Tag Archives: student

Helping the community with educational technology

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Many people associated with educational technology are driven by a passion for helping students learn using technology in a classroom setting. But I wonder if many ed tech people — either researchers or rank-and-file teachers who teach with technology — ever consider a slightly different role, voiced here by Seymour Papert:

Many education reforms failed because parents did not understand or could not accept what their children were doing. Remember the New Math? This time there will be many who have not had the personal experience necessary to appreciate fully the multiple ways in which digital media can augment intellectual productivity. The people who do can make a major contribution to the success of the new initiative by helping others in their communities understand the potential. And being helpful will do much more than improve the uses of the computers. The computers could be a catalyst for turning our communities into “learning communities.”

So true. So much of education falls to the immediate family, and yet often there are technological innovations in the classroom which fail to be supported at home for the simple reason that parents and other family members don’t understand the technology. Ed tech people can make a real impact by simply turning their talents toward this issue.

Question for you all in the comments: How? It seems that the ways that ed tech people use to communicate their thoughts are exactly the ones off the radar screen of the people who need the  most help — Twitter, blogs, conference talks, YouTube videos, etc. You would need to get on the level with the parent trying to help their kid in a medium that they, the parents, understand. How is that best done? Newsletters? Phone hotlines? Take-home fact and instruction sheets? Give me some ideas here.

(h/t The Daily Papert)

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Filed under Early education, Education, Educational technology, High school, Technology

Thoughts on the culture of an inverted classroom

I’ve just finished up the spring semester, and with it the second iteration of the inverted classroom MATLAB course. With my upcoming move, it may be a while before I teach another course like this (although my experiments with targeted “flipping” went pretty well), so I am taking special care to unwind and document how things went both this year and last.

I asked the students in this year’s class about their impressions of the inverted classroom — how it’s worked for them, what could be improved, and so on.  The responses fell into one of two camps: Students who were unsure of, or resistant to, the inverted classroom approach at first but eventually came to appreciate its use and get a lot out of the approach (that was about 3/4 of the class), and students who maybe still learned a lot in the class but never bought in to the inverted method. No matter what the group, one thing was a common experience for the students: an initial struggle with the method. This was definitely the case last year as well, although I didn’t document it. Most students found closure to that struggle and began to see the point, and even thrived as a result, while some struggled for the whole semester. (Which, again, is not to say they struggled academically; most of the second group of students had A’s and B’s as final grades.)

So I am asking, What is the nature of that struggle? Why does it happen? How can I best lead students through it if I adopt the inverted classroom method? And, maybe most importantly, does this struggle matter? That is, are students better off as problem solvers and lifelong learners for having come to terms with the flipped classroom approach, or is adopting this approach just making students have to jump yet another unnecessary hurdle, and they’d be just as well off with a traditional approach and therefore no struggle?

I think that the nature of the struggle with the inverted classroom is mainly cultural. I am using the anthropologists’ definition of “culture” when I say that — a culture being a system whereby a group of people assign meaning and value to things.

In particular, the way culture places value on the teacher is radically different between the traditional academic culture experienced by students and the culture that is espoused by the inverted classroom. In the traditional classroom, what makes a “good teacher” is typically that teacher’s ability to lecture in a clear way and give assessments that gauge basic knowledge of the lecture. In other words, the teacher’s value hinges on his or her ability to talk.

In the inverted classroom, by contrast, what makes a “good teacher” is his or her ability to create good materials and then coach the students on the fly as they breeze through some things and get inexplicably hung up on others. In other words, the teacher’s value hinges on his or her ability to listen.

Many students who are in that other 25% who never buy into the inverted classroom think that teachers using this approach are not “real” teachers at all. As one student put it, when they pay a teacher their salary, they expect the teacher to actually teach. What is meant by “teaching” here is an all-important question. Well, on the reverse side, if there were such a thing as a group of students who had only experienced the inverted classroom their entire lives and then entered into a traditional classroom, those students would think they are experiencing the worst teacher in the history of academia. The guy never shuts up! He only talks, talks, talks! We have to fight to get a word in edgewise, we get only brief chances to work on things when he is there, and we’re always booted unceremoniously out of the lecture hall (we used to call them “classrooms”) and left to fend for ourselves on all this difficult homework!

I’m convinced that bridging this cultural gap is what takes up most of the time and effort in an inverted classroom — forget about screencasts!

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Understanding “understanding”

This past Saturday, I was grading a batch of tests that weren’t looking so great at the time, and I tweeted:

I do ask these two questions a lot in my classes, and despite what I tweeted, I will probably continue to do so. Sometimes when I do this, I get questions, and sometimes only silence. When it’s silence, I am often skeptical, but I am willing to let students have their end of the responsibility of seeking help when they need it and handling the consequences if they don’t.

But in many cases, such as with this particular test, the absence of questions leads to unresolved issues with learning, which compound themselves when a new topic is connected to the old one, compounded further when the next topic is reached, and so on. Unresolved questions are like an invasive species entering an ecosystem. Pretty soon, it becomes impossible even to ask or answer questions about the material in any meaningful way because the entire “ecosystem” of a student’s conceptual framework for a subject is infected with unresolved questions.

Asking if students understand something or if they have questions is, I am realizing, a poor way to combat this invasion. It’s not the students’ fault — though persistence in asking questions is a virtue more students could benefit from. The problem is that students, and teachers too, don’t really know what it means to “understand” something. We tend to base it on emotions — “I understand the Chain Rule” comes to mean “I have a feeling of understanding when I look at the Chain Rule” — rather than on objective measures. This explains the common student refrain of “It made sense when you did it in class, but when I tried it I didn’t know where to start“. Of course not! When you see an expert do a calculation, it feels good, but that feeling does not impart any kind of neural pathway towards your being able to do the same thing.

So what I mean by my tweet is that instead of asking “Do you understand?” or “Do you have any questions?” I am going to try in the future to give students something to do that will let me gauge their real understanding of a topic in an objective way. This could be a clicker question that hits at a main concept, or a quick and simple problem asking them to perform a calculation (or both). If a student can do the task correctly, they’re good for now on the material. If not, then they aren’t, and there is a question. Don’t leave it up to students to self-identify, and don’t leave it up to me to read students’ minds. Let the students do something simple, something appropriate for the moment, and see what the data say instead.

This may have the wonderful side effect of teaching some metacognition as well — to train students how to tell when they do or do not know something.

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Five questions I haven’t been able to answer yet about the inverted classroom

Between the Salman Khan TED talk I posted yesterday and several talks I saw at the ICTCM a couple of weeks ago, it seems like the inverted classroom idea is picking up some steam. I’m eager myself to do more with it. But I have to admit there are at least five questions that I have about this method, the answers to which I haven’t figured out yet.

1. How do you get students on board with this idea who are convinced that if the teacher isn’t lecturing, the teacher isn’t teaching? For that matter, how do you get ANYBODY on board who are similarly convinced?

Because not all students are convinced the inverted classroom approach is a good idea or that it even makes sense. Like I said before, the single biggest point of resistance to the inverted classroom in my experience is that vocal group of students who think that no lecture = no teaching. You have to convince that group that what’s important is what (and whether) they are learning, as opposed to my choices for instructional modes, but how?

2. Which is better: To make your own videos for the course, or to use another person’s videos even if they are of a better technical or pedagogical quality? (Or can the two be effectively mixed?)

There’s actually a bigger question behind this, and it’s the one people always ask when I talk about the inverted classroom: How much time is this going to take me? On the one hand, I can use Khan Academy or iTunesU stuff just off the rack and save myself a ton of time. On the other hand, I run the risk of appearing lazy to my students (maybe that really would be being lazy) or not connecting with them, or using pre-made materials that don’t suit my audience. I spend 6-12 hours a week just on the MATLAB class’ screencasts and would love (LOVE) to have a suitable off-the-shelf resource to use instead. But how would students respond, both emotionally and pedagogically?

3. Can the inverted classroom be employed in a class on a targeted basis — that is, for one or a handful of topics — or does it really only work on an all-or-nothing basis where the entire course is inverted?

I’ve tried the former approach, to teach least-squares solution methods in linear algebra and to do precalculus review in calculus. In the linear algebra class it was successful; in calculus it was a massive flop. On some level I’m beginning to think that you have to go all in with the inverted classroom or students will not feel the accountability for getting the out-of-class work done. At the very least, it seems that the inverted portions of the class have to be very distinct from the others — with their own grading structure and so on. But I don’t know.

4. Does the inverted classroom model fit in situations where you have multiple sections of the same course running simultaneously?

For example, if a university has 10 sections of calculus running in the Fall, is it feasible — or smart — for one instructor to run her class inverted while the other nine don’t? Would it need to be, again, an all-or-nothing situation where either everybody inverts or nobody does, in order to really work? I could definitely see me teaching one or two sections of calculus in the inverted mode, with a colleague teaching two other sections in traditional mode, and students who fall under the heading described in question #1 would wonder how they managed to sign up for such a cockamamie way of “teaching” the subject, and demand a transfer or something. When there’s only one section, or one prof teaching all sections of a class, this doesn’t come up. But that’s a relatively small portion of the full-time equivalent student population in a math department.

5. At what point does an inverted classroom course become a hybrid course?

This matters for some instructors who teach in institutions where hybrid, fully online, and traditional courses have different fee structures, office hours expectations, and so on. This question raises ugly institutional assumptions about student learning in general. For example, I had a Twitter exchange recently with a community college prof whose institution mandates that a certain percentage of the content must be “delivered” in the classroom before it becomes a “hybrid” course. So, the purpose of the classroom is to deliver content? What happens if the students don’t “get” the content in class? Has the content been “delivered”? That’s a very 1950’s-era understanding of what education is supposedly about. But it’s also the reality of the workplaces of a lot of people interested in this idea, so you have to think about it.

Got any ideas on these questions?

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Salman Khan on the inverted classroom

Salman Khan, of the Khan Academy, sounds off on the potential of pre-recorded video lectures to change education in the video below. He calls it “flipping” the classroom, but around here we call it the inverted classroom.

I like especially that Salman made the point that the main effect of inverting the classroom is to humanize it. Rather than delivering a one-size-fits-all lecture, the lecture is put where it will be of the most use to the greatest number of students — namely, online and outside of class — leaving the teacher free to focus on individual students during class. This was the point I made in this article — that the purpose of technology ought to be to enhance rather than replace human relationships.

I hope somewhere that he, or somebody, spends a bit more time discussing exactly how the teachers in the one school district he mentions in the talk actually implemented the inverted classroom, and what kinds of issues they ran up against. Ironically, the greatest resistance I get with the inverted classroom is from students themselves, namely a small but vocal group who believe that this sort of thing isn’t “real teaching”. I wonder if the K-12 teachers who use this model encounter that, or if it’s just a phenomenon among college-aged students.

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Discussion thread: Student responsibilities

I’d be interested in hearing your thoughts on the following statement about responsibilities in college:

In college, it’s the student’s responsibility to initiate requests for help on assignments, and it’s the instructor’s responsibility to respond to those requests in a helpful and timely way.

Do you think this statement is true or false? If false, could you modify it so that it’s true?

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How the inverted classroom saves students time

Our semester is into its third full week, and most of my time (as you know from checking my Twitter or Facebook feed) is being spent, it seems, on making screencasts for the MATLAB class. I feel like I’ve learned a great deal from a year’s worth of reflection on the first run of the class last spring, and it’s showing in the materials I’m producing and the work the students are giving back.

The whole idea of the inverted classroom has gotten a lot of attention in between the current version of the course and the inaugural run — the time period I think of as the “MATLAB offseason” — through my blogging, conference talks, and everyday conversations at my work. One of my associate deans, off of whom I’ve bounced a number of ideas about this course, related a conversation he recently had with someone about what I’m doing.

Associate Dean: So, Talbert is using this thing called the inverted classroom.

Other person: What’s that all about?

AD: He puts the lectures all online, and instead of lecturing in class he has them do group assignments on various kinds of problems.

OP: Doesn’t that double the amount of time students have to spend on the class?

I’ve never encountered that exact reaction before, although I did mention once that the biggest negative comment from students last year in the MATLAB course was that it took too much time relative to the credit load (1 credit). I liked how my associate dean put the answer:

AD: Well… think about it this way. You are still doing both lecture and “homework”. But which part of that are going to need the most amount of help on?

OP: OK, now I get it.

Exactly. Students are going to need a lot more guidance on the difficult task of assimilating information than they will need on the relatively easy — incredibly easy, in fact — task of receiving a transmission of information. Both phases of the game need to take place in some form, but assimilation is harder, and the probability of sinking massive amounts of time into work that goes nowhere is a lot higher, than in transmission.

I’ve seen some great examples of where the inverted classroom method has actually saved students possibly hours of fruitless labor in the last two weeks.

Today, for instance, we were doing a lab problem set on command line plotting. In one of the tasks, students are asked to produce a 1×2 subplot illustrating the behavior of a two-parameter family of functions. One team was stuck because their M-file wouldn’t execute properly even though their code looked correct. The problem: They used a dash (-) in the title, which causes MATLAB to think that the stuff preceding the dash is a variable name, which wasn’t in the workspace. It’s an innocent error but not one that students with just two weeks of MATLAB under their belts could easily debug themselves. Had they run into this problem outside of class, who knows how much time would have been wasted getting nowhere? But inside class, it was solved in the amount of time it took for them to raise their hands and for me to come over and look.

Another example from today: A team had entered this code:

x = linspace(0,10);
y = 100 - exp(-2*x);
axis([0 15 90 105])
plot(x,y)

They had entered the code without line 3 already but didn’t like the look of the plot, so they added the axis command to try and change the viewing window. But nothing changed. Why? To the trained eye, it’s simple — you have to have something plotted first before you can change the axis. So just reverse lines 3 and 4. But to the untrained eye, again, who knows how much time would be lost in trying to figure this out? Instead I was able to instruct them directly on this, at the conceptual level (How is MATLAB thinking its way through your code?) and they got it. (It wasn’t just me telling them, “You need to switch lines 3 and 4.”)

So above and beyond being more instructionally effective, I’m realizing — and I hope students are too — that the inverted classroom makes student time a lot more efficient, and there’s a much higher success-to-effort ratio than in the traditional mode of teaching.

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How it all works in the MATLAB course

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I’ve put up a few posts and several comments about the inverted classroom this week. A lot of that is because the second iteration of the MATLAB course is coming around at the beginning of February (we have a January term, so spring classes start a little late for us) and that’s done entirely in “inverted” mode. There were a lot of comments in this post about the inverted classroom, and based on some of those comments as well as some questions I got at my Joint Meetings talk on this subject, I thought I’d say a little about how, exactly, this instructional method gets implemented on a day-to-day basis in the MATLAB course.

The MATLAB course meets once a week (Wednesdays) for 75 minutes. This sets up a once-per-week workflow that repeats itself every Wednesday. Here’s how it will go:

  1. On Thursday evenings, students are assigned one or more video lectures to watch in advance of the next week’s meeting on Wednesday. The videos are posted to the internet, so students can pause, rewind, and stop/restart at will, and most videos will be posted to YouTube for easy viewing on a mobile device such as a smartphone. Along with the videos will be given a list of actions students will be expected to perform with MATLAB before coming to class and a series of Guided Practice exercises to work through what they see in the videos. Students are expected to start early so that they can ask questions throughout the week as they come up.
  2. The Guided Practice exercises are turned in on Wednesday morning prior to the class meeting so that I can read through them quickly for any widespread issues that arise. It’s a light implementation of just-in-time teaching. (By the way: Read the page at that link. That describes something close to the inverted classroom idea.)
  3. In the first few minutes of the class meeting on Wednesdays, students take a short quiz designed to assess their completion of the tasks from the Guided Practice. Quizzes are open-MATLAB so they can check their work as they work. The quizzes are taken electronically so that grading is instantaneous (or near-instantaneous, anyway). The quizzes provide individual accountability on the basic competencies for the week.
  4. After the quiz, a brief question-and-answer session takes place in which I discuss any issues arising from the Guided Practice or Quiz, and students can ask brief questions as well. However: There is no lecture and no “re-teaching” during this time. The focus is on clearing up issues from student work. If a student asks, “Can you go over how to do ____?” and the blank contains some general topic (like “plotting” or “if-then statements”) I will generally say “no” because the student has had ample opportunities to ask those kinds of questions during the week. Well, rather than just saying “no” I will try to get at what the student’s real question is. “Can you go over plotting?” usually hides a small, good, targeted question on a single specific topic that can be cleared up in no time. Those questions are fine.
  5. The remaining time in class (about 60 minutes) is spent by students working in teams on authentic, problem-centered activities highlighting important ideas to be addressed in the course that week.
  6. Students turn in a partial draft of their in-class activity at the end of the Wednesday meeting and then turn in a completed draft by 11:00 PM on the following day (Thursday). At this point the cycle repeats itself with a new list of videos, learning objectives, and Guided Practice exercises.

This cycle is a bit different than what I started with last year, when I first ran the course. The in-class problem sets were supposed to be completely done by the end of class; that turned out to be ridiculously unrealistic. I let students turn in the finished products after 48 hours, which was nice for them except that some teams wouldn’t get far on anything during the meetings, intending to do it all outside of class, which then led to having to finish the week’s lab on top of the next week’s out-of-class assignments. To keep traffic moving better, I’m insisting this year that students turn in a reasonably complete rough draft by the end of the hour (I’ll have a rubric for that later) and then the whole thing before Thursday is done; at which point they should have no leftover work competing with the outside viewing and practice.

Also, the names have changed. Last year it was “homework”; this year it’s “guided practice” to emphasize that the exercises are intended to provide, well, guidance and practice. Last year it was “labs”; this year it’s “in-class problem sets” because there are significant differences between these problem sets and actual labs that science classes use. Last year it was videos; this year it’s “lectures”, to emphasize that it’s not the case that there is no lecturing taking place. Words mean a lot.

I estimate that students will spend no more than 1 hour  a week watching video lectures; between 1 and 2 hours a week working through the guided practice; and maybe 1 hour a week in a combination of reviewing old work, coming to office hours, reading and contributing to online discussions, and other class-related tasks. That’s about 3 hours a week, which is pretty typical for a 1-credit class that meets 75 minutes a week, and it’s even better when you consider the inverted model specifically relegates the least cognitively complex tasks to outside of class.

The number-one student complaint I heard last year was that, far from occupying 3 or fewer hours a week of time, it was taking 6, 8, 10 or even more hours a week to complete the out-of-class tasks. That concerns me greatly. Every now and then in any class you’ll have to spend more than the usual “3 hours of work for each hour in class” conversion formula. But if students are spending more than that much on a regular basis, I would want to see what they are doing. There’s no way what I am assigning will take that long, no matter what your background competency or comfort level or what-have-you are, unless there is some serious inefficiency happening in how the work is being done. That concern is manageable if addressed.

Your thoughts?

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The inverted classroom and student self-image

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This week I’ve been immersed in the inverted classroom idea. First, I gave this talk about an inverted linear algebra classroom at the Joint Meetings in New Orleans and had a number of really good conversations afterwards about it. Then, this really nice writeup of an interview I gave for MIT News came out, highlighting the relationship between my MATLAB course and the MIT OpenCourseware Project. And this week, I’ve been planning out the second iteration of that MATLAB course that’s starting in a few weeks, hopefully with the benefit of a year’s worth of experience and reflection on using the inverted classroom to teach technical computing to novices.

One thing that I didn’t talk much about at the Joint Meetings or in the MIT interview was perhaps the most prominent thing about using the inverted classroom model on a day-to-day basis: how students react to it and change as a result of it. I was actually quite surprised that nobody at my Joint Meetings talk asked me a question about this, because honestly, the inverted classroom sounds great on paper, but when you start to imagine the average college student walking in on the first day of class and having this method of instruction described to him, it becomes clear that a significant amount of work is going to have to be done in order to get students — who are already resistant to any change from their accustomed modes of instruction — on board with the plan.

Students do tend to resist the inverted classroom at first. Some forms of resistance are more benign than others. On the benign end of the spectrum there are students with little experience with the course material or its prerequisites who get bogged down on the basic podcast viewing (which takes the place of in-class lectures in this model) or the accompanying guided practice, and instead of actively seeking a resolution to their question will wait for the instructor to clear it up — in class. On the other end is the student who simply doesn’t believe I’m serious when I say there won’t be any lecturing, who then doesn’t do the work, assuming I’ll bail him out somehow — in class. But in the inverted model, students are held responsible for acquiring basic competencies before class so that the hard stuff — what we refer to as assimilation — is the primary focus of the class time.

I break this distinction down for students, but not everybody buys into it. Those who don’t will have to undergo a learning process that usually looks like shock — shock that I won’t reteach them the material they were supposed to have viewed and worked on, while the lab assignment based on that material is going on. This can get very ugly in ways I probably don’t need to describe. Let’s just say that you had better not use the inverted classroom model if you aren’t prepared to put out a constant P.R. effort to convince students of the positive benefits of the model and constantly to assuage student concerns.

I’ve often wondered why students sometimes react so negatively to the inverted classroom model. I’ve come to believe it’s the result of a invasive, false belief that can arise in students about their ability to learn things independently of others — namely, that they simply cannot do so. I have had students tell me this to my face — “I can’t learn [insert topic] unless you lecture to me about it in class first.” Clearly this is not true. Toddlers learn their native language without formal instruction, just by assimilating (there’s that word again) the language going on naturally in their background. We all learn things every day without sitting in a classroom; we may seek out training data first through printed instructions, worked-out examples, YouTube videos, etc., but it’s almost never in a classroom setting. Learning new things on our own initiative and without formal instruction in a classroom setting is as natural to humans as breathing. Indeed you could say that it’s the capacity to learn in this way that makes us human. But somehow many students think otherwise.

Where does this belief come from? I think that it comes from its own instance of assimilation, namely the assimilation of a culture of programmed classroom instruction that takes place from roughly the first grade through the twelfth grade in this country. Students have so few experiences where they pursue and construct their own knowledge that they simply come to believe that they are incapable of doing so. And this belief is propagated most rapidly in mathematics. I’ve been reading in Seymour Papert‘s book Mindstorms: Children, Computers, and Powerful Ideas, and this quote hits this issue right on the head:

Difficulty with school math is often the first step of an invasive intellectual process that leads us all to define ourselves as bundles of aptitudes and ineptitudes, as being “mathematical” or “not mathematical”, “artistic” or “not artistic”, “musical” or “not musical”, “profound” or “superficial”, “intelligent” or “dumb”. Thus deficiency becomes identity and learning is transformed from the early child’s free exploration of the world to a chore beset by insecurities and self-imposed restrictions.

That last sentence (emphasis added) sums it up, doesn’t it? Deficiency becomes identity. Eventually, if a student is robbed of experiences of self-motivated learning, the student eventually adopts a self-image in which she is incapable of self-motivated learning. It is a false self-image that is ultimately dehumanizing.

Which is why I put such stock in the inverted classroom model. I think this method of teaching, along with other learner-centered modes of instruction like problem-based learning, is on the front lines in reversing students’ negative ways of thinking about how they learn. Students may (will?) chafe at the inversion at first. But in the MATLAB course at least, something really cool happened at the end of the semester. I made up a slideshow for students called “Five myths about how you think you learn that CMP 150 has busted”. Among the myths were “I can’t learn unless a professor lectures to me” and “I can’t learn on my own initiative”, and I gave concrete examples of work that the students had done in the class that contradicted these messages. In the end I showed them that through this inverted classroom process they had taken majors strides toward being confident, independent, skill learners and problem-solvers rather than just people who can play the classroom game well. And even the most skeptical students were nodding in agreement. And I think that makes it all worthwhile for everyone.

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Misunderstanding mathematics

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Robert Lewis, a professor at Fordham University, has published this essay entitled “Mathematics: The Most Misunderstood Subject”. The source of the general public’s misunderstandings of math, he writes, is:

…the notion that mathematics is about formulas and cranking out computations. It is the unconsciously held delusion that mathematics is a set of rules and formulas that have been worked out by God knows who for God knows why, and the student’s duty is to memorize all this stuff. Such students seem to feel that sometime in the future their boss will walk into the office and demand “Quick, what’s the quadratic formula?” Or, “Hurry, I need to know the derivative of 3x^2 – 6x +1.” There are no such employers.

Prof. Lewis goes on to describe some ways in which this central misconception is worked out in our schools and in everyday thinking. The analogy between mathematics instruction and building construction, in which he compares current high school mathematics instruction to a building project where the scaffolding is constructed and then abandoned because we think the job is done, is pretty compelling. The whole essay is well worth reading.

I do think that it’s a bit too easy to lay the blame for the current state of mathematics instruction at the feet of American high schools, as Lewis does multiple times. Even if high schools do have flawed models of math instruction, certainly they are not alone in this. How many universities, even elite institutions like Fordham, have math classes or even entire curricula predicated on teaching math as rote mechanics? And what about the elementary math curricula? Pointing the finger at high schools is the natural thing to do for college professors, because we are getting students fresh from that venue and can see the flaws in their understanding, but let us not develop tunnel vision and think that fixing the high schools fixes everything. Laying blame on the right party is not what solves the problem.

Lewis brings up the point that we should be aiming for “genuine understanding of authentic mathematics” to students and not something superficial, and on that I think most people can agree. But what is this “authentic mathematics”, and how are we supposed to know if somebody “genuinely understands” it? What does it look like? Can it be systematized into a curriculum? Or does genuine understanding of mathematics — of anything — resist classification and institutionalization? Without a further discussion on the basic terms, I’m afraid arguments like Lewis’, no matter how important and well-constructed, are stuck in neutral.

Again coming back to higher education’s role in all this, we profs have work to do as well. If you asked most college professors questions like What is authentic mathematics?, the responses would probably come out as a laundry list of courses that students should pass. Authentic mathematics consists of three semesters of calculus, linear algebra, geometry, etc. And the proposed solution for getting students to genuinely understand mathematics would be to prescribe a series of courses to pass. There is a fundamentally mechanical way of conceiving of university-level mathematics education in which a lot of us in higher ed are stuck. Until we open ourselves up to serious thinking about how students learn (not just how we should teach) and ideas for creative change in curricula and instruction that conform to how students learn, the prospects for students don’t look much different than they looked 15 years ago.

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