There Is No Wrong Answer In Math

On Saturday November 22nd, I led a session at #edcampnj titled “There Is No Wrong Answer In Math”. As has been a common theme with this session, it quickly devolved into a #rantchat about what’s wrong with math education and we never really got around to the topic. Though I was able to make a few points.

For those unfamiliar with the edcamp-style unconferences, the session set up is in effect a large conversation. There is a session leader, however, the interaction is not that of presenter-audience.

The purpose of this article is to recap some of the conversation that took place, continue some of the threads that had been started, and to get a little deeper into what in the world I could mean by “There Is No Wrong Answer In Math”.

I started the session with the following question:

$$\mbox{What does } a^{2} + b^{2} = ?$$

And almost in unison, I heard “$$c^{2}$$” from the crowd. But I had drawn no triangle. I had given no context. Yet, the answer was somehow $$c^{2}$$. Why?

Herein lies one of our first problems. Context. Assumed context, implied context, unspoken context. $$a^{2} + b^{2} = c^{2}$$ without context. Would I have been wrong if I said that $$a^{2} + b^{2} = z^{94}$$? Of course not! There is no context. The “answer” is anything I want it to be.

My memory is a little fuzzy about the exact sequence of events, but it went something like this (and I will be paraphrasing heavily). I gave an example similar to this:

I was told of a story of students who, when asked to compute $$34 + 12$$, correctly answered $$46$$. However, when asked to compute
\begin{align} &34\\ +&12\\ \hline & \hline\\ \end{align}
gave the answer as 10! How is this possible?

Many of the audience members looked perplexed. And then after about 30 seconds or so, someone chimed in with $$3 + 4 + 1 + 2 = 10$$. Bingo!

And thus began the short-lived foray into “there is no wrong answer in math”. Is the student wrong to say

\begin{align} &34\\ +&12\\ \hline &10\\ \end{align}
?

We can easily argue that this is clearly wrong. The addition operator is not digit-by-digit operation; rather, it is supposed to work in a place-value context. Is it really though?

The plus sign “$$+$$” is an overused operator. We see it in marketing like $$Me + Math = Misery$$. We see it with string concatenation in many languages ” “hi” + “there” = “hithere” “. Heck, it is also used for list concatenation (eg, Python): “[0,1,2,3] + [1,2,3,4] = [0,1,2,3,1,2,3,4]” and languages like C++ and Python allow one to overload the plus operator. Why, we even see the $$+$$ operator used differently (and in multiple contexts) in the addition of matrices. For example $$\left( \begin{array}{cc} 1 & 2 \\ 3 & 4\\ \end{array} \right) + \left( \begin{array}{cc} 5 & 6\\ 7 & 8\\ \end{array} \right) = \left( \begin{array}{cc} 6 & 8 \\ 10 & 12 \\ \end{array} \right)$$

Notice that the $$+$$ has two simultaneous(!) uses — one is to denote the addition of matrices and the other (not shown) to denote the addition of the elements of the matrices. And of course, even preceding matrices, we have fraction arithmetic, which uses $$+$$ differently. We also have functional addition $$f(x) + g(x) = (f + g)(x)$$

So where is it, why is it, that stacked addition yielding an answer of 10 in the above example is wrong? It is absolutely not wrong.

The critical matter here is how do we discuss this with students?

The approach that many teachers take is to give “prescriptive correction”, meaning “Hey, that’s not how you add. This is how you add.” That’s straightforward, it doesn’t mince words and draws a clear line between right and wrong.

There is a time and place for this.

I offer a different solution. With emphasis on different.

What I have often found is that students are using logic — their logic — consistently to “solve” math problems. (I know, I know … students just randomly try things, or if they see a bunch of numbers in a word problem, they’ll just add all the numbers, or they’ll just write a bunch of things to hopefully get partial credit, etc. I’m not naive to this. And for those cases, there is a simple way of handling this to be discussed later.)

Let’s stick with the vertical stacked calculation
\begin{align} &34\\ +&12\\ \hline &10\\ \end{align}

If a student is adding in this way consistently, then rather than prescribe “no that’s wrong”, I prefer to logically lead the student into the correct method. It’s all about context, communication, and a common language.

I try to go as long as possible avoiding telling a student that their method is incorrect, especially when the student is consistently using the “incorrect” method. I prefer to steer clear of the word “wrong” or its synonyms because by the time students get to me (college), they are already terrified of math. The word “wrong” just adds another stab wound. I have to first undo their math trauma from high school.

Begin devolution of the session.

A high school math teacher chimes in and quips (I’m paraphrasing all these quotes), “I have to deal with the fact that they don’t know 8th grade math!”. An 8th grade math teacher interjects, “But my kids can barely add fractions! How can they do any Algebra?”. A 6th grade math teacher says, “But my kids freeze when they see fractions! They can solve problems with whole numbers, but can’t with fractions!”. Another teacher says self-assuredly, “The problem is that grade school teachers have math anxiety and are not well-versed in mathematics. They project their fears onto the students.” Then a first grade teacher (@btcostello) drops the bombshell, “I have students with math anxiety.”

Awww come on!! First grade! Math anxiety?!!?! How in blue blazes does that happen??? How is that even possible? But it is! And this runs counter to the claim that grade school teachers are the sole culprit (I know I’m citing one example, but I have a point somewhere in here).

I followed up with a story that a piano teacher told me. She told me that she asked an 8-year old student, “How many quarter notes can you play in the same amount of time it would take for you to play a whole note?” (same time signature). And the student looked back, somewhat defeated and said, “Is this … math?” to which the response was, “Yes.” And in a dramatic display, the student collapsed onto her bench wailing, “Oh nooooo!”. The saddest part about this story is that the mother of the student says from across the room, “Math isn’t her thing.”

Math isn’t her thing.

Think about that for a second. The girl is eight years old and her mother openly and plainly says “Math isn’t her thing.” I wonder where this girl is getting her math anxiety from / reinforced? Let’s suppose, for argument’s sake, that grade school teachers are math anxious and just muddle through the subject. I believe this happens. I have heard stories of first and second grade teachers not wanting to teach fourth grade because they wouldn’t feel comfortable teaching fourth grade math. Fourth grade math! If an 8/9/10 year old is supposed to learn this stuff, then so should any grade school teacher.

But the problem I have with pointing the finger at grade school teachers for our nation’s math woes is that it just seems implausible for it to be true on such a grand scale. The narrative is effectively saying that first, second, third, fourth, and fifth grade teachers, across the nation, overwhelmingly have so much math anxiety that students for five consecutive years, in practically any school district one goes to, get the joy of math beaten out of them. I think that’s an easy tale to buy into — and I am well-aware that there is research that supports the argument that grade school teachers are the (a?) root cause for math anxiety.

But let’s do some simple calculation here. Let’s say that first, second, third, fourth, and fifth grade teachers each have a 50-50 chance of having this pedagogically debilitating math anxiety. This means that there is a 1 in 32 chance that a school would have all five teachers with math anxiety. A 15.625% that four of five teachers would have math anxiety, 31.25% chance that there would be three of five teachers with math anxiety, 31.25% for two with math anxiety, 15.625% for one with math anxiety, and 3.125% for none with math anxiety. So what does it take to ruin a student in math, for the rest of his / her life? One bad teacher? The other eleven have no chance of fixing it? Two bad teachers and the other ten are helpless? Three years of rote math instruction defeats nine good years?

I don’t know. This smells a little fishy.

Here’s a table of the likelihood that there are $$N$$ math-anxiety teachers in first through fifth grade, given that the probability of a math-anxiety teacher is $$p$$, where $$p$$ ranges from 50% to 90%.

$$\begin{array}{c | r | r | r | r | r |} \mbox{Number of Bad Teachers} & p = 50\% & p = 60\% & p = 70\% & p = 80\% & p = 90\%\\ \hline 0 & 3.125\% & 1.024\% & 0.243\% & 0.032\% & 0.001\%\\ 1 & 15.625\% & 7.680\% & 2.835\% & 0.640\% & 0.045\%\\ 2 & 31.250\% & 23.040\% & 13.230\% & 5.120\% & 0.810\%\\ 3 & 31.250\% & 34.560\% & 30.870\% & 20.480\% & 7.290\%\\ 4 & 15.625\% & 25.920\% & 36.015\% & 40.960\% & 32.805\%\\ 5 & 3.125\% & 7.776\% & 16.807\% & 32.768\% & 59.049\%\\ \end{array}$$

Is this what research has found? I actually don’t know, I really am asking. If we had a 90% likelihood that first through fifth grade teachers had math-anxiety to the point that it would be harmful to the student, then a school district with one teacher per grade level, would have a 99% likelihood of having three or more math-anxiety teachers. And even if that were the case, are we saying that the remaining seven to nine teachers that a student interacts with (for math) are completely unable to help? Or are they also shuffling the problem upwards, unknowingly? Are we also saying that these three to five math-anxiety teachers affect their 15+ students all negatively to the point that there is no recovery? Even in what I consider as an extreme case of 90% chance of being a teacher with math anxiety, are we saying that there is a 100% chance that they will create math anxiety for every student? No one gets out alive? Maybe if you grade like this, but that’s an 8th grade teacher who doesn’t seem to show math anxiety, but rather math prescription.

I just have a hard time believing this; and if the unbelievable scenario is actually true, then if you are a superintendent, principal, teacher, etc., just get in touch with me. I will gladly offer my services as a mathematician and provide rock-solid, teacher training. I have plenty of teaching experience, I teach mathematics to adults (not just the 18-21 crowd — I also work with the 30-60 year old crowd. I know how to communicate math to them and I get them to teach it to others. That’s how I run my courses!), and I keep a foot in industry (so I can give “real” world, practical contexts, if we so desire). Get in touch.

Here’s what I think happens

I think that from first grade onwards, students are routinely told, “No, that’s wrong. This is how you do it. Now do it the way I just said.” Like I said earlier, in some sense, there is nothing wrong with saying this. It is clear. But what I am suggesting is to logically induce the “correct” approach.

Let’s go back to the stacked addition of $$34 + 12$$. If you and I agree that stacked addition is to be a short-hand for digit-by-digit addition, then you and I will have no qualms with the result of 10. However, with whom else will we be able to communicate? No one! We will first have to explain to anyone else that “hey, stacked addition is to mean digit-by-digit addition”. Published math papers in peer-reviewed journals do this. They have to define their notation and usage, especially if it is non-standard and in some cases, even if it is standard, but competes with another standard notation. Some math papers I’ve had to read in French would write $$]0,1[$$ to denote the open interval $$(0,1)$$. In fact, pick up any respectable math text and you will see definitions and examples of symbolism.

What we have in our math classrooms, especially, from 1st to 12th grade and even the first few years of undergraduate college is a fairly well-standardized (yes, “well”) math course curriculum with standard usage of symbolism. It is, in some sense, an amazing triumph of logistics (maybe linguistic oppression?) to have near-global acceptance of basic mathematical language.

However, much like the alphabet, the mathematical symbolism is new to (grade school) students. The difference, however, is that we show little, if any, patience with a student’s misuse of mathematical symbolism while relatively, there is an abundance of patience with learning the alphabet, spelling, grammar, etc. This unwitting lack of patience is, to me, a root cause of math misery. We are simply not teaching students how to read mathematics, how to write mathematics, let alone letting them explain their own context! Read more about that here.

There’s nothing wrong with $$34 + 12$$ yielding 10. Yet, what we really want is that students work out the addition in the context of a place-value system. Thus, when a student does give $$34 + 12 = 10$$ when the problem is written as stacked addition but will give $$34 + 12 = 46$$ when the addition is written horizontally, the problem isn’t a lack of understanding of what addition is, it is a lack of understanding of the symbolism! (I grant there may be other problems, but addressing the symbolism is paramount. The teacher may also want to address how clearly the student understands place-value.)

What’s interesting is that in the #edcampnj session, we never actually discussed how to address the problems that virtually all teachers of math have — “My students don’t know the prerequisites.” There were several, obviously frustrated (angry?) teachers present. I understand this. This is why I do #rantchat on Twitter! And this is why I let the frustration come out in the session. One teacher complained that students confused “hypotenuse” with “hippopotamus”. Another explained her plight when teaching Geometry — students didn’t know basic geometry facts. I complained that students can solve for $$x$$ in $$x + 3 = 7$$ but can’t with $$x + \frac{1}{3} = \frac{1}{2}$$.

Someone suggested why not just go back to basics? — Use blocks to teach fractions. This was immediately countered with “at some point, we have to stop hand holding and work with the abstraction. When should that happen?” To which the counter was “When they are ready.” And this, understandably, didn’t sit well with some (many?). Who has this time? Teachers blamed both their current students and their current students’ previous teachers and so in a circle we went.

There are a few simple questions to ask to a student who is working out math “incorrectly” before we blurt out “that’s wrong”.

• Who else will understand your method / notation?
• Why? (Why did you choose the operations you chose; why did you choose the mechanics you chose; etc)
• Under what circumstances can your method work? Is your method general, extendable?

Additionally, there are a few simple things for a teacher to do to drive home the point.

• Provide a counter-example to the student’s method.
• Provide an example that highlights the broader efficiency of the method taught in class versus the student’s method.
• Provide an example that shows that the student’s method is not scalable or works only in special cases (hence generally inefficient, but perhaps locally efficient).
• Say, the student is having difficulty with $$x + 3 = 13$$. And they keep getting $$x = 16$$ because for whatever reason the idea of subtracting three from both sides doesn’t make sense. Then “flip the question”! Give me an Algebra problem where $$x = 16$$.

The basic thesis here is that there is almost always some context where the student is correct and rather than pounce on their mistakes just keeping stepping back until you and the student are in a place where the student is correct or the two of you are axiomatically in opposition. This is what I mean by “logically inducing” the correct method. In a more abstract sense, this is called “proof by contradiction” — a standard logical method for showing absurdity. And proof by contradiction typically begins with “Suppose the statement is true …”

$$34 + 12 = 10$$? Ok, in what context? Ah, digit-by-digit addition. Do we mean digit-by-digit addition when we use $$+$$? If the student says, “Yes.” Then a simple answer is, “ok, if you want to use it that way, you can, but remember we have to try to communicate with the rest of the world. How does everyone else use $$+$$ in this context?” That ought to right the ship for all but the most recalcitrant of students. If the student says, “No.” then bingo! We’ve opened the door for a simple explanation of how it is used.

$$x + 3 = 13 \implies x = 16$$? Why? How can you be sure that you have the correct answer? Can you give me another problem where the solution is $$x = 16$$? And from here we can continue to walk them down the path until we have reconstructed the problem $$x + 3 = 13$$ and have shown that $$x = 10$$. Odds are, if you are the teacher, you’ve already explained several times that we need to take away three from both sides to obtain ten. But if the student is still making that mistake, it’s just craziness to try explaining it the same way! Because really what’s happening is that you and the student aren’t speaking the same language — the student doesn’t understand why what you are saying is correct, because he/she is still trying to reconcile why what he/she has done is incorrect. All the student hears is, “that’s not how” when they need to hear more of “that’s not why”. And more specifically, they have to hear it through their own speech.

Find the area of a circle with radius of 5 units. The student answers $$10\pi$$ units squared. Are they wrong? Absolutely! But what are they wrong about? did they use $$2\pi r$$ as the formula for the area of a circle (the wrong formula) or did they use $$\pi r^{2}$$ but thought that $$5^{2} = 10$$? If they used $$2\pi r$$ instead of $$\pi r^{2}$$, it’s a simple matter of changing the question to “Actually, can you tell me what the circumference of a circle is?” and wham! Most students will realize that they answered the wrong question! If they misunderstood how to compute $$5^{2}$$, then just ask a different question: “Actually, what is five times two?” and wham! again either you’ll ask the follow up question as “What is five raised to the second power?” or they’ll see what they did wrong. And you, the teacher, will have corrected the student without correcting them and you will also have snuck in a few other problems. It is this line of questioning that helps to sort out the facts that students have jumbled in their heads.

If a student confused “hypotenuse” with “hippopotamus” or “hypoallergenic” or “Hypollyta” or “hypotangent” or anything like that, that’s not a big deal! They’re actually on the right track. It’s altogether different if they confused “hypotenuse” with “telephone”. There’s a very easy way to correct this without belittling the student or getting yourself worked up. Just ask them to look the word up in the dictionary. By simply saying “no, it’s hypotenuse, not what you said” all we’ve done is fix their mistake for them. They’ve learned nothing. What if you were asked to recall the name “Mahalanobis” (there is a distance measure called “Mahalanobis” distance)? Is it outlandish if next week you tried to remember the word and said “Taj Mahal”? or “Anubis”? or “Mahalo”? They’re all close. No need to get angry at the student. See last week’s comic about closeness of words.

Are students confused about anything in the cross product of {fraction, exponent} $$\times$$ {addition, subtraction, multiplication, division}? First, if they simply have no idea how to do it, then, of course, show them. However, if they are consistently using an incorrect method, there are several things to try. Choose simpler, but similar problems and ask the student how they would work those out. If the method is different (and one of the methods is correct), then a simple compare and contrast of their own work against their own work shows the contradiction. If the methods are the same (and incorrect), then a few well-placed “why”s and some simple tangible examples help to get things back on track. Asking the student to construct a tangible problem where their method is correct is also valuable as the teacher can then (hopefully) easily construct an example where the method breaks down. Again, what’s important is that the student speaks the contradiction, not you, the teacher.

Sometimes students just guess and answer correctly! This too is easily handled with “Hey, how easy is it to extend your guessing method?” Most students, when they guess, know that they are just guessing. They know it isn’t extendable. But why are they guessing? It’s for points. It’s better than trying something, being wrong, and then having their method criticized. They will welcome the criticism on guessing because they are ready to accept that criticism — they already know it’s not how it’s supposed to be done and they too accept that that is actually wrong. I’ve often said to the student who guess / admits to guessing that “it is better to write only what you know / believe to be true than to just hope”.

I had a student who wanted to integer division his way. He always obtained the correct answer, but his method was inefficient (it was a somewhat long, convoluted counting method). So rather than try to force him to learn one of the standard division algorithms, I just constructed a slightly larger problem involving a four-digit number divided by a two-digit number. He finished fifteen minutes later obtaining the correct answer. Then, I showed him a standard long division algorithm explaining all the steps and I asked him if he wants to spend fifteen minutes or 30 seconds? He corrected himself.

Some of my Algebra students are stubborn about factoring. They will insist that to factor $$x^{2} + 4x – 12$$ they must conjure two numbers that multiply to negative twelve but add to four and eventually they find that the factorization is $$(x + 6)(x – 2)$$. So, I ask as a follow-up, how would we factor $$x^{2} + 4x – 13$$? Clearly, the same method can apply, but is it obvious how to find two numbers that multiply to negative thirteen but add to four? Some students gave up on their original method and learned how to complete the square as a general method for factoring quadratics. Others remained stubborn refusing to learn. Then we began to cover parabolas and more students converted with a few students remaining entrenched.

And why did some students remain entrenched? It’s because they had taken Calculus and had learned how to find the vertex by taking derivatives. Except they couldn’t remember how. Yet, they remembered it was “easier”. So, since they knew that an easier way existed, they refused to learn any other way and they made no attempt to go back through their Calculus notes to figure out what the easier way was. And some times, despite the best efforts of the teacher, some students just don’t want to learn. And this brings me back to the original commotion … whose fault is it anyway? Sometimes, it really is the previous teacher. Sometimes it’s the parents. Sometimes, it’s the students. Sometimes, it’s systemic. Let’s not fall into the classic trap of assuming that when there is a problem that it was caused by one and only one group.

The point of “there’s no wrong answer” is to show students a general framework for self-correction and self-confidence. “Plugging it back into the original problem” is a good way to check one’s work, but it’s completely useless if the method(s) used are consistently incorrect. I aim for positive reinforcement, by letting students explain their reasoning, showing alternatives, and letting them come to math on their own terms. It takes a little longer, requires more patience on my part, and a little more work on the student’s part, but once students learn to read and write mathematics in a standard way, learning mathematics gets a lot easier.

I merrily let my students go down the path they want to take and when things are “wrong”, my phrases are “almost”, “close”, “what happens if we make this small tweak?”, “suppose I asked this question instead …”, etc. Is there a time and place to simply say, “That’s wrong.”? Of course! It just can’t be done in a mean, hostile, belittling, or always prescriptive way. My recommendation is to save that phrase for when we’ve reasonably exhausted the (more encouraging) line of “tell me in what context you’re right, tell me how you can extend your method, and tell me if the rest of the math community can understand what your notation / symbolism is”.

[EDIT (11/25): Please check out @btcostello’s reflections.]

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3 thoughts on “There Is No Wrong Answer In Math”

1. Scott Richards

Enjoyed the read and the thinking it evoked. I appreciate learning that others are having and considering similar questions, even if I only have two remaining gray cells and one of those is blinking ylmodnar. I am looking forward to my next EdCamp in January to hopefully facilitate a similar math dialogue. Although I opening embrace technology, especially that which directly relevant to helping my Studetns learn, sharing a room with peers exploring these type of questions would be greatly welcomed.

Continuing the evoked thought,
Scott