Ep 6. Math Teaching Tips with Barry Garelick and JR Wilson
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You can listen to the episode here: Chalk & Talk Podcast.
Episode 6: Math Teaching Tips with Barry Garelick and JR Wilson
[00:00:00] Anna Stokke: Welcome to Chalk and Talk, a podcast about education and math. I'm Anna Stokke, a math professor and your host.
You are listening to episode six of Chalk and Talk. In this episode, we dive into the world of math teaching. I'm joined by two experienced teachers, Barry Garelick and JR Wilson. They have recently published a book called Traditional Math: an Effective Strategy that Teachers Feel Guilty Using. Throughout the interview, Barry and JR share strategies and practical advice that they've used in their own classrooms with great success.
We cover topics that will resonate with teachers, including how to get students excited about math, how to effectively use the “I do, we do, you do” method of teaching and the role of understanding in math. [00:01:00] We also touch on critical math topics that teachers should focus on, tips for teaching word problems, how to keep advanced students challenged, and how to help struggling students, among other things.
As a fellow educator, I found their tips and advice incredibly useful, and I'm sure you will too. So, without further ado, let's get started.
Anna Stokke: I am excited to have two guests joining me today: Barry Garelick and JR Wilson. They are co-authors of the book, Traditional Math: an Effective Strategy that Teachers Feel Guilty Using.
Barry Garelick is joining me from California. He majored in math at the University of Michigan and taught grade seven and eight math for 11 years.Teaching was a second career for Barry. He started teaching after retiring from a career with the federal government where he worked in environmental protection. In addition to the Traditional Math book, Barry is the author of [00:02:00] several books on education, including Out on Good Behavior, Confessions of a 21st Century Math Teacher and others.
I also have JR Wilson joining me from Washington State. He has more than 30 years’ experience in education. He's a retired teacher, having taught math in elementary, middle, and high school, and he specifically taught sixth grade math a lot. He was also a principal, a state curriculum consultant, and was involved in writing math and science standards. In addition to that, he served on the executive committee of a math advocacy group in Washington State called, “Where's the Math?”
I'm delighted that you are both joining me today. Welcome to my podcast!
[00:02:42] Barry Garelick: Well, thank you.
[00:02:43] JR Wilson: Thank you for having us.
[00:02:45] Anna Stokke: So, we're going to be talking a lot about the book that you wrote called Traditional Math. It's a great book and I highly recommend it. It has lots of tips for teaching math straight through from basic arithmetic up to advanced algebra. The book has two parts. Part one [00:03:00] covers key topics from K-6 math, and that was written by JR.
And parts two and three cover what are essentially the usual topics in grade seven and eight math and perhaps accelerated grade seven math. And parts two and three were written by Barry. So, let's start off with a question about the book title. What do you mean by traditional math?
[00:03:21] Barry Garelick: Well, let me take a crack at it. And JR can add as he feels necessary, but I guess a general definition of traditional math is teaching math using explicit instruction and combined with worked examples and doing so in a logical sequence of both skills, concepts, procedures, and so forth. And the method engages students while they developed a fact in procedural mastery algorithms, problem solving procedures, along with reasoning and understanding.
[00:03:56] Anna Stokke: So the subtitle of your book is “An Effective [00:04:00] Strategy that Teachers Feel Guilty Using.” Can you elaborate on that? Why did you include that subtitle?
[00:04:05] Barry Garelick: Well, and I'll point out first that, that JR says he never feels guilty about teaching using the traditional method. But he's been teaching for much longer than I have. And that may have made a difference. I think it comes from an attitude that is taught or projected in ed school that there's some disdain held for traditional math.
And in fact, I recently, I was reading an article that, that started, “have you ever felt an almost untraceable shade of guilt when wondering if we should have more explicit teaching in classrooms?” So, it's a pervasive feeling I find, probably with new teachers. And when I started teaching, even though I felt I was teaching in the way it should be taught, I felt I was doing something against the rules, perhaps even wrong.
And as I said, in Ed School we're taught that the [00:05:00] traditional method of teaching using explicit instruction was a failure. It, it, it was ineffective. It's much better to use discovery and student-centered methods and teachers should be facilitators, et cetera, you know, and things like telling students what they need to know to solve a problem is just handing it to them. It's “teacher talk”, “teaching by telling”, all these terms.
So, when we go in, when I went into it, even though I felt I knew how it should be taught I had this feeling of, gee maybe it's wrong, even though I knew it wasn't. And so, like I said, this article kind of encapsulated that there are other people who feel that way.
[00:05:37] Anna Stokke: I'm probably more of a traditional teacher. I am more of a traditional teacher, and I don't feel guilty about it. I think that, at the end of the day, we have to do what's best for our students. And if that means teaching using explicit instruction, then that's what we should do.
And, in fact, there's quite a lot of evidence backing up the types of methods that you [00:06:00] talk about in your book. And so I'm really happy that you're here today to give us some tips on teaching using explicit instruction or what you call traditional math.
When you're young and you're a new teacher, I think you feel like you want to try these new things and you know how to teach better than your colleagues. But I noticed that over my career I started adopting methods that were used by the more experienced professors in my, my department and I've developed into what I think is a good teacher.
Let's start talking about a few of these things. So, JR you mentioned at the beginning of the year in your sixth grade math class that a lot of your students actually hated math and they would've been happy if math class were actually canceled for the day.
And I think that I've heard this sort of thing a lot. But what you also mentioned is that a shift took place as the year progressed and students were actually excited about math class after a while. And what do you think caused that shift? Would you be able to give [00:07:00] us some tips for getting students motivated about math and, and how we can get them to enjoy math?
[00:07:05] JR Wilson: Yes, I think there are a lot of factors that really contributed to that. And it is true, my students at the beginning of the year when they'd come into sixth grade, and when I'd say it's time for math, the moans and groans just overrode any other sound in the room. And then somewhere about halfway through the year, when say it was time for math, you'd hear people yelling, saying “Yippy”, because they were glad it was time for math because they seemed to like it.
So, there's just, I think, a lot of things go into it. Having students set some goals. You obviously, they can't set goals for everything that you do, but there are things, lots of things in math that they can set goals for themselves.
Having some consistent daily practice that's of some short duration so that they know that, [00:08:00]this isn't gonna go on all day long kind of thing. It's just gonna be a couple of minutes and we'll be done. Having some leveled practice that focuses on one fact or skill at a time. Self-checking, and I'm gonna come back to the self-checking in a minute.
And getting immediate feedback, some record keeping so that they can see progress that they've made. And when I say record keeping where the students keep records of their progress. And the ability to move to a new level. Say if it's for timed drills my students would be able to move to a new level when they would reach an established proficiency goal.
Really gaining competence in basic math facts and being able to successfully apply their ability with those facts makes a lot of difference cuz then students aren't struggling, you know, to come up with the facts, which [00:09:00] sometimes if they don't know the facts, then, then that's a hurdle for 'em.
They, they trip all over themselves. Some, just some self-satisfaction of accomplishment and success. And even if the successes are small, you build on that so that they can grow and become bigger in an environment where it's okay to make a mistake and try to figure out why they, that they've made the mistake or how they made the mistake.
And I wanna come back to the self-checking because my students would, when they'd come into sixth grade, and I think this happens in a lot of classes, that when we would go to check a paper, they would not really pay attention to the mistake or the error that they might make if they missed something. But they'd hurriedly write the answer down, the correct answer down without any regard for why their answer differed.
And so I would have my students check their own paper, [00:10:00] and that was a shift from, something trying to please a teacher. And then it would eventually lead to them trying to figure out why they missed a problem instead of just trying to please me as a teacher. They were, they started getting curious as “why in the world did I miss that?”
They would look through their problem and see if they could figure out where they made their error. If they, if they couldn't readily find it, they, I, I encourage them to ask me if they had any, if there was any problem they wanted me to work through for the class so that they could see.
And through that they started, I think taking more responsibility for their learning. And it was kind of just an internalized thing that when they would be successful, they patted themselves on the back instead of having some, some external pat on the back.
[00:10:55] Anna Stokke: It's a lot about making sure that students are experiencing success, [00:11:00] so setting them up for success and helping students to achieve the goals that we want them to achieve. And so that's really useful information. And you talked a bit about the teaching method, known as “I do, We do, You do”. Would you mind explaining a bit how that method works?
[00:11:20] JR Wilson: Well I used this method for years. Eventually I began to see that there were staff development programs called gradual release of responsibility, which basically was based on the “I do. We do. You do” method. The “I do” would be where I would work an example of a problem and talk through it.
And I would cycle through working through several, several problems of the same kind of problem set for the students and give explanation as to why I was doing what I was doing, why it worked, how it worked, that kind of thing. And [00:12:00] then, then I would put up a problem like what I had just been working and ask the students to work that problem.
And that's the, “we do”, they would would be kind of guided practice. They would have the opportunity to try it. And then, after looked like most students had given it a try, then I'd come back and I would work through the problem for the class so that they could see you know, how they were doing and then, put up another problem and we'd cycle through that.
And again, working through the problems, giving explanation there. And then the you do part is independent practice where I would assign a set of problems, just like what we had worked through with the I do and we do cycles so that the students, they know what to do and they're getting it right after they've already had some practice and gotten some [00:13:00] feedback on how they're doing it.
I would use this for three things. I would use it to prep for current work and I would go through all three stages there. I would also use it to backfill deficiencies for skills that students should have when they, that I felt they should have when they come in.
But they, but if they didn't, then I would use this to back fill those deficiencies and I would use the, “I do, we do” part to pre-teach something that I knew would be coming up in, in a few days or a week, so that when we would actually get to that and it would be formally introduced, it wasn't completely foreign to the students and say, “oh, we've seen that before”, kind of thing.
All through this I'd be asking questions, “what should I do next? Why would we do that?” And, [00:14:00] I would encourage teachers to, to give this a try, and to tweak it to, to suit themselves, to have it work for themselves, for the sake of their students being successful.
[00:14:13] Anna Stokke: Let's shift our focus a bit. Let's discuss understanding and so regarding understanding you, you mention in your book that it's important for teachers to understand the math they're teaching.
So, for example, we teach students why a standard algorithm works. And for the listeners, if you're not sure what I mean by a standard algorithm, for example, I mean when we're doing addition by lining up the digits vertically and carrying, okay, so that's an example of the standard algorithm for addition.
And so, the teacher would explain why standard algorithms work or why division by zero is undefined, say. And these are all explained in your book, so that would be a great place to go to if anyone out there is looking [00:15:00] for the understanding behind the standard algorithms and that sort of thing. And what are your thoughts on student understanding?
To what extent and, and how did you emphasize understanding in your classes? What was the student's role in terms of understanding?
[00:15:15] Barry Garelick: Yeah, well, let me take a crack at this one. And this is getting back also to traditional math teaching and how it's viewed, and I've, I've had people tell me that traditional math teaching is all rote and it doesn't teach understanding. And my response, over the years has been, while we do teach it, we just don't obsess over it.
And that's brought me equal amounts of fan mail and death threats. But in all, all seriousness, it is important to connect students' prior knowledge to the concept you're teaching or the procedure being taught. Saying that, it's also important to keep the momentum going. And I'll give you an example. When I was working with my daughter when she was really young in math, and I'll be explaining something and [00:16:00] she'd say, “just tell me how to do it.”
And, I think this is the attitude, well, particularly when I taught seventh and eighth grade, you know, kids are more into procedures. Just tell me what it is I have to do. I want to know how to solve this problem. And so, you want to connect it to their prior knowledge, but at the same time, you don't want it to be, cover it in such detail with every step and nuance that it becomes a distraction and you're getting in the way of that momentum.
So, you have to work at it. It does work in an iterative fashion, procedures and understanding. And you do need both. And sometimes students are gonna get the concept and the conceptual understanding first. Sometimes the procedure comes first. For students in the elementary and middle school, my money's gonna be on procedures coming first.
That's where they're focused. So, I provide the connections to the conceptual understanding behind the formulas or the procedures, but take care not to go overboard and. I think Paul Kirshner has probably talked about this that young students are novices, so what they're learning and [00:17:00] what you're teaching is new to them.
As adults, we've been through it. So, you have to take care to not present it with your lens. Well, oh yeah, this is easy. They should be able to get this. No, they're not necessarily gonna be able to get this. What seems obvious to us is not gonna be obvious to them. They're, and at that age, they're highly distractable, concepts are gonna be viewed if you go into too much detail as extraneous information.
So, you really have to keep it short and sweet. So, I want to provide enough information about the conceptual piece, which allows students to see, or at least kind of see where I'm going with it. The key is I don't want to interrupt the momentum and even, okay, so when I was taking math in college, to give you an example, and there you do have to know the proofs and derivations.
If the professor was talking about a proof. If you were to ask me right then and there, okay, how does that proof work? I wouldn't have been able to answer it because I'm focusing on what he's saying. I'm taking notes. I'm gonna have to look at [00:18:00] it later to really fully understand it, because I'm trying to see where he's going with this and how we're gonna use it.
Maybe not to the same extent that a middle schooler does, but it's the same type of thing going on. So, you, you have to keep in mind where their focus is. And of course you can come back to these things later, but you want to get to the meat of it. How, how is this working? How are we gonna use it? What type of problems are we gonna solve?
[00:18:24] JR Wilson: I'd like to add a little bit to the excellent information Barry’s provided there. For me I would have students explain what they did and how they did it, but not like most people would think. Students showing their work is explanation. Math is a language and I would ask, I would have my students show their work and showing their work is explaining in the language of math step by step.
So, some math talk, and just some modeling takes place during the [00:19:00] I do you do, we do time. I would show students and model for students how I would expect them to show their work. That would give some opportunity sometimes to, point out where they can take some shortcuts.
And then I would let them know that anybody should be able to look at their work and follow it through from beginning to end to see what they've done, and they should be able to understand and follow that. If they haven't shown their work to that extent, then, then they're not really explaining it well.
But it can all be done, it's all done in the language of, of math rather than, writing an essay about how they added two numbers together. Just that doesn't make any sense to me.
And as well as showing their work that's an indicator of a student's understanding and it's also a big help if they come up [00:20:00] with an incorrect answer, they can oftentimes track back through their work and find where they made an error or if they're not able to it, it allows me, as a teacher to be able to, to spot where they're making the error.
[00:20:16] Barry Garelick: Yeah, I think his point is a good one, and it gets into the idea of this idea of checking for understanding, which is talked about a lot. How do you check for understanding? And I, I think the, what he said is the math itself, the steps that you write down is an explanation. And what I've done, - some teachers use mini whiteboards - I choose to use, have them work in their notebooks and go around the class and see what they're doing.
When I give a problem and I focus on students who I know are having some difficulties, you know who those students are after a while, the weaker students and you want to check their work and you could do a variety of things.
I always carry a mini whiteboard around with me as I go around the room and if [00:21:00]somebody's stuck, you know, I might write down the first step. And that might be enough to prompt him. “Well, what comes next?” Or as JR said they might have made a mistake and if you could pick up on what they're doing. “Oh, I see what you did. You might have done been thinking this. What, what do you think you did wrong? Oh. Oh, okay. I see it now.” So it's a really good way to check for understanding is this explanation. So, it's, it's kind of an iterative process.
[00:21:27] Anna Stokke: I'm a mathematician and, for me, understanding naturally means mathematical proof, like formal mathematical proof, which is usually not something that an elementary school student would be able to necessarily grasp.
But I have to admit, I was kind of surprised when my kids were in school and I saw the sorts of things that were being classified as understanding. And generally it was something like explain this in, in three different ways, or, or do this in three different [00:22:00] ways or, write some English words explaining how this works as JR mentioned, whereas for me, I would think of understanding for a young child as the same way JR thinks of it, that you should show your work and students should be good at showing their work. This is something that, it always sort of drives me crazy with my first year university students, not knowing, as you say, the language of mathematics.
So, for instance, I might see an equation and there are literally no equal signs. they, they've got this, this algebraic algebraic expression equal to this algebraic expression, equal to this algebraic expression. There are no equal signs. And I would have to explain to my students, you need to think of math as being kind of like English.
You have to make sure there are periods in certain places and, and capital letters and that sort of thing. So, so these things are really important for students to learn, being able to show their work and being able to show it properly. So, I really appreciate your [00:23:00] perspective on that.
And Barry, I wanted to ask you another question because you wrote another book Out on Good Behavior and, and you mentioned in that book that while conceptual understanding may come prior to procedures and vice versa, there are some students for which it, it actually may never come. Would you like to expand on that a bit?
[00:23:20] Barry Garelick: Yeah, I think there's this idealistic view that students have to understand everything. A hundred percent. And I don't think that's true in any discipline, whether it's history, geography, math, science. There's gonna be some things that are uh, just blank. But you go ahead with the procedure, you may not know exactly how it works.
I use an example of the invert and multiply rule for fractional division. I, I didn't know how that worked until about 15 years ago and I saw it written down. I go, oh yeah, that's so obvious. Well, I had a lot of tools with which to understand it. If you don't understand it, but you know the context [00:24:00] of what fractional division represents, what types of problems can be solved with it, you look at a problem, oh yeah, this will require dividing and, you know, the procedure - that that'll work.
And also I wanted to get into this idea of tools necessary. Yes, by the time I. Looked at this 15 years ago, I had a lot of math behind me, and also I had a lot of tools. Now I've, using this example again - the invert multiply. I, when you get to algebra, they, I feel okay, they have enough tools and I present a way of showing this.
And I've written, this is written in the traditional math book, and I'll, I'll show it. They'll follow it. Asked the brightest kids in my class, can you reproduce this? And it's sort of like me when the example of the teacher giving a proof and you, you say, okay, what am I doing? They had a, a difficult time following it.
I mean, they, they followed it, but they couldn't reproduce it right then there even a few days later. It, it, it was just an abstract thing [00:25:00] over time probably will sink in. You have to understand for some kids it may never come. But they might know the procedure. Some kids know how to do long division, but they can't explain to you exactly what's going on with the place value and all those things.
So, I think this idealistic view can be rather frustrating if you think well, everyone has to understand and they have to understand perfectly. It's just not possible. There are limits to what we can do in understanding any discipline.
[00:25:28] Anna Stokke: And to add to that, I also think that we should still make kids feel successful if it's just the procedure that they know really well, because the procedures are important. And I would also say that at the university level, so for instance, I've taught engineering students who said, “I do not want to know the why behind any of this.
I don't care about your mathematical proofs”, because of course we wanna prove everything because proves me a lot to us, and “I don't want to know that [00:26:00] I'm going to be an engineer. I need to know the techniques and that's it.” And this sometimes bothers me a bit, but okay. I mean, I can see the point. And they can get by with those techniques.
[00:26:11] Anna Stokke: Another good example is statistics. And so in, in a lot of professions, they use the statistics to analyze their data, but they don't know the why behind any of it. They only know the techniques. So it is, certainly, possible to be a person that knows the techniques really well, but doesn't know the why behind it.
You have a section in your book that you call “separating the wheat from the chaff.” And the idea is that teachers do need to spend more time on the most important topics. And given that math curricula can be quite dense though, and there are often a lot of topics to cover, how can a teacher know what are the most important topics, like which topics they should spend the most time on?
[00:26:59] JR Wilson: Well that's, that's something that a lot of elementary teachers aren't able to do. My experience being an elementary teacher and having worked with lots of elementary teachers is if you give them something to teach, they can do a great job of it. But in terms of figuring out what they need to teach may be a different story.
And I, in the book one example I use is Fibonacci numbers. I've watched a, a fourth grade teacher who had just come back from a conference and learned about fibonacci numbers, and she spent about two weeks with her class focusing on fibonacci numbers. Well, fibonacci numbers are very fascinating, but in my experience, I haven't found fibonacci numbers even mentioned in high school math textbooks.
Maybe occasionally [00:28:00] in a calculus textbook. So it's like, okay, could the time have been better spent? And if a teacher knows where the math that they are teaching at the level that they're teaching it at, if they know where it's going, they have a better idea of, what to shove aside, because that's not, that's not gonna be that critical down the road from my students.
And I've, I've always heard that it said that a teacher should know the math two years beyond the grade that they're teaching. And I really feel that it's, it's advantageous for elementary teachers to be familiar with at least first year algebra. And then they know that, some of the stuff that comes up in some of the textbooks, they don't need to really spend time on.
And, and they can, they [00:29:00] can sort out the wheat from the chaff and, and, really help their students. And one way to, to do that for, a teacher is get a good solid math textbook for the grade level that they're teaching. I'd suggest getting one from the fifties or sixties or a good Saxon math book or Singapore Math would be good.
Another way is take take a math class at a community college. And I'd say an appropriate level math class. An elementary teacher, I don't know how much good it's gonna do for 'em to take a calculus class, but a good refresher algebra class or, or just a, a good, math class would help them reach that point where they're, able to explain what's going on and they, and they know what stuff to teach and what stuff to let go.
[00:29:58] Anna Stokke: One great thing I would also add about your book is that you have this list of topics at the beginning and they're, it's broken down by grade.
And, actually parents often ask me about that sort of thing. “What should my child know by grade six?” Because they're worried about maybe their child isn't learning as much math as they, they think they should. And so that's actually a really good place to find that information. You mentioned fibonacci, numbers and, and fibonacci numbers are, are interesting, and that's, that's a really great enrichment topic. But that's, maybe just a short, fun lesson.
It can't be the main dish. If you're gonna spend two weeks on something like that, then other important topics that take a lot of time to practice are going to be avoided, or missed. At the end of grade six, for instance, what do you think the main math topics are that you would ideally want a student to be fluent with?
[00:30:58] JR Wilson: Well, you'd think with that question that it would be a very lengthy. But my answer I think is gonna be very short. You've got the four basic operations, addition, subtraction, multiplication, and division. Students need to be fluent, very fluent with all four operations, using whole numbers, decimals and fractions. And to be able to do that, I mean, they have to understand whole numbers. They have to understand decimals, and they have to understand fractions and be able to work with all of those.
[00:31:36] Anna Stokke: Absolutely. Students really need to be able to work with fractions. They're really gonna struggle with algebra later on.
Along those lines, at the end of grade eight, what do you think are the main math concepts that you would want a student to be fluent with?
[00:31:51] Barry Garelick: Okay. Well with grade eight, this might be different in the US than in Canada, but we have some students go [00:32:00] into an accelerated program where they're taking algebra in eighth grade. Others who don't qualify for it are just in regular math eight. Let's look at the, the regular math eight. They should, in addition to what JR said, you know, the basics they should know some elementary equation solving, proportions, ratios finding discounts, markups, that kind of thing.
Scaling, you know, scale models things like that. And then some geometric concepts. What is area, area of shapes. And then we get into volumes. You should have some fluency with that. At that level it's not a rigorous treatment of geometry, but just the, the basic, what are the formulas and, and how does one, how do we use them?
So, they should have a pretty good grounding in that. For the algebra students, they should definitely have, know equations solving very, not just the simple [00:33:00] ones, but the complex equations with, required distribution and so forth. And then fractional operations, algebraic fractions, quadratic equations things like that.
Some, sometimes, some algebra books have, a unit on trigonometry. I've never been able to cover that in one, one year. There's always something that they're gonna get hung up on, and I just don't have time for that. They should have a good working knowledge of working with powers, exponents, multiplying and dividing and so forth.
[00:33:32] Anna Stokke: What about things like factoring and working with complex algebraic expressions? Is that covered there in grade seven, eight, or is that later?
[00:33:44] Barry Garelick: That's Generally later. When before Common Core came around, California had some standards where actually multiplication of binomials was covered in, in seventh grade. But I haven't seen that. So, really it's in an algebra class [00:34:00] that you would get into factoring and, and multiplication binomials, and so forth.
[00:34:03] JR Wilson: Barry mentioned geometry and, and Trig and those are two topics that really are not covered in our book. And it's not that they aren't important there's definitely a place for geometry at the elementary level.
What foundation do students need before going to seventh grade? And, and we chose not to address geometry there. And you mentioned the chart or the table that I have that's laid out by grade level and, and topics. That's not something that's set in stone. Anybody looking at that to use it, should feel comfortable shifting things around to suit what will work for them.
And it's, it's pretty, I I'm trying to avoid using the word rigorous, but it's the [00:35:00] standards are set pretty high. on that table. So it's well within reason for some, for people to shift some things to the next grade level up.
Barry Garelick: Yeah, I wanted to just add in the seventh and eighth grade section of the book, there is a list at the very beginning of all the topics that should be addressed and in what order, the logical sequence, and that does include geometry. In that list, there's some topics that are bolded and that's, those are the ones that are covered in the book. So, a person who wants to know, "gee, what should a child of mine understand?" They could look at this list and see it just like they can in the early part of the book in JR section with the table.
[00:35:40] Anna Stokke: So, a lot of students have difficulty with word problems. Would you normally include word problems in your lessons? And what sorts of techniques have you used to teach students how to solve word problems?
[00:35:54] Barry Garelick: I definitely put some emphasis on word problems. I think it's a really important part of math. [00:36:00] And the unfortunate thing is that, well, one thing, they're not easy to teach. They're really not easy to teach. There's a lot that goes into it. So some teachers have difficulty teaching it. Some textbooks like algebra textbooks have a dearth of such problems.
And so they don't get, kids don't get a lot of experience with it. So, I start like in the seventh grade with word, word problems fairly simple ones. And algebra, of course, they get more complex. But I'll start each class with warmup problems, four or five. And these include previous material, but also some challenging problems.
And I'll include word problems in the mix and a key component is to know what's going on in, in, in the problem. And JR may have some advice on that cuz he, he looks at it, you know, the, having them read it three times. And this is true and, and algebra to know various algebra techniques and how to express English in algebra.
So, we'll start out with something like, how [00:37:00] do you say in math three more than two times a number. And so we work with that. Oh yeah, 3+2x and then what about three less than two times a number? Oh, that's gonna be 2x-3 because it's a little different. And so they have to get used to that.
And then there's types of problems that lend themselves to certain techniques like distance and rate. And so what I do. Let's just take distance rate problems is start with an arithmetic approach. Something that they know prior knowledge. Distance equals rate times time. Very easy formula. So, if you're going 60 miles per hour, how far have you gone in, two hours.
Okay. They can do that pretty easily. How about three hours? Oh, that's pretty easy. How about x hours? Some blank stares, but then put the pattern together. Oh, 60x. Ah, okay. So that's very important skill to have to use in, in the word problems. And then some simple arithmetic ones. If you've got two cars going in opposite direction, starting from the same point, one's going 50 miles an hour, the other's going 60 miles an hour, [00:38:00] how far apart will they be in two hours?
Fairly easy. Cuz they've had arithmetic problems before, so they figure it out. Now you could vary it. You're saying, well, in some amount of time they're gonna be 400 miles apart. How much time does that take? Okay, so what you're getting to is the 50x + 60x equals whatever the mileage is that you're giving in the problem.
So, you wanna work it up slowly, scaffold them so that they're using the prior knowledge, building it into algebra, and then getting into more complex problems. So that's kind of a snapshot.
[00:38:32] JR Wilson: Barry has a lot of good information there on problem solving, teaching kids problem solving. One of the things I picked up there that I probably would fail to mention is they need to learn to, to pay attention to the words in, that are used in a word problem so that they can tell, well, what operation do I need to use here?
So, they've gotta be familiar with the terms that, that might [00:39:00] direct them to the operation. I've watched students with word problems work a part of the problem and come up with a result and they're done and they don't go back and say, did what I find, answer the question that I was asked. So, I would always, coach my kids to, read the problem through three times.
First time you're just kind of getting familiar with, well, what, what is this about the second time you're going through and finding, well, what information is or here that I'm going to need to use, and is there some information in here that I don't need that may just be there to confuse me? And then, go back through and read it again to see, well, what am I being asked to find?
And then working the problem. And then once you come up with an [00:40:00] answer, go back and look at the problem again to see did I answer the question that was asked? Cuz a lot of times students will provide an answer that actually is, ends up being a part of the overall problem, but it's not the final answer.
[00:40:19] Barry Garelick: But I, I just wanna say that word problems are gonna be difficult and particularly for an eighth grader taking algebra. It's a lot of new stuff. It's a lot of new abstract ideas. And so again, this idea of, oh, everyone should have a hundred percent understanding, you're not gonna get a hundred percent of the kids being able to do all the word problems.
[00:40:40] So, what I aim for is can you do some of them and just have some idea of how these problems are set up cuz they're gonna get them again, I hope in a subsequent algebra class. And what I do on tests, I might have two word problems that are fairly straightforward that they should be [00:41:00] able to do, and then a more difficult one that I give for extra credit.
And so, the more capable students are gonna tackle. And there's no penalty to it. In fact, even everyone wants to try it. I find, and I have to advise some of the weaker students do the test first because you don't wanna spend your time on the extra credit. Cuz they want the extra points, you know?
But I think that's good. At least try it. Now not all the capable students get them right cuz they're gonna be more difficult problems. But that's my idea of a differentiated type of instruction. The instructions the same for everyone, but the reward system's a little different. And so the ones who are more capable and willing to take on a harder problem can do so and get some extra points for it.
[00:41:45] Anna Stokke: And I really like what you said about scaffolding the word problems. So start off with, basically you just have a basic sentence that you're going to change into an algebraic expression with maybe just one operation, [00:42:00] like addition or something like that. And gradually make it more difficult and then gradually have a full word problem. And probably we don't wanna put in too much extraneous information when students are first starting out because it's just going to confuse them.
But you mentioned something there that I did wanna follow up on too, because you mentioned that this is your idea of differentiating instruction. And I wanted to ask a bit about advanced students. So inevitably students are going to be working at many different levels in, in a class, and there will be students actually who will be able to get through material really quickly, who will pick it up more quickly than others for, for whatever reason.
And, and, and they'll be able to, to speed through the problems that you give them. While there may be other students who it may take a longer time to get through those problems. So, what do you do about the advanced students? How can you keep them challenged?
[00:42:57] Barry Garelick: I do it in a number of [00:43:00] ways. One way is through the warmup problems at the beginning of the class, cuz I'll sometimes include a more difficult problem in there and everyone takes it crack at it. The, the warmups don't count for a lot on your grade, so you know, they can afford to make mistakes and so it's kind of a no risk situation.
And I once had a, a student, a pretty good student in my algebra class and handed out the warmup sheets and he said, “oh, this is, this is my favorite part of the class.” Well, cuz he, he really liked the challenge. And another thing I dis discovered is in the lesson itself, after I did a few problems, worked examples, I would announce, “okay, I'm gonna give you a challenge problem.”
Now the challenge problem had some incentive to it because there's a reward attached to it, namely a Kit Kat bar. And so, in fact, students would, “are you gonna give us a challenge problem today?” Which is great. And the nice thing about it, I kind of discovered this by accident. I was a sub at the time and I was teaching a, like a geometry class in, in a high school.
And it was like the [00:44:00] area of a triangle, pretty simple. I said, oh, I suppose you had a triangle that's 30 square inches in the bases. 10 inches. What's the height? And I drew the thing on the board, no response. So I said, let me rephrase it. I said it in exactly the same way, but I held up a Kit Kat bar, and this girl just blurred it out. “Oh, well you divide the 10 by two and then you multiply. Well, it's gonna be six.” And then she said, “I don't know how I did it, but I just love Kit Kat bars.”
So I, I took away two things from this. One, never underestimate the power of a Kit Kat bar. Two, that kids really like the challenge and they like the competition and they like a no penalty, no embarrassment type of competition. You give 'em a problem that's gonna be, that they realize, oh, this is gonna be tough for everyone, so I may as well take a crack at it because not everyone's gonna get it right.
And I'm not gonna look foolish and I might get a Kit Kat bar out of it. So you get people to take part in it. The other thing is, as I said, the extra credit [00:45:00] problems on tests.
[00:45:00] JR Wilson: I'd add a little bit to that in terms of what I would do. I would wouldn't necessarily, call 'em challenge problems, but I'd put out a problem and I'd say, I'd tell the, my class said, okay, don't let me trick you on this one. Because it would, it would usually be something very much like what we'd just been working with, but with a slightly different twist, somewhere involved.
And, and sometimes it may be something that's gonna come up in the next day or two that's gonna be added to what we already know. And they knew that they needed to look at things carefully when I'd say, give this a try, but don't let me trick you on it. And boy, they would work like crazy so that they'd get it and wouldn't let me trick them.
[00:45:45] Barry Garelick: Once the answer is revealed, the thing I listen for is for somebody to say, “oh, I should have got that.” I love hearing that. Even if they got it wrong, it means that they had enough prior knowledge that they realized, oh, [00:46:00] I could have solved that. Now, at the same time, if I hear somebody saying, “there's no way I could have solved that”, that tells me, “oh, I gotta work on some stuff.”
They're not getting all this, this knowledge that they need. It's a very good thing to do is to, is to present a problem perceived to be beyond their range.
[00:46:15] Anna Stokke: So, on the flip side of that, how can you build confidence in weaker students? Do you have any advice for that?
[00:46:27] JR Wilson: I think those, all of those things, the setting goals, the consistent daily practice and, and self-checking, all of that, that in addition, helping students in getting them to like things. It, it helps build some confidence.
[00:46:47] Barry Garelick: There's a number of things that I do. One is the checking of notebooks. I try not to call on people who I, I think are, are weak students and who are gonna be embarrassed [00:47:00] because at that age they, they don't wanna look foolish in front of their peers. So you have to keep that in mind. So as I check around the notebooks and look at how they're doing it, and I work with the weaker students I'll say, I might tell one person, “yeah, you got that right.”
Then when it comes time, "okay. So and so, can you tell us the answer now?" He's confident cause he knows it's right and it gives him this boost of confidence that, Hey, I did it. You know, I'm, I'm not looking like a fool. The, the other thing, like I said, the challenge problems is this, no penalty, no risk type of thing.
You put them in situations like that where they're, where they're not embarrassed. Giving lots of problems and yes, you're right, working with them, giving them feedback, showing them where they made mistakes will increase their confidence.
[00:47:46] Anna Stokke: I also really love, that you spent a lot of time on things like factoring in, in the book. And the reason is because I teach introductory calculus a lot and generally what holds them back [00:48:00] actually in introductory calculus are things like factoring or simplifying complex algebraic expressions.
So, I really appreciated that. And there were actually a couple things in your book that I wanted to mention that maybe aren't always taught. So, one of them’s short division, which I love and, and I was taught short division as a student, but I don't think many people were, or, and I don't think it's taught very frequently. So, for the listeners, if you don't know how to do short division or if you haven't heard of it, I would suggest looking it up because it's, it's really useful. I actually use it all the time and it's quick. Do you think there's any merit to teaching that?
[00:48:39] JR Wilson: I do. But I have to tell you a little story about short division. I in, in an early draft I made that I shared with Barry, I made a comment about short division not knowing that that was actually a term that is used to [00:49:00] describe it. A method that I had learned in fifth grade and have used all my life and I, sure enough, I look it up and, and short division is a term, and I do think there's merit in, in teaching it.
Not all students necessarily are gonna pick up on it and run with it, but for those who have a good foundation with long division, they're gonna see that, oh, there are some division problems that I don't have to go through all of the steps of writing this out. I can take some, some shortcuts and, and really be doing the same thing just quicker and easier.
So, I think there is some merit and, and for those that pick up on it, great. For those that don't pick up on it, that's okay too.
[00:49:52] Anna Stokke: And another thing you mentioned that I found really interesting was that you, you sometimes teach splitting the middle for factoring quadratics. [00:50:00] So that's a, that's a method that helps with factoring quadratics when the coefficient in front of the x-squared term is not one.
I actually didn't learn that in school. In fact, I think your book is the first time I've ever really seen it written down anywhere. I've seen my students do it, particularly international students. And, and you've found that sometimes if students are struggling with factoring quadratics, that that's something that really helps them, right?
[00:50:26] Barry Garelick: Yeah, I will show it to them. Not everyone likes to do it and some, some do. So I'll show it to them, give them some practice with it, and the ones who like it will continue with it. Others like to do, you know, the trial and error and get the right factor.
[00:50:40] Anna Stokke: How much should a person tip in the United States, right now?
[00:50:44] JR Wilson: I'm not sure on that. It, it seems to me like it's been bumped up to about 18 to 20%.
[00:50:52] Barry Garelick: That's what I've been seeing.
[00:50:53] Anna Stokke: This is something you would always teach your students - how to quickly calculate a tip in their head and they would often impress their [00:51:00] parents with that, right?
[00:51:01] JR Wilson: It's something that I always would teach my sixth graders and, and I, I enjoyed doing it because it's, it's a mental activity. It's not something that they'd have to resort to paper and pencil and, and it was, it was something that I'd throw out there as kind of a challenge to see, well, who can get this and, and what do you do with it?
And it's, it's a very simple thing to do. I mean, they, once they learned decimals and, and that. You take the number to find 10% of it, you just move that decimal cuz you're, you're really dividing it by 10 and that gives you 10%. And if you want 15%, you take the, the 10% that you have and you take half of that and add it back to the 10% and you've got 15%.
If you need 20%, you just double your 10%. And for a lot of my kids, their parents [00:52:00] really did not know how to figure a tip. And so I would get lots of feedback with kids who were just thrilled because they'd gone out to eat with their parents and they figured the tip and, and their parents had typically, they'd tell me their parents just throw down extra money on the table when they're getting ready to leave without any idea of what's appropriate.
So I, it was, it was one of those things that I taught that I, that I got fairly immediate feedback on kids applying it in the real world, which, so much of what I would teach, I wouldn't get that kind of feedback.
[00:52:41] Anna Stokke: It's an absolutely great skill to have. So, last question. What advice would you give new teachers?
[00:52:47] JR Wilson: There's so much advice I, I'd give a new teacher. I, I think I'd start off with advising them to become familiar with the cognitive research findings [00:53:00] related to math and employ those findings as they work to find the, the methods and strategies that work for them in helping their students become successful.
And, and to me that's important. A teacher has to find what will work for them, what they're comfortable doing. I tried early in my career, I tried a lot of different things and I, I kept with those things that worked, that I saw that worked for me in helping my students be successful.
And the other things I thought, “nah, I can do it, I'm doing it in another way that seems to work better for me now.” How I do it, how I talk about doing things in the book might not be the best way for the next teacher. They need to find the ways that works for them.
[00:53:51] Barry Garelick: Yeah, just one thing that I would advise is, and this takes some experience, don't overload your students. And [00:54:00] the reason why this is hard to do is because textbooks are written in such a way that they overload. They put a lot of stuff into one lesson, and you really have to know how to break it up. And if you, the example I use is a, a lesson I saw in discounts where they show two ways of calculating the discount and then how to find the original price given the discounted price. That's three things in one lesson.
And just master things one at a time. Take three days to do what you're supposed to do in one day. That would be my advice. That pushback I get is “I've got a lot to cover and I can't take three days to do what's supposed to be done in one.” Well, you don't have to spend three weeks on Fibonacci numbers for one thing. And, I saw one class in eighth grade.
They spent three weeks on finding slope. It's not that hard folks. You don't need to spend three weeks. So, there are ways to cut some time and spend more time doing the more complicated things.
[00:54:55] Anna Stokke: And don't feel guilty for using explicit [00:55:00] instruction. If it works, you should use it. At the end of the day, we, we need the students to learn. So that's, that's what good teaching is all about. Well, thank you so much for joining me today and, and for sharing your expertise with our listeners. I really appreciate it.
[00:55:15] Barry Garelick: Well, thank you for inviting us.
[00:55:17] JR Wilson: Yes, definitely. Thank you. Enjoyed it.
[00:55:19] Anna Stokke: I hope you enjoyed today's episode of Chalk and Talk. Please go ahead and follow on your favourite podcast app so you can get new episodes delivered as they become available. You can follow me on Twitter for notifications or check out my website annastokke.com for more information. This podcast received funding through a University of Winnipeg Knowledge Mobilization and Community Impact grant funded through the Anthony Swaity Knowledge Impact Fund.