In one sense, I think the following is pseudocontext.

Some other mathematical / physical connections are: Volume of air in ball, volume of the air if it were out of the ball, mass of air in ball, density of air. Design an experiment to determine these. Volume of skin making up ball assuming skin is 1/16 th thick. Mass of skin. Density of skin. Design an experiment to determine skin density. Now recalculate the volume of air in ball taking into account the thickness of skin. What is definition of mass and weight. Does ball have water vapor or liquid water or both? What would this depend on?

These questions, by themselves, are potentially very good. For example, there is air inside of a basketball, and “How much air?” is a natural question to ask. One meaning of “how much?” is “what is the mass?” I think it would be very interesting to go through some way of figuring out how much mass the air inside of a basketball has.

HOWEVER, I think these questions have nothing to do with the game of basketball. Why would a basketball player care how much volume the air inside a basketball would have outside the basketball? The one related thing that would be of importance in a basketball game (the pressure of the ball) is not even mentioned.

I don’t care about pseudocontext that much. It’s a problem, yeah, but not the biggest problem I face.

However, thinking about the things a basketball player is likely to care about, that would improve his game. That led to a lot of ideas about reflections, geometry proportions, parabolas (of course).

I don’t know why everyone was talking about statistics. If I’m playing a sport, I don’t care about the probabilities. I care about the actualities.

I’m currently treading a line between maths-as-art to be fostered and nurtured on the learner’s own terms and maths-as-algorithms for external assessment and utility in higher study. I’m lucky to work in an Australian state which operates an school-based assessment system which is externally moderated by a panel of teachers from other school. This allows for assessment innovation at the school level as long as the moderating teachers can be convinced. This is in danger of being removed from us due to a hostile media landscape, a government which is looking to US models of school improvement and teachers like me who aren’t trained media communicators and who were taught from a textbook. I’m learning a lot more about maths as a result of WCYDWT and formal inquiry approaches, but I fear it’s a race against time. I can now always see the introduction to a topic involving WCYDWT/Play/Skills Assessment then moving into “these are the skills we need in this area to help, and to get the exam.”

I think checking against pseudocontextualiciousfulnessnity is a really good bridge between the two approaches. So what would matter to the basketball player here? If the “unit” is about modelling, then could you try to get handle on career trajectory, probabilities of selection, dimishing returns for effort etc. if you can find valid latent constructs to explore, and performance data for players who have gone before. What is the career equivalent of gravity that gives the player career its parabola? Is it a parabola? What is on the y-axis? You could certainly bring in relevant statistics like survivor bias, non-normal distributions etc.

What do you think?

Oops! I forgot to show my work. I’m calling this pseudocontext because any student who is interested enough in this problem to answer it probably already knows the answer, the years of each appearance, and the starters for each appearance.

“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”

We have a system that plays video clips over our network and associated documents (e.g. worksheets.) Hundreds of clips. The database broke “somewhere” and “some” videos have the wrong information attached. I asked a colleague to get our enrichment class to work out how to approach this (I gave her an algorithm that would have worked OK) but rather than 25 engaged kids finding the damaged records in a lesson, the poor library assistant was made to spend a week watching each video in turn. (Last time I checked on Friday she was starting to find errors at the end of the list *sigh*.

Anyhow, now that I’ve reread Dan’s exam instructions, the most pseudocontextual suggestion is the one about “how long would it take to get the whole team around the pitch (unless I’m missing some arcane basketball procedure)?”

i am a math lover – so please take this in that vein. and i love what Dan and everyone is doing here. i just watched Dan’s recording from saturday… great stuff.. take a listen if you missed it.. http://www.blip.tv/file/4347311

however… i have to nudge a little..
what if… we trusted more in the math… that it will just show up. (mathematical thinking vs some real life event we can paste a text book problem over) i absolutely love the 4 layered problems by the way..
what if.. we even quit talking about the math… let the kid just dive into the whatever (let’s say basketball here) let them notice they love basketball, imagine themselves going pro, or whatever, let them question/research/seek out what and who would they then connect with in order to do that… and then facilitate that. [interesting article here along those lines http://tinyurl.com/2ecp32b ]

after it’s all said and done… mathematical thinking will be there- no?

if we’re in an environment where we need to – yeah – go ahead and point out specific math topics – after the fact – or strategically along the way. but i’m thinking – preferably – let’s let the natural mathematical thinking happening all around us daily just be – just show up. maybe math goes nameless for a while.. maybe then intrigue in our art will just happen… like we all crave.. (as opposed to classroom management issues that arise while teaching math and a lot of time and effort to fit real life into math)

i know this is edgy and risky. no ill intent or disrespect. just seeking more humane ways – for all of us.

@Jon

All math taught in any classroom is pseudocontextual. Most kids solve problems by the easiest method possible… trial and error, and then only if the context in which they are working on the problem is relevant at the time.

It seems to me that you are calling any problem that can be solved by another technique pseudocontext. I think the distinction is whether or not the math generalizes to another problem.

With the UK and UCLA example, I cannot pick two other teams and use the same procedure to find how many times they’ve been to the NCAA tournament. A student would need to know the answer before they wrote the problem.

However, with the picture frame problem, the solution will generalize to other picture frames. You can watch a picture frame for a few minutes and guess how many pictures are on it. Also, the solution to the picture frame problem generalizes to other areas. Issues with sampling and the US Census are one example.

@Jon

I do not disagree that we need to frame out problems at the level of our students, and that leads to some problems that can be solved by non-mathematical means. I do disagree that every attempt to target problems at students can be labeled as pseudocontext.

Yes, we can look up the information on the NCAA basketball tournament on Google. My point is that the problem creates an artificial algebraic construct on top of the problem. This diminishes the students interest in the problem and reinforces the idea that learning math is only a pointless hurdle for them to finish school. That makes it pseudocontext.

The picture frame problem also puts an algebraic construct on top of the problem, but not an artificial construct. The technique I would use to solve the problem mathematically is a mark and recapture technique. You can use mark and recapture techniques to estimate the number of homeless people in New York City. That is a problem where it is impossible to measure directly. Because there is a one-to-one correspondence between what we are trying to measure and the techniques are the same, I would call the context of the photo frame as real as the context of the homeless in NYC.

(Would that be a homocontextmorphism?)

For the photo frame problem, the students can compare their results to the real answer. If their predictions are close, then they gain confidence in the technique. If their prediction is off, you can discuss with the students why there is a discrepancy. The parallel with the homeless in New York City is that the number of homeless people has an affect on public policy. There are political reasons for distorting the number of homeless, and informed voters need to know what numbers to trust.

I think that you are actually trying to argue a point similar to the one that I am making. Please be careful not to lump all problems together. Finding the attributes that make a successful problem is important. That is why we are worried about calling out pseudocontext.

If we in the mathematics education profession cannot show that there is value in how we teach, then there are people who are more than happy to impose their techniques. In the higher education world, where budgets are shrinking even though enrollments are increasing, administrators are looking for ways to get more people through classes. Placing students in front of computers and letting the computers teach the students is cost-effective, but not learning-effective. Educators need to show the value of our product. Diluting it with weak problems is not going to accomplish that.

I never took your response as an attack on me. I likewise am not attacking you.

Chris, I truly appreciate the exchange.

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.

Sorry, strike: Do students have to learn how to use algebraic techniques before we can offer them sensible problems?

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.

Dan
Sorry if I offended you.

Responding to Sue, it’s very difficult to find true “context” problems in which factoring quadratics is a key method. Even the context of gravity fails this since you end up with functions like

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?