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!