Your GPS Is Making You Dumber, and What That Means for Teaching

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Ann Shannon asks teachers to avoid “GPS-ing” their students:

When I talk about GPSing students in a mathematics class I am describing our tendency to tell students—step-by-step—how to arrive at the answer to a mathematics problem, just as a GPS device in a car tells us – step-by-step – how to arrive at some destination.

Shannon writes that when she used her GPS, “I usually arrived at my destination having learned little about my journey and with no overview of my entire route.”

True to the contested nature of education, we will now turn to someone who advocates exactly the opposite. Greg Ashman recommends novices learn new ideas and skills through explicit instruction, one facet of which is step-by-step worked examples. Ashman took up the GPS metaphor recently. He used his satellite navigation system in new environs and found himself able to re-create his route later without difficulty.

What can we do here? Shannon argues from intuition. Ashman’s study lacks a certain rigor. Luckily, researchers have actually studied what people learn and don’t learn when they use their GPS!

In a 2006 study, researchers compared two kinds of navigation. One set of participants used traditional, step-by-step GPS navigation to travel between two points in a zoo. Another group had to construct their route between those points using a map and then travel segments of that route from memory.

Afterwards, the researchers assessed the route knowledge and survey knowledge of their participants. Route knowledge helps people navigate between landmarks directly. Survey knowledge helps people understand spatial relationships between those landmarks and plan new routes. At the end of the study, the researchers found that map users had better survey knowledge than GPS users, which you might have expected, but map users outperformed the GPS users on measures of route knowledge as well.

So your GPS does an excellent job transporting you efficiently from one point to another, but a poor job helping you acquire the survey knowledge to understand the terrain and adapt to changes.

Similarly, our step-by-step instructions do an excellent job transporting students efficiently from a question to its answer, but a poor job helping them acquire the domain knowledge to understand the deep structure in a problem set and adapt old methods to new questions.

I’ll take that trade with my GPS, especially on a dull route that I travel infrequently, but that isn’t a good trade in the classroom.

The researchers explain their results from the perspective of active learning, arguing that travelers need to do something effortful and difficult while they learn in order to remember both route and survey knowledge. Designing learning for the right kind of effort and difficulty is one of the most interesting tasks in curriculum design. Too much effort and difficulty and you’ll see our travelers try to navigate a route without a GPS or a map. While blindfolded. But the GPS offers too little difficulty, with negative consequences for drivers and even worse ones for students.

2016 Jun 17. The two most common critiques of this post have been, one, that I have undervalued step-by-step instructions in math, and two, that this GPS study offers very few insights into math education. I respond to both critiques in this comment.

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

49 Comments

  1. Unfortunately, you can’t jump from:

    “So your GPS does an excellent job transporting you efficiently from one point to another, but a poor job helping you acquire the survey knowledge to understand the terrain and adapt to changes.”

    To:

    “Similarly, our step-by-step instructions do an excellent job transporting students efficiently from a question to its answer, but a poor job helping them acquire the domain knowledge to understand the deep structure in a problem set and adapt old methods to new questions.”

    It lacks a certain rigour.

  2. Certainly not without the supporting framework here:

    The researchers explain their results from the perspective of active learning, arguing that travelers need to do something effortful and difficult while they learn in order to remember both route and survey knowledge.

    I’m certainly interested in counterclaims that there’s nothing for teachers to learn from the satellite navigation studies.

  3. To stretch the analogy a bit further, when I get to a new city, one of my favorite things is to take off on foot and wander. I feel like I get a good feel to the layout, and in the future when I actually need to get somewhere, I’m more confident and efficient.
    I think the analogy holds when it comes to learning difficult math. Math play often can lead to deeper math learning. At least it does for me.

  4. Thinking a bit about this metaphor (or whatever it is we’re doing here) makes me think that learning a mathematical idea or skill is much more than knowing the route from one place to the next. It’s knowing your way around a neighborhood — you know good ways to get to the most important places, and you’re never really lost.

    I live in Washington Heights, NYC. Here are some ways that we might teach people to get around Washington Heights.

    1. Give ’em GPSs and spend a few days just following directions to go place to place.

    2. Same thing, but they have to go there with the GPS and go back without it.

    3. Give ’em a map and spend a few days asking them to go from place to place on it.

    4. Same thing, but there are five important trips that you use a GPS for.

    There are certainly people who would find each of these reasonable, but the reasonablest among us wouldn’t dream of doing 1 or 3.

    Where does that leave us? I wish these debates happened at the level of units of instruction instead of individual learning activities.

  5. Dan

    You will be aware that my case for explicit instruction is not based upon an argument about GPS. This wasn’t the context of my blog post about GPS which was a reflection on how people use this as an example of feedback.

    I don’t see how GPS tells us much about maths procedures. This is your non sequitur. You might recall an email exchange that we once had. I presented evidence from the kinds of trials run by Sweller on the worked example effect in algebra. You wrote that such studies were not ‘ecologically valid’ and would not necessarily replicate in maths classrooms. And yet here you are drawing from an entirely different domain and extrapolating freely.

    Have you changed your views on this?

    If you want to read something more relevant to the interplay between mathematics procedures and understanding then I suggest H Wu’s piece for AFT, Jon Star’s research on procedures or the evidence quoted in my ebook. ;-)

    Greg

  6. I have completely lost interest in this debate. Good teaching is not possible in a 100% direct instruction mode, or in a 100% discovery mode. Good teaching requires us to figure out how to combine the two given the complexities of different topics, different groups of students, different availability of tools and materials, and so on. See my article titled “Nothing Works”. http://www.mathedpage.org/teaching/nothing.html

  7. Henri, I agree.

    I find the GPS metaphor to be silly with regards to math and explicit instruction.

    Explicit instruction is not giving the student the answer which the GPS is essentially doing for a driver.

  8. I see this as a sort of existence proof. There exist situations where following step-by-step directions leads to less learning than (active learning/other strategies).

    I’m interested in whether we can specify the conditions under which this is true, and the situations under which it is not.

  9. The heated debate on this topic all stems from people who want to improve the learning of our pupils.

    Does anyone have some good ideas on how to encourage active learning so pupils can answer 5x^2-4x-2=0 successfully?

  10. This is a false dichotomy in my opinion. You needn’t do one or the other. You are free to do both for almost any topic. Further, within your own classroom, I would hazard that there are some students who would benefit from starting with GPS-like direct instruction followed up with exploration while others would benefit from wandering about first then explicit instruction leading to an algorithm.

    In my instruction, I prefer to let them wander a bit first then producing an algorithm together. I don’t see it as an either or proposition.

  11. Michael Paul Goldenberg

    June 17, 2016 - 6:28 am -

    https://youtu.be/76RWWl01JMc?t=2m18s

    REPO MAN, like the lyrics of Bob Dylan, contains the answer to every question ever posed and every question that will be posed in the future, thanks to flying saucers, which are really – yeah, you got it! – time machines.

  12. Paul Hartzer

    June 17, 2016 - 6:44 am -

    The first time I go somewhere I’ve never been before, I use the GPS app on my phone. If it’s somewhere I expect to go again, I try to pay attention to details around me. The second time I go, I might tell my phone where I’m going so it can help me out, but I don’t pay as much attention. The third time I go, I’ve learned the route and have started to pay attention to alternate choices. I currently know a good half dozen “basic” ways of how to get from home to work, in case there’s a traffic back-up.

    There’s a sandwich shop around the corner from the high school I teach at that all the students know how to get to. I have used that to explain different paths to the same goal, and students have a better understanding of what it means when the math teacher says that what matters most is a consistent approach to a problem, as long as the solution is “correct”, not always following the same steps.

    Point being: I think it’s important to show students a clear route, but it’s equally (if not more) important to reinforce to students repeatedly that most math puzzles have multiple routes. The GPS model of, say, WolframAlpha and similar sites reinforces the One True Way (although WA does provide alternate representations of values by default).

  13. “Designing learning for the right kind of effort and difficulty is one of the most interesting tasks in curriculum design.”

    Powerful learning is only possible when students are truly interested. Interest should be the foundation on which curriculums are written.

  14. Coming from a perspective of inquiry or bust, I think I am now interested in a broader, more inclusive view of instructional approaches.

    Some people may never feel the general confidence to wander around a new city without a map or any support at all. Those people may benefit from a GPS and I would not advocate for that to not be an option for them. I would hope to work with them to eventually not need it, but that would take some practice and gradual release. Conversely, there are some people, like Dan Anderson, who benefit from the freedom of discovery and inquiry to develop their own road map and understandings of new surroundings.

    In education, I think every student, teacher, and class is different (to butcher another metaphor, unique beautiful snowflakes in the blizzard of life), thus shouldn’t ALL pedagogical strategies be available to all students and teachers at all times?

    Need step-by-step explicit instruction to graph linear functions? We got you, here’s a worked example.

    Need the freedom of inquiry to develop your own conceptual understanding of variables? Here’s a balance scale with different shapes on it, lets figure out their values!

    I guess what I’m asking is, are these concepts mutually exclusive or can they both exist in the same classroom based on the needs of the learners within it?

  15. If you teach math in the same way a GPS gives you directions, and that is the only way you ever teach, then you suck at teaching. Even the kid most desirous of spoon-feeding and “Just give me a worksheet” will eventually rebel against your class.

    I think this analogy is badly flawed, though, because I don’t think any teacher teaches this way and it’s absolutely NOT “direct instruction”. I think we all hate “Monkey push-button” learning — giving explicit key by key directions with no explanations for why this works, what you’re doing, or what understanding you’re after.

    >> Shannon writes that when she used her GPS, “I usually arrived at my destination having learned little about my journey and with no overview of my entire route.” <<

    And that's because people use a GPS precisely because they don't want to know and don't care about the mess in between. It's like solving 5x^2 + 4x – 12 = 4 by typing it directly into Desmos, which does all the work and spits out -2.23 or 4.3 – no graph, no work, no understanding. Of course, you can use Desmos and learn a great deal, produce a graph and get a visual understanding of the solutions.

    Using a map is no guarantee of learning, though. Try to drive from the Back Bay to the Gaaden using a map. Along the way, I'll get trapped because I tried to drive the wrong way on one of Boston's multitude of one-way streets, or tried to access a road that seems to be there on the map but is in reality inaccessible from here. I might remember a notable building, but I doubt that the incredible mass of information is going to last particularly long. In the meantime, I missed the start of the game and there's no way I'm going to use that f*&%ing map again.

    You say that "Similarly, our step-by-step instructions do an excellent job transporting students efficiently from a question to its answer, but a poor job helping them acquire the domain knowledge to understand the deep structure in a problem set and adapt old methods to new questions."

    That misses the idea that the step-by-step is an efficient way to begin many things, allowing for a more exploratory process the second time around and avoiding the all-too-common problem of students exploring with no guide or direction, getting lost along the way … and if they happen across some bad information or an incorrect process that only works for a rare few instances, that false understanding is devilishly difficult to overcome. Google will give you millions of results but can you understand that anti-vaxxer "truth" isn't?

    26/65 reduces to 2/5 … but not because you cancelled the 6s.

    Direct Instruction doesn't rule out asking questions, making kids think, working on open-ended or open-middle tasks. If I had to give a reason for the benefit of DI, it's to avoid the costly wild-goose chases that result in many kids giving up … "Everything I come up with is false. I quit."

    If you already struggle in math, I hate to pile more failure on top of that. If you can't work with fractions and don't know your multiplication tables and your arithmetic understanding is limited, then wide-open exploration is not your friend.

    Sometimes exploration is great, but sometimes it's damned frustrating.
    Sometimes DI is great, but sometimes it's limiting.

    Only using one method is the true stupidity.

  16. Leigh Ann Mahaffie

    June 17, 2016 - 7:45 am -
    Paul Hartzner and Scott King– YES! It doesn’t have to be either/or. As a person who is prone to anxiety, I want to know I can get to wherever I’m going. Then I can relax enough to focus on the surrounding area. If I have an appointment somewhere, I want to get to the appointment. I don’t care about the rest, unless I’m going to need to be there again. But if I’ve gotten there successfully once, and have my GPS as a back-up, I know I’m less afraid to go search.

    Our kids are asked to be able to solve quadratics by a variety of methods, in isolation, many times without context or other outside purpose. They see this as the one-off “appointment.” Why should they care about searching around? In an ideal world, I can make connections to other things they know, and show them on “the map” how they connect and are juxtaposed.
    Some of this is knowing your audience. I work with students who are “mathematically fragile” so there is a lot of explicit instruction. But my explicit instruction isn’t just divide by 5. We’ve had a conversation that this is doable if the lead coefficient is 1. So what can we do to make that happen? I don’t just say “divide by 5” with no reason given. My kids have a set of generalized steps we’ve “discovered” together, and then we apply to a variety of situations. I guess I’m teaching them to look for landmarks, and then make a decision which direction to go. I tell my students I want them to not only be able to get an answer, but to be able to solve the problem.
    As to active learning – I don’t think the active learning starts with this problem 5x^2-4x-2=0. It started with quadratics, and graphs of them, and playing around with what and why they matter. We build the “map” there. Then, by the time we get to this problem, its one more location on the map.

  17. I like much of what curmudgeons said.

    In the original study, the incentive to acquire survey/route information is different for the two groups. The GPS folks can completely ignore survey/route information and still navigate whereas the map group cannot. We don’t know how much of the difference between the groups is due to the method (map or GPS) vs. the incentive.

    An actual GPS is a poor analogy for explicit instruction. The math version of this is more like instructing students to punch numbers into a calculator as a way of learning what “+” means. In this way, GPS is more like an extreme version of discovery.

  18. The analogy between navigating via GPS and learning math is very interesting. I understand the main point of the analogy is to emphasize the importance of active learning. Students “need to do something effortful and difficult” as Dan says. This is undeniably true.

    However, analogies are imprecise by their very nature, and I have trouble seeing exactly how the GPS analogy applies to learning math. For example, in the 2006 study that was cited, the researchers were testing for incidental learning as it says in the first line of the abstract. The subjects’ goal was just to get from one point to another. They were not studying the area in order to gain familiarity with it. In fact, the subjects did not even know they were supposed to try to learn the region. In the first paragraph of Section 2.3 of the article, the authors state, “Participants were not informed that they would be tested for route and survey knowledge.” These are great differences between the research study and a typical math class. In a math class, we are interested in purposeful, rather than incidental, learning. The students know (or ought to know) that the goal is not just to get from one point to another but rather to learn the material. In my opinion, the analogy between GPS navigation and math class would be better under the following conditions. One group of research subjects is given a GPS, another group is given a map, and a third group is given both a GPS and a map. All three groups are told to become as familiar as possible with the area and that their knowledge of the area will be tested. I hypothesize that the group that is given both a GPS and a map will outperform the other two groups on the test.

    The GPS analogy tells us that students should not be given step-by-step instructions. However, there are other analogies that can be made which indicate that students should be given step-by-step instructions. For example, when a student is learning to play a musical instrument, he or she almost invariably learns to read sheet music, which are step-by-step instructions telling a person which notes to play and in which order to play them. Similarly, a person learning circuit design is very likely to build a circuit by following the step-by-step procedure of a schematic diagram. Likewise, a person learning to cook will often follow the step-by-step procedure of a recipe. Obviously, an accomplished musician must do more than merely read sheet music, an electrical engineer must do more than merely read schematic diagrams, and a chef must do more than merely follow recipes. However, learning to read sheet music and schematic diagrams is an important step for a beginner. Because both sheet music and schematic diagrams, like mathematics, involve symbolic notation, learning to read them can be effortful and difficult, and so even following a step-by-step procedure is non-trivial and promotes at least some learning.

    Dan, I really enjoy your blog and always learn a great deal from it. I’ve never posted a comment on a math blog before, but for some reason I wanted to add my two cents today. Thanks for letting me.

  19. Julian Hallen

    June 17, 2016 - 9:34 am -

    Your point that moderate difficulty helps learning is one that grounds modern Cognitive Load Theory. If you read the literature on self-explanations, guidance fading, and process-oriented worked examples, you will see that your characterization of example-based instruction misses out on many key instructional findings. You seem to have jumped from the GPS research and related psych findings to guided problem-solving, which is not that effective.

    This study may be informative. https://goo.gl/RXdRS4

    Worked Examples and Tutored Problem Solving:Redundant or Synergistic Forms of Support

  20. Michael Paul Goldenberg

    June 17, 2016 - 11:55 am -

    I don’t disagree, Ihor. I’d suggest that we do curriculum in mathematics (and everything else) in an absurd and generally ineffective way, one that only “connects” with a limited percentage of students and serves most poorly (even if they buy into the game of schooling and “studenting.”

    For a more organic and meaningful approach, I recommend Marion Brady’s work, much of which can be found at http://www.MarionBrady.com

  21. I agree with this wholeheartedly. When I moved to a new city, I had GPS, and it took me a lot longer to learn the layout. It took having to bring students home by only the neighborhood name before I understood the spacial relationships. As a teacher, I can see how this all relates, HOWEVER, I’m a firm believer that this is not an all or nothing proposition. We need to give overall perspective AND tell them step-by-step options (especially if they get stuck). Keeping with the map metaphor, when I look at a map, I look at it overall for the general layout, then I develop a step-by-step plan based on the spacial relationships. Often I look at one or two other options as well so I can switch tacks if necessary. That methodology applies directly to learning anything in life. Study it, come up with a plan (and secondary plan), and apply. The actual lesson isn’t learned until the implementation phase.

  22. I fear you have severely over-reached from this GPS study.

    “Similarly, our step-by-step instructions do an excellent job transporting students efficiently from a question to its answer, but a poor job helping them acquire the domain knowledge to understand the deep structure in a problem set and adapt old methods to new questions.”

    Surface structure always comes first. Deep structure comes from lots of surface structure and practice. You cannot go straight into deep structure. This knowledge is now well established in cognitive psychology. Let’s not try and re-invent the wheel here.

    “But the GPS offers too little difficulty, with negative consequences for drivers and even worse ones for students.”

    How about GPS to start with and then transitioning to maps? Isn’t this the popular idea of “scaffolding”, that has been around since the 50’s? Perhaps that could be an efficient way?

    It’s funny you mention GPS. I moved to Germany last year. I tried the whole looking at a route on google maps thing before I drove anywhere, but always ended up getting hopelessly lost and having to pull over and use my phone, dangerously, for about 3 months. I didn’t seem to be learning routes very well, except for the 500m radius around my house.

    Then I got a sat nav and never got lost again. I even know quite a few routes off the top of my head now.

    It’s also interesting that you use the complete the square example.

    My teacher taught me this when I was 17, doing my A-levels. She even taught us a little song “Halve the coefficient of x and take away the square…do de da”.

    I have remembered it ever since those 10 minutes of instruction that she gave us, and have used it more times than I can count since then.

  23. The learners often push teachers towards this style of teaching as well. I’m personally for an eclectic mix and everything has a time and a place. But, often, students are most used to this style. In my experience, the highest achieving student, and parent, can buck against anything other than a GPS style of teaching. But I’ve seen other students thrive when their curiousity is peaked with puzzles and interesting problem solving tasks. First and foremost, a teacher needs to teach in a way that makes students wonder and ask questions…that’s where I try to hang out.
  24. Chester Draws

    June 17, 2016 - 8:26 pm -

    I think it is worse than a false dichotomy between GPS and no GPS. It also assumes every student needs the same thing.

    I don’t always tell my clever students how to solve a problem step by step. They need to work some out for themselves or they’ll never grow.

    I usually give my strugglers much more explicit instructions. Otherwise they are overwhelmed by the question so much that they shut down. To not do so is to hamper their improvement — because they do get better if led carefully.

    Interesting to use “closing the square” as an example. Does anyone expect their students to figure out how to do that by themselves in a logical and reliable manner? I bet we all teach that by an explicit process, if only so that they do it in the quickest way, rather than the awkward way that they otherwise find by themselves.

    (And I would usually teach my strugglers to go straight to the quadratic formula, since closing the square is something they find quite challenging.)

  25. Thanks for the feedback of all kinds. I’ve added highlighting to several comments that pushed my thinking along.

    I didn’t have enough time to highlight all of the people who wrote variations on “why can’t we do both step-by-step worked examples and … whatever the other stuff is?” Some called that other stuff “discovery.” Some called it “inquiry.” We don’t really have a common language about the other stuff.

    I realize I came down on worked examples to such an extent that it’s easy to assume I don’t think they’re ever valuable.

    On that point I’ll give the floor to Dylan Kane:

    I see this as a sort of existence proof. There exist situations where following step-by-step directions leads to less learning than (active learning/other strategies).

    The question that’s useless to us is “should we use [x] in helping students learn?” The answer for most values of x including worked examples is “yes.” The more interesting question to me is, “What kind of knowledge is easy and difficult to learn by way of worked examples?” And, “Under what preconditions are worked examples most helpful?”

    The answer to those questions for some of the traditionalists whose blogs I tune into now and then seems to be “all knowledge for all novices” and “no preconditions are necessary.” That kind of maximalism is pretty easy to falsify. (See Greg Davies‘ comment for an example: “Surface structure always comes first.”) Even one datum falsifies a universal claim.

    Whether you think this satellite navigation study is that falsifying study or too far outside of education, we can find similar evidence directly within education. Dan Schwartz and John Bransford and others have researched the preconditions question extensively, finding that students can and should learn the deep structure of a question space first, before learning a solution strategy through worked examples. He replicated those findings for the calculation of density and the calculation of variance. Without that deep structure – that understanding of how questions relate to each other within a concept – it’s very difficult for students to transfer their learning across contexts, though not impossible.

    So great! Use worked examples and the other stuff. (In Schwartz’s study, the “other stuff” isn’t inquiry or discovery but a particular instructional design called “contrasting cases,” similar to Christopher Danielson’s “Which One Doesn’t Belong?“) But the best research we have doesn’t recommend using them in any order we want or for some students and not others.

    Only some drivers need both route and survey knowledge in navigation, but all students need both deep and surface knowledge in math. We get both kinds of knowledge by surveying the map before we listen to step-by-step directions. Take the metaphor as far as you’d like.

  26. Michael Paul Goldenberg

    June 17, 2016 - 10:00 pm -

    @Chester writes: “Interesting to use “closing the square” as an example. Does anyone expect their students to figure out how to do that by themselves in a logical and reliable manner? I bet we all teach that by an explicit process, if only so that they do it in the quickest way, rather than the awkward way that they otherwise find by themselves.

    (And I would usually teach my strugglers to go straight to the quadratic formula, since closing the square is something they find quite challenging.)”

    I guess that “closing” and “completing” the square are the same thing.

    Now, first of all, let’s not make “discovery learning” into some insane approach where the goal and method is to demand that students “discover” for themselves (i.e., reinvent an already well-known and long-standing mathematical procedure for themselves from scratch or with little or virtually zero scaffolding before hand) at every turn. That’s simply not what I think is at the heart of any of the alternatives to teacher-centered, generally mini-lecture-based, direct instruction approaches, different teaching approaches that get often get contemptuously lumped together as “constructivist teaching” in some camps, but which comprise in fact lots of pedagogical methods that vary almost as widely as the instructors who employ them. (Of course, any teacher might CHOOSE to teach in that 100% discovery all the time, no scaffolding, no hints way that the parodists like to tell us is what goes on in all non-traditional classrooms. Just as there may well be teacher-centered instructors who never let students speak unless to answer what usually amount to “fill in the computation answer or next step HERE__________” inquiries. It’s possible at either extreme. But how likely? And in the case of the extremely “pure” (and to my mind terribly ignorant) “discovery” teacher, how much of that is fueled by the failure to think through or understand what such methods are about and how to make them work for given students and groups of students?

    But I digress. You want to see one way to get at completing the square that I think is rich and deep, and allows for all sorts of great questions from the teacher, thinking along the way to making sense of the method, and which, by the way, leads to many of my students both eschewing the usual way completing the square is accomplished and presented in the US, and to ignoring the quadratic formula for the most part? Try the video series James Tanton offers (free of charge) at http://www.gdaymath.com under “courses.” It’s the second subset of videos in the quadratics minicourse, and the first of the five is just a look at the use of the word “quadratic” to describe 2nd-degree equations, something that evolved exactly because of the method of “quadrangles,” and in particular regular quadrangles, aka, squares.

    I’ve used those videos for 2 1/2 years now, with a few hundred students, and most come away vastly enriched by the experience. They wonder why they weren’t given Tanton’s approach in high school. I wonder, too. It’s not “discovery” and it’s not “typical direct instruction,” and it’s not denying ANY student of any ability the chance to actually understand both why completing the square makes total sense, to SEE it geometrically and algebraically in a deeply interconnected way, to get some basic knowledge of elementary number theory (some of the little bonuses aren’t always explicitly in the videos but follow from them and are things I add in by pausing the videos to pose various sorts of questions) and to feel confident about being able to do the calculations and recall the procedures. And while Tanton points out both that the “famous quadratic formula is derived from completing the square,” he also states intriguingly but without going into a major argument that it’s unnecessary to learn that formula. And I add that like most formulas and procedures in my class, students can and likely should memorize less, understand and think more. And that’s what those videos help them do, along with, I believe, my provocative questions and their own – dare I say it? – explorations and discoveries.

    The notion that some kids are just too afraid (or stupid, to be blunt) to handle inquiry, discovery, or anything but step-by-step procedures handed to them on a plate to be repeated until “memorized” (at least until the exam) is wrong-headed and objectionable in a host of ways, at least to me. And I’m not working with kids in all-white upper-middle-class suburbs, but with working class students both in the inner city and blue-collar surrounding areas.

    There is so much potential variety in teaching mathematics. Some teachers seem to come into their first classes with a very definite way they believe all math teaching is supposed to be, and when that doesn’t actually suit some, most, or nearly all the students, they either get angry and double down on their one-size-fits-all approach (and blame the kids for things going poorly) or they slowly lower the bar in order to wind up not flunking out most of the kids (and then blame the kids – not entirely wrongly – for forcing them to do so).

    Either way, it’s never that the teacher hasn’t thought things through from enough different perspectives, hasn’t done enough really hard thinking about teaching and learning math, hasn’t been willing to try non-traditional methods (or if so, tries them once, half-heartedly and with a significant degree of self-sabotaging behavior, so that the self-fulfilling prophecy comes to pass and the teacher feels justified in returning to the tried-and-failed, completely teacher-centered, dull as dishwater model that we could all recite by heart if awakened at 3 AM and forced to immediately deliver).

    I can assure you that any horror stories you care to tell, I can probably top them. I’ve worked the majority of my math teaching career in Detroit, Flint, Pontiac and similarly impoverished and deprived communities, in both charter and regular public high schools, and from grades 3 to 12. (Oh, and many of the horror stories I have aren’t about the students). Math teaching, like doing mathematics, is hard work. It’s not for the weak. But my sense of what comprises weakness may not be yours. I think that denying some kids the opportunity to struggle (sure, with whatever particular scaffolding and individual support that one can REASONABLY determine might be necessary in the short-run, as long as one keeps in mind one of the wisest three words ever stated about mathematics teaching: be less helpful).

  27. Hi Dan

    I like the GPS metaphor and have used it to criticise some of the ways teachers give feedback. This was what Greg was mischievously responding to in his blog.

    Like all metaphors, they break down before long and the trouble with applying this one to explicit instruction is that it only works if you believe explicit instruction is simply telling students what to do. (And if you believe that, I have some terrific penny shares I’d like to sell you!) We’d do much better to consider Daniel Dennett’s application of Sturgeon’s Law and not waste time on the rubbish: http://www.learningspy.co.uk/featured/seven-tools-thinking-6-dont-waste-time-rubbish/ . So instead, let’s discuss good explicit instruction.

    When teachers use worked examples, they unpick the processes and the metacognitive strategies you might use to solve a problem and then set students challenging examples to embed the methods they have learned. This is more akin to consulting a map than following a GPS and as such negates most of what you’ve extrapolated in your blog.

  28. From the original analogy: “Another group had to construct their route between those points using a map.”

    Thinking on this in light of David Didau’s comments, is this really akin to discovery learning (at least, in its extreme)?

    Another way to look at the GPS example is in terms of two versions of DI: Straight-up cookbook mathematics (where students are provided a formula for finding the area of a polygon given the apothem length and number of sides, and a separate formula for finding the area given the side length and number of sides, and a separate formula for finding the area given the radius and the number of sides, and so on: For exercises 5-9, refer to example 10-1) and guided examples.

    When I was a lad (in the 1970s-80s), I don’t recall the spoon-feeding cookbook model, but mathematics was predominantly DI. We were trained to expand our strategies; in those days, “teaching to the test” consisted of creating strategies for entry points for problems we’d never seen before, not sitting down with the last five years of SAT/ACT practice tests and going over each problem.

    It seems to me that if we’re going to take the GPS analogy its full distance, the “cookbook” model is using the GPS instructions each and every time (refer to example 10-1 on p. 415), while the map-and-memorize model is closer to DI modelling where students are expected to apply general concepts to novel problems. After all, the respondents in the GPS study were provided with a map, and presumably tapped into their existing knowledge of how to read that map.

    A fully discovery based model would be more like putting someone in a zoo without a map but with the instructions, “The zebras are over there somewhere, close to the lions. If you get lost, call me and I’ll point you in the right direction.”

    And there’s call for that, plenty. That’s how new mathematics gets done, after all: By wandering, and exploration, and listening for the lion’s growls. Take a bunch of kids to the zoo, and some will want to just wander around to see what they find, some will want a map to set out a basic route, and some will want step-by-step instructions. If the goal is for everyone to see the zebras, and everyone sees the zebras, there’s room for multiple strategies.

  29. Paul, you’re absolutely right with this:

    “A fully discovery based model would be more like putting someone in a zoo without a map but with the instructions, “The zebras are over there somewhere, close to the lions. If you get lost, call me and I’ll point you in the right direction.””

    And you’re also right to say there’s plenty of call for that approach – for experts. The research into the reversal effect tells us that as we become more expert we benefit from less direction and structure as these interfere with our existing mental representations and produce extraneous cognitive load.

    These papers all make for interesting reading on the reversal effect:

    Kalyuga, S. (2009). Knowledge elaboration: A cognitive load perspective. Learning and Instruction, 19, 402-410
    Kalyuga, S., Chandler, P., & Sweller, J. (1998). Levels of expertise and instructional design. Human Factors, 40, 1-17.
    Kalyuga, S., Ayres, P., Chandler, P., & Sweller, J. (2003). The expertise reversal effect. Educational Psychologist, 38, 23-31.

  30. David Didau:

    Like all metaphors, they break down before long and the trouble with applying this one to explicit instruction is that it only works if you believe explicit instruction is simply telling students what to do.

    Re-iterating what I said above, I’m not trying to dismantle the explicit instruction project, simply to locate its value as an instructional strategy. What can and can’t the strategy do? When is it more and less effective? The evidence I quoted above indicates that worked examples aren’t an effective first strategy for helping novices develop certain kinds of knowledge (eg. deep structure, transfer). I don’t think the metaphor to survey and route knowledge is particularly strained.

    When teachers use worked examples, they unpick the processes and the metacognitive strategies you might use to solve a problem and then set students challenging examples to embed the methods they have learned.

    Here I wonder if explicit instruction proponents have committed themselves to a non-falsifiable position. If an EI condition performs poorly in a study, proponents claim that the study tested the wrong kind of EI. That these are the wrong kind of worked examples. No true worked examples would have failed.

    Neither side of this conversation has a monopoly on that kind of argumentation, of course. But EI has an extra layer of armor protecting itself. When a student in an EI condition performs poorly on an assessment item, the EI proponent claims that the students should have been told the knowledge assessed by that item.

    eg. If students didn’t understand the structure in a set of problems, why didn’t the teacher talk about the structure? If students didn’t transfer their learning across contexts, why didn’t the teacher talk about the other context? If students didn’t gain metacognitive skills, why didn’t the teacher talk about those skills? If travelers didn’t learn the survey knowledge, why didn’t the sat-nav system just tell them the survey knowledge.

    Interestingly, the sat-nav users in this study did receive survey instruction. Didn’t make a difference.

    Aren’t we interested in what students learn after we stop telling them things? What if the problems introduced by telling aren’t resolved by more or better telling?

  31. Paul Hartzer

    June 18, 2016 - 1:37 pm -

    Dan wrote, “When a student in an EI condition performs poorly on an assessment item, the EI proponent claims that the students should have been told the knowledge assessed by that item.”

    Personally, I wouldn’t argue this at all. I’m arguing for the opposite: Proper DI gives students tools for generalizing. If a student can’t find the area of a pentagon given its radius because they’ve only been taught to find the area of a pentagon given its side length, this is a problem with the instruction, not with the standardized assessment. A student ought to be able to recognize that the distance formula, the Pythagorean Theorem, and the equation of a circle are different framings of the same property, but how often do we encourage that transfer?

    I feel the current popularity of discovery learning, PBL, and related pedagogical theories is at least partly attributable to the spoon-feeding teach-to-the-test strategies that DI has widely become.

  32. Chester Draws

    June 18, 2016 - 2:25 pm -

    Michael,

    I went away and looked at that site of James Tanton’s.

    I have no problem with it at all. However I believe is almost purely Explicit Instruction, so no surprise there.

    He starts at the beginning. He works his way up in levels of complexity. He gives step by step instructions. He is clear in language and expectation.

    Sure, he adds some other bits on, like we all do. But he adds them at the end, when the students have some chance of following him. He doesn’t lead with that material, because he knows that will just confuse.

    Whether you need to close the square or use the quadratic formula depends quite a lot on the curriculum you follow. My students need to give surd answers to the likes of 0 = 3x^2 + 17x – 5 and there is no way I think that is wise to do that by anything other than the quadratic formula — which mine are given, by the way, so they don’t have to waste time memorising it. They also don’t ever have multiple choice, so get no prompts if they have blundered.

  33. What a great analogy! I think it is important to make the students think first before the step by step. Make them struggle and uncomfortable by navigating themselves .., then you help them make it more efficient by using the “gps” method. It all comes down to Dan meyer’s “create the headache” so you can offer the math step by step as the “Advil”

  34. Michael Paul Goldenberg

    June 18, 2016 - 2:55 pm -

    I think, Chester, that you are jumping to an easy but mistaken conclusion. The Tanton videos are not a class. They are a resource that can be used lots of different ways. So can really bad instructional videos (e.g., the things I’ve watched from Sal Khan, which I find poorly put together, to be polite), but the depth and richness of bad videos is primarily useful for helping teachers who want to see what not to do and why.

    If you look back at my previous comment, you might catch the point that I don’t assign my classes to watch these videos on their own for help or enrichment, etc., though that would certainly be a couple of feasible options. Rather, I show them in one class when we’ve worked through some basics of quadratics but haven’t hit the ones that don’t factor, even using the method where you multiply the quadratic coefficient times the constant, then split the middle term and factor by grouping (some texts call that the AB method; I taught from book in the ’80s to community college kids that called it “the Master Product Method,” perhaps after the authors watched TRON). In other words, the ones that need either completing the square or the quadratic formula or something other than factoring (a graphic/computing tool, for example).

    And I don’t have students just passively watching the short series of Tanton vids. I pose preliminary questions to getting them thinking about what might work. And I frequently (perhaps annoyingly so to some students) pause the videos to field questions, take comments, and pose more probing questions, e.g., “Why is it okay for him to do________? Why shouldn’t that change the results?”

    That’s not quite direct instruction as I experienced it. It’s not quite discovery, inquiry, etc. But it’s an intentional move to help students move along a continuum from passivity to more active learning. That’s the goal of my classroom. And I only have my students for about 8 weeks. So this is one of the better things I get to do and still take them from pre-algebra review through rational algebraic equations and the like.

  35. Chester Draws

    June 18, 2016 - 3:45 pm -

    But it’s an intentional move to help students move along a continuum from passivity to more active learning.

    Why do you think that my intention, as a person who tries to use Explicit Instruction most of the time, is not to develop more active learning?

    There is this idea that by using explicit instruction we are stopping our students from thinking. Nothing could be further from the case. I just try to avoid confusing my students, because confusion is clearly poor thinking. So I don’t cause them a “headache”. (It reminds me of the old joke about a guy banging his head against a wall. When asked why, he says “because I feel better when I stop”.)

    I want my students to think. And think hard.

    But I want them to already have the tools, at hand, to solve the problems I give. So I explicitly lay out the tools, show them how they are used, and when they are not used, and only then do I ask them to think.

    ——–

    You may chose to use Tanton’s videos in your way. But by themselves they are, to my eyes, mostly explicit instruction. Could you please show me how they, in themselves, are not.

    • Michael Paul Goldenberg

      June 18, 2016 - 4:01 pm -

      In the same way that a hammer is not a murder weapon until a human chooses to make it one.

      There is very good teacher- centered instruction but it is done by people who are likely chafing against the collar of its philosophical assumptions and inherent limitations. Done as I experienced as a math student c. 1955 to 1970, it is stifling for many.

      Any alternative approach can also fail. The question for me is: what methods get used in the classrooms of teachers who can bring the broadest spectrum of students closest to being able and wanting to learn as much math as they need and/or care to learn in a range of likely situations?

      I think that on balance, typical passive classrooms breed and reinforce a narrow type of passive student most of the time. We can and should do far better.

  36. I can’t help but notice that Dewey’s Child and the Curriculum also uses the metaphor of maps.

    “We may compare the difference between the logical and the psychological to the difference between the notes which an explorer makes in a new country, blazing a trail and finding his way along as best he may, and the finished map that is constructed after the country has been thoroughly explored. The two are mutually dependent. Without the more or less accidental and devious paths traced by the explorer there would be no facts which could be utilized in the making of the complete and related chart. But no one would get the benefit of the explorer’s trip if it was not compared and checked up with similar wanderings undertaken by others; unless the new geographical facts learned, the streams crossed, the mountains climbed, etc., were viewed, not as mere incidents in the journey of the particular traveler, but (quite apart from the individual explorer’s life) in relation to other similar facts already known. The map orders individual experiences, connecting them with one another irrespective of the local and temporal circumstances and accidents of their original discovery.”

    More here: https://archive.org/stream/childandcurricul00deweuoft/childandcurricul00deweuoft_djvu.txt

  37. +1 for Henri. Good teacher is the one who combines methods. And there a lot: Flipped classroom, project based learning, problem based learning, direct instruction, etc.

    But the tend is to pass from direct intruction (GPS-like method) to discovering instruction (“traveller method”).

    Perhaps we could discuss here how are the ways to go NY: by GPS, hick-hicking, … and what are the requeriments of these, their efficiency, etc.

  38. Dan, did you notice the following in the Schwarz density study:

    The direct instruction students were given a grossly incorrect definition for density (Exhibit A2a): “Density is how much stuff is packed into a space. Density can be how many people are in a room, the density of feathers in a pillow”. Density is the density of feathers in a pillow, got it kids?

    With a breathtaking disregard for bias, three of the four instructors in the study were researchers rotating across classes and treatments.

    The direct instruction group was clearly given a plug & chug approach (Exhibit A2a): “When working with density the trick is to use the simple equation”.

    It seems that direct instruction or worked example is being equated to plug & chug. In the above study, the students actually were not given direct instruction or a worked example ON DENSITY. They were given direct instruction on plugging & chugging. It would be interesting to see the results if the direct instruction group was given instruction on what density actually means as well as worked examples that did not involve using a formula.

    Makes you wonder who is trying to win an argument and who is trying to find the truth.

  39. Dan, you cite Schwartz & Bransford to show that there are times when learning deep structure before procedures is better. Then you ask which types of knowledge lend themselves better to rote fluency instruction vs. teaching for deep understanding.

    Are you familiar with Ken Koedinger’s work? He has been exploring your exact question for some time. He doesn’t have a definitive answer yet, but if you look at his 2012 article in Cognitive Science, you’ll see Table 4 , which provides his theoretical framework for that question.

    And if you want his summary of the empirical evidence so far, it’s in Table 6 with further explanation below the table.

  40. @Kevin, looking forward to digging into Koedinger’s work. Thanks for the reference.

    @l hodge, I hope you’ve managed to catch the breath that was taken from you by Schwartz’s piece!

    In any case, I don’t find your quotation unrepresentative of current practice. Check out what Sweller himself used as the experimental condition in his original paper. It’s more blunt than Schwartz.

    Maybe Schwartz’s calculation of variance article would be more to your liking. Maybe it won’t. In general this seems like an unfalsifiable position:

    Explicit instruction is always better for all novices for helping them develop new knowledge. If a study shows that another kind of instruction was more effective, that’s because they used the wrong kind of explicit instruction. You’ll know you’ve used the right kind of explicit instruction because the study will demonstrate that explicit instruction was more effective.

  41. Chester Draws

    June 20, 2016 - 8:02 pm -

    I think that on balance, typical passive classrooms breed and reinforce a narrow type of passive student most of the time.

    But that has nothing to do with Explicit Instruction.

    These straw men you set up are easy for you to knock down, but they don’t represent my case in the slightest.

    Why is it that giving a thoroughly worked example for 5 minutes, before expecting the students to start doing them themselves, is “passive”. Yet letting them watch a 5 minute video as part of a 3-act routine is “active”? Because, to me, watching a video is just as passive as watching a teacher. (No offence to the 3-act, because I can see it’s place — but not because it is somehow magically “active”.)

    My belief is that students learn Maths best by doing Maths. If they enter my room and get 60 Algebra problems done in a lesson, how is that passive? (And, no, not 60 of the same problems, they build in complexity and challenge.)

    Yet I get them on some “discovery” based learning and they spend 10 minutes confused as to what is required, then 10 minutes getting their heads around the first problem, and only 20 problems solved overall, and that’s “active”?

    There is very good teacher- centered instruction but it is done by people who are likely chafing against the collar of its philosophical assumptions and inherent limitations.

    I’d like some evidence that good teachers are actually teaching in a way they think is unsatisfactory — because, on the face of it, that seems unlikely. In my experience, it is falsified by an extraordinary number of teachers who use teacher-centred instruction quite happily and successfully.

  42. Michael Paul Goldenberg

    June 20, 2016 - 8:13 pm -

    Chester (and I’d be happier if you used your real name: this is supposed to be a professional conversation among colleagues, isn’t it?), I’m not making straw-people arguments so I can knock down empty (or hay-filled) suits of clothes. You might consider that my examples are drawn from the real world: classrooms where I was a student once upon a time between 1955 and about 1995); classrooms where I’ve observed student teachers and in-service teachers; classrooms where I’ve been a content area/instructional coach; and classrooms where I’ve taught less wonderfully than I wanted to.

    If the first three areas didn’t provide concrete (and repeated examples of what you claim is unlikely), my own practice unarguably did. I was, after all, the instructor; I did engage in reflective practice; and I did find my instruction to be wanting, and not just marginally so or so that I could say later that I was “self-critical.”

    If you want a conversation, I can readily flesh out the above with examples. If you want to “win” an argument, I won’t waste my time writing anecdotes that you feel compelled to ignore. Just let me know what you’re looking for. And consider doing it with a real name and putting some of your own practice out here for discussion. I’ve been teaching and working in education since 1973 and my record is anything but unblemished. But it’s all grist for the mill.

    Finally, please explain how the existence of teachers of “extraordinary number” who do something obviates the existence of those who have a different, perhaps conflicting experience. And let me know if my hammer example was too subtle for you or if there was another reason you chose to ignore it, as I fear you’re going to ignore what I’ve written here in yet another stab at “winning.”

  43. The variation study is interesting. A plug and chug question on the assessment that can also serve as a worked example for a different question.

    I am not criticizing the “Invention” approach. Just the opposite. And, I am certainly not arguing that DI is always better for … However, I do not see this study as providing evidence that Teach & Practice limits transfer. It is easy to think of content for the Tell and Practice lesson that would be of no help (calculating “r”) and content that would be very helpful (something just like the embedded worked example). How was the content for the Tell & Practice lesson chosen and how did that choice affect the results?

    It would be interesting to record the student ideas during the invention phase, and then provide some or all of that information to the other group using Tell and Practice. This way there is less confounding of the form of instruction by the differences in information.

  44. @l hodge, you’re asking whether inquiry-style activities expose students to multiple viewpoints or explanations that are often left out of Tell & Practice lessons but which, if included in Tell & Practice sessions, would make Tell & Practice as effective as inquiry.

    Here is one study that carefully controls for the exact content students are exposed to and still finds a benefit for inquiry.

    That’s from 2010, and I’m not aware of any prior studies that were as carefully controlled. I haven’t followed the literature as closely in the last 5 years, so I don’t know if this result has been replicated or refuted elsewhere. For those who want the tl;dr version, here’s the abstract:

    “Self-explaining is a domain-independent learning strategy that generally leads to a robust understanding of the domain material. However, there are two potential explanations for its effectiveness. First, self-explanation generates additional content that does not exist in the instructional materials. Second, when compared to comprehension, generation of content increases understanding and recall. An in vivo experiment was designed to distinguish between these potentially orthogonal hypotheses. Students were instructed to use one of two learning strategies, self-explaining and paraphrasing, to study either a completely justified example or an incomplete example. Learning was assessed at multiple time points and levels of granularity. The results were consistent, favoring the generation account of self-explanation. This suggests that examples should be designed to encourage the active generation of missing content information.”

  45. @Kevin, I don’t really see the connection between the the Schwartz study and the one you referenced.

    The difference in information in the study you referenced was a worked example vs a worked example that included explanations (didn’t appear to matter very much). All work was individual – some were asked to “self-explain” steps in the worked example & others to “paraphrase”.

    The difference in information in the Schwartz study was a Tell & Practice lesson for marking deviations on a histogram vs whatever ideas the students exchanged on how to decide which grade or which athlete did better.

  46. Bianca Lorenz

    June 26, 2016 - 1:23 pm -

    I really like this metaphor. I do think instructions for students are helpful, but at younger grade levels students do need to engage in some productive struggle. With that being said, only productive struggle may lead to a dead end. Students need to be engaged and focused. If you lose a students focus and they get lost along the way, then it can become even more damaging to attempt to try and retrace your steps to find your way back. I think there is a really important balance between giving student directions and letting them find their way.

  47. I think that “caligraphic”analogy is more descriptive than GPS because GPS you have to interpret the world: GPS gives you indications and you have to interpret them. Caligraphic activities (just follow the marked path) demands nothing from students. More exercises are caligraphic instead GPS-ed: eg resolve equation vs “what’s the highest point of the parabola y=-x^2+2x?”

  48. Obviously, this analogy is not perfect, and there are many more factors and considerations when applying it to students learning mathematics. However, I think it is still an effective metaphor, because my immediate thought was something that, to me, made a lot of sense. I’m the one guy out of my friends that doesn’t usually ask for directions to my destination or a certain known location nearby my destination in order to get there (if it’s local – of course). I tend to just say “Give me a street name,” or “Give me an intersection,” because, to me, that is the most effective way to do it. The moment I have that street name or intersection, I can think of multiple ways to get there, and if there is traffic or construction or some other impedance on the fastest route, I can change paths quickly. In my opinion, this is a skill that a math student should be striving towards and a math teacher should be wanting to teach. The key is not just to know a route, or even the fastest route, but to know multiple routes without hesitation, and to know which route is the best one to take. At least, that’s what I got from this.