? What have you seen or done that might constitute as mathematical play that secondary kids might be interested in?

Update (12/6): Within hours of publishing this post, my son had an idea for mathematical play (he did not call it that).

Mario Party is a video game for the Nintendo Switch. It acts essentially as a board game with little mini-games throughout. Characters roll dice and move around the board collecting things. What’s interesting and made this ripe for mathematical play is that each playable character has a different die. They all have six sides, but have non-standard values.

For example, the six values for the Luigi die are 1️⃣1️⃣1️⃣5️⃣6️⃣7️⃣. The six values for the Peach die are 0️⃣2️⃣4️⃣4️⃣4️⃣6️⃣. You can also have dice that give you coins instead of moves for some rolls. The goomba dice yields +2 coins, +2 coins, 3️⃣4️⃣5️⃣6️⃣.

For seemingly no reason at all, my son decided last night he wanted to tabulate the average (mean) values to determine the best character die. He also assigned commentary (“high risk, high reward”) to the dice. I do not know how he factored in the coin values.

He then sorted the dice into tiers – really good, okay, and bad based on the mean rolled value.

Why did he do this activity? Well, he’s not allowed to have screen-time during the school week, so this might have been his way of coping. But regardless, it was generally pointless, which, when it comes to mathematical play, is essentially the point.