Category Archives: Math

As part of preparing for our impending move from Indy to Grand Rapids, my family and I have made a couple of visits to the area. These by necessity combine business with pleasure, since our three kids (ages 2, 5, and 7) don’t handle extended amounts of business well. On the last visit, we spent some time at the Grand Rapids Childrens Museum, the second floor of which is full of stuff that could occupy children — and mathematicians — for hours. This “exhibit” was, for me, one of the most evocative. Have a look:

I asked this on Twitter a few days ago, but I’ll repost it here: In the spirit of Dan Meyer’s Any Questions? meme, what questions come to mind as you watch this? Particularly math, physics, etc. questions.

One other thing — just after I wrapped up the video on this, someone put one of the little discs rolling on the turntable and it did about a dozen graceful, perfect three-point hypocycloids before falling off the table.

Filed under Geometry, Math, Problem Solving

Targeting the inverted classroom approach

Image via Wikipedia

A while back I wondered out loud whether it was possible to implement the inverted or “flipped” classroom in a targeted way. Can you invert the classroom for some portions of a course and keep it “normal” for others? Or does inverting the classroom have to be all-or-nothing if it is to work at all? After reading the comments on that piece, I began to think that the targeted approach could work if you handled it right. So I gave it a shot in my linear algebra class (that is coming to a close this week).

The grades in the class come primarily from in-class assessments and take-home assessments. The former are like regular tests and the latter are more like take-home tests with limited collaboration. We had online homework through WeBWorK but otherwise I assigned practice exercises from the book but didn’t take them up. The mix of timed and untimed assessments worked well enough, but the lack of collected homework was not giving us good results. I think the students tended to see the take-home assessments as being the homework, and the WeBWorK and practice problems were just something to look at.

What seemed true to me was that, in order for a targeted inverted classroom approach to work, it has to be packaged differently and carry the weight of significant credit or points in the class. I’ve tried this approach before in other classes but just giving students reading or videos to watch and telling them we’d be doing activities in class rather than a lecture — even assigning  minor credit value to the in-class activity — and you can guess what happened: nobody watched the videos or read the material. The inverted approach didn’t seem different enough to the students to warrant any change in their behaviors toward the class.

So in the linear algebra class, I looked ahead at the course schedule and saw there were at least three points in the class where we were dealing with material that seemed very well-suited to an inverted approach: determinants, eigenvalues and eigenvectors, and inner products. These work well because they start very algorithmically but lead to fairly deep conceptual ideas once the algorithms are over. The out-of-class portions of the inverted approach, where the ball is in the students’ court, can focus on getting the algorithm figured out and getting a taste of the bigger ideas; then the in-class portion can focus on the big ideas. This seems to put the different pieces of the material in the right context — algorithmic stuff in the hands of students, where it plays to their strengths (doing calculations) and conceptual stuff neither in a lecture nor in isolated homework experiences but rather in collaborative work guided by the professor.

To solve the problem of making this approach seem different enough to students, I just stole a page from the sciences and called them “workshops“. In preparation for these three workshops, students needed to watch some videos or read portions of their textbooks and then work through several guided practice exercises to help them meet some baseline competencies they will need before the class meeting. Then, in the class meeting, there would be a five-point quiz taken using clickers over the basic competencies, followed by a set of in-class problems that were done in pairs. A rough draft of work on each of the in-class problems was required at the end of the class meeting, and students were given a couple of days to finish off the final drafts outside of class. The whole package — guided practice, quiz, rough draft, and final draft — counted as a fairly large in-class assessment.

Of course this is precisely what I did every week in the MATLAB course. The only difference is that this is the only way we did things in the MATLAB course. In linear algebra this accounted for three days of class total.

Here are the materials for the workshops we did. The “overview” for each contains a synopsis of the workshop, a list of videos and reading to be done before class, and the guided practice exercises.

The results were really positive. Students really enjoyed doing things this way — it’s way more engaging than a lecture and there is a lot more support than just turning the students out of class to do homework on their own. As you can see, many of the guided practice exercises were just exercises from the textbook — the things I had assigned before but not taken up, only to have them not done at all. Performance on the in-class and take-home assessments went up significantly after introducing workshops.
Additionally, we have three mastery exams that students have to pass with 100% during the course — one on row-reduction, another on matrix operations, and another on determinants. Although determinants form the newest and in some ways the most complex material of these exams, right now that exam has the highest passing rate of the three, and I credit a lot of that to the workshop experience.
So I think the answer to the question “Can the inverted classroom be done in a targeted way?” is YES, provided that:
• The inverted approach is used in distinct graded assignments that are made to look and feel very distinct from other elements of the course.
• Teachers make the expectations for out-of-class student work clear by giving an unambiguous list of competencies prior to the out-of-class work.
• Quality video or reading material is found and used, and not too much of it is assigned. Here, the importance of choosing a textbook — if you must do so — is very important. You have to be able to trust that students can read their books for comprehension on their own outside of class. If not, don’t get the book. I used David Lay’s excellent textbook, plus a mix of Khan Academy videos and my own screencasts.
• Guided practice exercises are selected so that students experience early success when grappling with the material out of class. Again, textbook selection should be made along those lines.
• In-class problems are interesting, tied directly to the competency lists and the guided practice, and are doable within a reasonable time frame.
These would serve as guidelines for any inverted classroom approach, but they are especially important for making sure that student learning is as great or greater than the traditional approach — and again, the idea of distinctness seems to be the key for doing this in a targeted way.
What are your suggestions or experiences about using the inverted or “flipped” classroom in a targeted way like this?

Filed under Clickers, Inverted classroom, Linear algebra, MATLAB, Screencasts

Speaking of the inverted classroom

On Wednesday, I gave a talk at Indiana University – Purdue Universty – Indianapolis (IUPUI, for short) to the teaching seminar for math graduate students on the inverted classroom. It was sort of a generalization of the talk I gave on the inverted linear algebra classroom back at the Joint Mathematics Meetings in January. Carl Cowen was in attendance at that talk and invited me to make the 20-minute drive from my house to IUPUI to do something like it, and I was happy to oblige.

Since putting the talk up on Slideshare yesterday morning, it’s gotten over 200 views, 2 favorites, a handful of retweets/Facebook likes, and is currently being highlighted on Slideshare’s Education page. So I thought I would share it here as well. Enjoy and ask questions!

A binary notion of “understanding”

Another great insight from Seymour Papert, via The Daily Papert blog. I put it up on my Posterous blog this morning but I thought it could go here too:

Many children who have trouble understanding mathematics also have a hopelessly deficient model of what mathematical understanding is like. Particularly bad are models which expect understanding to come in a flash, all at once, ready made. This binary model is expressed by the fact that the child will admit the existence of only two states of knowledge often expressed by “I get it” and “I don’t get it.” They lack—and even resist—a model of understanding something through a process of additions, refinements, debugging and so on. These children’s way of thinking about learning is clearly disastrously antithetical to learning any concept that cannot be acquired in one bite.

(Papert, S. (1971) Teaching Children Thinking. In Contemporary Issues in Technology and Teacher Education, 5(3/4), 353-365.)

And on the higher education end of the spectrum, all of the things that really matter are those things that take patience, time, and persistence to acquire. But these are the very things excluded by this binary notion of understanding in which many children are immersed and to which most college freshmen are completely habituated.

This also makes a good argument for insisting that students — particularly those in the STEM disciplines, but I would argue anybody — should learn computer programming as part of their studies. You cannot learn to program without engaging in the non-binary notion of understanding Papert is describing. Papert knows a thing or two about that subject.

(By the way, I must give Gary Stager many thanks for running The Daily Papert. It is a great resource. The month of March is ridiculously busy for me but once it’s over I am going on a massive Papert reading spree.)

Technology making a distinction but not a difference?

This article is the second one that I’ve done for Education Debate at Online Schools. It first appeared there on Tuesday this week, and now that it’s fermented a little I’m crossposting it here.

The University of South Florida‘s mathematics department has begun a pilot project to redesign its lower-level mathematics courses, like College Algebra, around a large-scale infusion of technology. This “new way of teaching college math” (to use the article’s language) involves clickers, lecture capture, software-based practice tools, and online homework systems. It’s an ambitious attempt to “teach [students] how to teach themselves”, in the words of professor and project participant Fran Hopf.

It’s a pilot project, so it remains to be seen if this approach makes a difference in improving the pass rates for students in lower-level math courses like College Algebra, which have been at around 60 percent. It’s a good idea. But there’s something unsettling about the description of the algebra class from the article:

Hopf stands in front of an auditorium full of students. Several straggle in 10 to 15 minutes late.

She asks a question involving an equation with x’s, h’s and k’s.

Silence. A few murmurs. After a while, a small voice answers from the back.

Every now and then, Hopf asks the students to answer with their “clickers,” devices they can use to log responses to multiple-choice questions. A bar graph projected onto a screen at the front of the room shows most students are keeping up, though not all.

[...]

As Hopf walks up and down the aisles, she jots equations on a hand-held digital pad that projects whatever she writes on the screen. It allows her to keep an eye on students and talk to them face-to-face throughout the lesson.

Students start drifting out of the 75-minute class about 15 minutes before it ends. But afterward, Hopf is exuberant that a few students were bold enough to raise their hands and call out answers.

To be fair: This is a very tough audience, and the profs involved have their work cut out for them. The USF faculty are trying with the best of intentions to teach students something that almost assuredly none of them really want to learn, and this is exceedingly hard and often unrewarding work. I used to teach remedial algebra (well short of “college algebra”) at a two-year institution, and I know what this is like. I also know that the technology being employed here can, if used properly, make a real difference.

But if there’s one main criticism to make here, it’s that underneath the technology, what I’m seeing — at least in the snapshot in the article — is a class that is really not that different than that of ten or twenty years ago. Sure, there’s technology present, but all it seems to be doing is supporting the kinds of pedagogy that were already being employed before the technology, and yielded 60% pass rates. The professor is using handheld sketching devices — to write on the board, in a 250-student, 75-minute long lecture. The professor is using clickers to get student responses — but also still casting questions out to the crowd and receiving the de rigeur painful silence following the questions, and the clickers are not being used in support of learner-centered pedagogies like peer instruction. The students have the lectures on video — but they also still have to attend the lectures, and class time is still significantly instructor-centered. (Although apparently there’s no penalty for arriving 15 minutes late and leaving 15 minutes early. That behavior in particular should tell USF something about what really needs to change here.)

What USF seems not to have fully apprehended is that something about their remedial math system is fundamentally broken, and technology is neither the culprit nor the panacea. Moving from an instructor-centered model of learning without technology to an instructor-centered model of learning with technology is not going to solve this problem. USF should instead be using this technology to create disruptive change in how it delivers these courses by refocusing to a student-centered model of learning. There are baby steps here — the inclusion of self-paced lab activities is promising — but having 75-minute lectures (on college algebra, no less) with 225 students signals a reluctance to change that USF’s students cannot afford to keep.

An M-file to generate easy-to-row-reduce matrices

In my Linear Algebra class we use a lot of MATLAB — including on our timed tests and all throughout our class meetings. I want to stress to students that using professional-grade technological tools is an essential part of learning a subject whose real-life applications closely involve the use of those tools. However, there are a few essential calculations in linear algebra, the understanding of which benefits from doing by hand. One of those calculations is row-reduction. Nobody does this by hand; but doing it by hand is useful for understanding elementary row operations and for getting a feel for the numerical processes that are going on under the hood. And it helps with understanding later concepts, notably that of the LU factorization of a matrix.

I have students take a mastery exam where they have to reduce a 3×5 or 4×6 matrix to reduced echelon form by hand. They are not allowed any technology on that exam. I’ve learned that making up good matrices for this exam is surprisingly tricky. My first attempt at writing the exam resulted in a nice-looking matrix whose reduced echelon form had mind-bendingly big fractions in it. I want the exam to be about row reduction and not fraction arithmetic, so I  sat down this morning and wrote this MATLAB function called easyRR.m which automatically spits out $m \times n$ random integer matrices whose row-reduction process might involve fractions but which aren’t horrendous:

%% Function to create an mxn matrix that is easy to row-reduce by hand.
% Basic idea: Construct this matrix by building an LU factorization for it
% where both L and U have small integer values.
% R. Talbert, Feb 15, 2011

function A = easyRR(m,n)

%% Create the L in the LU factorization. This matrix encodes the elementary
%% row operations needed to get A to echelon form.

L = randi([-10, 10], [m,m]);

% Replace diagonal elements with 1's:
for i=1:m
L(i,i) = 1;
end

% Zero out all entries above the diagonal:
L = tril(L);

%% Now create the U in the LU factorization, using smaller integers so that
%% the back substitution phase isn't too bad.

% This creates an mxn random integer matrix and zeros out all entries below
% the diagonal.
U = triu(randi([-5,5], [m,n]));

%% The easy-to-reduce matrix is the product of L and U.
A = L*U;



Here’s a screenshot:

The fractions involved here have denominators no larger than 25, which is way more doable for students than what I had been having them work with (sorry, guys).

And, if you happen to have the Symbolic Toolbox for MATLAB, you can add the line latex(sym(A)) to the end and the function will spit out the $\LaTeX$ code for that matrix, for easy copy/paste into the exam.

Anyway, I thought this was useful and so I’m giving it away!

Filed under LaTeX, Linear algebra, Math, MATLAB, Teaching

The inverted classroom and student self-image

Image via Wikipedia

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.

Bound for New Orleans

Happy New Year, everyone. The blogging was light due to a nice holiday break with the family. Now we’re all back home… and I’m taking off again. This time, I’m headed to the Joint Mathematics Meetings in New Orleans from January 5 through January 8. I tend to do more with my Twitter account during conferences than I do with the blog, but hopefully I can give you some reporting along with some of the processing I usually do following good conference talks (and even some of the bad ones).

I’m giving two talks while in New Orleans:

• On Thursday at 3:55, I’m speaking on “A Brief Fly-Through of Cryptology for First-Semester Students using Active Learning and Common Technology” in the MAA Session on Cryptology for Undergraduates. That’s in the Great Ballroom E, 5th Floor Sheraton in case you’re there and want to come. This talk is about a 5-day minicourse I do as a guest lecturer in our Introduction to the Mathematical Sciences activity course for freshmen.
• On Friday at 11:20, I’m giving a talk called “Inverting the Linear Algebra Classroom” in the MAA Session on Innovative and Effective Ways to Teach Linear Algebra. Thats in Rhythms I, 2nd floor Sheraton. This talk is an outgrowth of this blog post I did back in the spring following the first non-MATLAB attempt at the inverted classroom approach I did and will touch on the inverted classroom model in general and how it can play out in Linear Algebra in particular.

Both sessions I’m speaking in are loaded with what look to be excellent talks, so I’m excited about participating. I’d be remiss if I didn’t mention that Gil Strang and David Lay are two of the organizers of the linear algebra setting, which is like a council of the linear algebra gods.

I’ll give Casting Out Nines readers a sneak peek at my two talks by telling you I’ve set up a web site that has the Prezis for both talks along with links to the materials I mention in the talks. And if you’re there in New Orleans, come by my talks if you have the slots free or just give me a ring on my Twitter and I’d love to meet up with you.

Conrad Wolfram’s vision for mathematics education

A partial answer to the questions I brought up in the last post about what authentic mathematics consists of, and how we get students to learn it genuinely, might be found in this TED talk by Conrad Wolfram called “Teaching kids real math with computers”. It’s 17 minutes long, but take some time to watch the whole thing:

Profound stuff. Are we looking at the future of mathematics education in utero here?

Misunderstanding mathematics

Image via Wikipedia

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.