There is no narrative to the peaches problem. But Dan’s escalator video has it in spades.

Thanks for the response, Kathy! Your main point is what I’m questioning. When I think of what my students engage with outside the classroom, it’s video games, sports and TV shows or music about lifestyles that my students will never lead. These things don’t seem to be relevant to them, but they still engage. I’d be interested in knowing more about what led you to the conclusion that relevance is required for engagement.

My best guess is that students will engage if there is a challenge that stretches them but is solvable and they will also engage if there is excitement. I aim for the first one in my class because I think it’s provides more learning than the second.

DaoudaW: This is the pattern I try to bring to the classroom. I try to demonstrate by my own passion the joy of pure math, but also try to present problems with enough authentic context that students will see it as a potentially useful tool.

Passion: good. I wish we could just assume that about every math teacher.

Authentic context: good, but pretty ambiguous. As I develop curriculum, I’m finding the representation of that context to be one of the trickiest nuts to crack. It’s easy for me to see that paper is a really poor container for what exists Out There. It’s trickier to take any of the examples from your childhood and say, “what’s the best way to bring this context into the classroom? what’s the best way to reduce the extraneous load on the student while at the same time delegating to her the responsibility of making sense of the math?”

I have been a HS math teacher for 16 years, teaching everything from pre-algebra to calculus, and believe we have done a disservice to our students and our profession by our insistence that math is useful in everyday life. Be it history, science, english or math, most of what kids learn in high school is not immediately useful to them. I tell students that math is a game invented by mathematicians which occasionally turns out to be useful in real-life.

I grew up on a farm and learned us much math in that context as I did in the classroom. I watched my grandfather square up a gate and learned the pythagorean theorem. I tried to figure out an optimal low cost ration for high producing milk cows and re-invented the rudiments of linear programming. But in the evening after supper before bed-time, my dad taught me to play chess and I discovered the joy of pure logical thought.

This is the pattern I try to bring to the classroom. I try to demonstrate by my own passion the joy of pure math, but also try to present problems with enough authentic context that students will see it as a potentially useful tool.

The main problem I see with the “canned goods” problem above is that with a bit of modification, additional information it could become a rich for authentic problem-solving. Add in some nutrition facts, some recipes, a budget constraint and myriad of interesting, non-trivial problems emerge.

I dig the motivation/engagement angle that seems to be a major thread here, as in Dan’s “students get the sense that translating the world into math is a joke,” and Matt’s “They fail because they [the pseudocontextual problems] don’t matter to them [the students].”

But “context” is about more than motivation. It is also about using students intuitions about the world to productive mathematical gain. People do think quantitatively outside of math class, they just don’t do it formally. If the math classroom can harness that extracurricular knowledge, then students can build coherent and connected formal mathematics. Plus we know that memory is aided by narrative and by connections.

The “Young Mathematicians at Work” series of books by Fosnot and Dolk offers a really nice definition of “context” that works on this second argument in favor of context over pseudocontext (and offers some lovely examples of pseudocontext to boot-although without using the word, of course).

I’d also argue it’s a matter of efficiency and effectiveness. How many of us have given the psuedocontextual problems to translate to students, only to see them fail again and again on later assessments? They fail because they don’t matter to them, so they don’t learn the skills. It may very well be that there aren’t enough WCYDWT problems to cover every content area of math, but if infinitely many problems fail to consistently teach the students the skill in the first place, while WCYDWT problems tend to be more successful, what’s the issue with focusing our efforts on creating more WCYDWT problems?

Mark: I’m not seeing such a disconnect of context here. Of course you would never worry about the square root of a price at a store. But I can absolutely envision a shopper noticing that there are twice as many tuna cans as peach cans.

Noticing there are twice as many tuna cans as peach cans is certainly plausible. Noticing that same fact plus another similar fact all without noticing the number of cans is incredible.

But the square root problem triggers your pseudocontext sensor so I’m curious, Mark, where have you drawn the line? How have you calibrated that sensor?

Mark: What alternatives are there for practicing translation when you run out of WCYDWT problems?

Word problems derived from real context are fine also. (Of course, many of those can be refactored as WCYDWT problems.)

What do you think the point of learning to translate is? If the point is to eventually translate actual problems of your own creation from your own world into mathematical expressions, then whatever fluency is gained through pseudocontextual translation must be judged against what is lost when students get the sense that translating the world into math is a joke.

This is from the 2009 Texas state text for Grade 10 students.

Mandy bought a bag of peanuts to share with her friends.

Trisha received 1/2 of the peanuts in the bag.
Vince received 1/4 of the peanuts Trisha received.
Ray received 1/3 of the peanuts Vince received.

If Ray received 4 peanuts, how many peanuts were in the bag Mandy bought?

If you want to see a huge improvement in the types of questions we see in textbooks, get involved with The Partnership for Assessment of Readiness for Colleges and Careers (http://www.fldoe.org/parcc/) and/or Smarter Balanced Assessment Consortium (http://www.k12.wa.us/smarter/). These two organizations are developing assessments for the Common Core Standards. The second group provides e-mail addresses for representatives from each state, so you can make your voice heard.