What Silicon Valley Gets Wrong About Math Education Again And Again

The medium is the message. The medium defines, changes, and distorts the message. The words “I love you” mean one hundred different things spoken by one hundred different people. Those words convey different meanings spoken on the phone, written on a fogged-over bathroom mirror, and whispered bedside in a hospital.

YouTube videos, digital photos, MP3s, PDFs, blog posts, spoken words, and printed text are all different media and they are all suited for different messages. When you attempt to distribute mathematics through any of these media, it changes the definition of mathematics.

Silicon Valley’s entrepreneurs, venture capitalists, and big thinkers assume a shared definition of “mathematics.” They innovate around the delivery of that mathematics. CK-12 has PDFs. Khan Academy has YouTube videos. Apple has iPad apps. ALEKS and Junyo have computer adaptive tests. Very few of them understand that each of those delivery media changes the definition of mathematics.

Even worse, at this moment in history, computers are not a natural working medium for mathematics.

For instance: think of a fraction in your head.

Say it out loud. That’s simple.

Write it on paper. Still simple.

Now communicate that fraction so a computer can understand and grade it. Click open the tools palette. Click the fraction button. Click in the numerator. Press the “4” key. Click in the denominator. Press the “9” key.

That’s bad, but if you aren’t convinced the difference is important, try to communicate the square root of that fraction. If it were this hard to post a tweet or update your status, Twitter and Facebook would be empty office space on Folsom Street and Page Mill Road.

It gets worse when you ask students to do anything meaningful with fractions. Like: “Explain whether 4/3 or 3/4 is closer to 1, and how you know.”

It’s simple enough to write down an explanation. It’s also simple to speak that explanation out loud so that somebody can assess its meaning. In 2012, it is impossible for a computer to assess that argument at anywhere near the same level of meaning. Those meaningful problems are then defined out of “mathematics.”

Do you want to know where this post became useless to Silicon Valley’s entrepreneurs, venture capitalists, and big thinkers? Right where I said, “Computers are not a natural working medium for mathematics.” They understand computers and they understand how to turn computers into money so they are understandably interested in problems whose solutions require computers. Sometimes a problem comes along that doesn’t naturally require computers. Like mathematics. They may then define, change, and distort the definition of the problem until it does require computers.

Some companies pretend those different definitions don’t exist. They pretend that we all mean the same thing when we talk about “mathematics.” Khan Academy acknowledges the difference, though, and attempts to split it by saying, in effect, “We’ll handle the math that plays to our medium’s strength. Teachers can handle the other math.” So Khan lectures about things that are easy to lecture about with computers and his platform assesses procedures that are easy to assess with computers. Teachers are told to handle the things for which teachers are a good medium: conversation, dialogue, reasoning, and open questions.

That delegation only works to the extent that teachers and computers convey complementary definitions of mathematics. But the message from Silicon Valley and the message from our best math classrooms contradict one another more often than they agree. On the one hand, Silicon Valley tells students, “Math is a series of simple, machine-readable tasks you watch someone else explain and then perform yourself.” Our best classrooms tell students, “Math is something that requires the best of your senses and reasoning, something that requires you to make meaning of tasks that aren’t always clearly defined, something that can make sense whether or not anyone is there to explain it to you.”

I won’t waste any effort complaining that my preferred definition of mathematics has been marginalized. That effort can be better spent. Anyway, in every way that affects Silicon Valley’s bottom line, the Common Core State Standards have settled that debate. Mathematics, as defined by the CCSS, isn’t just a series of discrete content standards. It contains practice standards, too: modeling, critiquing arguments, using tools strategically, reasoning abstractly, and others. The work of mathematicians. Any medium that tries to delegate one set of standards to computers and the other to teachers should prepare for a migraine.

Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.

Has your ed-tech startup been struggling to demonstrate statistically significant gains on the California Standards Test, which features tasks like this:

That’s your home-turf. Simple, machine-readable assessments. It will never get any easier for you than that. How much worse will your results look when we assess the same standard in 2014 with tasks that connect mathematical content to mathematical practices:

The medium is the mathematics. How does your medium define mathematics and is that definition anything that will be worth talking about in two years?

Full Disclosure: I’m a doctoral student at Stanford University in math education. I was a high school math teacher. I consult with ed-tech startups as time allows. I also develop digital math curricula that I sell to publishers and give away online.

Comment Policy: My usual policy is to close comments on posts that mention Khan Academy because they get silly almost instantly. But Khan Academy is only a symptom of a sickness that’s gripped this valley for as long as I’ve lived here. That sickness interests me and your thoughts on that sickness interest me. I’m leaving comments open but I’ll trashcan anything that doesn’t enhance our understanding of that sickness. That includes “Attaboys,” etc.

2011 Feb 7. Neeru Khosla, founder of CK-12, responds in the comments.

2011 Feb 7. Josh Giesbrecht posts a useful reply at his own blog, focusing on technology as an assessment, not technology as a medium.

2011 Feb 7. Web Equation from VisionObjects does a fantastic job translating scribbles on the screen to LaTeX.

2011 Feb 7. Silicon Valley’s (unofficial) rebuttal to my post is at Hacker News. Let me excerpt a few responses.

jfarmer poses a very productive alternative to my thesis, “What is technology good for?” rather than “How does technology change mathematics?”:

In design, a skeuomorph is a derivative object that retains some feature of the original object which is no longer necessary. For example, iCal in OS X Lion looks like a physical calendar, even though there’s no reason for a digital calendar to look (or behave) like a physical calendar. The same goes for the address book.

This is what I see happening in online education. I don’t think it’s a case of “lol, Silicon Valley only trusts computers,” but rather starting off by doing the most literal thing.

Textbooks? Let’s publish some PDFs online. Lectures? Let’s publish videos online. Homework and tests? Let’s make a website that works like a multiple-choice or fill-in-the-blank test.

These are skeumorphs. There’s no reason for the online equivalent of a textbook to be a PDF, it’s just the most obvious thing.

For me it’s 1000x more interesting to ask “On the web, what’s the best way to do what a lecture does offline?” than to say “Khan Academy videos are the wrong way of doing it.”

Arun2009 offers a common view, that mathematics is many different things to many different people:

The trouble with trying to arrive at any single definition of Mathematics is that Mathematics is different things to different people. A research level Mathematician might see it differently (finding patterns, abstraction, theory – axioms and proofs) from an Engineer who has a purely practical interest in it (cookie cutter methods and formulas). For everyday use Mathematics is a set of algorithms for doing stuff with percentages, fractions, basic arithmetic etc.

He’s absolutely right, but if we’re pragmatic in the least, we’ll ask “which of those definitions does the most good for students?” and we’ll look at the Common Core State Standards, which is the de facto definition of mathematics for those students. (My appeal to the CCSS was an attempt to reduce exactly this subjectivity.)

Various hackers took me to task for claiming it’s difficult to represent mathematical notation using computers. ie. the square root of 9/4.


Square root ? “(3/4)^(1/2)” or maybe “sqrt(3/4)”. There’s no complexity in parsing that. I do agree it is not as natural as on paper but maybe tablets will find a way to improve that. Thats what innovation is here for after all.

japhyr rebuts convincingly, in my opinion:

He is writing about math education. The characters (3/4)^(1/2) make sense to all of us who have already learned math and know some programming languages, but that syntax is pretty confusing to students who are just developing a real understanding of exponents.


One could make the argument that any mathematical syntax is equally confusing for the novice–so why not start them on something they’ll be using later anyways?

That’s an interesting idea but unless we’re also positing a universe with utterly ubiquitous computing, we’d be better off preparing students to communicate in media that are readily available. What if a computer isn’t available for our students to code some LaTeX to express themselves?

davidwees illustrates my overall point well. Does this seem natural to anybody?

\png \definecolor{blueblack}{RGB}{0,0,135} \color{blueblack} \begin{picture}(4,1.75) \thicklines \put(2,0.01){\arc{3}{3.53588}{5.8888}} \put(.375,.575){\line(1,0){3.25}} \put(1.22,1.375){\makebox(0,0){\footnotesize$ds$}} \put(.6,.5){\makebox(0,0){\footnotesize$x=0$}} \put(3.36,.5){\makebox(0,0){\footnotesize$x=\ell$}} \dottedline{.05}(1.0,.575)(1.0,1.10) \put(1.0,.5){\makebox(0,0){\footnotesize$x$}} \dottedline{.05}(1.5,.575)(1.5,1.40) \put(1.5,.5){\makebox(0,0){\footnotesize$x+dx$}} \put(1.22,.65){\makebox(0,0){\footnotesize$dx$}} \dottedline{.04}(0.6,1.12)(1.25,1.12) \put(1.0,1.14){\vector(-1,-1){.45}} \put(.58,0.83){\makebox(0,0){\footnotesize$T$}} \put(.77,1.05){\makebox(0,0){\scriptsize$\theta(x)$}} \put(1.18,1.16){\makebox(0,0){\scriptsize$\theta(x)$}} \dottedline{.04}(1.5,1.41)(2.1,1.41) \put(1.5,1.44){\vector(4,1){.67}} \put(2.22,1.59){\makebox(0,0){\footnotesize$T$}} \put(1.95,1.45){\makebox(0,0){\scriptsize$\theta(x+dx)$}} \end{picture}

Symmetry, takes the conversation to Khan Academy:

As to why many people might want to defend Khan Academy, well, its because I think I would have been much happier with Khan Academy than the math education I actually had, and I would very much like it to be available to children like myself. I was bored stiff in math class in middle and high school, and being able to work at my own base, not bound by the slowest person in the class, would have been amazing.

This same sentiment crops up in the comments thread here and I think it’s utterly on point. Teachers are a great medium for lots of things that a YouTube video isn’t. “Conversation, dialogue, reasoning, and open questions,” as I put it in my post. If you, as a teacher, aren’t taking advantage of your medium, if you’re functionally equivalent to a YouTube video, you should be replaced by a YouTube video.

Sudarshan summarizes that elegantly:

Incompetent/bored math teacher < khan academy < better online learning platform < Good math teacher.

FWIW, I stopped by that thread and summarized my argument in three lines:

  1. There are different ways of defining mathematics, and some of them contradict each other.
  2. Silicon Valley companies wrongly assume their platforms are agnostic on those definitions.
  3. For better or worse, if you’re trying to make money in math education, the Common Core State Standards are the definition that trumps your or my preference for recursion, computational algebra, etc, and those standards include a lot of practices for which, at this point in history, computers aren’t just unhelpful, but also counterproductive.

2011 Feb 10. Zack Miller e-mailed. Zack graduated from Stanford’s teacher training program. He now teaches at the charter school where I’ve been consulting with my adviser and two other graduate students. He’s a proponent of blended learning, individualized instruction, and Khan Academy, in particular. He gave me permission to excerpt his e-mail. I’m going to post the entire thing, bold one line, and make a comment on that line at the end.

Students’ mathematical learning and sense-making has a much higher ceiling if the material they encounter follows a coherent mathematical narrative that is relevant to and keeps pace with their current individual mathematical understandings.

The above is not a reality in most math classrooms. Students’ math understandings are Swiss cheese, largely because time, not competency, was fixed in their math education. What’s standing in the way of the above becoming a reality? Resources and good data. 1 teacher must provide 100 unique learners with learning experiences, daily. Can our country’s median teacher (or best teacher) provide a learning experience that is in each student’s zone of proximal development every day (and also be an expert in classroom management, curriculum, parent communication, data analysis, etc.)? No. So we’ve looked for ways over the years to ameliorate the situation (tracking, heterogenous classes, differentiation, etc.).

Silicon Valley is trying to innovate because students’ mathematical learning has a much higher ceiling if curriculum always meets their ZPD, and perhaps now there are ways we can offer that. THIS IS BLENDED (not to be confused with computers replacing traditional classrooms with traditional video lectures). Yes, the medium matters — without question — but breaking from the ratio of “30 students:1 pace” matters more. Any tool that gathers relevant data and/or provides good learning experiences helps me individualize, and (like many of your commenters) I disagree that there is no computer-based tool that does this (Geogebra, for one, was great for today’s lesson that you and your team designed!).

I agree that, at present, Silicon Valley’s current tools are limited. Pedagogy is weak. But I disagree that doing this:

“We’ll handle the math that plays to our medium’s strength. Teachers can handle the other math.”

Implies also doing this:

Silicon Valley tells students, “Math is a series of simple, machine-readable tasks you watch someone else explain and then perform yourself.”

Isn’t it possible to frame computer-based tools in a way that supports your definition of math (a definition that I happen to closely align with)? Doesn’t procedural fluency (which will still be a large portion of Common Core assessments, I believe) support reasoning, problem solving, and sense-making?

And lucky for our students, people like you will help Silicon Valley with their pedagogical shortcomings and misguided definitions of math. As the quality and quantity of “learning experiences offered” improve, students can have a more individualized experience, and that high ceiling can be realized.

My point is that the tools that allow you to individualize math instruction are not neutral on the question, “What is mathematics?” My second point is that not every answer to the question, “What is mathematics?” harmonizes with every other answer. Sometimes they clang around against each other awkwardly.

2011 Feb 15: EdSurge offers a summary of this post and its responses over at Fast Company.

2011 Feb 15: Just noticed (way late) that Jacob Klein, one of the founders of Motion Math, responded here. Klein is hip deep in the culture I’m describing here and he makes games that help students practice mathematical skills I value, so give him a look.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. Thanks for putting this up, allows me to avoid my own posting. One thing that became clear to me when reading this debate (for example here https://news.ycombinator.com/item?id=3554719) is that too often “Silicon Valley” gets the wrong metaphor, even though they have a great one available. They want to think of mathematics as code, which you demonstrate leads to many problems. Instead mathematics is more like programming.

    This is even worth when you lift up to teaching mathematics. Trying to use code to teach mathematics, is like using code to teach programming, you might be able to get some of the basic technical skills over, but cannot give any understanding of the fundamental ideas.

  2. We homeschool our children, and this is a problem that we are very familiar with. What makes it worse is that neither myself nor my wife are able to really help with math because neither of us did well with it in school. We don’t know anything beyond basic algebra. My kids are there now, and struggling, and we can’t help much, so we try things like Khan and other software.

    Things will work for a week, then the newness wears off and the weaknesses come forth and we’re back to square one.

    What would you suggest for a homeschooled child with parents who don’t have math skills?

    Please don’t say “put your kids in school.” Let’s just assume that’s not an option.

  3. Your thoughts on assessment aren’t only (or even, specifically) applicably to computer-based assessment, but to any assessment regime that attempts to turn mathematics into a list of skills. Sadly, this includes just about every formal assessment in my country (the UK), at every level.

    Consider the way we assessment mathematical competency at 18, for example. In the UK, this is usually done via an ‘A-level’ examination, consisting of six 90 minute examinations taken in the last term of formal schooling. You can see sample assessment materials (in this case from Edexcel, a Pearson company) here:


    If we take a look at the first (and easiest) paper in the series, C1,


    we can see that many of these questions would be difficult or impossible for a computer to mark correctly (as they are all more than ‘did they get the correct answer?’), but don’t really test mathematical ability. Instead they are focused on basic mathematical process skills.

  4. @Tom In your situation things like Khan Academy can be a useful source of information and direction. As long as they come with the understanding that they are not anything more than that. Try to understand a technique and make your own videos explaining it in a similar style.

    In terms of helping break through basic algebra one secret is to go back to arithmetic. Start to play around with calculations there, playing with how to do multiplication for example. 7*8 is the same as 7*7 + 7, and 7*5 + 7*3 or even (5+2)*5 + (3+4)*(2+1). This will help to understand the rules that numbers use, that can then be considered for their own sake in algebra. Playing with number games, such as adding the sum of digits, casting out 9s, and considering prime factorisation can all help in this playing with arithmetic.

  5. Does anyone have good examples of things computers have supported that are part of a healthy definition of mathematics? For example, social networking, blogging, Tweeting, etc. have helped teachers improve their practice through the posing and responding too of good questions. Are there ways that networking kids in a particular environment with particular affordances has helped them learn to use all their senses to reason? That we could help Silicone Valley apply to math education?

  6. In this respect, does ‘Silicon Valley’ get *all* education wrong?
    There are many ways to write an essay, a paragraph or even a sentence – will a computer ever be able to assess the quality of what’s entered?
    Furthermore, what will happen when a student identifies a method that is better than all those which the programmer/developer thought of? What if they identify something new or previously undiscovered?
    Will a computer ever account for human judgement, or will it just return ‘syntax error’?
    I am not sure I have helped, and may have even missed the point, but just a thought.

  7. These edtech companies are filling a niche as they would with anything else. As long as education allows machine assessable problems to be the standard then they are happy to provide automated means to deliver and assess.

    Second, the mathematics changed the second we put it on paper or tried to communicate it to someone else. That goes back to our original “mathematicians”. To say that Algebraic notation is the purest embodiment of what I am thinking mathematically is silly.

    So while a video about the quadratic equation will reduce it down to its dead instructions or a adaptive learning software will only understand the string of characters “1.52” and have no understanding of what that means or if “1.5” is equally correct, computers are an excellent way to extend our understanding of math, like any tool it depends on how you use it.

    Someone has already programmed that software, but what if you are trying to teach a computer (via a programming language) to understand a fraction. Sure you will have to create constructs that may seem removed from our understanding, but our understanding of fractions is based on our own constructs and language limitations.

    Programming a computer, while certainly not any more pure than writing an idea on paper, affords new opportunities for students to stretch their understanding. As every teacher knows, if you can’t teach it, you don’t understand it.

    As long as educators are elevating the definition of what “mathematics” is as many who read (and write) this blog, then those companies will not be able to capitalize on the low hanging fruit.

  8. @Max

    > Does anyone have good examples of things computers have supported that are part of a healthy definition of mathematics?

    The AOPS (Art of Problem Solving – http://www.artofproblemsolving.com) site comes to mind. They have forums, online lectures etc. that students who are actually interested in improving their mathematical skills can use. AOPS is an example where mass delivery through the use of technology enables a large number of students access to expert coaching. Wish I had access to it when I was young!

  9. Geometry and fractions are not the natural turf for computers but that is not the sum total of mathematics nor the sum total of mathematics accessible & useful to highschool students, computers are a great boon to understanding recursion and logic, subjects which are just as fundamental and were sorely overlooked in my schooling 10 years ago.

    Geometry has been a staple of the curricula and was an early branch of math because measuring out shapes of plots of land was of keen importance for early civilization and has been baked into education ever since, they have garnered a level of prestige perhaps out of proportion to their utility to modern life.

    Recursion & discrete math ‘native’ to computing will be just as important to those starting their education today as geometry was to those arguing over land rights by the Nile 4000 years ago.

    It may be that the less a branch of math is native to computers the less empowering it will be to students. The utility of understanding probabilities, logic and recursion may be far more important than even geometry, which has had thousands of years of polishing.

  10. I’m a former middle school math teacher and doctoral student who is interested in assessing understanding. I fancy myself a pretty good interpreter of student thinking, and I find incorrect answers to be much more interesting than correct answers. That said, I often question computer adaptive stuff as being more inclined to replace wrong answers rather than identify and correct misconceptions. As a human teacher talking to a student in the moment, I can use my previous conversations with students, my pedagogical content knowledge, and the students’ responses to choose my next move. I learn something from every interaction with students that I add to my repertoire. Autotutor is a technological approach I’m cautiously optimistic about. It’s based on the use of latent semantic analysis (LSA) to make decisions like this as a student interacts with an intelligent tutor.

    Regarding Item #27 from a standardized test vs. the mosaic task. For a student, in the parallelogram task, it’s pretty clear what to do: find the values that work for a and b. (I would be disappointed in myself if my students didn’t try the answer choices and pick the pair that makes both sides work. I would be looking for an understanding that solutions make equations true, and the correct option for a and b is the one that makes opposite sides of the parallelogram the same value.) In the mosaic task, I’m not sure what is required. Do I just write as much as I know about the shapes? angles? symmetry? the ratio of squares to parallelograms? Something about the matching square and parallelogram sides? Wait, can I be sure those are squares just because they look like squares? Something about a circle? This guessing game is frustrating! Even if students have experience answering problems of this type, I’m not too keen on what the answers to this task tell me about what students understand or don’t. I’m sure there is background information and a learning goal for this item, but I cannot tell from how it’s posed, and I don’t have time this morning to make many guesses. Dan trains students to ask questions that are typically answerable. That’s very different from posing a task such as this.

  11. Tom,

    My recommendation is to find a friend who does get it in terms of mathematics, and trade services with them. You don’t need to put them into school, you need to find a teacher. It could be that you hire someone (if you can afford it) or that you find a friend who can help out. You do want to make it clear what your expectations are.

    Here’s another suggestion; don’t worry about the symbols that make up algebra. Focus on being able to solve real problems that require algebraic reasoning (which is different than algebra necessarily).

    Here’s an example that I came across just the other day. I’m leading a trip of students on an out of country trip. I have to figure out how much money each student needs to pay for the trip, given that they have to cover the costs of the chaperones, and that one parent paid for the flight with air miles. How do I calculate the cost of the trip per student? It turns out that algebra is a perfectly reasonable way to solve this problem, but that I had to reason out how to construct the algebra appropriately, so that I could explain my solution to others.


    There are some tasks for which computers are perfectly suited in terms of mathematics; they are excellent for the job of mathematical computations. I assume you’ve watched Conrad Wolfram’s TED talk; I participated in the computer based math summit last November (held in London), and I’m excited by the prospect. Can we change the curriculum we cover so that it is less focused on calculating things, and more focused on the uses of those calculations (or at least on things which we would consider mathematically beautiful? What high school curriculum includes fractals, for example? It is perfectly easy to explain the concept of fractals to students, but the computations are beyond their means; unless they use a computer).

    What you have suggested is that they are less than ideal for the quick communication of mathematics, and for deeper assessment of what mathematics students understand. They are terrible for the symbolic layer of mathematics as we designed it for paper, but my suspicion is, this will improve as touch screen interfaces improve. I saw a UI just the other day where one could enter a mathematical expression, and the computer could translate this expression into appropriate mathematical notation. One question; is it possible that what needs to change is the layer of symbols we’ve used to represent mathematics? Is there any fundamental difference between sqrt(x) and \sqrt{x}?

    I’ve taught my students LaTeX before (at least enough to communicate high school level mathematics) and nearly every student I’ve taught has managed to be able to use LaTeX efficiently enough to be able to post blog entries that contain mathematical expressions. It’s too time consuming for day to day mathematics though, and is only really useful for stuff that requires type-setting. It is fascinating to me though that a device that was essentially invented by mathematicians is so poor at typical day to day representations of mathematics. Our special codes are used because we, for some reason, never found an efficient way to input mathematics into computers.

  12. Unfortunately, in my public school district math classes were ALWAYS about watching the teacher solving problems on the white board and mimicking the teacher’s approach on the test (exactly the criticisms that have emerged recently of Khan Academy). I hated math, because I thought it was a tedious, brainless exercise. Computers were the only reason I found math enjoyable: when we learned prime numbers in class, my dad helped me build a prime number generator in matlab. When we learned about shapes, I coded up pictures using terapin logo.

    Describing mathematical concepts like a fraction or a square root may be difficult in Word, but they are used all the time in programming languages. The best computerized mathematics curriculum should teach simple programming as well, allowing students to creatively use the mathematics they learn to solve real world problems. This may change the definition of “math”, but in a positive way– by *increasing* the creativity involved in elementary and middle school math. The founder of Wolfram Alpha has a good TED talk about this topic.

  13. @Belinda The Mosaic Task seems like it’s the perfect opportunity for media to improve the message. As a standardized assessment task, it’s pretty “huh? What do they want me to do? What counts as a right answer?” If you imagine a bunch of tired math teachers who’ve been reading kids’ papers all day trying to rubric-score you, you start to want to just write as much as you can about shapes in hopes you hit the right combination of sentences to earn a 5. And that’s annoying.

    But the problem says, “you’re communicating with someone via a medium that doesn’t allow for pictures to be sent. Make them see what you see.” With computers, you can get kids on a computer writing to other kids in London or wherever, describing stuff. Have the London kids send back pictures of what they created. The assessment is from the kids’ own reaction to what the kids in London drew. Did I do it well enough to actually perform the task? Of course, you don’t really need computers or kids in London. All you really need is some cardboard or a binder. Kids in pairs, something to block their view, one has a picture and the other a pencil and paper, compass and straightedge. Go! The assessment is authentic: did I communicate the design so my partner’s drawing matched mine?

  14. @Brandon

    Interesting discussion but it exactly proves Dan’s point; as soon as I commented that I thought the Khan Academy should at least talk to an educator, I nearly immediately down-voted. I think that educators have been more than clear about their feedback for the Khan Academy, but I have yet to see the Khan Academy do anything about it.

  15. Blake Householder

    February 7, 2012 - 9:48 am -

    I think it’s a little odd that you describe how SV has math all wrong, but then give an example of a standardized test question as an example of what students should be able to answer.

    I look at that question and think “what a waste”. I don’t know anyone whose life has been enriched by being able to answer such a question.

    Do you think maybe there’s also a problem with teaching the wrong math? I think this parallelogram question does nothing to describe the beauty or usefulness of math.

  16. I applaud your stance, there is *FAR* too much emphasis on using technology as a panacea for problems in education, or just trying to convince people that there is a problem in education that technology is suitable for solving. As has been seen in studies for many years, education software usually ends up becoming less about education and more about entertainment, especially any of the mainstream Leapfrog junk. My mother was a teacher and through her I saw a very different side of education than most can contemplate, and as I watch my wife homeschool our two children I can tell that they learn *much* better with her sitting down to actually *teach* them than they ever do with any interactive gadgetry. And that’s not to say that the gadgetry has no place, it sure makes it easier to research planets & constellations when you can use e.g. a gyroscope & GSP-enabled iPhone app that shows you exactly what stars in the sky are which constellation, but that is far down the scale of priorities as compared to math, language arts, science, etc.

    BTW, for anyone looking for a math curriculum for homeschooling, try the Math-U-See series – it’s really well put together, the lessons are very easy, and our 8yo has done *really* well with it.

  17. Just as you stated in your opening paragraph, words are subjective in nature and open to interpretation. In my experience, people often interpret information according to their own beliefs, biases, and past experience. Education has even coined a term for how new learning is built upon long-term memory–schema.

    So Silicon Valley is just doing what it does best, based upon its own, particular schema. And my own experience using math ed-tech programs isn’t so great. What they offer in convenience, they seem to make up for in superficiality and arithmetic errors. From my perspective, they were never meant to be a replacement for all the components that make learning possible within an individual student.

    As far a CA State Standards in math education are concerned, don’t even get me started… To say that that’s a complicated issue is an oversimplification. (Please pardon the pun.)

    Suffice it to say, there is no panacea in learning.

    As someone who strives to practice critical thinking while using various learning modalities to access students’ abilities in math, I am really appreciative of what you are doing and your motivation and stamina to continue those efforts. I myself ran out of steam working within the system around graduate school. Congratulations on making it to your post graduate studies!

    The field of education needs people like you who are passionate and dedicated.

  18. Christian Bokhove

    February 7, 2012 - 10:26 am -

    The discrepancy between for example notation in tools and “actual” mathematical formulae is what is aptly described in theories on tool use and instrumentation. I agree with Dan that this is often forgotten in Computer Science. I must say that there are, however, better examples of computer tools than the ones he mentions (some working with, Apple forbid, java), expanding on inbetween-steps, geometry, checking these results and scrutinizing student results.

    I think an open source framework for recognizing maths like Joe shows could help in at least closing the gap “on paper” vs. “ICT”. The one with “our mind” will never be bridged. But that didn’t keep us from assessing on paper, any way, so why not at least try to find out what software and feature work and under what conditions. Keeping in mind that it will never replace a gooddiscussion IRL.

    One challenge would be to connect these generic tools with good educational software. There are many of these that are designed in conjunction with teachers but do not get a fraction of the Airplay of the Big Players in the field. In the end every succesful use of ICT depends on thoughtful design, use by the teacher etc.

  19. It is good to read an article that directly gets to the point, instead of ranting against Khan Academy and others, and loosing their main points in the noise.

    I was just wondering whether in all these discussions the people who are outside learning institutions are neglected? One can talk about all the modern pedagogical techniques, but they are not of much use if they don’t get to the hands of the lay man, is it?

    I have watched more than 700 videos on Khan Academy, over the past 11 months, and to me the single most significant impact of the videos is that it inspires me to learn more. I even use it to fight my medical condition. Stanford’s classes, Udacity, and perhaps even the early announcement of MITx, are to a great extent inspired by Khan’s work. I think my same feelings are shared by others in the several personal stories that they have sent to Khan Academy.

    It would be great for people like me if implementations of the various techniques that you talk of are made accessible to the common man, and implemented in a way that invokes the same level of inspiration that Khan Academy videos provide, and with the same quality of Khan’s videos. This of-course means that a good part of it must be freely available, even though I do not know if that is possible at all; maybe the content can be free, where as expert hands-on teaching can be provided for a fee.

    As far as I can tell, all the class room implementations of Khan Academy are directly supervised by teachers. By watching the countless interviews of Khan, it appears that every single feature of Khan Academy’s software come from teacher suggestions; almost all of the initial features added in the spur of the moment by Khan are being phased out. Khan’s summer programs, apparently, involve games that explore probability and expectations values, which I think would fit into your really great take on the definition of mathematics.

    My point is, can’t you work with Khan Academy, and the teachers trying to implement it in class rooms, to “convey complementary definitions of mathematics”? Can’t you use the excitement generated by Khan Academy to raise the standard of teaching and learning in class rooms, and else where? Over the last year or so I have read many educators say that many of the modern views on education have been there since 1980’s. Is it that they are not properly implemented in classes even now, because educators haven’t taken such moments to bring these to bear on the teachers working in class rooms i.e., the people in the trenches and not policy makers?

    From the way they are recruiting people such as Vi Hart, Brit Cruise, Beth Harris, Steven Zucker, and possibly in the future a famous american comedian (again, this is based on clues from Khan’s interviews/talks; I could be completely wrong), you shouldn’t find it hard to at-least have a discussion with them and the teachers. In terms of software development Khan Academy appears to be super “agile”.

    Please use the opportunity that Khan Academy is providing to take education and more importantly learning and appreciating the world around us, in the short time we have on this planet, to the next level.

  20. Dan, you are absolutely correct in saying that computers can’t replace (good) teachers. The view of ed tech startups seems to be (unlike Project Euler for example) that a math teacher is someone who taught you procedures and definitions to manipulate.

    However, there IS Silicon Valley software that presents math using Guided Discovery. In a visual manner that doesn’t hide the math. It also stresses introducing the conceptual BEFORE the computational as you do. The software is the online “ST math” program that has shown a respectable effect size under randomized controlled trials. While it only covers K-6 curricula and has room to improve, it is definitely worth discussing this program and not blanketing all math ed tech as bad. Can you comment on this?

    The info is…


    and the randomized control trail is…



  21. The best part about “The Standards for Mathematical Practice” ? They were previously known as “Life Skills”. Make sense of problems and persevere in solving them. Life skill. Construct viable arguments. Life skill. Reasoning. Life skill. Attend to precision. Life skill. This is essentially a repackaged version of the ‘school doesn’t teach you what to learn, it teaches you how to learn’ argument, which is how I view education. So don’t be scared of CCSS. If you view them as a set of tasks to be completed, then they can be the same vomit-inducing thing they were when they had a different name. If you really dig into them and realize that they are just as much telling you HOW to teach as WHAT to to teach, then they are much more palatable. A rationalization, perhaps, but who doesn’t get through the day with at least one juicy rationalization?

  22. This feels like a backhanded way of saying there’s no magic bullet in teaching and, contrary to a popular idea, computers are a subset of math; not the other way around.

    I work at a small rural school. I like Khan as a “substitute teacher” since we don’t have subs that can handle anything beyond basic algebra. But, as a one-size-fits-all teaching method it lacks any meaningful discussion for when I have my 90 minutes to give them something good. Sure they might walk away knowing the algorithm for dividing fractions but making the connection between that and what division really represents is a messy process that I’ve never taken the same route to twice in my 6 years of teaching. I guess what I am saying is I can use Khan for some pretty straight forward ideas but a deep understanding of math doesn’t come with railroad tracks that can be programmed into a computer or prerecorded for easy consumption.

  23. I’ve been trying to post a comment on the Hacker feed, but it keeps timing out with an error message. Hope you don’t mind if I post it here.

    “I respect that the Khan Academy has worked with some schools to have their program implemented and tested. No program should go untested.

    What I find frustrating is that the over-emphasis on the value of the data they’ve collected. I used the Khan Academy with my students (mixed in with a problem solving and project based approach) at the beginning of this year, and within 6 weeks I abandoned it (after two assessments I’d given to students external to the program) because I found no relationship between what my weakest students had “mastered” according to the types of questions the Khan Academy provides, and the students ability to use their calculations in any other context. They could do 30 problems in a row flawless related to the rules of exponents, for example, but not solve any problem directly related to the rules of exponents that they hadn’t seen on the Khan Academy videos or exercises. The transferability of what my weakest had learned was extremely low. By contrast, the approach absolutely worked fine for my stronger mathematics students, who were able not only to transfer what they had learned, but to work at a pace I could not hope to match.

    What I suggest is; sit down with some mathematics educators (and mathematicians, engineers, scientists, general public) and get some agreement first on what it means to do mathematics. As Dan suggests, “to do mathematics” has very different definitions, and we hardly ever articulate these in enough detail for it to be meaningful. I almost feel like the Math wars are a war not over how math should be taught, but what the word math actually means. Once we have agreement on this issue (or have split into separate distinct camps, each using their own words), then we can move forward with potential solutions. How can one define a solution without agreeing on the problem?

    I should also be clear, that the summer camps the Khan Academy are offering sound fabulous, and I fully support these. I think that more people should be offering similar camps all over the country because they are absolutely a step in a direction I support. These are almost exactly the opposite approach that they are offering with their videos and exercises, which are where my chief complaints lie.”

  24. @David Wees

    You make an interesting observation – KA is useful for the gifted students. Does this have something to do with Khan implementing a system that he himself would have liked in school (as opposed to weaker students)? Did he design a system for young Khan? These gifted students would already be the sort who can transfer knowledge and reason well, but are limited by the pace computation is taught. Hmmmm….

  25. Simon Levy:

    In this respect, does ‘Silicon Valley’ get *all* education wrong? There are many ways to write an essay, a paragraph or even a sentence — will a computer ever be able to assess the quality of what’s entered?

    The computer isn’t a natural working environment for mathematics but it is a natural working environment for composition. A computer may not be able to grade an essay but it can facilitate its creation in ways that a pencil and paper can’t match.


    As long as educators are elevating the definition of what “mathematics” is as many who read (and write) this blog, then those companies will not be able to capitalize on the low hanging fruit.

    I’m not following, Phil. Is this an argument against an elevated definition of “mathematics.”

    In any case, though we may all have different preferred definitions of mathematics, pragmatism demands we look at the Common Core State Standards and ask ourselves how a medium helps or hinders student progress towards that definition.

    Belinda, on the Roman mosaic task:

    Dan trains students to ask questions that are typically answerable. That’s very different from posing a task such as this.

    I posted the mosaic task because I think it requires much more comprehensive knowledge of quadrilaterals than the CST example. I don’t feel like debating its merits in this particular thread, but I’m happy to pass along sample student work [link] to inform your hypothesis that it’s too ambiguous.


    Unfortunately, in my public school district math classes were ALWAYS about watching the teacher solving problems on the white board and mimicking the teacher’s approach on the test (exactly the criticisms that have emerged recently of Khan Academy).

    What you describe is a teacher not taking full advantage of her medium. To paraphrase … somebody … if a teacher can be replaced by Khan Academy, they should be.

    Blake Householder:

    I think it’s a little odd that you describe how SV has math all wrong, but then give an example of a standardized test question as an example of what students should be able to answer.

    You’ve missed my point. I’m not endorsing the parallelogram problem. My point is that startups struggle to show statistically significant gains on simple, machine-readable problems that play right to their advantages. When the problems get better (ie. the Roman mosaic) those gains will be even harder to come by.

  26. I’m not a math guy but did a standard engineering cycle up through multivariant calc and engineering stats 25 years ago. Khan et al is great for reminding me of the things I forgot but I find I do not get deep understanding from such presentations. I would also point out that in the California Standards Test example you mentioned I did not even think of solving it using the method I was supposed to, having been forced to learn math without a calculator I solve multiple choice problems by simply putting in the four sets of results and seeing which is closer.

    Computer based multiple choice is also a problem.

  27. Hey Dan,

    I left this comment on Hacker News. I’d love your thoughts on it.


    I’ve been thinking a lot about this topic for the last month. The questions are: “What should education on the web look like?” and “What subjects are most suited to be taught on the web?”

    Since my formal education is in mathematics (B.S. University of Chicago ’06), whether and how math education is taught online really interests me.


  28. The problem you describe stems from the processes of communication which are automatic and therefore mostly unconscious in nature.

    Mathematics is the language of measurement. It is a visually oriented problem set. Coding it into language for transmission is explicit. That is the coder/sender is seeing the problem space and refers to it in order to create the language for the coded transmission.

    The receiver of this coded transmission has no such references to use to reconstruct those visuals into his own visuals. He must guess exactly how to do this.

    This dichotomy of explicit and implicit exists in all communications between people regardless of whether the topic is originating from any sensory references by the sender. Not just visual to sound (language).

    In addition even after the guessing of proper context on the general level of translation. When the local decoding in completed as well as can be done implicitly it must be integrated into the proper context of the persons world models, consisting of learned and accumulated beliefs. Which may be either correct or incorrect. This step enables the personal evaluation or understanding of the meaning of the coded transmission.

    Only then can the person decide the proper way to respond to this encoded transmission.

    This language process which we have all taken to be simple and efficient, creating understanding between the users is in fact an illusion. This illusion of successful communication and mutual understanding of the meanings we intend to share is the problem space you must solve to change the success rate % from near zero to some better percentage requires dialogue to insure success. Smoothing out the differences with each additional attempt, until mutual understandings of the individual meanings is accomplished. This is the actual nature of language encoding/decoding using different sets of coders decoders (individual minds) to accomplish the feat of true understanding of individual meanings.

    good luck! chuckle …..

  29. I would probably be more patient with arguments against Silicon Valley’s current teaching and assessment techniques if I hadn’t been taught math in just that manner my whole life. “Here’s the procedure for determining the circumference of a circle… Here’s your homework. There’ll be a scantron quiz tomorrow… and remember to bring a number two pencil.” I learned and retained more of what I do know from people who were busy trying to communicate other ideas like Newtonian physics and neural network engineering. I’ve dropped first semester calculus more times than I’ve completed mathematics courses… for lack of context more than any other reason. I understand the argument that their methodology is lacking, it’s just that real live humans aren’t doing any better. In my experience, “Our best classrooms” are a fiction.

    Communicating with computers… First, you need the right tool. A numerator and denominator button just sounds broken. Like a simple calculator understands the four key presses “4 / 9 =” just fine. Press the square root key and you’ve got the “complicated” answer. It’s an instruction that compares quite favorably to both spoken and written methods, and a practiced hand doesn’t even need to look. “But what if they typed 8 /18 =?” If the assessor (the person or group responsible the test specifications) determines that a fine point needs to be made of simplifing fractions, then the assessor needs to get with the programmer and fix it(maybe with the aid of a UI designer who knows that numerator and denominator buttons are a dumb idea). It’s doesn’t take one minute of AI, much less 40 years, to implement.

    The rigors of assessment don’t end with the student. Have a teacher evaluate thirty Roman mosaic responses. It isn’t as easy as posting to Facespace. It’s not even as easy as texting the picture to London (Look! It’s a picture of fossils crafting test questions!).

    The notion that computers are not a natural working medium for mathematics is just absurd. I don’t use a hand drill to put a hole in metal, when I have and know how to use an end-mill. Similarly, when I weary of calculating something in my head, I don’t turn to a co-worker and explain the problem. Putting it on a piece of paper may help if I’m determining _how_ to solve a problem, but I certainly don’t make an expository statement of it. Beyond that, I tell a computer, which sympathetically performs the drudgery from then on. Like the end-mill, the result is easier to achieve, highly accurate, and trivially repeatable. Also like the end-mill, it takes practice to employ efficiently.

    That computers aren’t a natural working medium for knowledge assessment is an implication of the manner in which the tool is being used or unreasonable expectations of the tool. Do you expect an assessment to evaluate the student’s talent for communicating a bit of logical analysis? I’ve noted that already isn’t happening in meatspace, and, without sentience, a computer will never comprehend the twisted logic of a human. I would argue that if it’s logic you want to assess, teach programming. The euler.net exercises, for example, make a determination of success based on one integer for some very complicated feats of logic, and the languages available are certainly no worse than the bastardized Greek of mathematics.

  30. Dan & other familiar with common core implementation in CA: I have heard through the grapevine that CA may not implement common core for 8th grade. What I have heard is that 8th grade teachers will still be responsible for teaching the Algebra 1 course and there will be an ‘addendum’ to make the course common core compliant. Do you know if the idea of a 2 year course which combines algebra and geometry (per common core) actually has hope of being implemented in CA? I have only heard 2nd hand rumors of the type: someone-heard-something-at-a-state-meeting.

  31. Steve:

    I understand the argument that their methodology is lacking, it’s just that real live humans aren’t doing any better. In my experience, “Our best classrooms” are a fiction.

    And how many math classes constitutes “your experience?” One hundred over your lifetime? Fewer? Balanced against the millions that exist outside your experience, your certainty about what happens in “meatspace” classrooms is hard for me understand.

    Similarly, when I weary of calculating something in my head, I don’t turn to a co-worker and explain the problem.

    Another commenter had the same sense that (in his words) “Mathematics is just a shortcut for computation.” If you can’t conceive of an end for mathematics education except to prepare people to compute, this post won’t make a lot of sense.

    Jesse, my response to your comment timed-out over at HN so let me say I appreciated the pushback and your citation of skeuomorphs. I’m working on some kind of review of iBook Author as it could be used for mathematics education and I keep getting tripped up on the bookness of it, which limits more possibilities than it opens, right down to the subtle page turning effect. You seem to think a lot of educational technology is in a transitory period right now, trying to sort itself out, hanging onto metaphors of the past. I may not be so optimistic but the possibility interests me.

  32. I’m going to leave aside the tech question for the moment, and focus instead on a different point you bring up. I am intrigued by the fact that the Common Core State Standards are going to require the teaching of mathematical habits of mind – in other words, the “practice standards” referred to above. I should be cheered by the thought that now that the Common Core exists and has been adopted by forty-splunge of the 50 states, that our students will become ever-deeper mathematical thinkers.

    Forgive me if I’m skeptical. I think a lot of the discussions that are going on around mathematics education are misguided – we are arguing about the “proper” role of technology, arguing about what the best type of curriculum is, arguing about standardized testing (will they get “better”? will they go away?), arguing about whether or not Khan is going to save or destroy us….when the reality is that our students are frequently getting turned off to mathematics at ever-earlier ages, and that no amount of technological wizardry, pedagogical brilliance, curricular innovation, or any other such combination seems likely to change that – as long as we remain trapped within the constraints of the existing system. What might change that is the recognition that there is not one royal road to the learning of mathematics, and that schools need the flexibility to use different approaches for different kids to take them where they want to go (and, that not all kids will be able to or should get to the same place in the end). What might also help is that our youngest students get a real exposure to real mathematics as well, by professionals who are quantitatively literate and know interesting places where math “lives” for that age group.

    Because right now, tech or no tech, CCSS or not, from where I’m standing I see first graders getting multiple math worksheets for homework every night (which, honest to goodness, include words like “rhombus”, “rectangular prism”, and other such stupidities). And it’s taking kids who are naturally curious and naturally – perhaps – mathematically able and turning them off. That’s what Silicon Valley in its infinite wisdom should be thinking about – how can we create real mathematics and use the technology we’ve developed to help teach it to all ages? Another app isn’t the solution.

  33. @ 28 Dan “The computer isn’t a natural working environment for mathematics…”

    While not as obvious as with composition, I think you undersell computers. For example, many of my geometry students see Geometer’s Sketchpad as a natural working environment for working with graphs and compass constructions, creating graphics, and even writing short mathematical arguments. They even use it in other classes of their own accord.

    Similarly, Mathcad has positioned itself as a potential replacement to a student’s mathematics notebook. Consider some of the demos here.

    Fluidmath is another great program that has potential to expand the role of computers as a natural working environment. The people at Exeter’s Anja Greer Math, Science, and Technology Conference have been touting such benefits for years. This is an exciting time for “digital natives” to be mathematics students.

  34. Morgan Warstler

    February 7, 2012 - 6:18 pm -

    the punchline to reality is:”I don’t have to outrun the bear, I just have to outrun you.”

    It might very well be that 5 or 10% of the student population needs to receive the hands on (read expensive) education that will solve for the Mosaic project.

    But the reality is we only need Khan style eud to outrun the 50% of teachers who are just horrible (so we can increase class size), scare 40% of them to work harder using Khan in the new bigger classes, and reward the last 10% with sizable pay raises.

    I spent years bouncing around inside Milken’s KU years ago evaluating edu tech, and am now pushing a project with the state of Texas… for the whole country.

    The plan is to release the work product of the for hire public school teachers under a Creative Commons license that allows for profit.

    This would allow for profit start-ups to outfit any given classroom of their choosing with cameras and record the entire semester / years lecturing of the very best teachers teaching any give textbook.

    As long as the students are autoblurred, and the textbooks are not shown as aids, we can layer multiple methods of teaching a given textbook (think multi-stream DVDs) and lay the tracks down against the pages in the book.

    The point is we can quickly augment and outperform Khan’s videos.

    I don’t mean this competitively, I mean this as for any given idea, there ARE multiple kinds of learners, from multiple backgrounds, and once you adopt a true autoteach system, you can spend your time coming up with more and more unique ways of exampling an idea to a smaller subset of learners.

    Passing information between brains is not nearly as complicated as tens of millions of marginal teachers would have us believe.

    Dan is the exception, and not the rule… if he doesn’t realize it, he shouldn’t be making rules.

  35. “Balanced against the millions that exist outside your experience, your certainty about what happens in ‘meatspace’ classrooms is hard for me understand.”

    I, apparently carelessly, extrapolated. I was under the impression that multiple-choice standardized testing had become de rigueur, since I was in school, the de facto, federally-imposed method of evaluation for math skills in all districts. I stand corrected.

    “Another commenter had the same sense that (in his words) ‘Mathematics is just a shortcut for computation.’ If you can’t conceive of an end for mathematics education except to prepare people to compute, this post won’t make a lot of sense.”

    Oh, I’ve no problem seeing multiple aims for learning math. It just seems a bit daft, given my limited experience, why you would expect such from a subject that was seemingly obsessed with one correct way to arrive at one correct answer. A computer seems a natural ally. Do you see how strange it appears, trying to press mathematics into the mold of applied sciences, with their attendant vagaries and imprecision? Classes where I learned how to _use_ tools with purpose and reason that the math teachers simply tossed at me?

    I agree with Jesse. Problems like bookness, outright bad interface design, and uninformed opinions about proper education (*cough*) will be overcome. There’s a momentum building, of late. Silicon Valley is involved, but isn’t the only force. Some eastern schools are starting to stir a bit more. There’s an obvious place in education for people, but there are a panoply of vested interests who are merely rent-seeking. They’re just starting to jockey for position.

  36. My two cents: I think one of your most salient points was that assessments need to attend to the process standards (or whatever else inhabits the same space)–and I worry that they will not.

    As a Math Wars vet, I weep for how promising CAP and CLAS were and how they sank; they had a chance to be models for how we could learn to assess the sorts of things we can’t figure out with forced-choice items.

  37. ny_entrepreneur

    February 7, 2012 - 7:47 pm -

    Yes, of course

    What Silicon Valley Gets Wrong About Math Education Again And Again

    since this is a special case of the simple fact that nearly all of education below the junior level of college gets math education wrong again and again. Here I concentrate on math at the junior level of college and beyond.

    Here’s why:

    (1) Math is by a wide margin both the oldest and the most solid of the serious academic subjects. From Bourbaki and more, the subject is polished to a level of astounding perfection.

    (2) Math is by a wide margin the most difficult academic subject.

    (3) Due to the difficulty, only a tiny fraction of the population has any meaningful understanding of math.

    (4) Math does make significant progress right along, with quite significant new results each 20 years.

    (5) Just what math is is rock solid and will not go away or be revolutionized.

    (6) How to learn math is very well established.

    (7) Essentially all of the software parts of practical computing and computer science have low competence in math. In particular, nearly all the chaired professors of computer science at the best research universities and editors in chief of the best computer science journals didn’t take much in math in graduate school and, thus, are not very knowledgeable about math. Really, broadly, the current Silicon Valley is hopeless at math.

    Learning math is not a spectator sport. Basically, to learn math, a student sits alone in a quiet room and studies a good book — on paper, via PDF, etc. So, see a definition, close the book, and try to guess what the next theorems will be. See a theorem, close the book, and try to prove the theorem. Once see a start in a topic, try to guess the major issues and results in the topic.

    The exercises are mostly proofs: For each, discuss changes in the hypotheses and conclusions such that make false results true or true results false (loosely from P. Halmos). The two pillars of math analysis are continuity and linearity (G. Simmons). Elegance counts: “The ‘elegance’ of a result is directly proportional to what can see in it and inversely proportional to the effort required to see it.” (S. Eilenberg).

    There’s no real challenge in math education: Typically there is excellent math education available at Harvard, Cornell, Courant, Princeton, Johns Hopkins, Berkeley, Stanford, and more. The tiny fraction of the population that has both interest and talent at math are doing quite well now. The rest should f’get about math.

    For Khan Academy, if some of those on-line lectures help some students in math in K-12 and the first two years of college, then fine. Still, those students would be well advised to study a good text alone in a quiet room. But, students should be warned: Math is too difficult for essentially all the K-12 system and the teachers in it.

  38. @Allison Krasnow – I was at a conference last week (with Dan actually) and a CA DOE guy sort of vaguely mentioned they didn’t know what to do about 8th grade and was considering the possibility that we’d give kids both the current Alg CST and whatever drops from SBAC. So…yay.

  39. Hi Dan,

    Awesome post. As an edtech entrepreneur I love your throwing down of the gauntlet; I fear your critiques of the new math games we’re designing, and it’s a wonderfully motivating fear. I certainly hope the CCSS practice standards you praise are well-enforced and assessed.

    But I’m confused as to why you think math products coming from Silicon Valley necessarily contain a message that they completely define mathematics. Does the marketing say that explicitly? Implicitly? Why can’t “a series of simple, machine-readable tasks” complement the more complex work you do in your classrooms, the way, for example, that footwork drills and jumping exercises complement playing organized basketball? That’s in some ways how we think of our games at Motion Math — a student should master number sense exercises so he or she can really play on game day, reasoning effectively, discussing, arguing, using tools, etc.

    The Roman Mosaic question looks great (I’d like to read student responses!) and I can think of many low level tasks that would help a student doing the activity:
    1. Define a right angle.
    2. How many degrees are there in a complete turn?
    3. What does it mean for shapes to “tile”?
    4. What is the length of a square’s diagonal?
    5. Think of a rhombus with two 45Ëš angles, two 135Ëš angles, and sides of length 1: what is the length of its longest diagonal? What’s its area?

    What role do you think these sorts of math facts and low-level calculation should play in a well-rounded math education?

    p.s. I speak for Stanford University

  40. I don’t think Khan et al. had to adapt the content to the medium. Most math taught in schools fits already with no effort the mechanizable scheme. As long as the schools keep making emphasis on memorization and repetition, Khan and his robotic friends would be the way to go.

    To me, what your post demonstrates, is that we’re teaching the wrong kind of math in the schools: the one that can be easily dealt with machines.

  41. Mr Meyer, thanks for writing this post. However, I disagree with the statement that computers are not a natural working medium for math. They ARE, in a big way, at least when you get out of middle school math. When’s the last time you took a log or a cos by hand? Fractions are a nonissue, with a basic understanding of calculator syntax.

    That’s the problem I’m running into with my digital curriculum — it puts math on computers, but not in the way that people who care about math would. They write a problem with fractions displayed in latex, and give the student a no-formatting textbox to write their answer. (I’ve seen some pretty dedicated space-formatting outta these kids) They say “one less than the dividend of seven and a number, added to seven times half the number.” when no self-respecting mathematician would turn his head to his co-worker and say this. There are plenty of nice online versions of manipulatives at the middle school level, and many students find it easier to express themselves by typing than any other way. Is it perfect? No. But digital natives feel comfortable on computers. It’s just a matter of time before math gets comfortable enough to feel happy there, too.

    Anywhere below 6th grade, I completely agree with you.

  42. It’s difficult to read comments written by people who either haven’t taken the time to read carefully what Dan has written or who, having “read” the piece, don’t understand it or willfully miss obvious points. So it’s frustrating to not be one of the very first respondents: by the time I saw this post today, the debate was already all over the place and I can’t really pick a spot to play off of. Instead, I’ll simply say that in my experience as a supervisor of secondary student teachers in mathematics, a coach of elementary, middle school, and high school mathematics teachers, and like Tim Erickson, a long-time Math Wars veteran, the vast, overwhelming majority of Americans, including mathematics teachers, and including professional mathematicians who should know better but pretend that they do not, view or profess to view mathematics as computation. They allow for reasoning to enter into the picture, heavily wrapped up in a strait jacket of limitations, in high school geometry, and then quickly put it back into the rubber room, as if letting it out into K-12 classrooms too long might lead to utter chaos.

    Dan’s points about SV are almost entirely correct, on my view. Keith Devlin has tried to address this issue in his latest book, Mathematics Education for a New Era: Video Games as a Medium for Learning (if he and his work are mentioned above, I skimmed to quickly to notice). I agree with the person who mentioned Geometer’s Sketchpad that there are some instances where software has added to the possibility of actually doing mathematics in meaningful ways – whether in school or not. But mostly, the tech folks have missed the boat because their idea of school mathematics is no different or very little different from that of the typical American (including the typical American K-12 math teacher).

    Lest someone accuse me of suggesting that there are no good K-12 mathematics teachers, or that all mathematicians are guilty of the sins I mention above, or that computation plays no significant role in real mathematics, let me simply say that I’ve said none of those things nor do I believe them, as I have ample evidence to the contrary in each instance. But the sad truth is that we MOSTLY miss the boat with mathematics and have been for a very, very long time.

    Finally, I’m not convinced at all of Dan’s apparent faith in the 2014 assessments that will roll out to support the Common Core, any more than I believe that many people will read or HEED the Process Standards, which read very much like things NCTM has been pushing since 1989, if not longer. Unless you’ve been living under a rock since then, you are aware that the philosophy of mathematics education embedded in those Process Standards is anathema to folks on the side of the Math Wars that LIKES school mathematics as computation and decries every attempt to do something besides that as “fuzzy.” Given the COST of implementing the sorts of meaningful performance tasks that Dan and others hope will come to dominate assessment in mathematics (I believe Mr. Obama has repeatedly paid lip-service to that notion, at least in general), IT AIN’T GONNA HAPPEN. And even if it did, briefly, the forces that lined up staunchly against the tests in California Tim Erickson mentioned above will attack anything that can’t be computer-scored “objectively” as more of the alleged “dumbing down” of US mathematics education. Mark my words on this: the Math Wars aren’t over, and the Common Core isn’t going to solve our problems. And as long as most Americans see mathematics exactly as their parents, grandparents, and generations before them saw it, we’re very far from making effective progress, and computers aren’t going to overcome blind prejudice, fear, and loathing.

  43. There is a vicious cycle here, and it’s the responsibility of math educators everywhere to fight like hell to break it.

    1. Teachers teach K12 math as a meaningless computation.
    2. Students gain a vision of K12 math as meaningless computation.
    3. Students grow up and advocate for ed reforms that sound great if you think that math is meaningless computation.

    And then Dan and other folks jump in and say, “Hold on, everybody! You’ve got a crazy picture of what K12 math is!”

    It’s too late. This battle was already lost 15-20 years ago.

  44. Morgan Warstler

    February 8, 2012 - 6:16 am -

    Michael, the purpose of edu.tech is to FIRE teachers.

    It isn’t like SV gears up to remake an industry, so MORE money can be spent doing the same thing a different way.

    We want to keep $1 for every $5 we save doing a better job than is currently done.

    And since you agree that the current situation is horrid, you make the very point Dan is arguing about.

    I made my arguments in 38, and they still stand.

  45. It’s sad to see technology pushing out good teaching. Technology is rightly raised up as potentially revolutionary, but too often iPads become fancy flash cards and interactive whiteboards become ways to add clipart to direct instruction. I think your point about medium shaping content is a great one, but it seems so far from the ed tech rat race to have the shallowest implementations of technology possible.

    The good news for me is that in its fullest form teaching math is full of all the depth and fun that it seems like so many people are working hard to drain from it.

  46. i’m thinking the only thing wrong with math ed is that it is compulsory. and not only compulsory, but in a compulsory order.

    i’m thinking learning is not linear.

  47. Great post… takes into consideration the constant interaction of constructed language(s) and the one thing that seems to be entirely concrete: math.

    Thanks for the post… keep writing!

  48. @monika-

    I think less should be required. Would solve more than a few issues. There should be a compulsory requirement that students reach a certain level (mastery of a certain numbers of concepts) but the order and time table it takes to get there should be flexible. If you want to take 4 years to get to the level, fine, if you want to get there in 2.5 semesters fine. Then they can either stop or continue for there. I think of all the kids that want to further their math knowledge that are slowed by the pace of the particular class they are in as well as all the kids who need extra time who can’t get it because of the need to stay ‘on pace’. Trying to serve all probably serves none, or at best only serves the middle 50%.

  49. yeah. again. non-linear.
    to me that means there are no basics.
    rhizomatic, so can jump in at any point, as long as the curiosity and drive is coming from within.

  50. Hi Dan, I love this post.

    “Now communicate that fraction so a computer can understand and grade it. Click open the tools palette. Click the fraction button. Click in the numerator. Press the “4″ key. Click in the denominator. Press the “9″ key.”

    Everything you do on paper, you should be able to do it easily on the computer. It’s not there yet, but that’s my kind of obsession. JP sent me this the other day:

    Should prove that progress is comming on the input front.

    “How does your medium define mathematics and is that definition anything that will be worth talking about in two years?”
    A computer contains more than one medium. Computers are also good for communication, not only for testing. If math input could be just a bit easier, I think open ended questions would be possible. And the fun thing: software change a lot in two years.

    “Any medium that tries to delegate one set of standards to computers and the other to teachers should prepare for a migraine.”
    Doing this properly is costly.

    Carl (from BuzzMath.com)

  51. I hesitate to comment as #55. Really, who’s still reading now?
    I have never sat in front of a piece of Math Ed software and felt it was valuable to me as a teacher. Dan, your foray into philosophy captures why. The medium changes the message and the definition of mathematics changes for each medium. This is irrevocably true.
    Computers have the potential to help users exploit the definition and meaning of mathematics that is appropriate to a context. But all of the Math Ed software that I have seen projects a given, generic definition that corresponds to the definition assumed in classes worldwide.
    Fractions can be written numerator first if the useer’s intention is to think in terms of measured portions. Denominators can be presented first if a partitioning action is the user’s first order of business. Maybe a comparison is in order and the user puts that idea out first before deciding what will be compared. Fractions have many more functions that I won’t even address here. The point is that the current computer medium demands conformity to a linear view of mathematics that does not adapt to the context and thinking of the user. How can they begin to assess actual student thinking when they can only impose their own thinking through the medium?
    For more on ossification of mathematics by an older medium, text, see Nick Jackiw’s mind-blowing Ignite talk at NCSM from 2010.
    “The Dynamics of Dynamic Geometry”

  52. Dan,
    I absolutely agree with you. Although I am a huge believer of innovation Silicon Valley makes possible, I believe, as you do, that learning in any domain is very complex and individual. It is more complex and complicated in math than we know, we also know that there are many things that help in learning and understanding – depending on what you are trying to understand and/or learn. The way we do math today is dramatically different for different people. Some people see it and understand it and others struggle forever and ever and eventually give the effort to understand or some just muddle through.

    Math on computers does not imply a rigid or one-dimensional pedagogy. Look at rich modeling environments — Geogebra, Octave Processing– that allow students to play with phenomena, building up understandings, intuitions, and the ability to transfer symbolic understandings to graphical representations to real-world manipulation. Look at all the data sets available online with which our students can ask and answer real-world questions.

    The power of practice has been marginalized. Practice helps us ensure that we don’t waste any energy on those very basic steps that go on to strengthen our work with complex ideas. The idea and the way practice is implemented have also been marginalized. One way to get practice out of the “boring and unfavorable” stand is to immerse the learner in many different modalities and that is where the power of computer comes in.

    In trying to present our ideas we have to be very careful that we don’t tilt the scale. This is why at CK-12 we have made it very clear that we don’t support any one way of learning or teaching. To that end we provide multiple ways of learning – print media, interactive as well as paper-based practice, project- and inquiry-based activities, etc. However, this challenge is worth undertaking because technology affords many modalities in one place while also allowing for human-to-human interaction. Perhaps the real reason is that we don’t want to lose the human interaction part.

    Last but not the least part is the power of collaboration with diverse and vast audience — as evidence by your presence on the web.

    I would love help in solving these problems. We would love to incorporate your ideas to improve what we have to whether we do it with and/or without technology.

  53. Three thoughts:
    1) Technology does seem to be effectively used to create virtual worlds and highly differentiated games. Could you have worlds or simulations involving financial transactions for example — choosing home loans, choosing to refinance or not, investing for retirement, etc. These things are fairly dull on paper, but if you can watch your virtual world develop (new car, new house, salary increases, credit card debt), and incorporate enough strategy into the game, it might be more interesting. Same sort of thing with the grade school project of planning a trip across the country — more interesting if you can see it unfold virtually with some of the unexpected things that come up on such adventures.
    2) The technology now seems to mainly drill single concepts in isolation. But, even if it could be done, I wonder if creating technology that allows for maximizing absorption & understanding of concepts is really that productive. Are the concepts more important than the experience of students and teachers working together, thinking, and figuring things out in a somewhat social environment?
    3) How do you directly test whether a student has the mathematical habits outlined in the standards of practice, and if you don’t test it, will teachers change their approach? How do you measure perseverance? Was the kid really smart and answered the question easily, or did they really wrestle with the question? That sweet spot between a problem that requires a tiny conceptual leap & one that requires a leap they just can’t make is a toughh target to hit in the classroom, let alone under test conditions with a heterogeneous population. It seems like, at the very least, you would need the test to be calibrated to a student’s level of mathematical intuition/intelligence (whatever that is).

  54. As #57 says, who is still reading! I find though, that putting these thoughts and reactions in writing is mostly only for my own benefit! In this case, it is beacuse, whilst I understand and sympathise with the general view being expressed, I think I actually disagree with the statement about ‘natural medium’! I read most of the responses and scanned the rest but the response from David Wees (#13) came closest to my reaction when he said ‘

    ‘There are some tasks for which computers are perfectly suited in terms of mathematics’


    ‘What you have suggested is that they are less than ideal for the quick communication of mathematics, and for deeper assessment of what mathematics students understand.’

    Regarding the first point….

    My relatively short teaching career (13 years) has spanned ‘almost no access to computers’ to ‘working with a one to one program at my current school’. There is no doubt in my mind that computers have had a hugely significant effect on the way mathematics can be taught and, more importantly, discovered, beacuse they provide a considerably more natural, able and versatile medium. A lengthy description of cases could follow, but I will limt myself to just a few…

    Dynamic geometry, as has been mentioned by some already. This tool has done amazing things for helping teachers to create opportunitites for students to make discoveries on their own and thus enage with mathematics. It can go beyond the teaching of geometry as well.

    Graphing software – largely by labour saving, but also through dynamic functionality – these tools as well have created new opportunities for exploring relationships.

    Data Handing – This has come to life through computers with access to real, live data, the functionality to collect it and the ability to process it. All this means that the nature of data handling tasks can now vary in new ways. (I will not say ‘more mathematical ways’ although that it is what I think.)

    As suggested, I could go on and will in my head!

    Regarding the second point….

    Yes I agree that progress is slow on more able and intuitive user interfaces for communicating mathematics. I think that this has worked in our favour as teachers though. For example, taking the fractions, modern calculators now make it much easier to input and work with fractions than it used to be and this may have resulted in a poorer understanding of what fractions actually mean. The fact that computers dont find it easy to accept fractions means that users have to think about what the fraction actually means in order to input it. A fraction is easily written on a piece of paper with no understanding of its meaning.

    Likewise, when programming with dynamic geometry (and I do consider constructions a type of programming), there is no ‘rectangle tool’, in order to construct one you have to know that a rectangle is made by two pairs of parallel sides intersecting at right angles. When you program it correctly it will always be a rectangle regardless of which points are moved. The process of drawing a rectangle on a piece of paper is not at all the same.

    In summary, dont get me wrong, I estimate that computers are used for about 50% of our lesson time and I am a committed believer in variety of tasks that range from the pencil and paper, to the practical, to the virtual. That said, I am a passionate supporter of what computers have done for mathematics education. I am also a relatively new blogger and always have a sense of fear when ‘submitting’ such responses. I think most bloggers understand that expressing your views and reactions is the best way to develop them, so thanks Dan for making me think! Apologies if I have missed the point somewhere along the line, I feel better for writing this down either way.

    As it was #58 I published this response as part of a blog entry on my own blog – http://www.teachmaths-inthinking.co.uk/blog-post/11751/natural-medium.htm

  55. Morgan, for what it’s worth, what makes you think I was disagreeing with Dan, if indeed that’s what you’re implying?

    My only disagreement with him is about the role the new assessments and the Common Core will play in this drama: I’m not optimistic at all.

    As for your arguments, sorry, but you don’t speak a language with which I’m familiar (at least not in too many places in what you wrote for me to be willing to wrangle with the entirety of it). So if you’ve made wonderful points that “still stand,” they were lost on me for the most part. I don’t know to what “Milken’s KU” refers. That probably makes me a bad person. To the extent that I understood your bear joke (an oldie) in this context, I think I disagree with your viewpoint. But my sense is that it doesn’t much matter to either of us whether I got what you were saying or not, and thus am content to let another day go by in ignorance of the particulars. Mea maxima culpa.

  56. Not sure if trackback thing is going to pop it up here anyway, but here’s my reply:


    Short version: I don’t think any of this is really about computers as a *medium*. I think this is about whether computers can do assessment for us.

    (Or, I’d rather separate those two, because the “computer as medium” argument isn’t convincing me at all while the “computer as assessment-processor” argument totally does.)

  57. For Michael: http://lmgtfy.com/?q=milken%27s+ku

    I’m mostly teasing (I’m finding Morgan’s posts rather difficult to parse myself), but hey! Look at that – instead of staying ignorant about something because it wasn’t explained clearly, we can now use all the new tools at our disposal to easily find explanations ourselves, to learn independently. Sounds kind of like what Khan and other SV ed-tech advocates are proposing, no? To push the analogy further, we’re having “conversation, dialogue, reasoning, and open questions” in this comment section, and the dialogue that’s happening here is really great. When a confusing aside like “Milken’s KU” is tossed out there, is it better for the author to clutter and confuse the thread with an explanation of a tangential statement that most might have skimmed over, thinking it wasn’t important enough to the main thrust of the post to question, or is it better for people who are genuinely curious to Google it themselves and explore whatever they find? And surely it’s a good thing to expect people to be able to learn things independently, rather than to wait to be told the answer.

    IMO, criticisms of that idea itself, that it’s probably a good thing to make additional resources available for students who are curious/confused so that they can explore and practice on their own time and allow the class to keep focused on the “main thrust” of the lesson, don’t really have a good justification.

    What can be justified are criticisms of how the idea is implemented, which I think is what Dan is doing here. The UI of math programs is a huge problem, Dan’s criticisms of the awkwardness of keyboard input compared to handwriting is spot-on. But like the link that Joe posted shows, companies are starting to develop ways to make this interface more natural, and as this technology continues to improve and as tablets continue to proliferate and become cheaper, I think that doing math on a computer won’t be much different from doing it on paper. But until the technology reaches that point, I think it’s somewhat unfair to criticize ed-tech for trying to put out products that do the best with what we have at the moment, since it’s rather silly to say that these products shouldn’t be available until they’ve perfected the interface; having products that are just okay has to be better than having no products at all. (The comments others have posted about the ways in which computers can be natural media for mathematics also have nice arguments, ones I agree with; I just wanted to provide a different sort of response to Dan’s point.)

    And as to the point that meaningful learning can’t take place on computers because computers can’t assess an argument at a meaningful level, while the point’s super valid and possibly the strongest criticism to ed-tech, two responses come to mind. The first is that even those in ed-tech understand this limitation of computers, hence the positioning of programs like Khan as supplements, not as replacements, to teachers and the realization of the need for teachers to be able to track their students’ progress on such programs so that they can make just-in-time interventions when a student isn’t quite understanding the task. The second is going to be more controversial, and isn’t necessarily a defense of computer-based learning, but how many teachers can genuinely give good, informative feedback to a student’s explanation of whether 4/3 or 3/4 is closer to 1 and integrate it into their lesson? This post is already getting long, so I don’t want to further digress into discussions of teacher quality in the United States (although I’d recommend Liping Ma’s book on this subject), but I don’t think it’s saying too much to say that teacher quality isn’t quite what we’d like it to be, especially in math, and while I’m sure that there are a ton of really good teachers who can really explain why you “invert-and-multiply” when dividing fractions, I’m equally sure that there are a lot of teachers who can’t. I guess what I’m trying to say is that a criticism of one alternative isn’t as strong as it might seem if it can also apply to the other alternative.

    We need strong teacher development, and we also need to rethink our curricula, along the lines of what Dan and other awesome teachers have been and are doing. But with all the potential to help improve learning that ed-tech can have, it would be a shame to simply criticize and condemn it for its current shortcomings and hold fast to the belief that all we need are good teachers and good curricula. Good teachers and good curricula by themselves probably would make math education what we all want it to be, but couldn’t GOOD ed-tech help us get there faster? All I’m saying is, whenever I read a blog post or news article about Khan or some other ed-tech venture, I always see the same criticisms of what ed-tech can’t do and comments bemoaning what bad ed-tech is doing or might do to our students, and all of these are valid, useful criticisms and comments. But how many of us have actually reached out to ed-tech rather than being content with preaching to the choir and complaining that ed-tech hasn’t reached out to them? If you’ve tried, that’s awesome, and if anyone has any stories about successes or (maybe more likely) failures trying to do so, I’m sure there are a lot of people who’d like to hear and learn from them.

    Otherwise, all our solid ideas are just trees falling in forests, with no one (in ed-tech) around to hear them.

  58. “For better or worse, if you’re trying to make money in math education, the Common Core State Standards are the definition that trumps your or my preference for recursion, computational algebra, etc, and those standards include a lot of practices for which, at this point in history, computers aren’t just unhelpful, but also counterproductive.”

    Dan, you really need to separate bad policy from (bad/good) computers. You are mixing oranges and apples.

    Ironically, the Roman Mosaic can be superbly described in a simple and readable language, Logo, by my 10 year old cousin, and he will have learned a *lot* doing it.

    Don’t know how to write in that language? Try to express your problem in Scratch, you will learn visually! http://scratch.mit.edu/

    You can share it, get it remixed, compare it, extend it, etc.

    My partner Steve had a kick reading your article and had the same idea as me without telling me. Like a kid, he did this version in the BuzzMath editor and had a kick discovering and expressing patterns:

    He did it there because it was *faster* to test multiple idea of how to do it than on paper and he could explore different ways quickly.

    I find it exciting that computers are *nearly* there. I find it boring that people use them on a large scale to automate the wrong things.


  59. I’m going to excerpt a long paragraph from Rollie because I think it grants a lot of dignity to two opposing viewpoints.

    And as to the point that meaningful learning can’t take place on computers because computers can’t assess an argument at a meaningful level, while the point’s super valid and possibly the strongest criticism to ed-tech, two responses come to mind. The first is that even those in ed-tech understand this limitation of computers, hence the positioning of programs like Khan as supplements, not as replacements, to teachers and the realization of the need for teachers to be able to track their students’ progress on such programs so that they can make just-in-time interventions when a student isn’t quite understanding the task. The second is going to be more controversial, and isn’t necessarily a defense of computer-based learning, but how many teachers can genuinely give good, informative feedback to a student’s explanation of whether 4/3 or 3/4 is closer to 1 and integrate it into their lesson? This post is already getting long, so I don’t want to further digress into discussions of teacher quality in the United States (although I’d recommend Liping Ma’s book on this subject), but I don’t think it’s saying too much to say that teacher quality isn’t quite what we’d like it to be, especially in math, and while I’m sure that there are a ton of really good teachers who can really explain why you “invert-and-multiply” when dividing fractions, I’m equally sure that there are a lot of teachers who can’t. I guess what I’m trying to say is that a criticism of one alternative isn’t as strong as it might seem if it can also apply to the other alternative.

    The argument that these innovations are all supplements to one another, and all of them to a teacher, and that none are meant as a complete solution is a common one in SV. But I think it is a mistake to assume that the different definitions of mathematics promoted by each of these different innovations will all harmonize with one another. I have tried to make the case that they can very easily contradict each other and contradict the messages sent by a good teacher, however rare those individuals might be.

  60. Carl:

    Ironically, the Roman Mosaic can be superbly described in a simple and readable language, Logo, by my 10 year old cousin, and he will have learned a *lot* doing it.

    I must be missing the irony. I never said these tasks are impossible to perform in a digital medium. I said that computers aren’t the natural medium for these tasks. And I stand by that.

    A written medium:

    A square of side length 10.

    In Logo:

    Forward 10
    Right 90
    Forward 10
    Right 90
    Forward 10
    Right 90
    Forward 10

  61. ‘A square of sidelength 10’ written vs logo

    Funny, I think I would use the same example (and did use a similar one above #60) to argue that computers were the more natural medium!

    I am now asking myself what my students would say! I think I know the answer. As you have said already, it comes down to the different perceptions/definitions of mathematics there are and how they work together! Again, thanks for making me think.

  62. “I have tried to make the case that they can very easily contradict each other and contradict the messages sent by a good teacher, however rare those might be.”

    I hope my english is OK on that paragraph: Isnt it more about competency vs knowledge/skills than innovations vs teacher?

    Currently, a lot of the “innovation” that you see in the U.S. education are targeted at assessments and crowd control, but they could be targeted at the opposite.

    Sound like the inverse of this story: in Quebec, they shifted from skills to competency based curriculum 10 years ago. The story did not include computers, but book content and the book editors were your silicon valley guys, getting $500 millions (Quebec as 400k students). In fact skills assessment was now officially evil. No job loss involved and the government was pushing competencies instead of skills. The produced/approved books were not very good at competencies. Teachers and parents asked for evaluation of skills to get back.


  63. Dan: sorry if I trigger the too-many-comments today :-)

    “I said that computers aren’t the natural medium for these tasks. And I stand by that.”

    Student will expect to build their stuff on the computer. I’m not a Logo mega-fan, but I did assist to grade 5-8 classes using it. They got engaged in good math discussions, more quickly. More time trying, understanding and communicating, less time drawing, and this was similar to the current task.

    You feel it gets in the way of the students. I have seen the opposite, students getting interested by the power of the tool (a protractor is quite limited).

    -The way the question is worded, Logo will not be the exact final answer. To my defense, in my Logo, I simply type “square(10)” which is quite competitive to your sentence. I did create a square function, and this is now part of my tooling, *forever*. This scale to more exciting stuff, like a polygon becoming a circle. I’m also very offended that you choose copy-paste instead of a loop :-)

    -“They may then define, change, and distort the definition of the problem until it does require computers.” can be complemented by Conrad Wolfram: “The problem is not that computers might dumb it down, but that we have dumbed down problems right now”.

    “Common Core State Standards are the definition that trumps your or my preference for recursion…”. Shouldn’t all teachers extend it?

    Building your own computer simulations, your own *networked* math world, is an interesting communication medium that is not developped enough and that we should explore more.

    P.S. kids don’t phone London from Spain, they take a picture and facebook it. :-)

  64. @ Dan 66: “A square of side length 10…”

    I think this is a misleading example, at least in so far is it makes language seem “natural” for mathematics. The only reason we can communicate anything with this sentence is because “square” encapsulates our complete understanding of a concept. We are *used* to using language to represent concepts because we do it all the time, but this is hardly unique to mathematics.

    Consider: My geometry students “know” what a polygon is, but writing an accurate definition is a hilariously difficult exercise. With Sketchpad, we might “define” a polygon as anything I can draw with a certain tool. While language is surely an important medium for mathematics — perhaps the most important — it is *not* a natural one.

  65. I read your blog, Mr. Meyer, because I am constantly searching for the answer to the question, “What do we really want our kids to get from their mathematics education”. I’m still searching. The medium through which mathematics is delivered to the students is important, yes, and I have enjoyed reading the responses to your article. But, aside from the delivery, what are we trying to deliver? Do we really expect all students to become mathematicians? Or do we want them to be able to manipulate numbers in an effort to solve problems that are inherent in other fields of study? Is mathematics the means to an end, or is it in itself the final goal?

  66. Dan – the “answer” here is pretty straightforward: people in the Valley are looking to create and sell a product.

    That’s the primary goal.

    Any educational benefit is tangential. The marketers can fill in the gaps with pretty brochures/press releases/fanboy blog posts/actively policing social media.

  67. I’m a programmer and I’d still side with Dan that the Logo example is less naturally readable to a random student than “A square of side length 10.” (Except I’d probably have to verbally rephrase that to “a square with sides that are 10cm long” to at least one student.)

    I do think that teaching computing and mathematics together would be a fantastic thing to do, or at least to try. But it *will* mean teaching an additional skill set above and beyond the mathematics. I think my ideal at the high school level would be to convince admin to book a double-credit course that gives credit for both math and IT / Computer Science. (Not likely but it’s a nice dream.)

    (At a middle school level, there’s already more flexibility and less of a crushing load of curriculum to cover so it’s probably more likely to just make it happen as a teacher.)

  68. Silicon valley seems to be focusing on how to deliver content not so much on what content is delivered. As has been pointed out by others, KA doesn’t have a single staff member with K-12 teaching experience. KA is developing assessments and online mastery teaching approaches, but still only providing dry-erase board style videos. Perhaps this is only a chicken and egg issue. Maybe with great delivery comes great content. But, it seems to me that KA is ignoring one whole side of the equation.

  69. Corey:

    As has been pointed out by others, KA doesn’t have a single staff member with K-12 teaching experience.

    Judging from its team page, Khan Academy has two former teachers out of twenty-one employees: a former Peace Corps math teacher and a former Teach for America member. (That page doesn’t include newcomers like Brit Cruise or Vi Hart, though, so who knows.) I think it’s fair to say that their talent portfolio values management, business, and engineering over teaching but it isn’t fair to say they don’t have a single staff member with K-12 teaching experience.

  70. Dan:

    >>(That page doesn’t include newcomers like Brit Cruise or Vi >>Hart, though, so who knows.)

    I can’t believe I had missed that addition! I stand corrected. That’s a good sign. I look forward to seeing what cross pollination might occur.

  71. A number of people have mentioned, and I think it’s central to all this…teaching tools are just that…tools. They don’t work all the time and they don’t work for all students. In defense of Khan Academy, it has successfully exercised the mechanical skills of mathematics. Putting those into proper context, whether physics, music, etc. is another thing.

    I suppose you can call KA a Silicon Valley operation, but Khan doesn’t come from the valley. He started a system that has grown a certain way because of his experiences, and it’s gotten backing from the Valley. It is a work in progress and like many other “systems” out there, it should evolve in its approach, and be adopted into larger methods of teaching. By no means should these replace teachers or class rooms entirely…that was never the intent.

  72. Roymond:

    A number of people have mentioned, and I think it’s central to all this…teaching tools are just that…tools.

    This is the attitude towards math education that prevails in the Valley and I’m explicitly disputing that here. The limitations of every tool constrain and distort the definition of mathematics. It is demonstrably untrue that all of those definitions complement one another.

  73. Ref. Jacob Klein’s comment above:

    My point isn’t that any Silicon Valley company has claimed it represents the complete definition of mathematics. I’m saying Silicon Valley companies are largely oblivious to the existence of different definitions of mathematics, or the fact that some of the different definitions contradict one another.

    I’m also not making a point here about basic skills v. conceptual understanding. They’re both important and I think the divide between them is over-emphasized. (ie. students should learn fast, skillful methods for adding two-digit numbers and they should understand those methods conceptually.)

    So basic skills are important. But the medium one company uses to assess fractions (let’s say) and the medium another company uses to assess fractions are different.

    One gives a fraction and asks the student to select the correct representation of that fraction from four circle graphs. It’s multiple choice. If the student’s answer is right, she moves on to a similar question. If it’s wrong, her answer is removed and she selects a different answer until there is one remaining and she selects it.

    The other company gives a fraction and asks the student to aim a bouncing ball at the continuous interval from zero to one. If she misses, an arrow indicates the direction of the correct answer, to the left or right. If she misses again, the interval is divided up into sub-intervals that match the denominator of the fraction. If she misses again, those sub-intervals are labeled. Et cetera.

    The point of this post is that both of those companies send different messages to students through their assessment media about what math is and how students can be successful at it. I hope you’ll agree, having spent a lot of time thinking through one of those assessment media, that those messages aren’t complementary, as some have hopefully suggested here in the comments. Silicon Valley, in general, calls both of those assessments “math,” without much distinction.

  74. As I see it, the real category killer in math is not going to be assessment, especially when states & school districts start running out of money and taxpayer-parents cry fowl on over-testing. We can assess until the cows come home, but the reality is that the assessments, however creatively designed, will continue to tell us that our kids ain’t got no number sense, and they can’t reason their way out of a (mathematical) paper bag with a pair of scissors. The assessments are the metric, but the real challenge is figuring out how to economically retrain the existing cadre of K-12 teachers in HOW to teach math, with the goal of creating students with the an effective overlap of number sense, computational skills, and problem-solving strategies that allow them to tackle non-standard problems. Only then will we see a plurality of kids who can perform consistently well on assessments, especially those involving non-standard/ real-world/ out-of-the-box questions. We don’t need more assessments to tell us the kids or the system are broken.

    As I see it, the huge stumbling block is that almost all American teachers, especially at the elementary levels, are themselves “textbook-taught”, and almost all existing math textbooks over-emphasis algorithms and abstraction at the expense of concrete models that facilitate conceptual understanding. The teachers who were “good” at math as students often have difficulty explaining how to do something to a child who doesn’t get it because they don’t actually own the concept themselves, although they know when and how to apply a specific algorithm (such as dividing two fractions). The teachers who weren’t “good” at math (and one of the huge embarrassments for me as an American is the social willingness on the part of parents, teachers & the general public to wear their innumeracy as a badge of honor rather than shame) understandably reach for a textbook to guide them when they struggle to present a concept or algorithm cogently to their students. Since the textbooks themselves are badly flawed, the end result is students with poor number sense and the ability to apply algorithms in only the very limited situations in which they have seen them demonstrated.

    So, is the solution fixing the content by making it interactive, modularized, and digitally delivered? That would be great, except the current crop of textbooks/ videos/ websites/ streamed lectures is, by-and-large, based on earlier paper textbooks, which were/are crap, continually repackaged for the latest math fad (don’t even get me started on when a certain large textbook publisher literally cut and pasted their algebra, geometry, and algebra 2 textbooks into a two-volume edition of Math A/ Math B for New York State’s hair-brained foray into a two-course high school math curriculum sequence, and sold them to a number of districts around the state, NYC being the largest, as the latest-and-greatest, when the districts ALREADY OWNED the algebra, geometry & algebra 2 texts). I have a copy of a three-year high school math textbook from 1939, and the only significant differences in content & presentation are dumbed-down/ less-rigorous language, the addition of graphing-calculator-driven modeling, the removal of log & trig tables, and an explosion of color, poor layout, and extraneous & tenuous connections to pseudo-real-world applications that, in the architecture world, is referred to as “treeing it up” (i.e. the design sucks & we hate it, but if we put in more shrubbery, the client will think it’s pretty and buy it). The only commercially available materials that come close to introducing math in a cohesive and accessible subject at the elementary levels are English versions of the Singapore national math curriculum, but even these materials are flawed when used without an understanding of their concrete-pictorial-abstract (CPA) pedagogical underpinnings, and can be extremely difficult to integrate with existing state math standards/ curricula. While Khan Academy does feature some videos based on CPA modeling, the bulk of their content and explanations reflects the tried and tired textbook approach.

    The common core is a good start on content changes in math, since it streamlines and deepens the notoriously “mile-wide, inch-deep” American curriculum. However, without a complete overhaul of HOW we teach math, starting with kindergarten and moving up from there, the common core is doomed to failure. And, while the battle rages over assessments, those of us in the trenches would love if some of the edutech entrepreneurs out there would stop drinking the assessment/ content-delivery kool-aid, step up and recast the battle in terms of cohesive and effective instruction, and start building/ creating tools we can use for TEACHING, rather than TESTING.

    Side note — the discussion about a “square with sides of length 10” being better/ worse than a Logo program to produce said square strikes me as a false dichotomy: They are two different ways of looking at the same thing, with the emphasis on the first being brevity, and the second serving as an illustration of the attributes & definition of a square (four sides of equal length, four right angles.) And one could consider a third approach, that of an image of a quadrilateral with geometric notation indicating four congruent sides of length 10, and four right angles. What I’d love is some software that’s versatile enough to elicit all three of these of these views (and more) from a group of students, and then facilitate a discussion among them regarding the benefits and drawbacks of each view, and then prompt them with real-world, non-standard problems that hinge on an understanding of squareness, and… Oh, wait, maybe what I actually want is a real live well-trained teacher…

  75. Holly, I’ll just quickly say that I’ve seen the solution to what you’re talking about, and it’s pretty simple: teacher training that actually *demonstrates* a great math class rather than just talking about it.

    I’ve seen this in my own training, and I’ve seen examples of it come up at workshops at least once. The best way to prove to people that there’s more than dull rote learning out there is to get them experiencing it in the role of student first.

  76. Although I’m late to reply, here is my good example of using technology.
    Using the computer program called “Graphing Calculator” by Pacific Tech helped us see how our students were defining and using functions. To get the computer to actually work the mathematics had to be defined well. There was actually a point to being careful about typically sloppy uses of functions and notation. There was now a point to function notation. “Graphing Calculator” allows students to use function notation because it doesn’t just have a y= button and unearthed a whole host of issues of understanding of this notation. The least of which involved students defining things like f(x) = 2w+3.

    For details see a paper I helped work on:

  77. I agree with what Dan says if the focus is on “instruction” in mathematics, but that is different from providing places for students to play with, explore, and learn mathematics. With this in mind I disagree with the claim that “computers are not a natural working medium for mathematics”. Computers and in particular the Internet can support such exploration environments. The key is the provision of a communication system that allows participants to talk about their emerging conceptions and collectively construct understanding.

    For the past 10 years I have been working on such online environments for mathematics collaboration and they do support the development of students’ understanding. See Math-Towers (www.math-towers.ca) for an example aimed at grades 6 to 10. More recently I and a teacher colleague have been working on a system that combines a wiki (PBworks), GeoGebra, and Jing videos to support student sharing and collaborative problem solving. See the PBworks wiki “Collaborative Mathematics Spaces for Senior Secondary School and University” (http://collabmath.pbworks.com/) for more information. Research does show that these environments work, but under very definite conditions that require a committed classroom teacher who sees mathematics as a humanly constructed way of thinking.