# I have a couple of questions about “Social Justice Math”

I have a couple concerns regarding “Social Justice Math” that I don’t think I’ve seen addressed. (If they have been, please let me know.)

From what I’ve read SJM is billed as a way to bring real world problems into the classroom with a “justice” lens. Problems related to climate change, economic inequality, racial equity, etc., would be used in class as frameworks for learning different math concepts. (Read more on that here.) In fact, it sounds a lot like Project Based Learning but with a more refined list of suggested issues to study.

The first concern I have is that, like it or not, “Social Justice” is associated with the political left.

Do those advocating for SJM openly say this is a political slant on mathematics and embrace it as such? (Let’s call this “motivation A”.)

Or do they argue they’re talking about social justice (fairness to people in general, without the political connotation) and not Social Justice? (Let’s call this “motivation B”.)

In the former case I’d have real concerns if I was conservative minded person and my child was in that class (or independently/liberal minded and concerned about one political viewpoint seeping into mathematics curriculum). In the latter case the perception will still almost certainly be taken as a leftward spin on math, again because “social justice” is attached to the political left.

The second concern I have is, what exactly is the “social justice” aspect of the math. Is it simply the selection of the topics chosen? Or is it in the conclusions that come from the students’ analysis? Will the teacher point out that social problems are complicated and that both the left and the right have something to say about their causes and solutions?

I can imagine a teacher trying to present these problems in an unbiased fashion and letting students arrive at a variety of remedies to the problems (motivation B folks). But I would bet money that many teachers implementing SJM will be pushing students to arrive at solutions from the political left (motivation A folks).

If they weren’t, then why call it “social justice” math? Why not call it “real world mathematics” or some other less politically charged title that still acknowledges you’ll be analyzing problems that humanity faces? (Again, this seems a lot like a political form of Project Based Learning.)

I fear “Social Justice Mathematics” is the title because they don’t want students to learn to take a dispassionate approach to the problems. They want students to take a certain, Social Justice approved, approach to analyzing the problems. If this is the case then I think we’d be right to push back against SJM, and if it isn’t the case then SJM will face a branding issue for the foreseeable future.

Are my concerns justified or am I way off base? I’d love to discuss it in the comments.

# 6 Reasons This is My Favorite Lesson

I want to share what might be the best lesson I’ve created and a few reasons why.

I actually wrote about this a couple years ago but since we’re doing it right now I thought it useful to reflect on and share it again.

This lesson came from the following problem I was struggling with:

I had spent a lot of time thinking about how to help students understand the connection between trig ratios on the unit circle and the graphs of trig functions on the Cartesian plane. Despite a couple activities and practice I was convinced, mainly through questioning, that they didn’t fully understand it.

My solution to this was to make a giant unit circle and cartesian plane and have students use them to work out problems. This would allow us to literally walk to specific angles and equivalent places on the cartesian plane. The hope was that this would help students solidify the connection between the two.

The details of how the activity works are in the original post and the materials are linked at the end of this post, so I want to emphasize the aspects of the lesson that really make it effective, in a convenient list.

### Assessment

The activity is broken into two parts. The practice portion and the assessment portion. The assessment requires students, working in pairs, to come into the hallway and work through five problems (like these). This portion is vital for the following reasons.

Like any assessment, it helps me know what they know.

It makes students take the practice seriously. The assessment mirrors what the practice rounds were like. They take it seriously and practice until they’re confident.

Students work in pairs, sometimes disagree, and then must convince each other of their reasoning. Tremendous mathematical conversations come from this time.

It puts students in a position in which the teacher is there, but can’t help. This is true of assessments in general, but the format of this one means students must convince themselves and each other that their answer is “their final answer”.

### No calculator. No notes.

On the assessment, and likewise on the practice, students cannot use a calculator, notes, their unit circle, or anything besides their brains and a whiteboard. This means students don’t have any crutches with which to rely on. These problems are not algorithmic. Each one is slightly different from the other ones. This means that the only way to be successful is to truly understand what is going on in the math.

### Engagement

This is my third year doing this activity and every year there’s nearly full engagement. Now, this is precalc and while I wouldn’t say that all of these students want to be there, it is an elective. But it’s difficult for me to get this level of engagement from them.

This is, in part, because they know there’s a test coming after they’ve practiced. But I think it’s also because each problem sparks at least a little bit of curiosity. “How do we figure this out?” Initially many students don’t have a clue about how to approach something like sec(2pi/3) with only their brains and a whiteboard. But with a good understanding of trig they can figure it out.

And figuring it out is satisfying. Students are proud of themselves when they solve one of these problems correctly. I love seeing high fives in my classes, and this is one of those activities where they happen.

### Embodied Cognition

I’ve written about embodied cognition before so I won’t go into too much detail, except to say that it’s incredibly valuable if you can incorporate it effectively. There is something fundamentally different from paper and pencil when you can stand there with a student inside of a unit circle and discuss these problems. It’s something that is hard to describe, but once you’ve tried it you clearly see the value.

### Purposeful practice without a book assignment

A few weeks ago students initially learned how to do these problems via a lesson and practice problems. If that was effective, then I wouldn’t have needed to do this activity. What ends up happening in this activity is that students end up doing a bunch of practice problems, that I never assigned! I just tell them they can do as many practice rounds as they feel they need. Then they work until they have convinced themselves they’ve mastered it.

### Partners

The test and practice require students to work in pairs. This is incredibly valuable as students are constantly conversing and helping each other understand. Once again, the knowledge that there’s an assessment plays into this, but who cares? From my observations students are rarely begrudgingly woking through these problems. They seem to enjoy them.

I probably see more learning and teaching happening between the students in this activity than any other lesson I do, for any class.

I understand that without seeing it happen it might be difficult for you to implement this. I’ve included some images below to give you an idea of the set up. Feel free to contact me with any questions you have. I’d encourage you to look for opportunities to use embodied cognition in your classes as I think it can be an incredibly useful teaching tool.

Here are the resources for doing the activity

Description Sheet

Possible Problem Bank

Practice Cards

Assessment Cards (Yeah, I’m not posting these on the web. I, shockingly, sometimes have students read my blog. But if you reach out to me I would be happy to email them to you and save you the time of making them.)

Assessment Rubric

X-axis “Tick Marks”

# Real Questions

Every year in precalc, two things happen. First, the exponential and logarithm unit gets squished (education jargon, I know). Second, we do the “student loan” blog post, which Steve came up with when we were first working on precalculus together. The former drives me crazy because I think that unit is more useful than some of the other things we spend time on. The latter I look forward to because it’s such a great learning experience for students.

The problem goes something like this:

Suppose you take out a \$5000 student loan every year for four years. How long would it take you to pay back the money? Please make sure to include research on government subsidized and unsubsidized loans and private bank loans.

Yeah, it’s vague. There’s no rubric, besides the standard blog rubric. The openness of the problem drives students a little crazy, which I’ve decided isn’t a bad thing. Let me explain.

The students in this class are, most likely, college bound. I don’t teach in a particularly affluent part of the state so most of my students are going to have to take out loans to pay for college. This means that the question, how long will it take to pay back loans, is painfully relevant.

Here are my main objectives for this assignment:

• I want students to understand the different types of loans and some of the verbiage they’ll encounter, if only at a surface level.
• I want students to understand the connection between exponential functions loans.
• I want students to understand how important interest rates are in total cost of a loan over time.
• I want students to realize that in this fairly common scenario, it’s not unlikely that they’ll be paying back loans for 20ish years.

I should also note that I don’t make students run these calculations by hand. I encourage them to use loan repayment calculators. I want them to understand that there is mathematics at play here, but I don’t want them to get too hung up in computations.

For many students this is an eye opening project. I’ve had students say they plan on paying it back in five years and they want to be a teacher. I had to gently explain to them that it was unlikely they’d be able to afford that. Steve has had students in tears doing this project. It has many great opportunities for students to learn about life, and also mathematics.

### Reflection

I wrote the beginning of this post when students were working on the project. I’m writing this part after grading them. Here’s a couple things I’m going to connect year to make it go better.

• Clear up the guidelines. There’s explanations in the video and a synopsis on the website. They’re both slightly different. I also plan on making them clearer.
• Listing things for students that are wrong and that students have done in the past. Things like only running one scenario, only computing a loan for \$5000, and not providing explanations for their numbers.
• I will include a question asking them to explain what they learned or gained from doing the projects.
• I will look up some useful repayment calculators as suggestions. I noticed that this can be a bit help or hinder ace depending on where google leads them.

Other than that, this activity will be used next year.

# I’ve underestimated the importance of vocabulary

I thought for a long time that I could get by teaching math while deemphasizing vocabulary. Obviously we would discuss the meaning of words, especially the ones that come up frequently. But I thought that if I was able to help students get a feel for the math, and show kids how to do math, without getting too caught up in what the new vocabulary meant, that would be success.

Part of this was time. Or rather to save time. Spending time helping students really understand vocabulary takes more time, especially if it’s something that is more easily shown/practiced. For example, I feel like one of my struggles with helping students understand domain and range is that I don’t do a good job at really helping them understand the words. In algebra II, if I present a new type of function to them and ask them to find the domain and range, they often struggle until they see a few examples. It’s as if they’re simply replicating the process for each type of function.

At risk of this turning into a domain and range post, let me explain a bit further. When we study quadratic functions I tell students the domain is always “all real numbers”. The student thinks, “Sweet. Whenever I see a question over domain on the quiz, I’ll just write ‘all real numbers’.” When we learn a new family of functions they have no understanding of how to find the domain, beyond “that’s something with the x values, right?”.

It’s not just that topic. In fact, the concept that propelled me to write on this topic was grading a quiz over factoring polynomials and finding zeros in polynomials. Way too many of my students don’t know the difference between factors and zeros and constantly get them confused. My most significant observation was that I find students are trying to get by with the least amount of vocabulary understanding, and I don’t think I’m helping things by demphasizing it.

Since I’m having this realization at this point in the school year, I think the fix going forward will be trying to find and develop small activities to help reinforce vocabulary. Simply emphasizing it more is a start. I’ve also done some activities, like concept maps and “functions back-to-back” which help with vocabulary understanding. Next school year I’d like to take a more systematic approach and deliberately build in vocabulary activities into each unit.

Drop your favorite vocabulary activities in the comments below or send them my way on Twitter. Thanks!

Image Credit: “Words” by Shelly on Flickr

After years of teaching how to factor quadratics and then getting in my car and banging my head on the steering wheel, I decided that enough was enough. I was going to spend some time finding a better method. I took to my favorite community of math educators, the #MTBoS.

Several different ideas were thrown my way, but the one that was most attractive was Mary Bourasa’s method, sent to me by Helene Matte.

Last year I had tried the “diamond” method, which worked a bit better than simply guessing and checking, which I’d done in previous years. The first problem I ran into was that I had trouble remembering what went in the top of the x and what went in the bottom. In videos I watch online teachers did it different ways. I think this was because of the second problem I ran into, which is where the hell did this giant “x” thing come from anyway? It’s not a “trick” really, but it does seem to have no connection to other things we do in math.

I might as well have said, “Today we are going to factor quadratics. Draw a random shape, fill it with numbers in the recipe I give you, then get your answer.” And kids weren’t that good at it.

### Enter Box Method (or area method, or whatever)

A few years ago I learned about the “box method” for multiply polynomials, binomials included. Put the first polynomial along the top, the second along the left side, multiple rows by columns. Very similar to multiplying actual numbers with this method.

Example with Numbers

Example with Binomials

The approach to factoring using this method is attractive because it feels like working the box method, in reverse. If students are familiar with the box method for multiplying binomials, it’s a natural extension to use this method to factor them (as I often talk about factoring as the reverse of distributing).

Step one

The first step in this process is writing down a*c (M), the coefficient on the middle term (A), and then finding two Numbers that multiply to give you M and add to give you A.

Step Two

Once you have your numbers, fill the box. The upper left corner and lower right corner have to contain the squared term and constant term respectively. Fill the upper right and lower left with the two numbers you found in step one.

Step Three

The last step is to factor out the GCF of each row and column. Then you’re done. You have the factors that multiply to give you the quadratic.

A couple notes

This method fails miserably if you don’t factor out any common factors at the beginning. For instance, if you have a 2 in each term that can be factored out, you have to do that first before using this method.

I still have to grade the quizzes that cover this section but the kids seemed to respond a lot better than they have in previous years. I’ll update this post once I know more.

# Thoughts on “GPSing our Students”

In June Dan Meyer posted Your GPS is Making You Dumber and What that Means for Teaching. In it he makes the argument that providing step by step instructions for math concepts results in students being able to get from point A to B, while not understanding much about the concepts they’re supposed to be learning. His argument can be summed up with this paragraph, and is somewhat inspired by what Ann Shannon wrote in what teachers should Look for in the CCSS Mathematics Classroom.

Similarly, our step-by-step instructions do an excellent job transporting students efficiently from a question to its answer, but a poor job helping them acquire the domain knowledge to understand the deep structure in a problem set and adapt old methods to new questions.

I would tend to agree. I do give students steps occasionally but it’s often in order to simplify concepts and, if I’m being honest, to some degree avoid students truly struggling and grappling with the concepts.

I’m curious as to what others think about his post and the notion that GPSing students leads to less learning.

In the last month I’ve become a Panera regular. We’ve been doing a lot of traveling and Panera was as close to fast food as we were willing to go. Until the last month however I’d only been there a few times in several years, without being particularly impressed.

image credit: nbcnews.to/29POCSz

Take a look at the menu above. This is what you encounter when you walk into Panera Bread for the first time. Seven sections of menu packed with different dining options. For the less food savvy among us, not only is the menu packed with stuff, but it’s packed with a lot of stuff that is unfamiliar. In my first couple of visits I stood there bewildered for a while, let my friend order, then picked something with turkey in it.

Because I was pretty sure I knew what turkey was.

Well, I was wrong. I mean, I think the thing had turkey in it. But everything else did not complement the turkey in a positive way.

The next time I went there with a few colleagues I ordered a caesar salad. Why? Because every time I’ve ever ordered a caesar salad in a restaurant I haven’t been surprised by what came out. The same was true for this visit.

Sweet. I found something on the menu without holding up the four people behind me.But after that I avoided Panera for a while. A long while. I could get a caesar salad from a lot of places. The notion of standing there in line trying to find something different felt as though it would only bring frustration.

Then my wife and I were looking for something quick and healthy to eat, so we decided, to my slight dismay, to give it another try.

This time however, on the way there my wife looked up the menu on her phone and read off several items, describing them from their website. When we walked in I knew several of the items on the menu, how it worked (that “pick 2” thing for example) and was basically ready to order when I walked up. We’ve been there a couple of times since and now it’s one of my favorite restaurants. Since I’m confident in my understanding of the menu, I try new items, ask questions about different types of food, etc.

### How does this relate to learning?

Think about the first time you encountered an unfamiliar topic and tried to learn it. This was difficult at first but I can certainly think back to math lectures (here’s lookin’ at you, Linear Algebra) in which there was new notation, vocabulary, and concepts and it all felt unfamiliar.

However, once I got with my peers and we began working through the problems (encountering the “menu” multiple times) the concepts began to feel more familiar. I realized that I knew more about them than what I thought (romaine and kale! Hey, I know what those are…). The more I worked individually and with my peers the clearer the concepts became.

If I avoided doing the exercises or only did it individually, that feeling of everything being foreign never really went away. Usually parts of the notation would be confusing. Or the instructions around a problem wouldn’t make sense. Or I could start a problem, but get lost in trying to solve it, etc.

A couple points can be pulled from this. First, a student’s first exposure to a topic is incredibly important. If you drop seven new vocab words and a gaggle of new notation on students at 8:00am on Monday morning you’re bound have a large group of students not wanting to come back to the menu you just presented them.

Even if they should know most of the food on there.

So we have to think carefully about how students first engage with content. The second point is helping students understand that multiple engagements with a concept will (usually) alleviate this feeling. I think many students never go back to the restaurant because they don’t want to be embarrassed for not knowing what their peers may already know. We have to help students be comfortable with this phase of unfamiliarity (my study group in college was a place I felt comfortable being wrong), and develop tasks that help them engage with the concept in safe, productive ways (investigating the menu on the way).

For a great post on how students see unfamiliar mathematics check out Ben Orlin’s recent post entitled What Students See When They Look at Algebra.

Alternative Titles

“What’s on that sandwich? Oh… will that be on the test?”

“I don’t even like lettuce so why should I be learning about salads?”

# My Precalculus Problem

This is Lucas’s first year in my school. He’s a senior in my precalculus class. After a few assessments his grade begins to tumble. I know little about his previous math education. I meet with him during lunch a few times to help on the content and see that he’s missing several foundational math and reasoning skills. He ends up with a D in my class.

Jennifer is bored in my class. She’s easily getting an A. I can tell that she knows most of the content because she remembers it from algebra II. At some point in the first trimester she asks me if the second trimester will be review as well. I said that a lot of this is not review and that there will less familiar content in the next trimester (which is mostly true). She ends up with an A-.

In the high school in which I teach there is honors precalculus and “regular” precalculus. I teach the regular level. This means that I get kids like Jennifer, kids like Lucas, and everyone in between. I have college bound students and students that will go into a trade. I have students that love math and some that are only there because their parents made them. I get a lot of students that have scooted by with A’s and B’s without trying and would prefer to continue not trying.

So, basically a typical high school class.

The last couple years I’ve struggled to differentiate for the diversity in this class. I’ve failed many of these students because the content either goes too deep or not deep enough. This summer I’m working on solving this problem, or at least minimizing it. The flowchart below is what I’m currently thinking, although I’m sure this will change as I continue to work on it.

My idea is to pre-assess over algebra II skills that are needed for the unit. If students have mastered most of those skills then they take a different track then those that haven’t. I haven’t worked out a full module yet but I’m thinking I have most of the track 1 materials made and need to make most of the track 2 materials.

Most of my direct instruction is on video which means not every student has to be at the same place at the same time. I just need them to be ready for the summative assessment on a certain day.

With this model I can patch the conceptual holes for the kids that need it, and push the kids that don’t.

What problems do you see cropping up with this idea? For example, I’m worried some less motivated kids might intentionally do poorly on the pre-assessment so they have the “easier” track.

Any feedback is welcome and appreciated. Thanks for reading.

# Three Arguments for a Mathematical “SSR”

I’m sure that at some point in your life you’ve either heard of or participated in sustained silent reading in school. The idea is that students simply spend a set amount of time reading anything they enjoy for an extended period of time. I remember doing it in middle and elementary school. Every English student in the high school in which I currently teach does it as well. In fact, since it’s implementation there has been a notable increase in our reading scores. This got me thinking….what would the mathematical equivalent of this look like and would it be valuable?

### Choice

I think it might have a few components. One of the main premises of SSR is you get to choose what you read. In the realm of mathematics I don’t doubt that many students would need guidance in this area for a couple of reasons. First, many students see mathematics through the lens of their math books and previous math books that led to their current math book. This means that they are sheltered from a lot of math they might find interesting. Second many don’t know what doing math is like. For instance, have a look at this video by Vi Hart (who has one of my favorite Youtube Channels) in which by doodling she makes parabolas incredibly interesting. This is an exceptional example of where simply playing with mathematics can take you. Now, I understand that her mathematical background allowed her to draw and discuss parts of the video that would be over many student’s heads. The point is that there are many access points to mathematics that are both playful and creative. The teacher would have to front load some of the explanation for what constitutes mathematics, to broaden their horizons.

Being able to choose the mathematics students work on gives them some ownership of the content, even if it’s only for a small part of the week. Math catches a bad rep. Even certain students in my AP calculus would hesitate to brag about their love of math and a number of them don’t like math. I’m not contending that after implementing some sort of mathematical SSR that everyone will be running around jumping up and down about how great math is. I’m simply contending that if students view of mathematics broadens into something they think is enjoyable, the subject in general might be viewed in a better light. I would also hope that there would be a “spillover” effect in math class. This would stem from the notion that, although “what you’re telling me now isn’t particularly interesting, I can see that there are parts of this subject that are.” The goal would be that students would be (even slightly) more motivated to learn other mathematics.

### Thinking

I constantly preach to my students that if you want to get better at something you have to work at it. No one wakes up one morning with the ability to shoot three pointers at 60%. Likewise, no one wakes up one morning with the ability to do and fully comprehend integral calculus. To this end, if we can get students thinking mathematically for a short period each week I believe that ultimately students would become better mathematical thinkers and problem solvers. Two of the critical components to the success of this is that a) students have enough time each week to make it worthwhile and b) students engage in activities that make them reason and use their logical thinking skills.

### Focus

I don’t think I’m alone when I observe that many students in my class are trying to do math with a computer sitting next to them, lighting up every 15 seconds. This makes any kind of extended focus and concentration difficult. How are students supposed to “make sense of problems and persevere in solving them” if their phone is constantly distracting them from what their work? To this end a mathematical SSR would be phone/distraction free. I’m not sure if English classrooms implement it this way, but I imagine they do. One of the goals of this would be that students get better at concentrating on problems for longer than a minute or two. My hope is that students would begin to see value in distraction free work. They might even increase their ability to focus.

#### Nuts and Bolts

A few things remain to be worked out. For instance, what are the guidelines for something mathematical. Vi Hart spent a bunch of time drawing parabolas but the result was much more mathematical than if most of my students did the same. Here’s a list of activities that, I think, would be fit this time nicely.

• Logic Puzzles
• Creating Desmos Art
• Sudoku, Kakuro, etc.
• Reading and playing games on Math Munch
• Something they find interesting from (gasp) the textbook
• Watching Youtube videos from approved Youtube channels (I’m not sold on this one…)
• Maker Stuff (Little Bits, Arduino, etc.)
• Logic Games
• Games (Chess, Guillotine, etc.)
• Coding
• Others (If you shoot me ideas then I’d love to add them to the list…)

This time would be explicitly not for remediation. I can think of no worse way for a student to spend this time than being forced to do math they don’t find interesting and are already struggling with. I can see the temptation for a teacher to fill this chunk of time with remediation but that completely misses the point.

#### Results

I have to believe that the end result would be better mathematical understanding in general. I also think that (another gasp) test scores would go up as a result. Many standardized test questions test reasoning more than given math skills anyway. I have no research to prove this, I just think that if students do more mathematical thinking, their math skills will improve. And to be quite honest, if the results are simply more students improving their reasoning ability and gaining a new appreciation for mathematics then I’d deem it a success.

On a final note, I think it’s important that the teacher does this with the students. This models what is expected and gives the teacher some time to explore the subject that they love. It would contribute to a culture of mathematics in the classroom and sends a message to the students that this time is valuable to the teacher as well.

This is just an idea that’s been pinging around my head for several months and I’m finally getting it out. I’d really love to hear feedback on this, including but not limited to “this idea sucks because…”.

# Is learning easy?

Something I’ve been thinking about for the last year is whether or not learning should be easy. I can think of times when I learned a great deal and it didn’t feel difficult at all. I can think of other times that learning was difficult and I didn’t feel like I learned very much. These are a few of the questions that bounce through my mind.

• Is there some kind of payoff for learning something that is difficult to learn, beyond simply the thing you learned?
• Is everything we learn ultimately worth learning, regardless of how difficult it was to learn?
• If we teach things that are consistently difficult to learn then how do make sure those learning experiences end up being valuable?

I try to think about these questions from my students’ perspective. For instance, I dragged my extended (slow pace) algebra II students through a unit on quadratics. Realistically speaking my students were never going to use most of the mathematical concepts that we covered, at least not directly. So why do we teach them these things that are so difficult for many of them to learn (especially at a conceptual level)? Or maybe a better question is what do we tell them about learning when we teach them concepts they’ll never use and find difficult? What message are we sending about learning? A lot of algebra II (a requirement for every student in the state of Michigan) is an absolute struggle for many students, so how do we make this struggle meaningful?

In my mind we have a couple of options. The first option is to try to reduce the curriculum to it’s simplest form. We give the students the tricks, shortcuts, calculator programs, and everything we can to get them to put the correct answer in the blank on the assessments. This way we can get as many kids through the curriculum as painlessly as possible. This method is fairly attractive and I know I’ve been guilty of it on several occasions. The glaring problem with it is that we are essentially wasting the students’ time. We are not creating opportunities for them to think critically or grow as learners (not to mention how this destroys the beauty of mathematics). Also, it’s been my experience that students don’t retain the concepts over the long term.

With this, I often consider another path. Maybe instead we take an approach that encourages critical and independent thinking. A model that allows students to construct the concepts within learning experiences that, although seemingly more difficult, allows them to grow as learners and mathematical thinkers. This route is more difficult for a number of reasons. First, developing these kinds of tasks is difficult. (Although, to be fair, it is getting easier. Consider the MTBOS search engine this list of Common Core aligned problem based curriculum maps or the power of online professional learning networks.) Second, students hate it. (Okay, maybe hate’s a strong word, maybe it’s not every student, and I think the culture of the classroom can make them hate it less, but I’ll have more on that in another post.) In addition, there is a concern that we won’t get through all the content. If you teach in trimesters, where a student might have a different teacher from trimester to trimester, this becomes especially important. From a teacher’s perspective this option can seem daunting. We are going to take kids that already (probably) don’t like a subject partially because they find it difficult and then we make it more difficult for them. For many educators this choice is simple. Go for option number one.

I would add one note about the second option. We wouldn’t be making the content more difficult because we are evil. We’d be trying to create valuable struggle. The idea would be that we help them build the concepts so students would be doing more thinking during class and the teacher would stop giving away the interesting stuff so frequently.

I think most educators, given infinite time and patience, would pick the second option to implement. So the big question becomes:

If the second option is more difficult for both the student and the teacher, does the payoff (if there is any) outweigh the difficulties in implementing it?

To be honest with you I don’t know the answer. My idealism pulls me hard towards the second option but my practicality pulls me in the other direction. Also, not having infinite time and patience is a big factor. I apologize if this post doesn’t feel like it has a resolution. It doesn’t, because I don’t. However I’d love if it started a conversation. I think this is something that all math teachers and departments should be having an open discussion about.