Is high school math failing students? The way math is taught today hasn’t changed much over the years. But along criticism for being being out-of-date it perpetuates inequity by limiting opportunities for many students. There are expectations for high school math to do a better job in preparing students for life in the 21st century.
Is high school math failing students? The way math is taught today hasn’t changed much over the years. But along criticism for being being out-of-date it perpetuates inequity by limiting opportunities for many students. There are expectations for high school math to do a better job in preparing students for life in the 21st century.
Major shifts in course organization, mathematical content, pedagogy, and assessment are long overdue. To initiate those changes, authors Steve Leinwand and Eric Milou have written Invigorating High School Math: Practical Guidance for Long-Overdue Transformation.
This conversation originally aired as a Facebook Live on the Heinemann Facebook page.
A transcript of this episode is available below.
Brett: Eric and Steve join me now. Steve, I want to start with you. What was the inspiration? This is a very bold book. It makes a very strong statement in the world, and it is an absolute pleasure to read. What made you want to write this book right now?
Steve: Very simple. Eric and I have both been talking and speaking about high school, about the problems of high school. And we worked with some organizations, and we worked with NSF to try and get some funding. We thought we could do a conference. We thought we could do a conference report. And for a whole lot of reasons, including the pandemic, none of those things panned out for a whole range of reasons.
And so Eric texted me about April of 2020, just as we are losing everything, and we are in the pandemic, and says, basically, "Screw all that. Let's just write a book." And I just went, "Duh, great. Give me 24 hours. I'll send you a table of contents and then you'll fix it." And then we talked to people at Heinemann and the rest is history. But why this interest? I mean, that's the motivation, except, I mean, I just ask everyone who's listening to play a little game with themselves. Eric and I say high school math. Ready? What's the first thing that comes into your mind? What word, what phrase? I mean, the fact of the matter is, for most people, the answer is deadly, useless, regurgitation, a bunch of answers that I didn't understand, little applications, not much fun. I mean, it is deadening and therefore it needs invigoration.
More important than that, I asked teachers to take any of the tests that they give their students and use the Google, Alexa, Desmos test, which is, if Desmos, Google and Alexa can get more than 75 on any test we give kids, then it's just clear proof that we are asking kids to regurgitate things without asking them to explain, without having them solve interesting problems, without having them show their work. And that's just insipid in today's world. And I think the last piece is, the pandemic has really shown us that without modeling, without statistics, we are losing it as a society. I mean, to listen to Angela Merkel, or to listen to Governor Cuomo back then talk about the Rs of nought, and the whole transferability of the virus, and know that people's eyes are glazing over. To see all of these graphs and know that most people either think that they're lies or ignore it. We, as people who care about mathematics have really won K-8.
The common core is in many ways manna from heaven. K-8 has changed. It's been improved. Why? It's internationally benchmarked. Why? It's fewer, it's clearer. There are progressions. Teachers know that they have less to do, but they have to go deeper. When we get to high school, none of those things obtain. High school is still, as we talk about in the book, driven by an 1894 report that basically lays out algebra and geometry. It's insipid. And so you put all of those things together and you have a desperate need to change what we do.
And Eric and I have argued that the common core hasn't helped, even NCTM's Catalyzing Change laid out a whole lot of things, but teachers have not been given guidance. And so we wrote this book to provide clear, specific guidance to engage people in discussions. We do not believe we have all the answers. We do not believe that you adapt anything in our book, hook, line and sinker, but we think that it is a starting place so that you can now create your own vision statement.
You can create your own goals, purpose statement. You can go and create an outline for these courses. You can begin discussions about detracking. You can talk about how these pathways are so critical. And you can acknowledge the fact that it's hard to get around all those excuses that it provided unacceptable inertia.
Eric: Let me add to Steve's passion here and go back to about 2016, when then NCTM president Matt Larson said, and I quote him here, he said, "Today, it seems as if nearly everyone agrees that high school mathematics needs to change. For far too long, high school math has not worked for far too many students." It's a bold statement. And it really was a call for action. NCTM took some steps. Catalyzing Change was released in 2018. And what Steve and I wanted to do was build on that momentum and just give some more steps that are actionable across the board for high school math. We wrote about the challenges and the realities that high school math now faces.
We wrote about the excuses, the excuses we are given for why change is not made. And then we've listed guiding principles, 14 guiding principles to guide this work. We've laid out a curriculum for grades 9 and 10 and integrate math sequence. And then we laid out pathways, rigorous pathways and alternatives to Algebra 2 for grades 11 and 12. But curriculum is not enough. And we couldn't just stop at curriculum. We not only wrote about those curriculum changes that I just mentioned. We wrote about pedagogy changes that are way overdue in high school mathematics. And of course we wrote about how assessment must change.
And the pandemic has exposed that more than anything. As Steve said, if Google math can do 80% of the problems on the assessment, then is the assessment any good today? What changes need to be made? Of course, no discussion of today's high school math curriculum is void of technology. What we call the enabling game changer. And of course the Catalyzing Change does make the point that the focus of high school math should be about modeling, the overarching 21st century competency. And finally, we give you an implementation guide, a five-year plan to put into place the changes that we suggest. That's what we wrote about. And we're hopeful that, finally, long-overdue changes, 50, maybe 100 years of high school math that has not changed. Maybe a start can happen.
Steve: I'm gonna follow up with the, where we end with implementation, and really provide five specific tables for the activities and the reasons and ideas for how to conduct those activities for a five-year, year by year by year plan. But the beginning of that chapter speaks to something that almost all teachers resonate with. And that is, we know about change. Michael Fullan has told us so much about change. We ignore most of it. And so it's important that we understand change takes time. This is a massive amount of change. When we've been locked into this for 125, 130 years. Too often, we're being asked to implement things prematurely before we get buy-in, before people are ready to do it. Change requires informed discussion. No change that's imposed top-down is going to make any difference.
And it's empowering as we want teachers to be. The fact of the matter is, when it comes to changing high school math, they've got to deal with a curriculum director, and assistant superintendent, a superintendent, and a school board that in fact wants to maintain privilege for their upper middle class white kids. And there's no nice way to say that, but we know that that means that it's gotta be done top down, bottom up. We know that change takes careful planning, and that's why this notion of this book is part of the first year, this book, along with Catalyzing Change. So look at the guiding principles and identify those that are not a problem. Identify those that you're not even close to. Begin to say, we have a year or two to begin to worry about this stuff. We all know that that change requires us winning over colleagues and decision-makers. We have not involved guidance counselors in any of these decisions, and we do that at our peril. We know that change requires mutual trust, and yet we don't have collaborative structures in most departments.
So, I mean, it's clearly not just the high school curriculum, but it is the culture in which teachers operate in, it's the culture of the schools and the whole change process that we think people can look at it and say, "Yeah," you know, "Steve and Eric are right. Yeah, you know, I've said that for a long time. Yeah, you know, maybe we need to stop and think about these are the three critical things we have to do this year, and then we'll take a breath, and then we'll reflect, and then we'll see if we're ready to begin to worry about some summer work to begin to pilot Integrated Math 1." And that's the only way that we see it happening.
Brett: In chapter two, you have a whole section dedicated to excuses. Steve, when we're doing change management, there's always that collegial conversation that we have when we get stuck talking about excuses, we go, "Oh, what are you going to do?" How can we in our conversations about excuses, you've debunked a lot of excuses, how do we get ourselves through those conversations of excuses so that we're taking actionable items that you've wrote about?
Steve: I think it's critical to understand that change is incredibly hard. Nobody really wants to jump into this, even when we're faced daily with the fact that so much of what we're teaching is obsolete. And we've got these ancient textbooks that ask kids to regurgitate things. People say, "Well, you know, we really can't do this because lots of kids don't have the basic skills." And we hear that everywhere. Well, I mean, the question is, do you backfill all these basic skills, which has never really worked? Or do you engage kids with the kind of problems where you are doing both the algebra or the statistics, you're engaging kids because it is about a speeding ticket, or is it about a basketball shot, or it is about something that kids care about.
In the book, we have the wonderful problem about at what point will your cell phone be totally charged? And all you have are four data points of the percent charge and the time. I mean, you can't help but want to graph that out, seeing that it isn't linear, make some extrapolations, but look at the basic skills we're doing at that point. And again, we talk about how many of the skills are today done by technology. And so we really talk about preparing kids for today's world, and being able to think and to reason and construct viable arguments, and justify and create and use models. Then basic skills become a less important thing. It's like, "Oh, well, they don't know what minus seven plus four is. Well, we know why that is. It's because they've never given them a reason. All I know is that all I have to do is say that Tom Brady lost four yards on the first down, and then he gained eight yards on the second play, and every single kid knows that he's four yards ahead. I just did integer arithmetic.
One of the excuses we hear all the time is kids are unmotivated. Well, open any math book. I mean, who's motivated by 12 examples of being told how to solve systems of equations when none of them are use technology to get a graph and interpret it. And what's really important then, every teacher knows this, what's really important and the reason why systems of equations are so hard, particularly given that the common core assets to do that in 8th grade as part of functions is how do I create the expressions? That's the critical skill that no calculator can do. I need to understand what the variables are. I need to understand what the constants are. I need to create those things, and when I have an interesting case about the speeding ticket in Vermont versus the speeding ticket in Connecticut, and the way they are linear functions based on so many dollars for every mile per hour over the speed limit, all of a sudden, motivation isn't a problem anymore. When I do my geometry with stacks of pancakes and the world record for pancake consumption, you know, there's other excuses, "I have to prepare them for the tests." Well, you know what? That is true. People talk about preparing for the test as if somehow we were doing such a great job on these tests. When you look at the raw score equivalent for a 22 on the ACT and a 500 on the SAT. Not exactly write-home-about scores. It's essentially half the items.
Eric: So in the book we talk about eight excuses, and Steve has mentioned several of them so far from the number one excuse, they don't have basic skills learned in middle school. Number two excuse, they aren't motivated. And of course we know that motivation reason is because the curriculum is stuck in the 1930s. I got to prepare them for the tests. These often irrelevant tests. It's what the colleges demand is excuse number four. And that excuse is becoming less and less of an excuse. As colleges begin to really make these changes, changes in their curriculum, changes in their placement criteria. In fact, colleges, I believe, are ahead of where high schools are as far as progressive math curriculum. Excuse number five, there's just not enough time. That's one where Steve and I agree, is there's not enough time to teach mathematics, and we absolutely say that we must have 60 minutes daily. Excuse number six, my textbook guides what I do. Oh boy, that one. Number seven, I don't have the technology needed to change.
And number eight, I don't have the training. We asked you to reflect upon which excuses are most familiar to you, and which excuses applied to your district. What other excuses that we have not listed do you encounter? And we ask you, how can you respond in a practical, reasoned way to the excuses we hear made? So the excuses don't become obstacles to the change that we talk about.
Steve: And we think the foundation is to stop and say, so we've got these challenges. They are real. We've got these excuses. They are real. Some of them are not our fault. Some of them are things that we have been, in a sense, deprived of. But what it does is it sets the stage for what we think are a set of 14 guiding principles. What I love about chapter three is we've agreed with Heinemann to have that available for free. All you do is you go to Invigorating High School Math, Heinemann's website, and you can download it. So there's no reason to spend any time talking about, you know, a 20-page chapter, when you have to have goals and purposes. It's one of the things that NCTM did a great job in Catalyzing Change.
What is the fundamental purpose of high school math? And when you answer that question, you realize that it is not exactly aligned with what we're doing. You have to have a vision. You have the vision of teaching and learning. Well, we don't just talk about it. The book has examples. The book has samples. The book has you start with this one, and then you say, "That one sucks, I did that and that's ridiculous." But this one really resonates. And we can try this one. You need to recognize that the whole bottom line is opportunity. And if there is not equity and access, there is not opportunity. And we have failed far, far, far, too many kids. We know we fail it. It's primarily an issue of mediocre teaching in too many places where it's not a teacher that gets blamed. There isn't a coach. There isn't guidance. There isn't a principal or an assistant principal that can come in and help those people. And so, you know, we need to know that teaching has to be more equitable pedagogy, and the curriculum has got to be different.
We talk about the culture. We talk about the scope of the curriculum that cannot continue to be 700-page textbooks with 14 chapters that are just not teachable. We talk about differentiation, which to me is one of the absolutely critical aspects. I'm working in a school district now about detracking and about fixing 8th grade and 9th grade so there's one level. And it all comes down to differentiation and enrichment and supplemental stuff and giving kids a reason to care. We believe that integration is no longer an option and it has to be integrated and we make the case for why. You've got to specify your content, the connections, I'm gonna stop at that point, except that the very last one is this critical notion of professional growth and collaboration, which is the essence of implementation, the essence of the strongest math departments that either of us work in. And what is absent in departments where we think kids are being unreasonably hurt. So, you know, we just say, go online, get it, get it downloaded. It's a beautiful PDF with all the color and we think that it is an incredibly good place to start. In fact, you mentioned that, you know, we built guiding questions in the end of chapter three.
Of these 14 domains, which ones do you think your department comes closest to meeting? In what ways is this the case? Yeah. Start on the positive. Start on the excitement. "Well, you know, we're working on this," or "We've actually got this." Right? Of these 14 domains, which ones do you think your department is farthest from meeting? Why do you think this is the case? And what can you do to begin to change that? And finally, of these 14 domains, and given that it's impossible to simultaneously address all of them, which two or three do you see you and your colleagues beginning to work on first? Practical.
Brett: In part two of the book, you then break it down, grade by grade, how to make these practical, how you're referring content courses and pathways. Can you talk a little bit about how you structured that practicality grade by grade?
Eric: So we fundamentally believe that as K-8 mathematics is, it's integrated all throughout, it's that we learn a little bit of algebra and geometry and probability statistics, especially throughout middle school. So we fundamentally believe that needs to continue in grades 9 to 10. It's that there should be an integrated mathematics courses that has combined algebra, geometry, data science, more than ever today in 9th grade and 10th grade. And then students. Based on their career trajectories, based on their, where they see themselves in the future, can choose different pathways, different pathways of mathematics that are equally rigorous.
Of course their traditional pathway towards calculus would still be there, but it's not the only pathway. We continue to fight this notion that calculus is the only rigorous pathway possible. We need to stop that. That a statistical pathway is just as rigorous and has not the pandemic exposed more than ever, that the understanding of data, the understanding of statistics is more valuable than ever in today's world? And then also a quantitative literacy pathway. That's been called for for over 20 years, that the quantitative literacy of the American citizen is so low, and that we've created three options for kids. These are pathways, completely different than tracking. The tracking of kids in America is insidious, where kids are in different sections of the same course, Algebra 2A or Algebra 2B or Algebra 2 Honors. The exact same course with different levels. Sometimes just slower and louder.
We are not advocating for such insidious tracking. We're advocating for rigorous pathways from quantitative literacy to statistics and data science to the traditional calculus pathway. Alternatives for kids in grades 11 and 12.
Steve: Everyone that knows Steve and Eric is sitting here listening and saying, "Oh my goodness, Steve's passion is just rubbing off on Eric." That's as passionate and as forceful as most of us have ever heard Eric, and it just warms our heart. You know, there's nothing else to say. I mean, he's defined the sensible way in which we simply move the goalposts so that the common curriculum goes through 10th grade because of how the world has changed, because of opportunities.
And when it's a common course with some accommodation for acceleration, but not a separate track, then we have this platform of K-10 common integrated mathematics. And then as we said, we talk about these pathways to begin to do things. I think that the most important and perhaps one of the most controversial proposals in this book is to recognize that the reason why we have tracking, the reason why we have acceleration is this mindless belief that the answer to my students, my kids, my son, my daughter's advancement is calculus in senior year, which means I have to do Algebra 1 because we have this thing. Well, good news.
The common core has done about a half of an Algebra 1 course now in 8th grade, when you do what's sensible. If we have an integrated program with the statistics and the geometry and plenty of algebra, Eric and I believe that when you look at the Intermediate Algebra, the old Algebra 2 outline in the book for the calculus pathway, it is a legitimate prerequisite to calculus. In other words, every single kid, without being accelerated, can move from 8th grade math to Integrated 1 to Integrated 2.
And then if you know you want to be a doctor, you want to be a scientist, you want to be whatever it is, you can do this Intermediate Algebra where we define it, and we talk about taking out the things like polynomial addition and synthetic division and trig identities and the list goes on, of things that have nothing to do with preparation for calculus, or things being important. They're great enrichment topics, but they ought not be what holds kids back. And then other kids are saying, "No, I mean, I know that I'm going to be a social scientist. I need to be able to handle statistics." Well, there is this other pathway, and I am incredibly excited about that.
And at the moment I'm actually working in a school district that has done it. They have done the system where every single kid is eligible to take calculus as a senior because they have combined the pre-calculus and the Algebra 2 into a coherent, meaningful, meaty technology-enhanced course.
Eric: And just to finish this discussion of curriculum out there. In the book, we will lay out the big ideas, the essential understandings and the units that are necessary for an integrated high school math course one and an integrated high school math course two. And then we will lay out the guidance and materials for a quantitative literacy pathway, quantitative reasoning pathway. The Dana Center has done tremendous work on quantitative reasoning courses. We will lay out a data science or statistical pathway. Jo Boaler and Youcubed have done great work on that. In addition to Introduction to Data Science, which was funded by a national science foundation and is run in Los Angeles County public schools. And in the state of California, that course Data Science, which is freely available on the internet, does substitute for Algebra 2 for all California State schools.
And of course, we will still continue to have a calculus pathway, but we will talk about how now, that we don't really need an Algebra 2 and pre-calculus two-year course. Is that we can move that to a one-year course because we have kids move into calculus in grade 12, and only kids who want to move, not all kids in that pathway. There are many obstacles here, and we would like you to think about those obstacles and what you see as the greatest obstacles of moving your school or district towards a common integrated set of courses in grades 9 and 10. And what are the greatest obstacles in creating pathways for your students in grades 11 and 12?
Steve: So believe it or not, all that we've discussed is the first half of the book, parts one and two. The question that everyone asks all the time when we're doing workshops, or we're writing stuff, or we're having discussions is very simple. The question is, yeah but what does it really look like? And so we have a chapter on pedagogy that speaks to equitable teaching, building off of NCTM's principles to actions and the teaching practices, and then some examples of what it looks like.
We have a chapter on assessment where, I mean, I frankly believe that the glue that holds all this together, in any school, and this actually is basically 2nd grade through 12th grade, it is a set of high-quality, balanced, meaningful, engaging unit assessments. You show me a school where in Algebra 1 or Integrated Math, or in 6th grade, or in calculus, there are assessments that are balanced across the depth of knowledge, that are balanced across some skills, some concepts, and some application.
You show me a test where there are no multiple choice items, and I will show you a test, and we've got a bunch of examples built into the book. We then turn to technology because you can't do all this stuff. You can't accelerate it. You can't engage kids without recognizing the technology, that it's everywhere. And there's some wonderful, wonderful examples about the technology and the rationale for it, and the fact that it's inexcusable that any kid is anywhere in high school, to say nothing of middle school, without that technology.
And then we end with mathematical modeling because it's the models that pull so much together. Whether it's the data science models, or it is the statistical models, or it is the functional models. And there are more and more three-act lessons and a set of units and tasks that allow people to do that. And having given you that picture, we then take a deep breath and say, "So how the heck do you do it?" And as we've said, we provide a five-year plan where there's a year of study, there's a year of initial implementation, there's a year of piloting the work. And again, doing it collaboratively, hopefully the school is big enough, or you can collaborate with another two high schools. And there are three of you that meet on Friday afternoons to sort of talk about how the week went and what would you do differently and how it's working and what, "Oh my goodness, I can't believe I found this amazing Desmos activity. It took us two days, but I believe that my kids understood that idea for the first time." Those kinds of interactions that speak to the collaborative structures and to the five-year, take a deep breath, let's do it right, let's do it iteratively, let's recognize that this is gigantic, massive change for people who care desperately about their kids and they don't want to hurt their students.
And so we think that five years and the outline that we provide allow us to move ahead with first, the Integrated 1, and then the 2, and we can start with some of the changes in the Algebra 2, and in some of the pathways in the second year of this as we begin to build out.
As Eric has suggested, some of the materials that are now available for the statistics pathway, or for the data science pathway, for the quantitative literacy pathway, all of those things already exist. And so it's a matter of stopping and saying, how do I adapt that? How do I take these two units from the lesson of math, these two units from the open education resources that is the current version of the math visions project. I think there are great things, but I couldn't teach every single unit of either of those programs. But I can pick and choose and go, whoa, I can make some minor adaptations after I tried teaching it for the first year. So again, rational, hopeful, and I just, I'm excited, I guess, you know, to personalize this in some small way, you know, so I'm 72, I've been doing this math ed leadership stuff for 40 years. I started in the classroom 50 years ago this month.
I am so proud of some of the things that I've seen change. I am so proud of, after all this time, we actually do have use of technology. We actually do have a K-8 program that I can be proud of, that I can watch as my grandkids have gone through a strong program. I am so proud about gender equity, that when I started teaching, I mean, girls were screwed in every possible way, female students, I mean, we did not have anything close to the kind of parity that we have now. I mean, it is something to look back on with great excitement.
Eric: Yeah.
Steve: But we fool ourselves, if looking forward, we don't also say that the tracking is still so disastrous, the instructional quality ranges from godawful to absolutely stellar spectacular, that high school is a problem in terms of curriculum and preparing kids, that the quality of assessment is just as varied, and there's no commonality to quality assessments.
And the professional isolation is probably bigger now than ever before. I can simply tell you that the pandemic, lots of reading, pausing and thinking about, I'm not ready to retire, but I'm also ready to sort of focus on some things. And I've shared this with Eric, but high school in detracking, I helped to write the ASSM, the Association of State Supervisors of Mathematics' detracking position paper. You know, building off of the National Council of Supervisors' position paper.
These are important issues. These are, as Jo Boaler wrote the other day, amazingly complex, complicated and controversial. But if I can use some of my experiences and use this book as a foundation, and we can invigorate high school and use that and our belief in equity and opportunity to do some detracking, frankly, I'll die happy. You know, all the awards and all the lifetime achievement stuff, they're nice. It's nice to be recognized. But you know what? That's not what floats my boat. What floats my boat is having people say, you know, "Your book, or your workshop, or your ideas," and I know I speak for Eric as well, because he's out there all over the country as well, and he's helping people see what you can do with this technology, what is available, what is free, and basically say to you, how can you not be using these things, these materials, to improve the quality of teaching and learning? And so that's really the motivation and the big picture from my crazy perspective.
Brett: Well, that energy, Steve, and that passion, Eric, it just, it leaps off the pages of this book. So I think we're gonna start to wrap things up there, Eric, I don't know if there's anything else you want to, if you want to have the final word?
Eric: Yeah, it's always tough to get the final word in with Steve.
Brett: That's why I thought I'd ask ya.
Eric: Right. But it is a pleasure to be able to be a co-author with him, and to be able to, I pushed him to want to write this book. And I believe what we have outlined is that the changes, we all agree, doesn't matter where you are with respect to your high school teaching career. We all believe it doesn't work for a large majority of our kids. And we've known that forever. We have sat there and refused to make the changes for a long list of excuses. And it's time to stop and say, we need to do this for the future of our students, but also for the future of the subject that we love. Gosh, so many of you, I think, listening to this right now, love mathematics like Steve and I do so much, but we know we have to make these changes. And we wanted to give you a roadmap to try to make these changes. From curriculum to pedagogy, to assessment, to technology, with an implementation plan and with resources. It's been a great honor. Thank you, Steve.
Steve: Thank you, Eric. Amen. So great. And I just end with the phrase that Eric, you know, has drawn my attention to, that comes from the higher ed community and from the mathematics societies, and that is, "The status quo is simply unacceptable."
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.
Eric Milou is Professor of Mathematics at Rowan University in Glassboro, NJ and co-author of Daily Routines to Jump Start Math Class and EnVision Math A|G|A. Eric served as President of the Association of Mathematics Teachers of New Jersey and on the Board of Directors of the National Council of Supervisors of Mathematics.
