# Ep 140: Cancel Simplifying

February 21, 2023 Pam Harris Episode 140
Math is Figure-Out-Able with Pam Harris
Ep 140: Cancel Simplifying

Have you ever noticed how many different meanings there are for the word simplify? Even if you limit it to mathematical meanings, there's a lot. In this episode Pam and Kim discuss how we can better communicate with students what we really want them to do when we say 'simplify.'
Talking Points:

• Asking students to 'simplify' can often be a guessing game as to what we really want students to do.
• Why not level the playing field?
• Some meanings of "simplify" can be contradictory.
• There are better ways to ask students what we actually mean for them to do when they see that word "simplify".

See episode 18 "This, Not That" for other examples of unhelpful words sometimes used in math teaching.

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Pam:

Hey, fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able! I'm Pam.

Kim:

And I'm Kim.

Pam:

And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But, ya'll, it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. We can mentor students to think and reason like mathematicians. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be.

Kim:

So, in today's episode, we are going to have some fun beating around a term that is often used in math classes.

Pam:

Yeah! So, Kim, I just got back from spending time in Utah, which is totally cool because I got to see my kids. Yay!

Kim:

Yeah.

Pam:

With some wonderful high school teachers. We've actually met a couple of times. And we had some great conversations. A super interesting conversation about some words we use in math teaching that have the potential to be super confusing because... Well, for many reasons. But I'm going to suggest today because they're not well defined.

Kim:

Mmhmm.

Pam:

Which is interesting for a group of people who are all about definitions.

Kim:

Right?

Pam:

Right? So, it reminds me of a story that I heard Dylan Williams say one time. Which I think "William" not. There's no "s", right? Dylan William?

Kim:

Yeah, no.

Pam:

Yeah. Sorry about that, Dylan. He's all about assessment. He's a good speaker. I've learned some things from him. And I really appreciated this story. He talked about the word "similar" in mathematics. So, the word "similar" in English has a very specific mathematical definition, but in everyday usage, we... (unclear). You know, Kim, you and I are similar in some ways, right? I mean, we live close, near each other. We both taught math (unclear). We're both women. We both have kids. I don't know. There's some similarities that we have. We both like chips and salsa. Though, you might like it more than me. So, we have some...there's this word "similar" in everyday language that means, you know, we're kind of alike. We kind of have some things in common. But in mathematical definition, it's very particular. It has a very particular meaning. And he suggested that...and I don't remember the country he was in, but I'm pretty sure it was an Asian country...that in that particular country, when you get to "similarity", and you're talking about that shapes can have the same shape but different size; that the angles are congruent on a triangle, therefore the triangles are similar, but they can, you know, have a small triangle that's equalateral and a large triangle that's equalateral but they're similar; That definition of "similar", similar shapes, is very specific definition. That in that Asian language, they made up a word. That they don't use a commonly known word to then have a specific different definition. They made up a word. And so, when kids in that country learn about "similarity", the mathematical definition of "similarity", they don't mess it up. That because there's this very. That's interesting, right?

Kim:

That is interesting.

Pam:

There's this very specific word that has a very specific definition, and it's only used in that way. Kids don't mess up "similarity" in the same way that they do when the language is not well defined. That's what I mean. Like, we have this word that can mean so many different things. "Oh, but in math, it should be this." But kids have a hard time parsing that out. And I don't know, maybe we as teachers could do a far better job of parsing out what the mathematical meaning of "similarity" is versus, you know, this sort of commonly used word.

Kim:

Yeah. That's interesting.

Pam:

Right? It's fascinating.

Kim:

Yeah, it is really cool. So, there are these words that we often say that have different meanings. And so, today, we'd like to focus on one of those words, which is "simplify". And we've actually mentioned this before because we did an episode... Oh gosh, way back when. I don't even know the number. It's like maybe (unclear).

Pam:

We'll put it in the show notes, yeah.

Kim:

Yeah. That we did a "This, Not That" episode, and we tackled some of the words in mathematics that people use that make us cringe just a wee bit. And one of the words that we said that we don't say is "reduce". And what I said at that time, and what you said at the time was, "We kind of use the word 'simplify'." But I mentioned it, and you were like, "Mmm..." You know, "Okay, but (unclear).

Pam:

Neither of us love it.

Both Pam and Kim:

Yeah.

Pam:

Yeah.

Kim:

So, we're going to talk about that today.

Pam:

Well, so to be clear. We for sure don't like the word "reduce" when we're talking about fractions.

Kim:

Yeah.

Pam:

Because if we're messing around with fractions, and we use the word "reduce" to mean "create an equivalent fraction where there are no common factors between the numerator and denominator", the word "reduce" feels like "get smaller". And often the numerator will be a smaller number than the previous numerator and the denominator will be a smaller number than the previous. I should give an example. We're being really esoteric here. So, if I was talking about twelve-fifteenths, and a textbook might say "reduce that fraction", then you might say, "Well, okay. Like, I can see that there's a factor of 3 in each of those. So, twelve-fifteenths. Divide a 3 out of the 12, and that's 4. And I can divide a 3 out of 15, and that's 5. And so, twelve-fifteenths is equivalent to four-fifths. And you might say, "Well, yeah. Like, 4 is smaller than 12. And 5 is smaller than 15. But the fraction four-fifths is not smaller than twelve-fifteenths." So, we hate the word "reduce" when we mean, "find an equivalent fraction". That maybe you've reduced common factors between the numerator and denominator. That's how some colleagues of mine once said, "No, no, no. 'Reduce' totally makes sense because you're reducing the number of common factors. You reduce. There's no common factors now between the numerator and denominator." And I'm like, "Yeah, okay (unclear)"

Kim:

But that's not what kids say.

Pam:

Not in any way, right. (unclear)

Kim:

Yeah.

Pam:

(unclear) explain why we use this horrible word to mean what it doesn't really mean. But we don't like it because it sends such a... What? (unclear).

Kim:

A confusing message. Yeah.

Pam:

A confusing message. Yeah, absolutely. And that's specifically when we're talking about fractions.

Kim:

Right.

Pam:

So, we said, "So, we can use the word 'simplify'..." But like you said, neither of us really love that. Because "simplify"? Does that really work when what we mean is "find an equivalent fraction". Like is four-fifths simpler than twelve-fifteenths? Yeah. Like, you could maybe make an argument that thinking about 4 pieces out of 5 pieces is simpler than thinking about 12 pieces out of 15 pieces. You could try that argument. But again, we don't really love the word "simplify" either when it comes to simplifying fractions.

Kim:

And at the time, we just kind of left it. We were like, "Ugh."

Pam:

Yeah. Yeah.

Kim:

But now you've had some more opportunities to think about that, right?

Pam:

I have. And so what's interesting is. Okay, I got to tell you, I've been trying to find the "This, Not That" episode as we've been talking, and so I'm a little distracted. It might be episode... No, that's not it. Okay. Oh, maybe I just found it. There it is. It's Episode 18. So, if you want to go back. And that was a while ago. So, episode 18. We talked about some of "This, Not That" things. But like you said, we've gained some clarity. Hugely, this last weekend when I was in Utah, and I was talking to some teachers, I gained some clarity about the idea of the word "simplify". When I was dealing with these high school teachers in Utah. It occurred to me, partly because they were suggesting it, that we don't just use the word "simplify"... And I know this. But we don't just use it with fractions. We use it in lots of other places to mean other things. Like, the word "simplify" is super confusing, not just because it doesn't relate well with when we're finding an equivalent fraction, but it also has other places where it doesn't... It's not well defined. It can mean other things. So, let me give you some examples. Let's stay with fractions for just a second. Notice, that I could have that twelve-fifteenths being equivalent to that four-fifths, but you lose information or you run the potential of losing information about the context. So, for example, let's say if I was talking about making 63 out of 119 free throws. You might be like, "Okay. Well, I kind of have a feel for that." But if I just simplify that, so that it's about 2 out of 4. And it's just about 2 out of 4 on that one. Then, you've lost some information. You don't know how many total free throws I was shooting. I mean, if I just said to the coach, "Hey, you know like, 2 out of 4 free throws" versus "63 out of 119." I might be like, Whoa, that kid shot a lot of free throws that day."

Kim:

Yeah.

Pam:

Versus... You know, like you run the potential of losing information. So, that's just one way of thinking about how it's not super with respect to finding equivalent fractions. But also, we might send the impression to students that if they say...back to my twelve-fifteenths example...if they say twelve-fifteenths, and I mark it wrong because they didn't simplify it, or they didn't write an equivalent fraction that's in simplest form. See, we don't have really good words for this. They didn't write a fraction where the numerator, denominator are relatively prime. That's kind of complicated way of saying it. Like, how could we say "Write a fraction that's equivalent"? "Rewrite this fraction equivalent, so there's no common factors in the numerator denominator," that's a way of saying it. Boy, super complicated. If we mark twelve-fifteenths wrong because they didn't write four-fifths, I worry that students think, "Oh, so twelve-fifteenths doesn't correctly represent the situation." When in reality, it might represent the situation better than four-fifths.

Kim:

Right.

Pam:

So, this is kind of a super tricky... Like, I understand that teachers often want to have four-fifths. They're like, "Yeah. But, like, I want to be able to see, can students find the equivalent four-fifths to twelve-fifteenths?" But I think we might want to be at least clear with students at least in a minimum. Teachers, may I invite you, be clear with students that twelve-fifteenths is not incorrect.

Kim:

Right.

Pam:

It's equivalent. It's just maybe not the answer you're looking for. And can we, then, maybe agree that it's convention that we're looking for that maybe easier to grade answer. That maybe we're actually looking... Sometimes. Now, maybe not all the time. Maybe we really do want to know if students can factor out those common factors, divide out those common factors. Maybe we are really. We do want to say that. But maybe we ought to say, "Rewrite that fraction and divide out all the common factors from the numerator, denominator." Something like that, but not give the impression it's incorrect. But could we at least admit that sometimes we're asking for that simplified version of the fraction because it's easier to grade? And at least admit that to students?

Kim:

Yes, yes.

Pam:

Kim:

Yeah.

Pam:

So, there you go. Here's another example. Let's say that you had... Now, I'll give you some high school examples. We'll give a couple of younger examples in minute. Let's say that you had 2x times the quantity 5x plus 3, plus 7 times the quantity 5x plus 3. You could say "simplify". And often, the instructions in a typical textbook would say "simplify" when we mean...and here's the instructions that they wrote..."Rewrite each expression as the product of two binomials." Now, I know that sounds like a lot of math speak to you, but you might notice that there was a 5x plus 3 that was common, And so you can sort of factor out that 5x plus 3, and you end up with the multiplication, the product of two binomials. And that's their sort of, "Can you un-distribute?" So, can you pull that 2x and that 7 out to get the quantity 2x plus 7 times the quantity 5x plus 3? And I know I'm doing a lot in the air here. You'd almost have to write that down to be able to sort of see what's happening. But it's a way of kind of. Typically, kids would take the 2x plus 7 times the quantity 5x plus 3, and then you could distribute that. You could think about that as 2x times the quantity 5x plus 3, plus 7 times the quantity 5x plus 3. And so, in this case, they're kind of undoing that distribution.

Kim:

Yeah.

Pam:

But you're being clear about what's happening. So, here's another.

Kim:

Yeah, you know what I love? Hang on one second.

Pam:

Kim:

You know what I love about that is that the mathematical vocabulary that they're using here is so simplified. They're saying "expression", and "product", and "binomials". Like, it gives the kids a window into what is happening, but also using correct mathematical vocabulary that we want them to understand anyway.

Pam:

Yeah. So, you might say, "Kids can't use that kind of vocabulary." And what we're suggesting is, as you just say "simplify" but expect kids to know that they're supposed to do this other thing... Each of these instances that we're using so far, kids are expected to do something completely different. And it's almost like we have the secret club, and if you've done 1 through 29 odd 29 times, you kind of get a feel for when "simplify" means "use the distributive property", and you get a feel for when it means "undistribute", and you get a feel for when it means "rewrite the fraction so that there are no common factors in the numerator, denominator". You get a feel for what "simplify" means. But if you don't ever get that feel? We haven't been explicit about what "simplify" means in these different cases. We're kind of expecting kids to guess what's in our head.

Kim:

Yes!

Pam:

We're kind of expecting them to just like know, "Oh, in this case, this is what it means. But not in that case." So, let's actually be clear. And I think these guys are doing a great job of at least admitting, "Hey, this is what we mean in this case. We actually mean that." And I think, yes, it might mean that in this case, we actually have to help explain those terms that these commands are sort of meaning. But that's better than saying, "Guess what we mean here by 'simplify'."

Kim:

Yeah.

Pam:

Yeah, cool. Alright, let me give you a couple other examples. Here was one where it was x times the quantity x plus 9, plus 2 times the quantity x minus two. And often, we could just say "simplify". It might... I've seen places where it would say "simplify and collect like terms". In this case they said, "Multiply using the distributive property, then add like terms." Giving a clear instruction about what to do. You know, I can almost picture some of the colleagues that I've talked with saying, "No, we're giving away too much. We're telling the student what to do too much. They should know that. They should be able to look at that and know what to do." I'm going to push back on that. I think it's actually... It's almost like we're saying, "If we do that, we'll level the playing field too much." How about if we level the playing field?

Kim:

Right.

Pam:

Like, I don't think it's necessarily a nice thing to make kids have to guess what we mean. Here's another example. "Rewrite each of the radical expressions to remove perfect squares from inside the radical." Now, if all those words are like crazy to you, let me give you an example. If I write the square root of 50. So, you can picture the square root symbol with a 50 inside. If I right square root of 50 and say "simplify", you might look at that and go, "It looks pretty simple to me." Like, you know, what does the word "simple" mean? And in fact, I probably could have said that, for all of those expressions that I just said above, it's kind of in the eye of the beholder whether one of the versions is more simple than the other. In fact, two of the examples that I gave a minute ago were the exact inverse of each other. So, if you say "simplify" when I mean "multiply them out", and I say "simplify" when I mean "factor them back to the other direction", then you've used the same term to mean "go one direction, and then reverse and go the other direction".

Kim:

Yep.

Pam:

So, being more clear. So, if I have the square root of 50, I might look at and go, "I don't know what you mean." Oh, except it just said, "Rewrite this radical expression to remove perfect squares from inside the radicals." So, then I got to think to myself, "Are there perfect squares in 50?" Oh, sure enough. Like, I can think about 25 times 2. So, now I'm thinking about the square root of 25 times a square root of 2. Square root of 25 is 5. So, square root of 2. There's no perfect squares inside 2. And so, then, I can say to myself, The square root of 50 is equivalent to 5 times the square root of 2. I have removed a perfect square from inside the radical. It's so much more clear, kids kind of know what to do. One last example that Travis sent me...Thank you, Travis, by the way. Super enjoy working with you...was "Rewrite each expression by performing the indicated operation. Each new expression should contain only one square root." And so, these were examples where there were lots of square roots happening, and that you could combine, so that you only have one square root. So, it's almost like you're kind of going the other direction. You're taking now a bunch of square roots, and you sort of squishing them all under the radical, instead of like that square root of 50 where you're taking one square root, and you're kind of like...or at least one number under square root...and now you end up with 5 times the square root of 2. Now, you have like multiple things happening. So, it's kind of, again, that sort of different direction. That's kind of important. Like, we can actually help students understand what we mean, what their task is by replacing the word "simplify" with what we actually want them to do. Okay, so let me think of a younger example. I think the best example I can think of with younger numbers... Younger numbers. Students that dealing with younger... Younger students. Would be simplifying fractions. I'm not sure I can think of. Kim, can you think of an example where? Would you write "simplify a division expression" or "simplify..."?

Kim:

Not off the top of my head.

Pam:

Yeah, really I can think of simplifying fractions, and then it kind of comes in more when we have variables. So, I could picture something like 2x plus 3x minus 5y plus y. And I could picture a textbook that says "simplify", when in reality what they mean is "collect like terms" or "rewrite the expression, so that..." How did they say that? "Rewrite the expression, so..."

Kim:

(unclear).

Pam:

Yeah, yeah. Or "Add like terms".

Kim:

Yeah.

Pam:

"Combine like terms"? "Add like terms"? Either way. Yeah. So, I think we can use phrases to say what we actually mean. You might have a textbook that is replete everywhere. It just says "simplify" all over the place. May I encourage you to at least begin you thinking about what it actually wants kids to do in that case? Like, are you clear? Yeah. Could you replace "simplify" with what you're actually expecting kids to do in that case? Ya'll, listeners, I'd love to hear some examples that you find as your teaching. Where do you see the word "simplify", asking students to do something? And then, either send us what phrase could you replace that with? Or ask. Let's ask the world. How could we all come up with better ways to ask students what we actually mean for them to do when they see that word "simplify".

Kim:

Right, right. Because if our gig is to make math more and more figure-out-able, we can't leave them guessing what's in our head.

Pam:

Exactly.

Kim:

Or what's in the textbook's head, right? So, let's cancel the word "simplify" and say what we actually mean.

Pam:

Nice, nice. Alright, ya'll, thanks for tuning in and teaching more and more Real Math! To find out more about the Math is Figure-Out-Able. Movement, visit mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able!