One of the things I try to get my (college) students to do is to start asking “what if?”.

Here’s how it works and we’ll start with something basic. There are several ways to teach single-digit addition — memorize those as facts, understand conceptually what addition is to mean, do drills, etc. And depending on your predilection you may agree with, despise, or tolerate one or more methods. My teaching tendencies are like an “all you can eat buffet”, a little bit of concepts, a little bit of memorization (through repeated use, rather than forced memorization through wacky mnemonics), a little of drills, a little bit of flailing, a little bit of guessing, a little bit of calculator work, etc. But if there is anything that I have a bias towards, it’s getting students to learn on their own by teaching them how to ask “what if?”.

It may sound simple — just ask “what if?”. And really it can be, but the hard part for a student is when they are on their own and they are left to contend with their own “what if?” questions. There are always resources online to help answer questions, but part of learning is being able to work on one’s own and solve one’s problems, be they be math problems, engineering problems, design problems, budget problems, grocery problems, relationship problems, etc.

To get a flavor of what “what if?” can do, let’s take something as simple as $$3 + 5 = 8$$ and begin to ask “what if?”.

• What if it were \(4 + 5\)?
• What if it were \(3 + 6\)?
• What if it were \(5 + 3\)?
• What if it were \(2 + 5\)?
• What if it were \(3 + 4\)?

With a “fact” as simple as \(3 + 5 = 8\), we can begin a first stage of inquiry by modifying the left-hand side to generate five new questions. You may have noticed, that the modifications were of the following forms:

• Increase 3 by 1 to 4
• Increase 5 by 1 to 6
• Swap the 3 and 5
• Decrease 3 by 1 to 2
• Decrease 5 by 1 to 4

One of the things that I learned while studying Electrical Engineering was “how to troubleshoot”. Our Engineering courses often had a lab portion where we would have to build some type of circuit. Inevitably, something would go wrong. We could have, literally, gotten our wires crossed, miscalculated the resistance and put in a wrong resistor, calculated the resistance correctly but put in a wrong resistor (because we misread the stripes), have faulty parts, had poor circuit design, etc. There were so many things that could go wrong, that if the circuit didn’t work as the specifications had wanted, one could easily get lost in troubleshooting.

Similarly, with programming, knowing how to debug one’s code was a skill in and of itself. As the software that one develops get larger not just in the amount of code generated, but the extra features that users request, it is easy to end up with tangled, hard-to-maintain code, which then becomes hard-to-debug code. Small changes can mean a colossal amount of code re-write if initial architectural designs were sloppy. But even with good architectural designs, people make mistakes, sometimes in the logic, sometimes in nuanced parts of the design, sometimes with communication with other team members, etc. But knowing how to efficiently resolve these problems can make the difference between having Frankenstein, spaghetti, yet working code and clean, readable, working code.

Mathematics is similar in this regard. When dissecting a math problem, statement, theorem, etc. we can, in essence, take two paths, one of uncontrolled flailing or one of controlled flailing. Uncontrolled flailing is just trying things at random with no reason why. Controlled flailing is asking pertinent “what if?” questions so that we can gain a deeper, more rich understanding of the object of study.

The “trick” if there were one, is to change one thing at a time. This is practically no different than how one would approach a controlled science experiment. Changing more than one variable at a time certainly has value, especially when those variables have correlated, non-linear effects, but that’s a complexity that one graduates to. Yet, the young student of mathematics (or any other subject), often takes an uncontrolled approach to problem solving. And unlike some other subjects where the feedback is in a real, physical, tangible form, in mathematics (especially in the non-applied cases), the feedback is abstract since we work with abstract objects.

If I made several changes to my code base, my circuit, my science experiment, etc., I would still be able observe the effects of those changes in the sense that whatever I am trying to fix is still broken, albeit in a different way. With mathematics, I will likely still have something broken, but the “it” is not really physical and I can’t just shake it to (hope to) make it work.

This is where it is helpful to teach students, young or old, how to debug / fiddle / play / modify their mathematical objects, in a scientific manner. This is asking “what if?” in a controlled manner. Uncontrolled flailing has its time and place, and later in this article I will explain how to use random modifications to one’s benefit rather than it becoming a source of math misery.

What if? with \(3 + 5 = 8\)

A few paragraphs above I gave some examples of what-if questions to ask for a statement as simple as \(3 + 5 = 8\). Here are those questions again, the modification we made in brackets, and their answers in italics.

• What if it were \(4 + 5\)? [Increase 3 by 1 to 4] We would have a result of 9 instead of 8
• What if it were \(3 + 6\)? [Increase 5 by 1 to 6] We would have a result of 9 instead of 8
• What if it were \(5 + 3\)? [Swap the 3 and 5] We would still have a result of 8
• What if it were \(2 + 5\)? [Decrease 3 by 1 to 2] We would have a result of 7 instead of 8
• What if it were \(3 + 4\)? [Decrease 5 by 1 to 4] We would have a result of 7 instead of 8

What I want to highlight with this basic example is one aspect of a general what-if process — namely, make one modification at a time from the original statement. What can we glean from this? We can see that if we increase (decrease) either the three or the five by one, our result increases (decreases) by one. Swapping the three and the five does not change the sum. No doubt, there are (many?) teachers who already do this in the classroom. The emphasis, however, is not about what \(3 + 5\) and its modifications result in, the emphasis is on getting students to do this by themselves so they know how to learn on their own — you know, how to actually derive value from homework (graded or ungraded, mandatory or optional) rather than it being an exercise in school bureaucracy.

Next, we can set ourselves up to perhaps hypothesize a theorem. We can identify the pattern that if we increase one of the values by one, then the sum increases by one. Is this always true for addition problems? Depending on what has been taught, the method of “proof” can vary. Though, in the strictest sense, we would want to insist on a mathematical proof like

If \(a + b = c\) then if we replace \(a\) by \(a + 1\) we would obtain \((a + 1) + b = a + b + 1 = c + 1\)

and even here, depending on how pedantic we want to be, we can complain about the intermediate steps that are omitted that allow us to drop parentheses and move the one around.

But this isn’t really the goal, for when working with \(3 + 5 = 8\) (the problem is so “basic” that either we are teaching it to students who are very young with mathematics or we are using this as an exercise in a formal course in mathematical rigor and analysis). If we were working a problem like this in an Analysis course, then we can harp on all the little details. But with students young at mathematics, we can get them to be convinced that indeed it is true that if one of the numbers on the left-hand side is increased (decreased) by 1 then the resultant sum is increased (decreased) by 1, through a bevy of examples (which sneakily gets the student to practice more of those “drills” that some educators despise in elementary math education). And yes, I know, proof by example isn’t proof.

Try this with \(4 + 5\), \(7 + 3\), \(9 + 9\), etc. A teacher, in fact, could give a basic template for modification (again depending on the student’s level of mathematical maturity, the subject matter, etc.) and ask students what conclusions they could draw. The educational follow-up would be in the classroom with the teacher.

What if? with \(5 \diamond 3 = ?\)

Previously, the modifications involved changing the numbers on the left-hand side and leaving the operator “constant”. What if we left the numbers on the left-hand side the same, but modified the operator? What if we changed \(+\) to \(-\), \(\times\), \(\div\), or exponentiation? What would happen to the output? Then once we’ve made the change to the operator, we can use the same technique of modifying one number at a time to see what happens with the result.

With subtraction we will find that where we increase (decrease) the number by one actually matters. Since we know that \(5-3 = 2\), then we’ll see that \(5 – 2 = 3\) (the result increased(!) by one) and \(4-3=1\) (the result decreased(!) by one). And clearly, swapping gives us a negative number! Now we can compare and contrast how subtraction is different from addition. We can also talk about what addition and subtraction mean (if we had already had that conversation, then this is a good reminder; if we hadn’t, then this is a great segue into that conversation) as well as those “big” words like associative, distributive, commutative.

Similarly, with multiplication we can see that if \(5 \times 3 = 15\), then \(6 \times 3 = 18\) (an increase of 3 from 15) vs \(5 \times 4 = 20\) (an increase of 5 from 15). And this can help to reinforce how multiplication works.

With division, things don’t work out as cleanly as they did with addition, subtraction, and multiplication, but that’s the whole point! Division is necessarily different and it is an operation that takes us out of the world of integers and brings us into the word of rational numbers. If we increase the denominator by one, do we have a larger value or a smaller value than the original problem? If we haven’t yet talked about fractions and are working with quotients and remainders, we can still compare the size of the new quotient and remainder against the original — for example, \(5 \div 3 = 1 r 2\) while \(5 \div 4 = 1 r 1\) where \(r\) is just the notation for “remainder” — thus, \(1 r 2\) is read as “1 remainder 2”.

Revisiting \(5 + 3 = 8\)

Once we’ve gotten a handle on changing one number at a time by one, we can see the effects of changing one number at a time by two, by three, etc. This can help to further generalize some conclusions that we came to when we increased (decreased) the left-hand side by one. We saw that with addition that increasing 5 by 1 increased the resultant sum by 1. What if we increased 5 by 2, by 3, by 4, etc.? Decreased it by 2, by 3, by 4, etc.?

And again, we can rinse and repeat this exercise for other operators.

Two moving parts

A next step in the what-if process is making multiple changes simultaneously. While the idea of “correlation” from a mathematical perspective may be too advanced to explain to students who are young in mathematics, the intuition behind it can likely be easily grasped. Correlation between inputs and outputs, in a loose sense, is observed “patterns” between inputs and outputs. What pattern(s) did we observe when we increased 5 by 1? decreased 5 by 1? increased 3 by 1? decreased 3 by 1? when considering addition? subtraction? multiplication? division?

For sake of discussion, I will stick with our addition problem of \(5 + 3 = 8\). A pattern we noticed was that if we increased (decreased) either the 5 or 3 by 1, the resultant sum increased (decreased) by 1. If change were by 3, then the resultant sum changed accordingly by 3. Thus, we can see that there is a correlation between changes in inputs (the 5 or the 3) and changes in outputs (the 8, initially).

Well, for a next layer of what-if, we can ask, “What if we increased 5 by 1 and 3 by 1?”, “What if we increased 5 by 1 and decreased 3 by 1?” (Incidentally, if we have two inputs and we are allowed to increase or decrease either one, how many different cases do we have? This is a great counting problem for elementary math students! Can they do this in an organized manner?)

If we omit swapping as a case to consider, and we keep only the cases where both inputs change, then we have

• What if we decrease 5 by 1 and decrease 3 by 1?
• What if we decrease 5 by 1 and increase 3 by 1?
• What if we increase 5 by 1 and decrease 3 by 1?
• What if we increase 5 by 1 and increase 3 by 1?

If you come from, say, an Electrical Engineering background, you may think in terms of Karnaugh maps and insist that an organized way of asking these questions would be that from one question to the next, the change should be by only one of increase or decrease but not both. While, perhaps someone from a Computer Science background will view the above sequence of questions as logical since we really just have a binary system (decrease maps to zero, increase maps to 1).

An Electrical Engineer, may have asked the questions in this manner:

• What if we decrease 5 by 1 and decrease 3 by 1?
• What if we decrease 5 by 1 and increase 3 by 1?
• What if we increase 5 by 1 and increase 3 by 1?
• What if we increase 5 by 1 and decrease 3 by 1?

A broad lesson on the side here is that if students can learn to ask questions in an organized manner, then they are more likely to be able to cover all scenarios. How many scenarios do we have if we have two inputs, and three possible states (no change, decrease, or increase)? How do we enumerate them in an organized manner?

If it is too early to talk about radix systems, then at the very least, we can help to build intuition about how it ought to work and worry about the formalities later. For example, in the three-possible-states scenarios, we really have a base three system, with “no change” mapped to zero, “decrease” mapped to 1, and “increase” mapped to 2. Thus, if we count in base three, we have

• no change 5, no change 3
• no change 5, decrease 3
• no change 5, increase 3
• decrease 5, no change 3
• decrease 5, decrease 3
• decrease 5, increase 3
• increase 5, no change 3
• increase 5, decrease 3
• increase 5, increase 3

Many of my adult students are not able to identify all cases in a problem like the above. This makes it harder to gain an understanding of what they are studying. Thus, in the absence of an organized approach, the student, adult or not, will, by default, ask random what-if questions and this results in confusion, frustration, and a general sense of insecurity in one’s exploration. Did I get everything? What if I missed a case? How do I know? And this eventually just becomes uncontrolled flailing.

Uncontrolled Flailing

One of the things I like to do when studying something new (often times mathematical), is that I like to make random, willy-nilly changes (uncontrolled flailing). I can derive great value from this because what I am really doing is I am just building a broad, superficial understanding of the object of study. I am not interested in understanding the minutiae just yet. What I am trying to get a sense for are things like “if I make big changes to the input(s), will my output(s) have big or small changes?”, “how is my computational performance affected?” (technically I can answer this question by basic algorithmic analysis, but in the case of something somewhat involved, I’m not terribly interested in this up-front time investment), etc. In other words, I am basically treating my object as a black box and “probing” it. This is how I fiddle. Once I feel that I’ve developed some comfort with the object of study, then I sit down to study it.

I encourage students to just “mess around”. But equally so, I encourage students to make a systematic study of what they are working with. For next week, I’ll write about the what-if process beyond just arithmetic. The principles, broadly are similar, but there are subtle nuances especially when working with theorems.

In any case, the point of teaching students how to “what-if” is so that they have some sense of control and organization when they work by themselves. The classroom teacher cannot / should not be omnipresent in a student’s academic engagement — it is just too much of a crutch. The student has to be able to self-correct, self-inquire, and self-diagnose in a learning context. In other words, by giving students an organized manner to tackle problems, we’re giving them an organized and hopefully efficient method of learning.

If you’ve been following along with my blog, you know that I’ve been learning to draw and have been working on a weekly web comic. When I first started to learn to draw, I was in my uncontrolled flailing stage. I was still getting used to working with pencil and paper in the context of drawing. Now, I’ve slowly but steadily begun to take a more systematic approach to learning to draw. Nevertheless, I still take some time to just doodle.

Here’s part of a doodle for this Friday’s comic: