[PS] Midterm

I keep wondering what a â€œBasketball Mathâ€ curriculum might look like for Tucker, one that would combine his serious interest in the sport with his growing interest in math.

Choose one:

1. Which of Will’s commenters has suggested a pseudocontextual problem?
2. Create a math problem in response to Will that would be pseudocontextual.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

1. Kristin

November 6, 2010 - 6:50 am -

I’m not a math teacher, but methinks it’s George’s comment that is psuedocontext because it begins with the assumed “fact” that a player has a 75% change of making a throw. Instead of watching actual footage, or having a player come to school and doing it, you use a deck of cards or a spinner.

2. Aaron

November 6, 2010 - 7:02 am -

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.

3. Z. Shiner

November 6, 2010 - 8:38 am -

From what I understand of how pseudocontext is being defined, I would have to agree with Aaron that the in depth analysis of the ball is an example of pseudocontext. I also think that these questions are also pseudocontextual:

How many times will a ball bounce before it stops (exponential decay). How long will a ball bounce before it stops (sequences and series)

That being said, it really depends on the frame of the problem. I can imagine a WCYDWT video where two seemingly identical basketballs are dropped from the same height and one bounces high while the other bounces low. I think questions about how many times a ball would bounce, what is the density of the air, and the relationship between those two would both naturally arise from such a video. I can also imagine a context in which students are tasked with ensuring that all basketballs are set to specific standards for NBA games.

I am still confused as to what pseudocontext is though. It’s been a while since I’ve read Jo Boaler’s book, but I was under the impression that pseudocontextual problems deny students’ sense of how the real world works. All of the above problems, though contrived and unrelated to basketball, are still consistent with the laws of reality. I tried to make sense of the different ideas on the context of problems in a blog post some time ago, but am clearly still wrestling with the concept.

4. hillby

November 6, 2010 - 9:27 am -

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.

5. Dan Meyer

November 6, 2010 - 12:50 pm -

@josh g. feel free to drop ’em here.

hillby: 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).

Is it your understanding that Kobe Bryant puts a lot of thought into reflections, proportions, and parabolas?

hillby: I donâ€™t know why everyone was talking about statistics. If Iâ€™m playing a sport, I donâ€™t care about the probabilities.

What if you’re paid to coach the sport? Or paid to comment on it? Or just interested in the sport like Will’s son?

6. josh g.

November 6, 2010 - 3:15 pm -

Ok, here goes. I took notes while I went outlining my process.

PSEUDOCONTEXT 101 Midterm, Josh Giesbrecht

STRATEGY: recreate the likely process textbook authors go through in the mad rush to publish first and get those education dollars

Step 1: Pick a math standard to target
– trig eqns

Step 2: Search stock photography in desperation
– source: http://flickr.com/photos/28816739@N04/2690796076/
(nba player dribbling ball)
(TODO: flickr user has tagged this as CC, and yet there’s a note in the comments about additional licensing? email publisher’s lawyers before printing)

Step 3: pull formula out of thin air and write problem

A basketball being dribbled down the court hits the floor every 0.7 seconds. The vertical position of the basketball can be described with the formula y = 0.5sin(2pi/0.7 x) + 0.5, where x is the number of seconds since the player started dribbling.

How long does it take for the ball to be exactly 0.75m above the floor?

Step 4: Justify why this is horrible.

No one in the history of basketball has ever replayed a video of a game to find out how long it took the basketball to reach 0.75m off of the floor.

There are all kinds of interesting questions around motion that you *could* ask in the context of a basketball game – eg. the classic “Will he make the shot?” WCYDWT. But I suspect most of the worst pseudocontext problems we see in textbooks are the result of writers needing to write n word problems for each topic on short order. This leads to scrambling to fit a context to the math, rather than what happens in real problem-solving – finding math that will fit the context. So, to try and recreate the results, I tried to recreate the process.

This also has the flaw of not being an accurate model of what a basketball’s vertical motion would really look like â€¦ but that’s an independent problem from the pseudocontext. Which, I dunno, maybe makes this a little unrealistic … if anything, when I was teaching this topic, my textbook was hyper-paranoid to a fault to try and use only things which were exactly sinusoidal. (Rather than include any discussion around when and why an inaccurate model may still be useful in real-life problem solving.)

Self-assessment:

Forced, artificial problem hook, check.
Likelihood to appear in a textbook … half marks.
I’d give myself a 3 out of 4 and suggest finding a less ambitious math-topic target for the next reassessment.

7. cnapi5

November 6, 2010 - 5:02 pm -

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?

8. Chris Sears

November 6, 2010 - 7:50 pm -

Since I am only auditing Pseudocontext 101, I will submit a problem from our Intermediate Algebra text for my answer.

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?

Source: Intermediate Algebra with Applications and Visualization, 3/E. Rockswold, Gary and Krieger, Terry.

[image attached – dm]

9. Chris Sears

November 6, 2010 - 8:15 pm -

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.

10. Bowen Kerins

November 6, 2010 - 9:40 pm -

I still want to know where I can get my share of these mad education dollars I keep hearing about.

That intermediate algebra problem is awesomely pseudocontextual. There is no way you would know the “combined 80 appearances” fact without the other two facts.

Not much basketball in our books, but several questions about free throw shooting:

“The probability that Todd makes a free throw is 0.642. In a game, Todd attempts 10 free throws. Assuming each free throw is an independent event, find the probability that Todd makes exactly 7 out of 10 free throws… exactly 8… more than 8.”

(Next question: complete this table for the function f(n) = nCr(10,n) * (0.642)^n * (0.358)^{10-n}.)

I think any sports questions run the danger of being too specifically targeted, using terms that only certain kids might be familiar with or requiring so much explanation that they become pseudocontext. Someone on that other blog mentioned fantasy sports, and that is one rabbit hole I would never go down with my students…

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…

11. cnapi5

November 7, 2010 - 2:28 am -

“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 intervalâ€¦) Itâ€™s interesting that itâ€™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)?”

12. I would have to guess that the math to all of these problems has been done previously. Any of the problems posed could be solved by doing a little investigative research. The answer to the question about the digital photoframe is: ask the owner. Surely, since you are looking at it, you know the owner, ask him/her. Problem solved.

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. Skateboarders don’t use physics or calculus to figure out how fast to go to perform a trick. They use gravity and numerous crashes and fractures to finally nail a trick. I don’t see any kids at the skate park with their calculators out determining the proper velocity to complete three cycles.

Pseudocontext is necessary to meet kids where they are mathematically — that is on paper. They are very smart and recognize BS, which I haven’t seen a textbook that was filled with it. No one example, no one rich investigation will spark a sense of relevance in every kid who reads it. It doesn’t take a genius to figure that one out. I stole one of Dan’s ideas, ask a simple question like… You and a group of students and adults are going to the movies, how much will it cost. Again pseudocontext, because you know how to eliminate many of the variables. The student task was to give an answer, and any bs answer they gave could have been justified as correct, OR write down as many things as you would need to know in order to answer the question. We then talked about the quality of questions in the book, how someone could more easily figure it out or know the answer with base knowledge of the conditions. It was then applied to the concepts we were studying.

That said. It made the student think about all of the other variables involved in arriving at an answer and look into the question (as dumb as it may be to ask) and reason out – a little more readily – how to set up and solve the problems using the method intended by the author.

Unless students are really interested in the mathematics behind the — personally relevant context inserted here — answers can be found to any given problem by using research that already exists.

13. monika hardy

November 7, 2010 - 6:31 am -

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.

14. Michael Paul Goldenberg

November 7, 2010 - 8:40 am -

So much great stuff here and at the site you refer to, as well as in the comments at both places and links provided. Seems like a whole curriculum on creating a curriculum could be found within. ;)

For my money, so far the most pseudocontextual problem I noticed in the comments at Will’s blog was the one about measuring area and perimeter of a soccer field with something called “knee ups” or words to that effect. I thought that to be classic pseudo-context, but perhaps on further review I’ll change my mind. For one thing, we already can find the dimensions by looking at any rule book on soccer. I suppose finding a non-standard unit that is grounded in a related activity has a level of interest and validity, but it seems very forced to me, the sort of thing a textbook author would come up with. What does putting a soccer field into the problem add to the math? What does bouncing a ball around a field have to do with the man in the moon?

Having played some high school soccer, my question would be: how many times will I have to run around the field before I wind up puking my guts out? Now there’s a unit of measure for you.

15. Chris Sears

November 7, 2010 - 9:33 am -

@Jon

All math taught in any classroom is pseudocontextual. Most kids solve problems by the easiest method possibleâ€¦ 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.

16. @Chris
The UK and UCLA problem can be solved by looking in an almanac or by using google. Where is the REAL need for mathematics other than a pseudocontextual construct? Again, the picture frame scenario can be solved by asking the owner. Why would I care how many photos there are, and why would I try to use some algebraic construct to solve the problem. Again, I argue that the relevance of the context is the most important constrain. One has to prioritize the context to make understanding the mathematics relevant.

I don’t disagree with anyone above. I do not see a differentiation between using another math and the pseudo context. Then again, the same applies to most language arts curriculum as well. I am an English Major and a Mathematics Minor and teach mathematics.

I believe we must help our students see there is a larger end to our efforts. Unless we are seeking something new, we are merely rewriting the essays of others, metaphorically. Logic transcends teaching mathematics. Solving problems from multiple perspectives, respecting other points of view, having mathematical reasoning to refute arguments — all are important to brain development. How many relevant problems and contexts did you research in your undergrad or grad programs. I would bet that it was a rehashing of old ideas under the premise of new understanding. I would argue a neo-understanding. I mean no disrespect to you or any others in this list.

I would argue that statistics and modeling leads to the best path to new, relevant context. Students need to collect data respective to a topic they find important. Sadly national standards lead us in the opposite direction.

Please understand my comments are not directed at you, personally, but your ideas. Once again, I argue that all pedagogical context is pseudocontext. It must be pseudocontext to allow students to grasp whatever content is taught. Students do not have the inbred understanding of the mathematics or concepts they can imply through their interaction and experience with the material world to be able to describe it mathematically. It takes mathematical geniuses to exspouse what they know symbolically.

17. Chris Sears

November 7, 2010 - 8:14 pm -

@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.

18. Aaron F.

November 7, 2010 - 11:07 pm -

@Kristin: To me, George’s free throw shooter problem seemed very true to life.

Iâ€™m not a math teacher, but methinks itâ€™s Georgeâ€™s comment that is psuedocontext because it begins with the assumed â€œfactâ€ that a player has a 75% change of making a throw.

If you’re a basketball team manager, wouldn’t free throw percentage be exactly the kind of statistic you’d have lying around? You certainly don’t have to look very hard to find, for example, a list of the top hundred free throw shooters in the 2009/2010 regular NBA season.

Instead of watching actual footage, or having a player come to school and doing it, you use a deck of cards or a spinner.

I agree that the activity might be improved here by adding a dose of reality. You could, for example, find footage of a game where Kevin Durant made 10 free throw attempts, and string all the attempts together into a short video. Students could predict how many baskets Durant would make, based on his stats from previous years, and then use the video to test their predictions.

I do think, however, that simulation is an important part of probability; in practice, it’s often the only way a given probability problem can be solved! For that reason, I’d say that cards and spinners should always have a place in a probability curriculum, as long as they’re not being used as a substitute for collecting actual data.

19. 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.

20. Sue VanHattum

November 8, 2010 - 7:12 am -

Seems to me there’s a big difference between:

Kentucky and UCLA have appeared in the NCAA Division I menâ€™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 sonâ€™s day care has a digital photo frame on â€œshuffleâ€ mode, where it randomly displays a picture â€” my son is fascinated by this and wonâ€™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 intervalâ€¦)

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.

21. Sue VanHattum

November 8, 2010 - 7:22 am -

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.

22. Dan Meyer

November 8, 2010 - 10:54 am -

Bowen: Hereâ€™s a question that came up â€œin contextâ€ the other day. My sonâ€™s day care has a digital photo frame on â€œshuffleâ€ mode, where it randomly displays a picture â€” my son is fascinated by this and wonâ€™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 intervalâ€¦) Itâ€™s interesting that itâ€™s a context, but if it had been my own photo frame, it instantly pseudocontexts since I know the real answerâ€¦

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 â€” that’s great â€” but because there’s nothing about wins and losses that lead inherently to solving equations.

23. Damon

November 8, 2010 - 5:58 pm -

I’m kind of perplexed at why we’re working hard to create pseudocontext. Seems like there’s a lot out there. I was drawn to this blog for good problems, not crummy ones. The beauty of the good WCYDWT problems, I think, is that they create a context that draws students in. Different problems do it differently, but the great ones create some kind of need to know in many people.

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

24. Bowen Kerins

November 8, 2010 - 9:33 pm -

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?

25. Numbat

November 9, 2010 - 2:56 am -

@Bowen I tried the sum and product thing last time I did factoring and while the kids got very good at answering those questions, very few of them then took the next step and related that to factoring… I can’t explain it but it really knocked my confidence in that.

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.