Hibernating + Open Thread

A reader via email:

Blog monster hungry. Feed me.

I’m not big on the retroactively “sorry I haven’t been blogging” posts. I’d rather proactively explain why I’m not going to be around here for the next several months.

It isn’t for lack of interest in math education or for lack of interesting things happening in math education. For instance:

But my dissertation hearing is scheduled for mid May. I’m in the middle of data collection with lots of writing and analysis ahead and I’m sure I need to become a bit more ruthless in managing my time and writing.

So I’ll see you on Twitter (can’t quit that obviously), at NCTM and other conferences. But I won’t see you around here for a few months.

Please use this as an open thread to talk about whatever while I’m off dissertating. Also here’s all the great classroom action I haven’t written about over the last twelve months. Plenty of food for the blog monster there.

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. PS. Why blog about not blogging? Why not just not blog? Fair question. But this is an actual, no-fooling community for me, more real in many ways than in-person communities I’ve joined. And one unspoken rule of good community membership is you don’t disappear for several months without an explanation.

    Plus, in ways big and small — from letting me borrow your students for pilot studies to just arguing with me a lot around here — you’ve made the process of completing my doctorate so much easier. I’m leaving to go extend the work we’ve been doing around here for years and you guys should know that. This could have happened without you … probably … but I don’t like to think about it.

  2. Love this quote – “I’m leaving to go extend the work we’ve been doing around here for years and you guys should know that. This could have happened without you … probably … but I don’t like to think about it.”

    I met you in person last May in Fresno, and have seen a couple talks as well as followed you on twitter. I appreciate how you’re humble and always acknowledge that math teachers opening up their classrooms across geographies and cultures has made us all better teachers (and humans?) for it! It’s clear you aren’t getting a doctorate simply for more money or fame, but to better the science behind the success you’ve seen in engaging kids with three-act tasks and the like.

  3. This is a random question and not really related to the post, but does anybody have any ideas to make inequalities interesting. I was given the honor of teaching pre-algebra to freshmen this year, and let’s just say it has been slightly challenging. Any comments would be appreciated–even if it is advice on how to manage a troubled crew of freshmen boys!

  4. @Brock: I remember someone doing an awesome inequality lesson on the point spread in sports books, but I can’t for the life of me find it.

    For instance, in the upcoming Super Bowl, some places have the Seahawks are a 2.5 to win, which means if you bet on the Seahawks you only win if

    Seahawks Score > Patriots Score + 2.5

  5. I have often read this blog to inspire my teaching and have used many of the 3 act tasks but I am wrestling with a few problems and want/need some help.

    I love the idea of a hook to get the students wanting/needing a certain mathematical tool (such as factorising quadratics) and I like the way the problems get stripped back to allow for independent thinking but here is my problem:

    If I have not taught the skill (eg factorising quadratics) then the students can not do the question without me leading them through it. This means in the lesson they have only a) watched me use the skill and b) seen one example. At the end of the lesson most of the students will still not be able to do the skill by themselves. Not only that, but once the resolution of the hook has been satisfied then the students I most want to hook disengage again anyway.

    If I teach the skill first then I am missing the hook when it is most needed. I want the hook to hook them into needing the skill not a hook to just let them use the skill once.

    I can dress most skills up in fun activities (like factorising quadratics snakes and ladders game) but I want to foster more independent thinking and really hook the disengaged students.

    So my question is this: when and how should/could I be using these 3 act tasks within the time pressures of the UK national curriculum?

  6. @G I think the problem is selling only the hook as engaging. We have to sell the pure maths as well, which should not be hard given the ubiquity of maths in decent professions (let alone the sciences that make our lives safer, easier, more entertaining).

    If I were back in the classroom I would follow the hook with “OK, the related of is “XYZ”. Let’s grow our maths “fu” by mastering quadratics by Friday. Now moving on…”.

    The sell here is on the pride of mastering a mildly complex process that is key to one of the most important humans arts: maths. Looked at another way, we cannot for a second buy into “why do we need this?”. Maths in and of itself simply rocks, and we maths folks should not apologise for our enthusiasm: we are right!

    The beauty of a good blended tool is that the kids can pretty much solo on those, leaving class time for applications and other enticements/motivations. On-line tools are extremely efficient because they are checked as they happen — no overnight turnaround waiting to find out how we did.

  7. Good luck with the dissertation, Dan! (I had an evil childhood dentist named Dr. Dan, though, so be careful where you use the honorific.)

    On factorizing quadratics: I think it arises really naturally if kids have a lot of experience with (and appreciation for) distribution.

    If you’re really soul-comfortable with (a+b)(c+d); if you can draw it with rectangles, and explain it as a double-application of a(b+c) = ab + ac; if you’ve used it in lots of settings, pure and applied; then factorization of quadratics is just doing this backwards. It’s a totally natural next question to ask.

    Of course, the curricula I’ve worked with almost never push the distributive property hard enough, so kids arrive at factorization unready. That’s why it has that artificial medicine flavor for so many.

  8. @Brock @G
    Throwing a few ideas on the table — discussion welcomed.
    I see both questions as related. On the one hand we are seeking ways of making aspects of mathematics interesting. Typically these are algebraic in nature (I sense an abstract/concrete discussion in here). I found G’s question particularly important, as it’s something we math educators all struggle with: “If I have not taught the skill (eg factorising quadratics) then the students can not do…If I teach the skill first then I am missing the hook.”

    I think it’s worth thinking about what we mean by the students able to “do?”

    If it’s factoring, and it’s framed as “factor this polynomial” then the likelihood of the students being able to do this without prompting is fairly low. But what if it’s, for example, getting 2 students to sit in pairs — one student has a graphical representation of a parabola, and the other is not allowed to see it. Their task is basically Pictionary — getting the other one to draw a parabola that’s exactly the same. Then through this we get them to recognize some of the key features of the parabola that perhaps is helpful for getting started.

    We can also consider the fact that in a factored form, we have basically a multiplication between the two. Then maybe starting them with the idea of areas of rectangles and have them play with tiles or any other forms of representations. E.g. give them algebra tiles which represent some sort of quadratics, and they have to put the pieces in a rectangle. Then talk about what they’ve effectively done….etc

    What do we mean by whether students can or cannot do a question? If they can’t do it without acquiring some sort of skill from us first, then maybe it isn’t a great question. What if it’s a question they come up with on their own? If we go with @Desmo’s Penny circle, they are doing lots of wonderful things with quadratics. If we frame it differently, these would be questions that they would come up with on their own, which we can then direct and explore.

    Besides, what use is any of these “skills” anyway, if not an exercise on being able to think better and reason effectively?

    I don’t think there is an easy answer. Like I said, I’m just throwing on some ideas for discussion purposes.

  9. I find decomposition is too much of a memorized process without meaning. I do the following:

    2x^2 + 5x + 3

    (2x 1)
    (1x 3)

    1. Write potential factors of the constant term vertically.
    2. a) Diagonally multiply (and add the results)
    b) If the sum matches your middle term, Bingo.
    c) If the sum does not match, retry with different factors.

    The diagonal multiply step(2a) is the two “arrows” from the distribution process, which if explained, makes the process very logical and meaningful.

  10. There’s a difference between making a subject comprehensible and making it interesting, right? Instruction manuals are comprehensible, though not often interesting.

    Also, offering students a job later on in life doing a subject they hate now has never worked out all that great for me. That’s aside from the fact that they can turn around and say, totally accurately, “My mom has a great job and she never does this math at all.”

    Also, Jim Pai has me changing all my passwords since he basically called my dissertation study blind.

  11. Hey Dan, I wish you success on your dissertation! Let us know when we need to call you Dr. Meyer. :)

  12. And one unspoken rule of good community membership is you don’t disappear for several months without an explanation.

    Uh, I do believe you have provided an explanation, which is particularly convincing.

  13. Good luck, Dan!

    @G: This past fall, I used this activity (http://bit.ly/1uE6Vlf), based on a TI activity (http://bit.ly/1JvvYrN) to link linear factors to quadratic functions through graphical representations. It worked pretty well at getting my students to think deeply about the connections, and what the factors really tell us about the graphs. It also gave us a reason to represent quadratic functions in factored form.

  14. I spoke to a woman at a Chicago selective enrollment school who had recently emigrated to the US. She let one factor guide her search for a school to teach at when she landed in Chicago… how much time the schools curriculum devoted to quadratic functions.

    Paraphrasing … what percentage of mathematical models and meaningful functions are quadratic? And among those what percentage are factorable?

    There’s a fun mathematical question.

  15. Thank you all for such a great response. I do want to clarify my problem though:

    My problem stems from the when and how to implement 3 act tasks and not from teaching factorising quadratics.

    Most of the times I have used the three act tasks I used them before the students could do a skill – as a way of introducing the skill and creating the “need” for the skill (not real world need but “I must know the answer to this really interesting question” type need). But this resulted in me essentially doing the problem for the students. This meant in the entire lesson the students have seen only one implementation of the skill and they have not completed the skill unaided themselves, so when we come to the next lesson or for homework most students will still not be able to replicate the skill themselves.

    I have used the 3 act tasks after teaching the required skill and this seemed to work better in terms of more of the students could actually do the problem unaided. It was also a nice revision tool coupled with the independent thinking skills that these tasks foster but using them this way feels like I am missing a trick with the hook though. I want the hook to engage the students to want to learn the skill but if I have already taught them the skill then the hook is not being implemented when I would like it most.

    So my question is this: when and how do people implement these 3 act tasks for maximum learning benefit.

  16. Reply to G: I enjoy using the 3-act task precisely because it gets students involved in the lesson – act two is the teaching of the material, but the students are listening, responding, making connections (many times connections that are unexpected and creative!) and building their own knowledge. It’s messy and very high energy, the pain of learning is tangible, and the importance of not giving them too much information is an important part. You comment that you either do the problem for them or you teach the lesson. Stick that lesson in the middle! The third act is also important, when, just as you clean up your workspace, you clean up their ideas and sum up the findings, and allow your kids to “debrief”, to organize their thoughts.
    One thing I do try to do before using a three act is thinking about what my kids already know and what I am hoping they will observe. Sometimes they are not ready for the ideas in the 3-act, at least not what I wanted them to see or wonder about. I am prepared to assist them from whatever starting point they pick. That is the teachable moment – the second act… It’s shouldn’t be one or the other. Be careful not to show them; let them struggle. That’s the beauty and the value of the 3-act task.

  17. If no/limited direct application of quadratics justifies not teaching it, let’s try to remove all such topics from math teaching. Do we need to teach addition and subtraction when calculators are around?

    In my mind, knowledge of a topic/skill is also an enabler for learning other topics that depend on the first one in some way.

  18. The content standards (and in this case factoring and quadratics) are the vessel in which to serve the practice standards. It is our obligation in an algebra course to serve the practice standards in the context of factoring (among many other contexts). I viewed a video by James Tanton this month that is at least tangential to this discussion. It is worth a look.

  19. (responding to Pawan Kumar): I don’t know if anyone is advocating against teaching quadratics at all (although personally, if I were designing US math standards, I’d strongly consider putting more about exponentials into the required-for-every-HS category and much of quadratics into pre-calculus).

    I believe the objection was to teaching methods of finding factors of quadratics that are separate from using the quadratic formula. For instance, the Common Core algebra standards include: “Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, completing the square, the quadratic formula and factoring.” That’s an example of how in the US, we talk about “factoring a quadratic” as being distinct from “find the factors of a quadratic” (which could be covered using the QF).

    I don’t teach high school algebra any more, but in my experience of it, we spent several weeks total on factoring quadratics; at the time, our state standards also insisted on doing it for equations whose x^2 coefficient was >1. Admittedly a lot of the time spent reinforced (or retaught) factors of numbers, +/- integers, etc., and factoring can also be a fun puzzle. But it drove me insane to see what a barrier this topic was, considering how seldom quadratics in the physical world are so easily factorable.

    If the quadratic formula can solve any problem that factoring can, and if a student can demonstrate knowledge of what the roots of a polynomial are, how to get them, and what they mean in the context of a problem, why do they need to know how to factor? In particular, is this really something we should let high school graduation hinge upon? Because right now it is a big topic in required high school algebra classes, and it takes a ton of time because students don’t get it, partly because they don’t understand why they need to know it.

    (The same arguments may apply to completing the square, but I haven’t thought that one out so well.)

  20. To be honest, *I’m* not sure anymore why we factor quadratics.

    There are plenty of countries that don’t even bother.

    I’d like to hear one too.

    Julie our state standards also insisted on doing it for equations whose x^2 coefficient was >1

    Well, in New Zealand this is not a problem, as we allow graphics calculators at this point.

    The skill then passes from factorising with the leading coefficient not one, which is just an application of a formula, to the much more important explanation of the solution.

    Knowing how to form, and when to factorise, quadratics is vital. Testing on plugging numbers into the quadratic formula is not.

    Usefully the questions stop being quite so obviously set up to work, as you can ask them to solve 1.2x^2 + 8.9x = 4.2 no problem.

    The use of the quadratic formula itself is tested, but via non-numerical questions.

    Graphics calculators are the solution to your problem. You still have to teach how to factorise using them, but at least you don’t have to spend ages working through the quadratic formula each time. You just have to persuade old-fashioned teachers that using a calculator is not cheating.

  21. Chester, New Zealand seems to have a good balance. I like how the emphasis is on how students would actually apply it with technology they’ll all have, and how they need to understand what the solution means and how to get it.

    However, if I were teaching high school algebra here (which I am not), I’d think long and hard about implementing your solution, even though I’m not an “old-fashioned teacher”. I’d be knowingly exposing my students to the risk of doing poorly on Common Core test questions (the tests allow built-in calculators only, and provide them only for certain questions). Since I believe a certain Common Core test score will be required for the main path to high school graduation, this could reduce their chance of graduating. Also, I’d be going against what my district tells me to teach, and not in a minor way.

    (None of that stops me from saying “I personally think factoring quadratics is great fun, but why on earth are we requiring it of all high schoolers?” at every opportunity, but nobody who could do anything about it is actually listening to me, so it’s more of a hobby discussion topic than an effective protest. I do keep hoping I’ll hear a good counterargument, but I haven’t yet.)

  22. Julie,

    Thanks for clarifying. I can see the problem now.

    I am not a teacher by profession. However, I have a great interest in math and in teaching math. I have taught math to my kids (the oldest one is in 7th grade, doing high school math) and plan to continue doing so at least through their AP courses.

    Thankfully, I am not bound by such external constraints (which topic and standard is to be taught for how many weeks), and I sympathize with teachers for having to serve multiple masters with different agendas. I voraciously consume what great teachers have been sharing online in this context. I like to expose my kids to relevant topics and approaches irrespective of the standards and syllabi. I have taught her about factoring quadratics more as an example of the relation of roots to the linear factors (Fundamental Theorem of Algebra). I want her to be able to see connections across apparently different topics and be able to look at a topic from multiple perspectives. I am not going to leave out topics just because a layman is unlikely to use it.

    My kid’s middle school has very good ranking and kids there score very well on tests. However, I am disappointed to say that her math classes at school start with, “The formula for ___ is ___ ,” and ends with worksheet handouts to plug and chug. While it gutted me initially, it also inspired me to start a Math Meetup in my community.

    I am really grateful for all the great teachers sharing their experiences, struggles, and successes online. Thanks for all you do.

  23. Julie, I wasn’t getting at you. It was more an aside that you can teach harder quadratics without pain if your local area allows you to.

    If your exam doesn’t allow graphics calculators, then you have to teach without them. You’d be doing your children a disservice otherwise.

    I find it more than annoying that school administrators and politicians in much of the world talk up a storm about getting technology into the classroom and preparing students for the modern world. Then don’t allow them to use relatively cheap purpose-built devices that allow that.

    Instead try to shoe-horn in the likes of iPads, which are expensive and not designed with students in mind.

  24. Chester, agree 100%, and I read you as criticizing me or (most) other teachers, but rather offering a sensible solution.

    This is an area I’m disappointed our Common Core didn’t overhaul more. They put such an emphasis on modeling and technology (as a tool) in the Mathematical Practices, but then they require students to factor (unrealistically pretty) quadratics by hand as a content standard.

    Must be because they need that skill for the work world! (Yes, that was sarcasm.)

  25. Thanks Clara, it is great to have some info and tips for 3 act tasks.

    How long are your lessons? – I get 35 minutes and to fit everything in feels really rushed.

    I do give them time to puzzle out their ideas but I find only the top end students actually doing this and only the very top get remotely close to the skill I want them to master. The lower end usually end up getting an answer quite quickly based on numerous misconceptions.

    Once I have collected all the ideas, addressed as many misconceptions as I have time for and then built upon the ideas that were in the right direction I find the lesson has gone; I have now demonstrated the skill (but only once in a specific situation) and most of the students would not be able to answer a GCSE style question on it which ultimately they will need to do.

    In terms of learning outcomes for this specific lesson, particularly at the lower end, not many have been achieved and certainly not any measurable ones. I recognise that the more I do these types of tasks the better at independent thinking my students will get but at the moment I feel that after a full lesson most of my students will still not be able to do a particular skill. Is this usual? Or, at the end of your 3 act lessons, can most of your students do the particular skill you chose the 3 act task to bring in? Are you seeing a measurable raise in the attainment for formal tests in your class for all students? Are you seeing a dip initially?

    Thanks for all the help. I really want to develop the independent thinking in my students but at the moment it is coming at the cost of achievement in formal tests. Am I approaching things wrong? More help is very much appreciated.

  26. @G
    Oh man, 35 minutes?

    I would recommend breaking that up into multiple days then. You definitely don’t want to skimp out on the valuable engagement and commitment time where they practice posing questions and thinking about questions. I would say start with that for the entire first day, then follow up on the next days.

    Time is always a concern, but I would recommend threading together multiple aspects of your course with the chosen task for engagement. e.g. a video of skateboarder hopping off a ramp or something… would allow for questions that eventually explore trig, quadratics, linear, function characteristics… etc. depending on what you pull out of the questions. Build the entire first day on harnessing their curiosities, and then have the days after that focus on the different aspects of what you’d like them to focus on.

    With respect to “coming at the cost of achievement in formal tests” I really want to say – “then perhaps these formal tests aren’t doing what they should be doing.” However, I recognize that’s not a solution to an immediate issue which relate to high stake incidents that (at least partly) determine futures. So I recommend perhaps building “mini-related problems” into the body of your 3-act.

    e.g. they find the skateboarder’s peak height as he hops off – then draw some related problems from whatever formal tests you are looking at and have them explore that as well. I find “what if” situations pretty powerful as well. What if I throw a rotten egg at this skateboarder and it happens to hit him when he’s at the top? What if the ramp was tilted differently? … etc.

    That way you build on their independent thinking and link to these external summative pressures.

    I am unsure if that helped at all… let me know!

    • @G, I agree with jPai – my classes are 55 min, but I often take more than one day to let the kids work on their questions, and the what-if scenarios will really let you know who has the understanding. I use the 3-act tasks to prime the pump – my students are more engaged because the questions we are answering are the ones they ask!
      Then I can throw in some of the challenging test problems, and they start asking questions about those problems, which allows us to examine the curious and convoluted world of the “test question” and how it relates to the stuff we are learning!

  27. @G
    Maybe the answer is in letting your students “discover” the process for factoring quadratics. If you take them to the computer lab with http://systry.com/wp-content/uploads/2015/01/Algebra-Tiles-factoring-trinomials-ws.docx and ask them to use the tiles to make rectangles, in my experience those who don’t think in abstract terms are empowered by the visual, kinesthetic nature of the activity. They end up being the ones who show the abstract thinkers how to factor. An additional part of the hook can be how does what we discovered yesterday relate to what we’re being asked to do today?

  28. I don’t see how that discovery process helps Stacie.

    They put blocks together until they get a rectangle. Then they work out, if they are clever, that the factors are the middle term added and the end term multiplied each time. No reason for it, it just is each time.

    They could spend quite a while with your rectangles before they factorise x^2 – x – 72. Or worse yet x^2 – 64.

    Show them:
    (x + a)(x + b) = x^2 + (a + b)x + ab
    and they have a reason, as well as a method.

    Even provided that they “discover” using rectangles, they still end up using in any later exercises that you try to find the middle sum and end product. You might as well teach them that up front — showing why, of course — and with the spare period teach them something new on top.

    I also notice that you put “discover” in inverted commas, because they aren’t actually really discovering anything. Do you think they don’t realise that?

  29. @everyone: to answer a question from earlier, while I haven’t done a comprehensive survey (maybe if I do grad school one day I’ll have the time/resources for that) but it seems to be Eastern Europe-ish that just uses the quadratic formula. (I suspect it’s “all of the former Soviet Union”.) Poland and Russia most definitely.

  30. Chester, you may be completely right. This may not be the most effective method.

    A bit of background – I taught high school Algebra to classes that were composed of roughly 50% Freshmen. Accelerated students in my district took Algebra in 8th grade and a very small number took it in 7th grade. Many of my 10th 11th and 12th grade students were taking Algebra for the second time – a few were on their third attempt.

    I was taught Algebra in the typical abstract approach (x +a)(x + b) = x^2 + (a + b)x + ab so when we were given Algebra tiles in my methods class, I had a hard time understanding why I would want to use them. I added them to my bag of tricks any way trying to differentiate instruction, but wasn’t convinced they helped until one year when the teacher next to me was using them the day I needed the one classroom set we had. A few days into solving equations by addition, subtraction, and multiplication and division things weren’t going as well; students didn’t perform as well as past classes. They didn’t get it that when a constant went to the other side of the equation it “changed colors”. After that, I made electronic ones so I could take them to the computer lab if the others were in use. I actually preferred the electronic ones because there is less maintenance if you can find an available lab.

    When teaching the factoring of trinomials, I led my students through factoring a number like 12 using blocks to remind them of how many of them first learned to multiply in elementary school. Then I asked them to see if we could factor trinomials in the same way. What would it look like if a trinomial could be factored? Quickly they came up with the rule that it had to form a rectangle. Then it was off to the lab where I integrated some MS Word skills like pressing Ctrl while dragging to copy or Ctrl + D for duplicate. They worked alone or in pairs to complete the lab.

    We convened back in the classroom the next day and debriefed. Who successfully factored the first trinomial? Does everyone agree? How can we check? How about the second? Are we seeing any patterns? 7 and 8 are more challenging. Who can take a stab at them? What are zero pairs? Why are they necessary?

    Finally we culminated with “is using Algebra tiles the most efficient way to factor these?” The class agreed they needed to be able to factor Algebraically because we couldn’t run to the lab each time one of these problems came up.

    I used the quotes around discover in my previous post because, left up to their own devices, few would get there. The examples chosen, the questioning process that led them to where they needed to go were very much thought out in advance. . However, I found they were more likely to be successful on subsequent homework, quizzes and the test on the topics because 1. some were still concrete thinkers and struggling with the abstract, 2. some were visual, enjoyed the field trip, like technology, you name it, and 3. they had ownership in their learning. I didn’t just tell them the pattern. They had to work it out for themselves. Did they know they were led to the discovery? I’m sure they did. Did they feel manipulated? I don’t think so.

    So is this the best method? Probably not with regard to the time it takes to get them to the same set of knowledge and skill set. Are there better ways, I’m sure. It depends on your audience, your teaching style, the resources you have at your disposal, the personality of the class. All can vary from year to year. We may find that it’s all a moot point and that we no longer need to teach this skill and should only be teaching the quadratic formula or that quadratics occur so seldom we devote way to much time on them. The beauty of teaching is that it’s a blend of science and art.

  31. Stacie,

    Thanks for taking the time to reply so fully.

    I confess that one of my weak points is teaching repeat students. Just doing the same things again very rarely works.

    So trying a new method, just because it is new, is likely to be more effective — even if it is not the “best” method.

    Fortunately the NZ school system has done away with punishing students who are poor at Algebra with having to do it year after year once they get to the last three years of high school, so I face different problems from you.

  32. That identity for factoring comes down to solving this question repeatedly:

    Find two numbers that add to ___ and multiply to ___.

    Students can do a whole lot of those: they’re fun, small puzzles. Armed with addition and multiplication tables, solutions can be found empirically, and all the numbers that add to ___ are on the same diagonal so you don’t really need the addition table. This concept can lead to questions like “What is the maximum possible product when two real numbers add up to 12?” and the method of factoring or solving by completing the square.

    For example, x^2 + 20x + 120 is unfactorable, because 120 is too big. Most students’ reaction to this problem is breaking into pairs that multiply to 120, instead of breaking into pairs that add to 20.

    I know a lot of teachers who love algebra tiles, but I’m not a fan, they give the impression that “x” is a positive number greater than 1 (by the length and width of the tiles) and almost all of the invented rules fall apart past x^2 … I’ve never seen an algebra tile for x^3, nor xy, nor x/2. I’d love to see more use of the “zero pair” concept beyond tiles. Instead, usually what kids learn with algebra tiles has to be un-learned in later courses because the fake rules invented for working with algebra tiles do not work for all numbers.

  33. Bowen-

    I have incredible respect for you as a mathematician and a pinball player, but your critiques of algebra tiles is off the mark.

    They’re not magic. They’re not even a distinct technique. I first encountered algebra tiles as a new teacher, I was impressed by how they helped my students build a conceptual bridge between the precise “factor puzzle” you describe and this alien looking thing in their homework.

    Algebra tiles as a manipulative have many limitations, although I have owned crazy cumbersome sets that did have x^3 cubes, along with xy tiles, xy^2, x^2y. I’ve never seen one that represented fractional coefficients, but I think that’s a problem that applies to most physical manipulatives.

    Along with the bridge betwee integer factoring puzzles and factoring polynomials, algebra tiles helped justify my preference for using area-model/generic rectangle factoring diagrams. This wasn’t something I learned as a student, but as a scatterbrained teacher I learned to LOVE the extra structure and written material, both to diagnose student errors and to restart problems that I had wandered away from.

    I’m not sure what specific “invented rules” you have in mind, but I still find tiles to be a good tool that helps students to enjoy the puzzle of factoring.

  34. In reply to Algebra Tiles. There is no need for made up rules, but I have seen them. Algebra tiles probably needs to go back and look at the work of Zoltan Dienes to refresh back to their intended use.

    There are negative sides to the tiles, and x doesn’t need to be thought of as any number at all. Of course any model has limitations, but that’s something to discuss. Base 10 blocks have limitations too.

    I have, and you can get (or create) y tiles and xy tiles and y^2 tiles. I have seen x^3 “tiles” but they were very expensive and probably discontinued.

    The area model is just that – a model, and like any other, has its uses and limitations. I believe that when you see “completing the square” as needing to take a shape and make it into an actual square, that has a lot of power beyond some seemingly arbitrary rules.

    It is possible to put a “rectangle” together with impossible length/width, so that is an issue with tiles, but one should choose examples carefully to avoid such issues.

  35. There are some things I really love about algebra tiles as a representation. For x^2+5x+6 = (x+3)(x+2) there’s not much better.

    I feel there are a lot of places where it falls apart as a representation. You can use algebra tiles to solve one-variable equations … as long as the solution is an integer. 3x – 1 = 6 is messy. -3x – 1 = 6 is messier, because there’s a rule that says if you multiply or divide by a negative number, you have to flip the tiles. Using algebra tiles to solve one-variable equations means fewer equations with non-integer solutions.

    Factoring something like x^2 – x – 6 requires you to remember that -x and -6 are a different color than x^2, that red is the product of black and red, and that you can add or take away zero pairs of any of the parts. And, you never need a zero pair of the units or the x^2, only the x’s, which you have to learn by example. All of this is only really valuable in the short term, and you’re playing by the rules of algebra tiles, not the rules of algebra.

    I feel very positive about the area-model / generic rectangle factoring diagrams. When using those, you get the same outcomes as with algebra tiles, except the facts you need to remember are the real facts (red = red x black -> negative = negative x positive) and you can do things that go beyond x^2 or involve fractions / decimals / more variables without additional effort. There are other things that work really nicely with a generic picture, like x^2 – y^2 = (x+y)(x-y) through geometry, that don’t seem possible with algebra tiles. Perhaps they are, but it would be a stretch.

    Algebra tiles are a good representation in the early going, and matches the 1-10-100 blocks kids used in elementary. But the representation gets stretched to the breaking point, at which time students end up having to re-learn or un-learn some things that they thought were true based on their work in the tiles. Meanwhile there are other representations (the area model) that stay consistent and power through to other uses beyond factoring quadratics.

    All just my opinion, and I respect anyone who disagrees!

  36. What about building up from arithmetic instead of starting with the symbols?


    6 = 3 x 2
    12 = 4 x 3
    20 = 5 x 4
    30 = 6 x 5

    What would be the 100th number in this sequence?

    If I want to continue this pattern, its easier to do when I notice the pattern in the second set of numbers.

    Now we look at another pattern:

    20 = 5 x 4
    30 = 6 x 5
    42 = 7 x 6

    Oh wait, this is the pattern from above shifted! Finding the 100th number should be fairly easy.

    Here’s another pattern:

    21 = 5 x 4 + 1
    31 = 6 x 5 + 1
    43 = 7 x 6 + 1

    Hrmm. That’s the same pattern but shifted.

    My thinking here is that you can build generalizable rules from the arithmetic rather than directly from the algebra. Or even if you do go back to the algebraic representations to build the rules, it is easier to show students, using the numbers, that the relationship exists.

    Aside: This also means you end up with plenty of opportunities in continuing to develop their arithmetic and numeracy.

  37. I’ve found Algebra tiles to be invaluable for some students, and not that useful for others. For some kids they just make perfect sense out of concepts they would really struggle with otherwise. Henri Picciotto manages to solve all of the issues with negatives in an interesting if cumbersome method. It’s worth reading:


    You can get the lab gear set from Didax with x^3, xy, y^2 (along with x^2 and y^2x I think also: http://bit.ly/1zQIl0Y

    They’re expensive, but worth it I think

  38. Bowen,

    I completely agree with your assessment of algebra tiles. Algebra tiles, like most physical manipulatives, are limited in their ability to help students access mathematical ideas at a deep, conceptual level. As mathematical concepts are indeed abstract, students need to learn to reason in the abstractness of mathematics.

    This must start early in students’ schooling experience. However, in the early years, teachers seem committed to meeting students at a concrete level and maintaining students’ positioning there. Teachers inundate students with distorting and restrictive manipulatives and models along with algorithms that prove to be little more that calculation matrices. Students spend little time learning to conceptualize and conjecture; thus, they gain very little in the way of transferable, connective knowledge. Furthermore, there is no coherence in the presentation of mathematics in schools, merely a listing of objectives that are random notions of the discipline. The aim of school math is for students do math – apply it through extensive practice of methods that they do not understand – rather than discover the mathematics that underlie the methods.

    I contend that we respect the numbers and their relationships, and seek to determine the ideas that they adduce. The understandings that we seek are found within the mathematics itself; math is the context that we need to explore. Contextualizing mathematics as money or even games – which essentially the algebra tiles create – will perhaps further illuminate the concept of number in part, but mathematical concepts exists independent of these contexts are far greater.

    • I agree that math is abstract, brilliant, breathtaking stuff. I want my students to experience the “knowing” so they feel empowered by maths, not made to feel stupid. I teach 11th grade, so they are cognitively ready for abstraction.
      I’ve been following the discussion here about quadratics, including the thoughts on using algebra tiles. They are limited and very concrete. They have their place in a classroom, however. I also agree that number sense – the ability to use the properties of numbers and their relationships is a valuable skill-set/understanding, which is, thankfully, now being taught from elementary upwards.
      I wish to address the issue of concrete vs abstract. Many studies in early childhood and elementary education (Piaget, Vygotsky, Erickson, Skinner, Bruner, Bandura, Chomsky…. And those are just the famous ones!) agree that children are concrete thinkers and learners. Math must be presented in a concrete way, and yes, this presents a challenge “down the road” as the child begins to understand and adapt to the abstract. That is why we teach, to help students make that leap. So I come back around to the use of manipulatives. As an adult, I see their limitations because I am capable of abstract reasoning. As a teacher, even of 11th graders who should be there… I have to meet my children where they are. I use all manner of ways to teach them the ideas, and I then guide them using imagery and language to allow them to bridge to the abstract. I do not want to teach tricks, but if children use them because of previous lessons, I scaffold and then show the students why they work. I want to impress upon those that decry concrete lessons, that they are necessary – hand in hand with number sense – for children through grade six or seven or even eight (essentially middle school!) until they are mentally ready. I am concerned, due to the wide readership of this post, that some who do not understand the concrete period of childrens’ development may feel they have some new criticism to level at elementary education. I love the changes regarding number sense. I can hardly wait to see those children in my high school classes! Let’s keep setting the bar high, but let’s set the bar with a clear eye towards the developmental ages (not the calendar ages) of our children.

  39. Nice to hear someone speaking up for pure maths.

    I am reminded of a counter-example, an ostensibly CCSS-conforming bit of PD on teaching division of fractions with several different forms of graphical modelling.

    An opening slide denounced using the rule that division was the same as multiplying by the reciprocal, so just invert and multiply.

    The simple cases they demonstrated looked great when done with models. When they got to the harder cases like 4/9 divided by 6/7 even I had trouble keeping up, but on they ploughed as if the new sheriff in town was visual modelling.

    Meanwhile, division is indeed just syntactic sugar for multiplication by the reciprocal.

    To me, the true mathematician proves that a divided by b = a times 1/b and then uses that truth on examples simple and hairy.

    But I like different ways of proving, so I would do enough simple examples with models to prove the rule, then leverage my new rule on the harder cases, perhaps checking one tricky one with models just to drive home that maths really does work.

  40. I see two questions raised by Kevin Moore. Or at least there are two questions implicit in what he wrote.

    1. Should mathematics be in contexts?
    2. Should students understand the mathematics they use?

    There is some interesting evidence about the first question based on 100,000+ people’s performance on these three logic tasks: http://www.philosophyexperiments.com/wason/Default.aspx

    If you don’t have the patience to do the tasks yourself, which I recommend, you’ll find that people found the first two tasks much more difficult than the third. However all three tasks are logically equivalent; the only thing that is different is the context. People reading this blog are more likely than the average population to be successful on these tasks, but if you tried out the tasks, you probably found the first two tasks took more mental energy than the third.

    My suspicion is that this effect holds for more than just this one logic puzzle, it might even be true for all of mathematics.

    Here’s another (potential) example. Which of the following two problems is easier?

    1. 129 mod 24
    2. If it is noon now, what time will it be in 129 hours?

    My experience with teaching students is that they need opportunities to extend their knowledge from their existing knowledge. Yes, we want them to think abstractly and withholding opportunities for them to do this is not right. While it is true that younger children tend to think more concretely, they can think abstractly in areas where they have more knowledge. So let them have opportunities to think abstractly as they grow their knowledge, but balance this against the intellectual need for what they do to make sense.

    It is possible to misuse contexts. If the ideas never leave the contexts, they never gain the generalization and power of mathematical reasoning. So while contexts might be good places to introduce mathematical ideas, they are probably terrible places to leave them.

    As for the second question, I’m not really sure why we wouldn’t them to both be able to use the mathematics and to understand why it works? Mathematics is both a tool we use and a set of processes that we can extend. Both are important. If you only teach children a set of mathematical tools they can use to solve problems, then I feel like they are missing out on learning about the processes used to develop those tools. Why would we want to withhold opportunities to learn these?

  41. David Wees,

    Thanks for sharing that Wason selection tasks. I tried and got them all. I found them to be of equal difficulty, but that’s because I could abstract them all to the logical representation of A implies B.

    I understand that the level of difficulty doesn’t necessarily relate to the the time taken to solve that problem by a particular person (then it depends on the person’s experience, knowledge, ability, expectation, approach, etc.). During the last year’s MathCounts “show”, the top 10 kids from the state were answering long geometry questions before I could read them on the screen. Then, on a simple expression with fractions involving one subtraction and one multiplication, they stumbled 4 times before getting the right answer. I did it mentally in 5 seconds. It would be hard to believe if anyone would consider the fraction problem to be more difficult.

    Having said that, I spent the most time on the first selection task because I wanted to be sure I understood what was being asked and I could filter out the incorrect approaches.

    I immediately recognized the second one to be an equivalent task.

    On the third one, I spent just a little bit more time appreciating how it would make sense to most people even without a formal education in logic.

    I really enjoyed reading your whole comment. It inspired me to start thinking about the role of abstraction in making connections across contexts. I immediately recalled the connection between earthquakes and US presidential elections – see http://en.wikipedia.org/wiki/The_Keys_to_the_White_House

  42. I made their point perfectly: strained on first two but figured I had it and got them wrong, had no problem with the third because I spend a lot of time watching bouncers work and the context guided me. Nice!

  43. Norman Sorensen

    February 6, 2015 - 2:12 pm -

    Mathematics can be a very abstract subject to teach young people, even for adults. Approaching a concept from a concrete way makes it easy for many to grasp. However much thought has to be given to the approach being use. Consideration must be given as how long to stay on one approach before moving. It is just like Kenny Rodgers song, The Gambler, you have to know went to hold and know when to fold.

  44. Furthering David Wees’s point on development, it must be clear that to deem knowledge or skill as developmentally appropriate has a great deal more to do with an individual’s experiences than with his /her age.

    Taking a reformed view of Piaget’s theory on intellectual development, this theory can be viewed as having emerged with Piaget heeding an era-specific social and cultural context along with presumptions of limitations present at biological stages. Given his observations within the context and considering biological limitations, he believed that it was highly unlikely for children to be confronted with critical ideas and pivotal situations at certain ages. And should they be, children do not possess a broad enough schema to make sense of them. Children’s narrow schema, he believed, was simply due to their lack of experience (or even perhaps negative experiences). He surmised that children’s schema is only broadened by a process of assimilation and accommodation. Thus, it can be extrapolated from Piaget’s theory that experience then, and the quality thereof, is of utmost importance to children’s intellectual growth, as many and varied experiences offers opportunity for a broadened schema.

    Teachers are responsible for creating situations to engage children in ways that provide a strong foundation for their success as abstract thinkers, broadening their schemas. Children can develop deep, complete understandings of complex ideas, through highly structured, defining experiences. These experiences should provide an adequate base of knowledge upon which to build. That is, children should be able to internalize, categorize, and activate this knowledge to make sense of other novel situations. Once these experiences are had, children are capable of moving from the concrete to the abstract with relative ease. This attributes to the fact that some children enter school better prepared to access the content of higher-level texts and make critical connections that facilitate their understanding of ideas in complex subject matters. These children have simply had deliberately rich experiences that broadened their schema.

    In the case of mathematics, the knowledge that students gained through their experiences must be principled and transferable in order for it to advance them as mathematicians. Building math ideas around restricting and distorting models does not meet this end. Such practices present mathematics as situational, fragmented, disjointed, completely stripping it of its coherence. How can students be expected to make sense of something that is incoherent? How do students assimilate fragmented notions into their stocks of knowledge? How is it accommodated? If learning to think mathematically, which is reasoning with mathematical ideas, is not the focus of mathematics instruction, what then is the aim of teachers of school math?

  45. Kevin,

    I agree with all of your points, but I just want to point out that because we mediate our understanding of the world through language and therefore our understanding of the mathematics through language, as well as use our existing understanding of the world to interpret what we learn in it, necessarily it is the case that our internal models for understanding mathematics are not true copies of the mathematics but instead distorted as you say.

    However if I look at an object from a variety of different perspectives and learn synonyms that describe an idea and even have some analogies of the most important characteristics of an idea, then it is more likely that my internal model of the idea is more similar in nature to other people’s internal models.

  46. Indeed, we interpret ideas as best as we can, considering our understanding of language and our presumptions that those with whom we communicate understand and use language in similar fashion. Thus, we are always filtering information through our experiences and presumptions, hence the risk of distortion. Yet there is reflection, an attempt to mirror through language (symbolic or verbal) that which we believe is the intended message of something heard or observed by some other means. This allows for the minimization and/or clarity of distortion.

  47. I love the way you are trying to change math instruction from print to digital. By introducing students to a more digital/ visual form of math, it should be easier for students to comprehend math more efficiently. The fact that you are doing more research on the foundation of math instruction and the teaching of math is going to be so rewarding for students of the future. I enjoyed reading your post February 10, 2015 at 11:20 p.m.

  48. These children have simply had deliberately rich experiences that broadened their schema.

    Or are born cleverer.

    I have to say that I agree with almost none of your analysis Kevin.

    I believe (as Piaget did before you “reformed” him to say the opposite) that children cannot think abstractly before they have the appropriate mental equipment. Merely giving more and better training will not improve a skill that is not present.

    Although we are no longer really allowed to say these things, everyone knows that some people are born with almost no ability at all for abstract thought. Others get it only very slowly, and even as adults struggle with abstract concepts. Whereas some children merely have to see a concept once and it is embedded.

    Teaching to the strugglers can be rewarding, and I often enjoy teaching them Maths — slowly. They can enjoy Maths, provided it is pitched at their level. But I object strongly to being told that it is my failure as a teacher that prevents them from being able to see abstract patterns like more gifted can.

    Your analysis effectively causes the blame for lack of progress in Maths to be laid completely at the feet of the teachers — since they have failed to provide the proper instruction. Politicians love this concept, because it allows them to brow-beat teachers for “failure”, when in reality there has been no failure.

    We can seek to improve Maths teaching, and I imagine everyone here wants to, but we can’t improve the material we work with — people. No amount of wishing children were blank slates, onto which we can write at will, makes it so.

  49. I do not only speak from a theoretical viewpoint regarding young children’s ability to learn abstract ideas, but I also speak as a practitioner. I have spent a number of years teaching young children mathematics; thus, I am not a novice in this regard. And in my experience, I have found that children – whether ill-prepared, struggling, lacking confidence, and/or unmotivated – are generally quite capable of understanding mathematics deeply.

    Nothing about math is concrete. It is an abstract subject matter. Thus, the teaching of mathematics can only be for the purpose of children apprehending that which is abstract. As teachers, we, then, must provide young students with experiences that lead them to abstract: use that which they know or observed in a particular instance and transfer or extend that knowledge to gain greater understanding of the idea or a new one. Certainly, physical and visual aids are part of the experience because of students’ narrow schema. Yet our aim with such aids is to teach students to think, teach them how to abstract.

    Again, I do not merely present this as a topic of debate, but as a sharing of my reality. I teach young children mathematics as interrelated concepts. My students are flexible in their thinking of numbers and operations, demonstrating reasoning by challenging me and each other, and confident in using mathematics vocabulary to present their ideas. This success has been due in large part to my deliberate effort to have students understand that mathematics is a study of ideas, calculated exploration of the ideas, and continuous encouragement to make connections between ideas.

    I am definitely not the first to take a more reformed view of Piaget’s ideas. His theory of cognitive development has been strongly contested since the 1960’s, and based on my experiences as a practitioner, I have come to understand the reason for reform of his ideas.

  50. I do not only speak from a theoretical viewpoint regarding young children’s ability to learn abstract ideas, but I also speak as a practitioner. I have spent a number of years teaching young children mathematics; thus, I am not a novice in this regard.

    I’d guess most of us here have taught Mathematics for a while. I teach older children, where the limits of their native ability become much more obvious, but it reaches back.

    And in my experience, I have found that children — whether ill-prepared, struggling, lacking confidence, and/or unmotivated — are generally quite capable of understanding mathematics deeply.

    I reckon I can teach almost any child almost any Maths skill. But getting them to understand it, no way!

    I’ve seen it in myself. I cannot follow post-graduate level Mathematics. It’s not that I’m stupid. Nor ill-motivated. Nor badly taught in the basics. But at a certain point I cannot follow sufficiently difficult Mathematics.

    For other people their level is lower. Much lower in many cases.

    I am definitely not the first to take a more reformed view of Piaget’s ideas. His theory of cognitive development has been strongly contested since the 1960’s,

    So about the time when the obvious — that some people are not very clever, by the misfortunes of genetics — became the unsayable?

  51. It appears that we are unable to divorce ourselves from the experiences that have shaped our beliefs.

    I contend that the reason that I cannot presently ascertain many advanced mathematical concepts, or medical, linguistic, psychological, biological, or even theological concepts, at a level of scholarship is due to: (1) my lack of interest at that level and (2) not having had a wealth of specific kinds of experiences – reading of certain literature, discussions, practicums, tutorials, research opportunities, lectures, etc. – that scholars or prospective scholars have had which allow them to access the information at such high levels. I am simply limited due my lack of interest and experiences.

    The reason that I decided to teach younger students was due to my experiences teaching older students. I recognized that they had not had certain kinds of experiences in mathematics in their most formative years, which was causing them to have difficulty understanding certain mathematical concepts in their adolescent years. I was spending a great deal of time helping adolescent students gain a more complete understanding of basic mathematical concepts in order to provide them a foundation upon which to build. Thus committing to work with young children, I was ensuring that students would not continue to move through their schooling experience with gaps in their understanding or simply being skilled with using standard algorithms. In doing so, I found it more valuable to teach mathematics as concepts from which there emerged an investment by students toward their development of skills that allow them to convey their thinking.

    My experiences will not allow me to view intellect as fixed nor am I able to approach my work with students from a deficit thinking perspective. For what then is the purpose of teaching?

  52. Instead of algebra tiles for factoring quadratics, consider an arithmetic approach. Begin with the understanding that every whole number is a unit of itself, composed of a specific number of ones. As such, each number can be viewed as a specific array of units of one, revealing its dimensions, it factors. This supports the idea of the set of square numbers. And although there are whole numbers that are square numbers, not all whole numbers are squares. Yet all whole numbers can be compared to square numbers to better make sense of their dimensions.

    Six, for example, being a composite of 6 units of one is also a single unit of 6^1 = 6 = 6 ● 1. Understanding the numbers as possessing dimension and being able to reason proportionally, more precisely inversely so, creates arrays of equal units of one and varying dimensions. Thus, 6 ● 1 = 1[6(1)] = (⅙)[6●1](6) = [(⅙)6][1(6)] = 1 ● 6, and:

    6 ● 1 = 1[6(1)] = 2 (½)[6 ● 1] = [½(6)][1 ● 2] = 3 ● 2
    6 ● 1 = 1[6(1)] = 3 (⅓)[6 ● 1] = [⅓(6)][1 ● 3] = 2 ● 3

    However, when comparing six to 4, a square number,
    4 (¼)[6 ● 1] = [¼(6)][1 ● 4] = 1 ½ ● 4,
    six can be understood as 1(4) + ½(4) = 4(1 + ½) = ½(4 (1 + ½))2 = [½(4)][(1+ ½)2] = 2(2 + 1).

    These are all important understandings that would aid students in factoring algebraic polynomials and especially quadratics. For example:

    x^2 + 7x + 12; let x = 5 → (5)^2 + 7(5) + 12 = 25 + 35 + 12 = 72

    Eight of nine is the expression of 72 that compromises whole numbers and presents 72 as close as it can be to a square. When 8(9) is compared with a square of 25 units, as 5 was substituted for x, is (5 + 3)(5 + 4) — the length of 5^2 is extended by three units and the width by 4 units. Thus, the factors/dimensions of x^2 + 7x + 12 are (x + 3)(x + 4).

    This also works with negative integers:
    x^2 + 7x + 12; let x = -2 → (-2)^2 + 7(-2) + 12 = 4 + -14 + 12 = 2; thus (-2 + 3)(-2 + 4) or (x + 3)(x + 4).

    Another example:
    2x^2 + 11x + 15; let x = 2 → 2(3)^2 + 11(3) + 15 = 18 +33 +15 = 66

    Eleven of six is the expression of 66 that compromises whole numbers and presents 66 as close as it can be to a square. When 11(6) is compared with a square of 9 units, as 3 was substituted for x, is (2(3) + 5)(3 + 3) — 3^2 is doubled and the length extended by 5 units and the width extended by 3 units. Thus, the factors/dimension of x^2 + 11x + 15 are (2x + 5)(x + 3).

  53. While on my way to work one morning, a sight caught my eye. A bicyclist was on a trail with a flashing light on the spoke of one of his tires. I wanted to pull over and film him/her but decided not to cause an accident knowing my camera phone wouldn’t do justice at that distance anyway. Every day since I have watched for that rider with my DSLR and telephoto lens on the seat beside me, ready to pull into a parking lot should the weather be promising enough and I get lucky enough to time it right. With trepidation I watched the sun rise earlier and earlier but the temps weren’t budging, keeping it too cool to encourage riders. Then the time changed and I dared to hope again.

    My patience wore thin so I finally decided to take matters in my own hands and bought some LED lights. I enlisted my son since helping him with a pre-calculus angular velocity problem was what inspired me to try to film this in the first place. The result is a derivative of Dan’s “Three-Act Math Tasks” style questions.
    Angular Velocity to Linear Velocity – http://systry.com/angular-velocity-to-linear-velocity/ Suggestions for improvement are welcome and encouraged.

  54. I love the video. I must admit it made my head spin a bit – there’s so much going on and so many questions it could spur. What software do you use to transpose the dots onto the video?

  55. Sean Ashburner

    April 11, 2015 - 7:32 am -

    Hey Dan.. Looking forward to seeing you at NCTM. Do you think you’ll have time to make your top picks for this year’s conference?