Let's Talk Math! Purposeful Math with Kent Haines and Steve Leinwand

What do we want students to get out of math class? Proficiency? A love of math? Job prospects?

Below is a transcript of this episode. 

Steve Leinwand:

First and foremost, we break down the isolation within which almost every teacher I know works and operates. So few professions allow people to remain as isolated as education allows teachers to remain isolated. We don't provide adequate coaching. We don't provide adequate support. We don't provide adequate time. We don't allow teachers to grow together, and I think that it's best to flip it around to the positive side.

When I walk into some of the most powerful schools that I spend time in, in every case, you have a case where people are collaborating, where people are working together. And I think that that's what's missing. We operate in isolation and there's no way that when you are isolated in a business that is inherently social, that you can be as powerful as you need to be.

We create moats around our classroom. I'm in schools where I'm told, "Oh, so we're really glad you're here. Just so you know, you are really not supposed to go into room 220 and 274." Well, you know. I mean, I don't give a damn. I mean, those are the rooms where there are problems. Those are rooms where teachers don't want to be observed. Those are teachers who in general are hurting kids that everyone else in the department has the following year.

You know what I do? I write them down and the first thing I do is I walk into room 220 and it's this embarrassing of, I'm not supposed to be here. I said, "Oh, are you teaching math?" And the kids will go, "Yeah, this is math." And I go, "Good," and I just sit down. I mean, what are they going to do? Throw me out? But seriously, I mean, you need to know what's going on in a school. And when people say, "You can't come into my room. I don't want to be coached," you have a dysfunctional school.

Kent Haines:

I certainly agree that this isolation is really inhibiting my development as a teacher because I have so few opportunities to hear from another adult who knows what they're talking about. Hey, I love how you did this. Did you notice that the kids missed this element of the class? Or, oh, have you thought about structuring your warm-up this way, or what have you? Is this something that other countries have a better model for, a more effective?

Steve:

Absolutely. I mean, this goes back to some of the original work that was done with the TIMSS, the Trends in International Mathematics and Science Study, where we had videos of Japanese classrooms and every one of those classrooms had five adults in the back of the room. You saw people learning from each other. I mean, number one, it just makes people more accountable. But more importantly, it's an incredible feedback mechanism.

Kent:

So let's say, all right, we're waving our magic wand. My school, every school in the country, we get that common planning time and also the attitude among teachers that it's okay to come in my room. It's okay to give me feedback, positive, negative, constructive feedback, I should say. What should we be talking about? What should we be focusing on?

Steve:

I love this. I think it is as simple as a 20-minute collegial discussion at the end of every observation. I mean, if I have been observing you, I have a responsibility to give you my thoughts. If you're being observed, you are sitting there going, "Well, what do you think? I mean, can I grow from this experience?" And so it always starts positive because psychologically that makes sense, but it also sets the tone that there's always something that's positive.

I've done hundreds of these discussions afterwards, and sometimes it's hard to find something positive, but you can always find, I mean, something micro often. I got to tell you, the high point in this lesson was when Emma, was that her name? And they'll go, "Yeah, I mean, the one sitting in the side of the room." And I go, "Yeah. When Emma raised her hand and you handled that discussion and her confusion, I go, that's when I just said, this teacher really gets it. And every kid in the class benefited from that interchange.

And I imagine you worry about spending too much time on it. It always starts with something positive. And the second point is, so here are the questions I have. I mean, I wasn't sure about, or can you give me more information as to why you did this? It's not critical. It's simply a matter of, so help me explain some of your actions. And then the discussion always ends with the observer saying, "On the basis of this discussion, on the basis of the observation, this is the one thing I'm going to try to do differently tomorrow."

So it's action oriented. Those are the three things. Anyone can go to my website. There are a whole bunch of slides that are there. My presentations are posted. There are probably 20 different presentations, and there is one about professional development. And the one about professional development says that it's not professional and it doesn't develop the way in which we are currently doing it. And then the alternatives are, among other things, these collegial visits.

Kent:

Let's shift a bit and talk about assessment because I did something very scary. A few weeks ago, I sent you a couple of tests that my colleagues and I had given to our students, and you wrote back with a ton of excellent feedback about which questions you liked, which ones you thought I could do without, and what you thought was missing. So if you don't mind putting yourself back in that mindset, when you opened up my test for the first time, what were you looking for?

Steve:

Great. So let's put this in context. We have to live with a whole bunch of tests. I think we live with too many tests. I think that the quality of the state assessments is mediocre in many cases. I think that when we lost the Park and Smarter Balance and all that came along with the Common Core, we lost a tremendous amount. And so we've returned to multiple choice drivel that just diminishes the quality of teaching that doesn't tell us nearly enough. So I have come to believe that, first and foremost, every teacher needs to practice formative assessment.

It is always asking questions. It's listening to kids. And there is an exit ticket or an exit slip every single day, every single classroom. Now, having said that, it means that we both know that one out of five days, I don't need an exit ticket because the last task the kids did conveniently served as my exit ticket, and I didn't realize that was going to happen. But I knew everything was going on and I knew the kids were successful and I didn't need to go any further so I could spend time on it.

I also know that one out of every five classes, the class sucks. It didn't work. I blew it. There was an interruption, and I don't do an exit ticket. So I start with that's instruction, that's not assessment, but it is assessment built into instruction. The second piece is I believe that the very best programs are driven by a set of high-quality common unit assessments. It means that beginning in second grade, there are eight units and there are eight unit assessments.

In the best world, there are two forms, a form for us to use and then a clone form that we use for a retest or a makeup or for an opportunity to show that you didn't get it the first time, you can get it the second time within five or six days or something like that. But these unit assessments are done collaboratively. They answered the question. We are successful as a unit grade level team, as a course team when our students do well on this unit assessment. And when they don't, we know we need to build in some reteaching before we get to the final.

It's really common sense. It's, again, why less is more. We have to have time for the reteaching. We have to know what our focus is. I just think that a unit assessment is a place that not just answers the question, were we successful? Did the students learn? We know, but it's my fundamental planning guide. And so when I looked at your assessments, I have looked at literally thousands of assessments of that sort, the first question, is it balanced? That is, does it ask for the core basic skills that are non-negotiable for that unit and for any future further learning of mathematics?

Two, does it get at conceptual understanding? Does it, in other words, move from depth of knowledge one, the core, the recall, the vocabulary, the understanding that is enabling? And then does it ask, do I have evidence that students understand the key concepts, that they are not falling into the misconceptions? And then thirdly, do I have knowledge that they are able to do some reasoning and solve some interesting kinds of problems? I mean, that to me is an assessment that also differentiates.

This idea of differentiation says we need to build in some of the enrichment. It says that differentiation to me, great teaching is, and we'll talk about this perhaps under instructional quality, but to me, differentiated instruction says we ask kids why. How do you know? Can you explain? How did you picture that? Who did it differently? Those are those key guiding questions.

Kent:

So it's a collaboration among colleagues, and there's obviously attention where we want this to be an assessment that everybody feels comfortable, that sort of thing. And so a common response that I hear from my colleagues and feel frankly myself is we have to put a grade from zero to 100 in for the assignments that we have at our school. That's not negotiable with our district. If I have a test with 10 questions, my students are like, "Oh, I missed one question, now I'm down to a 90. I missed another question. I'm down to an 80. I missed it," how do you respond to that idea?

Steve:

Why is everything worth the same amount? I mean, there are some things you get wrong that are only worth a point. That's all. It's not a big deal. Okay? I can reteach it. You didn't remember it. Why am I going to zap you for something you didn't remember? A serious conceptual error. Yeah. I mean, you know what? There's a problem there, but I respond to that, first and foremost, by saying we ought to have a system of retesting.

Every single kid ought to have an opportunity to go over that test, to figure out what they didn't know, what they couldn't do, and have a chance within the next five days to come back and retake the test. That's how I deal with it first. That's just fairness. That's just common sense to me. This idea of one shot. So I'm working in the highest performing high school in all of New Jersey a couple of years ago. Well, how come you only have this one shot? These kids are so pressured in all those ways.

And I said, all you do is contribute to that. What do you think? We should give every kid a trophy? I mean, that's the kind of mindset that I run into. You could hear a person who's a good teacher who cares, thinking about, well, I'm just making it too easy. Anyway, I think that there are lots of ways to respond to that, and they require us to say, "We're going to do it differently. That we are about serving our kids and we lose nothing and the kids lose nothing. Everybody gains by having a second shot at it."

And then the whole point structure is really key. I mean, I think that we ought to have a point system that it's not 10% for everything. But you know what? Here is a 40 point test. And we have marks, so we have points. It's so clear. This is one point. This is one point. This is two points. This is four points. This is five points with partial credit, and all those kinds of things. And then we simply convert the points, gee, we're math people, to a number that we need. I mean, you're going to tell me that there's a difference between one student's 89...

Sorry, let's be fair. One student gets a 79 by making lots and lots of careless errors and loses all those points, but hits it out of the park on the two biggest, most important consolidating items. Meanwhile, the students sitting next to that person gets the 81. They hit it out of the park on all the mindless, all the stuff. They can get all the skills. They were regurgitate beautifully, but they really cannot come to it. And on the big items, they get low partial credit. And we sit there and say, "Well, I mean, 81's better than that."

No, I mean, we've got to find a way to be able to have our point structures so that we're not differentiating between 79 and 81. We're differentiated between a B minus and a C plus.

Kent:

That's interesting. I had a system that I used for several years here. I didn't implement it this year because it's my first back at this school, and it takes a lot of work, as you'll hear. But I did a standard space grading system where I would break out, okay, well, unit three has essentially six major ideas. And so I would have a line at the top of the test and it says I can and basically each of the six major ideas are standards.

And then I would give them a test, and then I would grade it item by item, but then I would just go back and holistically say, do I feel that this student has mastered this standard? And sort of treat it a little bit more I wouldn't say arbitrarily, but more holistically. Just trying to say, do I think that this student understands and can apply the Pythagorean theorem, let's say?

And so I could actually give the same tests that I've been giving or the types of tests that I sent you or what have you, I just assessed it in a different way. And then the benefit that I found from that is my students knew where they needed to study for the reassessment, and then we could just reassess that item as opposed to, I have to retake the unit three test.

Steve:

I think that we have some real problems and confusions with the term standards-based. Part of it is the teachers are saying, "What are you talking about? All my tests have been standards-based. Every one of my items is linked to a standard. So it's standards-based." So now what do you want me to do with that stuff? The language is confusing and it doesn't motivate me to change.

I think that we ought to talk about less standards-based than more balanced, common, collectively reviewed, great task system of unit assessments. And so we look at it from the perspective, as we said, of balance, and we do it together and we review it and we look at the student work. One person is a day ahead of everyone else, and so we can make some minor adjustments on the test so quickly.

But it is a set of items that are aligned with a set of standards. And then there is the opportunity to say, okay, so in terms of the basic skills and the vocabulary, very strong. You did well on items one, two, and three. That's standards-based. That's giving you feedback like your three-year-old gets. In terms of the conceptual understanding, there are some issues here.

You were not able to explain. You were not able to show how these things relate. So I think that we can accomplish what ideally standards-based is about when we construct our tests with deliberate attention to what we're measuring.

Kent:

So we've talked a little bit about how to help teachers improve their instruction by collaborating. We've talked about how to help teachers improve their assessment. And I really like those because those are things that I have a handle on, things that I have at least the possibility of having an impact on. It definitely requires a bigger shift than any teacher's capable of implementing personally, but it could be very interesting. So let's just start quickly. What do you think is wrong with high school mathematics in America?

Steve:

High school math teachers have not been given anywhere close to the level of useful guidance as K-8 teachers have been given. When I look back on my 50 years, I can tell you that I am proudest of the fact that there have been some real successes. What I've seen at K-8 in the last 40 years is just unbelievable shift from worksheets and computation and little more than that done in 30 minutes a day to really teaching mathematics and engaging kids.

And the same thing applies to technology, the use of technology and its ubiquitousness. Even when it's used poorly, it's still there. But I see a lot of incredibly good use of technology. I see interactive whiteboards being used in ways that are clearly supporting learning in so many ways across all disciplines. I only tell you that because when I look at where I failed, when I look at where there's been no change, we still track and group in ways that are just nefarious.

We screw more kids by telling them, "You can't do it. You don't have a math brain. You don't have a math gene. You can't do it." And all too often those are kids of color or kids of lower economic class, and it is just disgusting and pathetic the way in which the system systematically preserves privilege at the detriment of the entire society. And high school sits there still with an 1894 committee of 10 model of algebra and geometry. That it is about getting kids ready for calculus today.

That the idea of a common curriculum for all kids and I think without levels and without all kinds of tracking and all that that entails is the right way to push the quality of common, appropriate, important math for citizenship and for the workplace through the end of 10th grade. That means that Algebra 1 that is currently in eighth grade, thanks to the Common Core, continues, it's important, but it means that Algebra 1 has to change and it hasn't. I mean, if it's taught right in eighth grade, then we review it and we build on and we expand it in ninth grade.

The idea of this Algebra 1 with quadratics and simple manipulation with an Algebra 2 course that every teacher says is a nightmare to teach unless you just love telling kids how to do something without thinking or reasoning or applying it, that's Algebra 2. And then you talk to the calculus teachers who say, "I don't know why they're doing synthetic division. I don't expect kids to do that even when they're doing Newton's quotient in calculus." So we've got a screwed up, obsolete curriculum that has to change.

Kent:

Thank you so much for sharing that, and what I really appreciate about this is in all these areas we discussed today, you're not just focused on identifying the problem, but you're giving us ideas for a solution. So that even if we don't necessarily implement exactly what you're talking about, it's much easier for me to think about how to improve something if I have a framework to look at.

Steve:

If we've got three seconds, I just want to say that, I mean, obviously there is a soft spot in my heart for Heinemann. It's been a wonderful relationship, but of all the books that I've written and the one that I understand second only to Tom Carpenter's Children's Mathematical Thinking, the Accessible Math is the book that I turn to first, because before we even talk about curriculum, what happens when you close your classroom door?

And I think that we didn't spend a whole lot of time on instructional quality today, which is perfectly fine, we can't do it all in an hour, but the first thing you asked me is, what do I look for in a test? The real question is, what is it that every coach, every principal, every parent who visits a school and every support consultant needs to look for in a classroom is, how do you get the kind of engagement and the kind of thinking that is just so critical and that it's nowhere close to uniform?

I'm just reminding people that the real solution, the way in which we prepare kids for all these things is we're doing cumulative review. We're asking kids why. We are using context. We are doing the kinds of things that we know research tells us make a difference. Just an unabashed, shameless plug for a book that's still out there called Accessible Math.

Edie:

Thank you for tuning in today. For more information and a full transcript, please visit blog.heinemann.com.

Kent Haines is a National Board Certified middle school math teacher in Birmingham, Alabama. He has spent 10 years in the classroom, as well as two years as a visiting instructor at the University of Alabama at Birmingham. Kent is a 2016 Heinemann Fellow and has helped develop curriculum for The College Board's Advanced Placement program, A+ College Ready and Citizen Math. He writes about math games for parents and kids at Games for Young Minds. 

Steve Leinwand is Principal Research Analyst at American Institutes for Research in Arlington, Virginia and the author of Accessible Mathematics and Sensible Mathematics, and coauthor of Developing Numerical Fluency. Steve served as Mathematics Supervisor in the Connecticut Department of Education for 22 years and is a former president of the National Council of Supervisors of Mathematics. In 2021, he was awarded the National Council of Teachers of Mathematics' Lifetime Achievement Award.