Mark Zuckerberg's foundation has pivoted from a huge investment. He misunderstood the nature of good teaching but he also misunderstood the nature of mathematics itself.
This is just a great post. And what's also so cool about Noticing, Naming, and Using Patterns is that those are cross-disciplinary skills. My best lessons in History were always the ones that focused on a certain primary source - "What's up with that?" was the question I hoped would connect with them when we uncovered something odd about it.
I do agree that there's something rather unique to how this plays out in Math, though, because there *is* a final answer and a there *is* a need for computational accuracy that doesn't exist in the quite the same way in any other discipline. I'm watching it play out with my sons (7 and 8), who are both very quick thinkers able to do lots of sums and computations in their heads. It took a great teacher about a half year to get my eldest to stop saying that the reason for his answer was "I'm good at math" and to start actually explaining his thinking. Without her, and without a school running a curriculum that emphasizes process as well as product, I don't know he would have been able to make that shift. But I sure am grateful!
> I do agree that there's something rather unique to how this plays out in Math, though, because there *is* a final answer and a there *is* a need for computational accuracy that doesn't exist in the quite the same way in any other discipline.
Just to push a bit here - what is the final, computational accurate answer to the question, "What do you notice about the pool border?" I agree that attributes like accuracy and correctness play a more prominent role in mathematics than in many other subjects, but those aren't the _only_ attributes that define math.
Oh I agree with you. But I'm thinking about how little kids start to learn early on about right and wrong answers. So much of early years involves decoding, spelling, basic sums, and things like multiplication tables. Even if a teacher works hard to focus on processes and noticing type questions, a first grader also clearly can tell that there are right and wrong answers.
Or maybe it's just that fixed mindset stuff is genetic and my boys have inherited it from me ...
I teach a Math Reasoning course where we give more credit on the exams for "Explain how you got the answer" than we do for the answer itself. I explain to my students that increasingly machines will do the math, but our value as humans is in translating and explaining that math to other humans. I also tell them that this is something the employers who hire our students (I teach at a community college) really want to see.
You can diminish the interestingness of math when it is not creative or social. Explaining answers is important for developing problem solving skills. When I was in the class, students liked getting incorrectly solved problems and finding and correcting them. When students showed significant work, but arrived at an incorrect answer, I'd ask them to create a new question that would lead to their "incorrect" answer.
You know my shtick on this Dan -- there's a science of how we learn, and personalization ain't it. Humans are such social creatures and there's more and more evidence that our cultural practices shape not just what we think about but *how* we think. Why is philanthropy so scared of the social side of learning?
It is always ironic that people who design sophisticated computer programs believe that the most important thing in math education is to create poor imitations of the sophisticated software that can solve any standard high school math problem. To them, the height of mathematics is the procedural work that is most easily done by a computer.
They don't take the time to understand effective pedagogy or cognitive functioning despite lots of approachable material on the subject. They don't understand their users' problems. I assure you it is not a lack of questions to answer.
Your take on mathematics is clever and appropriate. I'd like to invite—or possibly challenge—you to take an equally deep dive into the social roots of the issue. I do not think the people you are calling out sincerely believe mathematics to be that shallow. That would have been hard to address, but doable. Instead, we have one of those social messes on our hands: https://en.wikipedia.org/wiki/Wicked_problem
One of the wicked elephants in the room is the following: A lot of people believe that it's perfectly okay for "math for the masses" to be primitive and shallow. The technologists' solutionism champions shallow math because they don't care about the people involved, not in the way you describe and wish for. It's okay with many decision-makers if "other people's kids" go on to "study" ratio in an afternoon and then grow up to flip burgers. If you think teaching the funders deep mathematics and sound mathematical pedagogy is challenging, wait till you try to teach them social consciousness.
I like Dan’s mountain peak metaphor. Here’s another metaphor that seems appropriate for this discussion. I used it many years ago when talking to students studying to be math teachers at a women’s university in Mumbai, India.
I asked my Indian audience to close their eyes and imagine mathematics as a beautiful 3-D landscape with mountains, rivers, plains, and varied topography, all representing ideas and concepts that blend and morph as one hikes and and wanders through the terrain—in Dan’s words— the lands of Terra Mathematica. I asked them to look for places they knew mathematically. Then I asked the, to imagine someone digging a ditch two meters deep across their landscape.
I started my talk by pointing out that, too often, schools and school curricula around the world march students from one end of the ditch towards the other. The ditch represents school mathematics. Some kids make it to the end of the ditch and are rewarded. Most never make it. No one is tall enough, whether they reached the end or not, to see the real world of mathematics around them.
Most of the computerized systems for teaching mathematics developed by “successful” technologists are simply efforts to march students along the ditch more efficiently or more quickly. Not only are the systems they create disasters from a social perspective and ignore all we know about good teaching, they don’t even let students get a glimpse of the beauty, awesome nature and reality of mathematics.
The number of students who are anointed with a badge of success at then end of their school experiences in mathematics but who cannot carry a conversation about mathematical concepts is just short of uncountable. :-)
If you ever do get the well-meaning, deep pocketed folks together ask if they might try to leverage their resources in helping teachers with diverse (in so many ways) classrooms of kids help get our students to talk productively with one another. For now, I'm off to work to attempt that very effort. My experience is the richer the content discussion among students the more they all get out of their work. The best I can do is attempt to set up the environment and ask well-placed questions for my particular group - a task I don't see technology well-suited for given I'm trying to account for personalities, likes, interactions between different students, et al.
You are my hero. In my experience as a math teacher there is nothing better than a math class where students debate, disagree, brainstorm, collaborate, defend their point of view, question each other's assumptions, try again using new ideas, make discoveries, and share success.
This reminds me of one of my favorite recent quotes, from Jordan Ellenberg's "Shape," commenting on the student's role in this dynamic: "Yes, some students have no trouble picking up the basic rules of algebraic manipulation or geometric constructions. Those students should still be asking questions, of their teachers and of themselves. For example, I have done what the teacher asked, but what if I'd tried to do this other thing that the teacher didn't ask of me, and, for that matter, why did the teacher ask for one thing and not the other? There's no intellectual vantage from which you can't easily sight a zone of ignorance, and that's where your eyes should be pointed, if you want to learn. If math class is easy, you're doing it wrong."
Jordan Ellenberg's great. I think of my own experiences when I first started teaching algebra and started designing my own questions and the choices I made to turn a given question into a "good" question. (Like maybe it brings out into the open a certain misconception the students tend to have.) There's a lot of deep thinking there, even on the simplest questions, I'd love for my students to have that experience but haven't yet figured out how to bring it into a classroom.
"Instead, the regions where you find yourself noticing, naming, and using patterns with people you respect and like, are simply too interesting for students to content themselves with life in the arid regions of Terra Mathematica."
I love this quote... I've always found that learners who connect new ideas to their previous experiences have a much better chance of developing new knowledge. The power of social learning is exponential since the number of connections explodes when students get to hear the thinking of their peers. Noticing, naming, and using patterns is such a great way to tap into someone's perspective, and creating an environment where perspectives are valued just as much as the right answer is the kind of classroom I want to be in.
What if this community of math educators and the community of individualized computer learning proponents are seeing the same problem of disengaged, frustrated, or bored students from two different sides?
Of course the ideal solution to this problem is that every child have an inspired, creative, passionate teacher like Liz. There are thousands out there! Those immersed in this field see the problem from this angle. However, what if advocates of individualized computer learning see the other side of the coin - the classrooms where students continue to be frustrated and discouraged and bored year after painful year? Is computer learning better then? Maybe it depends on how discouraged or bored you are.
Rather than assuming those who comment about bored students twiddling their thumbs are "lost" or don't understand the breadth and depth of math, perhaps we can acknowledge that some (many?) teachers do not have the "curriculum, pedagogy, and technology" needed to consistently create the classroom you describe. It might then be possible to work together with these funders to invest in other kinds of technology that would support more teachers in consistently creating low floor/high ceiling spaces that are both personalized and communal, eliminating the perceived need for computers to do so.
> Rather than assuming those who comment about bored students twiddling their thumbs are "lost" or don't understand the breadth and depth of math, perhaps we can acknowledge that some (many?) teachers do not have the "curriculum, pedagogy, and technology" needed to consistently create the classroom you describe.
Me, personally, I recognize the truth you're describing here. And I find that lots of the program officers at these foundations understand the height, depth, and breadth of the task at hand. But I find the people whose names are on the front door of those foundations see student boredom and apply a strongly technosolutionistic approach to the problem, rather than calling for the basket of social, pedagogical, curricular supports that you are.
This is incredible, and very hopeful to hear when so many new ideas seem to be focused on personalized learning. My question is though, what happens when there aren’t such skilled teachers? Is personalized learning a “good” solution when the best (collaborative learning) isn’t possible?
Great post. I'm publicizing it on my Linkedin, where it's getting a lot of reaction.
Having worked at a company that created personalized learning games for teaching mathematics, I've seen the forces that cause smart people to come to such narrow-minded conclusions. There is, of course, some value in personalized / adaptive solutions to math education. but it completely ignores the social aspect of learning.
The company I worked for did experiment in social learning but they shut down those projects because they did not fit their delivery platform, which was software as a service, and it did not fit their business model, which did not include paying lots of individual people. in other words, like the technologists you mentioned, they were blindsided by digital technology and a desire to create scalable solutions. the more I participate in education, the more I conclude that education is not fundamentally scalable, although portions of it are.
The reason I listen to you is you have a background as a classroom teacher. unfortunately most technologists do not, AND you have the will and insight to imagine something better.
Here's an interesting response to my LInkedin repost of your article, from my friend Roy Leban:
--------------------
I designed the adaptive engine underlying DreamBox Learning, so I completely believe that adaptive and individualized learning has merits. I could not have designed it without working with and learning from some great teachers. It is important that systems like DreamBox are not intended to replace all education, or classroom education. That's a key point that some of these big projects have missed.
I love your metaphor of mountain peaks that are actually connected by footbridges. I think of the thousand or so problems in any algebra textbook that say "Simplify" and then you never do anything with the simplified form. If this particular "simplify" has any use within mathematics, why not use it right then and there?
I think this has to do with context, and we often hear of the need to provide students with "real-world context", but even when we fail to do that, there's no excuse for failing to provide some *mathematical* context. How is this work useful and connected to other math? OK, we solved the equation, can we do more than check the answer key and move on to the next equation? Couldn't we at least check it against a graph? (I'm always surprised at how difficult many students find that, probably because their experience is that equations live on one mountaintop and graphs on another.)
I think I shared this on a previous post, but I just think it fits so well with your thesis here, and I think every math teacher needs to read this. How do we fix this?
I think you should realize a lot of math teachers have read it and haven't come to the same conclusions you have, and yet still teach a version of math that would win your approval.
As countless people have observed, did Lockhart never hear of applied math?
That's not his argument. He ignores applied math entirely.
But in all honesty, that's only one of the problems I have with Lockhart's Lament. Just observing that I'm a good math teacher, by no means the drill and kill variety (I have a whole blog you can check to confirm) and Lockhart's Lament is not something I recommend to beginning math teachers. And a lot of wannabe math teachers who find it meaningful are going to be really disappointed when Lockhart's world doesn't map to reality.
Well, sure, if I took it literally I'd be filling in my retirement paperwork tomorrow morning. I took it as a polemic, a wake-up call, an attempt to provoke discussion. Nobody should take it as "Here's how you teach math." But I do think he makes a lot of good points about how you DON'T teach math.
This is just a great post. And what's also so cool about Noticing, Naming, and Using Patterns is that those are cross-disciplinary skills. My best lessons in History were always the ones that focused on a certain primary source - "What's up with that?" was the question I hoped would connect with them when we uncovered something odd about it.
I do agree that there's something rather unique to how this plays out in Math, though, because there *is* a final answer and a there *is* a need for computational accuracy that doesn't exist in the quite the same way in any other discipline. I'm watching it play out with my sons (7 and 8), who are both very quick thinkers able to do lots of sums and computations in their heads. It took a great teacher about a half year to get my eldest to stop saying that the reason for his answer was "I'm good at math" and to start actually explaining his thinking. Without her, and without a school running a curriculum that emphasizes process as well as product, I don't know he would have been able to make that shift. But I sure am grateful!
> I do agree that there's something rather unique to how this plays out in Math, though, because there *is* a final answer and a there *is* a need for computational accuracy that doesn't exist in the quite the same way in any other discipline.
Just to push a bit here - what is the final, computational accurate answer to the question, "What do you notice about the pool border?" I agree that attributes like accuracy and correctness play a more prominent role in mathematics than in many other subjects, but those aren't the _only_ attributes that define math.
Oh I agree with you. But I'm thinking about how little kids start to learn early on about right and wrong answers. So much of early years involves decoding, spelling, basic sums, and things like multiplication tables. Even if a teacher works hard to focus on processes and noticing type questions, a first grader also clearly can tell that there are right and wrong answers.
Or maybe it's just that fixed mindset stuff is genetic and my boys have inherited it from me ...
I teach a Math Reasoning course where we give more credit on the exams for "Explain how you got the answer" than we do for the answer itself. I explain to my students that increasingly machines will do the math, but our value as humans is in translating and explaining that math to other humans. I also tell them that this is something the employers who hire our students (I teach at a community college) really want to see.
You can diminish the interestingness of math when it is not creative or social. Explaining answers is important for developing problem solving skills. When I was in the class, students liked getting incorrectly solved problems and finding and correcting them. When students showed significant work, but arrived at an incorrect answer, I'd ask them to create a new question that would lead to their "incorrect" answer.
You know my shtick on this Dan -- there's a science of how we learn, and personalization ain't it. Humans are such social creatures and there's more and more evidence that our cultural practices shape not just what we think about but *how* we think. Why is philanthropy so scared of the social side of learning?
This is what I taught too, social aspect of learning among others.
It is always ironic that people who design sophisticated computer programs believe that the most important thing in math education is to create poor imitations of the sophisticated software that can solve any standard high school math problem. To them, the height of mathematics is the procedural work that is most easily done by a computer.
They don't take the time to understand effective pedagogy or cognitive functioning despite lots of approachable material on the subject. They don't understand their users' problems. I assure you it is not a lack of questions to answer.
Your take on mathematics is clever and appropriate. I'd like to invite—or possibly challenge—you to take an equally deep dive into the social roots of the issue. I do not think the people you are calling out sincerely believe mathematics to be that shallow. That would have been hard to address, but doable. Instead, we have one of those social messes on our hands: https://en.wikipedia.org/wiki/Wicked_problem
One of the wicked elephants in the room is the following: A lot of people believe that it's perfectly okay for "math for the masses" to be primitive and shallow. The technologists' solutionism champions shallow math because they don't care about the people involved, not in the way you describe and wish for. It's okay with many decision-makers if "other people's kids" go on to "study" ratio in an afternoon and then grow up to flip burgers. If you think teaching the funders deep mathematics and sound mathematical pedagogy is challenging, wait till you try to teach them social consciousness.
I like Dan’s mountain peak metaphor. Here’s another metaphor that seems appropriate for this discussion. I used it many years ago when talking to students studying to be math teachers at a women’s university in Mumbai, India.
I asked my Indian audience to close their eyes and imagine mathematics as a beautiful 3-D landscape with mountains, rivers, plains, and varied topography, all representing ideas and concepts that blend and morph as one hikes and and wanders through the terrain—in Dan’s words— the lands of Terra Mathematica. I asked them to look for places they knew mathematically. Then I asked the, to imagine someone digging a ditch two meters deep across their landscape.
I started my talk by pointing out that, too often, schools and school curricula around the world march students from one end of the ditch towards the other. The ditch represents school mathematics. Some kids make it to the end of the ditch and are rewarded. Most never make it. No one is tall enough, whether they reached the end or not, to see the real world of mathematics around them.
Most of the computerized systems for teaching mathematics developed by “successful” technologists are simply efforts to march students along the ditch more efficiently or more quickly. Not only are the systems they create disasters from a social perspective and ignore all we know about good teaching, they don’t even let students get a glimpse of the beauty, awesome nature and reality of mathematics.
The number of students who are anointed with a badge of success at then end of their school experiences in mathematics but who cannot carry a conversation about mathematical concepts is just short of uncountable. :-)
The ditch does some really important metaphorical work here. Really helpful.
Dan,
If you ever do get the well-meaning, deep pocketed folks together ask if they might try to leverage their resources in helping teachers with diverse (in so many ways) classrooms of kids help get our students to talk productively with one another. For now, I'm off to work to attempt that very effort. My experience is the richer the content discussion among students the more they all get out of their work. The best I can do is attempt to set up the environment and ask well-placed questions for my particular group - a task I don't see technology well-suited for given I'm trying to account for personalities, likes, interactions between different students, et al.
You are my hero. In my experience as a math teacher there is nothing better than a math class where students debate, disagree, brainstorm, collaborate, defend their point of view, question each other's assumptions, try again using new ideas, make discoveries, and share success.
This reminds me of one of my favorite recent quotes, from Jordan Ellenberg's "Shape," commenting on the student's role in this dynamic: "Yes, some students have no trouble picking up the basic rules of algebraic manipulation or geometric constructions. Those students should still be asking questions, of their teachers and of themselves. For example, I have done what the teacher asked, but what if I'd tried to do this other thing that the teacher didn't ask of me, and, for that matter, why did the teacher ask for one thing and not the other? There's no intellectual vantage from which you can't easily sight a zone of ignorance, and that's where your eyes should be pointed, if you want to learn. If math class is easy, you're doing it wrong."
"If math class is easy, you're doing it wrong." Wonderful. Thanks for sharing the excerpt.
Jordan Ellenberg's great. I think of my own experiences when I first started teaching algebra and started designing my own questions and the choices I made to turn a given question into a "good" question. (Like maybe it brings out into the open a certain misconception the students tend to have.) There's a lot of deep thinking there, even on the simplest questions, I'd love for my students to have that experience but haven't yet figured out how to bring it into a classroom.
"Instead, the regions where you find yourself noticing, naming, and using patterns with people you respect and like, are simply too interesting for students to content themselves with life in the arid regions of Terra Mathematica."
I love this quote... I've always found that learners who connect new ideas to their previous experiences have a much better chance of developing new knowledge. The power of social learning is exponential since the number of connections explodes when students get to hear the thinking of their peers. Noticing, naming, and using patterns is such a great way to tap into someone's perspective, and creating an environment where perspectives are valued just as much as the right answer is the kind of classroom I want to be in.
What if this community of math educators and the community of individualized computer learning proponents are seeing the same problem of disengaged, frustrated, or bored students from two different sides?
Of course the ideal solution to this problem is that every child have an inspired, creative, passionate teacher like Liz. There are thousands out there! Those immersed in this field see the problem from this angle. However, what if advocates of individualized computer learning see the other side of the coin - the classrooms where students continue to be frustrated and discouraged and bored year after painful year? Is computer learning better then? Maybe it depends on how discouraged or bored you are.
Rather than assuming those who comment about bored students twiddling their thumbs are "lost" or don't understand the breadth and depth of math, perhaps we can acknowledge that some (many?) teachers do not have the "curriculum, pedagogy, and technology" needed to consistently create the classroom you describe. It might then be possible to work together with these funders to invest in other kinds of technology that would support more teachers in consistently creating low floor/high ceiling spaces that are both personalized and communal, eliminating the perceived need for computers to do so.
> Rather than assuming those who comment about bored students twiddling their thumbs are "lost" or don't understand the breadth and depth of math, perhaps we can acknowledge that some (many?) teachers do not have the "curriculum, pedagogy, and technology" needed to consistently create the classroom you describe.
Me, personally, I recognize the truth you're describing here. And I find that lots of the program officers at these foundations understand the height, depth, and breadth of the task at hand. But I find the people whose names are on the front door of those foundations see student boredom and apply a strongly technosolutionistic approach to the problem, rather than calling for the basket of social, pedagogical, curricular supports that you are.
This is incredible, and very hopeful to hear when so many new ideas seem to be focused on personalized learning. My question is though, what happens when there aren’t such skilled teachers? Is personalized learning a “good” solution when the best (collaborative learning) isn’t possible?
Great post. I'm publicizing it on my Linkedin, where it's getting a lot of reaction.
Having worked at a company that created personalized learning games for teaching mathematics, I've seen the forces that cause smart people to come to such narrow-minded conclusions. There is, of course, some value in personalized / adaptive solutions to math education. but it completely ignores the social aspect of learning.
The company I worked for did experiment in social learning but they shut down those projects because they did not fit their delivery platform, which was software as a service, and it did not fit their business model, which did not include paying lots of individual people. in other words, like the technologists you mentioned, they were blindsided by digital technology and a desire to create scalable solutions. the more I participate in education, the more I conclude that education is not fundamentally scalable, although portions of it are.
The reason I listen to you is you have a background as a classroom teacher. unfortunately most technologists do not, AND you have the will and insight to imagine something better.
Here's an interesting response to my LInkedin repost of your article, from my friend Roy Leban:
--------------------
I designed the adaptive engine underlying DreamBox Learning, so I completely believe that adaptive and individualized learning has merits. I could not have designed it without working with and learning from some great teachers. It is important that systems like DreamBox are not intended to replace all education, or classroom education. That's a key point that some of these big projects have missed.
I love your metaphor of mountain peaks that are actually connected by footbridges. I think of the thousand or so problems in any algebra textbook that say "Simplify" and then you never do anything with the simplified form. If this particular "simplify" has any use within mathematics, why not use it right then and there?
I think this has to do with context, and we often hear of the need to provide students with "real-world context", but even when we fail to do that, there's no excuse for failing to provide some *mathematical* context. How is this work useful and connected to other math? OK, we solved the equation, can we do more than check the answer key and move on to the next equation? Couldn't we at least check it against a graph? (I'm always surprised at how difficult many students find that, probably because their experience is that equations live on one mountaintop and graphs on another.)
I think I shared this on a previous post, but I just think it fits so well with your thesis here, and I think every math teacher needs to read this. How do we fix this?
http://worrydream.com/refs/Lockhart-MathematiciansLament.pdf
Thanks for this, I'm afraid if you asked most students what mathematics is, they'd say it's the study of how to properly arrange symbols on paper.
I think you should realize a lot of math teachers have read it and haven't come to the same conclusions you have, and yet still teach a version of math that would win your approval.
As countless people have observed, did Lockhart never hear of applied math?
That's interesting, I thought his argument is that mathematics is most beautiful and interesting when it's NOT applied?
That's not his argument. He ignores applied math entirely.
But in all honesty, that's only one of the problems I have with Lockhart's Lament. Just observing that I'm a good math teacher, by no means the drill and kill variety (I have a whole blog you can check to confirm) and Lockhart's Lament is not something I recommend to beginning math teachers. And a lot of wannabe math teachers who find it meaningful are going to be really disappointed when Lockhart's world doesn't map to reality.
Well, sure, if I took it literally I'd be filling in my retirement paperwork tomorrow morning. I took it as a polemic, a wake-up call, an attempt to provoke discussion. Nobody should take it as "Here's how you teach math." But I do think he makes a lot of good points about how you DON'T teach math.
Amen. Your concluding paragraph says it all!
Bravo!