[00:00:05.470] - Pam
Welcome to the Math is Figure-Out-Able podcast! I'm Pam Harris, a former high school math teacher, university lecturer, writer, speaker, and founder of Math is Figure-Out-Able. At Math is Figure-Out-Able, we are on a mission to improve math teaching. We know that algorithms are amazing human achievements, but they are not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop. In this podcast, we help you teach the mental actions that mathy people actually use when they math, where students grapple with and use mathematical relationships to solve problems, and in the process, learn what it means to math. Where the goal is not just repeating steps to get answers but teaching deliberately and intentionally the mental actions and mental maps, so that students are math-ing to their full potential. In this episode, I'll be continuing to discuss a white paper that we have co-written with Dr. Catherine Beals who is currently the Director of Special Programs for Blaine County School District. And before that, she was the Mathematics Coordinator at the Idaho Department of Education for seven years. Like the last episode, I'll be reading parts of this white paper and commenting on the main points. Cathy Beals and I met when she presented a session at NCSM in 2025 comparing the Science of Reading and mathematics education.

[00:01:51.930] - Pam
We decided to work together on a document to get that information out to more people, especially to practitioners, teachers, and leaders who want to understand. At the time this episode drops, we will have given it out at the national NCSM and NCTM conferences, and you can download it at the link in the show notes or at bit.ly/mathedreading, bit.ly/mathedreading. The document is called Comparing Mathematics Education to the Science of Reading. In the last episode, I discussed the parts of the paper that acknowledge that reading educators call evidence-based teaching practices the Science of Reading, and mathematics educators call them evidence-based teaching practices or high-leverage teaching practices. We also discussed that while reading educators have consensus around a definition of their discipline, mathematics educators do not have that same consensus. Mathematics educators disagree. And lastly, that language arts literacy is the comparison to mathematics.

[00:03:09.600] - Pam
Reading is a subset of language arts literacy and does not have a narrow part analog in mathematics. Therefore, what refers to one may not simply transfer to the other. In this episode, I'll focus on two sections of our document. The first one answers the question, what is the general consensus? First, think about the consensus around the meaning of reading. Quoting from our paper. Quote, "When you say 'reading', people understand and agree as to what it means to read. To read means understanding language in written form." Unquote. That makes sense, right? Most, if not almost all, people read. Many for pleasure. Many to get information and other reasons. When you say reading, it means something, and we can all generally agree on what that means.

[00:04:07.990] - Pam
Quoting again. Quote, "This consensus means that the public generally grasps the broad arguments about reading proficiency and assessment." Unquote. I think you would agree that broadly speaking, the public generally can follow discussions about reading and reading proficiency. Now, I'll read from the math section of our paper. Quote, "When you say math-ing, people interpret that to mean different things. There is no consensus in the general population on what it means to do math. Some define doing math based on their experiences in school, following the steps their teacher showed them to get answers to math questions. Others define doing math as noticing and using patterns, generalizing relationships, and using mathematical connections in flexible, efficient, and creative ways to solve meaningful problems." Unquote.

[00:05:11.600] - Pam
Let's dive in here some more. What do you think about what it means to math? If you're like most people, you were in a classroom where, as you were a student, it was all about getting answers, following the steps. You might have been using what you know--finding patterns, reasoning, logicking your way to an example--or you might not have. You might have been more like me who just did what the teacher told me to do.

[00:05:38.510] - Pam
I waited until they showed me the steps, and then I repeated their steps. I got the right answer, and we moved on. What could it look like to use what you know, finding patterns, reasoning, logicking your way to an answer, using the mental actions of math-ing? Well, here's a quick example. If I were to ask you 99 plus anything. I don't know. 99 plus an icky number like 47. 99 plus 47. What are you thinking as you solve that problem?

[00:06:06.990] - Pam
Well, you might line up the numbers, and then do some digit calculations, thinking only about digits as you add them, and then read off the answer in the opposite order that you just found those digits to say the answer. You could do that. But could you also say, "Hmm, 99 and 47? 99..." That almost nudges you to think about, if you weren't already, 99, 100. I've already added 1 now, but I was supposed to add 47. Okay, what is 100 and 1 less than 47? 146. Bam! That's an example of using what you know to logic your way to 99 plus 47. Could you logic your way to 99 plus anything? 99, 100. You've added 1, so you better now add what's left, 1 less than the number you were supposed to add, and you've got 99 plus anything. You might say, "Pam, That just only works for 99." 999? 98 plus anything? 297 plus anything? 297 plus anything. Could you add the thing but just 3 less? Bam! You could even add the thing, and then take off the 3 at the end. Logicking your way using what you know.

[00:07:27.360] - Pam
What about 5 times anything? Could you think to yourself, "Mmm, how about 5 times 27?" Well, if you know 10 times 27 is 270, and 5 is half of 10, can you cut 270 in half? 135. Is that 5 times 27? Anybody want to check real quick? Could you use reasoning instead of 5 times any number, multiply it by 10, and then cut that number in half? Now, we might have to work on halving numbers, but did you know mathy people often play with doubles and halves. Those are some of the mental actions that we can build to math the way mathy people actually math. And the good news is we now know how to help students do that.

[00:08:17.290] - Pam
Is it possible that how you were taught math influenced what you believe it means to do math? What if doing math like mathy people did when they were young and grew up doing is more like what we just did. What would that mean for how you engage in discussions about teaching math, teaching the mental actions of math-ing? Back to our document. We could also briefly address the question, what do you test? From the section on reading, I'll reread a sentence and then keep going. Quote, "This consensus about reading means that the public generally grasps the broad arguments about reading proficiency and assessment.

[00:08:59.230] - Pam
Reading is much more than just saying written words, and the ability to decode words is an essential, fundamental component of reading fluency and comprehension. Reading instruction throughout the grades recognizes this and has reached some consensus regarding methods to assess both decoding and comprehension." Unquote. Now, I'm not a reading expert, and I don't pretend to be. But to me, this means that because decoding is such a fundamental prerequisite for more things that need to happen for a student to be proficient in reading, it means that we can test for that. And reading education has reached some consensus on methods to test for decoding and comprehension. This generally makes sense to people. They read. They know that to read and comprehend, they need to decode. It makes sense that we should teach that to students. If my student isn't doing that well, please help them.

[00:10:03.480] - Pam
Is the same true for math? From the section on math answering the question, what do you test? Quote, "Unfortunately, conceptual understanding and procedural fluency are often pitted as competing priorities rather than working together. There is disagreement about whether students simply answering mathematics questions counts as proficiency. Because conceptual understanding can be difficult to measure, students are often assessed solely on reproducing procedures and correct answers." Unquote.

[00:10:40.040] - Pam
I personally don't even like that we've separated out conceptual understanding from procedural fluency. In real math-ing, they are so intertwined, so connected, using the mental actions that mathy people actually use. But the general public's view of what it means to do math gets in the way here. If a person's idea of doing math is to do the steps you've been shown to get answers to math questions, then just assess if the student repeated the steps correctly, and if they got the answer correct.

[00:11:17.240] - Pam
And often, they believe that a correct answer is enough. It doesn't matter how you were thinking about it because to do math just means to get answers the way you were shown to. But if a person's idea of doing math is learning to do the mental actions that mathy people actually use, thereby making us all mathy people, then assessing that is quite different. The real rub comes in that the general public either thinks that doing math is just following steps to get answers, then just assess that. But now we're leaving your students behind, treating them as mere mimickers.

[00:11:56.490] - Pam
Or a smaller subset of the public knows what it means to math because they actually use those mathy mental actions, but they may never have seen good, effective ways of teaching or assessing what they've actually been doing. Without public consensus about what it means to do math, assessment is super messy, tricky, and hard to discuss. That doesn't mean that there are not good ways of assessing. It means we need to do a better job of helping everyone understand what it means to math. The good news is that we can.

[00:12:33.880] - Pam
Another section of our paper addresses the question, what is the role of explicit instruction? Quoting from the section on reading. Quote, "Explicit instruction in reading is well-defined and includes demonstration and guided practice. This is an effective teaching practice that helps children enunciate sounds correctly, blend them together into words, and learn to spell." Unquote. This is true, in part, because so much of language is social convention and does not follow a pattern. We have to tell kids that "there" has 3 different meanings and is spelled differently. There is a dog. Their shoes are blue. No, they're actually red. And to the store, too many, and two pizzas sound the same, but do not mean the same thing. Similarly, that sometimes G sounds like game, gorilla, going, going, gone, but other times sounds like gem, giraffe, and gym. And there are rules that we could memorize to help us know which G sound to make, but even those don't work all the time. Is this same patternless cacophony also found in mathematics? I would say not nearly as much.

[00:13:50.370] - Pam
Yes, there are some things that are social convention, like what the number 3 looks like and sounds like, notation and vocabulary. But much, much, much more of mathematics is internally consistent. Full of rich, amazing patterns that hold true and that make math figure-out-able. We can use what we know to figure out more math. And those actions of figuring out? Those are the actions of math-ing.

[00:14:19.750] - Pam
Quoting from the section on math. Quote, "Math educators disagree on the meaning and role of explicit instruction. Some interpret explicit instruction as telling students what steps to do when, how to replicate procedures. I do, we do, you do. They believe that drill and practice are essential and timed assessments are necessary to assess student recall." Unquote. I respectfully suggest these are people who either don't math themselves or don't know how to teach the math-ing they are doing. Back to the paper. Quote, "Other interpretations of explicit instruction focus on creating clear understanding of student and teacher roles, clearly defining mathematical goals, and working towards mental clarity of mathematical relationships and strategies. Practice has a role in building mathematical fluency, but rote replication of procedures does not help students develop mathematical habits of mind." Unquote.

[00:15:25.830] - Pam
I suggest these are people who math themselves and know how to teach others to math, to use the mental actions of math-ing. I invite you to consider my story. I grew up hearing teachers say, "Repeat these steps. Memorize when to use them. Get correct answers." It didn't really relate to my life. I wasn't sure why we were doing that in school. I did the thing. I played the game of school, and I got them right. And give me another one. And I repeated those steps. I got that one right.

[00:15:57.950] - Pam
I began to do 1-29, odd, repeating those rules and procedures, getting them correct. I rote memorized and mimicked all of the things that my teachers gave me and became a high school math teacher. Now, sometimes people will say, "Pam, you mean you weren't thinking at all?" Oh, I was thinking a lot. But it was always about someone else's thinking. Now... Now, I math quite differently.

[00:16:21.580] - Pam
How did that happen? Well, one major influence on me is when I began to student teach. Graphing calculators had just come out, and I landed in the amazing Pam Giles classroom as a student teacher. And immediately, I saw kids thinking and reasoning, playing with the math. I saw functions alive, graphing on the screen. I could tweak the parameters and watch the math change. I could math. Then also my young children began to grow up and my son hit school and thought about math differently than I ever had. He would think and reason using what he knows to solve problems. And I don't know. It had something to do with the fact that he came from me.

[00:17:07.820] - Pam
I birthed him. That it gave me the courage, the impetus, the motivation to listen and ask him how he was thinking. And I began to wonder, could I think and reason that way? Could I get my high school students to think and reason that way? So, I dove into research and found that there is a real math, math-ing way to teach. Now, I see a different universe.
There are patterns everywhere mathematically. I can think and reason mathematically. You might hear that story and say, "Yes, but that is because you memorized all of that easy stuff, and so that then you could build on it." I respectfully invite you to consider not so with my own personal kids, nor the thousands of kids that I've worked with since. I have literally seen thousands of students math from the get-go, never getting caught in the myth, the misconception that math is about rote memorizing and mimicking, simply using what they know with very deliberate, intentional teaching, helping them build more and more mental connections, so that they are using the mental actions of math-ing.

[00:18:18.180] - Pam
Respectfully, consider if you're asking, "Why did math change? Why can't we just teach the way we were taught? Just directly tell kids stuff, have them practice it just like we did?" I'll ask you to consider when that was happening, what were the mental actions you were doing? Did you believe the system that told you to math meant to rote memorize and mimic steps to get answers?

[00:18:46.550] - Pam
Or did you use relationships and connections to logic your way to answers despite being told to rote memorize and mimic? If you bought into the myth that to do math means to wait until the teacher tells you what to do and then rote memorize and mimic that, I invite you to consider that to math is actually quite different. The good news is you're probably actually pretty good at real math-ing. Alternatively, if you were math-ing despite the system being set up to tell you to rote memorize and mimic, I invite you to consider that we now know how to mentor students to do what you did. You kind of did it on your own.

[00:19:28.950] - Pam
We now know how to deliberately and intentionally teach math-ing. In this podcast and the last one, I've been discussing our paper, Comparing Mathematics Education to the Science of Reading. Join me in our next episode where we'll discuss the rest of the paper with major takeaway points and a call to action. You can download our paper at the link bit.ly/mathedreading,

[00:20:02.930] - Pam
bit.ly/mathedreading. Download it and share it and come back next week to hear about more. That's bit.ly/mathedreading. Thank you so much for your interest, for tuning in, and for teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Together we can spread the word that Math is Figure-Out-Able!