Ep 74. Mailbag II: Math standards, teacher content knowledge, and more with Jonathan Regino
This transcript was created with speech-to-text software. It was reviewed before posting but may contain errors. Credit to Canadian Podcasting Productions.
In this special two-part Chalk & Talk mailbag series, Anna Stokke is joined by Jonathan Regino, Pre-K–12 Supervisor of Math at Interboro School District, to answer questions submitted by listeners.
In Part 2, Anna and Jonathan tackle topics such as calculators in IEPs, math fact fluency, teacher content knowledge, the importance of mastering fractions, the role of NCTM in math education, and what evidence-informed math instruction looks like in classrooms. Drawing on their extensive experience in mathematics education, they provide practical, research-informed insights for teachers, school leaders, and parents.
This episode is available in video at www.youtube.com/@chalktalk-stokke
Olivier Chabot’s Notebook LM: https://notebooklm.google.com/notebook/c97a098a-7c1f-4e80-aa02-0ff73164b8e8?addSource=true&pli=1
TIMESTAMPS
[00:00:22] Introduction [00:02:42] Is too much content contributing to lack of math mastery?[00:07:11] Prioritizing critical math content [00:09:13] Common Core and issues with focussing on multiple strategies
[00:14:47] Questions about standard algorithms [00:18:19] The role of NCTM
[00:26:11] Helping teachers improve math content knowledge
[00:33:23] What to focus on after math facts
[00:41:38] Myths about math facts and neuroscience
[00:44:17] Do calculators help with students who have IEPs?
[00:50:36] Final thoughts
[00:00:04] Anna Stokke: Welcome to Chalk & Talk, a podcast about education and math. I'm Anna Stokke, a math professor and your host. Welcome back to another episode of Chalk & Talk.
And welcome to part two of our mailbag special with Jonathan Regino. Today we continue the conversation with even more listener questions on topics like calculators in IEPs, math fact fluency, teacher content knowledge, fractions, the role of NCTM in math education, and what evidence-informed math instruction should look like in classrooms. Once again, I really want to thank everyone who submitted questions.
One of the best parts of doing these mailbag episodes is just seeing how thoughtful so many teachers and parents are about improving math instruction and helping students succeed. If you have a question for a future mailbag episode, you can submit it using the form on my website and your question could be featured on a future mailbag episode. I'll also again link in the show notes to the Chalk & Talk virtual assistant created by longtime listener and teacher Olivier Chabot, which lets you search previous episode transcripts topic and question.
And of course, a huge thanks to Jonathan Regino for coming on and helping me answer these questions. His expertise and knowledge are invaluable. Now let's get into it.
This one is from Robin, and Robin writes, I love your podcast. I have recently started teaching middle school math. Many of my students lack basics.
And so even on topics like fractions, I am reteaching concepts I would have assumed they mastered years ago. So, as I said, another sort of version of that same question about gaps. I am wondering to what extent the breadth of the standards is causing teachers to move from topic to topic too quickly without adequate practice to move from novice to master.
For example, I am wishing that I could exclude any equations information at all from sixth grade if we haven't truly mastered fractions. I'm hoping you can find some guests to discuss whether the standards might be moving too quickly through topics and depriving students of mastery. I wonder if the state reorganized the system, simplified it, and reduced it if we might get stronger results.
Thoughts, Jon?
[00:02:42] Jonathan Regino: There's two different directions you can go with this. So number one, the reason why you're doing review is because the students aren't getting the content to long-term memory. The way most textbooks are designed, even the ones that are secular and keep coming back, they aren't doing interleaving or space practice correctly.
I teach a lesson the next day, I move on to the next concept, or I add the next layer onto that concept. It's not enough time for a student to get that stuff into long-term memory. It's not enough practice to get it into long-term memory.
If you find yourself always reviewing or when the state test comes, you shut down and for a week, you're practicing for the state test. Chances are that the kids aren't learning the material well enough to get it. So, it's in that long-term stage.
There's not much from the teacher level that you can do. If your curriculum is not designed for that, then you are essentially rewriting your own curriculum. And depending on the rules in your school district, you might be allowed or not allowed to do that.
If you are doing a ton of review, you're going to have to remove content. In Pennsylvania, we have 180 school days and that's it. And you take in the testing days, the state testing days, the assemblies and everything else.
You have way less than 180 days. So if I'm going to spend two, three weeks reviewing, I'm going to have to remove two, three weeks of content. One way you could do that is think of math as patterns.
Math builds off itself. And if you teach math in a version of patterns, you can save yourself a lot of time. So a lot of Asian countries teach math this way.
So for example, in America, we teach one digit plus one digit, and then one digit by two digits, then two by two, and then two by three, and then three by three, and so on. And a lot of Asian countries, they teach that as a pattern. So after I get to that two by two stage, it's assumed that the kids know the patterns and they can do a three by three and a four by four without actually having to spend a lot of time practicing or teaching that.
So if you think about the content that you're teaching and you think about the overarching ideas that happen within the content and you connect those dots, you can teach a lot less content and still cover the same amount of content over time. I think this is a version of what Chris Bolton is trying to do with atomization, looking at the bigger structures in math and teaching their structures so you can connect the dots in multiple different places. So if I was in your shoes and I wasn't allowed to really rearrange the curriculum, I might try to figure out how do I teach in a more patterned version of math and see if the kids can pick up on the patterns instead of assuming that the kids can't do three by three after doing the two by two.
For us and districts I've worked with in Pennsylvania, we've gone through the standards and there's about a third of stuff that just it's there for the sake of being there and not necessarily important to get to algebra. And especially when you think of kids who are really far behind and they had to make up a lot of content over a short amount of time, you really have to go through and figure out what specifically is the most important content. There are websites that have done this depending on your philosophy of math, but you can also do this looking across the K-8 spectrum and you really need to have an algebra teacher in the room.
But if you sit down and kind of talk through what is important for the next grade level and all the way important to get to algebra, you can go through the content as a school entity and figure out what do we actually need to teach and what can we remove from the content and still cover everything over time. But I think you need to fix that idea of I have to review. That's the problem.
Then we need to fix that. Everything else is just a band-aid until you fix that issue. And the way you're going to fix that issue is teaching the way that we know students learn, doing the interleaving, doing the space practice, remembering that forgetting curve and bringing back content over time so they can't forget it.
And it does get to long term. And a lot of that comes down to making the time to do practice, practice each step, master each step and not just move on until the kids master. But if you're not going to fix the original problem, you're just going to have to band-aid until somebody gets into leadership that decides they want to fix that problem.
[00:07:11] Anna Stokke: Yeah. And I think a really important thing is to focus on the critical content. That's what you said.
The problem is a lot of times teachers may not know what the critical content is. I've given sessions to teachers and I'll say fractions are really important. And they'll say, you know, sometimes they might email and say, like, what aspects of fractions?
So that part can be tough. I think I will post something from the National Math Advisory Panel, the Critical Foundations for Algebra. The main thing is you want to prepare students for algebra, right?
If you're in those earlier years and to prepare students for algebra, they need to know fractions. And there's actually research that tells us that students' fluency with fractions in early years, so about grade five, six, predicts how well they're going to do in math later. Like fractions are really important and it's where a lot of students fall off the math curve.
But I kind of want to bring in just another related question because I think this may, you know, sort of shed some light on what's going on with some of these standards. But this one was from Bobby and Bobby was asking about the Common Core Standards. So similar issue and mentions that in elementary years, teachers are encouraged to refrain from teaching the standard algorithms, but to introduce several methods of, say, dividing.
When my kids in seventh grade come to me, several of them haven't even seen the long division algorithm. They didn't use that method at all. So, Bobby is asking what are our thoughts on Common Core Standards and how they prepare students for higher math.
And I want to say something about this multiple strategy stuff. So my understanding is that's kind of baked into the Common Core State Standards. Is that true?
[00:09:13] Jonathan Regino: Yes.
[00:09:13] Anna Stokke: Yeah, same with the curriculum. We call it a curriculum here in Canada. This is a big problem.
When you teach someone new, you don't want to teach them a whole bunch of different strategies. OK, it's going to introduce cognitive overload. Also, who has the time to do all this?
And how could we expect that students are going to master each one of those strategies? And also, some of those strategies are not generalizable. So, for example, you mentioned how you look at the patterns and you can easily go if, like, if you have a good method for adding two-digit numbers, that method should also work for adding four-digit numbers, right?
That's why we teach the standard algorithm. That's why it's the best method to teach. Here is a quote from Direct Instruction Mathematics, that book I was talking about.
Some commercially developed mathematics programs suggest that students generate a number of alternative strategies for the same problem. Rather than developing a conceptual foundation that highlights mathematical relationships, this confuses instructionally naive students. Teachers should select the most generalizable, useful, and explicit strategies to teach their students.
And that is precisely correct. If you want to introduce multiple strategies, I would argue in a lot of cases it's not necessary or helpful, but if you want to, the stage to introduce those would be the adaptation and generalization stage. And I'm also going to link to an article by Amanda Vanderheiden about the perils of using multiple strategies at the acquisition and fluency stages.
So, I don't know if you have anything to add to that, but I just thought that that is something we should mention.
[00:11:03] Jonathan Regino: Going back to your original, does the Common Core say this? So, for example, in fourth grade, there is a standard that ends with illustrating explain the calculations by using equations, rectangular arrays, and or error models. So it's baked in.
It does also say to teach the algorithm, but publishers took the Common Core and kind of created their own version of expectations. And that's what many schools are using now. The other piece is the standards were supposed to be a benchmark, not the highest point where kids should achieve.
It was like the lowest version of what they should achieve at. It kind of got flipped over the years. And this is the expectation now instead of this is the bottom of the expectation.
But also at the end of the year, at least in America, we have our state tests and the majority of state tests don't say to use a specific way of solving a problem. It gives you a problem and then asks you to solve it. And even though the standards say, you know, teach these multiple methods, you're not outside of like you personally or your curriculum personally testing those individual ways of solving the end of the year, they're just being asked to solve a problem.
And most state tests don't ask a specific way to solve a problem because as you move on different programs and different curriculums emphasize different methods. So you can't ask a specific method if multiple school districts in the state are using different ways to solve a problem. So even if you go rogue and you just teach the algorithm, your kids are still going to be able to achieve on the end of the year assessment.
All the conversation around should we or shouldn't we, you could teach the standard algorithm and your kids are going to be able to do all the check marks that you need to in a school year. It's not going to hurt them. I agree that standard algorithms should come first and then we should be able to use that knowledge to find shortcuts and find other methods.
But I should be a master of the standard algorithm first. I share the example of 8 times 4. In my head when I was little, I struggled remembering that 8 times 4 was 32.
And I always did the doubling. I would do 8 times 2 is 16 and double 16 is 32. But I knew the standard algorithm.
I knew all the multiplication problems that get to that point. And then I found the shortcut to help me remember that that 8 times 4 is 32. But the knowledge of how to do the problem came first.
And then I was able to manipulate the problem to make it easier and more rememberable for me. And I think going that direction solves a lot of the issues that we're having. If you teach five, six different ways of solving a problem and we know cognitive load, we can only really handle three to five things at a time, a kid's got to go through and rattle off all the different methods and figure out which method works.
And then write off all the steps in that method. And now you're taking a problem that you could do in one or two steps. And you're making it even longer and more difficult.
Eventually, yes, that could be the easier way to do it. But at that very beginning, when they're first learning, the standard algorithm makes it the simplest way of learning how to do these things. Get that solid.
And then we can expand if you have that ability to expand or you're ready at that stage to expand. And I agree that happens in the adaptation and generalization. It shouldn't happen in the acquisition or fluency.
There's too much information if you're doing multiple methods to be able to get out of acquisition and into fluency or just stick to one, become a master at it and then expand.
[00:14:47] Anna Stokke: Exactly. And I'll just address something else because over the years, you hear all the arguments about why standard algorithms aren't great or whatever. And I've never heard one that actually is really legitimate.
So as an example, this is what you'll hear. Oh, but have you ever seen the student who uses a standard algorithm to do 1000 minus 999? OK, novice learners do stuff like that, right?
The point is that the standard algorithm, it actually will work for that problem. But it's not the best technique to use for that problem. But you develop the skill, you develop fluency with the standard algorithm.
And then after you start saying to students, OK, let's just think about that. Do you really think that was the best way to solve that problem? You know, given that 999 is like one less than 1000, right?
We don't want to use it all the time. But it works for every set of numbers you give them, right? That's why it works relatively well.
That's why it's the best strategy to teach. You know, that's why we call them standard algorithms. But you do want to later, you want to sort of point out when it's not the best to use in a particular situation.
But I will mention one other question that came in about standard algorithms from Kristen. And Kristen asks, in the Ben Solomon episode, Ben mentioned teaching standard algorithms a few times. And I wanted to clarify what he meant with the new math program that my district here in Michigan just purchased.
Students are taught up to four ways to multiply and divide. So here we go again, you see. And when he talks about teaching standard algorithms, does he mean just focusing on the traditional way to do it?
Or does he mean to teach many different ways to compute, but teach them explicitly? Thank you for your time. Kristen, I know exactly what Ben Solomon meant, because we've talked about this a lot.
He meant just teaching the traditional standard algorithm. So that's addition with a carry, subtraction with a borrow, the usual vertical array for multiplication and traditional long division. So I hope that clears that out.
Next question is from Brian. And Brian writes, good evening, Dr. Stocki. I am a math teacher in San Diego.
I stumbled upon your podcast in a quest to find answers to doubts I have had as a 14-year high school math teacher. I think something that might be worth discussing is the role of the NCTM in all of this. So often when pushing for education myth-busting or evidence-based instruction, we run up against the advice of NCTM, who are seemingly deeply committed to inquiry-based instruction.
I don't know if there are experts who can touch on the history of NCTM or if there are alternatives. I also don't know if this idea is even helpful. It is helpful, Brian.
Thank you for your podcast. For many math teachers, it feels like a tiny light of truth in an otherwise dark and grim PD world. Keep up the great work.
Well, thank you. Thank you, Brian. So first of all, I do want to say I have an episode on NCTM and that is with Tom Loveless.
So you will want to check that out. I'll maybe talk a bit about what he said, but I actually like to hear Jon's thoughts on this first.
[00:18:19] Jonathan Regino: It's interesting to like dig into the history of it. So after the Tom Loveless episode, I went back and started reading more about NCTM. And, you know, when I was teaching, I was a member, you know, when you know better, you do better type thing.
And coming out of college, you join NCTM because that's what you're supposed to do. And you followed what you were taught in college. I think faith in NCTM, I lost it.
When you start to read the history and you learn that, you know, it started off with the purpose of why it started. Like it had a good start, right? As many things do.
And then over time it devolves and becomes something different. And that during the Cold War and during Sputnik, what they stood for really changed. And instead of being the focus of supporting teachers and helping teachers be better in math and have a voice in the math world, it changed to becoming more of advertisement for the things that the military and the government wanted out of math.
When the Cold War happened, we wanted more people to go into math and science so we could do better than the countries we were against. And that meant that there were people who thought that math needed to change in schools. And NCTM was very important piece of convincing schools to change the way that we taught math and what should be taught so we could produce better mathematicians in their eyes.
But what they wanted were nuclear physicists and anybody who's worked with little kids knows there's a long distance or far distance between what we teach in school and getting a kid to be a nuclear physicist. And the idea of changing even K-8 math to, with the idea that they're going to become these mathematician scientists that were going to work on nuclear world, it didn't really focus on the student and how to teach them correctly. And we kind of jumped in that progressive side and then the math wars came out of it, like Tom Loveless shared, and it just hasn't come back.
For me, NCTM is really pushing one side and they're doing everything they can to push that one side, but they're coming from a history of being more of a marketing firm than anything else. So when you put those pieces together, it's really hard to take what they say at face value. And then when they start putting out what they call evidence and you actually read the research and the evidence is very flaky, you can't put your faith into an organization like that.
I shared before that I don't push out research or evidence without actually knowing what the research or evidence says. So I do my own evidence and research at this point. I don't trust the national organizations or just anybody who's putting out information.
I think anybody in leadership should be able to defend their decisions. And if you're using a program that's not evidence-based and somebody calls you out on it, you better have a reason of why you're using it and you better understand that evidence. And when it comes to NCTM, I haven't had a membership in a long time.
It's not something I share with my teachers. We talk about evidence and we look at the evidence and we talk about what's good evidence versus not good evidence. And we do it in-house now.
[00:21:42] Anna Stokke: And if they have a membership fee, you don't have to get a membership. You don't have to support a group that is perhaps promoting things that are not evidence-based. The problem is people will assume that they are promoting things that are evidence-based.
I know this because I'll have people write to me and say, but NCTM says this. And I'll say, well, you know, that's not evidence-based, right? For example, I believe they released a joint position statement with the Council for Exceptional Children.
And we talked about that with Sarah Powell, that that was not an evidence-based position statement. And that's just the tip of the iceberg. I'll mention a few of the things that Tom Loveless said about NCTM, just for history and just as a recap.
So for example, a big thing were these 1989 NCTM standards that they released that were enormously influential. And they really helped launch what became known as reform math and then the math wars. And those standards in 1989 de-emphasized memorization of basic facts.
They de-emphasized standard algorithms and explicit teaching by telling while promoting calculators and discovery learning. So we know these things are not in line with the evidence about how students learn and what they need to learn math. Okay, so it started with that.
And those ideas, they weren't new even at that time. They're part of a long tradition of progressive education dating back to the 1920s. And they treated arithmetic as outdated because calculators could perform those computations.
And we know that hasn't worked out well, right? We have lots of students who are unable to do basic calculations. So the calculators, they didn't seem to, that idea didn't seem to work.
And the other thing I would note is that the NCTM did not take the recommendations of the National Math Advisory Panel. And the National Math Advisory Panel, that was a group of experts commissioned to look at the best available evidence at that time. That was 2008, I believe.
The best available evidence on how to teach math. They looked at around 16,000 research articles, many of which had to be essentially thrown out because the methodology was so poor, which is what you get in education. But they came up with a list of recommendations based on the best available research, which was essentially ignored by NCTM.
So I do think we need to be careful with that group. And just because they have National Council for Teachers of Mathematics, and it sounds like they should be giving evidence-based advice, they're not necessarily doing that. Just like if you hear someone is a professor at Stanford, who cares?
You know, like you're a professor at Stanford, great. But what are you actually saying? And is it evidence-based?
That's what we need to look at, right? Not what the title is, you know, what the person, the substance of the argument. Let's move on to this question from Tim.
And I really like this question. It's something that's kind of near and dear to my heart because I teach at a university. And so I don't teach in a faculty of education.
I teach in a math department, but I have a lot of students in the education program. And Tim writes, can you do an episode on increasing teacher content knowledge for middle and high school math teachers? I could do one, but I've got Jon here.
So we're just going to give some advice. And Tim says, I'm looking for professional development workshop ideas or books to work through with teachers. For example, if I have a sixth grade math teacher who teaches pre-algebra content, how much more math should that teacher know?
Should she know algebra, algebra two, pre-calculus? I love this show. And I've listened to almost every episode.
I'm using your podcast with the I do, we do, you do guest this week. I think that would have been Anita Archer in my PLCs in my public school district. Thank you.
All right, teacher content knowledge. Jon, what do you think?
[00:26:11] Jonathan Regino: You have to think of capstone courses, right? So elementary is building up to algebra one and algebra one is the gatekeeper to all the high school classes and the college courses that you want to take. So a teacher in K to eight or K to whatever grade you teach algebra in should know all the way up to algebra.
And I would emphasize that they should know algebra content as well. There are many times, even at the kindergarten level where a teacher doesn't know something that they're teaching connects all the way through to algebra. And if they might decide, oh, this isn't important and they skip it and you're already breaking the very first leg of a piece of content that goes all the way through, right?
So a teacher at the elementary level at least in where I'm at only has to take one math course. And it's usually not even how to teach math. It's more of a, like a philosophy driven math class.
So a lot of our K-5 teachers come out and they really haven't had math in a really long time. They really struggle in the content but they should be able to get to the point where they can do the middle school content at least. And then middle school teachers should definitely be able to do the algebra content.
High school teachers should be able to teach and at least solve all the way through calculus because it's the capstone of the high school level. We are finding, as I mentioned our elementary teachers really don't have to take a math class. So we are finding that a lot of them are coming out and they actually don't know the math well enough to teach it.
And if there's anything that we should be able to agree on in education is if you don't know something you can't teach it. So, I mean, we can fake it, we can pretend but like if you really don't know your content you're not going to do a very good job of teaching it. You're going to make a lot of mistakes.
I've been in classrooms where a teacher had a misconception that many of our kids have and then they teach that misconception and then you've got, you know, 30 plus kids in the classroom that we have to undo that misconception at a later time. I mean, no, once you start practicing something it is really hard to undo a mistake or a misconception. So we have, when I started at this district we started giving math tests to all of our incoming new teachers to determine what they know and what they don't know.
It won't necessarily stop them from getting the job but at least gives me a starting point of, you know on PD days, I'm going to pull you aside and we're going to work on the math content that you don't actually know because I don't want you in front of my kids if you don't know the content that you're supposed to teach. For teachers with a lot of missing content I have suggested them and bought them licenses for math academy and throw them on math academy and have them do the courses that I think they need to do. So if I have a middle school teacher who doesn't know their content I will ask them to take the algebra one math academy course and go through that course and build up your knowledge or we'll pay for you to go to a local university and take a face-to-face course with a professor to learn that content.
But we've gone too long allowing teachers without the knowledge of the math to teach the course. And it's not okay. If it was my own kids' teachers I would be upset knowing that they were teaching the content wrong or they didn't know the content if my kids were in their class.
So I take that as a parent. I'm going to expect that same thing for the teachers that work with me.
[00:29:28] Anna Stokke: Yeah, I agree. And I want to emphasize this is like a systems problem. You know people will take a job that they're offered, right?
Because they need to work. So if someone gets offered a job and it involves teaching math they're going to take that job even if they don't know the math. And as you said and as I like to say you can't teach what you don't know.
And so first of all the universities shouldn't be graduating students in the education program to teach math if they don't know the math. That's the first thing. So and I'm well aware that this is a big problem.
So the kids and those teachers are being failed by the universities. And then the second thing is they're getting put in classrooms when they don't have the knowledge to teach that math. And I think it's possible that out there in the public that people think if you've graduated from grade 12 you must know the math to teach grade six.
And I can tell you that that is absolutely not true. I think people would be quite surprised to see just how little sometimes high school graduates know. And you know our scores are getting worse and worse over time at the K to 12 level.
And some of those people are becoming teachers. And so what I like is your suggestion that you are giving your teachers a test. My goodness you're doing so many great things in your district.
I just can't get over it. Like people really need to listen to this. You know you're doing sort of the more ability grouping.
You've got all these great little professional development things going on for your teachers so that they know what to use. In addition to that you're giving a content test. And then if the teachers don't know the math you're giving them math academy.
Like that is exactly what you could do. Get math academy for the teacher and just have them go through it and learn the math. They'll be better off for it too.
Like do people really want to be in front of a class when they don't know the material? And how's that going to work out? So let's try to do something about it.
And I'm really glad that Tim brought this up because I think it's an important issue that does need to be addressed.
[00:31:43] Jonathan Regino: Yeah as far as a book I would go back to that direct instruction book. Like you could work through that especially for the K-5 teachers. And you could work through direct instruction or Barry Garelick's traditional math.
And just work through those books because you're going through the content they're going to teach. And you can have a conversation about how you should teach it. And then you can practice.
Practice pretending to be a teacher. Practice being a student and work through those problems. You can come up with a list of misconceptions that are going to pop out.
So every teacher is ready to teach the content. And just by going through those books as a book study.
[00:32:16] Anna Stokke: Yeah absolutely excellent advice. Let's take this one from Nisha. I've really enjoyed listening to Chalk & Talk and your insights on the importance of mastering math facts.
My daughter is in grade 5 and my son in grade 3. They both know their addition, subtraction, multiplication, and division facts up to 12 by 12 automatically. At school the math program focuses more on exploration and problem solving.
So we do most of the structured practice at home. Both of my kids are strong in math and enjoy it. And I'd like to keep building their skills in ways that will have the greatest long-term impact.
Once a child has mastered their basic facts, what would you say gives the most bang for the buck in the upper elementary years? Should we focus more on fractions, problem solving, algebraic thinking, or something else that best prepares them for high school math? Are there any particular curricula or resources you'd recommend for this stage?
What should I prioritize? Thank you for your time and for all the wonderful advice you share on your podcast. Okay, so what are your thoughts, Jon?
[00:33:21] Jonathan Regino: Yeah, so if you came to me with that question, I would tell you to move on to fractions. Fractions has a direct correlation with success in algebra. And if your kids are at the point where they know their basic facts, that's the next step.
One thing that I like to do is start with decimals first and then move into fractions because that conceptual piece kids tend to struggle with the idea that a fraction is a number. They see it as two pieces of a number instead of a number and the decimals kind of help them understand that. I think here in America, that's because our money system allows them to connect the dots a little bit easier.
It doesn't work everywhere. The other thing is problem solving versus word problems. Those words are used interchangeably all the time and they are two different things.
So I define word problems as having a solution already. We're just helping the kids work through the steps to get to that solution. Whereas problem solving is an unknown.
We don't have the end solution yet. There could be multiple ways of getting there. And both of them have different skill sets that you need to work on.
But the reason why I bring that up is because a lot of middle school and high school teachers will tell you that their kids still struggle with word problems and problem solving. A lot of IEPs have problem solving written into it. It's an area to focus on.
And both those have separate directions you want to go. So if it is problem solving, you're going to need a lot of content knowledge. You can't solve a problem without content knowledge.
Like if I wanted to find a cure for cancer, I better know how cancer works. You know, cancer has an unknown. I need a lot of content knowledge to be able to work on that problem.
I can't brute force my way into a solution there. When it comes to word problems, you're looking for structure. A lot of times teachers teach keywords and underline the question.
That doesn't work long-term. It's very much a band-aid. In math, I think it's in fourth grade, you're up to like 400 vocabulary words in math.
Dr. Powell has exact numbers. Her team's done a lot of research on that. And a lot of math words have double meanings, right?
So I would talk about on our state test, there used to be a problem where there was a picture of a shirt with two pockets. And the left pocket and the right pocket both had money in it. And the question was, how much money is in the left pocket?
But because the kids circled the word left and that was the keyword of subtraction, they would subtract the two pockets and get the answer. So if you actually focus on the structure of a word problem, you don't fall into that trap of the keywords and making those types of mistakes. Word problems are made up of language comprehension, which are the words, the reading comprehension.
So the understanding of the sentences and then the content knowledge. And if you need all three pieces to be able to do word problems successfully. And if you're missing any of those pieces, you're really going to struggle with word problems.
Dr. Powell has Pirate Math, which does a really good job of the structure of word problems and teaching that. And it's designed for 30 minutes, three times a week. But that structure is called schema.
And if you can get the schemas of word problems down, you can use those schemas across every grade level in all content areas. And it works a lot better than the idea of circling and underlining and those types of methods that we typically teach word problems.
[00:36:43] Anna Stokke: I second that about word problems. Word problems are important. Like students do need to be able to do word problems and also some problem solving.
And then of course, fractions, very important. So I think Nisha is saying that the kids know all their addition, multiplication, division facts. Up to 12 by 12 automatically.
So of course, the next place to go from there is just whole number arithmetic, right? Being able to multiply, divide, whatever larger numbers. And then absolutely fractions.
Fractions are imperative. As you already said, they do set students up for success in algebra. And then I'll just mention, because you mentioned about sometimes people don't know what a fraction actually is.
And just to be clear, like fractions live on the number line. They live on the real number line, right? And so direct instruction mathematics, that book by Stein et al, they actually start with the number line.
They start with showing you how we represent those numbers on the number line that are the fractions by dividing each whole into the same number of parts. And, you know, the denominator is giving us the number of parts in each whole and the numerator is essentially telling us how many parts have been used, etc. It might be worth people looking at that too, because I do think sometimes people are a bit confused about fractions because the books tend to start out with pictures of circles and it sort of removes what a fraction really is.
It sits on the number line, right? You know, it may be worth looking at that direct instruction book if you're thinking about that. But the goal always, always, always has to be to become fluent with symbols because that is the thing that predicts later success.
It's not drawing pictures, although, of course, we do that with fractions because we do need to be able to interpret things that way and represent, you know, a figure with a fraction and vice versa. But we want students to be able to add, subtract, multiply and divide fractions very fluently with the symbols because that's what they need to be able to do in order to succeed with later math. Let's move on to Regina.
OK, this one's fun. All right. So that's a great question.
But from Regina, I often hear the work of Jo Boaler being recommended to K-12 teachers. When I first began teaching elementary math, I was astonished to find that Boaler claims children don't need to memorize multiplication facts and claims she herself never memorized them. She seems to encourage inquiry and diminish explicit instruction.
I've become skeptical of her approaches, but she claims to base them on neuroscience. What do you make of Bowler and her methods? Do they have a place in the science of math?
And if so, what is it? Maybe I'll start with this one, Jon. Let's talk about this.
So first of all, it is simply not true that students don't need to memorize their multiplication facts. They absolutely do. I would argue that it's one of the most important things a primary school teacher can give their students.
I don't care what grade level you're at. If a large proportion of students in your class do not have their multiplication facts known off by heart, I would implement a program like Brian Poncy's Facts on Fire across the board every day until they do, because it will never get better. And those students will always be crippled because they don't know their multiplication facts.
Now, whether Boaler memorized her multiplication facts or not, we'll never know because it's impossible to know. A person can say they didn't memorize them, but how would we ever know, right? I mean, that's up to her, right?
But that doesn't mean the rest of the world doesn't need to know their multiplication facts. Now, she's also not a neuroscientist. There have been a lot of claims about neuroscience and neuroscience is a complicated thing.
You know, I don't know much about neuroscience myself, and I probably wouldn't be making claims based on neuroscience because I'm not a neuroscientist. But a lot of people have challenged some of those claims she makes about neuroscience. And I think we will link to some of those pieces.
So, for example, by Greg Ashman and whether, you know, everybody has something, I'm sure, to offer and not all things she says are going to be false. But I'm just addressing the things that you said here. So, for instance, the claims based on neuroscience, I'd be very careful about those.
And also the claim about multiplication facts, which I absolutely do not agree with. So, Jon, do you want to add anything?
[00:41:38] Jonathan Regino: Yeah, I think Kelsey Piper and Greg Ashman did a really good job of essentially answering this question in their sub stacks recently. The other thing, so let's go back to the classroom. Go up to any high school and pull your high school teachers and ask them what their biggest needs are in the class and what they've seen over the last 20 years and the difference of the students that are coming to them now versus, you know, 10, 15 years ago.
They're going to bring up that kids don't know the facts. They don't know fractions and they can't solve word problems. And you can go to as many high schools as you want to ask that question.
You're going to get some other things, but that's going to be a common theme across most high schools. And if that's the truth, then that's a systematic issue. If we are sending students up there and high school teachers are struggling with kids who haven't mastered things that they should have learned and mastered by the end of elementary school, that's a problem.
And if they feel as teachers feel like they can't teach the content that they're supposed to teach, that they can't teach the algebra to the pre-calc, the calculus, trigonometry because the kids are lacking in these skills. It doesn't matter whether neuroscience or evidence says like we should or shouldn't learn our facts. The fact that those teachers are struggling to teach their contents because the kids don't know their facts should be more than enough for school leaders to say, we need to do better and we need to mandate that our kids know the facts before they get to high school.
Roots on the ground are more important than what's happening. You know, the fight, the math wars that are happening aren't as important than what's happening in the classroom. And if your high school teachers are saying this is a need, you need to fix that need.
You can't ignore it.
[00:43:17] Anna Stokke: We want to make sure that students are able to succeed in later math. So we definitely have to make sure students know their math facts. Let's end with this question from Lisa.
And this is about calculators and calculators in IEPs. I am a math supervisor in the US and my special education students always have calculators as a modification in their IEPs. Is there any research basis for this?
Many don't know their facts and can use a calculator well. I find a multiplication table much easier to use and a bit more helpful. At least the student can start to see patterns and make connections.
But neither is a replacement for learning as many facts as fluently as possible. How can I help my special ed team get calculators out of IEPs? Is this really what's best for students?
And Jon, you're really the best person to answer this question. So I'd love to hear your thoughts on it.
[00:44:17] Jonathan Regino: There is a time and place for calculators. We definitely introduce them way too soon and we rely on them way too heavy. But if a kid didn't know their facts and we were teaching content like we're teaching one-step equations and the kid didn't know their facts and the cognitive load of subtracting from both sides or multiplying is going to take away from actually learning the steps of solving the equation, then sure, give them a calculator.
Let them memorize the steps for the equation. And then you spend time later on helping them achieve and master the facts that they're missing. But to give a kid a calculator just for the sake of giving the calculator, there's a huge problem with that.
Anytime you give a kid a scaffold and you don't have a plan for removing that scaffold, you are taking the thinking process away from the kid. So if I give a kid a calculator because they don't know their facts, that kid is never going to learn because they know I have this tool that's going to give me the answer without thinking. And we know the more steps that we give a kid later on in math, the harder that's going to be to keep all those steps in line.
And we know calculators aren't always correct because the way we put them in don't always come out the way that we want them to. So things like giving them the multiplication table and giving them a number line and giving them things like TouchMath, like all these scaffolds are okay, but we need to have a plan to move them off those scaffolds as quickly as possible. They should be there to help them get over a hump until we have the time to explicitly teach them and intervene in the correct manner.
And then we need to move them as quickly as possible off those scaffolds before they become the crutch that becomes that long-term thing for them. This happens with TouchMath where a kid learns how to count using TouchMath and add and subtract and all the different pieces with TouchMath. And then you can see them as they get on in high school and they're touching their paper in the same pattern as they were with the TouchMath.
Like they never come off that crutch. It's the same with finger counting. Like finger counting is good, but we need to figure out a way to get them past the finger counting onto better ways and more efficient ways of combining numbers.
So every scaffold to give a kid is fine, but if you're going to leave it there, you're taking the thinking off of them. And if you're taking the thinking off of them, you're pretty much guaranteeing that they're not going to get to the long-term memory and they're not going to be able to use that content later on. So have a plan in place.
And then going back to that calculator, if you can get away with not giving them a calculator, don't give them a calculator. If it's absolutely necessary, you're thinking middle school, high school, where it is taking away from the on-grade level content, then sure, but only at specific times and for specific reasons, not in general. And that might mean that they have to get up out of their seats to get a calculator and not just have it on their person at all times.
And then with IEP kids, it's the same thing. If you're going to give a kid with an IEP a calculator, you are saying that you're okay with them not learning their facts and not memorizing their facts. Like whether you're saying that out loud, you're saying that by the wording that's in the IEP of giving them the calculator.
So be careful with giving a scaffold that you're not ready to remove in a short sequence.
[00:47:31] Anna Stokke: Yeah. And sometimes you've got to do the hard work, right? Of addressing that prerequisite skill.
And in this case, you've got to address the fact that the kid doesn't know their math facts. And it actually shouldn't take that long, particularly if you have the resources for someone to work one-on-one with that student or to work in small groups, right? You can use flashcard techniques like incremental rehearsal, right?
I've got a little podcast episode on evidence-based techniques for learning math facts. I'm going to link to that. But you definitely should address the prerequisite skill.
It's not going to get better, right? Like I get that if you're in a situation where you need a student to work on a particular topic and that topic requires knowing the basic math facts and they don't, they need something. Otherwise, they're not going to be able to do the topic, right?
But in the meantime, like Jon says, you've got to have a plan in place for addressing the prerequisite skill.
[00:48:37] Jonathan Regino: There are places all over the country that are taking the kids with the highest needs, no matter what the disability, and moving them. It's not 100% of kids, but the majority of kids with the right type of teaching and the right type of intervention can obtain. So, a kid struggling with facts, there are places out there that are getting the kids following their facts in two weeks.
There are science-based, evidence-based interventions to get kids to know this stuff. They're just missing in schools. Precision teaching, the behavioural side, does a really good job of nudging kids with processing disorders to be able to process things faster and obtain things at a quicker pace.
There are different academies and tutoring places that do a really good job of getting kids to learn their facts in a really short amount of time. So it is possible. And it's not hard to take the things that are happening in these private environments and bring them into public schools.
There are schools like us that are bringing those things in and figuring it out. I am fortunate enough to work with a superintendent that allows me a lot of free reign to bring these things in and adjust schedules to make it work. But it is possible to have things come in that we know that work and can move kids a lot faster and you don't have to rely on those scaffolds.
If you look at an IEP and over time, the SDIs and the scaffolds keep getting bigger and bigger and bigger, you're not doing the IEP correctly. Over time, there are scaffolds and their supports should be pushed onto the student and they should learn how to cope with them. So over time, there's SDIs and the things in the IEP that support the kid start to lessen over time because the goal is when they graduate from high school, they're going out to the real world.
They're not going to have all those scaffolds and all those SDIs as they move out into the working world and you want to set them up to be successful there. So determining a way to help their students is our goal. So, and those things exist.
Let's just get them into the classrooms.
[00:50:36] Anna Stokke: Absolutely. You're doing it. So other districts can do it too, right?
Great advice. And I'll just, before we close off, I will mention that I have teachers writing me from all over the U.S. about illustrative math. And so we didn't address illustrative math today, but I want to mention I actually do have an episode planned on that in the future.
And then I have teachers all over Canada writing to me about a program called MathUp and they're all complaining about it. And it's on my list, okay? I have some initial thoughts on it, but I don't want to talk about it until I've had a chance to review it properly.
But it is on my list of things to talk about at some point in the future. It's just, there's a lot on my list. So we'll get there.
And I want to thank you so much, Jon, for coming on and helping me out with these questions. It's just wonderful to have you. And I hope you'll join me again for more questions.
I hope this becomes a regular thing because I think we're a good team. I think it works really well. I really appreciate it.
You're doing such amazing work in your district and you're so knowledgeable. And I really appreciate you sharing your knowledge with us.
[00:51:54] Jonathan Regino: Thank you. I really enjoy answering the questions and I got into this line of work because I want to help people. And this is my way of helping families and teachers.
I'm all for it. And anybody can reach out, ask questions. I love learning and I love supporting.
So, reach out.
[00:52:12] Anna Stokke: You're doing great work. OK, thanks, Jon. Thank you so much for listening.
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