I haven’t tried algebra squares yet so next time i plan on trying them or the more visual / manual presentation as shown in the first video on this blog.

http://letsplaymath.net/2010/11/08/how-to-be-a-math-genius/

f(x) = -4.9x^2 + 20x + 6

While it’s possible to invent situations where this quadratic factors, as you say, they’re jury-rigged; almost all such situations would be best solved via the Quadratic Formula, which (in my opinion) shouldn’t be the first method presented to students.

There are some other possible contexts, but I don’t really feel any of these are true “grabbers”…

– Use the historical origin of quadratic equation solving: problems like “Add 10 roots to one square and the sum is 39” (equivalent to solving 10x + x^2 = 39). This can help students understand the origin of the phrase “completing the square”.

– Key on students’ equation-solving skills (and lack thereof). An equation like x^2 = 10x – 21 can’t be solved using any means the students know about yet, and factoring is the key. I feel that students should learn the Zero Product Property before they learn factoring; most books do this the other way, but using ZPP to solve an equation is the place factoring is truly useful.

– Relate factoring to factoring. The factoring x^2 + 5x + 6 = (x+2)(x+3) is really, really close to the factoring 156 = 12*13. There are lots of these, and a large number of multiplication “tricks” boil down to clever use of algebraic factoring, especially difference of squares. For example, 33 x 27 = (30+3)(30-3) = 900 – 9 = 891.

– Solve “sum and product problems”. Again, not a true context, but kids seem to really enjoy answering questions like “What two numbers add to 10 and multiply to 21?” Spending time on these, without the algebraic overhead at first, makes the algebraic side much easier and helps students picture what is happening when they expand (x+3)(x+7). Also cool: ask students to add and multiply things like (6+sqrt(2)) and (6-sqrt(2)), which can help them understand why the Quadratic Formula has to look the way it does.

– A stretchy context is area and perimeter problems: find the dimensions of a rectangle with perimeter 20 and area 21. I don’t like this as an introduction since the perimeter number hides the “sum” and some students don’t quickly see the correspondence to sum 10 and product 21.

– Algebra tiles can provide another context / representation, but beware: the tiles do not work well for fractions, negatives, or large numbers, and they do not work at all for unfactorables or for anything larger than quadratics.

I hope this helps. I’ve found students are plenty willing to play with mathematics in areas without a true context, as long as the thinking and work is interesting and rewarding. What other ideas do people have?

Maybe there’s an underlying premise that I’m missing. By trying to create pseudocontext, we’ll avoid it?

Bowen: Here’s a question that came up “in context” the other day. My son’s day care has a digital photo frame on “shuffle” mode, where it randomly displays a picture – my son is fascinated by this and won’t leave the day care. We watched a number of pictures go by, and the 20th picture (estimated or OCD, you decide) was the first one that matched a previous picture. So the question: about how many pictures are in the shuffle? (Harder: find a 95% confidence interval…) It’s interesting that it’s a context, but if it had been my own photo frame, it instantly pseudocontexts since I know the real answer…

Love the problem. I just want to register my vehement disagreement with a theme that’s recurring throughout this thread, that if the answer is known or can be observed or attained by trial and error that the problem becomes pseudocontext. Indeed, it’s extremely satisfying for students to commit to an interesting problem and find out that their solution agrees with the actual answer, that math is consonant with the real world. That satisfaction is only irrelevant if we teach math not for an understanding of the undergirding structure of the world, but to solve some logistical problem for a student. The latter is important but giving it exclusive focus in our math classrooms cheapens math.

I love Chris’s pseudocontext example above but, again, it isn’t pseudocontext because the answer is known in a table or a newspaper – that’s great – but because there’s nothing about wins and losses that lead inherently to solving equations.

What I meant was: Do we really have to show them story problems in beginning algebra? It’s hard to find good ones there, and perhaps we should wait until they have enough tools for a realistic problem to be solvable?

I am teaching factoring polynomials right now. I started by throwing chalk in the air, and talking about how gravity is an acceleration, making the x2 term. I’m eager to get to the step where we solve by using factoring.

The book has a whole section on ‘problem solving using factoring’, and it’s abysmal. There is one problem in the whole bunch (of about 40) that I’d want to use.

I don’t think you can provide them with a variety of problems at this stage. Gravity is the only sensible one I know of, and even that has to be tailored just right to be factorable (which I discussed with my students). Looks like I have a lot to say here. Maybe it’s time to post my own pseudocontext example.

Kentucky and UCLA have appeared in the NCAA Division I men’s basketball tournament 80 times, with Kentucky appearing 8 more times than UCLA. How many times has each team appeared in this tournament?

and

My son’s day care has a digital photo frame on “shuffle” mode, where it randomly displays a picture – my son is fascinated by this and won’t leave the day care. We watched a number of pictures go by, and the 20th picture (estimated or OCD, you decide) was the first one that matched a previous picture. So the question: about how many pictures are in the shuffle? (Harder: find a 95% confidence interval…)

I cannot imagine anyone really asking the first question. Perhaps only math nerds like me (and Chris) would ask the second, but we do ask questions like that, and then answer them, because we are personally intrigued by the interaction between randomness, number of elements, and repetition. Probability is deep.

Is it possible to come up with a bunch of simple questions that help students with translating from problem to simple algebraic equation, where the problems are not pseudocontext? I don’t know. I agree with Dan that pseudocontext kills any belief that math is truly useful. Do students have to learn how to use algebraic techniques before we can offer them sensible problems?

I’d rather offer puzzles than allegedly real problems.

I think the largest percentage of students see any context problems as fake. It’s kind of like selecting an audience for an writing assignment, but the only person to read the work is the teacher. Kids see through that. I know mine do (both children and students).

I think we need to explain to students that yes, some of the problems are asked in a silly manner; but also allow them to contemplate and identify the lurking variables and more complex mathematics underlying such a simple problem. By understanding what other variables exist within the context, the students can more readily accept the weak context created by the authors to see that it is what it is to meet them at their level and provide a strategy or set of strategies upon which they can build.

My experiences with online math instruction is that it works for a very small population of learners. The lack of support necessary for lower level learners is its downfall. Students involved in any credit recovery programs available in my area (Michigan) do poorly. The students show less of an interest in the online work than they do with classroom instruction. Even those who like gaming contexts tend to “click” though just to finish.

We need to keep dialogue like this alive to push math education forward.

I would also like to see a change in the types of questions national testing companies ask. If these companies had questions that required more than rote work, justifying, clarifying, predicting… then instead of halting reform of math education (since federal and state dollars are tied to test performance) there would be the nationwide push to change the what and the way we teach math. We math teachers know that we need a change, but until the policy makers see a need to teach and think differently about math instruction, we will be stuck in the 50’s. We really didn’t land on the moon yet did we? Sorry about going off topic.