Incorrect but useful might be under the category of ‘howler in math’ google it.

Ouch.

I just did this.

I hate it when you all cause reflection.

(I also hate being under time pressure to teach too much content, and kids that are used to just memorizing steps rather than getting deeper understanding)

Re: Incorrect but useful

I have long been a fan of deliberately sabotaging a solution to something that I might be doing on the board so that somewhere down the road things become obviously wrong. This is so students can start to develop strategies for what to do when this happens.

Many will tell you that it’s important for students to make mistakes (in fact, that they learn the most when they do). But that sometimes runs counter to what they see in class. That is, a teacher demonstrating flawless execution of mathematics. Even some of our best students often won’t even attempt a problem unless they are sure they will get it correct. If they are ever going to become comfortable with making mistakes as part of the normal process then we have to include managing those mistakes as part of our day to day in class.

Incorrect but useful: possibly estimation? When I teach problems like “how do you add the integers from 1 to 100?” I’ve found it useful to ask kids to estimate their answer, even though it’s incorrect to just take a guess. If they’re within 1000 of the correct answer, I tell them that I’ll give them the solution. This gets them to thinking about how to estimate well, and “99 is close to 100, so…” usually gets them closer to thinking about how to get to the right answer the “correct” way. It’s an incorrect but useful step.

Also incorrect but useful to estimate things like area problems, in order to find out a ballpark figure and check if you’ve done the math right. Like, find the area of the circle inscribed in this square. Using the area of the square to estimate gives an upper bound, so that if students do something like set the radius equal to the side of the square (happens WAY too often!) they notice that their answer must be wrong.

All I can say is the types of “incorrect but useful” math that David, Cathy and Moana mention are awesome examples of steps I want my students to do when they are working on problems.

Under “incorrect but useful”, I’d include Fermi estimates.

Indeed. Doing math is a bit like writing a poem. Just like there are rules about what rhymes with what and what scans correctly, there are rules about how you can manipulate quantities and expressions. But putting together a couple of lines that rhyme doesn’t make you a poet. Stringing together a couple of legal transformations doesn’t make you a mathematician.

Poetry is about expressing meaning by uniting form and purpose. Mathematics is, similarly, about expressing meaning by uniting form and purpose. Both are about choosing which of the legal possibilities best unifies the form to the purpose.

Most scientific models are incorrect but useful. Like maths, it’s tough to get students away from trying to memorise “correct” answers in science.

One obvious example is modelling electricity as the flow of positive charge around a circuit. Of course this is almost never “correct” but is incredibly useful in predicting behaviour.

Re: Incorrect but useful

Math history is riddled with examples of incorrect but useful steps. In fact, many great advances wouldn’t have taken place without them if people first waited for the correct formalism to be in place. Discoveries in Math didn’t occur in the order that the concepts are taught in the class.

Specifically, I am thinking about discoveries about the infinite series and calculus. The unproven assumptions made about existence of certain numbers by Euler (a very large number such that its product with a very small number is finite) and not worrying about convergence of series and working with the infinite polynomials by Newton, Euler, Taylor, and their contemporaries turned out to be immensely useful. Surely, this led to some incorrect results and some of their steps couldn’t be accepted in proofs today, but on the whole they led to massive advancements.

Today, I think that we have a formalism for handling some such situations using conjectures. If this conjecture is true … then these proofs hold. What this implies is that all your work based on this assumption could go to the trash can if the conjecture is disproved. It’s still a risk that many people take.

Another instance that comes to mind is the discovery of non-Euclidean geometries. Gauss was so scared of jolting the mathematical world with a statement like “the sum of the angles of a triangle doesn’t have to be 180 degrees” that he kept the work in his desk for decades until Bolyai and Lobachevsky came to the same conclusion independently. At the time, going against one of Euclid’s postulates was definitely an “incorrect” step.

Most people would poke fun at a physicist who let his work be guided more by mathematical beauty than anything else. It would definitely be an “incorrect” step. Not so for a theoretical physicist named Paul Dirac, who discovered half the universe via one beautiful equation. His equation predicted antimatter, which was an unknown concept at the time.

Budding mathematicians probably won’t score highly in our classrooms because they would be bored with what they are required to do there. In addition, it is probably a disservice to the students if we don’t empower them to bring creativity, imagination, and appreciation of beauty into learning of Mathematics.

PS: I didn’t mean to write an essay here but couldn’t hold it back :)

The following 2 problems come from Philips Exeter Academy’s Mathematics 1 text. They relate to this post in a couple ways.

Firstly, they utilize the idea of mathematically useless steps to broaden students’ understanding, which is the main theme of Dan’s post. Secondly, they relate to solving systems of linear equations using the elimination method, which is the leading example in Dan’s post.

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Problem #1:
Lee was asked to find the coordinates of the point that belongs to both of the following lines:

4x + 3y = 20
3x − 2y = −5

Lee took one look at these equations and announced a plan: “Just multiply the first equation by 2 and the second equation by 3.” What does changing the equations in this way do to their graphs?
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Problem #1 Answer:

It doesn’t change the graphs at all. Students almost invariably do not realize this until they actually do the multiplication, and then begin going through the process of graphing both, which usually involves transforming the equations into y=mx+b form and realizing that they are the same. When they get the same y=mx+b form, most students think they have made a mistake somewhere. Only after they trace their steps a few more times do they start to see that they haven’t, and in turn start to realize that multiplying both sides of an equation doesn’t change it’s solutions. “After all, when I’m transforming the equation into y=mx+b form, all I’m doing is adding, subtracting, multiplying, and dividing both sides of the equation by the same number, and that doesn’t change the graph. Oh yeah, that makes sense. I get it” thinks the student.

In this problem, students engage in two sets of steps that are useless to the process of solving a system of linear equations using the elimination method: namely transforming the equation into slope-intercept form and graphing the lines. But both of these activities serve to put one of the steps of the elimination method into a larger family of activities, namely and respectively multiplying both sides of an equation by *any* number leaves the solutions unchanged and in turn leaves the graph of that equation unchanged. Having students solve this problem and then discuss the results afterwards is sufficient to solidify and justify this idea for them in an understandable way.

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Problem #2

Start with the equations 2x − y = 3 and 3x + 4y = 1. Create a third equation by adding *any* multiple of the first equation to *any* multiple of the second equation. When you compare equations with your classmates, you will probably not agree. What is certain to be true about the graphs of *all* these third equations, however?
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Problem #2 Answer

The graphs of all of the third equations created by you and your classmates will go through the *same intersection point* as the original two. The students find this out by generating their third equation and then graphing it, finding that theirs does go through the same intersection point. They usually stop there and assume (correctly) that it couldn’t be a coincidence. The point is made more clear and memorable, however, when we begin to add multiple students’ third graphs to the GeoGebra sketch at the front of the classroom, all of which go through the original intersection point.

The big idea: when we solve a system of equations using the elimination method, adding the equations and solving for x doesn’t get us x-coordinate of the intersection point directly. Instead, those steps give us the equation of a new line that we know goes through the original intersection point. Since this line’s equation is in the form x=a, we know it is a vertical line and can then conclude that the x-coordinate of the intersection point must have the value a. It’s a remarkable realization.

—————————————–

These two problems serve as excellent examples of the perspective-expanding principles mentioned in this post. They fill in the conceptual holes that are present in most treatments of solving systems of linear equations using the elimination method, creating a stunning, beautiful and complete argument that opened my eyes wide and made me gasp with awe and satisfaction the first time I saw it.

Howard Phillips makes an excellent point about definitions. In many maths classrooms, a “correct” step is the next one in the predefined path towards solving a problem. Therefore, any step that isn’t the next one in the official algorithm is considered “incorrect”. Of course, many of these steps are very useful for improving students’ understanding even if not the most efficient path for solving the specific problem.

I think that Dan’s question about “incorrect” steps is aimed more at “arithmetically incorrect” as the above category are often “correct but not useful”.

Arithmetically incorrect steps that are useful are hard to find outside of deliberate approximations used in estimation and similar. sinX=X is another one that is often used even though technically incorrect because it is useful at low values of X.

The only examples I can think of where arithmetically incorrect steps can be useful is when they highlight to a student why that arithmetic step was incorrect. One example which comes to mind is the “proof” of 1=2 that involves cancelling 0/0 as the arithmetically incorrect step.

Incorrect and useful:

– treating differentials as variables
– assuming every vector space is basically R^2
– approximating a function with the first few terms of its McLaurin expansion
– using a big number on your calculator to approximate infinity (or a really small number to approximate the “limit” at zero)
– in “long” Pythagorean triangles, the hypotenuse and long side have basically the same length
– in geometry, drawing an accurate diagram (with ruler and protractor) proves nothing, but shows you what might be true
– “consider a spherical cow…”

Incorrect but highly useful: Reductio ad Absurdum

In Geometry it is sometimes convenient to assume the negation of what you are trying to prove, then follow a logic chain until you reach a contradiction.

It is incorrect to assume that the square root of 2 is rational. But this leads to a logical inconsistency, enabling us to prove that the square root of 2 is irrational.

Incorrect but useful? (Neither of these mine.)

19 / 95, cancel out the 9s, leaving 1/5.

log(3x+2) + log(4x+1) = 2 log 11, cancel out the word “log” leaving (3x+2) + (4x+1) = 22, giving the correct answer.

Developing Pawan Kumar’s response: Euler could perform magic with his intuition. Some of the things he did would be, by today’s standards, totally non-rigorous, and some, when taken at face-value, are just incorrect.

The most popular example these days is surely his computation of values of divergent infinite series, like (4)-(7) here:

http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/

These are the “p-series” from calculus with p=0, -1, -2, … We learn in calculus class that p-series only converge with p>1. The formulas are thus incorrect.

To see the utility, we can consider a function on (1,infinity) whose input is p and whose output is the sum of the convergent p-series. This is the Riemann zeta function, and the formulas (4)-(7) suggest that it may be possible somehow to extend the definition of the function to non-positive numbers. In fact, Euler did just that and found a remarkable “functional equation” satisfied by the function that makes the assignment of values (4)-(7) much more reasonable.

It seems most or all examples of “incorrect but useful” given herein concern either approximation or intuition. A third category that might encompass or exceed both: examining incorrect deductions. These can often lead to new useful definitions or theorems. For instance, incorrect proofs of Fermat’s Last Theorem use unique factorization in number systems that resemble the integer. This leads to the idea that Unique Factorization is something that needs to be proved in such systems to be used. Or in making calculus into the more rigorous “real analysis”, certain arguments lead to the notion of a “uniform bound” being needed in some statements (the “epsilon” used needs to be independent of other choices made ), leading to definitions like uniform convergence and uniform continuity.

Incorrect is always useful for error analysis. Students and learners should always be more interested in the problems they get wrong, because that’s were there’s learning to be done. My students mistakes interest me because the often reveal common pitfalls, ambiguities, and confusions. “Why did I get that one wrong,” should be a question posed with feelings of curiosity and determination, unfortunately it’s too often accompanied with shame, frustration, and anxiety over “being wrong”.