Recent posts on the Khan Academy — including an e-mail I posted from founder Sal Khan — sparked a lot of interest and reaction from readers.

One post, titled “Khan Academy: The hype and the reality,” by Karim Kai Ani, a former middle school teacher and math coach, and the founder of Mathalicious, took issue with the way Khan Academy videos deal with the concept of slope. Sal Khan sent in a response to the critique, which you can find here.

I asked readers to weigh in and following are several of the responses I received. The positions are different, showing how complex something that seems simple can be.

Here they are:

*This was written Raymond Johnson and Frederick Peck, Ph.D. students in mathematics education at the University of Colorado at Boulder and the Freudenthal Institute US. Raymond and Fred each have six years of experience teaching Algebra 1 and are engaged in research on how students understand slope and linear functions. Raymond blogs about math education and policy at MathEd.net and Fred shares his research and curriculum at RMEintheClassroom.com. This post first appeared on MathEd.net.*

The Answer Sheet has recently been the focus of a lively debate pitting teacher and guest blogger Karim Kai Ani against the Khan Academy’s Salman Khan. While Karim’s initial post focused mainly on Sal Khan’s pedagogical approach, Karim also took issue with the accuracy of Khan Academy videos. As an example, he pointed to the video on “slope.” Specifically, Karim claimed Sal’s definition of slope as “rise over run” was a way to *calculate* slope, but wasn’t, itself, a *definition* of slope. Rather, Karim argued, slope should be defined as “a rate that describes how two variables change in relation to one another.” Sal promptly responded, saying Karim was incorrect, and that “slope actually is defined as change in y over change in x (or rise over run).” To bolster his case Sal referenced Wolfram Mathworld, and he encouraged Valerie Strauss to “seek out an impartial math professor” to help settle the debate. We believe that a better way to settle this would be to consult the published work of experts on slope.

Working on her dissertation in the mid-1990s, Sheryl Stump (now the Department Chairperson and a Professor of Mathematical Sciences at Ball State University) did some of the best work to date about how we define and conceive of slope. Stump found seven ways to interpret slope, including: (1) Geometric ratio, such as “rise over run” on a graph; (2) Algebraic ratio, such as “change in y over change in x”; (3) Physical property, referring to steepness; (4) Functional property, referring to the rate of change between two variables; (5) Parametric coefficient, referring to the “m” in the common equation for a line y=mx+b; (6) Trigonometric, as in the tangent of the angle of inclination; and finally (7) a Calculus conception, as in a derivative. (See below for the reference to Stump’s work.)

If you compare Karim and Sal’s definitions to Stump’s list, you’ll likely judge that while both have been correct, neither has been complete. We could stop here and declare this duel a draw, but to do so would foolishly ignore that there is much more to teaching and learning mathematics than knowing what belongs in a textbook glossary. Indeed, research suggests that a robust understanding of slope requires (a) the versatility of knowing all seven interpretations (although only the first five would be appropriate for a beginning algebra student); (b) the flexibility that comes from understanding the logical connections between the interpretations; and (c) the adaptability of knowing which interpretation best applies to a particular problem.

All seven slope interpretations are closely related and together create a cohesive whole. The problem is, it’s not immediately obvious why this should be so, especially to a student who is learning about slope. For example, if slope is steepness, then why would we multiply it by x and add the y-intercept to find a y-value (i.e., as in the equation y=mx+b)? And why does “rise over run” give us steepness anyway? Indeed, is “rise over run” even a number? Students with a robust understanding of slope can answer these questions. However, Stump and others have shown that many students — even those who have memorized definitions and algorithms — cannot.

This returns us to Karim’s original point: There exists better mathematics education than what we currently find in the Khan Academy. Such an education would teach slope through guided problem solving and be focused on the key concept of *rate of change*. These practices are recommended by researchers and organizations such as the NCTM, and lend credence to Karim’s argument for conceptualizing slope primarily as a rate. However, even within this best practice, there is nuance. For instance, researchers have devoted considerable effort to understanding how students construct the concept of rate of change, and they have found, for example, that certain problem contexts elicit this understanding better than others.

Despite all we know from research, we should not be surprised that there’s still no clear “right way” to teach slope. Mathematics is complicated. Teaching and learning is complicated. We should never think there will ever be a “one-size-fits-all” approach. Instead, educators should learn from research and adapt it to fit their own unique situations. When Karim described teachers on Twitter debating “whether slope should always have units,” we see the kind of incremental learning and adapting that moves math education forward. These conversations become difficult when Sal declares in his rebuttal video that “it’s actually ridiculous to say that slope always requires units*” and Karim’s math to be “very, very, very wrong.” We absolutely believe that being correct (when possible) is important, but we need to focus less on trying to win a mathematical debate and focus more on the kinds of thoughtful, challenging, and nuanced conversations that help educators understand a concept well enough to develop better curriculum and pedagogy for their students.

This kind of hard work requires careful consideration and an open conversation, even for a seemingly simple concept like slope. We encourage Sal to foster this conversation and build upon what appears to be a growing effort to make Khan Academy better. Doing so will require more than rebuttal videos that re-focus on algorithms and definitions. It will require more than teachers’ snarky critiques of such videos. Let’s find and encourage more ways to include people with expertise in the practice and theories of teaching mathematics, including everyone from researchers who devote their lives to understanding the nuance in learning to the “Twitter teachers” from Karim’s post who engage this research and put it into practice. This is how good curriculum and pedagogy is developed, and it’s the sort of work that we hope to see Sal Khan embrace in the future.

**Sal’s point is that if two quantities are both measured in the same units, then the units “cancel” when the quantities are divided to find slope. As an example, he uses the case of vertical and horizontal distance, both measured in meters. The slope then has units of meters/meters, which “cancel”. However, the situation is not so cut and dry, and indeed, has been considered by math educators before. For example, Judith Schwartz (1988) describes how units of lb/lb might still be a meaningful unit. Our point is not to say that one side is correct. Rather, we believe that the act of engaging in and understanding the debate is what is important, and that such a debate is cut short by declarative statements of “the right answer.”*

Schwartz, J. (1988). Intensive quantity and referent transforming arithmetic operations. In J. Heibert & M. J. Behr (Eds.), Number Concepts and Operations in the Middle Grades (Vol. 2, pp. 41–52). Reston, VA: National Council of Teachers of Mathematics.

Stump, S. L. (1999). Secondary mathematics teachers’ knowledge of slope. Mathematics Education Research Journal, 11(2), 124–144. Retrieved from http://www.springerlink.com/index/R422558466765681.pdf

-0-0-

*This was written by Martin Weil, a physicist who happens also to be a brilliant colleague of mine at The Washington Post who usually manages to suppress his views on major public issues.*

What I would say, at the outset, however, is that if I wanted to take issue with Khan, it would not be over his definition of slope. This may add to your understanding of that narrow issue. It would not be worth going into any detail, were it not that slope and the concept of slope is at the heart of calculus.

Khan’s definition of slope is a good approximation. It introduces the concept and gives an intuitive feel for it. Calculus depends on a refined version of that rough and ready idea.

In a sense, everybody knows what “slope” is. If you drive your car uphill, and you start at sea level and rise in one mile to a height of 100 feet, then the slope is 100 feet per 5280 feet....which is about 1/50. We would say that the road has a 2 per cent slope, or a two per cent grade. That is a concept that needs little explanation. If the road rises steadily from the bottom to the top, we can say that at every point in the route, the slope is 2 per cent. Or one in 50. At every point. (note that no units are required) It’s the rise over the run. That’s the slope

At every point of the route, you are going up a 2 per cent grade. A 2 per cent slope.

Calculus deals with more sophisticated and complicated problems. In these, the slope may change at every yard, every foot, every INCH along the way. Knowing the precise i rate at which things are changing, with time, with distance, with some other variable, makes it possible to solve a variety of significant and important problems.

The basic concept is indeed “ rise over run” . Nothing wrong with that. But rise over run where? at what point on the road? Well, the example I have given is an easy one. Because rise over run is the SAME at every point. It’s 100 feet per mile. 100 feet per 5280 feet. In this case of an unvarying slope, it is at every point on the road, 1 foot peer 52.8 feet. and so on.

But calculus is applied to more complicated problems. You can not solve a calculus problem by looking at it and repeating to yourself “rise over run.” In calculus problems the slope changes constantly. No matter how brief the rise, how short the run, the slope differs at every point. Every inch of the way. At every 10th of an inch. At every 10000th of an inch. Until the intervals of rise over intervals of run become infinitesimally small.

Considering the slope at every point along a roller coaster gives an idea of what the problems are. On a roller coaster, you start at the ground level and you end up back at ground level. So if you simply use a “rise over run” rule for calculating the slope, you will get ZERO. The coaster goes up and down and up and down, but at the end, you have risen no higher than you were at the start. So the rise is zero, no matter what the run might be. So a rough application of “rise over run” to get the slope of the roller coaster route will give you ZERO.

Even though it is obvious that at (almost)every point the track is sloping up or sloping down.

So this SUGGESTS that there is more to slope than a mere application of “rise over run.”

Finding the slope at any point on a roller coaster demands answers to questions such as these: what is the rise at a single point? What is the run? Can there be such a thing? Can there by a rise at a single point????? A point has no dimensions. How can it have a “rise” How can it have a “run” ?

An answer exists... It takes a little thinking about. But it can be understood. And it is at the heart of calculus.

It requires an appreciation of the fact that at any small segment of track the so called “run” can be made smaller and smaller and smaller.And the same for the rise.

Then specify a point at which the slope is to be calculated. Specify the “run” involved at that point. Specify the “rise.” Do this by making approximations. A good approximation is to place the point in the middle of a small interval of distance. Calculate the slope at a point which is half way between the beginning and end points of that small interval. That seems like a pretty good way to approximate the “run.” And, it is!!!

Make the interval, the run, one inch. The point then is half way between the zero inch mark and the one inch mark. Then bring the beginning and end points of that interval closer and closer to the point in question. Keeping the point half way between them. Let the point be one ten thousandth of an inch from the start of the interval; that will be a better approximation to the run at that point. Then reduce the distance to one 1 ten thousandth of an inch.. Then one millionth of an inch. Then one billionth of an inch. This leads inevitably to an interval that can be considered almost infinitesmally small.

And all the while, for these increasingly tiny intervals, there is a corresponding rise, also increasingly tiny. And all the while, as these intervals are shrinking, the ratio of rise over run is being computed, until rise over run becomes the ratio of an infinitesmally small rise over an infinitesmally small run.

And finally you have obtained the “rise over run” for a specified point. Even though a point in itself HAS NO RISE or RUN.

This serves as a plausible and persuasive process for finding the slope at ANY specified point along ANY curve. (With certain limitations t hat need not be gone into) Calculus employs techniques for computing that infinitesmal rise over infinitesimal run. That ratio is what is understood in calculus to be the “slope.”

This would not be of any great practical value if not for the fact that knowing the slope at any point on many curves points the way to solving many problems of a practical nature.

The definition: “Rise over run” may not take all of this fully into account. B ecause as we have said, if we expand our interests beyond straight lines and such regular and symmetrical figures as circles, then the concept of rise over run at the dimensionless point of geometry, does not seem to have meaning. Yet, it is he essential principle at the heart of the more sophisticated calculations of calculus

So in this sense it is correct, and challenging it is not the best way of taking issue with some school of pedagogy that may put it forward.

-0-

* This was written by Peter McIntosh, a high school math teacher at Oakland Unity High School in East Oakland, California.*

In his recent criticism of Khan Academy Karim Kai Ani suggested “that there’s nothing revolutionary about Khan Academy at all. In fact, Khan’s style of instruction is identical to what students have seen for generations.” He went on to echo the concerns of many educators by criticizing Khan’s approach to content delivery and pointing out flaws in his videos.

Christopher Danielson and Michael Paul Goldenberg suggested [in this post] that “rather than revolutionizing mathematics teaching and learning, Khan’s work adds a technological patina to a moribund notion of teaching and learning mathematics.”

Messers Ani, Danielson and Goldenberg are obviously experts in their fields, and they make any excellent points. However, I believe that their thoughtful analysis is misdirected.

When an answer to a problem remains elusive after decades of effort by legions of passionate people, perhaps it is time to consider whether we are asking the wrong question. I believe these authors, like many American educators, are mistakenly looking for better approaches to content delivery and have missed the real problem in our math educational system, content reception.

Teachers are not failing because of ineffective content delivery; they are failing because they are not effectively addressing the character deficit in many of our students. We have spent years looking for better ways to deliver content to students who are increasingly uninterested in receiving that content.

Rather than address this root cause of educational failure, it is used as an excuse to explain the patchwork results of classroom reforms. It makes sense that the academic scores in East Oakland are lower than the scores in more affluent school districts; or that children from families where education is a priority do better than those from less focused backgrounds. However, educators need to stop using these character issues as an excuse for failed educational initiatives, and start making character their real focus.

To better understand how teachers can address this character deficit let’s examine how it developed. By the time students reach the high school classroom many of them are far behind grade level in basic math concepts such as fractions. This content gap has lingered for many years. Students became accustomed to being unable to do the problems and they rationalized their constant failure. They concluded that they could not do these problems because they had a poor teacher or that they were simply not smart enough. They soon began to see each homework assignment as just more evidence confirming that they had a poor teacher or that they were not smart.

Because of these rationalizations students developed the habit of never doing homework, deepening the spiral. And this problem was compounded by the total lack of consequences. Despite not knowing the material and not doing the work, many of these students were passed on to the next grade, often with A’s and B’s.

These students enter the high school classroom with a content gap and a seriously skewed view of education. They have heard fractions explained dozens of times, and they have A’s and B’s on their transcripts. Consequently they greet much needed review lectures with false confidence. “I know this!” they say as they tune out brilliantly delivered content. Then their habit of not doing homework ensures that they do not absorb this material.

False confidence – laziness – lack of responsibility – What is this if not a character deficit? I believe we need to shift our focus from improving content delivery to helping our student repair their character. And I believe that we can do that in the classroom. But we first need to accurately name the problem.

Discussions of flaws in Khan videos are an example of misdirected focus on content delivery. The videos do have flaws, but the genius of Khan Academy is the pause button. Students have control. They can watch the videos if they choose, and they can stop any video. What is totally missed in these criticisms is the effect Khan has on student habits. And that effect is not based on the videos!

Khan Academy is an ensemble performance. Something about the design of the math exercises engages students. And the first effect is often the confrontation of the false confidence so prevalent among these students. It is amusing how many students call me over to complain that Khan has the wrong answer to a problem. But they are engaged as I explain why the Khan answer is correct.

The second part of the ensemble is the availability of “hints.” For any problem students can request the detailed steps, just one to get started, or the entire solution.

The most powerful part of the ensemble may be the very natural tendency of students to help each other. They actively listen as another student explains a problem, and they become intensely focused when they are the one providing the help.

The teacher takes on a different role in a Khan classroom. The casual observer will see a teacher providing one-on-one coaching to students receptive for that guidance as they struggle on a specific problem, or providing brief explanations to small or large groups struggling with a difficult concept. The more focused observer will see a leader: defining objectives and encouraging the students to take advantage of this full ensemble of resources.

What is most interesting is the engagement. Disruptive behavior fades when the computers come out. The coaching screens and reports display amazing persistence, with students patiently working through dozens of problems until they master a topic and complete a string of correct exercises.

These students have begun to take responsibility for their education. They ask for help from peers or the teacher, or they use hints from the system. Sometimes they refer to notes to use prior problems as a model. They find a way to solve the problem in front of them and then move on to the next problem.

Oh, and some students watch videos.

Responsibility – Effort – Confidence. Real confidence based on accomplishment. A willingness to persist on difficult problems because of that confidence. Autonomy to seek help from a variety of sources. Students in a Khan classroom exhibit significant changes in their character traits. And they learn.

Interestingly, this resurrected character makes it much easier to engage the students in challenging word problems or hands-on projects, and their learning is deepened by their strengthened skills and persistence. Importantly, Khan frees teachers from more routine preparation to facilitate these deeper learning experiences.

The critics are correct on one very important point. We do need great teachers. But we need their focus to be on leadership rather than just content delivery. And we need them to use tools such as Khan Academy to reach students in ways traditional teaching cannot.

The real issue is changing the question being asked. Decades of focusing on content delivery has resulted in arguments but little success. We need to start focusing on repairing character. Khan Academy cannot replace teachers in this effort, but it provides a tool that can leverage the skill, energy and love these professionals bring to class every day.

-0-

*This was written by Ben Tilly, * *a computer programmer with a master’s degree in math and a nearly completed doctorate, who has taught Calculus, multi-variable Calculus, linear algebra, etc.*

In mathematics there are many, many cases where there are multiple equivalent ways to define things. You can choose one, and then the others are all theorems. Math doesn’t care which we choose to be our definition, and therefore we should choose the simplest to understandand work with. Many high school math teachers seem to believe that whichever one they happened to be taught is the “right” one, the others are “incorrect,” and that the distinction between the two has some fundamental importance. Speaking as a mathematician, absolutely nothing could be farther from the truth.

Sal Kahn used the widely used definition “rise over run” which is simple to calculate and easy to visualize. This is an excellent definition to use.

Karim Kai Ani used the definition, “slope is a rate that describes how two variables change in relation to one another.” I find this definition unclear, abstract, and hard to visualize. I want students to be able to understand that slope does that, but this is NOT how I want beginning Calculus students to understand slope. The derivative is a complicated enough thing for them to understand as it is, and starting with an unclear picture of slope will just make it worse.

Both definitions can be made to work. Both appear in textbooks. I prefer Sal’s.

-0-

*This was written by Evan Turner, an engineer and former instructor. *

I just wanted to note that the explanation Khan is giving for slope is not particularly useful. His counter-example of calculating memory size per cost and saying it’s the “inverse” slope isn’t wrong, per se, but it’s pointless.

From a top-level perspective, the concept of slope in elementary algebra is really just the derivative of a linear function. That is, a rate of change of a straight line. Hence, the informal definition of a derivative at a point is “the slope of the tangent line”.

For example, a ball dropped and falling toward the earth will have a position of ‘h - 1/2*g*t^2’, where ‘h’ is the height (in meters) it was dropped from, ‘g’ is the gravitational constant (9.81 m/s^2), and ‘t’ is how many seconds the ball has been falling. If you graph that function, where your x-axis is “time” and y-axis is “height”, you get a parabola that looks like an upside-down ‘U’ (everything to the left of the origin can be ignored, since we don’t care about the ball’s velocity before we dropped it).

The derivative of position is velocity, so the slope of a line drawn tangent to any point on this parabola tells us the velocity of the ball at that instant of time. Divide the rise (change along y-axis) by the run (change in x-axis) and you get the ball’s velocity. Points farther to the right, i.e. at later times, have steeper tangent lines. As the slope continues to be more and more negative, the ball is falling faster, which we know intuitively from the experience of dropping things from low and high heights.

What Khan seems to be missing, and what Karim didn’t specify, is that the importance isn’t that slope is change between two variables, but that it is the rate of change for a dependent variable vs. an independent one. In the case of the ball, the independent variable is time — because we can’t affect its progress — and the dependent variable is position -- how far the ball has fallen after t seconds. Khan’s statement about putting memory size on the y-axis and price on the x-axis is irrelevant, because the dependent variable is price and the independent variable is memory size (”how much will I pay for X gigabytes of memory?”). When you shop for an iPad, the store has signs that list the price of an iPad with 16, 32, or 64 GB of storage. This makes sense intuitively, because we want to know what we have to pay to get an iPad with the amount of storage that we want. The inverse of this would be, “I have $599, what size iPad can I afford?”, by swapping the dollar and GB axes.

Regarding units, if both variables have the same unit, the units cancel out (e.g. sales tax: for X dollars worth of items purchased I pay Y dollars total, Y dollars over X dollars causes the unit of dollars to cancel out, which is why sales tax is always given as a percentage). However for velocity, we have meters fallen divided by seconds in the air, so our unit is m/s.

What is important to a student isn’t just to repeat a mnemonic or formula, such as ‘rise over run’, but to understand what these quantities mean, why we’ve selected our axes a particular way, and whether the slope (rate, derivative) we’ve calculated is useful. In my example of the ball, it gives us a way to quantify an intuition: that dropped objects fall faster the longer it takes before they hit the ground. Telling a student “subtract y1 from y2, and divide that by x1 subtracted from x2” shows them how to get a slope, but teaches them nothing of what the slope represents.

-Evan

P.S. My background is in engineering (now a software developer), but I have spent time instructing at various levels.

-0-

*Charles McLane, a former teacher, sent emails that said in part:*

It also strikes me that you are shooting the messenger in your protests about the Kahn Academy. A pathetic state of mathematics education is implied by the Kahn Academy tutorial videos: despite rather mediocre quality, the videos have achieved notable popular acclaim from struggling students, explaining to them what was not understandable in their classes. One can only speculate as to how bad those classes must be. It misses the point to criticize Kahn because he did not pay proper homage to “lesson plans” or “pedagogical intentionality....”

...and...

Slopes and rate-of-change are related, but distinct, concepts with slopes leading us gently and intuitively into the concept of rate of change. Building on both of these concepts leads us toward the beauty of calculus. Mathematics first defines its terms and then critically proceeds. It is the process of critical thought starting from whatever given definitions that defines mathematics. Kahn’s definition of slope is both common and age-appropriate, if not the only possible definition. Appeal to authority for “correct” definitions is illustrative of what’s wrong with math education as it totally misses the spirit of mathematics.

--

Here are the pieces I have published recently on the academy:

The first piece was called “Is Khan Academy a real education solution?” and the next was titled “Khan Academy: The hype and the reality.” This elicited a response from Khan, which you can find here. Then there was a piece called “How well does Khan Academy teach?”

-0-

*Follow The Answer Sheet every day by bookmarking www.washingtonpost.com/blogs/answer-sheet .*