Ep 187: Getting Answers to Fraction Problems is Not Enough

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
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
Instagram: Pam Harris_math
Facebook: Pam Harris, author, mathematics education

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
Instagram: Pam Harris_math
Facebook: Pam Harris, author, mathematics education

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

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

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

Kim  10:35

Okay.

Pam  10:35

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

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

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

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!