wish I had a classroom full of kids. That sort of rhyme between how the original problem – or your hypothetical statement of it – eliminates all the real world stuff in order to teach “math” and how you’re isolating the “math” in order to (maybe; hopefully) show how the real world stuff absolutely inflects it is pretty darned elegant.

Would it be helpful to have physics classes? I have a small sample size, but I’ve got them until Tuesday.

I don’t have a class full of kids, but I have a prediction that none of the four sufficiently turns on the “No, really, I want a _real_ answer” framework and turns off the “Ignore everything but the statement” filter.

The last one, maybe, but it would take a smart-aleck.

I’m pretty sure that most of my students would interpret all versions of this problem as the equation in disguise if the context was math class.

In a physics class, the same students would tell me that there’s no sensible numerical solution as one would need to make additional assumptions regarding the velocityvs time function.

Sadly, math class remains the only place where people buy 60 water melons and no one wonders why.

I have a group of seventh graders who will do it Wednesday, June 8.

Same issue I addressed here: http://www.maa.org/devlin/devlin_05_10.html
We all need to keep pressing this point.

My grade six class in Calgary will take a stab at this. Hopefully by mid-week.

I’ll pass this out for one of my Math 7 classes tomorrow.

Give me until this weekend and I will have it.

I’ll get more detailed info to you soon, but on the first go-round:

19 students
10th and 11th graders, mostly 10th
very low achieving Alg 1 class
average grade in class C- (I used SBG or whatever you want to call it)
1 out of 19 factored in decreasing mile time over 26 miles, everyone else came up with 156 min, except for those who had multiplication errors

I don’t have many days of class left with my 5th grade students, but I’ll do my best to get it in. I’m interested to see the results.

I have loved reading (and receiving tips) from your website! I will definitely do this with my 6th graders tomorrow.

Dan, will you accept scanned copies of originals in PDF form or do you really need the originals? Thanks.

After the magic that “Graphing Stories” created in my class this is the least I can do for you! And I know how much work Graphing Stories is for you now so consider us even! :)

I’m glad I read this tonight – I have the last class tomorrow – I’m on it!

Thanks!
Julie

I will do this with my 24 kids this week after we come back from overnight camp (gahhhhhh…) on Wednesday. I’m very eager to see what they come up with.

My 6th graders will take this on as well! I’ll have them by the weekend for you!

Hi Dan,
I’m in the southern hemisphere, so I’m on break at the moment. In two weeks I’ll be back in the classroom and I think this is a great thing to start the class with. I’ve got 120 students in different classes from 7th grade to 10th grade. I can easily turn the answers into a PDF and send it to you. My questions are:

Would getting the results on the 23rd or 24th of June be too late for you to be useful?

Do you care about results from other parts of the world? If you’re specifically after US data then I won’t bother.

Hey Dan, happy to try this out with my class.
They’re a class of 22 12yr-olds studying Maths in the Scottish system. Is that ok for your trail?

Why not make it plain that the context is not the issue, just ask them questions like:

Popeye lifts a pebble with one finger, how many fingers does he need to lift a train?

I think the real issue here is to make them question the question instead of answering it. Mathematical or otherwise. I would love if my son would ask me why the coyote orders all that elaborate stuff to catch the road runner if he could just as easily order a pizza.

I will get you about 15 of them asap :)

Dan,

Would grades without names solve the privacy issue and still give you the data you want? You don’t really need the students’ names, right?

If I had just ONE more week of school I would have time to do this… but it looks like you found plenty of volunteers. I’m curious and anxious to hear the results of your analysis.

As someone who is athletic, this question bothers me a little bit. I have looked at enough lap times for cyclists to know that times are not strictly increasing (that is, velocity is not strictly decreasing). While I recognize that a marathon will be run more slowly than one’s fastest mile, the dynamics of human performance are complicated and we are all going to make assumptions that simplify the problem to one we can deal with. I’m not sure that I’m even convinced that realizing that you should take your fastest mile time and multiply it by some factor (which will be different for every runner) to get your average per mile time for a 26 mile race is all that mathematically interesting.

I suspect that at my urban charter school few students would consider fatigue as few students are involved in endurance athletics. To conduct an ad hoc study and not consider the context (school and students) in which the study is conducted seems like a grave mistake.

What if you asked the same question but with a car traveling on a local road that has significant traffic?

You’ll be getting 30+ from my 7th grade classes in Minnesota. It was interesting discussion afterwards. Amazing how little they apply reason to real life situations.

From me, you’ll be receiving 5 sections total:
2 of 8th
1 of 7th
2 of 6th

I’ll have them all finished up by next Tuesday, 14 June, and e-mail them. Keep us posted on results!

I think the problem with this type of question is that while it’s “real world” (meaning it happened to someone), it’s not “real world for the students.” What does that mean?

Consider buying loaves of bread a local baker. A loaf costs $3.50. How much would 15 loaves coast? First, how often do people buy 15 loaves of bread at once and if they do, wouldn’t they at least ask about a volume discount? People do buy bread, but this type of problem discourages rational thinking and results in relatively meaningless computation.

Likewise, the running problem removes the element of personal experience (except for the students who are also runners and get the fatigue factor). I think a different approach might be to take students outside and have them run a mile and then estimate how long it would take them to run 2 miles, 5 miles, a marathon, etc. After doing that, maybe they can then address the 6 minute mile question with both additional questions (are we assuming constant speed or should we factor in for fatigue, etc.) and active thinking. And the kid that points out that s/he would die before the end if they tried running a marathon gets extra credit for bringing reality back to math.

I realize that’s doesn’t answer your question about the medium, but maybe the fact that it’s still a worksheet is the problem we should be addressing first. :)

There are lots of interesting mathematical modeling problems. I am not sure that students who have not made the leap from repetitive addition to multiplication are ready for real world models, which are more complicated.
I don’t have a problem with “add some” for the marathon. A mathematical model would have some secondary function multiplying the primary function. The “add some” idea is more like statistics, where you take a statistical wild-ass guess (SWAG) and then see how likely you are to be right.
I still think we are discussing the difference between pure and applied mathematics. Students need to learn the rules of mathematics, just as they need to learn the rules for written English (if they want to be able to use them). They might want to be able to move on to applied math, or pure math, or figure out the mortgage, just as they might want to write a novel or a scientific paper or leave a note for the milkman.
It’s the reason why “calculator solutions” are not germane. You haven’t learned how to use the math, you haven’t learned to model a real application – the exercise had no value. IMHO.

Sorry if this comment is reaching you a bit late Dan.

It seems like your hypothesis is (a) changing the medium matters and (b) the mechanism by which changing the medium will matter is that fewer students will fall back on learned habits of ignoring entropy, gravity, friction, etc.

It’s not clear to me that your experiment tests hypothesis (b). Also, wouldn’t any arbitrary method of “alienating” the student (in the theatrical sense of alienation: http://en.wikipedia.org/wiki/Distancing_effect ) produce a similar effect? Put a different way, does the medium only matter because it’s different/novel compared to the norm?

Well I gave these out to my 38 8th grade math students last week (will work on the scanning this week and will then get to you) and over all only 2 students had any thoughts about the runner slowing down, but only wrote about it and did not calculate with it. Neither are my top students either. I was so proud of them.

I think they are so used to these types of problems from books and standardized testing that they did not even think about the true context of the situation… something I plan to remedy for next school year.

Two classes of grade 7 students (52 kids in total) and NOT ONE student dared to go beyond the basics of 26×6=156 minutes. No one even mentioned the idea of the runner slowing down – and there are a few kids in each class who just took part in a cross country running event!

Perhaps a separate post will provide some background.

In a few studies of the NAEP math items in recent years, mathematicians found about 20-25% of the items mathematically incorrect. Errors ranged from missing information, through unstated assumptions, to simply wrong answers. Yet psychometrically these items “performed well” — proficient students provided “correct” answers (i.e., what item writers deemed correct), less proficient student performed less well on them, etc.

This seems to me a clear demonstration that to a degree we train students to guess what we want them to answer, rather than value correct thinking.

I’ve read at least one if the NAEP studies you refer to…for some context, most of the “mathematically incorrect” answers are more “mathematically overly simplified” answers. If my memory is correct, something like a question on a 7th grade tests ignores possible imaginary answers. I see this as a very different kind of “mathematically incorrect” than the question posed in this post. To use an extreme example, no one believes we should be teaching 5 year olds set theory in order to allow them to conclude that 1+1=2. That said, it seems that most people would agree that students of all ages should read this question and go, wait, I don’t have enough info to answer this question precisely. The point is that ‘mathematically correct’ is a human construct.

Avery,

Rereading my first post, I tend to agree with you. It does come across as hostile. I was annoyed with the underlying assumption of this exercise, and with what I presumed — admittedly always risky — will be the “lessons” drawn, with little or no justification.

First, the facts. We are given four variant of a problem that needs trivial math skills, and whose difficulty stems solely from the non-mathematical ambiguity of the question. More specifically, two variants ask explicitly about what seems like the *number of minutes,* and two others are a slightly less restrictive when they ask “how long would it take,” although the form still implies a number.

Now, essentially all kids above age10 know that one cannot run at a given speed forever. If you don’t believe me, ask any whether it will take him to run 200 yards exactly twice as long as it takes him to run 100 yards. Kids run these distances in PE by then, and they know the answer. Some responders already said as much.

Yet the kids mostly answer 6*26. Why? They clearly know the operation needed. They clearly know that one slows down over longer distances. They know the don’t need any fancy formulas here. So why?

Any normal kid will look at the question and come up with the feeling the question — in all its variants — expects a *precise* answer. Well, there is no other data to provide a precise answer, except 6*26. So, their thinking goes, the teacher is an idiot for asking a clearly unrealistic question here but, what the hell? If that’s what he wants, that’s what he’ll get. End of story.

If a kid is a bit more “brave” he will — at best — say “longer than 26*6” based on his non-mathematical experience of real life. But there is no obvious place-holder for this answer in at least two variants, so the kid know he’s pushing it. If the teacher is smart, perhaps he’ll get it. If not, he’ll be marked down. Why risk it?

Now I read the some of the comments and I find “students not using reasoning” and similar. Bull. They did. It is the form of the question that dictated their responses.

So my annoyance was that people may conclude that “students can’t apply their math.” Perhaps. But this badly designed experiment certainly can’t tell us one way or another.

But I shouldn’t have let my annoyance show. Sorry.

@68 Avery: No, many of the errors are of a very simple nature, nothing fancy, nothing to do with complex numbers. Often one of those pattern problems when they expect you to guess the Kth member based on some unstated assumption of a linear (or quadratic) progression. If I recall, one even had some relation between (young) age and weight exhibiting some such pattern, asking about a table entry few years ahead. If you were to take them at their word, you would believe that when you get to the age of 40 or 60 you would weigh many hundreds of pounds (smile). There was a NAEP validation study circa 2008-9 that had some examples. Tom Loveless (Brookings) also had a similar study some time back.

@71 Russel: I find value in not trying to over-analyze flawed results. It consumes enormous amount of time and the analysis will be speculative anyway. It is much easier to design a study properly up front. In our case, mixing a study of the impact of variant posing, with the study of student’s ability to apply & interpret math in terms of reasonability, was a bad idea. Each could have been studied much easier in isolation.

Russell mentions Vladimir Chaloupka. Vladi is truly a wonderful man. You should listen to this song about what Vladi has to day about the universe:

What I think is neat about Ze’ev’s post is that he seems to be calling into question the idea that some contexts don’t elicit reasoning.

Seems that we’ve got a bit of a dichotomy set up:
answer “26*6” = not reasoning
answer “definitely more than 26*6” = reasoning

The idea then is that context supports kids to reason if it’s good context, and hampers their ability to reason if it’s not (and by context I mean the whole milieu of math class).

But the timing of this conversation is perfect… Check out this article in the NYTimes: http://www.nytimes.com/2011/06/15/arts/people-argue-just-to-win-scholars-assert.html?_r=1

The premise of the article is that reasoning is not an evolutionary adaptation to get closer to the truth or accuracy or the best ideas. It’s a social adaptation. We use reasoning to win arguments.

By that standard, kids are reasoning no matter what, thinking what answer will get me what I need in this context. In math class that would be a good grade. In the real world, it would depend on why you wanted to estimate his time.

So I’d argue that it’s plausible that in both cases (straight multiplying and multiplying & adding a slow-down factor) the kids are reasoning, along the lines of:
“what counts as a good answer in this context? how can I best generate it?”

That’s a mathematically valid question. We want kids to answer this question differently if they’re in charge of handicapping the marathon runners, if they’re placing bets on the marathon, if they’re estimating what time to get to the finish line to see the kid cross it, etc. So “in math class” is just another one of those contexts, and it’s hard to say what would be a better “math class” answer than 26*6.

If the question is, “what media help kids do the sorts of mathematical reasoning about estimation and what’s close enough that they will need to do everywhere that’s not math class?” then I think this is a decently-formed and interesting experiment.

If the question is, “what media helps kids reason mathematically” then Ze’ev’s post raises the question, how can we say that 26*6 is not a well-reasoned answer? Jean Lave’s research about how kids see learning math in math class speaks to this idea…

For teachers who wonder what to do about this: a teacher friend of mine invented the “star fishy” a little star symbol to draw on your paper when you’ve answered a question but you know “something’s fishy here…”. We’ve noticed that kids with wrong or unreasonable answers often know “something’s up” but without being able to recover from that fishiness to get a answer they’re more confident in, they stick with the wrong answer and cross their fingers. I wonder how many kids would give this problem a star fishy?

Now I read the some of the comments and I find “students not using reasoning” and similar. Bull. They did. It is the form of the question that dictated their responses.

I agree, and think this is important. In fact, kids almost always use reasoning to come to their answer–however wrong it is. The interesting part of teaching is getting to the root of faulty reasoning. The interesting part of this conversation is deciding which is the faulty reasoning (if it is inn fact an either/or), reasoning that you’re in math class, so 26*6 must be the answer the teacher is looking for OR reasoning that the runner will slow down so the answer will be greater than 26*6.

So my annoyance was that people may conclude that “students can’t apply their math.” Perhaps. But this badly designed experiment certainly can’t tell us one way or another.

I didn’t read this. I never saw this as being about whether students understand the mathematical content d=rt, but instead about the students playing (or not playing) the school game.

How’s this for a study design:
I call N randomly selected people. Half get the following question: Hi. I’m hoping you could answer the following question. Michael Benson ran a mile in six minutes – his fastest time ever. How long will it take him to run a marathon? The other half hear the following: Hi. I’m a math teacher and was hoping you can answer the following question. Michael Benson ran a mile in six minutes – his fastest time ever. How long will it take him to run a marathon?

@78 Avery:
I never saw this as being about whether students understand the mathematical content d=rt, but instead about the students playing (or not playing) the school game.

Well, I obviously did not. Nor did some of the responders. However, you may be right. Let me try to roll with it.

You suggest two variants:

(a) I’m hoping you could answer the following question. Michael Benson ran a mile in six minutes – his fastest time ever. How long will it take him to run a marathon?
(b) I’m a math teacher and was hoping you can answer the following question. Michael Benson ran a mile in six minutes – his fastest time ever. How long will it take him to run a marathon?

Both forms seem to me identical in the critical element How long will it take him to run a marathon? and both have the uncertainty reduced to the same “I’m hoping that,” which anyway comes across more as a form of address than as introducing uncertainty. So, in my judgement, those are no better than the original forms suggested by Dan and at best may try to tease out empirical, but non-essential (i.e., psychometric), differences.

If I were to design such, I would use something along the following three forms (two open ended, one MC):

(a1): Michael Benson ran a mile in six minutes – his fastest time ever. How long will it take him to run a marathon (26 miles)?
(b1): Michael Benson ran a mile in six minutes – his fastest time ever. Can you estimate about how long will it take him to run a marathon (26 miles)?
(c1): Michael Benson ran a mile in six minutes – his fastest time ever. How long will it take him to run a marathon (26 miles)? (i. 26*6 minutes; ii. less than 26*6 minutes; iii. more than 26*6 minutes; iv. can’t say*.)

Incidentally, I suspect that forms b1/c1 will have somewhat similar distribution of responses, while form a1 will have mostly 26*6 as an answer. The difference between b1/c1 responses will be able to tell us something about how substantive is the difference between CR and MC forms.

And I would need at least a dozen or more of this type of item-sets before I’d dare to draw any conclusion, for dissertation or for my own use.

* I would probably also experiment with two MC variants, one with “can’t say” as the fourth choice, and one with something truly unsupportable like “26*6*3.” “Can’t say” is, after all, a somewhat reasonable answer.

@79 Christopher Danielson – I was going to write a response about the ‘didactic contract’ issues (Brousseau) here but you have articulated them beautifully.

I would only add a few clarifying points for those who haven’t been following the French-language and U.K.-based investigations of this and its importance for this discussion.

The didactic contract is the socially defined agreement between teachers and students about what kinds of knowledge, learning, and understanding will – or will not be – acceptable in the math classroom.

As Nunes and Bryant point out (they are developmental psychologists in the U.K. who study children’s mathematical development), we are discovering that children’s social and emotional conception of mathematics has a HUGE impact on how kids approach mathematical situations. The math classroom itself is a social situation, and most kids define this socially constructed world as having an inside and an outside which rarely, if ever, meet.

In the inside world – the life they know inside the math classroom – they attempt to apply seemingly arbitrary rules from a bewilderingly large catalog in an attempt to please the authority figure and avoid humiliation, pain, or emotional devastation. This amounts to an attempt by discouraged math learners to imitate certain ritualized actions demonstrated by the teacher in an attempt to please the authority figure (“succeeding”) and avoid the pain of not-succeeding (“failing”)

Meanwhile, in the world outside the math classroom, in their daily lives, many of these same kids will often exhibit an openness to becoming curious, to relying on their own powers of observation, reasoning, and representation, and to being receptive to exploring and experimenting with mathematical thinking that they check at the door of the math classroom.

In other words, for most kids, the social and emotional dynamic of the math classrooms comes down to two things: seeking pleasure and avoiding pain.

And this dynamic finds expression for many kids in a conditioned attitude of ‘learned helplessness,’ in which they abandon their own trust in their own powers of observation and thought in favor of trying to recall the rules they believe the teacher wants them to apply.

To them, the social system of the math classroom sends kids the clear message that math class is the place where they should check their powers of observation and reasoning at the door in favor of whatever memorized procedure will please the authority figure within (i.e., the teacher).

Those who succeed in pleasing the teacher will do well — and may also learn some mathematics.

Those who do not succeed in pleasing the teacher will be trapped between compliance and confusion, locked in a cage and subjected to a barrage of questions whose answers must be drawn from a bewildering catalog of rules/formulae/procedures that must be correctly reproduced in order to receive a food pellet and avoid devastating electrical shocks.

So basically, SBG, WCYDWT, and ANYQ’s are necessary (but not sufficient) tools and conditions for renegotiating the existing ‘didactic contract.’ And that will require that we also find ways to (a) help kids notice their conditioned responses of panic and learned helplessness when those feelings arise and (b) help them to rebuild trust in their own powers of observation, reasoning, and representation so that they can get back in the game and learn mathematics in ways that make sense to them — i.e., in ways that are grounded in their own reality-based observations, experience, and intelligence, rather than in various educated non-experts’ opinions and ideas about what children must “learn” in order to achieve mastery over mathematics.

For more information presented in a non-embattled way, see Nunes and Bryant, Children Doing Mathematics (1996) and Anne Watson’s fabulous articles, “Key Understandings in School Mathematics” (parts 1, 2, and 3), published in Mathematics Teaching 218, 219, and 220.

We are not training parrots here. Not on our watch.

– Elizabeth (aka @cheesemonkeysf on Twitter)

I’m curious about how other people would “solve” this problem. My approach was to look up the record times for the marathon and the one mile run (124 and 3.72 minutes, respectively) and solve the proportion x/124 = 6/3.72.

Oh! I think I see what’s angering you, Ze’ev. The straight-text version of Dan’s problem, if it were presented on a test, would leave many test takers feeling tricked. That’s a horrible feeling, thinking, “If I knew you wanted me to estimate, I would have done that. I thought you /wanted/ a stupid answer!*”

As someone who probably has a lot of stake in writing assessment items that cue kids to do what we want to assess, I can see how the feeling that Dan is deliberately trying to trap kids into doing the “wrong” thing would be upsetting.

BUT! Reading your thoughts on pictures made me realize… the care that we put in to writing perfectly formed test questions with no “distractors” could be part of what contributes to kids being in this terrible state of math class being only (as cheesemonkeySF so eloquently put it) avoiding pain and seeking pleasure by trying to memorize and guess.

When we are being incredibly careful to write questions that follow a particular set of rules, that’s what kids learn to attend to and memorize and guess based on. It’s brilliant learning on their part — if they learn the rules of the test writers, they avoid pain and get good grades. They learn: I need to ignore the diagram, I need to look for key words, I need to cross out distractors, etc. They don’t learn, I need to pay attention to the context, I need to make reasonable estimates, I need to use what I know about this kind of situation to check my thinking. Sadly, what they learn is how to “do tests” and not how to “do math.”

So there need to be some times when we explicitly break the rules with kids. Say, “this is not your standardized-test-writing-grandaddy’s math!” In this kind of math you need to pay attention to pictures and stories, use what you know, attend to friction and air resistance and getting out of breath and stopping to tie your sneaker and… In this kind of math Dave’s method of solving a proportion is as reasonable as he can justify it to be, and so is Dan’s method of saying, “definitely more than 26*6,” and questions like, “why do we care?” are fundamental questions, not distractions.

It would be great news if a picture could do that. I would be sad, but not totally surprised, if kids have learned (from taking too many well-written tests of the very kind that you are arguing for, Ze’ev) that pictures have nothing to do with math. And so they try to extract Dan’s text-based question from the picture.

2nd BUT! I also think that when we are assessing kids on standardized tests (something which the National Research Council has finally set is not an effective driver of school reform; maybe this is one part of why) we DO have to be really careful not to write mean trick questions. The text-based question Dan posed would be an evil one to put on a test, and I hope no one here is arguing for its validity!

It’s just that we do need to learn to ask different kinds of questions off of tests, and we need to do experiments just like this one to learn what helps kids to cue in that this is a different kind of question. It’s kind of like how we had to learn to write good test questions by studying how kids answered them, we have to learn how to write good THIS IS NOT A TEST questions by studying how kids answer them.

Executive summary: neither question Dan asked is the ideal one for assessing estimation and “real-world math” skills on a standardized test. They aren’t designed by the rules of that game. They shouldn’t be judged on how well they do that. Instead, they are a test to see if kids can get out of standardized test question mode when the question explicitly breaks the rules of that “language game” and how far the question has to go to break those rules before kids switch thinking caps.

*Though estimation is something we haven’t done such a great job teaching kids to do thoughtfully. Many kids, when told to estimate, do the problem and then change trailing digits to 0s

I agree that part of the reason students are answering 26×6 is because they think that is what is expected for an answer. If their life depended on it, they would give a different answer. Perhaps different presentations, such as a time lapsed video of Benson running in this case, will effectively prompt students that we are truly looking for an answer that makes sense in terms of the real world context.

However, I don’t think that the lack of sensible thinking is unique to application problems. How many times have you received answers that made no sense in the context of the mathematics involved — like an algebraic solution not making any sense in terms of the corresponding graph — or a positive value when the answer must clearly be negative? If you ask the student whether the answer makes sense, many times they can tell you that it doesn’t. Why didn’t they ask themselves if the answer made sense in the first place? Do students feel that they are expected to reflect thoughtfully on their answer and the mathematics involved? Or do the feel they are expected to write down some equations to show work and put a box around an answer? How do we prompt students that we are really looking for sensible answers to all questions?

I think that perhaps one fundamental flaw in Mathematics education that is clearly exposed by this “experiment” is that students are not exposed to the possibilities of solutions that occur in such “real” mathematics. Speaking of my own students, they enter my class believing that there can be only one “right” answer AND that the answer is the ONLY thing that is important and not necessarily the solution or process of developing a solution.

They fail to connect the math that they do with something in the real world that is being modeled by that math and therefore never consider the unreasonableness of there solution, though computationally correct.

At my local university, I have heard that one of the early grade math workshops that they conducted a local study of the effect of the current California State Mathematics Standards for K-2 and found that the focus is teaching students to rely too much on formulas and algorithms and not enough developing the fundamental concepts that support those formulas and algorithms. As a result many more students are leaving elementary school without knowing their multiplication facts or even recognizing that 2×3 can be represented also by 2+2+2 or 3+3 or any other way. Such important connections are not being made by students, locally; perhaps elsewhere.

So, by not having the experience of connecting the math that a student dose with modeling something tangible, how can we expect a student to think outside the box? It doesn’t fit in their formulas and algorithms, which was definitely evident in the data that I provided Dan from my classes, then they will never see a different solution or recognize that their solution is unreasonable or doesn’t make sense, and therefore they conclude that math is hard or confusing. In reality, math is much more than just “number crunching!”

@GregM
“Possible solution: Consider having the kids in the class run around the track/school (a measured distance that has them reach anaerobic threshold or max effort) 4x without stopping for recovery and time each lap. They will feel the effect on their body’s and see the time for each. Ask them if they ran 22 more times what they think would happen. You can also collaborate with the gym teacher for this activity and connect the dots later or next day in the classroom.” I love this and could easily make it work at my school – plus, my classroom faces the soccer field so it’s very convenient.

As far as the ! goes, I am the crazy, !!, energetic type so !’s appear often in my life and classroom. I would not be frustrated with the 26 miles at a 6mph pace answer as it does require deeper though, but AM often frustrated when I get (5)(0.5) = 25 and such. But, I am frustrated with myself, not my students because they know what I have taught them. It is those times that I think, how am I teaching wrong and how can I fix it? For them to not think, “Is my answer reasonable?” means that I have not deeply instilled in my students the correct way to think about math. What I would like to know is how as a teacher do I do this with 6th grade students? Because just telling them to do it (over and over and over) obviously isn’t working.

One of the things we want to discover by asking this is which question students will answer: “What answer do you think your math teacher wants you to give?” or “What do you really think?”

But if they choose to answer the second question and say it takes 26*6 minutes will we assume they answered the first question? How will we know that they really don’t understand the context of the problem?

Our goal should be (and I know Dan’s is) to work toward the day when students don’t expect their teachers to ask questions that are answered with a single operation.

Did/will we ever find out your research results on this?