Two Warmup Questions, Worlds Apart

February 7, 2008:

I bought my car new on January 15, 2006. Today, February 7, 2008, it has 37,846 miles on it. On what date will I need my 120,000-mile tune up?

February 20, 2009:

On what date will my car need its 120,000-mile tune up?

In 2008, my students proceeded admirably through a challenging problem, successfully navigating proportional reasoning, but let’s not pretend I did anything for their ability to see the world through a mathematical framework.

In 2009, my students had to ask themselves, “what do I need to know in order to answer this question?” a line of inquiry thoroughly absent in 2008, a line of inquiry thoroughly absent in my textbook, which supplies only relevant information and, in some cases, “helpfully” suggests a route to the solution.

As my (patient) readership has no doubt realized, the impotency of our textbooks to do anything but teach procedure has recently whacked me over the head. Part of this, I realize, is fundamental to the print medium, which doesn’t permit a layered application of mathematical structures, but part of this is the inexcusable lack of imagination of publishing houses, whose bundled supplements are both costly and unhelpful, who don’t understand that they need to help students less:

Dan bought his car new on January 15, 2006. It’s a four-door sedan with 16-inch wheels. Today, February 7, 2008, the car has 37,846 miles on it. He lives 24 miles from his job and drives, on average, 48 miles per hour. The weather in his hometown ranged from 23° to 107°. On what date will he need his 120,000-mile tune up?

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. What constitutes a successful student answer? Are you looking for students to lay out a plan of attack in a paragraph, a good student discussion, back of the envelope estimates, or something else entirely?

    Also, do you give any verbal instructions with this written problem that help set them on the right track?

    –Interested in details because I love the idea.

  2. I’d be looking for a strong argument based on relevant information and mathematical process. The fact that the kids have to ask how long have you had it, how far have you driven it, how far do you drive each day… blah blah blah. I like the fact that this seems in line with the visual ideas you’ve been posting, but you don’t need pictures to accomplish a similar environment.

  3. Thanks to Mitch Weisburgh for introducing us (in a previous comment) to the Pittsburgh Science of Learning Center (PSLC). While exploring their website, I found this gem:

    “Does Solving Ill-Defined Physics Problems Elicit More Learning than Conventional Problem Solving?”

    Sample Ill-Defined Problem:
    “Regina is practising skateboard tricks. She grinds her board along a horizontal rail and falls of the end onto a mattress she placed there. How fast is she travelling just before hitting the mattress?”

    You’ll be glad to know that it *does* improve learning, but only with the proper support and scaffolding.

    One more gem from PSLC via Mitch:
    “The ideal relationship between worked examples and problems to solve is one to one. This is completely at odds with what we do in homework now in math and science. Ninety percent of what is on a homework assignment is problems to solve. Half of the activity should be to study solutions that illustrate each skill, and possibly ask students to explain them.”

    That last sentence is brilliant. (And quite obvious in hindsight, as I think about how I learned skills like programming, etc.)

    I’m going to be trying these out in the next few weeks!

  4. I should point out even the Feb. 7 version of your question opens up questions students should be asking, like: is the amount of miles consistent throughout the year? Do you go on road trips? Are you planning a road trip but have never been on one before? Is anything in your life circumstances causing the addition of a second driver to the vehicle?

  5. You’re coming down a bit hard on your textbook. I bet it’s pretty hard to develop your curriculum without one.

  6. Why would a student want to care about when a car needs its 120,000 mile tune-up? What does a 120,000 mile tune-up consist of? If it is just an oil change could I push it to 160,000 miles? What is the risk of damage done to the engine if I neglect it for so long? What does such a service cost? Would knowing the date help me save money?

    Additionally, are we taking into account rising gas prices, leaving the vehicle in non-operation for any extended period of time, a failing economy where the owner of the vehicle may lose his/her job or perhaps be forced to ride his/her bike or find some alternative transportation, the fact that the owner may downsize or upgrade their vehicle or get into an accident that totals the car before the odometer reads 120,000 miles?

  7. The difference between the two questions seems to be this: are we interested in teaching students a particular skill (or set of skills), or are we interested in giving students a conceptual understanding of mathematics and where it fits into their lives?

    The first scenario leads students down the path, essentially screaming ‘proportions!’ as they go. The skill has been identified for the students and now they just need to apply it.

    The second scenario requires an understanding of the concept of proportional reasoning and when and why it is applicable.

    Unfortunately, most students and parents (and even some teachers) are happy to take the path of least resistance to higher standardized test scores, which by their very nature must assess skills and not concepts.

  8. How many of the standards we are directed to meet are tested by procedure-ish problems and how many by open-ended application problems?

    I know my year end tests are procedure driven with really badly written word problems sprinkled throughout.

  9. I taught Algebra through Pre Calculus for many years and always told my students to show their work on any assignment. I did that not so much to check their process against what the book said was the correct one (and only!) but to be amazed at the unique and wonderful ways some kids would arrive at an answer. (AN answer, not necessarily a right one. :-)

    In math, and many other areas of school, we need to help and encourage kids to create their own path rather than insist that they use the one in the book.

  10. Ben, in terms of acceptable outcomes, I am looking for a single, succinct answer followed by, you know, a-page-and-a-half of rigorous justification.

    I didn’t preface this problem with any verbal instructions. I acted a little taken aback when they first claimed the problem was unsolvable. I then doled out information to students only as they asked for it.

    Nick, good eye. This sort of questioning is, I think, a trait of good teaching – whether founded on digital media or not.

    Frank, thanks for the links.

    Kenny, no, it isn’t. And I’m not saying my textbook is useless or that I never use it. It’s just a lousy tool for developing conceptual understanding.

  11. Dan, I agree with your disdain for most textbooks and especially agree that we are over-focusing on procedures. Most of the big publishers (Holt, Glencoe, McDougal-Littel, Prentice Hall, etc) just pump out these massive, 1000+ page textbooks bloated with fancy bells and whistles that really aren’t so fancy (websites with lame animations, online computarized procedural tutors, test generators that generate tests filled with procedural questions and–even worse–word problems that are very poorly written, like bcarrera said).

    What do you think of IMP ( EduGroups that love procedures (like reeeeeally hate IMP, which in my mind is one of the best compliments that can be given.

  12. Dan, do you see an inherent contradiction in your assessment methods and your quest for a highly conceptual framework? A strong grasp of the individual skills does not imply a fundamental grasp of the concepts involved. Do you include conceptual questions on your skills assessments?

  13. @Touzel, next on my to-do list is to track down editions of IMP and CPM. I have a great deal of respect for the president of CPM but what little I recall of both texts is an overarching, chapter-long scenario framed by a visceral question (“will the pendulum etc. …?”), all of which sounds like it ought to thrill me.

    But the rest of my recollection has students proceeding through a section fairly schematically from question a) through question f), each building on the last. This used to impress me but now I wonder if that doesn’t offer students too clear of a road map when, really, the students ought to be deciding on the road map for themselves.

    @Clint H, this is only a contradiction if you imagine a class exclusively driven by procedure or exclusively driven by concept. I want my kids to aim for the conceptual high-hanging fruit but I won’t fail them if they can’t reach it. However, if they can’t handle the basic procedures that comprise Algebra II (for example) I won’t pass them on to Math Analysis. That is the division of labor between concept & procedure, assessment & digression, and like I mentioned in a previous post, the two factions have teamed up in some cool ways.

  14. Yeah, I think you would totally love the beginning of most IMP units. There are five units in each year and most start with some hook (like the Pit & the Pendulum, or Fireworks, or Cookies) and the first assignment of the unit involves the students trying to figure out what it is they need to know in order to even be able to continue.

    There are two units in Year 1, however, that don’t have a central problem, but there are many other good ones.

    Here are some of the units (with the unit question paraphrased):
    Year 3: Small World (how long until there are so many people in the world that they are all pressed up against each other?)
    Year 2: Do Bees Build It Best? (is a honeycomb the best arrangement for maximizing volume while minimizing surface area?)
    Year 1: The Game of Pig (what’s the optimal strategy for the dice game “Pig”?)
    Year 2: Cookies (how much of each type of cookie should a bakery make to maximize profits?)
    Year 2/3: Fireworks (where will the fireworks land?)
    Year 1: Pit & the Pendulum (will the prisoner escape or will he die?)
    Year 3: Pennant Fever (what are the odds–with a week left in the season–that a given team wins the pennant?)

  15. @Touzel I teach IMP. I’m currently teaching one of the units that doesn’t have a central problem – “The World of Functions”. I think this may be my favorite unit of all of them. The depth of understanding my students are displaying (mostly seniors) in this Year 4 unit is blowing me away.

  16. Jackie, Touzel, I checked out IMP years one through four from my school’s bookroom (which kept some copies around even after the community here ran the curriculum out at the end of a pitchfork) and I’ve gotta say I was really, really impressed. Really impressed. I need to blog a review with page scans and everything, and I need to take another pass at CPM, but for now this is as close as it gets.

  17. Hell yeah, Dan. I knew you’d like it.

    I look forward to reading your take on it–I’m sure you’ll have plenty of ideas I never even considered.