January 16, 2024
Pam Harris
Episode 187

Ep 187: Getting Answers to Fraction Problems is Not Enough

Math is Figure-Out-Able with Pam Harris

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Math is Figure-Out-Able with Pam Harris

Ep 187: Getting Answers to Fraction Problems is Not Enough

Jan 16, 2024
Episode 187

Pam Harris

How many people do you know that shy away from fractions? In this episode Pam and Kim discuss why teaching students to use step by step procedures to solve fraction problems actually leaves them not able to think about fractions at all.

Talking Points:

- We love hearing your stories!
- What kind of reasoning do fractions engage?
- What happens when students learn to mimic steps rather than reason about fractions?
- What does reasoning about fractions look like?
- Are greatest common denominators really that important?

See Episode 5, 67, 68 for more about Development of Mathematical Reasoning

Check out our social media

Twitter: @PWHarris

Instagram: Pam Harris_math

Facebook: Pam Harris, author, mathematics education

Linkedin: Pam Harris Consulting LLC

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How many people do you know that shy away from fractions? In this episode Pam and Kim discuss why teaching students to use step by step procedures to solve fraction problems actually leaves them not able to think about fractions at all.

Talking Points:

- We love hearing your stories!
- What kind of reasoning do fractions engage?
- What happens when students learn to mimic steps rather than reason about fractions?
- What does reasoning about fractions look like?
- Are greatest common denominators really that important?

See Episode 5, 67, 68 for more about Development of Mathematical Reasoning

Check out our social media

Twitter: @PWHarris

Instagram: Pam Harris_math

Facebook: Pam Harris, author, mathematics education

Linkedin: Pam Harris Consulting LLC

**Pam **00:00

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

**Kim **00:07

And I'm Kim Montague.

**Pam **00:08

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

**Kim **00:35

Hi.

**Pam **00:36

Hey.

**Kim **00:37

How's it going?

**Pam **00:38

It's going.

**Kim **00:40

I was just thinking about NCTM, and how when we did a podcast, there were a couple of people there, and it was super funny and enjoyable to like talk to you while there were people that were like nodding, and laughing, and whatever.

**Pam **00:56

Yeah, yeah. Joining in. Mmhm.

**Kim **00:57

It would be super fun to do a live when sometimes. We should talk about that. That would interesting.

**Pam **01:02

That would be. Alright.

**Kim **01:03

I'm going to make that happen.

**Pam **01:04

Yeah. Well, maybe at next year's NCTM or something.

**Kim **01:07

Could be.

**Pam **01:08

Yeah.

**Kim **01:09

Okay, so I know that we usually like jump right in and we super talk fast, but I'm going to slow it down for just a second because I read something that I think a lot of people can relate to. And I know you don't know what this is, but...

**Pam **01:22

Surprise me.

**Kim **01:23

Yeah, spiritfilledkansasgirl left a kind of lengthy review, but I want to I want to read it because... Yeah, I think a lot of people can relate.

**Pam **01:36

Fun. Okay.

**Kim **01:37

So, she says, a homeschool friend of mine shared your website with me, and I searched for the podcast! Wow, I wish you had both been my teacher!"

**Pam **01:47

Aw!

**Kim **01:47

"As a parent and home educator, I'm feeling so much more equipped. I'm learning so much from each podcast, and I just started listening. Pam, I love how you never make people feel dumb for the way they think through a problem and your genuine interest of how they solve even the simplest steps. It shows a humbleness that we often don't see with people who are great with numbers." (unclear).

**Pam **02:09

Aw.

**Kim **02:09

I know, isn't that sweet? "When I was younger, often the teacher would explain how to do a problem mostly through memorization and nothing on relational numeracy. I could do math well on paper and I got good grades, but I realized my numeracy skills and strategies lacked. For fear of being slow to answer or thinking a problem in the wrong way, I often didn't answer the questions to build my own math knowledge. On tests, I would basically just do long algorithms to get the solutions with no real understanding of how they correlated with the numbers. I especially love the most recent parent tips..." The white paper that we put out, download. "I especially love the most recent parent tips, giving my children time to answer, patience, and allowing them to take ownership. So smart! I think because of my own baggage, I thought the solutions to my problem was if only I had memorized better, so I could say the problem faster. I have put that pressure on my own children. Now, I realize it isn't so much speed as the relational numeracy. Personally, I'm more excited about solving math after learning from you both. Kim, you often challenged me in ways I don't think about answering a problem, and I write your way down to see if I can gain some better math strategies. Thank you for this podcast! Every math teacher or anyone wanting to improve their math skills should listen."

**Pam **03:27

Wow!

**Kim **03:28

I know.

**Pam **03:29

Wow!

**Kim **03:29

It's so thoughtful.

**Pam **03:29

You are spirit filled, Kansas girl. That's awesome.

**Kim **03:33

So, the reason I wanted to share this one is because I think of a couple of things. A, I think so many people can relate. Right? It isn't until their adult years that they hear something differently that they go, "Oh, maybe that's what math is really about." And they reflect on their earlier years in education, and they feel like they missed out in some way.

**Pam **03:55

Mmm.

**Kim **03:56

I mean, I feel like that's probably what you have shared.

**Pam **04:00

Yes, for sure.

**Kim **04:01

Yeah.

**Pam **04:01

Absolutely, yeah.

**Kim **04:02

And... Go ahead.

**Pam **04:03

Well, I was just going to say, I really appreciate and I'm honored that you would sort of talk about how I don't make people feel dumb. And yay. I'm glad that's coming across. Because to be clear, I don't want to make me feel dumb. Like, I was the one who had no numeracy. You know, I joke about how I had negative numeracy because I was so just a rule follower.

**Kim **04:12

Yeah. Yeah.

**Pam **04:28

So, yeah. What kind of person would I be to then turn around and not just support and empower everybody? That is my goal is to just give everybody what we all kind of missed out on. Yeah.

**Kim **04:38

Yeah. Super cool.

**Pam **04:39

Cool.

**Kim **04:40

Anyway, so I...

**Pam **04:42

Thanks, Kim.

**Kim **04:42

I want our listeners to feel like they're not alone. And, you know, it's super cool to chime in and learn some more. And I super, super love when people share their stories. So, listeners, will you continue to share those with us? It's meaningful to us, and it helps people see that they're not alone on their journey.

**Pam **05:01

Yeah. And when you throw a review like that into the wherever you hear your podcast, it helps more people find the podcast. So, we appreciate you rating and giving a review. That's awesome. Thanks.

**Kim **05:13

Yeah. Alright, so we are talking about something really important today, as if not... Everything we talk about (unclear) is important But this a...

**Pam **05:20

This is a big one.

**Kim **05:21

You know, fractions is one of those things that people sometimes shy away from. And they don't think they can reason about probably because they didn't really make sense of things early on. But that is not so.

**Pam **05:32

In fact, Kim, I am reminded of a quick... I don't think she'll mind me telling this story. But a super good friend of mine several years ago. Several years ago. Because I was writing Lessons & Activities for Building Powerful Numeracy. And I was creating As Close As It Gets problems where I give you a problem, and then I just give you four answers. And none of them are the correct answer. And you just have to get as close as you can. So, you reason through the problem, and then you choose one of the responses. And I was kind of testing out. I had written the problem, and I'd written the four responses. None of them are correct, right? You can just get as close. And I wanted to get somebody's take on whether they were good options. You know, because sometimes if you don't write it well, then the right answers too obvious. You know, the closest ones too obvious and stuff. And so, I kind of want to get a feel for that. So, this friend of mine was leaving. I don't know if she dropped by for whatever reason. And we were chatting at the door. And as she's walking out the door, I said, "Hey, hey, hey! Can I show you this?" And I held it up. And it had decimals in the problem. I don't even remember what it was, but I just remember the decimals. And I'll never forget, she goes, "Oh, I don't do decimals." And I was like, "No, no, no. Like, this is the kind of problem that you don't have to remember the rules or whatever. You can just, you know, do the thing." And she goes, "No, no, no. Really, I don't do decimals. And she kind of was like walking backwards out of the door. And I said, "Oh, okay. Well, how about this one?" And I showed her one with fractions. And she goes, "Ah! I really don't do fractions." And she left. I mean, it was, it was so interesting to me. And then, now later in life, she's... You know, we're still good friends and everything. She's like, "Wow, I really did have this major block against fractions." And I think many of us do. They're so rule bound when we teach fake math. Yeah. Of course, it has been.

**Kim **07:10

Yeah. Well, and if we keep teaching, you know, the way that fractions have traditionally been taught, I think we run the risk of not helping people feel like they can. And, you know, it doesn't have to be done at whatever grade you finish school. There's still always opportunities to learn more about fractions. So, we're going to spend a couple of episodes back on this topic that we think more people need experience and more confidence.

**Pam **07:36

Yeah, let's do it. So, we've had lots of conversations about developing mathematical reasoning. And as I say that, I'm making concentric ovals. So, if you have seen the graphic that we've created, Developing Mathematical Reasoning, that starts with counting strategies, and then we want to build on that to additive thinking, and we build on that to multiplicative reasoning. And I'm making these ovals with my hand. And we build on that for proportional reasoning, and that we build on that for functional reasoning at the high school level. So, all of those kinds of reasonings are ways that we want to reason and grow our brains, to build more capacity, to be able to reason about more and more complex, complicated, sophisticated ideas. So, check out how if we just do rules for fractions, how you can actually use way less sophisticated reasoning, get correct answers to fracture problems, and then unfortunately be stuck not being... We haven't built your reasoning to reason about fractions. You can do some rules, but what we haven't done is help you build the capacity to actually reason about these animals. Because what are fractions? So, if you think about... Now, I really wish everybody could have that graphic in front of you. So, if you don't have it in front of you, you could just go to mathisfigureoutable.com and scroll down a little bit until you see these concentric ovals. You can kind of follow along. Where do fractions fit? Are fractions things that you count one by one? Not typically, right? Unless you're counting whole number fractions. Are they additive? Well, like we add and subtract fractions. Is that were fractions live? Are they multiplicative? Well, we multiply and divide fractions? Is that where they live? Are they proportional? Well, I'm going to say that fractions themselves are not functional. So, they're not the high school. We definitely deal with fractions as we deal with functions, and relations, and graphs, and tables, and transformations. But they're kind of inside. And I'm going to suggest that fractions are multiplicative-proportional animals. That's interesting. If they are multiplicative-proportional animals, then what are we doing with fractions at younger grade levels, where kids are really still developing from counting to additive thinking and from additive to multiplicative reasoning? We do have students doing some fraction things. So, our goal is to help kids actually build their fraction sense at the same time as we're building these other kinds of reasonings. Because in order to reason about fractions, we got to be reasoning multiplicatively and at least somewhat proportionally. So, often when I talk about developing mathematical reasoning, I'll give examples of how algorithms can keep students. Students can be getting correct answers, but they're actually using less sophisticated reasoning than that kind of problem is asking for. So, let's do that with fractions. So, if I were to give you, Kim, a problem like three-fourths plus two-fifths. Three-fourths plus two-fifths. Now, don't do it yet. So, we're going to kind of talk through this a little bit

**Kim **10:35

Okay.

**Pam **10:35

Let's start with it's a fraction plus a fraction. So, three-fourths plus two-fifths. So, we've got kind of this additive thing happening. And you might say, "Well, that's going to require additive reasoning." To which I would agree. It's definitely we're going to require at least additive reasoning. But we're adding these things that are multiplicative-proportional things. These fractions that are at their heart multiplicative. And you're like, "What? How's a fraction multiplicative, Pam? Three-fourths. It's 3 out of 4." Well, if the only way you're thinking about a fraction is 3 out of 4, then yeah, you could be using counting thinking, counting strategies, that least sophisticated thing on the on the DMR graphic. You could be like envisioning 1, 2, 3 out of 1, 2, 3, 4. You're literally counting by ones to think about three-fourths. And you could also be envisioning two-fifths as 1, 2 out of 1, 2, 3, 4, 5. You're thinking about one at a time, and you're seeing sort of 3 shaded out of 4 and 2 shaded out of 5. So, you could literally be thinking using counting strategies to just kind of envision what those fractions are.

**Kim **11:38

Can I jump in for a second? Because I think sometimes people say, "Why do my kids say that that is 5 out of 9?"

**Pam **11:47

Ah, the sum.

**Kim **11:49

If your stuck in counting. Right. If you're counting and you have these like two pictures, then kids, very early, will say, "Then if I have 3 out of 4 and 2 out of 5, then now I have 5 out of 9."

**Pam **12:01

Yeah, exactly. They sort of add the two numbers on the top. And could we agree that they could be like, "Okay, 1, 2, 3. 4, 5." They could literally be counting by one to do that, right? And so they could, "1, 2, 3. 4, 5." And they put the five in the numerator of the answer. And then, they could be like, "Okay, 4. 4, 5, 6, 7, 8..." And they could count by ones, and they could get that 9. So, they could be doing this additive reasoning thing, where we're adding two things together by using... Sorry. An additive reasoning thing, where we're adding multiplicative-proportional animals, but they could literally be counting by ones. Now they're not getting.

**Kim **12:38

And getting it wrong.

**Pam **12:39

Yeah, they're not getting a correct answer. But if we only have students thinking about fractions as part-whole, and that's the only thing that they ever. And you're like, "What else is there?" Oh, there's so much else we need to do. So, a thing that can happen is kids could just be reasoning incorrectly, counting by ones, and getting wrong answers. What's another thing that kids could be doing if they're... Maybe they're getting correct answers, but that how could they be stuck in less sophisticated reasoning? Well, so again, we've got three-fourths and two-fifths. Could a kid say, "Alright, this is what I'm supposed to do. I know that I've memorized that in order to do fraction addition, my teacher told me I got to find a common denominator. I don't even really know what that means, but I know I'm supposed to do this thing." So, they write down three-fourths down below, and they say to themselves, "My teacher told me that now I need to do all the fours. So, off to the side, they go 4, 8, 12, 16, 20." And as they're doing they're best skip counting. They're like, skip counting through all the multiples of 4. You're like, "Pam, why are you doing that?" Bear with me. And then, they go, "Okay." So, maybe I'll do a couple more 20, 24, 28." So, a whole bunch of fours. And then, they go over and they go like, "Now, I need the 5s." Because remember, it was three-fourths plus two-fifths, so they're thinking about that fifth. And so, they go over, and they write down a bunch of 5s in there. 5, 10, 15, 20. 20." And they just... Their teacher told them, "Just write down a bunch of 5s." Now, that they've got a bunch of fours written down and a bunch of fives written down, they see where they have a common number. And so, in the 4s, I had a 20, and in the 5s I had a 20. Notice, I got there by skip counting. Skip counting is additive reasoning at best. If I'm sing-songing the song for fours and fives that isn't even reasoning at all. So, additive reasoning at best. So, then, next to the three-fourths, I wrote down off to the side. I write down 20 next to the 4. And then, I say, "Okay, so how many 4s does it take me to get to 20?" I go back over to my list, and I count. "It took me 5." I count down the list of fours. And it was five to get to 20. So, then, I go, "5 times 3." Because the 3 is above the three-fourths. So, now, I go five 3s. 3, 6, 9... And I skip count five 3s, and that's 15. So, all of that work I've just done, at best with additive reasoning, I now have an equivalent fraction to three-fourths, which is fifteen-twentieths. Now, I do the same thing for the two-fifths. I write down the two-fifths. I go over there. I know it's going to be 20. I say, "How many of those 5s did it take me to get to 20?" I count. I'm using counting. Now, I'm counting down the list. It was 4 of them." Then, I do 4 times 2. I need four 2s. I skip count those four 2s. And again, an equivalent fraction of eight-twentieths. Now, notice, I've gotten the correct equivalent fractions. I've got fifteen-twentieths and eight-twentieths to be equivalent to the three-fourths and the two-fifths. So, now, I add those two together. So, now, I've got fifteen-twentieths plus eight-twentieths. And now, I go, "Alright, I need to have 15 plus 8. 15, 16, 17, 18,. 19..." And I count by ones to get 15 plus 8. And that's 23/20. And then, maybe I turn that into an improper fraction. I don't even know at this point. But would you agree with me that I could have gotten the correct answer of 23/20, but the most sophisticated reasoning I use the entire time was additive. If I was skip counting those multiplication facts.

**Kim **16:10

And if you were skip counting them correctly. Making a long list. I mean, if you're doing a elevens and twelves.

**Pam **16:15

Oh, yeah.

**Kim **16:16

That's a pretty heavy skip count.

**Pam **16:19

Or sevenths and eighths, I'm going to have to get all the way up to 56 for both of them. Yeah. So, I could be doing what my teacher told me to do. I'm following the rules.

**Kim **16:29

Yeah.

**Pam **16:29

But I'm using super less sophisticated reasoning. I'm not reasoning about three-fourths and two-fifths at all. I'm following a series of steps. Notice also that at the very end, I actually counted by ones, right? Like, that's the least sophisticated thing in that development. And nowhere did I use any multiplicative reasoning. Now, I could have used some multiplicative reasoning. I've seen people probably yelling at their car right now, their car radio. Radio? Nobody listens to the radio anymore. You're listening to your phone through your car. And you're saying to yourself, "Pam, Pam, Pam. When my kids are finding the equivalent for three-fourths, they're saying 4 goes into 20. And they're thinking about multiplication facts. And they're saying 5. And then, they're going 5 times 3, and they're thinking multiplication. It's 15." But they're following a series of steps to get the equivalence for three-fourths. They're not actually thinking about the relationship of three-fourths to twentieths. Or, heavens, the relationship to three-fourths to two-fifths. They're not thinking about those relationships. They're following a series of steps, where maybe they're thinking about multiplication facts but they're not thinking multiplicatively about fractions. That's noteworthy. Oh, there's something else I was going to say. Hang on. Let me remember. Let me remember. Oh! And I've also seen some people, after the kids have written down those those... Help me. The skip counting. The list of multiples of those numbers. And they say, "Okay, so how many of those? It was 5 of them." And then, they do 5 times 3. When they do 5 times 3, then they go over and look on a multiplication table. Ah! All the things, where we're not actually reasoning at all with fractions. So, Kim, how are... If I were to say to you three-fourths plus two-fifths, what's one way that you could reason about that problem? Three-fourths plus two-fifths? Just one, Kim. don't give us more than one.

**Kim **18:21

Can I use percents?

**Pam **18:22

You could do whatever you want.

**Kim **18:24

Okay, okay, okay. When, I see three-fourths, then the first place my brain goes is 75% because I know that one-fourth is 25%. And so, then, I have three 1/4s, which is 75%. And then, I also happen to know that one-fifth is 20%, so two-fifths would be 40%, two 1/5s.

**Pam **18:53

So, two 20 percents is 40%. Okay. So, so far on my paper I wrote 75% plus 40%. Okay.

**Kim **19:00

Yeah. And so, then, I'm grabbing the 25% out of the 40%.

**Pam **19:07

Mmhm.

**Kim **19:08

And I'm thinking about the 75%, which is the three-fourths, and 25% to make 100. And then, I have leftover 15% from that 40%. So, 40% is split into 25% and 15%.

**Pam **19:20

Okay.

**Kim **19:20

And so, then I have 115%. But you're probably going to tell me I can't keep it in percent. So, I know that 100% is 1 whole. And 15% is the same as fifteen 1/100. So, then, I'm going to actually scale that down to be three-twentieths. So, 1 and 3/20.

**Pam **19:49

So, you got 1 and 3/20, which is similar to my 23/20 that I got before. Nice. I like your percent reasoning. Let me point out a couple of things that you did that are particularly nice that are very multiplicative, and proportional. So, you said you're going to think about three-fourths as three 1/4s. Notice, how multiplicative that is. It's 3 of something. 3 groups. 3 lots of that thing. Three 1/4s. That's a very multiplicative way of thinking about three-fourths. It's super different than thinking about 1, 2, 3 out of 1, 2, 3, 4, where I'm just using counting and that part-whole relationship. Instead, we're using a multiplicative relationship. Three 1/4s. And then, you do the same thing with two-fifths. Two 1/5s. Very nice. And then, of course, you went to percents, which is super proportional. Percents is very proportional. And so, you didn't have to necessarily go to percents. But I love that you did. What if I restricted you and said you can't go to percent. Do you mind if I said please don't go to percent?

**Kim **20:52

Can I use money?

**Pam **20:53

Absolutely.

**Kim **20:55

Okay. I mean, do I need to have different monies? So, I could think about different coins where 100 Is my denominator. So, like, in a way, I'm making them equivalent by thinking out of 100. So, I would still say 3 quarters is $0.75.

**Pam **21:19

And so, are you thinking about that is 75 pennies out of 100 pennies?

**Kim **21:26

I kind of wasn't. These problems are kind of hard for me because I just know them already.

**Pam **21:31

Mmm, okay.

**Kim **21:31

You know like...

**Pam **21:32

So, if you were to say to me, "You're thinking about three-fourths of 100 as 75," I would I would write down 75 out of 100, even if you didn't.

**Both Pam and Kim **21:41

Okay.

**Pam **21:41

Okay.

**Kim **21:41

That's fair. So, then, two-fifths would be 40/100.

**Pam **21:45

And how do you know?

**Kim **21:48

One-fifth is 20/100. So, two-fifths is 40/100.

**Pam **21:51

It's like a fifth of 100 is 20.

**Kim **21:54

Is $0.20. Mmhm.

**Pam **21:55

Okay, okay. So, now you got 75/100 and 40/100.

**Kim **21:58

Mmhm. So, that's 115/100.

**Pam **22:02

And we're kind of back to the 115.

**Kim **22:04

Mmhm, yeah.

**Pam **22:07

1 and 15/100 or 1 and 3/20.

**Kim **22:10

Yeah.

**Pam **22:10

I just had to write it down. Mmhm.

**Kim **22:13

What I like about money is that you could use different coins. So, you could think about that same 20 denominator that you were talking about earlier.

**Pam **22:23

Mmhm.

**Kim **22:24

So, you can think about nickels in that way. So, three-fourths is the same as 15 nickels. So, I wrote down fifteen-twentieths.

**Pam **22:35

How do you know that?

**Kim **22:36

Because 1/20 is 1 nickel.

**Pam **22:39

Twentieth of a dollar is a nickel. Mmhm.

**Kim **22:43

Mmhm. And so, how do I know that 15 nickels is?

**Pam **22:49

$0.75 is three-fourths of a...

**Kim **22:51

$0.75.

**Pam **22:51

...dollar. Yeah. Do you just know that?

**Kim **22:54

Because 5 nickels is $0.25.

**Pam **22:56

Yeah.

**Kim **22:57

So, you could go 5 nickels, 5 more nickels, 5 more nickels. But really, I would go. I would scale from 5 nickels to 15 nickels.

**Pam **23:03

Oh, and I totally thought you were going to say. When you said it was 5 nickels was $0.25, I thought you were going to say, "So, back $0.25, back 5 nickels from 20."

**Kim **23:11

Oh, wait, that's... Of course, I would have done that. No, I didn't. I didn't.

**Pam **23:14

Oh, that's awesome.

**Both Pam and Kim **23:14

That's very Over.

**Kim **23:15

No, I didn't.

**Pam **23:16

Look at me getting Over-ish. I love it. I love the fact that I see Over more and more and more. I love it.

**Kim **23:21

Yeah. That's cool. So, then, same kind of thinking. If I wanted to think about nickels.

**Pam **23:26

Two-fifths for nickels. Mmhm.

**Kim **23:27

Mmhm.

**Pam **23:29

Well, so, 1/5. Can I do it out loud? 1/5...

**Kim **23:31

Yeah.

**Pam **23:32

...would be... I got to think for a second. 1/5 of 100 is $0.20. So, that would be 4 nickels. But we need two-fifths, so that's 8 nickels. So, 8 out of 20.

**Kim **23:47

Yep.

**Pam **23:48

That's how you got the 8 out of 20. So, now you have fifteen-twentieths plus eight-twentieths. Cool. And then, there's the 23.

**Kim **23:53

And that's you're 23.

**Pam **23:54

23/20.

**Kim **23:54

Yeah.

**Pam **23:54

So, several different ways. And I'll just point out that two of the ways that Kim did it did not have a Least Common Denominator. And I'm just going to boldly say we do way too... We make way too much of a big deal about Least Common Denominators, LCMs, Least Common Multiples, and the Greatest Common Denominators. And bleh! How about if we actually helped kids reason using denominators that make sense? And I'm going to give Cathy Fosnot credit for giving me that idea of the very first time as I was doing a lot of her materials in her Context for Learning, where she made a big deal about, "Nah, it's not about a specific denominator, it's about denominators that makes sense." And using money can be super helpful, using denominators that makes sense with time for factors of 60 can be super helpful. And all of those can help kids really reason about fractions. And sometimes people will say, "Yeah, but if you do that, you're only going to get certain denominators." To which, Kim, I've been thinking about this a lot lately. And you and I have talked about it just a little bit, where we as a society have gotten kind of hung up. Kind of. We've gotten totally hung up on having a generalized solution that can solve any problem. Where you're like, "Well, Pam, that won't work with sevenths." Or, "Pam, what are you going to do when it's elevenths, and, you know, crazy numbers that don't work for for time or money?" Or, you know, "Kids are going to have a hard time reasoning about those." And more and more, I'm getting more clear that it really isn't as important to get that generalized solution as it is to get kids reasoning using the most important relationships. If we get them reasoning using the most important relationships, the generalized stuff either will come, or you ask Siri, or ChatGPT, or like pick your AI. That it's really much more important that we are generalizing and thinking, building students capacity to know and do more.

**Kim **25:52

For sure. You're reminding me when you said that people are concerned that we only use some denominators in that way. You're reminding me. I haven't thought of this in forever. But I used to do a thing with my kids that we just called "Also Known As."

**Pam **26:07

Oh!

**Kim **26:08

And so, when... I mean, it wasn't like a real big thing. But when we would talk about fractions, just as a way to continue to talk about equivalencies, then when they would say something like you know a sum or a difference, then I might say to them, "Also known as?" And somebody would supply an equivalent one that kind of makes sense. Like, if we were talking about money, then they would, you know, they would give a different coin or whatever.

**Pam **26:35

So, kind of like our 115%, also known as 1 and 15/100, also known as 1 about 3/20.

**Kim **26:42

And I remember I had one kid in particular who was super into saying it back like when I would say something. And so, they would be like, "Also known as?" But that gave me the opportunity to say one if there was something that repeatedly didn't come up, like my kids was shying away from something, then I would supply, "Also known as." I haven't thought about that in forever.

**Pam **27:06

Nice. I like it. We might have to do a whole episode where all we do is, "Also known as?" Alright, ya'll, what would your students need to own to think and reason through problems like we just did, like Kim just did? Well, tune in because we're going to do some of that in next couple of episodes. We're going to dive into some important things that we can develop in our brains and our students brains, so that we can think and reason with fractions. So...

**Kim **27:33

Perfect.

**Pam **27:34

...thank you 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. And keep spreading the word that Math is Figure-Out-Able!

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