Teach Students Correct But Useless Steps

Christopher Danielson and Megan Schmidt have both written recently and compellingly about the trouble students have when taught that math is a series of correct “steps.”

Danielson, doing his best Howard Beale:

THE STEPS WIN, PEOPLE! The steps trump thinking. The steps trump number sense. The steps triumph over all.

Let me add to the conversation the category of “steps that are correct but useless.” These are great. They come from a conversation I had, like, fifteen minutes ago with a teacher named Leah Temes here at NWMC 2014.

Leah teaches Algebra II. We were talking about solving systems of equations. It’s really easy to teach the solution to a system like this as a series of correct, useful steps:

2x + 3y = 10
5x – 3y = 4

  1. Add the second equation to the first one.
  2. Solve for x.
  3. Substitute x in either equation to solve for y.
  4. Check that pair in the other equation for full credit.

Leah said she was tired of seeing her students mimic those correct steps without understanding why they worked. So instead of showing her students steps that were useful and correct, she asked them if she was allowed to add the following two equations:

2x + 3y = 10
5 = 5

To get:

2x + 3y + 5 = 10 + 5

Is everything still correct? Yes.

Was that useful? No.

This experience awakened her students to a category of steps in addition to the correct and useful ones they’re supposed to memorize and the incorrect and useless ones they’re supposed to avoid – correct and useless steps.

Alerting your students to that category of steps may make math seem less intimidating and more interesting. Math isn’t any longer a matter of staying on the right side of a line between the incorrect and correct steps. There’s another region out there, one that’s a bit less tame, a place for explorers, a place where the worst thing that can happen is you did something right but it just wasn’t useful. That category of steps also requires justification – “how do you know this is correct?” – which can help bend the student away from memorization and back towards understanding.

BTW. All of this implies a fourth category of steps – incorrect but useful. Can anybody give an example?

Featured Comments

Cathy Yenca:

I do a similar thing when solving equations in one variable by asking students if I can add 1,000,000, let’s say, to each side of an equation… or if I can subtract 27 from both sides… or divide both sides by 200… etc. etc. We talk about what is “legal” (have we followed the rules of algebra and the concept of “balance” and equivalence?) and what is “helpful” (have we done something “legal” that helps us isolate the variable so we can solve this thing?”) Exaggerated examples like adding 1,000,000 to both sides seem to make an impression on kids.

David Petro:

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.

Moana Evans:

[It’s] 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.

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I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

37 Comments

  1. This is great! I do a similar thing when solving equations in one variable by asking students if I *can* add 1,000,000, let’s say, to each side of an equation… or if I *can* subtract 27 from both sides… or divide both sides by 200… etc. etc. We talk about what is “legal” (have we followed the rules of algebra and the concept of “balance” and equivalence?) and what is “helpful” (have we done something “legal” that helps us isolate the variable so we can solve this thing?”) Exaggerated examples like adding 1,000,000 to both sides seem to make an impression on kids.

    Incorrect but useful… that is an interesting idea. It’s useful to know that something doesn’t work.

  2. 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)

  3. Dan,

    I would really be interested in what we are classifying as incorrect steps, but useful. It could be classified as all the of “quirks” in math operations that lead students to think an operation proceeds one way- but in reality it only works for one special case. 2^2 or √4 are such creatures, it leads students to believe to multiply by two or divide by two because the get the correct answer.

    If we are classifying incorrect as not making progress towards an answer, then I would have to disagree with Jim, drawing or using a model is typically very useful for students to conceptualize and solve problems (not I say typically).

    So, do we need restraints on “incorrect but useful?”

  4. 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.

  5. I was paraphrasing (and should have cited) statistician George Box’s aphorism “essentially, all models are wrong, but some are useful.” In this context, I would argue that no sequence of steps will ever lead you to a completely correct and error-free model, but some steps might lead you to very useful models. On the other hand, to argue that such steps are truly “incorrect” would require an absurdly standard for correctness.

  6. 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.

  7. 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.

  8. Couldn’t you say that “incorrect but useful” is a perfect description of the very first steps in teaching integration as a tool for measuring area under a curve? Here, look – we’ll draw these big clunky boxes and that sort-of gets us kinda in the right realm for an answer, but not exactly.

  9. I don’t think estimation is necessarily incorrect but useful. It is an estimate, not intended to be presented in the final formulation (except maybe as an aside) as far as I can tell.
    My best guess for “incorrect but useful” would be abuse of notation, overloading operators, and similar things.

    We also sometimes let students get away with things that we should correct…but it lets the critical thinking happen, which is very useful but incorrect. It’s the more naturalistic version of the sabotaging mentioned by David.

  10. 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.

  11. Paraphrase of a memorable estimation problem from one of my physics lectures in college:

    What is the mass of the earth?

    Let’s call the earth a sphere (even though it’s not). We can compute the volume of a sphere if we know the radius. We can get the radius from the circumference…what’s the circumference of the earth? Hm…there are three hours of time zone difference between California and New York, and they are about 3000 miles apart. Since there’s 24 hours in a day, let’s say the circumference is 24000 miles. Then the radius is the circumference divided by 2*pi, and we’ll say that pi is about 3 (which is not exact), so the radius is 4000 miles. Convert to meters by multiplying by 1500 (also wrong): 6×10^6 m.

    Then the volume of a sphere is 4/3 pi radius ^3. We can cancel the pi / 3 (not exact) and say the volume is 4 * (6×10^6) m^3 which is about 8×10^20 m^3.

    What is the density of the earth? The core of the earth is iron (density 7000 kg/m^3), so we’ll say the density of the earth is about 5000 kg/m^3, since it’s not all iron.

    Multiplying the density with the volume gives 4 x 10^24 kg. The right answer is about 5.972 x 10^24, but we got the right order of magnitude (10^24) by making lots of inexact but fast calculations.

  12. 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.

  13. My personal favorite illustration of correct but useless steps in solving equations is to multiply both sides by 0.

    For correct and useful, but not always obvious (is that a new category?) steps: I think it was Lipmann Bers who told some graduate students at Columbia in the ’60s that a significant amount of mathematics entails adding convenient forms of zero and multiplying by convenient forms of one. I believe that’s a bit deeper than the usual “stuff” math teachers ask students to memorize and imitate.

  14. 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 :)

  15. Terminology, or the language of math:
    Correct really means valid AND useful, or just sensible.
    There is confusion between “Have I done the operation correctly?” and “Was that the correct operation to do?”.
    The example given
    It’s really easy to teach the solution to a system like this as a series of correct, useful steps:
    2x + 3y = 10
    5x – 3y = 4
    if you treat it as an example where the numbers could have been anything.
    But the “correct, useful steps” given are woefully inadequate, as they are limited to this special case.

  16. Harry O'Malley

    October 11, 2014 - 7:54 am -

    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.

    ————————————–
    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?
    ————————————–

    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.

    ————————————–
    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?
    —————————————–

    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.

  17. Two thoughts about incorrect but useful. Plugging in small numbers into an equation can give you a sense of whether you are going in the right direction, but is rarely given as a sanctified “step” Alternatively, in proofs, one can often assume an incorrect conclusion to prove something- indirect proof. I suppose anytime you approach a problem indirectly instead of directly you are taking a purposefully wrong step to establish a right direction…but of course if it is useful, in the end it wasn’t incorrect.

  18. 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.

  19. David Srebnick

    October 12, 2014 - 4:27 am -

    One incorrect but (arguably somewhat) useful example might be dividing by a variable instead of factoring, such as, when solving some variation of x(x-3)=x.

    It exposes the solution x=4 quite easily by solving x-3=1. If you then remember that this is valid only for x not equal zero, then you can just test x=0 as a solution. The first time my students are introduced to this type of problem, they have trouble seeing x=0 as a solution, although if I tell them to test x=0 they can see that it works.

  20. 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…”

  21. To go with the systems example, I often ask S’s what we should do next. Even if what they say is “wrong” I follow through with the move and let them catch the mistake. Either right then, or later on. I have a question I always ask after we manipulate an equation: Did doing that help us, not change anything, or make things more complicated? — In the long run I want them to see that we can do anything, as long as we maintain equality, but not everything is beneficial.

  22. 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.

  23. Coming up with steps is similar to other “tricks” that we sometimes teach students (such as FOIL, cross multiplication, etc). It happens with the best of intentions… we as teachers look for the patterns and organize what needs to be done and presents that information. We try to make sense of what needs to be done and then pass along that understanding to the students. The problem is that all the interesting work (looking for the patterns) is done by the teacher, not the students, so the steps seem to lack meaning and purpose.

  24. 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.

  25. David Srebnick

    October 15, 2014 - 1:07 am -

    to Bowen Kerins: I disagree with your assertion that your incorrect cancellations are useless. The trick for 19/95 also works for 16/64, which could lead to some great number theory problems, like: are these the only two examples of these “incorrect” fraction cancellations that work for two digit numbers? Can you find other examples or show that they don’t exist? If you work with three or four digit numbers, are there any such incorrect cancellations that produce a correct result?

  26. 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.

  27. Chris Stephens

    October 16, 2014 - 9:00 am -

    “Incorrect but useful” is actually my favorite of the four zones you mention. I think “incorrect but useful” is an area the mathematical problem-solver lives in a great deal of the time.

    If you’re only doing things that are both correct and useful, then you already knew how to solve that problem. That’s fine, but it’s not interesting.

    If you’re doing things that are neither useful nor correct, then I’m not sure what you’re doing (or even why).

    If you’re doing things that are correct but not useful, then you’re generating trivia. Granted, this does happen sometimes in problem solving. The solver might think, “I know this is legal, but I’m not sure whether it helps…let’s follow it for a bit and see if it turns into something useful.” I personally find this to be more of a bland problem-solving process. It often works, but is more procedural than creative.

    But things that are useful but not correct–that’s where creativity lies, in my experience. “This is not true, but I wish it were; because if it were, then I could solve this problem like so… It’s not true, but I wonder if some part of it is true? Or almost true? Or if something kind of like it is almost true in a partial way?…”

    The deliberate pursuit of statements that are useful but not correct can lead to beautiful, surprising solutions.

  28. 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”.

  29. Just noticed a link to CMC14 in “Recent Comments”. Glad to see “My Favorite No” from the Teaching Channel in there. That is one of my favorites for building a culture where incorrect steps can be useful.