Professional site of Sylvia Libow Martinez
Math education is failing too many students.
We are owners and participants in an entrenched school system that is tough to change, and we could all list the thousands of reasons why. Big systems have a lot of inertia, like the Titanic heading towards its icy fate.
But proving that the system is working by showcasing the few students who make it through is like saying that the Titanic was a success because some of the passengers made it to New York.
Sylvia
From NCWIT (National Council of Women in IT) –
Did you know that a recent study using data on 15,000 students from the National Center of Education Statistics found that teachers consistently rate girls as less good at math than boys, even with similar grades and test scores? Researchers in the study found that while on average teachers rate minority students lower than their white male counterparts, these differences disappear once grades are taken into account. However, they found patterns of bias against white girls that can’t be explained by their academic performance. According to one of the study’s authors, the misconception that white girls can’t handle math persists “Because the idea that men and women are different in this regard is considered natural, and not discriminatory.” At the same time, teachers may be more aware of race and ethnicity – and the problems of racial discrimination – than they are when it comes to gender.
Why are High School Teachers Convinced that White Girls Can’t Do Math? – Forbes.com
The research (the abstract is free at least) – Exploring Bias in Math Teachers’ Perceptions of Students’ Ability by Gender and Race/Ethnicity – University of Texas at Austin
From the press release:
Wolfram Offers Next Innovation in Education Technology with Wolfram Education Portal
Champaign, Illinois–January 18, 2012–Wolfram today announced the launch of the Wolfram Education Portal, providing teachers and students alike with a new way to integrate technology into learning.
The Wolfram Education Portal, available at education.wolfram.com, comes equipped with dynamic teaching tools and materials such as an interactive textbook, lesson plans aligned to the common core standards, and many other supplemental materials for courses, including Demonstrations, widgets, and videos, all built by Wolfram education experts.
“Wolfram has long been a trusted name in education, as the creators of Mathematica, Wolfram|Alpha, and the Wolfram Demonstrations Project,” says Crystal Fantry, Senior Education Specialist at Wolfram. “We have created some of the most dynamic teaching and learning tools available, and the Wolfram Education Portal offers the best of all of these technologies to teachers and students in one place.”
The Education Portal, currently in Beta, contains full materials for Algebra and partial materials for Calculus, but will continue to grow and improve. Wolfram plans to expand the Education Portal to include community features, problem generators, web-based course apps, and the ability to create personalized content.
Wolfram developed the interactive textbook by working with the CK-12 Foundation, a nonprofit organization with the mission to produce free and open-source K-12 materials aligned to state curriculum standards and customized to meet student and teacher needs. The available Algebra textbook takes CK-12’s Algebra I FlexBook and makes it dynamic with Wolfram technologies, including Wolfram|Alpha widgets, Wolfram|Alpha links, interactive Demonstrations created in Mathematica, and the Computable Document Format (CDF).
Sample widget
Thanks to everyone who commented and tweeted about my recent series of posts about Khan Academy and the questions it raises regarding pedagogy, learning theory, and how we teach math in the U.S.
Here are the links all in one place.
The original post – Compare and contrast: using computers to improve math education This post compares the vision of math education of Sal Khan and Conrad Wolfram in their TED Talks. There was so much commentary on this post I decided to delve deeper.
Part 1 – Khan Academy and the mythical math cure. This post is about how we believe certain things about math that are not true, but we keep on doing them anyway.
Part 2 – Khan Academy – algorithms and autonomy How math instruction tries to help students but may actually be undermining student confidence and basic numeracy.
Part 3 – Don’t we need balance? and other questions. A conversation with myself about how Khan Academy is often justified, and why it’s being hyped as a “revolutionary reform” in math education.
Part 4 – Monday… Someday. Teachers face a dilemma – even if you agree that math learning and teaching need to be different, it’s not going to change overnight.
Sylvia
Sylvia
Note: This is part 4 of 4 of a series on Khan Academy.
Part 1 – Khan Academy and the mythical math cure
Part 2 – Khan Academy – algorithms and autonomy
Part 3 – Don’t we need balance? and other questions
Part 4 – Monday… Someday (this post)
In the previous post, I ended with a question about the inch-deep, mile-wide math curriculum in the U.S. that essentially requires teachers to force-feed their students so they can “cover” the material and pass the tests.
This is the Monday… Someday problem – the fact that even if a teacher changes everything in their classroom, nothing else in the system will change. How can one argue for a long term (Someday) overhaul of math curriculum, pedagogy and assessment when you know even if it does change, it’s going to be long time from now, and you have kids coming in on Monday who need to pass a test on Friday that will depend on them memorizing a bunch of facts and skills?
What good does it do to fight when the system not only doesn’t care, but will slap you down for it.
Even the students will likely complain – “why don’t you just tell us what we need to know?” They’ve been conditioned over the years that they aren’t supposed to really understand; they have figured out the rules of the game and don’t have any reason to believe that the game will change, even if one teacher insists it will.
So… what DO I do on Monday?
Even if I thought I had that answer I wouldn’t have the hubris to tell a teacher that I had a magic formula. I respect teachers too much to think that I could prescribe a “solution” that would fit everyone, every context, every kid, and every other major variable in the teaching/learning equation.
All I can do is present alternatives to what I see are these math myths that are so pervasive in American culture. To speak about things that often go unquestioned. To point to great thinkers and research that have changed my own thinking. That’s my whole purpose here with these posts on Khan Academy.
Seymour Papert said, “My answer is that if you have a vision of Someday you can use this to guide what you do Monday. But if your vision of where it is going is doing the same old stuff a bit (or a lot) better your efforts will be bypassed by history.” – Technology in Schools: To Support the System or to Render it Obsolete.
Working on “a vision of Someday” requires doing some thinking about what you believe about learning, and how different teaching models align with those beliefs. Without having an “eye on the prize” it would be difficult to steer your own classroom towards anything new. In a similar vein, Alvin Toffler, futurist and author of the book Future Shock said, “You’ve got to think about big things while you’re doing small things, so that all the small things go in the right direction.”
Someone else who has influenced my thinking on the “what do you do Monday” is Gary Stager. He says that anytime you go to “help” a learner, pause and think about whether you are taking away an opportunity for them to learn it themselves. He summarizes this as “Less us, more them.” His recent TEDxNYED talk showed examples of what at-risk students could do in science and math when given the chance.
A Monday solution pretending to be a Someday solution
Unfortunately, Khan Academy is a simplistic “what do I do on Monday” solution that is being hyped as a Someday solution. If you have a long-term vision that in any way aligns with more open-ended, more constructivist learning, Khan Academy is not a step on that path. It’s a “more us, more us” solution.
You can’t expect an instructionist solution like Khan Academy to pair with, or even more implausibly, magically morph into a constructivist solution.
Instruction begets instruction.
Well, if it’s not Khan, what does Someday look like?
I’m going to go back to theory for this and talk about constructionism (the learning theory developed by Seymour Papert.)
“The word constructionism is a mnemonic for two aspects of the theory of science education underlying this project. From constructivist theories of psychology we take a view of learning as a reconstruction rather than as a transmission of knowledge. Then we extend the idea of manipulative materials to the idea that learning is most effective when part of an activity the learner experiences as constructing a meaningful product.” – Seymour Papert (Wikipedia article on constructionism)
“Meaningful product” – now there are a couple of important words. Is it possible to shift math education to “constructing meaningful products”? What does a classroom look like when students are engaged in developing meaningful products?
Good news — there are lots of answers out there that are well researched and classroom tested. Bad news — these models are virtually ignored in most American classrooms. Call it constructivist, project-based, progressive, inquiry-based… there are a lot of good books, websites, and help out there to change the way we teach math. Gary Stager has collected a starter kit of constructivist books in the collection at The Constructivist Consortium.
One of Papert’s answers was to invent a programming language for children called Logo. This language allows very young people to construct things on the computer, or to control physical objects like robots. There are thousands of versions of Logo today that offer all kinds of constructivist learning opportunities on the computer. Scratch, free from MIT is one such Logo offspring.
Even when people acknowledge that some topics in the math curriculum are not needed in the real world anymore, we hear, “oh, but it’s good for developing logical thinking.” Really? How about some real world math that teaches logical thinking… programming. Where is that to be found in math curriculum? In my original post about the difference between Conrad Wolfram and Salman Khan’s TED talks (Compare and contrast: using computers to improve math education), Wolfram also mentions that programming is a way to teach procedural, logical thinking.
This is an idea whose time has come.
Here’s a video of a keynote speech that Seymour Papert (by the way, another protégé of Piaget) gave in Australia in 2004 where he makes some incredibly good points about these things.
Seymour Papert in Sydney, Australia 2004 from The Daily Papert, Gary Stager
Why I keep coming back to Papert
Seymour Papert has been an enormous influence on my life and my thinking about math learning. He’s also the reason I do what I do. If you read the article I linked to above (Technology in Schools: To Support the System or to Render it Obsolete.), one of the things he talks about is “Kid Power” and Generation WHY – which was the original name for the non-profit of which I’m now president, Generation YES.
Seymour says in this article that there are changes coming:
“A hundred years ago John Dewey was showing the faults of the curriculum-driven, non-experiential ways of teaching favored by schools. But all his work had only a marginal effect on what schools do; they have changed in some details but most are not essentially very different from those which Dewey criticized way back then.
Critics of school reform (including Todd Oppenheimer) are fond of quoting the failures of past movements as evidence for the extreme difficulty of changing school and hence casting doubt on the likelihood that revolutionary change is likely to come this time round.
But the critics are misled by their failure to look below the surface of what is happening to the learning environment. If they did they would recognize three aspects of a profound difference between the present situation and anything that has happened in the past.
Each of these takes the form of a reversal:
The strategy for overcoming the last obstacle brings us full circle to my opening paragraph:
for those of us who want to change education the hard work is in our own minds, bringing ourselves to enter intellectual domains we never thought existed. The deepest problem for us is not technology, nor teaching, nor school bureaucracies.
All these are important but what it is all really about is mobilizing powerful ideas.
And there it is…
“We have met the enemy and he is us” – Pogo
Khan Academy is the system’s pushback against real curriculum reform. In “Reversal #2” that Papert talks about, we see he has explicitly predicted Khan Academy. The system likes the status quo and things that support the status quo have a predictably comfortable feel to them. Then we get to call it a revolution without changing a thing.
Moving beyond Monday to Someday involves mobilizing powerful ideas that might not feel comfortable. It’s tiring to constantly rethink everything you “know” about learning, but rewarding in the end.
For technology in education advocates, it means constantly pushing the envelope towards learner-centered technology and away from teacher-centered technology. So although I said that the discussion about Khan Academy is not about educational technology, it should be, because it’s the perfect example of a wolf in sheep’s clothing “technology revolution” that really supports the status quo.
Some last thoughts
To teachers – 1) Keep your skeptic hat on tight. Anything that “solves all problems in education” probably doesn’t, and any “revolution” probably isn’t. 2) Anything that you use in your classroom should align with your theory of learning. Find one and drink deeply. Strangely enough, this is exactly the same two pieces of advice I give about using games in the classroom.
To students – Math isn’t what’s found in textbooks. Sorry you have to slog your way through a lot of crap to get to the good stuff. There are lots of people, videos, and sources out there to learn about vibrant, useful, beautiful things that also happen to be numerically interesting. Use the Google. PS- If you love math you aren’t weird.
To parents – The math your children are learning should not be about worksheets and lectures, even if those worksheets and lectures are on a computer. Demand that your children not be sacrificed to the gods of standardized testing. Your children’s teachers need you on their side to fight for something better.
To everyone else in the system – Don’t mandate to teachers or replace them with videos. Include them in long-term learning communities and conversations that support change, growth, and better teaching and learning for everyone.
Sylvia
Part 1 – Khan Academy and the mythical math cure
Part 2 – Khan Academy – algorithms and autonomy
Part 3 – Don’t we need balance? and other questions
Part 4 – Monday… Someday (this post)
Note: this is Part 3 of a 4 part blog series on Khan Academy and math education. This post is an imaginary Q&A about what I’ve said in Part 1 about math myths and learning theories and Part 2 about algorithms, practice, and autonomy. The following questions are made up from what I’ve heard people say about Khan Academy. I am solely to blame for the answers.
Isn’t it best to offer a balance of all different kinds of learning opportunities for students?… Can’t we have open-ended problem-solving AND show the kids how to do the hard parts when they get in trouble?
Now, I would never tell a teacher what to do, it’s too easy for me to type a bunch of words and I don’t have to be there every day. But I think you have to consider the unexpected consequences of striving for balance between two opposing theories of learning – instructionism and constructionism.
To illustrate this, let’s imagine a playground game of hide and seek. On Monday, when everyone has hidden and the seeker finishes the count, he or she looks up… and at that moment, the teacher steps in and points out where everyone is hiding. On Tuesday, the teacher stands back and says nothing. On Wednesday, the teacher helpfully points out the hiders, on Thursday, says nothing.
What do you think happens on Friday?
I’m pretty sure that the seeker would immediately look to the teacher and ask where everyone is hiding. Or maybe everyone would just refuse to play since there’s no point to it. On the previous days, the teacher has trained them how to get the answer. Even with the “balance” in game play, one outweighs the other. There is no balance possible, because the teacher’s authority causes the balance to permanently shift. It’s the very essence of disempowerment. Teacher power and authority is the 800 pound gorilla siting on the end of a see-saw.
I believe that for many of the same reasons, the attempt to explicitly show students how to solve problems becomes a roadblock when you suddenly turn around and demand that they figure things out for themselves. It just sounds like a trick, and if they wait long enough, you will give them the answers and move on. Children are pretty pragmatic about these things.
I still think you need balance…
I could almost go along with the “balance” argument if the world of U.S. school math weren’t so unbalanced. I would guess that 95% of all math taught in all classrooms across the US is direct instruction aimed at the “skill” level and memorizing the right algorithm to solve problems most likely to be found on standardized tests. So there’s no balance there to start with – the only way to achieve “balance” is to do more open-ended, student-led inquiry about math, solving real problems (not textbook or test prep problems), not telling students what the right answers are, etc. And do LOTS more of it. Then we can talk about balance.
But at least the ability to stop and replay the video gives the student control – isn’t that what we always look for in student-centered learning?
Here’s the tradeoff – is student control over the pace worth losing student control of the entire process? They get to choose how quickly they are force-fed someone else’s representation of a process instead of creating their own representation in their heads. Asking the student to give up control of their own thought process to absorb a one-size-fits-all delivery of information requires a large degree of compliance on the student’s part. In my book, the ability to control the pace pales in comparison. I think a teacher would have to weigh these very different kinds of control and whether the trade-off is worth it.
Why shouldn’t we teach students a good way to solve a problem, what’s the point of letting them fumble around?
When we tell a student the “right way” – we are really telling them that math ability is primarily about compliance. This is about power, and we lose a lot of students in these power struggles.
Margaret Mead said, “emphasis has shifted from learning to teaching, from the doing to the one who causes it to be done, from spontaneity to coercion, from freedom to power. With this shift has come… dry pedagogy, regimentation, indoctrination, manipulation, & propaganda”. (thanks to Ryan Bretag for this quote)
What we call “good students” are compliant students who don’t call this power structure into question. (By the way, this was me – even when I saw other ways to solve problems I knew not to say anything. I amused myself by solving problems in alternate ways, then would write down the answer the way I knew the teacher wanted.) If you don’t think students are acutely aware of the power structures in school, you are underestimating students.
Students “fumbling around” is actually where the learning happens – and there’s no shortcut for this process.
Why waste time letting students “discover” everything. They aren’t going to re-invent the Pythagorean theorem by themselves.
It’s a straw man argument about inquiry-based, constructivist education that it’s “illegal” to lecture. Whenever I hear this I imagine a scene where the constructivist police burst through a classroom door and wrestle a teacher to the floor who was just explaining to a student how to do something. The difference is that explanations should serve to naturally move a problem-solving process along, not be the whole lesson.
In this kind of classroom, the teacher’s role is crucial – by posing problems that lead to big ideas and steering a class as they solve problems. By “being less helpful” as Dan Meyer says. (He doesn’t say don’t help at all!) This is not wasting time, it’s letting the students build the knowledge in their heads and acknowledging the fact that this takes time. It also takes time to learn how to teach this way. It’s not the case that the teacher is off taking a smoke break while the kids do this on their own. The teacher’s role is crucial – it’s difficult work and takes years to master.
This exact question is discussed by Piaget as related in a brilliant essay by Alfie Kohn – What Works Better than Traditional Math Instruction from his book The Schools Our Children Deserve. (I can’t improve on his explanation of why traditional math instruction is failing our children – please read this essay.)
So isn’t this the “flipped classroom” that Khan Academy proposes?
People are associating Khan Academy with the “flipped classroom” – something I talked about in this post (‘Teach Naked’ and complacency natives). In a so-called flipped classroom, the lecture takes place outside the classroom and classroom time is spent on discussion and problem solving. Students might watch the video at home (or in the car, bus, or anywhere) and then there would be a lot of classroom time freed up for discussion, working on individual problems, or whatever else needs to be done. That’s the theory, anyway.
So, first off, do you believe:
But I’m willing to let all these assumptions slide so we can move on. Let’s pretend that most of the students will listen/watch a math lecture on their own time.
Can you disconnect the lecture from the problem solving? Khan Academy videos have no context outside of class – other than that they match the standardized tests. As Derek Muller points out (see Part 1), these videos may have the unintended consequence of cementing incorrect models as students assume that they understand, thus making the teacher’s job that much harder.
Swapping the timing of certain teaching practices seems a minor logistics issue, at best. Moving the timing of the lecture doesn’t change the fact that it’s still a lecture, and not even a lecture about interesting stuff. Most of these “lectures” are simply worked out example problems. Do we think that a student who doesn’t “get it” in the classroom is more likely to “get it” on the bus? The main issue is the reliance on information delivery to trigger understanding.
This also assumes that you are replacing one lecture with no feedback with another lecture with no feedback. That’s pretty insulting to LOTS of teachers. I won’t assume that ALL teachers who lecture are bad, or that there aren’t a thousand ways to intersperse lecture with checks for understanding. There are no raised hands in the Khan Academy, no questions, no teachable moments, no interesting asides. You have one interaction, and one interaction only — the ability to play, stop, and rewind.
If I were a huge fan of making videos about how to solve problems, I’d certainly try to make it more student-centered by allowing students to make the videos. The process of figuring out how to clearly explain a concept would give a student time to reflect about the process in depth. They say teaching is the best way to learn, so why let Mr. Khan have all the fun!
But seriously, here’s a conundrum — the art of leading a productive learning discussion is much more difficult than lecturing. Are we to expect that the teacher who couldn’t even do the lecture part is suddenly going to be able to lead a productive discussion about math? It would seem to me that the teachers most likely to see Khan Academy videos as a good substitute for their own lectures are also the least likely to be able to take advantage of the classroom time for any substantive discussion that would help students.
Let’s not even talk about what happens if 3, 4 or 5 teachers each assign a 40 minute lecture to listen to every night – so if this model actually works… it’s impossible. Don’t you love models of teaching where successful adoption assures failure?
Nothing like this has ever existed before, it’s so exciting!
Really now? Didn’t you ever watch Donald Duck in MathMagic Land or Sunrise Semester? The amount of acclaim for Khan Academy is, in my view, way over the top and only reflects our acceptance of math myths as drivers for pedagogy and wishful thinking that there is a easy answer for learning.
I’m just glad to see that technology is finally useful in education.
I’ve seen Khan hyped as a transformative use of technology, but. I can’t even begin to understand how turning the computer into a VHS player is seen as transformative. I know, I know — he’s got quizzes too. Answer ten questions and you can resume playing the video. Brring, brrring… 1988 called and they want their CAI (Computer Aided Instruction) back.
But the Khan Academy videos show students how to solve the math problems that will be on tests – don’t we want students to do better on tests?
That is the heart of it – do we care about kids learning math or doing well on tests? They aren’t the same thing.
These videos have millions of hits on YouTube – it proves that students need this help and are searching for it.
Yes, it does. It shows that many students really do want to do well, and doing well is defined as passing tests. We have a nation where lots of students are working their hardest to do something that matters little. Imagine if we asked students to do math that was actually useful and interesting!
My teacher is terrible and these videos help me.
I’m sorry. I’m glad you’ve found something that helps. Nobody is trying to take away something that is helping you.
Salman Khan is a master teacher and shouldn’t everyone get the best teacher?
Salman Khan obviously has a gift for clearly explaining how he understands complex computations. Being a teacher, however, is more than explaining stuff. When a student has misconceptions, they often need to talk through them, and a teacher SHOULD be an expert in recognizing those misconceptions and steering students through those rough waters. There SHOULD be a lot of listening involved. I’m not excusing bad teaching practice – far from it.
You’ve cherry-picked your research and sources.
Absolutely true. I said at the beginning this wasn’t going to be a literature review. I’ve included a few quotes and references that influence my thinking. Kamii, Papert, and Kohn appear often. Between them they have decades of work, dozens of books, and research to support it all. If you disagree, I hope at least you’ll read further. Their ideas form a connective network with other great educators from Piaget to Dewey to Vygotsky to Freire and many more.
Just to pile on, I’m looking forward to Alfie Kohn’s new book, Feel-Bad Education . . . And Other Contrarian Essays on Children & Schooling. I’m also loving his recent column, What does education research really tell us? He relates new research about how studies done in the short term often support the use of traditional teaching practices (like direct instruction and homework for practicing skills). However, as these studies are refined and the students followed for longer periods (months or years instead of weeks), these traditional practices have zero, or even negative results. Yup, I <3 Alfie.
I like teaching in a more open-ended way – but no one understands.
Many teachers struggle with these math myths and the cultural expectations of how math should be taught. Even if they want to teach in more open-ended way, they are often alone, facing off with parents, colleagues and administrators. Any attempt to teach math as less skill-based is met with skepticism, if not outright hostility. Even research is met with a “… yes, but, I believe it’s important” as if it’s a matter of opinion. It’s almost impossible not to give into that pressure, and as a consequence many teachers give up.
I for one would never encourage a teacher to martyr themselves in a no-win situation, especially with the overemphasis on standardized testing and current punitive politicized atmosphere.
As far as parents go, though, I think that most parents really do want what’s best for their children and many can be convinced. Teachers may find allies among parents who are at their wits end with battles over math homework or with parents who watch their children go into school natural learners and come back hating it. Some parents are going to buy fraction flashcards for their kids no matter what you say or do, that won’t change. Try showing them this: Finland’s Educational Success? The Anti-Tiger Mother Approach
Find allies wherever you can. Teachers are doing amazing things all over the US and around the world. These days, it’s possible to develop colleagues who you may never meet in person, but might be your pedagogical soulmates.
You must not know much about real schools – haven’t you seen the list of standards that math teachers have to meet? The expectations for the test? The 400 page textbook? We have to get the kids through this stuff and there’s just no time for exploring, discovery, or anything else. Hoping that things will change someday doesn’t help me or my kids today.
You are right – the need for Khan Academy is completely fits the way we assume math has to be learned and taught. The “if it’s Tuesday it must be exponents” model is failing us. That has to change.
I’ll say a bit more about this Monday… Someday dilemma in my next (and last) post of this series.
Part 1 – Khan Academy and the mythical math cure
Part 2 – Khan Academy – algorithms and autonomy
Part 3 – Don’t we need balance? and other questions (this post)
Part 4 – Monday… Someday
This is part 2 of a 4 part series on Khan Academy and math education, specifically American math education. Part 1- Khan Academy and the mythical math cure set up the context, my point of view, and a bit of learning theory. It also discussed one prevalent myth of American math instruction, that math is a discrete set of sequential skills. It wrapped up with some research on effective multimedia in math and science instruction. And of course what all these things have to do with Khan Academy.
I’ll continue with another American math myth — that math is best taught by having students practice step-by-step procedures that lead to the right answer.
The prevailing theory goes — experts figure out the best set of steps to solve any problem, we show students these steps, then they practice the steps until they can easily solve problems. If you believe this myth, it follows that if students don’t learn math: 1) it’s the teacher’s fault for not being clear enough, or 2) it’s the student’s fault for not practicing enough.
Khan Academy fits this myth perfectly. Here’s a quick (and even better, free) way to help with both of these. Replacing or supplementing a teacher with a video solves #1. It solves #2 by saying that these videos should be watched outside of class, thus freeing up time for more student practice in class.
But here’s a question… What if it’s not the teacher’s fault or the student’s fault? What if the assumption that people learn math by watching and practicing the pre-determined steps is wrong?
Alfie Kohn has famously said that you can’t practice understanding. The confusion is that we think math is similar to tennis or other skills that demand muscle memory and reptilian-brain reaction.
“By contrast, when students are simply told the most efficient way of getting the answer, they get in the habit of looking to the adult, or the book, instead of thinking things through. They become less autonomous, more dependent. Stuck in the middle of a problem, they’re less likely to try to figure out what makes sense to do next and more likely to try to remember what they’re supposed to do next – that is, what behavioral response they’ve been taught to produce. Lots of practice can help some students get better at remembering the correct response, but not to get better at – or even accustomed to — thinking.” – Alfie Kohn Do Students Really Need Practice Homework?
And worse, assuming that practice creates proficiency backfires in the worst way with students who are furthest behind. Students who understand the material and made to complete a lot of practice will be, at worst, bored. But students who do not understand are being drilled into desperately guessing, never quite sure why they get some answers right and some wrong. It develops into a feeling of dread, of never being sure that they are doing anything right, but mostly that they just aren’t cutting it, and never will. Students who develop a deeply-held belief that they are not “good at math” may never overcome this.
The trouble with algorithms We double-down on the assumption about “learning by practicing” by breaking problem-solving into bite-sized chunks. We teach children specific ways to solve types of problems – tricks, mnemonics, and step-by-step processes (algorithms) like borrowing, carrying, or FOIL. We prompt students to look for clues in word problems, like if you see the word “more” it means to add. At the end of the day, this trains kids not to think, but to quickly try to guess the hidden rule and move on. All the help is well-intentioned, but reinforces a guessing game approach to math.
“Algorithms are harmful to most young children for two reasons: (1) They encourage children to give up their own thinking, and (2) they “unteach” what children know about place value, thereby preventing them from developing number sense.” Constance Kamii and Ann Dominick – The Harmful Effects of “Carrying” and “Borrowing” in Grades 1-4 (also in PDF, sometimes the Google doc doesn’t seem to work..)
This quote is from a research study that found that teaching carrying and borrowing to children significantly damaged their ability to solve addition problems. This is a must-read from 1998 that points out that despite numerous research studies that confirm the damaging effects of training children in carrying and borrowing algorithms, we continue to do so in most U.S. classrooms.
“The Harmful Effects of Algorithms in Grades 1-4” was published in 1998, four years after Kamii (1994) had published even more data. But 15 years later, most curricula still include the teaching of “carrying” and “borrowing.” When educators use research to inform practice and teach mathematics as a sense-making discipline, we will have a much better chance of helping all children be successful in mathematics.”
Our beliefs, even when refuted by research, allow us to continue to hope for magic wand solutions that make our beliefs real. Math myths keep us on the lookout for an easy answer that isn’t there. When something doesn’t work, myths allow us to ignore evidence and keep doing the same things because we “believe” in them. (If practicing isn’t working, practice more!) It makes us less willing to do the hard work of actually dealing with students individually and grappling with deep and difficult questions about how best to teach math. It’s all too easy to say, let’s push play on the video! Hurray, all our problems are solved.
The problem with “problems”
Additionally, we confuse solving problems with answering test questions and textbook exercises. Khan Academy deals with the later – specific steps for finding the right answer to “problems” that students are mostly likely to find on tests and in textbooks. (Some good examples of the differences between the two can be found in this blog post: Khan Academy is an indictment of education by Frank Noschese, a physics teacher and blogger.) This tricky word swap is confusing, because we DO want kids to have good problem-solving skills, but we certainly mean more than just answering textbook exercises. If we break a student’s confidence by imposing someone else’s problem-solving algorithm, when they encounter a real problem, one that isn’t made up for a test, they lack the confidence to explore their own solutions once they’ve gone through the list of algorithms we’ve had them practice with such fury.
The curse of the right answer
Math is viewed by many people as being logical, somewhat cold, and very rigid. Math is seen as the one subject where there are cut-and-dried right answers. But we forget that there are many ways to the right answer, and exploring these different paths helps strengthen existing mathematical understanding. Instead, we give kids lots of problems to work on so they can show us that they can get right answers quickly. It becomes about the product, not the process.
What ends up happening is that we spend a lot of time telling kids they are wrong, hoping that they will “get it” and start being right. Constance Kamii, an eminent math educator and a protégé of Piaget, says that this is completely the wrong approach – that if you destroy a child’s sense of autonomy and self-confidence, they will never recover that. She says that you should allow children to solve problems and LISTEN as they do, preferably in a group setting, as they discuss their answers. Let them convince each other based on their own observations and problem-solving ability. Let them defend their answers – even when they are wrong. Because it is destructive to tell a child they are wrong, but constructive to let them move from their first answer to an answer they come to like better.
I wrote a post about seeing Dr. Kamii do professional development with math teachers using this model – Questioning assumptions with Constance Kamii. Constantly telling children they are wrong creates a sense that right answers are simply mysteries that appear out of nowhere, and some people can guess them and some people can’t. And if you fall into the “can’t” pile, you are doomed forever.
This is not a made-up, one-off fantasy
You may be thinking, well, I’m sure a few teachers here and there do this, but not at the scale we need in this country! (You can also read my thoughts about the scaling question – Big problems require small solutions.) Take a look at this one national example (there are others). The New Zealand Numeracy Project encourages flexible strategies for solving numerical problems, and discourages reliance on standard computational algorithms. The project supports teachers with professional development, resources, and coaching. It gives parents information so they understand why their children aren’t being taught the same problem-solving rules they were taught.
Here’s just one evaluation done on it – The Algebraic Nature of Students’ Numerical Manipulation in the New Zealand Numeracy Project showed that, “…that students who participated in the Numeracy Project solved numerical problems that required manipulation with more success than did students who had not participated in the project.” Why are they doing this? Because New Zealand decided to pay attention to research about how children learn, not myths.
Autonomy shmatonomy
Americans have a bi-polar view of youth autonomy. We want them to be empowered AND to do what we (adults) tell them to do. We want them to find their voice AND sit still and listen. We want them to think outside the box AND bubble in the right answers. We are fooling ourselves if we believe that we can tell children that math is fun and creative, but only if you do it MY way. We must be able to answer “why do I have to learn this” with something better than “because you’ll need it in grade n+1” (where n = the grade they are in now.) This just reinforces the message, “… shut up and do what I say.”
When we think about how students learn math, it’s all too easy to discount how they feel about themselves as math learners and users. We want them to just do the work, pass the test and move on. There are students who are compliant and do just that. But there are many many students who get caught in extended power struggles with teachers, parents, and the school system. Some of these power struggles are overt, some quiet, but it’s a waste of potential all the same. (I can’t think of a better book about this than Herb Kohl’s I Won’t Learn from You and Other Thoughts on Creative Maladjustment.)
Constance Kamii has this quote front and center on her website, “A classroom cannot foster the development of autonomy in the intellectual realm while suppressing it in the social and moral realms.” Why would a math educator care so much about autonomy? I would encourage anyone exploring this question to take a look at some of the videos on her website that show classrooms where great care is given to this question.
In other words
Teaching math is not like Teach Me How to Dougie.
Part 1 – Khan Academy and the mythical math cure
Part 2 – Khan Academy – algorithms and autonomy (this post)
Part 3 – Don’t we need balance? and other questions
Part 4 – Monday… Someday
My recent post about the differences between Salman Khan and Conrad Wolfram’s TED Talks (Compare and contrast: using computers to improve math education) brought a lot of traffic to the blog, some great comments, and more than a few Twitter conversations about how to teach math.
So I’d like to get more specific about what I think is wrong about the Khan Academy approach by writing about things I see as wrong with the way we teach math in the US.
No matter if we agree or not about Khan Academy, I’m fairly certain we can agree math learning is not going as well as we’d like (to say the least.) Too many people are convinced by the system that they “hate math”, and even students who do well (meaning, can get decent test scores) are often just regurgitating stuff for the test, knowing they can safely forget it shortly afterward.
There is plenty of blame to go around… locked-in mile-wide inch-deep curriculum, focus on paper and pencil skills, lack of real world connections, assessments that are the tail that wag the dog of instruction, a culture that accepts “bad at math” as normal, teacher education programs that have don’t have enough content area specialization, … you can probably add to this list.
I can’t tackle all of these. But if you are interested, I’d like to share my thoughts about Khan Academy and a few epic math myths that are relevant to a discussion of the Khan Academy. In America, these myths are so pervasive that even people who were damaged by the way they were taught themselves accept them and insist that their children be taught using exactly the same methods.
I think these myths explain both the widespread acceptance of Khan Academy as a “revolution” and also why in reality it’s not going to change anything.
Myth: Learning math is about acquiring a sequential set of skills (and we know the sequence)
I think people have a mental image of math that looks something like a ladder. You learn how to add single digit numbers – rung one. You learn 2 digit addition – rung 2. You learn 3 digit addition – rung 3. In this model, you get to rung 3 by throughly learning rung 1 and then rung 2.
The myth continues with the idea that the march up the ladder goes faster if we tell children exactly how to do the problems step-by-step. In the language of math instruction, these step-by-step processes are called algorithms. Some kids “get it”, some don’t, but we accept that as a normal way that learning happens, and “help” the ones who don’t get it by drilling them harder in the step-by-step process, or devising additional tricks and supports to help them “remember” how to solve the problem.
If they don’t learn (meaning pass tests), we take this as evidence that they haven’t practiced the steps well enough, and prescribe more of the same.
Khan Academy plays perfectly into this myth. Here are a convenient set of videos – you just find the one you need, push play and the missing rung in your mental math ladder is filled in.
A corollary to this myth is that we can test students for these discrete math skills, see which “rungs” are missing, and then fix that problem with more instruction and practice on that specific skill.
Let’s diagnose how we think about learning a simple math skill
When we teach 2-digit addition, we immediately introduce the algorithm of “carrying”. You should know, though, that the U.S. form of carrying is just one of many addition shortcuts, not handed down on stone tablets. It’s not used world-wide, nor is it something that people naturally do when adding numbers. But it’s cast in concrete here, so we teach it, then we practice that “skill”. With our ladder model in mind, if a child can’t answer the 2-digit problems correctly you do two things: 1) Do more practice on the rung under it, and 2) do more practice in the algorithm, in this case, carrying.
The problem is that if a student has simply memorized the right answers to rung 1 without real numeracy, reviewing carrying will not increase that understanding. In fact, it will reinforce the memorization – because at least they are getting SOMETHING right. They are like the broken watch that’s right twice a day. This issue gets worse as the math gets more complex – the memorization will not be generalizable enough to solve more complex problems.
A different vision of learning
“Some of the most crucial steps in mental growth are based not simply on acquiring new skills, but on acquiring new administrative ways to use what one already knows”. “Papert’s principle” described in Marvin Minsky’s Society of the Mind.
If this is true, and since these administrative skills are not sequential, it makes it less likely that we really learn math in a sequential way. I think we’ve all had similar experiences, where a whole bunch of stuff suddenly makes sense.
This different vision of how people learn is called “constructivism“. It’s a theory of learning that says that people actively construct new knowledge by combining their experiences with what they already know. The “rungs” are completely different for each learner, and not in a specific order. In fact, rungs aren’t a very good metaphor at all.
“…constructivism focuses our attention on how people learn. It suggests that math knowledge results from people forming models in response to the questions and challenges that come from actively engaging math problems and environments – not from simply taking in information, nor as merely the blossoming of an innate gift. The challenge in teaching is to create experiences that engage the student and support his or her own explanation, evaluation, communication, and application of the mathematical models needed to make sense of these experiences.” – Math Forum
Learning theory? What’s the point?
We need to talk about learning theory because there are different ones at play here. And to be complete, we are also going to need to talk about teaching theory, or pedagogy, along the way. Constructivism doesn’t mandate a specific method of teaching, but is most often associated with open-ended teaching, constructionism, project-based learning, inquiry learning, and many other models. Most of these teaching models have at the heart an active, social view of learning, with the teacher’s main role as that of a facilitator.
However, the teaching theory underlying most of American math education is instructionism, or direct instruction – the idea that math is best taught by explicitly showing students how to solve math problems, then having students practice similar problems. Direct instruction follows when you believe that math is made up of sequential skills. Most American textbooks use this model, and most American teachers follow a textbook.
This is important distinction when talking about Khan Academy. Khan Academy supports teaching by direct instruction with clear (and free!) videos. If that’s your goal, you’ve found the answer…. but wait…
Is clarity enough?
Well, maybe not. Even if you believe in the power of direct instruction, watch this video from Derek Muller, who wrote his PhD thesis on designing effective multimedia for physics education. Really, if you are pondering the Khan Academy question, you must watch this video.
“It is a common view that “if only someone could break this down and explain it clearly enough, more students would understand.” Khan Academy is a great example of this approach with its clear, concise videos on science. However it is debatable whether they really work. Research has shown that these types of videos may be positively received by students. They feel like they are learning and become more confident in their answers, but tests reveal they haven’t learned anything. The apparent reason for the discrepancy is misconceptions. Students have existing ideas about scientific phenomena before viewing a video. If the video presents scientific concepts in a clear, well illustrated way, students believe they are learning but they do not engage with the media on a deep enough level to realize that what was is presented differs from their prior knowledge. There is hope, however. Presenting students’ common misconceptions in a video alongside the scientific concepts has been shown to increase learning by increasing the amount of mental effort students expend while watching it.” – Derek Muller, Khan Academy and the Effectiveness of Science Videos
Derek makes an interesting point – clarity may actually work against student understanding. Videos that slide too smoothly into an explanation do not give a student a way to process their misconceptions and integrate prior knowledge. The very thing that makes the videos so appealing – Khan’s charisma, sureness, and clarity may lull the viewer into comfortable agreement with the presentation without really absorbing anything (Research references and Dr. Muller’s PhD thesis on this subject)
Hooks, not ladders
This goes back to my original point. People learn by reorganizing what they already have in their head and adding new information that makes sense to them. If they don’t have a “hook” for new knowledge, it won’t stick. The tricky part is, though, that these hooks have to be constructed by the learner themselves.
Wishful thinking about downloading new information to kids is just that – wishful thinking.
There is no doubt that Khan Academy fills a perceived need that something needs to be fixed about math instruction. But at some point, when you talk about learning math, you have to define your terms. If you are a strict instructionist – you are going to love Khan Academy. If you are a constructivist, you are going to find fault with a solution that is all about instruction. So any discussion of Khan Academy in the classroom has to start with the question, how do YOU believe people learn?
I have more to say about Khan Academy and math education in the US — this post turned into 4 parts!
Part 1 – Khan Academy and the mythical math cure (this post)
Part 2 – Khan Academy – algorithms and autonomy
Part 3 – Don’t we need balance? and other questions
Part 4 – Monday… Someday
My context for these posts: I fully admit I’m not an expert in math or math teaching, just an interested observer of K-12 education in the U.S. In my work, I have unique opportunities to see lots of classrooms in action and talk to lots of teachers. It means I get to see patterns and similarities in classrooms all over the country. I don’t intend to do a literature review or extensive research summary in these posts. This comes from my personal experience, my master’s degree in educational technology and draws from a subjective selection of research and sources that have had a deep impact on my thinking about learning. Finally, I am NOT trying to tell teachers what to do. I’m not in your classroom — that would be silly.
Compare and contrast these two approaches:
1. Conrad Wolfram: Teaching kids real math with computers
2. Salman Khan: Let’s use video to reinvent education
Wolfram talks about how computers should be used to advance the understanding of math the way it’s really used in the real world.
Kalman talks about using computers to deliver traditional math instruction and gold stars.
Kalman reinforces the “teaching” paradigm; Wolfram blows it up and insists we look critically at what’s being taught.
Both these talks are about “improving education with technology” – but they couldn’t be further apart in world view.
But the TED audience applauds them both. This is why conversations about reforming education are difficult.
Sylvia
PS Wolfram makes a great point at the end of his talk about how, if we think that learning to calculate teaches procedural thinking, we have a much better way to do it by teaching programming.
Last week I was interviewed by Ihor Charischak for Math 2.0, a weekly webinar about math education and how it’s evolving (or not, as the case may be!) Ihor is an old friend and we framed the conversation using the stages of my own career and how math fit in. I’ve been a math geek, an electrical engineer and programmer in aerospace, a game designer and producer (educational and regular games), a parent, and now work with schools evangelizing a more student-centered, technology enhanced way to learn. All these experiences have shaped what I know and how I feel about math education.
I’m shocked at the trend that sees math as an endless series of skills, diced smaller and smaller into unrecognizable (but seemingly testable) nonsense. It’s a problem that many people only see arithmetic as math. I see too much vocabulary and not enough comprehension. I’m at a loss to understand why math is continually portrayed as not creative.
I’m becoming even more sure that we are doing damage to students when we teach them to narrowly find answers quickly and “the right way.” In the name of helping students “be confident with math skills” I think we are unwittingly teaching students to be less sure of themselves. I’m afraid that this math-phobia will have disastrous long-term consequences for students and society.
But like I said to Ihor, I can’t make any other choice than to be positive about the future of math education and work to expand and explain alternatives. This is where the kids are!
Ihor says in his blog CLIME Connections: Math 2.0 Linchpin Interview: Takeaways, “I had a fun time talking about the trials, tribulations and hopes for the future of math education with Sylvia Martinez last night.” I agree, it was fun, and we could have gone on for hours!
You can listen to the recorded session here. (This will launch the full Elluminate session with the recording, whiteboard, and chat window. Your browser may ask for permission to launch something!)
Sylvia