I used to teach math at a high school. Then I taught math at a high school using Project Based Learning (PBL). This blog is dedicated to brainstorming interesting and dynamic math problems and projects.
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).
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.
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.
[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
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)
Chicken Schwarma Thighs ($4.99/lb)
Dried sweet mangoes
Either the pollo asada or beef bulgogi, whichever I’m feeling for later in the week
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 bloggeda bitabout 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
Games of chance
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.
This is a post in the ongoing Emergent Math mini-series: Routines, Lessons, Problems, and Projects.
As we stand on the balcony and gaze out at our own version of the MCU (Math Class Universe) that consists of Routines, Lessons, Problems, and Projects, we must be sure we’re not missing the crucial ingredient that stitches it all together: caring. More specifically, active caring.
Many, if not most teachers demonstrate passive caring. Such teachers show a general, blanket kindness to their students. They’re open to students’ questions. They typically like their students and certainly don’t show unkindness. Often, the student-teacher relationship hinges on the student’s academic aptitude or natural charisma. A teacher might have a decent relationship with all his students, but truly special relationships only with students who excel in their classroom or have otherwise magnetic personalities.
A teacher who is actively caring cares for each student as an individual and views each student as a mathematician. He reaches out to students individually, not broadly. Consider the difference between “if anyone needs help come and see me” versus going to each student to see if they need help. Think of the difference between welcoming the class all at once versus greeting students individually, by name, at the door. These individualized acts of kindness and care are as essential as the task at hand – the routine, the lesson, the problem, or project. Well thought out curricula and tasks are nice, but active caring will ensure that they land for each student in your classroom. Active caring often involves a disruption of social or academic norms: students who typically don’t engage in math receive the same level of care as students who do.
To be sure, active caring is a challenge for a teacher who may see upwards of 100 students a day (or more). It’s difficult to get around to each student in such a compressed amount of time. Don’t beat yourself up if you’re unable to. But make an achievable goal: perhaps every two days you’ll have a personal conversation or check-in with each student. It’ll require a level of intentionality that might seem forced at first. You may have to print out a class roster and check off your interactions with each student as they come. But in the end it’s worth it. An excerpt from a, uh, certain book:
Briana is a 10th grader, talking about her middle school math experience. “I was invisible to the teacher,” she begins. “I always got my work done. I never got in trouble. I would raise my hand to ask a question but my teacher would never call on me. It got to the point where I would ask my friend to ask a question for me so I could get something answered.” Briana is soft-spoken, but clearly motivated. It’s tragic and understandable how she would feel “invisible” to her teachers. In the hustle and bustle of a noisy middle school classroom, soft-spoken students get short shrift.
Recently an administrator I know took part in a “shadowing a student” challenge, in which the administrator identified a student and followed her around for an entire day. From the moment she got off at the city bus stop in front of the school until the moment she got back on it at the end of the day, the administrator followed the student around to each class, every passing period, even lunch. Debriefing the experience, the administrator was stunned by how little teacher-interaction the student received. Other than a greeting here or there, the student received few words from her instructors.
Shy students, or students who don’t have as much academic status, or who are still learning the English language can easily become invisible in a school day, for weeks at a time. Make sure this doesn’t happen. Try some of these strategies:
Document your interactions with students to ensure you’re having conversations with everyone.
Demonstrate vulnerability by sharing details from your personal life.
How do you demonstrate active caring for your students? Let us know in the comments.
Your daily classroom has a lot of moving parts. I’ve attempted to categorize those parts into Routines, Lessons, Problems, and Projects, acknowledging that these are imprecise buckets and you might go between them several times throughout a day. Holding these all together is an atmosphere of active care for each student.
As you think about the upcoming school year, which of these are you curious about? Which do you want to get better at? Do you want to try a project this year? Would you care to create an assessment structure of using “Portfolio Problems” for students’ portfolios of understanding? What’s the right ratio of routines, lessons, problems, and projects?
In addition, what will you do in the first couple weeks of school to demonstrate active caring? How will you touch each student and make sure they’re welcome at the table of our oft-uncaring discipline?
I hope you enjoyed this mini-series. As much as anything it was a think-aloud for myself to wrap my head around all the different ways of being for a math class. I may update the posts going forward as new resources come across my radar. As always, feel free to share insights and ideas.
This is a post in the ongoing Emergent Math mini-series: Routines, Lessons, Problems, and Projects.
I graduated high school twenty years ago this year. What’s remarkable is how little I actually remember about my classes. I remember certain feelings I had towards particular teachers or classes, but not the actual classroom action itself. There are three exceptions. There are four distinct activities I remember from my classes and they’re all projects:
In my combined Physics/Calculus course we divined the accelleration due to gravity based on an experiment me and Eric Durbin concocted. And we were pretty close! We’ll call this theLearning Project.
In AP Stats we conducted soil testing and surveyed the neighborhood to determine whether they cared about this issue. We’ll call this Project-Based Learning.
In English we had to recreate various scenes from The Lost Horizon. We’ll call this the Dessert Project, for reasons which will become clear.
In Biology we had to collect a bunch of leaves. I don’t remember why, but we had to do it. We had to get certain kinds press them is a special way. I hated it. We’re not calling this project anything other than The Leaf Project
That’s it. That’s all I remember about my classwork in High School. Don’t get me wrong: I remember other stuff too, like that time the time my friend Ash was talking so the teacher made him get up and teach the class,I recited “Shaft” in English class (“Who’s the black private dic that’s a sex machine to all the chicks?”), and my creepy Algebra 2 teacher making about ten too many jokes about “french curves.”
But by and large I remember the projects. Don’t get me wrong, there’s other stuff in there. I graduated college and everything, partially with some knowledge I acquired in school. But I only remember these actions.
Projects are an opportunity to illustrate how crucial your discipline is to the world or our understanding of it. They’re also an opportunity to waste several days or weeks of class time and force students to jump through imaginary hoops concocted by the teacher. In both cases, students will remember.
Projects apply mathematical knowhow to an in-depth, authentic experience. A project occurs over the course of two to four weeks. Ideally, projects are outward facing, community based, and/or personally relevant to students.
Let’s take a look at three types of projects. As with this entire mini-series, I’m painting with a broad brush and I’d happily concede that what I call one thing, might actually be another in another’s eyes.
The Dessert Project
I’ll withhold why it’s called the “dessert” project for now. These are typically given at the end of a unit intended to sum up the content. These often occur as a retelling of the content, such as my Lost Horizon example above. We read the book, we identify crucial scenes, and then we reenact them. We’re barely doing any analysis, let alone synthesis.
The best Dessert Projects take what a student has learned and unleashes it on an appropriate real-world scenario. Now that we’ve learned the content, we’re going to see how it looks in a different context. Most end-of-chapters offer this kind of project.
A Dessert Project
The Learning Project
In a Learning Project, we learn something germane to the topic at hand through the use of an in-depth investigation. The structure of a Learning Project is more-or-less dictated by the teacher, but there is enough agency awarded to the students to experiment on their own. The WHY and HOW are often provided and the WHAT is relatively self-contained.
In my gravity example above, we were given the task (calculate the accelleration due to gravity), the materials (a video camera that allowed you to fast forward one frame at a time – this was the 90’s mind you), and the format of the product (a lab report). We had
It was a deeply memorable and engaging task. Unlike Dessert Projects, we are asked to actually find out something new, rather than repackaging information. Despite the fact that Learning Projects may not have a community partner, a public presentation, or a shiny final product (ingredients of Project-Based Learning which we’ll get to in a moment), we construct or deepen our understanding of some new knowledge or knowhow.
I’d suggest these as other examples of Learning Projects:
PBL has gotten the most headlines lately. Schools across the country want to provide deep, authentic, and motivating experiences for kids in all subjects. And to be sure, the best of PBL absolutely achieves that. Students are given a open ended, authentic challenge and students develop and present a solution. Through this process, students acquire new mathematical knowledge and skills.
In PBL (like Problem Based Learning), the task appears first and necessitates the content. Students learn the content in order to achieve their final product. Often – if not always – PBL occurs in groups.
Project Based Learning
But don’t be fooled: quality PBL entails a lot more than just giving the students and letting go of the process entirely. The teacher/facilitator crafts the daily lessons and activities to support the process. The following graphic is taken from the New Tech Network, my employer. It explains well the various phases of a project and a menu of options for lessons, activities and assessments throughout a project.
The project launch occurs at the beginning of the unit. It kicks off and drives the unit. In this case the project is the “meal” as opposed to the “dessert” (recall from early Dessert Projects). The project is how students will learn the material.
Project Launch: Have students read the Entry Document (the letter) and collect “knows” and “need-to-knows”.
To the students of Akins New Tech High School,
The US presidential election of November 7, 2000, was one of the closest in history. As returns were counted on election night it became clear that the outcome in the state of Florida would determine the next president. When the roughly 6 million Florida votes had been counted, Bush was shown to be leading by only 1,738, and the narrow margin triggered an automatic recount. The recount, completed in the evening of November 9, showed Bush’s lead to be less than 400.
Meanwhile, angry Democratic voters in Palm Beach County complained that a confusing “butterfly” ballot in their county caused them to accidentally vote for the Reform Party candidate Pat Buchanan instead of Gore. See the ballot above.
We have provided you the county-by-county results for Bush and Buchanan. We would like you to assess the validity of these angry voters’ – and therefore Al Gore’s – claims. Based on these data, is the “butterfly” ballot responsible in some part to the outcome of the 2000 election? What other questions do the data drum up for you? And what can we do to ensure this doesn’t happen again?
We look forward to reading your analysis and insight, no later than May 5.
Your county clerk
Example project pathway – the 2000 Election
In this example, we still retain many of the lessons and workshops that we would typically teach during this unit, but notably they occur as students are working towards various benchmarks and the final product. The lessons inform the student products.
There is enough to write about PBL to merit its own miniseries. Structures and routines are crucially important in a PBL Unit. Assessment must look different. Managing groups becomes an entirely different challenge. How the authenticity of the product and the external audience enhances the quality of student work.
For now, here are some other tasks that adhere to PBL.
The genesis of this entire mini-series stems from questions I receive about Problems and Projects. Mainly, “how often should I use problems? How often should I use projects?” The unstated part of that question is, “when do I actually, y’know, teach?” (Actually, sometimes that’s stated). I’ll save another post for “putting it all together” or “adjusting the levels” but know for now that’s why I put together this framework of Routines, Lessons, Problems and Projects.
As for the “how many projects?” question, I’ll give a squishy answer and a non-squishy answer.
My squishy answer: design a project whenever (A) the standards uniquely align such that you can create multiple lessons around one scenario and (B) when you can identify a project scenario that will maintain momentum over the course of several weeks.
My non-squishy answer: One or two a year. Most standard clusters don’t lend themselves to multiple investigations around one, single context. But some do! Content clusters around things like Data and Statistics, Area and Perimeter, and Exponential Growth and Decay are ripe for real-world scenarios that can be analyzed through the lens of multiple content standards.
As challenging as it is to design and facilitate projects, and as little time we have as educators to carve out the time for it, we don’t want to deprive students of the real-world insight math can have. We want to provide these experiences that will live on in students’ minds as the power of mathematics, whether or not they go into the field. So be on the lookout. Look for news articles and community opportunities that might embolden students to use math for maximum impact.
This is a post in the ongoing Emergent Math mini-series: Routines, Lessons, Problems, and Projects.
Ah problems. I have to reveal my bias here: I love problems. Problematic problems. Problems are where I honestly cut my teeth as an educator. If you’re reading this blog, might have stumbled across my Problem-Based Learning (more on that specifically in a second) curriculum maps. I’ve blogged about Problem-Based Learning (PrBL) a bit. I’ve learned so much from teachers and math ed bloggers about what makes a good problem, how to facilitate a problem, what kinds of problems are out there. Some of that I’ll share here. Let’s just start with Problems.
The questions on voluminous review packets? Not problems. My first resource on problems, problem-based learning, and problem solving is NCTM’s research brief on problem-solving, Why is teaching with problem solving important to student learning? (2010). In it, it hints at the “what really is a true problem” question:
Story or word problems often come to mind in a discussion about problem solving. However, this conception of problem solving is limited. Some “story problems” are not problematic enough for students and hence should only be considered exercises for students to perform.
This brings us to my personal, current definition of Problems: Problems are complex tasks, not immediately solvable without further knowhow, research, or decoding of the prompt. Problems can take anywhere from one class period to three or four class periods.
So when I say “problems” I mean problems that are genuinely challenging to the problem solver. Even the difficult, toward-the-end-of-the-section questions may not be problematic enough for some students. Also, a problem ought not to be so obtuse or convoluted as to not be accessible for all students. Just because something is real hard doesn’t necessarily mean it’s a problem. If someone were to challenge me to make the U.S. gymnastics team, I wouldn’t consider that a problem; I’d consider it futility.
I like to think of good math problems like this: a good problem is accessible enough so students a couple grades lower can attempt it, yet challenging enough so students a couple grades above have to think about it. I actually think this of all mathematical tasks, but it’s particularly apropos of problems.
I like this problem for many reasons. One, it combines two not-often mathematical things: lines and quadratics. In most curricula you have your unit on linear functions and your unit on quadratics. Why aren’t these two things combined more often? I have no idea. Most textbooks presents lines in one unit and quadratics in another, as if they’re in a different universe. It’s like we’re reading a geography textbook about pre-Columbian South America and Europe. But back to our discussion of problems, it’s the confluence of these concepts that makes this such an interesting, challenging, and worthwhile problem.
There’s straight up problems – just give students a prompt and facilitate as you see fit.
There are countless other modes of problems, here are a few.
Would You Rather? problems
I’m not sure of John Stevens is the first “would you rather” problem designer, but he certainly codified it with his stellar website. A Would You Rather (WYR) provides students two possible choices and students must decide which one makes more sense to choose: which one is cheaper? which one is better? what deal gives the greatest value? etc.
There are several things that make this format incredibly appealing: 1) Providing students an initial choice naturally facilitates guesses and estimates at the beginning of the problem. 2) Making it a choice makes CCSS.MP3, making arguments and critiquing the reasoning of others, a necessary part of the task. 3) In many cases, either answer may be correct, depending on how it’s interpreted, the desired outcome, or the input variables (in the WYR above, the answer may depend on how far away one is from the airport, how much airport parking is, etc.). And 4) there’s something delightful about the “would you rather” framing. Maybe because it reminds me of the “what’s worse?” scene from So I Married an Axe Murderer.
Dan Meyer gave us this format years ago and countless of math teachers have built upon it sense. Following the narrative structure of movie, in act 1 the “conflict” is established and we’re drawn into the plot of the movie/problem. In act 2, our hero / students go questing for the solution. In act 3, we come to a resolution.
Most often these act 1’s kick off with a video or picture to pique the interest. What do you notice/wonder? What do you think will happen? In act 2, students will work through the scenario presented in act 1, sometimes provided with additional information or knowhow that might be useful to solve the problem. In act 3, students make their final answer and we come to some sort of resolution (often by playing the last part of the video).
I’m sure I’m missing others. Please let me know in the comments who I’ve missed.
Like WYR, there’s something inherently appealing about a narrative structure that we’re already used to. We’ve all seen movies, plays, TV shows, and read books. If you can provide a successful hook, we’ll want to see how the movie ends.
Just straight up puzzles
While sometimes challenging to align directly to required content, give students mathematical puzzles. NRICH has a great library of puzzle-like maths, or perhaps maths-like puzzles.
Let’s take a slight birdwalk into the practice of Problem-Based Learning or PrBL. It uses problems as a means to teach new concepts or knowhow. The problem creates a need (and in the best cases, a desire) that requires the intended content knowledge, additional information, or mathematical dispositions.
I suppose in some ways this may not differ much from just giving students the problem and teaching as-needed, as you go. In PrBL there’s an intentionality (and even predictability) with how the problem is posed and how the learning is facilitated (for instance, you prepare the lesson beforehand, rather than just winging it).
Facilitating a problem
One of the biggest mistakes teachers make when using Problems for the first time is that they think that by posing a clever enough problem, students will intrinsically work their way through it dilligently, testing out different methods along the way. And to be sure, it’s understandable to think that when you watch a presentation on problem solving in math or participate in a conference session and the participants or audience dilligently work their way through a problem. But here’s the dirty little secret about conference sessions: the audience is entirely composed of adults who are excited about math and presenters are showcasing their absolute best problems. It’s easy to present engaging problems as a panacea when the audience is entirely bought in and the presenter gets to cherry pick which problem or lesson he or she gets to present. So it’s easy to walk away from these experiences thinking that – just like in that session – I’ll present this super-cool problem to my students and they’ll collaborate, problem-solve, and stick to it just like at that conference.
It’s never that smooth. Rather than – like Carrie Underwood – letting “Jesus Take the Wheel” – you need to keep your hands on the wheel and your foot on the pedal (and sometimes the brakes as well). Problems should be facilitated, not tossed in like a hand grenade. So how do we facilitate a problem?
Use routines. The biggest tip I can provide for facilitating problems is something we’ve already covered in this mini-series: provide routines. Routines to get started on the problem, routines to facilitate discussion in the middle of a problem, and routines when students are sharing their solutions.
Consider this sample Problem facilitation agenda:
Introduce the problem
Facilitate a Notice & Wonder routine
Identify next steps and let students begin working
20 minutes later, take a quick problem time out and have groups do a gallery walk routine to see how and what other groups are doing
Give a problem “time in” and have students continue working toward a solution
After finishing the problem, have students show appreciations to one another via a routine.
One problem, three routines. And who knows? If students are struggling, you may want to hold a small workshop lesson in there as well. We’re starting to see our Routines, Lessons, Problems, and Projects framework become a set of nesting dolls.
Provide consistent group roles. Assuming students are working in groups, provide consistent, well-understood group roles.
When you think of a math lesson, you probably conjure up an image of a teacher in front of the classroom demonstrating mathematical concepts. While that certainly qualifies as a lesson, I’d like to broaden your mental image. Consider a “lesson” any facilitated activity where students are building or practicing their content knowledge. In addition to our imagined lecture, let’s also consider activities such as card sorts, investigations, practice time, and other structured times in the classroom.
Lessons include any activity that involves transmitting or practicing content knowledge. They can vary from whole class lectures to hands-on manipulative activities.
Lessons probably make up the bulk of your course. Students walk in to your room, you teach them some stuff, the day ends. That’s a lesson. How you teach offers endless possibilities. Let’s look at some of these possibilities.
There’s nothing inherently wrong with a lecture. I’d suggest it’s not always the best way to engage students. But oftentimes it is the most efficient way to transmit information, provided you are lecturing effectively. How does one lecture effectively? Despite being perhaps the most oft-used instructional approach, little time if any is spent in pre-service teacher programs in how to do lecture well.
Things to consider:
How will you ensure all students are engaged throughout the lecture, not just an eager few?
What’s the shortest amount of time you could possibly do the talking? Go with that. And maybe subtract a few more minutes.
Are you incorporating visual elements into your lecture?
When you’re lecturing, you want to stop and prompt discussion often, perhaps every 3 minutes or so. Rather than asking a question and waiting for a student to raise a hand, consider utilizing some of our general discussion routines from the previous post. The more you can make your lecture feel like a conversation the more successful the lecture will be.
When you’re lecturing, try to get students in the mode where they’re talking to one another rather than to you. See this blog post on various lecture models.
Some additional tips for lecturing:
Start your lecture with pizazz. Bring in a recent news article that pertains the the topic. Start with a memorable or funny quote. Post a picture or diagram and ask a question about it. For example, launch a lecture on horizontal asymptotes with the following graph and the prompt “Do you think these lines will ever intersect? Turn and talk to your neighbor and explain your reasoning.”
Create a hook that will grab students’ attention. A picture plus the Notice and Wonder protocol works extremely well.
Question authentically, not putatively. Questioning to get to deeper understanding is a skill that takes years to hone. It’s important to get genuinely curious about students’ ideas. As much as possible try to avoid the punitive, I-bet-he’s-not-listening questioning. Of course we want students to be paying attention, but we don’t need to “gotcha” students by asking them to derive the quadratic equation on the spot when we’re actually trying to make them feel foolish for zoning out during our boring lecture on the quadratic equation.
Talk slower. Every human talks 30-40% (not precise calculations) faster in front of audiences than they do in normal conversation. I’m not sure why, but it just is. Slow down. You need natural pauses and a good cadence, otherwise your words will morph into that of Charlie Brown’s teacher. I found this potentially effective technique:
Mark a paragraph / in this manner / into the shortest possible phrases. / First, / whisper it / with energetic lips, / breathing / at all the breath marks. / Then. / speak it / in the same way. / Do this / with a different paragraph / everyday. / Keep your hand / on your abdomen / to make sure / it moves out / when you breathe in / and moves in / when you speak.
Before you whisper each phrase, take a full bellyful of air and then pour all the air into that one phrase. Keep your throat open, and don’t grind your vocal chords. Lift your whisper over your throat. Pause between phrases. Relax. Then, take another full breath and whisper the next phrase. Whisper as if you were trying to reach the back of the room.
As a fan of the Discovering Mathematics series of math textbooks, investigations were a staple in my classroom. These lessons involve an intentionally structured activity that reveals some new mathematical truth.
Using tools or manipulatives
As an example, here is an activity on Triangle Inequality and dried spaghetti:
Kids use their hands and dried spaghetti to determine the triangle inequality theorem: the sum of two sides of a triangle must be greater or equal than the third side.
Discovering Geometry was big into patty paper activities. These were excellent, cheap ways to get kids using their hands to make discoveries.
In this activity students (much like in the spaghetti activity) identify mathematical concepts through purposeful questions. In Running from the Law, it was the connection between the distance formula and Pythagorean’s Theorem.
The questions are carefully ordered to point out possible discoveries hidden in the mathematical weeds. In some ways these activities mimic a quality activity debrief.
The desmos team – and many of their contributors via the activity creator – use the clean interface to construct lessons that allow for students to construct their own understanding through carefully designed activities.
Students begin by interfacing with a challenge, notably without any discernable mathematics. Throughout the activity, students are prompted to identify what information would be helpful to solve the challenge. Eventually, we build enough knowhow to write expressions that help us out. Each slide presents an additional prompt intended to get students to think mathematically about the scenario.
The Card Sort
Another general type of lesson is The Card Sort. Teachers provide students materials that need to be matched up or ordered in a specific way to make the puzzle work. The most common type of card sort is matching. Students match two or more like items, typically in the form of paper or card cuttouts.
A twist on the matching card sorts I quite like is that of “dominoes.” It’s like card sorts in that there are cuttouts and students are asked to arrange them in the matching order. But in this case each cuttout has two “things” on it and they match with another “thing” from another card. The result is a circular matching activity:
I like it because it offers an immediate check: the “dominoes” should circle completely around and there shoulndn’t be any gaps.
Things to consider:
Card sorts take a little time to build. It’s helpful if you have a template. Here’s one: Card Sort Template
Card sorts take significant time to cut out and put into plastic baggies. However, if you do it once – and have students place them back in the baggies at the end of the period – you’ll have them forever. I’ve had some card sorts in baggies for almost ten years now.
The Practice Problem(s)
Some classes and class days incorporate a lot of practice problems, packets even. That’s ok. We can work with that. A packet of a few high-quality problems can be an effective means of deepening understanding. I’ll go ahead and re-emphasize it for ya: a few high-quality problems. Now that we have that out of the way, we can hone in on effective means of teaching on a day – or a time of day – with a lot of student practice. I’ll offer two strategies that make the Practice Problem lesson an effective one.
Same problem, same time
Assuming students are progressing through practice problems in groups (which I recommend), make this a norm in your class: “same problem, same time.” This means that group members cannot proceed to the next problem or next page until all their group members are ready and have demonstrated understanding. Every group and every group member ought to be on the same problem so they may discuss it when it becomes challenging. You should never have a student call you over to ask about a problem that they’re working on and their groupmates aren’t (either because they left him in the dust or vice versa).
What are the norms of groupwork you want to see in a given problem work time? Make those public and identify when those moments are happening – or not happening. This can easily be achieved through a document camera or anything that’ll project a document.
In this case, the teacher identifed “plusses” and “deltas.” Or, positive behaviors or phrases students are..