Happy holidays! Pam and Kim invite you to revisit Ep 87, it's super important! How do you navigate teaching math that is figure-out-able if your standards include the teaching of algorithms?
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Hey, fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.
And I'm Kim.
And you are in a place where math 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, where you notice patterns, and reason using mathematical relationships. We can mentor students to think and reason like mathematicians do. Ya'll, not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keeps students from being the mathematicians they can be.
It's holiday season!
And almost your birthday.
Oh, sure enough. That's coming up. Yay. Hey, and this is a particularly good birthday because I have been 29.
And I told my kids, like just since I've been 29. Pretty much I just stayed 29. And I told my children, "As soon as you give me grandchildren, I will be 39.
So, this year, I'm 39.
You're turning 39. Fantastic.
I am a grandma twice. And yeah, it's been a great year for.
Super, super good year for your family.
Loving it. Yep.
So, we're wrapping up this year and looking forward to 2024. And this week, we decided to do a throwback to a really important episode that we want to be sure that you have not missed. If you've already tuned in, give it a listen anyway because sometimes hearing it twice is fantastic. We're all learning and growing, and you might hear it differently this time. So, this one is about a real argument or concern that many teachers have.
Yeah. Many, if not most teachers, have standards that say something like, "Students will use algorithms and strategies based on place value and properties..." blah, blah, blah "...to multiply and divide (unclear)."
Or something like that. So, it's a real question. How do you navigate that if I'm suggesting, we are suggesting at Math is Figure-Out-Able, that students do not ever need to be taught algorithms as a thing to do, how do you navigate that standard?
Yeah. And that's a really important distinction that you just mentioned. It's as a thing to do. So, does that mean that students are taught them some other way?
Kim, it's almost like we're on the same page. Yes! And you are going to want to listen to this episode to hear all about that. So, here is episode 87. But, "Pam, I'm required to teach algorithms!" Alright, ya'll, enjoy. Hey, fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able! I'm Pam.
And I'm Kim.
And we make the case that mathematizing is not about mimicking steps or rote memorizing facts. But it's about thinking, reasoning, and about creating and using mental relationships. We take a strong stance that not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keeps students from being the mathematicians they can be. We answer the question, "If not algorithms and step-by-step procedures, then what?"
In the last episode, we talked about how pouring algorithms into kids heads, and then having them use them is not mentoring mathematicians. It's not really helpful in teaching math. But, Pam, teachers are required by standards to teach algorithms. So...
I mean... Okay, nevermind. So, you're in a place where your standards say, "Students will use strategies and algorithms to solve multiplication problems." Okay. Then, I cry uncle. Go ahead. I'm kidding. No! No! No! So, how do we make sense of that? Like, am I telling you to just, you know, forget your standards? Ignore them? Your kids are going to bomb your high stakes tests? Is that what we're suggesting? And I think this is so important, that we're going to dial down on this today. So, we're going to really attack, if your standards call for a formula, a step-by-step procedure, an algorithm, and you're thinking to yourself, "Well, then I can't do the thing." Like, "There. I'm going to have to do that. And I'll do that with other areas of math, Pam. That will be fun. We'll make other areas fun." It's not about fun. It's about real math, what mathematicians actually do. So, what does that mean? Well, here's the way we attack it. If you were to have looked in Kim's third grade, fourth grade, fifth grade math class, you would have seen all year long working on helping students make mental connections, increase their sophistication of relationships, learn strategies that are based on all those relationships, and kids getting more and more sophisticated by solving more and more sophisticated, more difficult problems, by looking at all the different ins and outs, by comparing strategies, kids getting more and more confident, learning intuition, that when they see a problem, they're like, "Oh, like based on those numbers, that structure, oh, I'm using that strategy." Where another student might go, "Well, actually. How about this strategy?" And they were, "Oh! Wow! You're right. That one's so good!" And it's this conversation and learning and growing about which strategy to use when because of which relationships are inherent in the problem, which things are pinging for kids on a certain day at a certain time, and how we can use that to make sense and figure out problems. And, ya'll, pick a grade. Let's pick fourth grade. All year long. Kim is doing that with addition, addition and subtraction of harder numbers than they did in third grade. Multiplication. Really because the third grade starts it, but fourth grade is really where you dive into it. And so, Kim is spending a lot of time in fourth grade really developing multiplication, and multiplicative reasoners, and kids that can think about not just naked number problems but problems in context. At the same time, she's really developing area, and what it means to have length and width, and how that creates the square unit of measure. Which is so weird for kids because they finally got used to, "Okay, I'm going to measure this linear thing, this kind of like meters, and feet, and inches, and centimeters, this miles, and kilometers. This linear measurement. Now, you want me to measure it in squares? What?!" You felt this, listeners, when you've ever ordered carpet or tile. When all of a sudden they're giving you these crazy measurements, and then they're talking about how, "Well we'd have to, you know, make a seam here." Like, if you don't understand area, have fun with that because it's a different sort of measurement to have this kind of square unit. And then, if you've ever ordered topsoil or any kind of like. If you've ever landscaped and you're ordering, I don't know, mulch or whatever, and they're going to sell you a yard? Now, that's in the states. I probably should have looked up what it would be in metric. But if they bring you like this measurement of topsoil or some sort of cubic measurement? I will never forget. I was like, "Three yards. We're going to need more than that. Like, we have a whole space to fill." And the guy looked at me, and he's like, "What's your space dimensions?" and treated me like I was a dingdong. Because in that moment, I was acting like one because I didn't have the sense of what it meant to have this volume, this three-dimensional measurement. So, in fourth grade, Kim's really building this sense of a square unit of measurement, like really dealing with area. Fifth grade, she's going to do volume. At least for our standards. All year long, that kind of stuff is happening. Then, kids are going to take the stupid high stakes test. And now, you know how I feel about the test. They're going to take it, and they're going to do really well because her students have been thinking, and reasoning, and figuring, hitting new situations over and over again. Used to that when they get a story problem or word problem they've never seen, alright we dig in. We've been doing it all year. We just dig in, and we figure it out. It's not reaching in rote memory. "Okay. Is that formula? Is it the one we did on Tuesday? Maybe it's the one we did on Wednesday." And the teacher's like, "No! It's Thursday, for heaven's sakes. It's the formula for Thursday!" None of that. None of that. Not that thing. Kids have been reasoning all year long. So, now, when they hit that high stakes test, Kim's students are going to dive in, and they're going to think and reason. They're going to use their most sophisticated strategy that they've developed because that's what they do every time they solve problems. And nothing's going to surprise them because they're constantly in this place of just trying new things, and learning to struggle productively, and working to use what they know. Like, they are so clear, it's about using what they know. Test goes by. Kids do great. Kim has not only the sort of passing standard. Well, Kim, I know she's hating that I'm doing this. But it's true that her passing standard was like pert near perfect. Like all of her kids passed. And then, whatever that score is that means these are the kids that are scoring better than just the passing scores. So, whether it's "commended" or whatever you call it in your state. She has the best scores of that. Like, so many of her kids are doing so much better than just that passing standard. And so, the test is over. Kids aren't freaked. Kids aren't stressed. Like, it's not this big panic. Now, she works hard to help them not... What? Ingest that panic from all the other classrooms around the school because everybody else is freaking out. But she's like, "No, we're good. We just do our thing." Test passes. It's the last week of school. And Kim's like, "Hey, guys, we got a new strategy for you. Check this out." And then, not as a series of things to do. I might say this four times. But as a study in why it works, Kim will put up an algorithm. So, say it's fourth grade. She'll say, "Hey, have you ever seen anybody do this?" She'll write down 47 times 99, and she'll line them up, and she'll put the little x in the line. And then, she'll go, "Sometimes people have learned to do this. You might have seen your parents do this. Maybe your older siblings. Where they'll take this number over here, and then multiply it by that number." So, I don't know. What did I just say? 47 times 99. So, they'll take that 7 times the 9, and we know that 63. "And so, they'll write down just the 3 here, and they'll put that 6 up here. Weird, right? Anybody have any ideas why they would do that?" And the kids will kind of fuss. And they may or may not come up with anything brilliant at that moment. I mean, they're like, "Well, we see the 63. There's the 3. A little weird they put the 60 over there. That's kind of weird." And then, Kim might not spend too much time there because they need a little bit more to kind of see what's going on. And then, she'll sort of do the next step. She's not saying, "Now, students, all do this next step." No, no. She's saying, "Somebody might do this." And so, I just had to write it down because I can't keep track of the algorithm in my head. What did I say? 47 times 99?
So, now, they're going to think 9 times 4, right? And so, she'll say, Now, they'll say 9 times 4, and then they'll get 36, and they'll add 6 to that." So, 9 times 4 is 36, add 6 to that, it's 42. "And then, they'll just write that 42 down next to the 3. Whoa! That's weird. Ya'll, what are they doing? Does that work? Why does that work? Draw an array. Find those pieces that just happened. Like, can you find something that is 400 and... Help me. 23? Can you find like some chunks of area that would add to 423? Where does that 6 when they just do the 9 times 4. And then, the kids, because they have great place value, because they have been developing place value all year long, look at that, and go, "Oh, it's like they're sort of doing 9 times 4 to help them get 9 times 40. But they know they've got that extra 60 hang along. But they're doing 9 times 4 and adding the 6. Wow, that is like 9 times 40 and adding 60. But wow, the meaning is kind of hidden behind that. I mean, that's really cool. Woah, I wonder who figured that out. Okay." And then, Kim will show the next step. And the kids, because they have great place value, will be able to figure out why it works and how it works. And by the end of sort of showing some steps, the kids will say something like, "Whoa, that is interesting. Like, all those... I mean, okay. Like, all the meaning is kind of behind the scenes. And it's kind of hard to like feel the place value. Like, somebody has to be really brilliant to figure that out. But are you saying people actually do that every time? Bleh! Like, for that problem, wouldn't it just be easier to find one hundred 47s and subtract a 47? I mean, that's a lot of steps! With all that meaning that's kind of hidden and all the... Wow, that's really. Like, why would anybody do that if they could just think about it?" Now, you might be like. Listeners, right now, you might be like, "Pam, really. Her kids said that?" Well, I'll tell you what. Really, my kids said that. My personal kids that had Kim as a teacher literally would look at an algorithm after seeing some steps, and they would literally be like, "Whoa, that is really intense. All that meaning sort of shoved behind the scenes. You can't really tell what's going on. I mean, I guess you could do that. But why would you if you can just think about it? Why would you stuff your memory full of all that stuff to rote memorize? Why not just think about the numbers? Just use what you know to solve the problems. Okay." And then, they would just move on. And it doesn't bother them because that point there's such good thinkers that it's just one more sort of thing to make sense of, and then kind of move on. And it doesn't become this crutch. But it doesn't become this sort of thing that takes over their life. Now, listeners, you might be like, "Well, Pam, if that's true, I'll just show them the algorithm way earlier in the year, and then they'll just have the same reaction."
Both Pam and Kim 13:34
Do Kim and I hang out or what? So, the earlier you show them the algorithm, the more you run the risk that they say, "Oh, I can just turn off my brain here."
Like, "Thinking is hard." Thinking is hard. Like, thinking takes effort. Building your brain to think more and more sophisticatedly takes some struggle and some sense making, and that takes effort. Like, you have to work at it and some sweat. And if they see the algorithm too early, and it becomes sort of a quick scapegoat, then you run the risk that they then just do that. Especially if you haven't quite got the knack yet of making it not about answer getting and making it about thinking and reasoning in your class. Which we're going to talk about answer getting in an upcoming episode soon, so make sure you pay attention. Tune into that one. But if you do it too soon, you run the risk of that it just becomes this thing to do and could stop thinking, and that's not what we want in math class. That's not our goal.
So, you might be... So, let's see if we can recap just a little bit. Teach thinking and reasoning all year long. Get kids really developing their brains, so that they can look at any problem and go, "Oh, yeah. I got something. I have relationships I can use to solve that. Yeah, no problem." Then, I'm totally okay that you're going to meet your standards by putting that algorithm in front of them, dissecting it, studying it. They're going to understand it far better than anybody who's ever just doing the steps. You have now met the standard. You've got kids that are doing great on your high stakes tests because they're thinking and reasoning, and everybody wins. We have kids that are mathematizing. We have teachers that feel satisfied and fulfilled because good stuff is happening in their class. Kids are... I was about to say "happy to be there". We want them happy to be there because of the relationship they have with you. But also, like satisfied to be there. It's a satisfying thing to really use what you know, to think and reason. And everybody's... What did I miss in that list I just did? We've met your standards. We've got good scores. Kids are happy. Teachers are fulfilled. Bam! (unclear). There you go.
That's a good place to be.
That's a good place to be.
And, you know, I think one thing that you left out while you were talking was that teachers spend quite a bit of time showing the steps of an algorithm, and then having kids practice them.
Over and over.
Over and over. And then, sometimes say to me, "I don't have time to focus on these alternate strategies, and like relationships, and stuff." So, one of the things that you and I have come back to several times is that this idea that if you can remove that time where you're practicing, and teaching, and over and over again the steps of the algorithm, and replace that with the thinking and reasoning, then when you get to that last week or two, and you're exploring the algorithm, then kids really do go, "Oh, that makes sense. Okay. Well, I'm going to use it or not. But it makes sense." But you've given them all the time they need in your year to make sense of something that actually matters.
Absolutely. And another place you're going to save time. You taught me this, Kim. Everybody in... This was so new to me. Everybody in my kids school. What? A month, and then it was like two months, and then it was like... I mean, the time just kept growing before the stupid high stakes test. Everyone would stop teaching. They're like, "Oh, no. We have to get everything taught by February because then we're going to review until the test." And I'm like, "What?! No! No, that's terrible!" Let's just keep building all year long until the test. Like, if you feel like you have to do this huge review, maybe would you consider that is a symptom of sort of not teaching real math. That's a symptom of that you're treating math like a bunch of things to memorize, and so we better go over and over and over and over it again. Because that's what you have to do for things that are rote memorized. But if you're just continuing to build sophistication, then we can use that time to continue to build sophistication. Sorry, what were you going to say?
And, well, I was going to say, we're not saying that there aren't things that you don't want to review or circle back to, right?
There are some. Yes.
There are some things.
But not the steps of the multiplication algorithm.
Yeah, yeah. So, you have mentioned state tests a few times. And I want to speak to... I can hear teacher's saying (unclear).
Hey, before we go there. I think I know where you're going to go. Sorry. So, briefly, can we mentioned some things that you might want to review? Like...
...vocabulary. Is that where you were going? Were you going there?
No. Go ahead.
Okay. So, vocabulary. That's a rote memorize thing. That's a social, conventional thing. We're going to memorize vocabulary. Notation might be a thing. "Hey, guys, on the test you might see a division problem that looks like this. Remember, we've done that. You might see a..." I mean, help me. Let me get to other grade levels because I've been really focusing on fourth grade. "You might see a proportion setup like this. You might see the equation of a line written like this. You might see a quadratic equation written like this." Like notation. Now, you should have been dealing with that notation. But maybe you dealt with one more than another. That's a good moment, actually, when you're looking at your high stakes test. That's a decent purpose for high stakes test is to remind you of things that you might have de-emphasized. "Oh, that's right. They should see this division symbol. They should see that form of the equation of an line. They should... Oh, okay. Next year, we'll do more to actually develop that and keep it in our repertoire as we go." But especially vocabulary is a thing you're going to review. Okay. Sorry, Kim.
I hope you didn't forget where you were going.
It's okay Well, so I'm the teacher, right now, listening, saying, "You don't know my state test. These algorithms that you just said teach..." Teach. Whatever. "...show in the last week or two. They're on my state test, and so I am required and want to be responsible as a teacher in helping my kids make sense of them because they're tested on them.
Yeah. And so, let me honor the fact that you want your students to do well. You don't want anybody to get to a test and be shocked. "Oh, my gosh. I've never seen this. I don't know how to do that. I feel terrible now." Your teachers. You have a teacher heart. Of course, you want your students to do well and feel good about what they're doing. And so, if you're, right now, thinking, "But, Pam, you haven't seen my state test. My state test, the high stakes test that my students..." So, we're saying "state", but we have listeners around the world. If you have a high stakes test, a test that students have to take at some point that matters for them to move on. That high stakes test asks questions that demand that students perform the steps of an algorithm, then I can totally see why you're bothered right now. I can totally. Like, you are like, "But I have to get the kids ready for those questions." I just took a deep breath. May I invite you to relook at any items you've ever seen released from your high stakes test, and ask yourself, "Do your students really need to be able to mimic the steps of an algorithm in order to pass that item?" If you can find an item like that, please send it to me. I have offered this around the world. Please send me an item that your students have to. Like if they don't show their steps, if the steps they show aren't correct or they don't find the error correctly, or whatever, that you're saying to me, "Kids must have drilled the algorithm, have learned it, and own it enough they'll be able to pass this item." Please send me those items. So, I've had several people take me up on that, and they have sent me items. And to this day... And I've looked at many, many state tests. I have looked at item after item. I've looked at the major test writers and their items. I've looked at countries tests and items in countries tests. I have yet, today, to find one that that's true. Now, you might be like, "Pam, I've seen it." Okay, okay. I've gotten some items. But, ya'll, every time I get an item that actually tests the steps of an algorithm, those items have not been on a high stakes test. They've been in a test prep material. In other words, some company has said, "Hey, we'll help your students get ready for the test. Here's our test prep material. Use these questions, and your kids will do great." And that's where they're guessing about how a standard will be tested. And their guess is showing the steps that you have to fill in. It's the guesses in a textbook, in test prep material that are making kids fill in steps, that are making kids sort of have the algorithm steps rote memorized, and they're able to spit them back out. That's where I'm finding those items. Not actually on a high stakes tests. So, if you find one, I'd love to see it. And boy, then will put some pressure on the writers of that test to say, "Not a good item." If you happen to find one, I'm almost going to... I feel very confident in guaranteeing you, you might find one. You're just not. Like, the preponderance of that test, most of the test, is not going to be that. Maybe there might be one. And how much time are you spending on drilling those algorithms? Like, even if there's one. Which I'm not going to even grant you there is. But even if there's one item on a high stakes test that demands that kids know an algorithm. It's one. It's one item out of the entire test. Again, I'm not granting it's there. But even if there was it's one. And consider how that weighs up against all of the time that you're spending sort of drilling those things. I'm looking at the time. Some other time ask me about a colleague in a different country that showed me some. Okay, I'm going to say it right now. Showed me the scoring rubric that students are graded on in that high stakes test to say, "Look! Look! They have to do the right steps or they don't get credit." But when we actually sat down and looked at the scoring rubric, even she smiled and said, "Oh, whoops." Missed the line that said, "So, students will show..." something something "...or correct mathematical relationships." Like, as long as what the student did was correct mathematically, they got credit for the steps. Like, in other words, there were marks for showing how they got to the answer, and it said sort of, "Do the traditional thing or whatever they show is mathematically correct." The students got marks. So, our point is, whether it's demanded in your standards, it's going to show up on your test...which we don't think it will...you can teach algorithms the way that Kim did. You can meet your standard by exactly what we've just said. I'm not going to repeat it all. What we said in this episode. You can get there that way. And get your kids thinking, and reasoning, and confident, and being able to do all the other things because you've spent your time doing that and not just prepping for kids to be able to mimic algorithms.
So, in summary.
Please. Teachers, we're saying relationships over algorithms. Yes?
And we're going to argue that you spend the time that you have building thinking over practicing procedures.
Bam! Nicely said. Oh, I knew I liked you. So, if you want to learn more mathematics and refine 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.