Math is Figure-Out-Able with Pam Harris

Ep 99: If Not Algorithms, Then What? Finale!

May 10, 2022 Pam Harris Episode 99
Math is Figure-Out-Able with Pam Harris
Ep 99: If Not Algorithms, Then What? Finale!
Show Notes Transcript

In the last 5 episodes we've laid out the most important strategies for students to be able to solve for all four operations. So what? In this episode Pam and Kim solve a gnarly problem to demonstrate how mathematizing can empower you and your students, and open up to the world of Real Math.
Talking Points:

  • How to use the BIG downloadable e-book
  • A gnarly problem solved using Real Math
  • It's all about efficiency and developing more sophisticated reasoning
  • Partial strategies are a great starting point but not the ending place
  • Kudos to Pam's research and thoughtfulness
  • Why does Pam seem to focus on computation?
  • What believers are saying
  • Other ways to learn with Pam
  • We can make higher math more accessible
  • BONUS: Kim's failed attempt at the standard algorithm

Get the BIG download here! https://mathisfigureoutable.com/big

Pam Harris:

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

Kim Montague:

And I'm Kim Montague.

Pam Harris:

And we make the case that mathematizing is not about mimicking (unclear). Let me try that again. And we make the case that mathematizing is not about mimicking steps, or rote memorizing facts, but it's about thinking and reasoning about creating and using mental relationships. People don't realize we record this every time. We take a strong stance, and not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keep students from being the mathematicians they can be. We

answer the question:

If not algorithms and step by step procedures, then what?

Kim Montague:

Okay, so today, we're wrapping up our series on the major important strategies for each of the four operations. And, listen, we have been giving you this amazing download, this free ebook that has all kinds of goodies in it. And we want to take just a minute to mention how we anticipate, how we think that you will use that or not use that. So what we've said before, add that we want to reiterate is we don't think that you're going to run off a copy of each of these for your students. We don't think -

Pam Harris:

Please, please don't.

Kim Montague:

Please, please don't. We don't think that you're going to -

Pam Harris:

In fact, Kim, before you go on, sometimes when we have a teacher handout at a workshop, I will kind of jokingly say, "So right up on the top of this write: Not for student use."

Kim Montague:

Yeah.

Pam Harris:

And it always cracks me up a little bit. Because people will dutifully 'write not for student' - I just say that every time you know, like, here's your hint, this is not for student use. This is for teacher use. This is teacher learning, teacher understanding, and then you're going to use it as you teach students.

Kim Montague:

Yeah, so absolutely. This is really about your learning. And it's really an opportunity for you to study and to say, "Man, which of these strategies do I find myself regularly using? Do I understand each of the strategies? Could I find problems where I would use each of the strategies?" I really feel like a way I would use it is I would study it. I'd put it away. And I would say to myself, "I'm going to dive into MathStratChat, I'm going to dive into looking for problems in the world." And then after a while, I think I'd pull that booklet back out. And I would say, "Oh, I use 5 is Half of 10 a lot. I use the Over strategy a lot. But I'm not really finding myself using Equivalent Ratio a lot. Let me re-familiarize myself with that. Let me put it away. Let me focus on that particular strategy." Go do some strings. Yeah

Pam Harris:

Go do a lot of Problems Strings with that with your students, so that you get used to numbers that it makes sense to do that particular strategy. You build those relationships in your brain, you're building in your students' brain. Ah, all of a sudden, that strategy is going to ping for you because those relationships are created now mentally.

Kim Montague:

Yeah. Another note is that for teachers and leaders, we want you to build yourselves alongside students. this is not the thing where you have to own every single strategy. And then you can teach strategies through Problem Strings. You are -

Pam Harris:

Not that. Not that.

Kim Montague:

Not that. Right? It's not be perfect, know all the things before you can actually get started. But there is a little bit of some relief maybe about knowing just a bit more, at least owning the fact that there are alternative strategies so that you can be open to listening to your students, and figuring out which relationships are worth chasing down.

Pam Harris:

Yeah, that's really well said, because when you know that these are the major ones, then you have sort of a sense as you're listening to students, Ah, yep, I have I, you're gonna be able to listen, you're gonna be able to hear what they're doing the relationships they're using a little bit easier because you have created those relationships. And let's say that you don't have them all, yet, as you listen to students, you're going to create them better in yourself. But you're starting with this notion that there are these major relationships. And like Kim said, then you'll have a better sense and you'll learn more as you go which strategies you want to chase down, which ones are going to be more fruitful for everybody to compare to and listen to and construct in class.

Kim Montague:

Yep. And if you're listening for the first time, and you're like, "Wait, I don't know what you're talking about." You are gonna want to go right now to download that ebook, that resource, that fabulous download at mathisFigureOutAble.com/big because it is incredible.

Pam Harris:

Because it is big and incredible. And yeah, B-I-G. Great place to go. And if you'd like more experience in creating these mental relationships in your own head, because that's our favorite way to do it then join us, ready? Tomorrow. If you're listening to the podcast on the day it came out on May 10. That's when this podcast drops. Then you can join us tomorrow and then you can Change Math Class Challenge begins May 11, 2022. You need to register. Register at Mathis Figure-Out-Able dot com slash change mathisFigureOutAble.com/change is where you register for the You Can Change Math Class Challenge on May 11, where we will continue to help you develop the most important strategies for students and you to learn so that computation becomes a natural outcome and just naturally let things ping in your brain and bam, you are computing with the best of them. Okay, Kim, we thought it would be fun today to launch a fairly gnarly problem. And I'm when I say 'gnarly' it's one of those that if you look at your like blaaah, like the numbers just don't look particularly, maybe at first glance, don't particularly look all that like lovely. You know, if I say, "Hey, Kim, here's a really hard problem. 100 times 49." You're going to be like, "What? Like, that's easy." or like, it's readily. So these are problems are not like that. They are gnarly, you look at them, you're gonna have to, like, think for a second, like, what could be your strategy. And in doing that, we thought we would parse out the fact that all of the major strategies we've been talking about for addition, subtraction, multiplication, division, how they sort of all come together. We've solved lots of problems on the podcast, but we've been a little less explicit about all of the things we're thinking about when we solve them. So today, we're going to attempt to be really explicit.

Kim Montague:

Okay. Say all the things.

Pam Harris:

Say all the things. Yeah, like get our, I have the other meaning of explicit in my head right now. What am I?

Kim Montague:

Metacognitive.

Pam Harris:

What do you say?

Kim Montague:

Metacognitive? And then like, say it out loud.

Pam Harris:

Yes. Be very metacognitive. Think about our thinking. Alright, so, Kim, here's your question. How would you solve this lovely, because it is a lovely, a lovely problem. It took me a minute to come up with this problem, because I really wanted one that the people are going to listen to and go blah. 49 times 67. Tadaa.

Kim Montague:

Okay, I just just wrote it down. Oh, yeah, 49 times 67.

Pam Harris:

Oh, yeah. Because Einstein said paper is for the things you need to remember. And your brain is for thinking.

Kim Montague:

I'm thinking. Like, I'm looking at it for a second.

Pam Harris:

Yeah.

Kim Montague:

So um, this is me saying that. Okay.

Pam Harris:

And that's legal, right? Let's be really clear, it is legal. I remember, while you're thinking, when I was the math student, I knew in my heart of hearts that what I was supposed to do was see a problem and instantly know what am I supposed to do? And we're saying: not true. In fact, what mathematicians do as mathematicians look at a problem and they consider, they play around with things. So Kim, what were you just thinking?

Kim Montague:

I was looking at 49 and 67. And I was wondering which one I want to mess with?

Pam Harris:

Say more.

Kim Montague:

So I don't love 67.

Pam Harris:

When you say 'mess with', I'm going to be really explicit. You were wondering, do you want to find forty-nine 67s or -

Kim Montague:

Or sixty-seven 49s.

Pam Harris:

Thank you.

Kim Montague:

And 49 is near some things that I like. So I'm thinking that I want to mess with forty-nine 67s rather than sixty-seven 49s because I can alter that 49 to my liking. Like I can mess with it a little bit more readily.

Pam Harris:

Okay.

Kim Montague:

So I'm gonna go with forty-nine 67s. Okay?

Pam Harris:

We shall let you.

Kim Montague:

Okay, thank you. So I'm actually thinking, I want to do a hundred 67s. And I'm going to tell you why in just a second if you'll let me. So I know a hundred 67s because I know how to scale up by 10, and then by 10, again. I'm gonna say that a hundred 67s is 6700. So is that clear enough? Good on that one?

Pam Harris:

Yeah, that's good.

Kim Montague:

So then the reason I went to 100, is because I wanted to eventually get to fifty 67s. And so I went way over, so that I could do 5 is Half a 10. I get fifty 67s.

Pam Harris:

That makes sense.

Kim Montague:

So then I know that I need to halve the 6700.

Pam Harris:

Because you halved 100 to get to 50.

Kim Montague:

And I actually went from left to right there. And I halved 6000 to get 3000. And I halved 700 to get 350.

Pam Harris:

So half of 6700 is -

Kim Montague:

3350.

Pam Harris:

So if I can pause you for just a second, because I wonder if listeners might find it noteworthy that as you're talking, I have written down a ratio table. So I have one to 67.

Kim Montague:

Okay.

Pam Harris:

100 to 6750 and 50 to 3350.

Kim Montague:

Interesting.

Pam Harris:

What do you have on your paper?

Kim Montague:

I wrote down equations actually.

Pam Harris:

I can see that. I've seen you do that before. So you have like 100 times 67 equals 6700.

Kim Montague:

And then 50 times 6. Yeah. Okay. And as I'm saying what I'm saying I'm underlining to make sure that I've included the words.

Pam Harris:

Alright, so you're now at fifty 67s.

Kim Montague:

Yeah. And so then I know that I have fifty 67s and I'm going to just go down a 67, I put down on my paper.

Pam Harris:

Why?

Kim Montague:

I'm going to subtract because I need forty-nine 67s. And I already have 50 of them. So I just need 49 of them so I can subtract one 67. Yeah.

Pam Harris:

From?

Kim Montague:

From 3350.

Pam Harris:

Cool. And how are you going to subtract 67 from 3350.

Kim Montague:

Yeah, at that point, I subtracted 50. To Get to a Friendly Number. So I started with 3350. And I subtracted the 50. I actually wrote it on a number line, subtract 50 to get to 3300. But I still need to subtract, my total to subtract was 67. And I did 50. So I still need to subtract 17. And so I'm at 3300, and then subtract 17. And at that point, I just knew it was 3283.

Pam Harris:

And how do you know?

Kim Montague:

Well, yeah, so I know my combinations of 100 really well, because I play I Have, You Need a lot. But I could have subtracted 10 and then subtracted seven if I wasn't sure.

Pam Harris:

So 3300 subtract 10 would be 3290. Subtract 3 more would be 3283. And you also could have subtracted from that 3300. You could have subtracted 20. What is that?

Kim Montague:

You want me to do that? So 3300 minus 20, will be 3280. And then because I've subtracted too much, I did 3 too much, then I could add 3 to the 3280 and get 3283.

Pam Harris:

So you can use sort of an Over subtraction strategy to do that?

Kim Montague:

Yeah.

Pam Harris:

So you're saying that forty-nine 67s is -

Kim Montague:

3283.

Pam Harris:

And in solving that you used the 5 is Half a 10 multiplication strategy.

Kim Montague:

Over.

Pam Harris:

Over multiplication strategy. And then you use partners of 100 to do the subtraction. But you could have done an Over subtraction strategy.

Kim Montague:

Well, first I did Remove to a Friendly Number.

Pam Harris:

True.

Kim Montague:

Well, I actually, interestingly, because of what I had, I Removed a Friendly Number that also Got me to a Friendly Number.

Pam Harris:

Sure enough, yep. Yep.

Kim Montague:

And then we could have done Over subtract, or I knew partners of 100. Yeah.

Pam Harris:

Yeah. So I don't know if anybody finds that interesting that we just kind of wanted to do a problem where we sort of parsed out all the things that we're thinking about. But Kim, let's be clear, like if you hadn't had to say all that stuff?

Kim Montague:

Oh, yeah.

Pam Harris:

Do you think? How many?

Kim Montague:

It would be -

Pam Harris:

Go ahead.

Kim Montague:

It would have been much quicker. Not that speed is everything. But you slowing me down to make me say all the things might feel to our listeners like, "Wow, that took a really long time."

Pam Harris:

You really had to slog through all that. Wow, that was a lot of effort?

Kim Montague:

But no, it would not have been.

Pam Harris:

So if I may, in actuality, you looked at a problem and you said, "Forty-nine 67s. I'd rather do that than sixty-seven 49s." Because you knew 49 was almost 50. And bam, 50 is half of 100. So you found 100 of them, halved it to get 50 of them, took one of them away. And you're done.

Kim Montague:

Yeah, yeah.

Pam Harris:

And in reality, that was fairly, it was very efficient, and far fewer steps than if you would have done all of the single digit multiplication that would have been needed to do the traditional algorithm, with all those opportunities for errors. And also the opportunity that you could have been using Additive Reasoning, or at best Multiplicative Reasoning with single digits. And then instead, you're using Multiplicative and Additive Reasoning as you quickly or efficiently and using sophisticated strategies, were able to solve what I think is a pretty gnarly problem.

Kim Montague:

Yeah.

Pam Harris:

Nicely done.

Kim Montague:

Can I tell you that you know you just said less steps? I think that if you had said, "You have to do the traditional algorithm." I would have been very uncomfortable, not because I don't love it. But because it has been so long since I have, that I would not have known if my answer was correct. Like, I feel so sure about the fact that because I'm thinking about the entire amount that I'm dealing with.

Pam Harris:

The magnitudes involved.

Kim Montague:

Yes, I would be very unsure whether my answer was correct using an algorithm.

Pam Harris:

So it's not like you couldn't have done it. You could have done the steps and done all the things or whatever. But at the end of it, you're like, "If I would have made just one little error in any of that." The magnitude of your answer could be completely crazy off.

Kim Montague:

Right.

Pam Harris:

But instead, you're like, "No, I'm really clear. 6700, 3350, just one less 67. We're done.

Kim Montague:

Yeah.

Pam Harris:

Yeah. And that's fascinating to me that sometimes teachers get so upset about how their kids' answers aren't reasonable. And my suggestion is we've taken them out of reasoning land to begin with. If we stay in reasoning land, we're so much more confident that the answer is reasonable. Because we're reasoning the entire way along. Nicely done. Anything else before we do this next part?

Kim Montague:

Nope, I think I'm good.

Pam Harris:

So one of the things I wanted to emphasize as we sort of wrap up our series on this idea that there are these major relationships that if a person has developed the relationship so that these strategies are, they own these major strategies, then they can solve any problem that's reasonable to solve without a calculator. That, as we do that, I just want to bring out the point that too many places, what we see are people stop with the partials. We have programs out there and whole whole sort of fields of thought where, "Okay, give kids some conceptual understanding that we're going to do partial sums, partial differences, partial products, partial quotients." And once we've done those, "Okay, now our next step, bam, is the algorithm." And we either then see kids doing these really inefficient partials. Partials are good, and they're unnecessary starting place, but they can't be the ending place. But also, the ending place is not the traditional algorithm. The ending place is, really I should even say 'the ending place', because we just keep getting more and more sophisticated, that after we've got the partials, then we continue to build kids sophistication. We continue to help them think about bigger and bigger jumps, bigger and fewer jumps, bigger and fewer chunks. And as they do that their brains literally have more and more neural connections that then we can build on as we want to think even more multiplicatively as we build Proportional Reasoning.

Kim Montague:

Yeah.

Pam Harris:

So we cannot stop with just the partials. Go ahead.

Kim Montague:

Yeah, and I want to give you a lot of props, a lot of credit to my thinking, and to a lot of people's thinking in that. Because I know that you spent a lot of time looking that at all that's, the kind of put out in the world. And that you have been very clear for a very long time that either the work was conceptually based and just kind of stopped at partials, or that it was all about the algorithm in the end, right? And I remember you, I mean, you've come to me and asked me how I've solved stuff. And you know, I've given you kind of my gut response. But then you have taken that and you've asked, like all the people. And you've known that we need to generalize and create a system and like this cohesive plan, so that there's no holes. Because your expertise, you know, started with high school. And so you knew and have known that, like all this work that we do, and the elementary has to be carried through. It's gotta make sense in higher math. And, you know, I appreciate what you've given to me, but also to the world in that arena.

Pam Harris:

Yeah, I was really clear that we had to think

through:

Is this just about getting kids kind of some conceptual stuff and now they just repeat these memorized steps? Or can we leave kids in partials and call that good enough? And I mean, I knew instantly that wasn't the answer, because I knew where higher math went, like I knew we needed to get more sophisticated than just partials, that we were going to have to end up at algorithms. And so thank you. I had to look at and do all the research to make sure. Like, what are those major relationships? And if we, could we really, and sure enough we can. We can identify the major relationships that lead to these major strategies. And then we can give students any problem that's reasonable to solve without a calculator. And not only can the kid solve that problem, as efficiently as a traditional algorithm most of the time more efficiently than any of the traditional algorithms. But more importantly, they are building the relationships and connections so that they are thinking and reasoning more sophisticatedly, because then upon that we can then reason, even more sophisticatedly. If we don't have the system that we leave kids either too unsophisticated, and then we're stuck at higher math. We can't build on what they didn't build earlier. And so I feel like that's really important. And Kim, as you bring that up, I want to mention that there's a follower that we have on Twitter, I might slaughter your name Tad and sorry, Tad Watanabe, I think, is fantastic. And we really appreciate the fact that he listened to the podcast. He writes a blog post back. And because it pokes on some of the stuff that we say we have these great conversations on Twitter. I love how deeply you're thinking about all the things. One of the things that he asked is why we are emphasizing computation so much. And I kind of push back and I was like are you saying that I'm only about - In fact, he said, "Calculation strategies as nice and useful for and nice and useful for mental calculations are only a small component of mathematics. Students need to learn." in his opinion. He said, "in my opinion." And I said back to him, "Are you are you saying that you think I'm only about computation?" Because that was kind of surprising to me, because I'm not so only about computation. But then he said back to me, "I wouldn't say that you're only about computation. But a vast (question mark), majority of your podcast time is about computation." I had to think about that.

Kim Montague:

Yeah.

Pam Harris:

And can we talk about it? And you know what? Sure enough, it is. And let me tell you why. So there is a reason that upon first contact with me, you will hear me talk so much about computation. Because if we don't, if I can't convince you that it really isn't about just get some conceptual stuff, and then go to algorithms that that's not mathematizing. If I can't convince you of that, then teachers are gonna, and this has been my experience, teachers will look at and say, "Well, Pam, your stuff is all really nice and everything, but it's not necessary." And then good people look at really good work. They look at all this stuff that's out there about building conceptual understanding, and I don't blame them. And good people will say, "I mean, sure, but I don't waste my time doing that stuff. Because I'm really clear that kids need algorithms to compute. And so since they need step by step procedures to compute, I got to spend time doing it. Because if I don't, kids aren't good at them. And so I spend the time doing them, I show them to kids, make sure that kids can do the algorithms." And here's what we get. What we get is that students who are immature decision makers, immature choosers, in that moment will choose to do the algorithms. Because to be clear, those step by step procedures are easier than thinking. They are easier than students grappling with these big ideas. Because the students grapple that's hard, and it's uncomfortable, and they have to, like literally create new neural connections. And that work takes effort. Immature students will say, "I'd rather just do the step by step stuff over here." And then they will choose them. And unfortunately, that unwittingly cuts those students off from higher math. Students will immaturely make a decision going, "Well, I'd rather just do this easier step by step stuff rather than, ah, you're making me think over here miss. Quit making me think." And they'll unwittingly cut themselves off from higher math, and then less and less math will make sense to them. And we will continue to get what we've gotten. I feel so strongly that we need to help everybody understand what mathematics actually is. It is not just repeating these steps. Because if we just repeat steps, we then cut ourselves off unwittingly, from access to understanding higher math. And then all we can do is we're left with steps and we can only solve problems that we've seen before. Kim, one of the funniest things that is happening lately, we are right now in the middle of a workshop cycle. So we have online workshops, where that's where we get really in dive deep into building these relationships. And I love hearing from workshop participants, because I'm active in the message boards. And I love hearing. I hear so often, where participants will say things like my brain is changing. This is the change we need. Now we can get about teaching differently, because we think differently. In fact, Sarah just said, and I'm quoting, "I love the idea of saying goodbye to the traditional algorithm. The more I move away from the algorithm in my class, the more successful and confident I am seeing my students become." And Robert just said, it was so cool. He just said, "It's still not hard. (No, sorry)." He said, "It's still hard to not revert immediately to a standard algorithm. But" and in all caps he wrote, "BUT the knee jerk reaction is wearing out. And my new math mine says, 'what relationships do I know that can simplify this problem to something requiring less for formulaic thinking', plus," I'm still quoting Robert, "finding relationships is more fun than any standard algorithm. To me, it's more like a puzzle to solve. Not a series of dry steps." Oh, it's so much fun to listen to people who are beginning to do the transformation that I did, where all of a sudden, I realized, wow, mathematics is this other world and it is so cool, and we want to invite you into it. And that's why we're making such a big deal about computation and about these particular strategies. Because if I can convince you here, then we can open up the rest of the world of Real Math. Hey, speaking of workshops you just mentioned, we would love to invite everyone to take what we call DMR the Developing Mathematical Reasoning workshop. Totally free.

Kim Montague:

Totally free. In fact, it is at mathisFigureOutAble.com/freeworkshop. It's so fantastic. You'll love it. And if you've already done DMR then you should join Journey, which is your oh gosh, what do I say.

Pam Harris:

Our signature implementation support product. That is where we join with teachers and leaders all around the world as we are all working together to teach more and more Real Math.

Kim Montague:

Yeah, you can find that at mathisFigureOutAble.com/journey.

Pam Harris:

And we sure appreciate all of our Journey members. We have a blast with them. Alright y'all in conclusion, it is about looking for and making use of patterns and that creates mental relationships. And this is the most important part that as we create those mental relationships. And we get more and more clear on using those relationships to solve problems. And as we get more clear, then we call those strategies. And as we define and describe those strategies that helps cement and make those pathways stronger. It is not about direct teaching strategies. It's about developing brains to own more and more connections and relationships, becoming more and more dense with those relationships and connections and then solving more and more complicated problems. And that process just continues on and on. And that is mathematics. So if you're interested, don't forget to download the big resource, this ebook that we have written that contains all this about models and strategies and strategies versus algorithms. And what are the examples of the major important strategies for all of the operations at mathisFigureOutAble.com/big. And join the You Can Change Math Class Challenge at math is Figure-Out-Able.com/change, which starts tomorrow, May 11. So make sure that you register. Join us on a You Can Change Math Class Challenge. It's a blast. I love meeting people from all around the world on that challenge. So if you want to learn more mathematics, and refined your math teaching, so that you and students are mathematizing, more and more, then join the Math is Figure-Out-Able movement and help us spread the word that Math is Figure-Out-Able.

Kim Montague:

This is like a bonus that we get.

Pam Harris:

We shut off the episode and Kim keeps talking. She says to me, go ahead, are you, can you talk? You are crying laughing.

Kim Montague:

I am. I said, "Do you want to know what I was doing while you were talking there at the end? I wrote down 49 times 67 and did the traditional algorithm and -

Pam Harris:

You just decided to have fun. You're just gonna play with the algorithm for a little bit, because it is not really fun.

Kim Montague:

Because to be fair, it has been - (pauses) - Holy cow - 15 years?

Pam Harris:

A few years.

Kim Montague:

I mean, maybe longer since I literally wrote a traditional algorithm for multiplication. So I'm out of practice. And I wrote 49 times 67 and got it wrong.

Pam Harris:

Then I said, "We got to record this." So I hit the record button. Because I'm like, Kim, I want to hear. What did you do wrong?

Kim Montague:

Here's what I did wrong.

Pam Harris:

You don't remember a fact correctly? You wrote the zero in the wrong place?

Kim Montague:

I did every algorithm, well, two things that

I did wrong. One:

when I said 9 times 7 is 63. In my head, I was thinking of the value 63. And you know what I wrote first? The six. I was thinking 63. And so I put six and I carried the three and in my mind right now I'm thinking, I wonder if that's why kids do that. I wonder if they're thinking 63. And they write 60, carry the three. I've never considered that before.

Pam Harris:

That's fascinating. Yeah, absolutely. What was the other thing you did wrong? Um, so then when I wrote the zero, whatever, moved on to the next step, I carried a five. And you know what I did? I counted. I tapped it five times. Back to Counting strategies. So you didn't really do that wrong, but you found yourself doing something less sophisticated.

Kim Montague:

I found myself like reverting to, I'm putting like, thought behind. 'Did I do that correctly?' and I went back to counting strategies. Absolutely.

Pam Harris:

I think that's fascinating.

Kim Montague:

Yeah. So I got 3256 for my answer, and that would have been a big fat red X.

Pam Harris:

And no credit for all that you did right.

Kim Montague:

And no understanding of what I did right or incorrectly.

Pam Harris:

Yeah. No understanding that. Yeah. Your teacher would have just been like, "Try, do it again."

Kim Montague:

Yeah.

Pam Harris:

Yeah. And also, without any relationships, you wouldn't have had an idea if you were reasonable or not. Right. Alright. Well, I don't know if you guys were listening to the end of this episode, because we kept talking. There's your bonus to the end of that episode. Standard algorithms stink, because they don't build mathematicians.