Ep 76: The 'As Close As It Gets' Instructional Routine

Pam Harris  00:02

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

Kim Montague  00:09

And I'm Kim Montague. 

Pam Harris  00:10

And we make the case that mathematizing is not about mimiking steps, or rote memorizing facts. But it's about thinking, about reasoning and creating and using mental relationships. Y'all, we take the strong stance that not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keep students from being the mathematicians they can be. And let's be clear, probably teachers too. We answered the question, if not algorithms and step by step procedures, then what?

Kim Montague  00:42

If you've been around for a little while, you know that we absolutely love Problem Strings as a really great routine. 

Pam Harris  00:51

Favorite routine.

Kim Montague  00:52

Definitely favorite, but there are other things that we love so much.

Pam Harris  00:56

Yeah, we've got some other favorite routines that we really like, that we have sort of got from other people, and we've kind of tweaked them and made them our own, that we like to use, not near as often as Probably Strings. But for sure, as definitely instructional routines that can happen in the classroom, to help students really reason and stop from just knee jerk performing steps. Because we want students thinking, looking to the numbers first, letting the numbers influence the way they choose to solve a strategy, not just knee jerk reactions. So when we started working together, Kim, a long time ago, and I started working with the teachers in my kids district, you were one of those teachers. And we found this routine in the book 10 Minute Math, and we sort of tweaked it a little bit. And we've kind of made it our own. But we thank 10 Minute Math for the good work that those folks did to give us the inspiration. And we call it 'As Close As It Gets'.  Alright. So As Close As It Gets is an instructional routine, which means that we want to do it often enough that it sort of becomes routine. And the purpose is to help students think, to let the numbers influence what they do. But without just knee-jerk, "Oh, I must compute." For example, Kim, let me just give you what I did as a student, whenever I was asked to estimate, or to round off an answer. I never did anything except do the steps of the algorithm, get the right answer, and then round it off, and then do an approximation and then say, "Yes, my approximation is reasonable." And it was so purposeless for me. I was like, "Why am I doing this? Like, I'm just gonna solve the problem." 

Kim Montague  02:33

Yeah. 

Pam Harris  02:33

Because I didn't own anything else. You're laughing because Kim never would have done that. 

Kim Montague  02:38

It's a real thing. 

Pam Harris  02:39

I mean, it was totally real.

Kim Montague  02:40

I mean, I've seen students do that. Yeah.

Pam Harris  02:42

And I was so frustrated. I was like, "Really, why are we doing this? This is the dumbest question ever. Just tell me to solve the problem. I'll just do the thing. I'll just do the steps to get the answer." So instead of that, what kinds of questions could we ask so that kids can't have that reaction? That will actually get them thinking and reasoning.

Kim Montague  02:57

Yep. So let's give some examples. 

Pam Harris  02:58

Alright.  I'm going to give you some examples so that you can share your thinking about how you decide which was the correct As Close As It Gets. What solution would you give? Alright, so here's your first example. You ready? You want to write this down?  Yes. Okay. Go. 

Kim Montague  03:13

So you have your pen. I know.  I do, actually. Right next to me.  And I have my pencil. Okay. So your first question is four fifths times 10 ninths.

Pam Harris  03:25

Okay, 4/5 times 10/9 is the question.

Kim Montague  03:28

Right so that's the question, so let me give you your choices.

Pam Harris  03:30

Okay. So it's like a multiple choice question.

Kim Montague  03:33

Yeah. Again, you're not you're not solving, you're thinking. So zero. 

Pam Harris  03:37

Okay.

Kim Montague  03:38

 0.5, or half. One, or two.

Pam Harris  03:42

So I can only choose those four choices.

Kim Montague  03:44

Correct.

Pam Harris  03:45

Like if I work out the problem, it's not - I don't have to do that. Because I'm looking for the answer that's as close to the correct answer. Yeah, cool. And it's not about rounding or estimating, it's about me like making sense of the problem. Alright, let's see four fifths of ten ninths, four fifths of ten ninths... Four fifths feels like almost one. Little bit less than one. So I want almost one of 10/9. 10/9s is a little bit more than one. So I want almost all of a little bit more than one. So to me, that sounds very much like one. So I'm choosing one as my strategy.  Or as my answer and that was my strategy. Cool. 

Kim Montague  04:26

Nicely done.

Pam Harris  04:27

Would you think about it any other way? I'm curious.

Kim Montague  04:29

Yeah. You know, I like percents. And so when I see four fifths, I just know 80%. So I would think 80% of a little more than one. And that's going to be about one.

Pam Harris  04:41

Gotcha. Cool. Yes, you are the percent girl. It's true.

Kim Montague  04:43

I do like percents. Okay, you ready for another one? 

Pam Harris  04:45

Yep.

Kim Montague  04:47

What if I asked you 45/46 - 

Pam Harris  04:52

Holy cow. 

Kim Montague  04:53

That ones hard to say. 45/46 divided by two.

Pam Harris  04:57

Okay, so the fraction 45 out of 46 divided by two. So fractions -

Kim Montague  05:05

You need choices. 

Pam Harris  05:06

Yeah, yeah, yeah.

Kim Montague  05:07

Zero; 1/4; 1/2; 3/4.

Pam Harris  05:11

Of course, the answers are all fractions. Okay. So I could choose to say to myself, "Well, it's fractions," so I could reach into rote memory and do a bunch of steps, or I could just say to myself, "I've got something divided by two." Well, that's just like cutting in half. Bam. So 45/46 is almost one. That is hard to say. And so I could just think about one - a little bit less than one - cut in half is about a half. So I'm saying a half. One half. Nicely done.  Did you think about it any other way?

Kim Montague  05:46

No, I thought about it the same way actually. 

Pam Harris  05:48

Alright, there you go. Cool. Bring it on. 

Kim Montague  05:50

Let's do another? 

Pam Harris  05:51

Yep.

Kim Montague  05:52

Alright, here we go. 5,003 minus point five.

Pam Harris  06:01

Okay, there's this -

Kim Montague  06:03

Choices, choices. And so I don't know that we mentioned, but when you show kids As Close As It Gets, when you experienced this routine, I have this question for you. Do you put the problem and the solutions all up at once? Or do you put the question, and then the answer choices separately?

Pam Harris  06:28

No, I do just what you did, which is not let me start thinking about it until I see my answer choices. And there's a pedagogical reason for that. Because I'm trying to not have kids just do the computation. I want to give them those choices so that they have the sort of sense of, "Oh, I could reason about those choices," without just like, starting to do a bunch of steps. And y'all we want to train kids this way for when they take stupid high stakes tests, like ACTs and SATs. Now you know how I feel about those tests. Because we want them to look at the question and the answers before they just knee-jerk start doing a bunch of steps or trying to remember a formula that somebody gave them before - but not even before instead of - like, we really don't want them trying to like reach into rote memory. We want them just reasoning about what's happening. So we can start that here. So if you were to show me this in class, I would see the problem 5003 - .5, but I would also see the answer choices at the same time. And you were about to give those.

Kim Montague  07:25

Yes. Okay, here we go. 5,000. 

Pam Harris  07:28

Okay. 

Kim Montague  07:28

5,001. 

Pam Harris  07:30

Okay. 

Kim Montague  07:30

1,000. 

Pam Harris  07:31

Okay. 

Kim Montague  07:32

4,999.

Pam Harris  07:34

4,999. Okay, so let's see, I'm noting that the 5,000, and the 5,001 and the 4,999 are all really close together. Which means I might have to get actually fairly like close with this answer. I can't just say it's about 5,000. Because I mean, is it really about 5000? I'm looking at 5,003. And I'm thinking it's about 5,000. And I'm subtracting 1/2, point five. So that's not very much at all. I could say, well, that's about 5,000. But I've got three numbers that are around 5,000. So I think I better get a little bit more picky. So 5,003 minus a half doesn't even get me to 5,002. So the largest choice you gave me was 5,001. So I'm calling it 5,001 is the best answer. I'm smiling a little bit at your choice of 1,000. Because I've seen for sure students where they would take a problem like this, and they would line up the numbers. And so it would be 5,003 minus five, but it would look like 5,000. And they would end up with like more like 1,000 or something. So yeah, absolutely. Good answer choice.

Kim Montague  08:44

I could absolutely see this routine being something like how would somebody get one of the other three choices?

Pam  08:50

Oh, nice. Nice iteration.  

Kim Montague  08:54

Alright, last one.  Ready?

Pam Harris  08:55

Yep, go.

Kim Montague  08:57

  992 + .3.

Pam Harris  08:57

992 plus point three. So you just really want me to sort of think about these relationships. What are my answer choices?

Kim Montague  09:09

993; 995; 1,000; 1,200.

Pam Harris  09:17

Okay, so let's see. Yeah, this would be good too - maybe I'll have you do that. You can talk about how all the answer choices -

Kim Montague  09:24

Oh, gosh, okay. 

Pam Harris  09:26

Okay. So 992 plus something that is not very big at all. Like it's less than one half. It's less than one, its less than one half. Is just about - I was about to say pertineer -its pertineer 900. That's a joke on Texas. 992 plus just a little bit is about 992. And my choices are, is it closer to 993 or 995? Well, I didn't even make it up to 993. So I think that's got to be my choice because it won't be as big as 995. Alright, So Kim, the 993 and 995 I think makes sense to me -

Kim Montague  10:02

But the 992, if somebody wasn't looking carefully at that, that's three tenths. And they called it three, then 992 plus three that's a pretty common error. That would be 995. I'm thinking again, if you're not thinking about place value, then you're thinking 900 plus 300. You're lining that point three up underneath the 900. You're gonna get something close to 1200.

Pam Harris  10:32

Bam. Yep, yep. And I'm also thinking 1,000 comes from if you just look at 992 and you say, well, that's almost 1,000. Then you can sort of round to the 1,000. Cool. Alright. We like As Close As It Gets, it can help students get out of the mode of this knee jerk reaction. So that they can really think and reason. So Kim, I gotta tell you, when we were thinking about doing this podcast episode, we were looking at problems in my book, the book that I wrote, Lessons and Activities for Building Powerful Numeracy. I realized just how much I've grown. Like, I was like, "These are the questions like those are are too easy." And you were like, "No, no, no." And I was like, "Kim, those are too easy." And you're like, "Pam, you've grown a lot." It was really interesting to me to feel that growth. I mean, I wrote the book, I wrote these problems. And now it's kind of cool. So we can all learn and grow more and more as we do more real math.

Kim Montague  11:29

Yeah. So where can we find these Pam? If people want to do As Close As It Gets, I'm going to tell you where you can find them. You can check them out on our website, we have an instructional hub that has problems that you can do. That would be found at bitly/instructrout.

Pam Harris  11:54

It's like 'instructional routine.' That's what I was thinking, don't make fun of my Bitly link. We'll definitely put that link in the show notes. But y'all, you could just go to the website, MathisFigureOutAble.com go to 'Learn Now', and under 'Learn Now' you'll see 'Instructional Routines'. And you can see more than just this one, but for sure this one's there. If you go to the instructional routine page, you'll see As Close As It Gets. And then lots of examples that we have put out already that you can just flat out, take off the website, use with your classes, you can also click on each individual one and go see what people have said on social media about each particular question with its answer. So we like that routine. So if you want to learn more mathematics and refine your math teaching so that you and your 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!