A Train Leaves Chicago Traveling At Who Cares, Ctd.

Some of the responses to my last post focused on narratives to hang around the old math problem – lovers meeting, trains nearly crashing, heroes stuck on train tracks, etc.

Forgive me. I lack faith.

First, I don’t think these efforts engage students (same goes for using student names in boring math problems) but I admit engagement is a fickle, subjective thing.

Second, my broader, more objective criticism of these efforts is that in every one of these cases you will eventually have to create a math problem, to fabricate numbers and dimensions for students to work with and operate on after you’ve (apparently?) piqued their interest in those operations by showing them a clip from that recent Denzel Washington movie. Then, when your students get an answer, you will tell them, “Yes, your math was correct and your answer is correct and the world and your math verify one another.” This isn’t worthless. Your students are practicing operations that require practice. But as a sales pitch for math’s connection to the world outside the math classroom, this is worthless.


Imagine you are going door-to-door selling knives. You get a foot in a door and tell a prospective customer, “These knives are sharp enough to cut through a soup can.”

The customer says, “It so happens I have a soup can right here. Let’s try it out.”

Then you say, “Just take my word for it. These babies are incredibly sharp.”

The customer says, “Are you kidding me?”

Then you say, “Okay, well if you won’t take my word for it, have a look at this pamphlet. Right there. See? ‘Cuts through soup cans.'”

The customer says, “But your company wrote that pamphlet.”

You: “So?”


All the time.

We ask our students all the time to take our word on the presence of math in the world.

Or we ask them to take the textbook’s word on it.

You wouldn’t buy the knives. Why would your students buy the math?

A better sales pitch? Give them a video from which they can draw their own measurements, on which they can operate mathematically, at the end of which they can verify for themselves whether the trains meet at 2:37.

They’ll buy that. But don’t take my word for it. Try it yourself.

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. Joe Henderson

    July 12, 2011 - 11:00 am -

    At the end of the day though, both of these methods involve students occupying the role of information consumer, don’t they? Sure, one is definitely more inquiry oriented and technologically slick, but how authentic is it to them as the learner? I suspect that this is your point though.

    Or, imagine this, they have a role in the production of the problem itself. I’m not really interested in beating a dead horse here, but you and I have been down this road before…

  2. Dan,

    The modeling instruction curriculum I use in my physics classes tackles this very problem by deploying every model the students develop through a lab practicum. For the constant velocity particle model, the students use a small buggy to develop algebraic (x=vt+x_0), graphical (x vs. t and v vs. t) and diagrammatic (motion map) models explaining the behavior of the buggy. Then, once they feel confident in their knowledge and have a strong understanding of the model, we challenge them by pairing groups whose buggies move at different speeds. They are asked to predict where the buggies will will collide when aimed at one another.

    They are responsible for determining how to use the models that they’ve constructed in the activity and what additional measurements they need. Some students will solve for a common position while others will graph the two functions. Some will do both. They aren’t allowed to run both carts at the same time until they have marked the physical location of their prediction. And then, we test it. Let me assure you, there is no more exciting or nerve wracking moment for my physics students up to that point, than when they slowly watch those two cars approach one another. The tension in the room is palpable and the cheers of success fill the entire science hallway.

    You could easily adapt this to predict when they will collide, especially if you chose to start the cars at different moments. Additionally, when students are slightly off, this leads to a great discussion of uncertainty, though I’m not sure if that’s a topic covered in math classes. I don’t think tying math or physics to the real world necessarily has to be grand to be authentic, so long as students are asked to do something with the knowledge they have developed.

  3. Dan, does it HAVE TO BE video? Would a picture work just fine or even the text as long as the experience fits into your three act theory? How do you propose a teacher who does not have the talent to make videos or time to make them? It would be nice to have a central resource where teachers can download these videos for free that is very user friendly.

  4. Christine Lenghaus

    July 12, 2011 - 8:58 pm -

    Hi Dan,
    We only have one train line out in the country until we reach outskirts of the city. It is realy important to know when these trains pass each other (going to/from the city) as this can only happen when the trains meet at station which has two platforms or one of the few sections of rail in between that has a double line. These are problems that I imagine the people who create timetables must look at. Also with different types of trains (intercity, fast and stopping all stations) going in the same direction on a line, when they pass each other is important if the number of lines are limited.
    If you graphed it up it would be a great example of simultaneous equations – how far do you take the “real” analogy: first attempt- use average speed between stations, then include things like slowing down and speeding up as a train approaches a station, time waiting at stations …

  5. Jan van Hulzen

    July 13, 2011 - 4:56 am -

    How about this one:

    An isle just off the coast is a popular destination for tourists. The ferry connecting the isle to the mainland can hold 50 cars and needs 30 minutes to get to the isle, unload and get back. The tourists arrive between 8 AM and 5 PM at the rate of one car per 30 seconds. What is the maximum waiting time a tourist has to endure?

  6. I don’t think this is your point, but it sounds like you are making the case that video can save most math problems. And that video is the ONLY savior.

    In other posts you talk about developing a story to engage and hook the student. This can be done by other means than video. Ask anyone who has ever picked up a Harry Potter book way before any of the movies came out.

    In an area of the country where train accidents are in the news, I would expect articles related to these accidents would provide a hook that may be even more intrinsically motivating and engaging than any video.

  7. I don’t think this is making the case that video is a required component of this type of inquiry. However, video is an authentic representation of the precision of reality. If a math problem claims to make a connection to reality, there must be a way of drawing in that reality without just stating that the answer is correct.

    In the end this is really getting into the art of teacher-craft. Reminds me of an Chekhov quote: ‘Don’t tell me the moon is shining; show me the glint of light on broken glass.’

  8. How much of the math in an Alg I or II text does relate to the real world? Maybe 10% if lucky? I do not count academia as the real world, which is where most of that math is used. Being an Alg I and II teacher I really cannot answer the “where do I use this in real life” question for most chapters in the books. I have never had to factor a polynomial, or solve a trig identity, or solve for the roots of a polynomial in the real world. But there are careers out there that require this math knowledge as a background, and if the student does not learn it in high school they are definitely not going into that field. I can do a pretty good job of justifying my stats and discrete math courses.

  9. @Michael not that it has to be or should be, but the resources and knowledge to make a video tutorial or example are quite available. Patrick and I have in depth instructions on how to teachers can create professional videos with a webcam and computer and the software is free and open source!


    We have spent a lot of time refining this because Dan as well as others have demonstrated how useful videos can be in the classroom. While a picture can do the job as mentioned in some of Dan’s earlier posts, videos can tell a story and make for a more dynamic mathematical problem. I hope you will pass this along to anyone you feel is in need of support in making videos.

  10. Joe: I don’t think this is making the case that video is a required component of this type of inquiry. However, video is an authentic representation of the precision of reality. If a math problem claims to make a connection to reality, there must be a way of drawing in that reality without just stating that the answer is correct.

    This, exactly.

    The words I used were “accurate representation.” In a lot of cases, a photograph is accurate enough. In cases where the question is “how long will it take for [x] to happen?” though, video is essential. How do you convey the passage of time in a photograph?

    Michael: How do you propose a teacher who does not have the talent to make videos or time to make them?

    I don’t. I think you should use the best resources at your disposal and feel no guilt from me about it. This is aimed at the teachers and, particularly, the publishers who have the resources and time at their disposal and who continue to produce the kind of sales pitch that has students slamming the door on their teachers.

  11. I don’t believe time must be conveyed visually in order to engage a student in a meaningful problem. Again, I point to stories. They convey time with no visuals at all and I don’t think any less of them for it.

    Take a look at Mathalicious.com. They’ve done a great job of setting up lessons full of meaningful problems. Most are conveyed via slides. Some have videos, but I haven’t seen one yet that was used to convey time.

  12. If we want help students look at the world through the window of math, instead of putting whatever we think kids think is cool on the curtains, we need to work on the view (and also the window is connected to a door).

  13. Marc Stephen

    July 14, 2011 - 6:27 am -

    What if they have no interest in your steak knives?

    Text book, picture, video–it is all the same if it doesn’t engage the student. So often I hear that we should connect math to the real world to engage students, but the real world is only fascinating to some of the students.

    We need to connect it to the students’ world…and trains will inspire a small percentage, the soldiers some others, lovers meeting still others, and who can resist crashing cars in class? So do we make a problem for each student? Not practical, so…

    What’s harder, more fun and engaging for all the students when it comes to word problems? How about giving the students an answer and have them design the problem?

  14. Joe Henderson

    July 14, 2011 - 7:14 am -

    @Marc: Yes. Thank you.

    Of course, what you suggest involves ceding some authority to students in the problem-creation and knowledge construction process.

  15. @Shari, I think you misread my comment.

    @Marc, thanks for the suggestion. This goes a long way to solve the problem of student interest. It re-opens the problem of faith-based math, though. If I understand you, you’re giving students a model, an answer, and asking them to construct a question that leads to that answer. But what if they don’t have faith in that model? They’re just doing what you ask, taking your word that the math will do what you say it will. A problem that begins and ends with an accurate representation of itself, on the other hand, goes a long way to solve that problem. Certainly, I don’t see an issue using both techniques – one for instruction, the other for practice – but you and Joe H. might. It’d be helpful for me, I suppose, if you could elaborate on how the students learn the model in the first place.

    @Joe H, do you do any professional development around your pedagogy? I’m curious if you’re this condescending in person, with other teachers, or if it’s just here on my blog. If you’re in a workshop, for instance, and a teacher asks a question about implementation, do you advise them to have more courage, have more faith in their students, to cede a little authority? Has that been effective?

  16. So here’s an interesting sideline: NCTM has an applet here http://www.nctm.org/standards/content.aspx?id=25037 that simulates a “when will they meet” situation. Bets can be placed, models can be constructed, estimates can be tested, reality can intervene before, during, and after math thinking. All hallmarks of good contexts for math.

    On the other hand, it’s a simulation. They’re up front about calling things “seconds” that aren’t really seconds. The math behind the model is hidden. The context is silly and clip-arty.

    Good tool for inquiry? For practice? Neither?

  17. Joe Henderson

    July 14, 2011 - 8:10 pm -

    Dan, it was, and is, not my intent to be condescending here, and if that’s how you’ve interpreted it, then apologies are in order. Sorry.

    However, I still stand by the crux of my comment.

    To engage the meat of your question. I work with pre-service science teachers. We do engage in professional development around inquiry-based, student-centered science teaching, although not in the typical one-off PD variety that has been shown as ineffective over and over again. Here’s an excellent resource along those lines:


    Instead of one-time PDs, the experiences for learning the inquiry science teaching model are built into our teacher education program and scaffolded over time. At the beginning of the program we intentionally engage our teachers in an open-ended scientific investigation where they have to come up with their own research question, the methods for hypothesis testing, and ultimately the production and defense of knowledge claims based on their work. In this process both the research question and the answer are unknown. The students are responsible for determining their question of interest.

    I teach this first class. Yes, I do cede authority in the creation of the question to the students. It’s uncomfortable, as that’s largely not how I was taught myself. And while it sounds squishy, I have to have faith in their ability to run with the messiness. And I know that each student/group will take ownership differently. And I scaffold along the way depending on the needs of the class (mainly by introducing tools or technology at specific times, but also by withholding information other times, and frequently by engaging them in metacognitive stops – making sense of what they’re doing and what they know at a point in time). Again, flexible depending on the cultural needs until they don’t really need me anymore and I recede into more of a facilitation stance as the course progresses.

    And yes, it’s amazing how effective that is. Seeing students move from a stance of knowledge consumption to knowledge production, and the liberation that happens when they take ownership over problem solving is something that has been so powerful that I will never go back to a more “traditional” teacher-centered classroom model. And I’ve said this before, but it’s not all or nothing here either. The teacher has a role in facilitating these experiences.

    If you want to read more about the philosophy and pedagogy of our program, I recommend this piece from one of my mentors, April Luehmann:


    I’d love to know what you make of it. Incidentally, this will be out soon as well:


    April’s work is fantastic and largely the reason that I found your blog in the first place. We’re trying to also understand teacher PD via the social processes of blogging, but that’s another time.

    Anyway, it’s getting late and I need to go to bed. I’ll leave with this for now.

    Instead of constantly trying to sell knives, I simply ask you to think about what the students might try to sell to you? And how can we, as teachers, draw that out in a way that makes the learning more authentic, engaging, meaningful, etc.?

    Dan, I deeply, deeply respect the work you’re doing here, which is why I want to push you. Again, I really hope I’m not coming across as condescending. Please tell me if I am.

  18. Joe H.: Instead of constantly trying to sell knives, I simply ask you to think about what the students might try to sell to you?

    I think you’re right: that is simple.

    I’m probably just the wrong receptor for this kind of signal, which prompted my curiosity about how other audiences in other fora perceive your koans and aphorisms about giving up and letting go and etc. I just don’t envy the task of scaling your intervention without a large satchel of practical exercises of the kind Marc offered here. A non-trivial fraction of the millions of teachers in the world are going to be put-off by your implication that their biggest challenge is sociocultural rather than practical. I am one of them.

    Any time I think about engaging you on this ground, I remind myself I’m just gonna get some trip about how I must be power-mad or insufficiently interested in what my students are interested in, and restrain myself. Since I couldn’t leave well enough alone this time, here’s a question for the road:

    If only a handful of teachers online can pull off this WCYDWT / 3ACT curriculum design, how is assigning that same task to students anything but cruel and unusual?

  19. I’m with you, Dan, about practicality being incredibly important, and that the more we can initially provide good questions that build on students’ understanding of reality (rather than explicitly asking them to ignore reality) the more chance we have of teaching well.

    I also think that students (like their teachers) can learn to get better at being curious about their world, and eventually figuring out for themselves what else they would need to know. I imagine you saw that in your classes… the wonderings converged. The requests for information became clearer and more useful. Maybe they started mentioning times they saw something in the world and felt like they were in your class. They thought what you were doing was powerful and began to see a role for themselves doing it, and even what it would take to get better at it.

    Focusing on getting the teachers good at making really powerful questions to explore has to come first, but I’m still holding out hope that when all their teachers are this good, by the time they get to high school they’re coming in with a (digital) folder of stuff they’ve captured that they want to know more about.

    As Shawn Cornally illustrates in his TED video (http://www.youtube.com/watch?v=gPeKdXhGcZQ&feature=player_embedded), kids can ask good questions and learn from them when we can make institutions get out of the way, and we eventually have to trust that there is actually math in kids’ real world that they will be curious about. But right now we have to start by selling them that idea since they’ve had over a decade (if they’re high school) to learn otherwise, and we have to train ourselves first to get over what we thought we knew about “word problems”

    So I would like to offer the idea that we can have a vision of students getting better at noticing and wondering about math at the same time that we work really hard to equip teachers to do it for students first.

  20. Joe Henderson

    July 15, 2011 - 8:11 am -

    I think Max echoes something that I raised in an earlier comment, and I commend him on his clarity (something I’ll admit I’m really struggling with right now). He raises the notion that moving toward a more open-inquiry learning experience needs to be scaffolded over time, with a move from teacher driven to student driven.

    We intentionally build that into our course structures so that the ownership in the learning process gradually moves from teacher to student, at a number of scales. But that move seems to be the key, and that’s where I see the success in my own teaching. I worry about teachers that don’t make that move. And I worry more about students that never get to experience that type of ownership in their learning (resulting in the ever more passive consumption of knowledge).

    I think the WCYDWTs definitely disrupt the notion of teacher as sole authority in the learning process. That’s really commendable and is probably the reason the method has gained so much attention. Rightfully so. I don’t want to lose the fact that we’re largely arguing about a matter of degrees along the spectrum. Max adds the temporal aspect to the debate, and that’s really important for us to consider, especially in course design.

    Regarding some of the other issues you raise. I guess I think that any curricular innovation will fail, or at best remain inadequate, unless it attends to, and is somewhat shaped by, the sociocultural reality of its implementation. That is, curriculum doesn’t really exist until it’s embodied in practice. That’s a highly localized process. The history of educational reform is littered with “generalizable” silver bullets that fall short in their implementation. How are standardized tests working out in practice these days? What’s the carnage that results from the assumptions about learning that are embedded in that particular practice?

    Also, I hesitate to position the sociocultural against the practical, as you seem to be doing above. I would argue that they’re intimately related. What’s practical is socioculturally responsive.

    What I sense Dan, and I’m not going to go too far into the PhD weeds here for the sake of the readers, is that we just have different epistemological stances about how teaching and learning occurs. Perhaps this is the source of our struggle?

    I’m also not sure if I’ve answered your last question at all either. I need to think some more about that. And thank you for this conversation. Know that it’s pushing me as well. And I’m still going to buy you that beer at some future AERA.

  21. You raise some excellent points. I am currently studying to become a teacher, and have been wondering how I can get my students interested in math. I remember doing these types of problems, figuring out at what time two trains will meet, as a student, but never really saw the point of them. They never piqued my interest. I think it is important that we connect to things that are interesting to the students. But with students having such a wide variety of interests, how can we do this?

  22. Marc Stephen

    July 17, 2011 - 4:48 am -

    Joe, Dan and others,
    A great discussion on this topic. I’ve cut and pasted a summary of the posts (credit included) to hand over to my 6th grade students in August so they can start their own discussion.
    Thanks for a lively debate that gave me some answers and even more questions…a great place to build from.

  23. Joe, on behalf of the readers, I want to thank you for not veering too far into the PHD weeds. There’s only so much a core audience can take. I for one almost unsubscribed when I saw the word ‘epistemological.’ Nonplussed, I frantically right-clicked the word and looked it up in my dictionary. Three seconds later, I felt better. As an educator, I never want to have go through that again. Thanks again for looking out.

    One important note. Remarkably, you seem to equate WCYDWT to a traditional curriculum. One that is described in the paper you cite (Luehmann) as a ‘transmission model where “teachers
    are there to tell and students to listen.”

    If part of my professional development is tightening what WCYDWT is and where it fits, I think I’ve made some progress.

    Because that quote describes exactly what it isn’t.

  24. I tend to agree that paper is not the most compelling way to connect math to the real world. For similar reasons, I would also argue that paper is not the most compelling way to connect curriculum and pedagogy to the real world.

    I would like to see tapes of several classroom lessons — not staged lessons — but typical day to day lessons. What does it look like for the ownership in the learning process to gradually move to the students? What does it look like to use WCYDWT lessons on a regular basis?

    There are many claims about knifes that are sharp enough to cut through disengagement. I realize there are obstacles to releasing videos, but it would be awfully nice to see some demonstrations instead of a bunch of folks arguing about who has the sharpest knife.