# Tag Archives: math education

## Student (mis)understanding of the equals sign

Interesting report here (via Reidar Mosvold) about American students’ misunderstanding of the “equals” sign and how that understanding might feed into a host of mathematical issues from elementary school all the way to calculus. According to researchers Robert M. Capraro and Mary Capraro at Texas A&M,

About 70 percent of middle grades students in the United States exhibit misconceptions, but nearly none of the international students in Korea and China have a misunderstanding about the equal sign, and Turkish students exhibited far less incidence of the misconception than the U.S. students.

Robert Capraro, in the video at the link above, makes an interesting point about the “=” sign being used as an operator. He makes a passing reference to calculators, and I wonder if calculators are partly to blame here. After all, if you want to calculate 3+5 on a typical modern calculator, what do you do? You hit “3”, then “+”, then “5”… and then hit the “=” button. The “=” key is performing an action — it’s an operator! In fact, I suspect that if you gave students that sequence of calculator keystrokes and asked them which one performs the mathematical operation, most would say “=” rather than the true operator, “+”. The technology they use, handheld calculators, seems to be training them to think in exactly the wrong way about “=”. What we have labelled as the “=” key on a calculator is really better labelled as “Enter” or “Execute”.

In fact, the old-school HP calculators, like this HP 33c, didn’t have “=” buttons at all:

That’s because these calculators used Reverse Polish Notation, in which the 3 + 5 calculation would have been entered “3”, then “5”, “+”, then “Enter” — and then you’d get an answer. What HP calculators label as “Enter”, on a typical modern calculator would be labelled “=”, and in that syntax lies a lot of the problem, it seems.

The biggest problem I seem to encounter with “=” sign use is that students use it to mark a transition between steps in a problem. For example, when solving the equation $3x - 2 = 10$ for x, you might see:

$3x - 2 = 10 = 12 = x = 4$

The thought process can be teased out of this atrocious syntax, but clearly this is not acceptable math — even though the last bit of that line (x=4) is a correct statement. If the student would just put spaces, tabs, or even a semicolon between the steps, it would be a big improvement. But many students are so trained to believe that the right answer — the ending “4” — is all that matters, they have little experience with crafting a good solution, or even realizing that a mathematical solution is supposed to be a form of communication at all.

What are some of the student misconceptions you’ve seen (or perpetrated!) with the “=” sign? If you’re a teacher, how have you approached mending those misconceptions?

## Calculus and conceptual frameworks

I was having a conversation recently with a colleague who might be teaching a section of our intro programming course this fall. In sharing my experiences about teaching programming from the MATLAB course, I mentioned that the thing that is really hard about teaching programming is that students often lack a conceptual framework for what they’re learning. That is, they lack a mental structure into which they can place the topics and concepts they’re learning and then see those ideas in their proper place and relationship to each other. Expert learners — like some students who are taking an intro programming course but have been coding since they were 6 years old — have this framework, and the course is a breeze. Others, possibly a large majority of students in a class, have never done any kind of programming, and they will be incapable of really learning programming until they build a conceptual framework to  handle it. And it’s the prof’s job to help them build it.

Afterwards, I thought, this is why teaching intro programming is harder than teaching calculus. Because students who make it all the way into a college calculus surely have a well-developed conceptual framework for mathematics and can understand where the topics and methods in calculus should fit. Right? Hello?

It then hit me just how wrong I was. Students coming into calculus, even if they’ve had the course before in high school, are not guaranteed to have anything like an appropriate conceptual framework for calculus. Some students may have no conceptual framework at all for calculus — they’ll be like intro programming students who have never coded — and so when they see calculus concepts, they’ll revert back to their conceptual frameworks built in prior math courses, which might be robust and might not be. But even then, students may have multiple, mutually contradictory frameworks for mathematics generally owing to different pedagogies, curricula, or experiences with math in the past.

Take, for example, the typical first contact that calculus students get with actual calculus in the Stewart textbook: The tangent problem. The very first example of section 2.1 is a prototype of this problem, and it reads: Find an equation of the tangent line to the parabola $y = x^2$ at the point $P(1,1)$. What follows is the usual initial solution: (1) pick a point $Q$ near $(1,1)$, (2) calculate the slope of the secant line, (3) move $Q$ closer to $P$ and recalculate, and then (4) repeat until the differences between successive approximations dips below some tolerance level.

What is a student going to do with this example? The ideal case — what we think of as a proper conceptual handling of the ideas in the example — would be that the student focuses on the nature of the problem (I am trying to find the slope of a tangent line to a graph at a point), the data involved in the problem (I am given the formula for the function and the point where the tangent line goes), and most importantly the motivation for the problem and why we need something new (I’ve never had to calculate the slope of a line given only one point on it). As the student reads the problem, framed properly in this way, s/he learns: I can find the slope of a tangent line using successive approximations of secant lines, if the difference in approximations dips below a certain tolerance level. The student is then ready for example 2 of this section, which is an application to finding the rate at which a charge on a capacitor is discharged. Importantly, there is no formula for the function in example 2, just a graph.

But the problem is that most students adopt a conceptual framework that worked for them in their earlier courses, which can be summarized as: Math is about getting right answers to the odd-numbered exercises in the book. Students using this framework will approach the tangent problem by first homing in on the first available mathematical notation in the example to get cues for what equation to set up. That notation in this case is:

$m_{PQ} = \frac{x^2 - 1}{x-1}$

Then, in the line below, a specific value of x (1.5) is plugged in. Great! they might think, I’ve got a formula and I just plug a number into it, and I get the right answer: 2.5. But then, reading down a bit further, there are insinuations that the right answer is not 2.5. Stewart says, “…the closer $x$ is to 1…it appears from the tables, the closer $m_{PQ}$ is to 2. This suggests that the slope of the tangent line $t$ should be $m = 2$.” The student with this framework must then be pretty dismayed. What’s this about “it appears” the answer is 2? Is it 2, or isn’t it? What happened to my 2.5? What’s going on? And then they get to example 2, which has no formula in it at all, and at that point any sane person with this framework would give up.

It’s also worth noting that the Stewart book — and many other standard calculus books — do not introduce this tangent line idea until after a lengthy precalculus review chapter, and that chapter typically looks just like what students saw in their Precalculus courses. These treatments do not attempt to be a ramp-up into calculus, and presages of the concepts of calculus are not present. If prior courses didn’t train students on good conceptual frameworks, then this review material actually makes matters worse when it comes time to really learn calculus. They will know how to plug numbers and expressions into a function, but when the disruptively different math of calculus appears, there’s nowhere to put it, except in the plug-and-chug bin that all prior math has gone into.

So it’s extremely important that students going into calculus get a proper conceptual framework for what to do with the material once they see it. Whose responsibility is that? Everybody’s, starting with…

• the instructor. The instructor of a calculus class has to be very deliberate and forthright in bending all elements of the course towards the construction of a framework that will support the massive amount of material that will come in a calculus class. This includes telling students that they need a conceptual framework that works, and informing them that perhaps their previous frameworks were not designed to manage the load that’s coming. The instructor also must be relentless in helping students put new material in its proper place and relationship to prior material.
• But here the textbooks can help, too, by suggesting the framework to be used; it’s certainly better than not specifying the framework at all but just serving up topic after topic as non sequiturs.
• Finally, students have to work at constructing a framework as well; and they should be held accountable not only for their mastery of micro-level calculus topics like the Chain Rule but also their ability to put two or more concepts in relation to each other and to use prior knowledge on novel tasks.

What are your experiences with helping students (in calculus or otherwise) build useable conceptual frameworks for what they are learning? Any tools (like mindmapping software), assessment methods, or other teaching techniques you’d care to share?

## What does academic rigor look like?

I got an email from a fellow edu-blogger a couple of days ago asking for my input on the subject of academic rigor. Specifically this person asked:

Is the quest for more rigor an issue for you? Is it good, bad, meaningless? What does rigorous teaching look like in your classroom?

I hope she doesn’t mind my sharing the answer, because after writing it I thought it’d make a good blog post. I said:

For me, “rigor” in the context of intellectual work refers to thoroughness, carefulness, and right understanding of the material being learned. Rigor is to academic work what careful practice and nuanced performance is to musical performance, and what intense and committed play is to athletic performance. When we talk about a “rigorous course” in something, it’s a course that examines details, insists on diligent and scrupulous study and performance, and doesn’t settle for a mild or  informal contact with the key ideas.

Example: A rigorous course in geometry goes beyond just memorization of formulas, applications to simple geometric exercises, and “hand-waving” attempts at proof. Instead, such a course treats details as important, the ability to explain on a deep level the truth of formulas and results as a key goal for students, and sets a high bar for the exactitude of mathematical arguments. Euclid’s “Elements” for example is the prototype of the rigorous treatment of geometry. It’s not a difficult work to understand, necessarily; in fact one of the enduring qualities of the Elements is the clarity and precision of not only each individual proposition but also in how the overall collection of propositions fits together. By contrast, many modern books on geometry are highly non-rigorous, omitting details, putting theorems out of order, and defining a proof as a “reasonable explanation” only.

Is rigor good? It depends on the audience and the goals of the class. When I teach a geometry course for junior and senior Math Education majors, rigor is of the utmost importance because I want those pre-service teachers to go into their classrooms with tough, precise minds for the sake of their students. If I were to teach a geometry class for fifth-graders, on the other hand, I think rigor would obscure the subject, and I would depend a lot more on intuitive explanations and perhaps constructivist techniques for discovering key ideas in geometry and save rigorous proofs for another day. Similarly, when I teach calculus at my college, the audience is about 50% business majors, and so we designed the course not to cover much theory. This is not a rigorous treatment of calculus, but it is more effective for the students than if we included the epsilon-delta proofs and what not.

The quest for more rigor is most important in the post-calculus courses I teach (geometry, abstract algebra, and introduction to proof). These are subject areas where precision and detail-orientation are essential for a complete understanding of the material. Students are not allowed to give examples when a proof is called for, and I nitpick every little thing in their proofs up to and including the choice of punctuation and prepositions. [If any of the five who took this course from me this past semester are reading this, feel free to chime in with an “Amen.” – RT.] At the calculus level and below, I lay off on the theory but the rigor in the course comes from getting details of mechanical calculations right. And this is a big issue, because students in high school are generally taught only to produce a correct answer, not a clear and detailed solution. I am on a mission to make sure students can not only get right answers but also communicate their methods in a clear and audience-appropriate way, and that’s what “rigor” looks like there.

[After-the-fact note: To clarify, in calculus I insist on details in mechanical calculations but also on the details of processes and in paying attention to nuances in solving application-style problems. For example, students know that if you just set $f''(x) = 0$ and solve for $x$, that this doesn’t give you an inflection point; and in an optimization problem you can’t just find the critical number of the model function, you must also test it with the First or Second Derivative Test to see if it really yields a maximum. Or at least, they don’t complain when they forget to do it and I take off points!]

I have two kids, ages 3 and 5. (There’s a third one on the way in three weeks, but that’s another story!) I’m pretty rigorous with them, too — when the 5-year old says “Mimi comed to our house this weekend” I correct her grammar, and she gets it right the next time. You have to do it in a gentle way, but getting details right now will help them get the more complicated things right later. If I were to project myself out of higher ed and into the K-12 sphere I could see my teaching being “rigorous” in that kind of way — insisting that kids get the details right and not gloss over things, but doing it in a lovingly persistent way. I wish more K-12 teachers would do this, though, because it’s obvious from my freshmen in the last 4-5 years that this isn’t happening (or at least it’s not sticking).

[Final note: That last sentence isn’t a slam on either my freshmen, who were really quite excellent this year in calculus, or their teachers. It’s an observation, and I stand by it. I can show you their work at the beginning of the semester if you don’t believe me. Why this kind of “rigor” is not sticking with them is something I can’t fully explain because I don’t know what was going on with them in high school. Is it them? Is it their teachers? Is it the system? Is it the preponderance of standardized testing, which makes rigor more or less irrelevant? Comment!]

Filed under Calculus, Education, Geometry, High school, Teaching

## Straight talk on constructivism

Hat tip to Darren at Right on the Left Coast for this article, which starts off saying in a plainspoken way:

Here are two of the clues to America’s current mathematics problem:
1.”Student-centered” learning (or “constructivism”)
2.Insufficient practice of basic skills

The article then goes on to say, of constructivism:

In small doses, constructivism can provide flavor to classrooms, but some math professors have told me the approach seems to work better in subjects other than math. That sounds reasonable. The learning of mathematics depends on a logical progression of basic skills. Sixth-graders are not Pythagorus [sic], nor are they math teachers.

That’s right. Constructivism, when used with the right kinds of students and in the right ways, can be quite effective. But it’s important to remember that not all students are ready for this, and not all material is taught effectively this way. When I teach geometry to junior and senior math majors, it’s almost entirely constructivist, because the process of mathematical investigation and discovery is precisely what I am trying to teach them (through the medium of Euclidean and non-Euclidean geometry). But I’d be crazy to try constructivism at that level on, say, a precalculus class full of students who have little skill in and absolutely no taste for math at all. Those students aren’t dumb, but they need structure and guidance a lot more than they need the supposed thrill of mathematical discovery.

And then, about drill and practice:

Another problem in math classrooms is the lack of practice. Instead of insisting that students practice math skills until they’re second nature, educators have labeled this practice “drill and kill” and thrown it under a bus.

I wish I had a dollar for every time I heard that phrase. It’s a strange, flippant way to dismiss a logical process for learning. Drilling is how anyone learns a skill. […] Everyone drills – athletes, pianists, soldiers, plumbers and doctors. Drilling is necessary.

It isn’t good or bad – it’s simply what must be done.

I’ve said it before here: No human being can do meaningful creative work until they are completely fluent in the rudiments of what they are working with. Musicians, athletes, and skilled workers all know this. For some reason, there’s no outcry among music educators that we need to just hand new musicians a saxophone and try to get them to discover how to play it all by themselves. This fact — that drill and mastery precede creative work — is so painfully obvious that I feel a little embarrassed for my colleagues in math instruction who don’t seem to get it.

Constructivism and drill/practice are pedagogical tools, not religions. You look at your class, your students, and the material to teach, and then choose the right combination of tools for the job. To hear some proponents, and opponents, of constructivism, you’d think that you’re supposed to choose sides and swear undying allegiances instead.

Filed under Math, Teaching

## Technology in proofs?

We interrupt this blogging hiatus to throw out a question that came up while I was grading today. The item being graded was a homework set in the intro-to-proof course that I teach. One paper brought up two instances of the same issue.

• The student was writing a proof that hinged on arguing that both sin(t) and cos(t) are positive on the interval 0 < t < π/2. The “normal” way to argue this is just to appeal to the unit circle and note that in this interval, you’re remaining in the first quadrant and so both sin(t) and cos(t) are positive. But what the student did was to draw graphs of sin(t) and cos(t) in Maple, using the plot options to restrict the domain; the student then just said something to the effect of “The graph shows that both sin(t) and cos(t) are positive.”
• Another proof was of a proposition claiming that there cannot exist three consecutive natural numbers such that the cube of the largest is equal to the sum of the cubes of the other two. The “normal” way to prove this is by contradiction, assuming that there are three consecutive natural numbers with the stated property. Setting up the equation representing that property leads to a certain third-degree polynomial P(x), and the problem boils down to showing that this polynomial has no roots in the natural numbers. In the contradiction proof, you’d assume P(x) does have a natural number root, and then proceed to plug that root into P(x) and chug until a contradiction is reached. (Often a proof like that would proceed by cases, one case being that the root is even and the other that the root is odd.) The student set up the contradiction correctly and made it to the polynomial. But then, rather than proceeding in cases or making use of some other logical deduction method, the student just used the solver on a graphing calculator to get only one root for the polynomial, that root being something like 4.7702 (clearly non-integer) and so there was the contradiction.

So what the student did was to substitute “normal” methods of proof — meaning, methods of proof that go straight from logic — with machine calculations. Those calculations are convincing and there were no errors made in performing them, and there seemed to be no hidden “gotchas” in what the student did (such as, “That graph looks like it’s positive, but how do you know it’s positive?”). So I gave full credit, but put a note asking the student not to depend on technology when writing (otherwise exemplary) proofs.

But it raises an important question in today’s tech-saturated mathematics curriculum: Just how much technology is acceptable in a mathematical proof? This question has its apotheosis in the controversy surrounding the machine proof of the Four-Color Theorem but I’m finding a central use of (a reliance upon?) technology to be more and more common in undergraduate proof-centered classes. What do you think? (This gives me an opportunity to show off WordPress’ nifty new polling feature.)

## School choice and streamlining

BusinessWeek’s TechBeat blog has this article about the federal panel report on K-8 mathematics instruction that I blogged about here. It’s good to see this report getting attention in the blogosphere and MSM. It needs more. One thing from the BusinessWeek article that needs a slight bit of correction, though — it says:

The sad thing about the report that despite the unanimity on a panel that represents a broad spectrum of the mathematics and math education communities, it will take a decade or more for its recommendations to be implemented. It simply takes that long for curriculum guidelines to be recast, textbooks to be rewritten, and teachers to be trained or retrained. And in that time, a lot more damage can be done.

That may be true of traditional public schools, where red tape and opposing political forces must be overcome at every turn, but it does not have to be true of private or charter schools where reaction times to changing pedagogical climates can be much faster. I think this report, and the dire consequences of ignoring it, create a prime situation for charter schools and private schools to lead the way to better education for our kids. If governments would open up schools to market forces to a greater degree, we might even get the traditional public schools on board once parents choose to send their kids to places that are getting on the ball faster.

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Filed under Education, Math, School choice, Teaching

## Streamlining and upgrading math instruction

A federal panel examining K-8 mathematics education in the USA has made some forthright recommendations, according to this article in the NYT today. Unlike many federal panels, this one has an uncommon amount of common sense in its conclusions. For example, this finding that is striking in the way it refrains from choosing sides in the math wars:

Parents and teachers in school districts across the country have fought passionately over the relative merits of traditional, or teacher-directed, instruction, in which students are told how to solve problems and then are drilled on them, as opposed to reform or child-centered instruction, which emphasizes student exploration and conceptual understanding. The panel said both methods have a role.

“There is no basis in research for favoring teacher-based or student-centered instruction,” said Dr. Larry R. Faulkner, the chairman of the panel, at a briefing for reporters on Wednesday. “People may retain their strongly held philosophical inclinations, but the research does not show that either is better than the other.” […]

“To prepare students for algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency and problem-solving skills,” the report said. “Debates regarding the relative importance of these aspects of mathematical knowledge are misguided. These capabilities are mutually supportive.”

Say what? An appeal to actual research rather than anecdotes and personal biases when thinking about effective math teaching? Amazing. And this shocking discovery:

[T]he panel found that it is important for students to master their basic math facts by heart.

“For all content areas, practice allows students to achieve automaticity of basic skills — the fast, accurate, and effortless processing of content information — which frees up working memory for more complex aspects of problem solving,” the report said.

Dr. Faulkner, a former president of the University of Texas at Austin, said the panel “buys the notion from cognitive science that kids have to know the facts.”

“In the language of cognitive science, working memory needs to be predominately dedicated to new material in order to have a learning progression, and previously addressed material needs to be in long-term memory,” he said.

Why, it’s almost as if they think that mastery precedes creativity or something. And finally:

The report makes a plea for shorter and more accurate math textbooks. Given the shortage of elementary teachers with a solid grounding in math, the report recommends further research on the use of math specialists to teach several different elementary grades, as is done in many top-performing nations.

The article goes on to give some of the panel’s recommended benchmarks for mathematical skills in grades 3-7. There’s also a link to the panel’s report.

All I can say is that I hope math educators, prospective teachers (especially prospective elementary school teachers), curriculum designers, ed schools, school boards, and everybody else who is a stakeholder with some influence in this process are listening. We’ve got 2 years until our oldest starts kindergarten and she needs teachers and curricula who get math right.

[h/t God Plays Dice]