Ep 6: The Development of Mathematical Reasoning part 2: Functional Reasoning

July 28, 2020 Pam Harris Episode 6
Math is Figure-Out-Able with Pam Harris
Ep 6: The Development of Mathematical Reasoning part 2: Functional Reasoning
Chapters
Math is Figure-Out-Able with Pam Harris
Ep 6: The Development of Mathematical Reasoning part 2: Functional Reasoning
Jul 28, 2020 Episode 6
Pam Harris
Many people feel that reasoning and numeracy come to an end in highschool. How could we mathematize algebraic concepts? What does it mean, and what does it look like to have functional reasoning? Listen as Pam explains how functional reasoning builds off of proportional reasoning and empowers students to analyze and understand patterns of related value sets. Sounds complicated? Don't worry, Kim does her best to keep Pam on track and grounded.

Talking Points

• How functional reasoning builds on proportional reasoning
• It's not about being able to visualize the graph, it's understanding the relationship between the variables
• How to tell if someone is using functional reasoning
Many people feel that reasoning and numeracy come to an end in highschool. How could we mathematize algebraic concepts? What does it mean, and what does it look like to have functional reasoning? Listen as Pam explains how functional reasoning builds off of proportional reasoning and empowers students to analyze and understand patterns of related value sets. Sounds complicated? Don't worry, Kim does her best to keep Pam on track and grounded.

Talking Points

• How functional reasoning builds on proportional reasoning
• It's not about being able to visualize the graph, it's understanding the relationship between the variables
• How to tell if someone is using functional reasoning
Pam Harris :

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

Kim Montague :

And I'm Kim.

Pam Harris :

And we answer the question, if not algorithms, then what? Ya'll, we couldn't be more excited because we have just launched our free online workshop all about the development of mathematical reasoning. This completely free mini workshop is based on my most often requested in person workshop, we decided to make it available for free. Don't worry, if you've missed registration this time, head on over to sign up for the waitlist and you'll be notified about it the next time that it runs. So exciting.

Kim Montague :

Very exciting. So today, we're going to talk about something a little different. Sometimes we talk about math for teaching, and sometimes we talk about just the math itself, right? Today we're going to do some higher math stuff. Parents of high schoolers, this is going to be especially exciting for you Because we're going to help you make sense of some stuff that your kids are doing: functional reasoning. But listen, younger grade teachers don't tune out - math is figureoutable. If it gets a little too much, let us know where we lost you, and we can work on that. So, Pam, I'm gonna be real honest with you for a minute here. When you and I talk about the development of mathematical reasoning, like we did last week, from counting to additive, multiplicative, and proportional reasoning, I'm with you, 100%. I totally get it. Okay, but here's the deal. When you mentioned functional reasoning, I kind of gloss over a little bit. I have to admit, I'm not even sure I know what you mean by that, because I didn't teach high school. And frankly, I know I'm not alone.

Pam Harris :

Yeah, you're totally not alone. And to be honest, right back at you. I'm still working on some functional reasoning. And based on the comments we've been getting from listeners, there are plenty of people in the same boat. So let's focus on that today. Now, I know you said that you reason proportionally but if it's okay, let's do some proportional reasoning ish a little bit here because in order to discuss functional reasoning, I want to take you from what you know, some proportional reasoning stuff and connect it to the functional reasoning, okay?

Kim Montague :

Yeah,

Unknown Speaker :

Yeah. So, proportional reasoning has everything to do with two things varying in tandem. Both things are varying at the same time. So when you buy more pounds of apples, you pay more. But what about when you start with a ratio of five pounds for \$2? That's not as easy as a unit rate of \$1 for one pound. So if it's five pounds for \$2, what happens if you just add one more pound? Do you add two more dollars? Well, no, because you can get five pounds for \$2. Yeah, so thinking about those two things varying in tandem: five pounds for \$2. What if you only want one pound or what if you have \$3? Those kind of problems deal with proportional reasoning. When you're trying to build a ramp and by ratio the law is 1 to 12 or 1 to 20. Those are typical ratios out there. What does that even mean? So that means if it's the ratio of one to 20, that means for one inch of vertical rise, then you have to have 20 inches of horizontal length or run. Those two vary in tandem. So if you've got a certain height that you need to build a ramp for, then you have to consider how long that ramp has to be. Or what if you're trying to get a job done? Say that you know that it takes 20 students 30 minutes to pick up trash for the whole playground, you know that you've got this experience, kids are all working. It takes 20 students 30 minutes to get that whole job done. But today, you've only got 15 students, how long is it going to take? That's another kind of relationship. You'd want to use proportional reasoning. All of these scenarios, deal with rates and not just unit rates, but non unit rates, right. It's not just a rate to one or one to something but it's got two numbers varying in tandem, and that is the change from elementary school to middle school.

Kim Montague :

Yeah.

Pam Harris :

When you go from unit rate to a non unit rate. Go ahead.

Kim Montague :

So I feel you on the unit, right? Because I feel like if you say there's 32 students on a field trip needing to go in eight vans, were really focused on four kids to one van, right? We don't really call it the unit, right? We're focusing on the division. Or if we say there's six apples in one basket, we have kids use multiplication to figure out their 30 apples and five baskets is scaling up by five is that is that kind of like what you're

Pam Harris :

Yeah, totally. So in a big way, in third, fourth and fifth grade, when you guys are dealing with multiplication and division, often you have rates like you just said that you'll have I think you said eight kids or four kids in a van. That's four kids per van, that's a rate for kids per van. But you also had a rate where you had 32 students per eight vans, but you don't really think of it that way. Typically in elementary school, we're just sort of focused on, like you said, the multiplication or the division. So then when we move up to middle school, we're still focusing on multiplication division. But with these non unit rates, then it's a bit more complicated. So like five pounds for \$2, 4 slices of pizza for \$5. If you look at the tag on a bulk food order, you know, when you go down the bulk food aisle and stick a bunch of food in the bag, you know, don't touch that stuff, right? Use the scoop. And then when you look at the tag, after you've weighed it, you might have gotten 6.2 pounds for, I don't know, \$10 and 20 cents, that's a non unit rate. 6.2 pounds for \$10 and 20 cents, maybe you drove 420 miles with a tank that holds like 16 gallons, how far could you drive if you only had five gallons of fuel? Those are all two things varying at the same time. That's complicated because there's a bunch of stuff happening simultaneously. So let me tell you a quick story. We actually have a family in our church and he rides a scooter to get along. And in order for him to use that scooter, effectively, he needs to be able to get in and out of his house. And right now there's no ramp for his house. So some guys in our church are getting together, they're going to build a ramp. And so in order to do that, they had to look up the necessary dimensions for that ramp in order for them to make sure that they had enough for that horizontal run for that rise because the rise is a given when we know how tall it is from the driveway up to the doorstep. And so they have to figure out now how they're going to build a long enough ramp for it to be accessible for his scooter. So that's all dealing with proportional reasoning or proportional reasoning things.

Kim Montague :

Yes. Okay. I'm with you on that. Right. I did a ton of talking about proportional reasoning with my middle schooler this year, it feels like that's kind of in the land that he was living. And and we had a lot of fun talking about that. Um, but pretty sure soon he's going to get into content that I have not reasoned through yet.

Pam Harris :

Right? Because when you were in high school I think we've talked about before a little bit about that was a point in your life where, yeah, boys came along and you were like, yeah, math was fun. Now I'm just gonna kind of memorize it, spit it out and you got other things to do. Sure. So totally know you're not alone there.

Kim Montague :

So here's what I'm gonna do. I'm gonna sit back and just take some notes while you school me a little bit. You said there were three things that you're going to talk about today.

Pam Harris :

Yeah, so today I'm going to talk about three ways of reasoning functionally. I'm going to talk about slope, specifically rate to change, we're gonna talk about transformations. And then lastly, we'll talk about rational functions.

Kim Montague :

Okay, so you're going to talk about slope first. Let's go there.

Pam Harris :

Kim Montague :

So I can see how my proportional reasoning will help me better understand slope when I consider that it's not just one number. Yeah, nice. Cool, cool. Okay, so you said three things. Tell us about number two.

Pam Harris :

Kim Montague :

Sure, and it sounds like you can almost hear the difference between the student who's thinking functionally based on how they're describing what's happening to the parent function.

Pam Harris :

Yeah, absolutely. It's a great way to put it. Yeah.

Kim Montague :

Okay, so we talked about slope a little bit and talked about transformations, you said there was one more thing that you want to talk about.

Pam Harris :

Kim Montague :

Well, that's -

Pam Harris :

For you high school teachers out there, I hope you can sort of picture if you know something about the long run behaviors of polynomials, then you should be able to use what you know about ratios to think about the ratios of those longer behaviors to inform the long run behavior of the rational. Let me finish with one more thing. Also, those polynomials have short run behaviors what's happening in the short run like right around zero usually, well, I can use what I know about the short run behavior of those polynomials to also help me think about the short run behavior of the rational function. And you know that rational functions have this funky behavior. Kim, you probably remember from high school, where we have these asymptotes and things are like, going like they were sort of sucking up to these vertical asymptotes, all of that vertical asymptote. And the removable discontinuity behavior that's happening in a rational function is all based on the short run behavior of the polynomials, because that's the short run behavior of the rational function. So I'm not gonna go on too much more about that. But that is a way that you can sort of understand what I'm talking about with functional understanding functional reasoning, in high school math.

Kim Montague :

You know, and I'm sitting here thinking about how it's really clear to me that the stronger of a proportional reasoner you are, the better you're going to be able to make sense of functional reasoning because you're able to relate so much of it to that type of reasoning. I have written so many notes, and frankly, I wish that I had you for my high school math teacher because, while they were really nice and, and explained well, they I don't know, were tapping into what I made sense of already. So listen, I'm gonna ask you to expect some phone calls when my kids get to high school, because we're gonna need some more talking.

Pam Harris :