mathematics – Page 2

Example analysis from DeltaMath

With so much problem-based learning happening this year, I’ve been mixing in plenty of algebra by example-esque problems. They work really well because they get kids to analyze math work on their own and then use it to solve a similar problem.

I’ve been writing some of these problems from scratch (horribly), but DeltaMath has shown up on the scene and helped out in unexpected ways. At the beginning of the year, I originally intended for DeltaMath to be a review of the problems/topics we learned in class. I assign them one big assignment that’s due the day before the next exam and they do it over time as we explore ideas in class.

That’s happening, yes, But what I’ve found is that the kids are also using the DeltaMath to learn the new ideas by means of the examples, not just review them. They’re independently leaning on their own analysis of DeltaMath examples to learn rather than on me to hand-hold them through examples in class. Independent learners, yay!!

The result is that someone regularly comes to class saying “…on DeltaMath I learned that…,” when presenting a problem we’re discussing in class – even when its an introductory problem on a topic. And, more often than not, this opens the door for a complete student-led class discussion around the problem.

For example, take this “Factor by Grouping Six Terms” problem that I assigned earlier in the year:

When they click the “Show Example” on the top, a worked-out example appears:

Students can even filter through different types of examples of the same problem by clicking “Next Example.”

Factoring trinomials by first rewriting them

So back in December, I gave this problem to my students:

To my surprise, it got a lot of traction with the kiddos. We spent the entire period talking about it. The idea was for them to see how rewriting a trinomial with four terms helps us to factor it. I’ve used this approach, often called the “CAB method” and used with a large “X” to organize the product and sum of the A and C terms, to factor trinomials for the last several years and I really like it for two reasons:

It doesn’t matter if a is greater than 1.
It naturally integrates factoring by grouping. Traditionally, grouping is learned after factoring trinomials. But with this approach, I teach grouping before we even see trinomials. Yeah:

So, yeah, this is all great, but as I was explaining this approach to a colleague, she asked me why it works. It was in that moment that I realized that I had no idea.

Well, it turns out that later that day she went ahead and wrote up a proof of the method.

I read a quote somewhere or heard someone say that the real usefulness of algebra is the ability it affords us to rewrite things in order to help reveal their underlying structure. This method surely epitomizes that idea.

My first (and second) memory of learning math

Some time ago Wendy Menard got me thinking about my first memory of learning mathematics. What was it?

Its two things, actually. Both happened in fifth-grade. My teacher was a redheaded man with a great beard, Mr. O’Discoll (a.k.a Mr. O). Great guy, great energy. He made learning fun. He even played me and a couple of my friends in basketball at the end of the school year in the school gymnasium. We lost 100-98. I’ll never live it down.

Anyway, I digress. Back to learning math. The first vivid memory I have of learning math is the multiplication worksheets that Mr. O would give us. He would time us. I don’t recall it ever being a race or competition to finish, but I do remember being pressured by time constraints.

The second memory comes from an exam that I took in his class. I don’t remember the math that was on it, but before the exam, I remember him telling us to always check our work after answering the problems. Well on this particular exam, I remember following his advice for about 3/4 of the exam, finding and fixing several mistakes, but then stopping — thinking that I had already done a great job. I was presumptuous. When Mr. O handed the exam back, I had a perfect paper — up until where I stopped checking my work. I had so many errors in the unchecked portion of my exam. I distinctly remember a comment he wrote directly on the exam: “why did you stop checking your work, Brian?”

Sometimes I think about how these two distant moments from my childhood have impacted how I teach mathematics.

Firstly, I teach mathematics the way I was taught math. I think this is the norm for so many teachers regardless of the subject — and it’s not a bad thing. It’s reality. In my case, drill-in-kill was what I experienced early and often, like in the case of Mr. O’s multiplication worksheets. This experience brainwashed me equate math with speed and correct answers…and this is very evident today in my teaching. I try hard to combat this, but I am not the most inquiry-based math teacher. I struggle to move beyond test-prep style learning. Its a product of the culture in which I teach, yes, but its also a direct result of the math education I received. This bothers me.

Secondly, through the years I have always been prone to mistakes when it comes to learning and teaching math. I consider myself a slow thinker, but I don’t want to be. Thanks to my fifth-grade class (and others no doubt), I want to get it on the first attempt. Sometimes I feel like I have to get it on the first attempt. Whether it is typos in handouts, mistakes in grading, or my blunders in planning thoughtful mathematical experiences for my algebra 2 kids, I always find errors that could have easily been edited had I not been too lazy or overconfident to dig deeper. Heck, even my typo-laden tweets are evidence of this. Mr. O’s exam and his advice are always in the back of my head. I do my best to follow his advice, but I fail much more often than I succeed.

My experience at Phillips Exeter Academy

To build upon my experiences this summer at the Exeter Mathematics Institute and to improve the newfound problem-based classroom, yesterday I paid a visit to the renowned Phillips Exeter Academy in Exeter, New Hampshire.

I observed six mathematics classrooms, had a private meeting with some students and had lunch with a few Exeter teachers. In between all of that, I also had some time to roam freely about the leafy campus, spending a good chunk of time at their library. I was on campus from 8am to 5pm.

I tend to process experiences pretty slowly. I say that because I know that I won’t be able to fully synthesize this visit for at least a few days — especially after I’m back in my own classroom. Nonetheless, I’m going to get out my immediate reactions with what else? Isolated bullet points whose main ideas are scattered and disorganized. Clearly, I still blog for myself.

This place is very old. Some of the classrooms looked like they hadn’t been renovated since the 1800’s (see photo above). The look and feel aren’t for everyone, but I found it charming.
The teachers were so welcoming. Each one mentioned my presence in the room and had every student introduce themselves. I shared the purpose of my visit and thanked them all for allowing me to share their space for the day. I got the vibe that they are accustomed to having visitors almost every day, but I still loved their transparency. One of the teachers valiantly tried all period to get my last name right until the moment I walked out of his classroom. It was a small thing, but I really appreciated that.
The students were highly motivated. I fully expected this. Maybe what I didn’t expect was how helpful and respectful they were. I got lost several times while on campus and each time I was politely helped and redirected. They also gave me some student-driven advice on how to encourage buy-in from my own students in this type of learning environment.
Most all of the students I spoke with came from a traditional learning setting and they all enthusiastically preferred the problem-based, discussion-based environment that Exeter has pioneered. Their families are also paying upwards of $50K a year for tuition, so yeah, there’s that.
In terms of instruction, I saw the same thing in every class. The period opens up with kids spending about 10 minutes putting up the homework problems (~7) on the boards around the room. For the rest of the period, the students present their own (or someone else’s) work and/or solution and the class discusses and draws conclusions. The onus was put on the students to push the lesson forward. This confirmed what I’m doing in my own classroom.
Every teacher spent a good amount of time sitting at the Harkness table with the students. I don’t have a Harkness table nor would I want one (give me couches and coffee tables instead), but actually sitting amongst the students during class has been a game changer for me.
With that said, just like in any class, there was some variation to how teachers enacted this structure. Some teachers assigned students to certain problems when they walked in by having their names on the board. In others, students openly chose their own problems. In some classrooms, students could not present their own work; they had to present someone else’s.
In a couple of the classes I visited, when the class got stuck, it felt like the teachers wanted to lecture — and sometimes they did…for like 15 minutes. Maybe it shouldn’t have, but this was surprising given the completely student-centric classroom that Exeter pushes.
This made me think about the problem sets. Every Exeter mathematics teacher uses them and they all did while I was there. If the need for direct instruction was as evident as I witnessed, are the problems scaffolded enough? How much flexibility do the teachers have when it comes to class time? Must it always be problems, problems, and more problems? Or can they filter in occasional days of enrichment based on the concepts learned from the problems?
Desmos was widely used in the class discussions around the problems. Most all of the classrooms had a slick setup with an Apple TV and Airplay where students could easily toggle between whose laptop/tablet screen was displaying on the projector. Other than that, there was no sign of using Desmos Activity Builder or any other structure to help maximize its obvious benefit. Maybe a problem requiring Activity Builder to answer it?
A few teachers used doc cams for student work. Nice.
I constantly saw kids taking photos of the boardwork with their phones. Since my kids can’t use their phones, this affirms why I now have a class iPad and a volunteer that snaps photos of the boardwork and emails it to everyone at the end of each class.
I only spent one day on campus, but if I’m honest, I felt a gulf between the teachers and students in the classrooms I visited. The focus at any given time (even at the onset of the period) was overwhelmingly on the standardized problems and less on the individual students in the classroom. Shouldn’t the problems be supplemented with other materials/resources for different classes based on the needs of the kids? Again, my sample size is incredibly small, so I may be way off.
From what I saw around campus, Exeter seems to be in touch with the revolution that is happening in our country right now around race, gender, sexual orientation, and other social issues. The library was exceptional on this front. At the same time, students of color were disappointingly scarce both on campus and in the classes I visited.