I love this. It seems many of the problems you post use mathematics to describe “real” situations, but they’re situations that embody great math, not “real-world applications.” I don’t think anybody would seriously use the orb on their iPhone maps to figure out how fast they were going, but it opens a space for some really interesting math!
i have 3 kids that are venturing out daily to capture real world stuff in the name of math. their idea.
they were so psyched their first trip out, got some great footage, made a cool video. worked on it during our 2 snow days even. but doing something they would never do in real life, in order to show.. they did the math.
so – we’ve talked a lot about, if they’re looking for math, the math they have been taught, they probably won’t find real stuff. but that if they ignore the “math hunt” they can’t not find math/mathematical thinking/real life science/etc. it just won’t look like they are doing school math. and it won’t be doable by the math standards – as we do them in school. it will be much messier. and it will involve a ton more thinking.
i can’t wait to see how this shakes out.
> Math is only interesting in its applications to the world
Sometimes I wonder if all application problems send this message. If a student only perks up when I give them something that relates math to the real world, then isn’t it natural to conclude that math is only interesting in application?
A likely response to this is “A good application problem reveals that the math itself was interesting.” I’m not sure about that, though. First, we usually have to teach the math before the application. Second, I think what we really could be doing is convincing students that there are good applications of most of our math, but, umm, yeah, a lot of the subjects we teach in high school math are resistant to great application problems.
A likely response to that is “Then I think we really need to reconsider what we’re teaching in high school.” That’s fine as far as it goes, but I’m a teacher and I work with what the state gives me.
I think that we might have to move past the idea that “applications” involve “real world” applications. There are some genuinely cool scientific and pure mathematical stuff that we could be exposing our students to. Applications don’t have to be about things that you could encounter in the course of daily life.
I understand that this is a feature of WCYDWT problems. A major feature of WCYDWT is that by being part of the “real world” a student can have a stake in the answer and math will be defanged. But (a) this can be disconnected from the notion of application problems and (b) I’m convinced that we could still invite guesses and intuitions with scientific or purely mathematical applications.
I recently had students interview professionals about what math they use. We, as a class, saw a few trends:
1. We hardly ever use estimation in our math standards, but we use it daily in mental math. Yet, estimation was a skill used often (same with checking to see if the math “makes sense”)
2. Analyzing graphs and visuals in comparison to the math itself. In other words, we found professionals had check visuals and graphical displays to see if they were accurate or misleading.
3. Numerical predictions and statistical analysis. Often, jobs that we hadn’t considered “statistical” jobs used more statistics than we had assumed, which leads to the next point.
4. Every job had math, but it was often areas outside what we would expect. Kids thought that mechanics would use geometry, when in fact, they often solved algorithms, used estimation and analyzed data. Meanwhile, the custodian could rattle off the equations of volume and surface area, because he’s used them in some incredibly random situations.
5. There were some really interesting math skills connected to life skills that are never taught (which type of mortgages exist, for example).
6. Every professional (from doctor to sweeper) could talk about math, but just about all of them said that the math they learned in school was useless (a little hyperbole there).
It makes me think that textbook companies could create decent word problems by simply interviewing people at their jobs. It’s how screenwriters and authors learn context and ultimately a word problem is essentially a story (conflict, characters, plot, setting and hopefully a common theme of relevance.)
I understand why pseudocontextual problems are especially bad, though. The student says “Why on Earth are we wasting our time with this problem if it’s stupid?” and moves from there to “It must be because this is the only sort of thing that will make it interesting.”
In contrast, a problem with good context doesn’t motivate the question “Why are we learning this?”
You might not use the orb, but your car-rental company does. They’ll assess a fee if the time between points calls for an average speed above the speed limit.
To the picture: That is unfortunate and unfortunately, typical, but probably speaks to the need to get another woman into the textbook to keep up with all the quotas.
To the general group: One reason that math seems so dry is the level we’re teaching. These kids are just beginning and are learning the moves.
Simple problems devoid of “interest” and limited in their solutions are where they need to begin. Trying to jump right in to the deep end can be a way to teach, but you will lose a couple to drowning, a couple more to a life-long fear of water and most will flounder around for much longer learning and practicing bad habits than would if you showed them the reason why and then taught them how to swim in the shallow end.
Applications require that you have some knowledge and ability and can choose a method that works.
Real-world applications have details and decisions that are quickly eliminated or decided by the real-world solver, details that would overly confuse the issue for the beginner. What remains is, of necessity, rather dry.
Insisting that ALL problems be interesting is a short journey to madness. Understanding that the dry, mechanical problems must lead to more interesting ones, and knowing when to make that transition, is the key to education.
Of course, it doesn’t help that textbook writers and editors are, by and large, not in the fields they portray in pictures. Attempting to formulate a real-world problem in a field you know nothing about is difficult. If those authors could speak to the custodian, lawyer, etc., put all the problems they present into a database and then sort by required pre-knowledge, it might be a good thing.
>If a student only perks up when I give them something that relates math to the real world
I must not have the same students. Mine only perk up if the problem is interesting, regardless of whether it has a real world connotation or not.
I think Psuedocontext is the result of what would normally be a rookie teacher mistake: trying to directly respond to “When am I ever going to use this?” They don’t really care about the answer to that – they’ve just learned that it’s a tool they can use to get the teacher off track. And they’ve learned to use it so successfully that they’ve managed to get the textbook companies off track too.
This is so true. I recall some high school textbook projects I worked on in the past where the pedagogy of the book required a “real-world application” in every section. This is silly, and the consequenses are disastrous for thoughtful students, as your post so eloquently explains.
Are there practical applications of lterature? Sure there are, but we don’t emphasize them when we teach Shakespeare. Literature, art, music, and poetry enrich our lives, and we learn them for that reason. Mathematics and science, taught well, also enrich our lives. We must stop teaching vocationally, and focus on teaching so that our students can engage with the mysteries of the world more deeply.
Philosophically, I’m with you. Practically, I struggle mightily with this.
Dan, with what percentage of your students were you successful with a “mathematics that enriches our lives” approach (to borrow Santo’s phrasing)? And, can all of us here agree that Dan is likely to be more successful than the mean math teacher? (pun partially intended)
I think all of us would agree that there’s more to mathematics than simply practical applications, just like all of us would agree that students should have the opportunity to take art classes. But would we agree that all students be required to take four years of art in high school and demonstrate proficiency by way of a standardized test?
I guess my point (if there is one) is that the “Lockhart’s Lament” argument about high school (in particular) mathematics seems to me to ignore the students that are sitting in front of us and the fact that they are required to take our classes. Is it possible that for many of our students math *is* only “interesting in its application to the world?” I’d like to see less discussion about pseudocontext and more discussion about whether our students should be required to take this stuff at all (sorry MBP, but that’s where I’m at).
(As a side note, it’s kind of fun to be on the other side and be the “yeah, but” guy for a change. Well, it was fun until I hit “submit” and you guys rip it apart. And, yes, I’m probably having a mid-teacher-life crisis.)
You can’t learn to swim without learning the strokes to be sure, but swim lessons would suck pretty hard if you weren’t ever allowed any time to play around in the water.
The notion that simple and dry are always and necessarily equivalent is pretty weak, and in the classroom, attacked with two prongs:
With real world applications to simple problems, students already have lots of intuition to build on (or dismantle, if the intuition is false) and thus success in these areas is easier to engineer, and the level of difficulty should be raised as the lesson progresses. That’s what most of the WCYDWT entries are about–real-world-related content appropriate to the audience.
Problems that just don’t lend themselves to real-world applications until much later (i.e. polynomials) can be explored with meta-cognitive and summarization activities that include group work and class discussions. (A lecture about “this is a polynomial” can seem much less dry if augmented by a really good activity–more than just a worksheet or problem set–that lets the students explore the topic on their own.) Here the level of difficulty can and should be fine-tuned as well. This stuff is well-covered by the article Dan shared the other day.
Back to the analogy, you have to agree it would be worse for them if they were never even expected to swim for themselves. Let’s be clear that throwing swimmers in the deep end on the first day is absolutely not what’s being suggested on this blog.
To bring in another analogy, a teacher is expected to both provide the training wheels and oversee their removal ASAP.
I agree. I am teaching a class right now (OK, it’s physics, but the same kinds of issues arise), and I do my best to make the material accessible; a big part of that is to help students make connections to their every-day world (which includes all the gadgets they use). But I also (when I can) bring in the “deeper mysteries” discussions. I know many students don’t care about this, but some do, and I do believe that it is worth doing.
So we all do our best in the meantime, but ultimately the system must change. Forcing students to sit through 12 (or more) years of mathematics does not make any sense at all to me. Have them study something that may be of greater interest, and probably of greater value to most. For instance, most students graduate high school with very poor reasnoning ability, and can’t write very well.
There is some “research” showing that student’s reactions to pseudocontext problems is negative. I would even argue that they may harm students views about science and math. Take these problems for example:
Be sure to check out Mazur’s entire presentation. IIRC, there’s a part which contradicts Dan’s notion of “be less helpful” when it comes to pictures, diagrams, and photographs. If the diagram is TOO detailed, eye-tracking devices reveal that students are not focusing on the relavent aspects.
Anyway, I don’t know what’s worse: measuring dog bandanas, pulling penguins, or drawing underwater?
I’m interested in the connection between #1 here and your WCYDWT problems. I’m also interested in how there doesn’t seem to be an analog to WCYDWT for #2. Especially in Algebra 1, where so much feels like things you will “never do in real life” I’d love some ideas on how to inspire interest in the math for math’s sake.
Killer discussion; wish I had more time to engage.
Karl: I think all of us would agree that there’s more to mathematics than simply practical applications, just like all of us would agree that students should have the opportunity to take art classes. But would we agree that all students be required to take four years of art in high school and demonstrate proficiency by way of a standardized test?
No room to pile on here, Karl, but it just seems to me that we should engage separately the issue of whether or not math should be elective and whether or not math can be interesting to students apart from its applications to the world.
Katie: I’m also interested in how there doesn’t seem to be an analog to WCYDWT for #2. Especially in Algebra 1, where so much feels like things you will “never do in real life” I’d love some ideas on how to inspire interest in the math for math’s sake.
Mr. K: Mine only perk up if the problem is interesting, regardless of whether it has a real world connotation or not.
Let me try to add some color to “interesting.” If I can introduce a student to a cognitive conflict, then the student will be interested in the work. A cognitive conflict occurs if I can introduce a problem — real-world or not — that nails the boundary between a student’s zone of current development and a student’s zone of proximal development.
Let’s say a student is an ace at finding the next term in an arithmetic sequence 4, 7, 10, 13 …. “Find the next one.” She’ll do it. “Find the next one.” She’ll do it. She’s super confident. Then you ask her to “find the 100th number.” You’ve tapped into a cognitive conflict that supplies a lot of interest and motivation.
Mr. K: I think Psuedocontext is the result of what would normally be a rookie teacher mistake: trying to directly respond to “When am I ever going to use this?” They don’t really care about the answer to that – they’ve just learned that it’s a tool they can use to get the teacher off track.
It’s also (and I’ll argue it’s largely) a response to powerlessness. If a student feels competent, she won’t ask that question — even if she’s analyzing end behavior of rational functions.
“Imagine traveling down the interstate with no police in sight — but being charged by your rental car company for speeding. That’s what happened to customers of a Connecticut car rental company, and state officials don’t like it.
“I feel that my privacy was invaded by being tracked across seven states,” says James Turner, who was charged $450 for allegedly speeding three times. “And now I’ve got a car rental company acting as a state trooper.”
After that lawsuit, many states jumped to legislate against the practice, but others allow it as long as the fine print mentions it. Regardless, the math is there.
As for the bad habits. A student can get the wrong idea if the result seems reasonable and a dimly remembered idea is reinforced by a coincidence. That wrong idea can maintain a fierce hold. 26/65 = 2/5 because you can cancel the 6s? If you let it go then you will get (x+1)/(x+1) = 1/1 because you can cancel the x. Then you’ll get that (x² – 1)/(x-1) = x+1 because the x cancels and -1/-1 is +1.
Which is great if you only look at the answer but not the method. And you never give them a problem that doesn’t work that way.
Another one I dealt with was “cross-cancelling”. Even if the terms are on different sides of the equal sign? In the beginning of the year, I consistently got that reason for a problem like: 12x/39 = 13/24. They cancelled the 13 and 39, the 12 and 24, got x/3=1 so x=3. They confidently told me that their previous teacher told them to always “cross-cancel” so I had to be wrong.
>I must not have the same students. Mine only perk up if the problem is interesting, regardless of whether it has a real world connotation or not.
Just to save face a bit: I wasn’t saying that my students only perk up if the problem is a real world application. All I was saying was that the “Math is only interesting in its applications to the world” misconception isn’t limited to pseudocontext. Rather, it’s a result of relying too exclusively on real world applications in general, even if they are totally appropriate.
My experience with my students completely coincides with yours.
Thank you. I’m sending your post to my sister who’s taking a prerequisite math course for a program she’s applying to, and in her words, learning to “factor polynomials on demand” and manipulate algebraic expressions such as “x/ay” which represent bogus quantities such as the number of hours spent babysitting.
Being a former math teacher and a math enthusiast, I am forced to apologize for my profession thus:
Her: Why are the problems so stupid?
Me: I don’t know. I’m sorry. I’m sorry. I’m sorry!
Did WordPress auto-post here (above) because I linked to this page in my blog post? I was coming here to thank Dan and Curmudgeon for the conversation that put me on to that speeding case, which I think is a pretty compelling problem for a high school student. Then I saw that something had already been posted here. I think I am failing miserably at technology use.
I’m sorry, do other people not shove pigs down slides? How do you have a good time then?
Seriously, these have to have come from the book my daughter used in college. She took my suggestion that the students use their cell phones to record water balloons being lobbed at the professor, and proposed it in class but sadly he would not cooperate.
We did go to the store at 10pm to get a helium balloon, then did donuts in the parking lot ( it’s pretty empty at 10:30pm on a Wednesday), but once again when i sent the balloon in with her, he had no interest in either trying the experiment as a demo for students or in encouraging the students to try it themselves.
I love hands-on physics, and my daughter described her Physics course as “watching someone do math on the board.” So sad.
In high school, physics=algebra applications = physics. If we combined them we would get some good work done. So far I have had no takers, because we do biology first ( it does begin with B).
“Everything is said by an observer.” -Humberto Maturana
What is interesting? What is practical? What is real(-world)? I can read most all of the comments on this page without considering Maturana’s statement and be equally puzzled by the dilemma captured by @Curmudgeon, “Insisting that ALL problems be interesting is a short journey to madness.”
But Maturana reminds us that we must first get beyond our own eyes when defining interesting. Of course, what is interesting to us is not universally interesting. And further, their probably is no “universally interesting.” (I make this claim on empirical evidence from my classroom, dead certain I had found the perfect problem to pose to my students.)
Maturana reminds me that we must see the world through the eyes of each one of our students to consider what might be interesting, what puzzle might create a desire (need) for the student to solve it. So, the power in a WCYDWT sort of approach, to me, is that the student’s identify the puzzle/problem they wish to pursue. The teacher corrals this energy and sneaks in the “mathematics” that must be “covered.”
And finally, applying Maturana’s comment to mathematics, and maybe “high school mathematics” as @Karl describes it. I would say it is worth noting the difference between “high school mathematics” and the “mathematics of a high school student.”
The mathematical ways of knowing of a teenager is different from our own. To say, “These kids are just beginning and are learning the moves” seems to assume we know them. Who is to say we to are’t just beginning to learn the moves.
Trying to bring this esoteric approach to my comment toward something tangible, I suggest teaching mathematics in such a way that we pay attention to what we know, as “keepers” of the discipline, but we strive to make our HS math courses about the “mathematics of our high school students.” Our two primary roles would be to encourage the emergence of this sort of mathematics, and then to listen well, and work to connect it to the ways we know mathematics, what we consider to be the Discipline, that is “high school mathematics.”
Psuedocontext: agreed @Raj, brilliantly concise deconstruction of this lazy approach to curriculum publishing.
@Dan: a problem, naming psuedocontext and tearing it down makes us want to ask then, what then is real context? We seem stuck to defining “real context” in the moment/context of the learning environment. And as such, can curriculum only be created after it has been experienced? Which at best then we are stuck with a prefabbed sequence of psuedocontextual problems, psuedocontextual for any student thereafter?