# Guess-Check-Generalize and the Scrubbing Calculator

Several other blogs have been talking about Bret Victor’s Kill Math website, including its Scrubbing Calculator.  I’d like to talk about how the Scrubbing Calculator is both very similar to and very different from an approach to solving word problems we call “Guess-Check-Generalize”.  Here’s a graphic from a sample problem solved Scrubbingly.

The challenge is to find the height of each bar, given the information about other heights.  When I first taught Algebra 1, my approach to this would be to get students to “translate” the problem into algebra, trying to get them to write an equation that would be true for the right height.  And the results were a mixed bag, for a lot of reasons that might be good for a different post.  I think there’s something inherently challenging about trying to write a fully symbolic statement immediately from a problem situation.

The concept of guess-check-generalize starts by changing the nature of the problem.  The question to start with changes:

from What is the correct bar height? …

to Is 100 the correct bar height?

Here, 100 could have been any number at all, it’s a total guess.  (Some teachers using this method ask students to write down their first guess before even presenting the problem, since students may be afraid to guess incorrectly.)

Now we see if the guess is right.  Up until now, I agree completely with the philosophy of the Scrubbing Calculator: make a guess at the bar height, then see if it’s right.  This is where things get interesting, because there’s more than one way to check the guess.  The most conventional way is to add up the heights on the right side, and a student might do this:

60 + 100 + 20 + 100 + 20 + 100 + 20 + 100 + 20 + 100 + 20 + 100 + 20 + 100 + 20 + 100 + 20 + 100 + 140 = nope

It doesn’t actually matter what that equals, as long as it doesn’t equal 768.  Guess-check-generalize is about determining a process you can use to check any guess; then, the process you’ve described becomes an equation to solve.  And the process can evolve from one guess to another, as students realize they’ve used the same number 8 times or that this thing is twice that thing.

So 100 was wrong; take a second guess.  It doesn’t have to be a better guess, because you’re not trying to nail the numeric answer, you’re trying to nail the process of checking a guess.  Let’s guess 36.  Checking this guess a student might notice they could combine some terms from before:

$60 + (9 \cdot 36) + (8 \cdot 20) + 140 = 684, \text{nope}$

No more guessing.  The third guess is $h$, a variable.  (Students may need more guesses, especially at first; eventually some only need one or zero guesses.)  Take all the places the guess was found and replace them with the variable, noting that the correct guess yields 768:

$60 + (9 \cdot h) + (8 \cdot 20) + 140 = 768$

Solving that equation and bringing the answer back into context are still issues, but I always found the largest difficulty with the dreaded “word problem” is an inability to take the situation and make a mathematical statement about it.  When almost every real mathematical situation an adult encounters is a “word problem”, this is a major issue that needs to be addressed.

Here’s why I think guess-check-generalize is a good way of dealing with word problems.

• The method is general in nature.  The method presented here works equally well for linear and nonlinear situations, for problems with a variable on each side of the equation, for rates, coin values, painting houses, counting beans, whatever.  This is a general-purpose tool that is useful over many years, including some surprising topics like generating the equations of lines and circles.  (More on this some other time.)
• This is what people do with problems.  When a problem is new or overly complicated, picking a few cases and following them through leads to an understanding of what happens in general.  Traditional word problem methods expect students to have the generalization at the ready, and it just doesn’t work that way in reality.  The concept of generalizing from repeated example is a fundamental one that all students should learn, not just those heading into STEM careers.
• Students have a simple place to start from.  By asking students to guess at the answer, the difficulty level of word problems can be reduced by 2 or 3 grade levels immediately.  Students with language difficulty can learn what is happening by calculating with numbers, connecting the new language to the calculations they know, then advancing to symbols when appropriate.
• There are no black boxes.  Students construct equations and can understand where they come from.  Multiple equations with the same answer can be found from different techniques used on the same problem, leading to good discussions about the basic moves of algebra and how different equations and formulas are related.
• Connections between arithmetic and algebra are reinforced.  Bret Victor says this: “We are accustomed to assuming that variables must be symbols. But this isn’t true — a variable is simply a number that varies.”  I’d like this to change.  Too many students only see variables as symbols for manipulation, and not as numbers that vary.  Students make mistakes with variables they would never make with numbers.  When this happens, it is because they don’t see that the symbol represents a number.  Since arithmetic is at the heart of guess-check-generalize, students are asked to solidify their number skill and sense.  Students begin to guess “nice” numbers, like a multiple of 3 when they see that dividing by 3 will be part of the process.

It is on this last point that I disagree deeply with the philosophy of the Scrubbing Calculator; students don’t really do any of the calculating.  In the end, a student might see that the answer produced by Scrubbing works, but if there is more than one answer, there’s no way for a student to discern this.  If the problem changes slightly from its original form (say, to a 1024-high screen), the Scrubbing solution method is to start from scratch, which doesn’t help students generalize toward functions and formulas (in this case, a relationship between the screen height and the bar height).

What if the correct answer to the equation is $\sqrt 2$ or even $\frac 2 3$?  I don’t see how the Scrubbing Calculator could get these answers.  I agree that too many students don’t see the real meaning of a variable, but this is no reason to ditch symbolic algebra, this is a reason to make the connections between arithmetic and algebra as strong as possible, as often as possible.

The Scrubbing Calculator’s method is an opportunity for students to make deep connections between arithmetic and algebra, between real problems and symbolic algebra.  I’m disappointed that its intended purpose is to remove symbolic algebra altogether, because it could be pretty cool.  What do you think?

For homework, solve this problem using guess-check-generalize or come up with a better one.  No scrubbing, please!

Nancy takes a long car trip from Boston.  In one direction she drives at an average speed of 60 miles per hour, and in the other direction she drives at an average speed of 50 miles per hour.  She’s in the car a total of 38 hours for the round trip.  How far from Boston was her destination?  (Bonus: what city did she drive to?)

Bowen is a mathematics curriculum writer. He is a lead author of CME Project, a high school curriculum focused on mathematical habits of mind, and part of the author team of the Illustrative Mathematics curriculum series. Bowen leads professional development nationally, primarily on how math content can be taught with a focus on higher-level goals. Bowen is also a champion pinball player and once won \$1,000 for knowing the number of degrees in a right angle.

### 11 Responses to Guess-Check-Generalize and the Scrubbing Calculator

1. jim says:

I say Atlanta at 1045 miles, but I’m just guessing.

2. Sean says:

Great stuff here.

Some questions:

1. How do the majority of students get from guess 2 to step 3 without explicit guidance? Most students- when guessing and checking- continue doing so, unaware that there is something to generalize. When it’s modeled like step 3, some students check out. Their intuition and number sense has been preyed upon. Now it becomes a ‘math class’ problem.

2. Why does the first guess have to be wrong? When the first guess is right – through coincidence or refined number sense- a whole new set of questions come out like: ‘Is that the only answer?’ ‘Can we prove that it’s the only answer?’ ‘Can you provide answers that are wrong?’ ‘Is there a visual representation for why it’s the only answer?’

3. Bowen Kerins says:

Thanks for the questions!

1. The reality is that it will take more than two guesses for the first such problem. Ask students to continue focusing on keeping track of their steps, getting the “rhythm of the calculations” (as Al would say). What I’d want to hear is a student who gets sick of guessing and says “Stop it already! Whatever number you use, it’s just going to be…” That kid is ready to generalize. I have seen it taught explicitly by asking students to take three guesses, then the fourth guess is a variable. I feel it should be up to the student to decide when to generalize, which may at first take many guesses.

You can also “preload” this behavior by using the same tactics when building expressions for things like “3 less than a number” — do “3 less than 20” then “3 less than 50” until “3 less than n” makes sense.

2. These are great, great questions, and well worth asking. I also feel that if a student can determine the correct answer by some means, they shouldn’t be required to also create an equation to solve the same problem. It’s math for no purpose at that point. It’s good to present a mix of problems that have ‘nice’ answers and ones that deliberately do NOT have nice answers. The first problem we present using this method is one from Benjamin Banneker, we call the “four fours” problem:

There’s this number. When you add 4 to it, subtract 4 from it, multiply it by 4, or divide it by 4, you get four different answers. That’s not too interesting, but the four different answers add up to exactly 60. What is the number?

(Banneker’s original phrasing flips the question: Divide 60 into four such parts that the first being increased by 4, the second decreased by 4, the third multiplied by 4, the fourth part divided by 4, that the sum, the difference, the product and the quotient shall be one and the same number.)

I like this problem because it’s simple to take a guess, students should eventually be generalizing, and the equation is relatively necessary to find the answer. Plus, the problem is over 200 years old!

4. Another advantage of this method is that it puts metacognition right into the process from the beginning: because students start by asking the question “is this actually right?” and maybe “how far off am I?” I think they would be more likely to ask those questions at the end, too. Good stuff!

5. Christopher says:

Chattanooga. She wanted to ride the Choo-Choo.

Here’s my concern with the scrubbing calculator-different from yours, I think.

I don’t get how the scrubbing calculator helps the student who is struggling with the set up. In the original post I read about it, Bret writes:

This is a simple problem, but it’s not obvious. It typically would require either writing out and solving an equation:
2910-x=426+x
or recognizing the “trick” that we have to split the difference.

Then he sets up two equations, 2910-1000=1910 and 426+1000=1426 and “scrubs” until they are equal. I’m fine with this, but I find that for this problem, the same insight is required for either technique. Namely, that we need to add to one person’s total and subtract from the other one.

This seems like a fundamental structural insight. In my experience with community college developmental mathematics, this is the insight students struggle with and not so much the solving of the equation. So let’s just be clear about what the scrubbing calculator does and what it does not.

When Bret alludes to the calculator being an alternative to using division and subtraction to solve an equation, I agree. And I’m curious about the consequences of the tool from that perspective. But he seems also to imply that the scrubbing calculator is an alternative to setting up an equation. And I disagree. I think we need the same structural insights into the problem to set up a scrubbing solution as a symbolic algebra one. Not that there’s anything wrong with that. I’m just not sure the scrubbing calculator really solves the pedagogical problem it claims to.

6. Bret says:

@bowen “If the problem changes slightly from its original form (say, to a 1024-high screen), the Scrubbing solution method is to start from scratch, which doesn’t help students generalize toward functions and formulas (in this case, a relationship between the screen height and the bar height).”

You may have missed the section on unlocked numbers, which addresses this very issue. If scrubbing bothers you, take a closer look at unlocking to see how it allows you to turn any number into a variable, and solve for it without scrubbing at all.

@christopher “I’m just not sure the scrubbing calculator really solves the pedagogical problem it claims to.”

Believe me, I have never claimed to address any pedagogical problems whatsoever. My interest in these tools is purely practical. Most people I know (adults, solving problems they care about) have no trouble with the insight that we have to add to one person’s total and subtract from the other — that’s what it means to pay for something. But these people won’t go near anything with an “x”, and don’t know or care about “moving terms to the other side of the equation”.

7. Bowen Kerins says:

Thanks for the comments, Bret. My first thought is that philosophically we are a lot closer than I thought! I agree that there is a disservice to a large number of students when they are taught only symbolic algebra. I feel that many of the real purposes of learning algebra are ones that can and should apply to adults solving problems they care about: generalizing from examples, looking for structure and similarities between different kinds of problems, reasoning about and picturing calculations, and more. These are what an algebra course should really be about.

I feel that symbolic algebra, and the connections between arithmetic and algebra, are critical to a deep understanding of algebraic habits of mind — and that these habits of mind are what school mathematics should be about. You posted a link to a paper by William Thurston, and this paper contains a quote we (at CME Project) frequently cite when we introduce the philosophy of the program:

“What mathematicians most wanted and needed from me was to learn my ways of thinking, and not in fact to learn my proof of the geometrization conjecture for Haken manifolds.”

I feel this should still be true if you replace that last part with “the quadratic formula” or even “the basic moves of solving equations”. Mathematics education should be preparing students for their adult lives, and courses that are purely about symbol manipulation do not accomplish this.

I was unclear in my comment about relationships between variables. I meant that there doesn’t seem to be a way to find the overall relationship between two variables when using the Scrubbing Calculator. For example, if a proportional relationship emerged between two variables, it can be observed through several specific cases but not generalized. Similarly it would be difficult to identify when variables were in an inverse, quadratic, or exponential relationship.

There’s more to say about Guess-Check-Generalize (more posts some other time) but I think there is a lot to talk about, educationally, as a result of these types of tools. I am hopeful that school mathematics courses can better serve students by targeting high-level thinking goals, the habits of mind that Thurston talks about, instead of just being about content goals and “mindless manipulation” of symbols.

But I still think the symbols of algebra are necessary to accomplish those higher goals…