Thanks for the clarification! I love all the stuff you’re doing on here, but I wondered if this trend would get us too far away from the “math for math sake” artistic side where the beauty lies within and not through some extrinsic motivation of “application.”

I think one of the primary values of math is the habits of mind that it develops (persistence, precision, reasoning, concentration, etc.). When taught well, mathematics affects all areas of life because of the mental processes it forges.

Hey Dan, I feel like there is an element missing from this whole discussion on Real World Math, Pseudocontexts, and Pseudoteaching. Paul Lockhart wrote an amazing article on math education that the MAA has published on their website (http://www.maa.org/devlin/LockhartsLament.pdf). In the article he argues that the true heart of mathematics isn’t finding the solution to a problem, it’s the creative process by which the problem was solved. The job of a mathematician is to take a problem and through a creative process find a way to view the problem so that its solution becomes evident. The real mathematics is in the creativity, not the ultimate solution. Whether the problem contains physical “real world” elements or is completely abstract is irrelevant, it’s the creativity that is key.

I agree with Lockheart about what real mathematics is. I also think that viewing math in this way gives us insight into why problems with physical application yield such success with young students. At the highest levels, the tools of mathematics are essentially definitions, axioms, and theorems. When a mathematician encounters a new problem, she approaches it much the same way that a painter approaches a blank canvas. The mathematician must engage in a fundamentally creative process. She uses her tools, like paint, to give the problem structure and form… to give it scope. Weaving together her creative medium (definitions, axioms, and theorems) she creates a portrait that, when complete, shows the whole picture in a new light that makes the solution clear.

At the elementary, middle, and high school levels mathematics is the same thing. The difference is that at these levels they don’t have the same tools. These students aren’t at a place where they can truly grasp definitions, axioms, and theorems. Sure, they can get a taste of them. But, they don’t have the mastery necessary to use them to paint beautiful landscapes. They can really only manage stick figures. With geometry proofs for example, students often greatly struggle just to keep the concepts and theorems within their minds. The amount of energy required to truly grasp the content and concepts within four theorems at once is substantial for students at these levels. There is little energy left over for the creative process. It’s like we are asking them to hold fifteen paint brushes at once when they really haven’t mastered using one yet.

It is in this sense that I believe “real world” problems can truly turn a math classroom around. Kids experience the “real world” every day. By giving students physical “real world” problems, we are handing them a creative medium that they can instantly grasp with mastery. Students don’t need to exert any effort to keep the concepts of basketball and gravity clear in their minds. By employing concepts that they are familiar with we are allowing them to focus all of their energy on the creative process. They don’t need to keep reminding themselves what the definitions are, what the concepts mean, what their goal is. The question becomes simple “Does the basketball go through the hoop?” This is where the true power and success of “real world” problems lies. Students can thrive in the creative mathematical process without also needing more short term memory then should reasonably be expected from individuals their age.

Now how does this play into pseudocontextual problems? Well, I would like to propose that we have been focusing on the wrong elements of math problems when we look to determine whether they are of a pseudocontext. Most of the time when I hear someone label a problem pseudocontextual, they are looking solely at the “story” behind the problem. If a problem involves aliens, tarzan, or is worded in some overly fanciful way, the problem gets labeled psudocontextual. Also, if the problem appears to have no application it also gets labeled. I don’t believe that the story behind a problem, or how applicable it is, should be part of the conversation about its worth. A problem’s worth should be based on the level of creativity it demands from the student. Does the problem give them room to engage in the creative process of mathematics? If mathematics IS a creative process, then no math can be accomplished in the absence of creativity or creative potential. Are we handing students a blank canvas that they can express themselves on? Can they be proud of their work of art? Or, are we handing them a paint by numbers picture instantly killing any chance for creativity and robbing them of ownership and pride that they could, and should, feel in their work?

For example consider the following problem: “Is it possible to create a sequence of points such that the number of lines containing exactly four collinear points each, is greater than the total number of points squared?” This is a tough problem… It would require a lot of creativity to solve this problem! I would argue that this is a real math problem that has real value to the student that attempts it. However, I could also ask the problem like this “Benny is an alien who has a spaceship. Benny’s spaceship can fire missiles that can destroy planets! One day while flying along, like aliens do, Benny accidentally fired a missile that destroyed four planets that were in a straight line right next to each other. Since Benny is a very curious alien, he wonders if there is a pattern that he can arrange ALL of the planets in such that the number of times that he can hit exactly four in a straight line is greater than the number of planets squared? Can you help Benny?” OK, so at first glance I bet that anyone who read this problem would pull out their psudocontext stamp and go to town. But, this is the same problem listed above, and does actually have value. There is a lot of creative potential in this problem. Should we throw it out just because of the super dumb story that it’s wrapped up in? Should we unpack it before we give it to students? I don’t have the answers, but I believe that we need to be more careful when labeling a problem to be pseudocontextual solely based on the story that it’s presented within and not on the level of creative value that it holds for the student.

For example, you recently posted a project that you created that involved burning toast. I thought the project was great. It required a lot of creativity on the part of the students. It was wonderful mathematics. However, because studying burnt toast didn’t seem very applicable or of much value, some of you Blog Patrons criticized you for creating a problem of pseudocontext. That’s a bunch of bull! You can’t label a problem pseudocontextual because it’s wrapped up in a story involving burnt toast!

Here is one last example at the other end of the spectrum. This is a problem that is absolutely of pseudocontext! “Jane and Kim are friends. Jane is twice as old as Kim and the sum of their ages is 30. How old are Jane and Kim?” Why is this such a terrible problem? What is its defining characteristic that makes it bad? I have heard people claim that it’s bad just because it’s so unrealistic. In what world would you have access to this kind of information about Jane and Kim but not their ages? Asking students this kind of problem will almost certainly elicit the response “Why don’t you just ask them their ages?” (Note that in answering the question this way the students have indeed found a more creative and elegant solution!) I would like to argue that this problem isn’t bad because it’s impractical, although that is a fair criticism. It’s a bad problem because it involves absolutely ZERO creativity! When the story is removed the problem boils down to “Solve the following system, J=2K and J+K=30.” If we want students to solve systems of equations, give them systems of equations to solve. If we insist on giving word problems, then we need to find meaningful problem that really do require solving systems.

How can we turn the above problem into a problem that requires creativity? How about this “Jane and Kim are friends. You are allowed to ask them two questions. Your goal is to find out what their ages are. You are not allowed to ask them their ages directly. What two questions could you ask, and how would you use their responses to ascertain their ages? What if you could only ask them one question? Would it still be possible to find their ages?” This problem gives plenty of room for the creative process, and thus real mathematics!

I hope that the idea of creativity being at the core of mathematics can begin to show up in the discussion of pseudocontext and pseudoteaching. I think that there is so much value in considering the creative potential of what we’re asking our students to do. I also think that there is value in considering the story that we wrap our problems up in. Can we give a problem a story that students can instantly grab hold of, like basketball? Can we call “Real World Math” math that requires creativity, and “Fake Math” everything else? I hope that my ramblings were of some value. I truly enjoy reading your posts and I appreciate all of the work that you do! Thanks for listening!

@Michael: I like what you’re saying about pseudocontext —

“OK, so at first glance I bet that anyone who read this problem would pull out their psudocontext stamp and go to town. But, this is the same problem listed above, and does actually have value. There is a lot of creative potential in this problem. Should we throw it out just because of the super dumb story that it’s wrapped up in?”

I think keeping the alien story (in your example) in the problem, though, cheapens it. Students know they’re not likely to be on an alien ship destroying planets. They groan because someone is trying to force a context that doesn’t exist on a problem that doesn’t need one. This is harmful to mathematics.

When we try to give everything a context, we end up making up the context to fit a problem, rather than the other way around. Students aren’t stupid, and if they see that every problem has some context, but some are real and some are fake, they’ll throw out the real math that is done with the fake context.

So in your example, I would say don’t get rid of the problem — it does ask for a bit of creative thinking and there are good things in the process to solving it. But scrap the context. Just ask the problem, as is. We don’t need to lie to our students to trick them into liking math with phony contexts. It doesn’t work.

I read this article by the way, it was what guided me to this blog in the first place. I think we as educators (not just Math teachers) strive to be “real-world teachers” because we are dealing with students who aren’t going to learn for learning’s sake. I think that might be the bigger issue. I feel like my students constantly need to be learning to build an engine in order for any of my lessons to help them “outside” of class. However, explaining to them that Math is a piece of the puzzle, not the entire project is hard for them to conceptualize.

I like the idea of putting Math into real context, problems that the students could reasonably solve. However, I don’t think that a student who wants to farm needs all farming examples, or a art student needs all art examples. I think that the world is too vast for a high school freshman to have decided their entire path in life.

My point here is that this is not just a conversation for Math teachers. This is an educational problem. Real world is good, but context is better. We need to give work that solves problems, problems that the students value. This is why I like the approach on this blog, students rarely are going to need to measure the speed at which a team counts beans, but the math inside is valuable. We just need to give a context that students care about to help them to learn the valuable math inside.

I have a bigger concern that mathematics is devolving into engineering. Mathematics is not arithmetic. Mathematics does not have to have an application. There are pure and applied fields of mathematics, and pure mathematics is not of less value than applied mathematics. Moreover, engineering, divining formulae to find arithmetic answers to known scenarios, is perhaps the least creative of the mathematics/science/engineering trio.
I cannot argue that this is perhaps beyond the scope of high school “math,” but I am concerned that students in high school, and many of my fellow teachers, seem to think that mathematics means calculation.
The overwhelming adoption of “graphical solutions” in algebra, bypassing the logic of algebra, seems to emphasize engineering over mathematics. The point of doing the algebra is to learn the way the math works, not the way to design a bridge.

The truth is, in life, there is a lot of variety to what I’d consider genuine mathematical activity, and I believe we should help students to make contact with that variety. No one owns mathematics, and can say with great certainty what it is and what it is not. The purists and the realists are fighting the wrong fight.

I tend to find psuedocontext annoying when it seems to be trying to be a “real world” application of a problem when it clearly is not.

I also place a lot of value of math itself being a context.

However, that being said, I do see value in “dressing up” problems now and then for fun, especially in testing situations.

I see two advantages of doing this:

1) It can be more entertaining for the students and help alleviate stress. If there is no pretext that a problems is supposed to reflect a real situation there are no sore feelings at psuedocontext. The students (in my experience) understand it is just for fun.

2) Sometimes a narrative, even a silly one, can add clarity in reading the premise of a problem just by the way our brains work.

For example, Michael Lomuscio’s alien problem is a good example. I found his alien narrative easier to read, just because there was a narrative around it. And it was mildly entertaining.

I’m not suggesting that we train kids away from thinking math can be fun for it’s own sake, but sometimes it’s fun and helpful to be a little silly.

I know on my last test for 8th graders I had a really silly context. You could hear a giggle go around the classroom as the students got to that problem. I got a number of comments written on test showing that the students appreciated the comic relief.

In that particular problem the premise would have been fairly confusing and difficult to make clear without some sort of narrative. There was no “real world” narrative that I could think of that would be forced, so I went for fun and silly, but adding clarity at the same time.

Maybe I’m doing something wrong here and perpetuating bad habbits, but I’m really not quite sure how I could have done it differently in this case.

On top of all of these interesting points, is the fact that the ideal balance between “real world” and “abstract/pointless” is different for different ages and types of students. Most children don’t have any difficulty seeing that the basic arithmetic functions are or could be useful in their daily lives (which doesn’t mean that they don’t resist the practice necessary to master them), once they are shown a few examples of how that usefulness plays out. Once we’re on to Algebra and Geometry and beyond, the fact is that students have to apply themselves even when the utility begins to look cloudy, or they’re not sure their future studies will require that much math. Or they are sure, but the math will serve as a credential rather than a constantly needed skill, At that point, we just have to assume that some students will have the guts to keep on keeping on, but others won’t. We do our best to keep the classes well-taught, but some students weigh the costs and benefits and then opt out.

I see a number of people equating “abstract” math with the “pointless” exercises kids do, and I think that’s common in educators’ minds. When I teach, I try to teach real problem-solving by giving the students real problems like the four points/planets mentioned above. I have found that when I pose an interesting, accessible problem, abstract or concrete, the students get completely absorbed and forget themselves, and never ask “when will I use this.”

Proud of you, cover boy.

I’ve been thinking about this recently – good contrived non-real world math problems are “MAPS.” And like geographical maps, what they lose in detail, they gain in breadth.

Real world problems which involve mathematical problem-solving are typically very particular, and the skills one uses in solving one of them do not necessarily transfer to other problems.

Contrived, textbook problems are typically designed to embed a collection of problem-solving skills which would be broadly applicable in solving any one of 100 “real-life” problems.

So – complaining that textbook problems are not the same as “real-life” applications is equivalent to complaining that you can’t see your mother waving to you on a map of the United States. True, but that’s sort of the whole point of a map!

Rich, the map example is very terrific, and I have to say that much of what you say is genuinely true. However, I still have this problem of presenting these problems to students as “real world”. A problem that a student isn’t going to use outside of a classroom is not necessarily a bad thing in the right context. However, I have told my students many times this year that presenting them information that they won’t use and telling them is “real world” (like my textbook does) makes a liar out of me. I believe a classroom relationship is created out of a mutual trust, and I feel like I, the teacher, have a lot to lose in how I present information.

In short, I think application problems need to really be real application problems. You can have word problems that don’t apply, just don’t label them as such. Students need to be prepared for their future classrooms and future lives. I think we can prepare for both.

Hi Dan and Joshua,

Thanks for responding to my comment…I agree with you that there is certainly room for improvement in the modeling component of math education and that honesty is a vital aspect to classroom management.

I recently read a study by Susan Gerofsky of University of British Columbia in which she interviewed 50 math students and math teachers and not one of them saw any point to contrived math problems. This viewpoint sometimes works to drive these problems out of the curriculum – but to be replaced by what?

Any time I have been involved in “real-life” applications, I recognize the situation as some combination of various aspects of the contrived problems I studied in class. The work I did on the contrived problems then provide me with the tools I need to solve the “real” problem. Keep up the good work!