Total 24 Posts

## [Makeover] Systems of Equations

Here is the oldest kind of math problem that exists:

Some of you knew what kind of problem this was before you had finished the first sentence. You could blur your eyes and without reading the words you saw that there were two unknown quantities and two facts about them and you knew this was a problem about solving a system of equations.

Whoever wrote this problem knows that students struggle to learn how to solve systems and struggle to remain awake while solving systems. I presume that’s why they added a context to the system and it’s why they scaffolded the problem all the way to the finish line.

How could we improve this problem – and other problems like this problem?

One group recommended we change the adjectives and nouns. That we make the problem more real or more relevant by changing the objects in the problem. For example, instead of analyzing an animated movie, we could first survey our classes for the movie genres they like most and use those in the problem.

This makeover is common, in my experience. I don’t doubt it’s effective for some students, particularly those students already adept at the formal, operational work of solving a system of equations through elimination. The work is already easy for those students, so they’re happy to see a more familiar context. But I question how much that strategy interests students who aren’t already adept at that work.

Another strategy is to ignore the adjectives and nouns and change the verbs, to change the work students do, to ask students to do informal, relational work first, and use it as a resource for the formal, operational work later.

This makeover is hard, in my experience. It’s especially hard if you long ago became adept at the formal, operational work of solving a system of equations through elimination. This makeover requires asking yourself, “What is the core concept here and what are early ways of understanding it?”

No adjectives or nouns were harmed during this makeover. Only verbs.

The theater you run charges \$4 for child tickets and \$12 for adult tickets.

1. What’s a large amount of money you could make?
2. What’s a small amount of money you could make?
3. Okay, your no-good kid brother is working the cash register. He told you he made:
• \$2,550 on Friday
• \$2,126 on Saturday
• \$1,968 on Sunday

He’s lying about at least one of those. Which ones? How do you know?

This makeover claims that the core concept of systems is that they’re about relationships between quantities. Sometimes we know so many relationships between those quantities that they’re only satisfied and solved by one set of those quantities. Other times, lots of sets solve those relationships and other times those relationships are so constrained that they’re never solved.

So we’ve deleted one of the relationships here. Then we’ve ask students to find solutions to the remaining relationship by asking them for a small and large amount of money. There are lots of possible solutions. Then we’ve asked students to encounter the fact that not every amount of money can be a solution to the relationship. (See: Kristin Gray, Kevin Hall, and Julie Reulbach for more on this approach.)

From there, I’m inclined to take Sunday’s sum (one he wasn’t lying about) and ask students how they know it might be legitimate. They’ll offer different pairs of child and adult tickets. “My no-good kid brother says he sold 342 tickets. Can you tell me if that’s possible?”

Slowly they’ll systematize their guessing-and-checking. It might be appropriate here to visualize their guessing-and-checking on a graph, and later to help students understand how they could have used algebraic notation to form that visualization quickly, at which point the relationships start to make even more sense.

If we only understand math as formal, operational work, then our only hope for helping a student learn that work is lots and lots of scaffolding and our only hope for helping her remain awake through that work is a desperate search for a context that will send a strong enough jolt of familiarity through her cerebral cortex.

That path is wide. The narrow path asks us to understand that formal, operational ideas exists first as informal, relational ideas in the mind of the student, that our job is devise experiences that help students access those ideas and build on them.

BTW. Shout out to Marian Small and other elementary educators for helping me see the value in questions that ask about “big” and “small” answers. The question is purposefully imprecise and invites students to start poking at the edges of the relationship.

## [Makeover] Marine Ramp

Previously

Makeover Preview: Marine Ramp

The British Columbia Institute of Technology explains the origin of this task in their Building Better Math project:

Many high school students are avoiding math and cutting off pathways to exciting technical careers before they even know about them.

Their solution? More real world problems. Specifically, job world problems, problems that relate to “areas of geosciences, health care, engineering, renewable resources, oceanography, forensics, architecture and other industries.”

The BCIT has a very shiny coin here. They know better than anybody else – better than most teachers and curriculum developers, certainly – where our mathematical models are useful. I was blind to the mathematical modeling essential to the construction of a ramp at a boat dock, for example. BCIT helped me see it.

The BCIT knows that “trigonometry lives at the boat dock!” but without very careful curriculum development and very careful enactment by teachers, students will only experience the opportunity to calculate at the boat dock! This context offers many other opportunities to think mathematically besides calculation.

Here is one way to exploit them.

I begin so many of my applied tasks with video not because “kids love their YouTubes” but because multimedia allows me to de-mathematize a context that has already been heavily mathematized, leaving information, formulas, and other scaffolds to be revealed at an appropriate moment, and involve students in that process.

A: Without a ramp from pier to dock we can’t get on the boat.

Then ask students, “Which of these four ramps is best? Which is worst? Why?”

A: The shortest one is lousy because it’s too steep to safely cross. The longest one is lousy because, while it’s safe enough to cross, it’s longer than it needs to be, which is wasteful. The best is probably one of the other two and there may be one that’s even better.

This is an important moment for student learning and for student interest.

Learning. There are cognitive gains to be had by showing students contrasting cases of the same question and asking them to invent a measure to describe them. Here is an example from Schwartz and Martin (2004).

One group attempted to invent a measure and another group simply received instruction on the canonical measure. (“Variance” in this case.) Both groups then saw a worked example, after which the “invention” group outperformed the “tell-and-practice” group on a battery of measures. The invention activity helped students transfer in knowledge that prepared them to learn from explicit instruction later.

These multiple contrasting cases also allow me to ask students, “What measurements stay the same in every case? What measurements change?” That sets us up to assign variables to the changing measurements and quantities to the fixed measurements. The original problem offers only one case – one single ramp – offering us none of those cognitive gains.

Interest. As I summarized earlier, Sung-Il Kim’s research predicts that students will find this makeover more interesting than the original. Rather than explicitly stating the question and all of its relevant information, we’ve shown something incongruous and stated just enough that students will have to make the inferences that drive interest.

We should mathematize the context further now, assigning quantities to the measurements we know. (The distance the boat dock drops and the distance from the dock to the pier.) We should tell students the crucial constraint that the ramp can’t be any steeper than 18° as it meets the dock. We should model for students how a mathematician takes a context full of useless noise (eg. the color of the water, the shape of the hills) and draws a new version that includes only the useful details.

The problem is now where we started, fully mathematized. The goal of our previous work was to expand student access to the mathematics and also broaden that mathematics to include more verbs than just “calculate.”

Let’s not stop there. Let’s head to Chris Lusto’s Boat Dock Generator (source code).

This allows us to extend the existing problem. Hit the refresh button and get a new boat dock. Another one. And another one. Can students turn their one correct answer into a method for quickly calculating the best ramp length for any boat dock? Can they write it in algebraic language?

Concluding Remarks

I realize the new problem is more difficult to implement than the old. This new problem requires the teacher to involve herself in the posing of the problem and not just the assignment of the problem. It’s relatively easy to say to students, “Head over to this link and do the problem. I’ll be around to help if you need it.” It’s rather more difficult to embed yourself in that problem, to see yourself as an agent in the posing of that problem and the development of its question, even if the upside is better learning and more interest. This makeover is high reward at a high cost. At the moment, the reward interests me more than the cost.

You can download the problem at 101questions, but my main intent here wasn’t to create a problem we could use in the classroom. The point of a math problem isn’t just to get an answer, it’s to learn about math. And in the same way, the point of a math problem makeover isn’t just to get a better math problem, it’s to learn about learning.

What You Recommended

I have also been rolling this same problem in my head, but I didn’t know about the Vancouver version. I teach on an island in Maine, where the tide swings are larger, and these kinds of contraptions are everywhere. I’ve thought about making a three-act type problem, but can’t wrap my head around the best application. I was thinking of doing it for more advanced trig in precalculus: Here’s the ramp, here’s the dock, and for what portion of the day will the ramp be usable? For walking up and down? For hauling a hand-truck? For a wheel-chair? How could you change it to make it usable for more of the day? How might the harbormaster foil your plans? This is a great problem for my context, because many of my less mathy students know more about harbor restrictions and practical “dockery” than I do.

Justin Brennan offers a word of caution about these job-world applications:

After spending 8 years as an engineer prior to teaching, I always felt that I’d include all kinds of stuff from my engineering life into teaching. However, now that I am slightly wiser and more humbled, that stuff is too specialized, only interesting to me and maybe 2 other kids on a good day.

I appreciate Justin’s testimony that “math + jobs = fun!” is too simple an equation. But rather than give up the “jobs” part altogether, I have attempted here to bring students into the job in a particular way. Not all job math problems are created equal, in other words.

Jonathan Newman made a simulator in Desmos. My concern with every simulator is that the person who made the simulator uses more math than the students do. Scaffolding questions around the simulator to simulate mathematical thought, as Jonathan does, is no small task.

## [Makeover Preview] Marine Ramp

I have been rolling the same math problem around in my head for the last two months. Here is a link and a PDF.

“Obsessed” wouldn’t be too sharp a description. Not with the math, which isn’t more advanced than high school trigonometry. Rather with the problem itself, and the opportunities it offers students to think mathematically.

In its current form, those opportunities are limited. In its current form, the problem asks students to read given information (and a lot of it), recall a formula, and calculate the result. That’s important mathematical thinking but hardly the most important kind of mathematical thinking (a statement of opinion) and not the only kind of mathematical thinking the context offers us (a statement of fact). There are more mathematical opportunities, and more interesting ones, than the problem offers in its current form.

So change that! How would you makeover this problem and help students experience all those interesting opportunities to learn mathematics?

On Monday, I’ll offer my own thoughts, along with a collaboration with Chris Lusto.

## [Makeover] Central Park & These Tragic “Write An Expression” Problems

tl;dr. I made another digital math lesson in collaboration with Christopher Danielson and our friends at Desmos. It’s called Central Park and you should check out the Walkthrough.

Here are two large problems with the transition from arithmetic to algebra:

Variables don’t make sense to students.

We give students variable expressions like the exponential one above, which they had no hand in developing, and ask them to evaluate the expression with a number. The student says, “Ohhh-kay,” and might do it but she doesn’t know what pianos have to do with exponential equations nor does she know where any of those parameters came from. She may regard the whole experience as one of those nonsensical rites of school math which she’ll forget about as soon as she’s legally allowed.

Variables don’t seem powerful to students.

In school, using variables is harder than using arithmetic. But what does that difficulty buy us, except a grade and our teacher’s approval? Meanwhile, in the world, variables are responsible for anything powerful you have ever done with a computer.

Students should experience some of that power.

One solution.

Our attempt at solving both of those problems is Central Park. It proceeds in three phases.

Guesses

We ask the students to drag parking lines into a lot to make four even spaces. Students have no trouble stepping over this bar. We are making sure the main task makes sense.

Numbers

We transition to calculation by asking the students “What measurements would you need to figure out the exact space between the dividers?” This question prepares them to use the numbers we give them next.

Now they use arithmetic to calculate the space width for a given lot. They do that three times, which means they get a sense of the parts of their arithmetic that change (the width of the lot, the width of the parking lines) and those that don’t (dividing by the four lots).

This will be very helpful as we take the next big leap.

Variables

We give students numbers and variables. They can calculate the space width arithmetically again but it’ll only work for one lot. When they make the leap to variable equations, it works for all of them.

It works for sixteen lots at once.

Variables should make sense and make students powerful. That’s our motto for Central Park.

2014 Jul 28. Here is Christopher Danielson’s post about Central Park on the Desmos blog.

Featured Comment

In thinking further about your complaint about “Write an expression” I think what is also going on in this app is a NEEDED slowing down of the learning process. The text (and too many teachers) are quick to jump to algorithms before the students understands their nature and value. Look how long it takes to get to the concept of an appropriate expression in the app: you build to it slowly and carefully. I think this is at the heart of the kind of induction needed for genuine understanding, where the learner is helped, by scaffolding, to draw thoughtful and evidence-based conclusions; test them in a transfer setting; and learn from the feedback – i.e. the essence of what we argue understanding is in UbD.

One reason I like this activity so much is that it hits the sweet spot where “What can you do with it?” and “What does it mean?” overlap.

## [Makeover] These Tragic “Write An Expression” Problems

We need to do something about these problems, which recur all throughout school Algebra.

The original title of this post called them “horrible,” but they’re truly “tragic” – the math education equal to Julius Caesar, Othello, and Hamlet – full of potential but overwhelmed by their nature.

Here’s the thing about variable expressions: they’re used by programmers and students both, but those two groups hold variables in very different regard.

Ask programmers what their work life would be like without variables and they’ll likely respond that their work life would be impossible. Variables enable every single function of whatever device you’re reading this post on.

But ask students what their school life would be like without variables and they’ll likely respond that their school life would be great.

What can we do?

The moral of this story isn’t “teach Algebra 1 through programming” or “teach computational thinking.” At least I don’t think so. I’ve been down that road and it’s winding.

But in some way, however small, we should draw closer together the wildly diverging opinions students and programmers have about variables. Ideas? I’ll offer one on Monday.

2014 Jul 25. I appreciate how Evan Weinberg has thought through this makeover (now and earlier).

Featured Comment

Dylan Kane restates our task here in a useful way:

In terms of making these problems a little better, students should feel a need for the expression. I think this question stinks in part because the expression it’s asking for is so trivial — it’s extra work, compared to just multiplying by 3/4 or doing some simple proportional thinking.

Jennifer offers an example of that kind of need:

I like to introduce the idea of expressions by having the students playing the game of 31 with a deck of cards!their goal- play until they can predict how they can win every time! This will take less than 15 minutes, and a whole class summary of verbal descriptions on ‘how to win’ are shared. Verbal descriptions become cumbersome to write on the board, so ‘shorthand’ in the form of clearly named and defined symbols are used to make the summarising more efficient. the beauty of this is that the idea of equivalent expressions presents itself.

i think the ability to generalize and write a rule with variables is really important, but you can come to that through lots of nice activities and investigations as well.

for example, i did dan’s “taco cart” with my students with a few notable changes. instead of telling the students how fast dan and ben walked, i had each group decide on the speed of the two men themselves and list that along with other assumptions they made in the problem.

when we did the whole class summary, i told them that i had written down a formula on my paper that would allow me to check if their answers were correct and that i needed that since everyone used their own speed. i should’ve asked them all to take a minute to try to come up with the formula i used, but instead i elicited it at that moment and one student gave me the correct formula. the need for two variables (speed on sand and speed on pavement) was obvious.

in part two of taco cart, when the students were trying to figure out where the taco cart should be so that the two men reached it at the same time, one group did seemingly endless guess and checks. i suggested to them that this wasn’t a good method and asked if maybe it wasn’t better to write an equation with a variable. again, they could see the need. once they started with a variable, the rest of the problem started to fall into place.

here’s the thing: these “write an expression” problem want to train students to learn to generalize and write a rule. they want them to be able to see a situation that would best be tackled with a formula of some sort, write said formula with various variables, and use their formula to solve complex problems. but the issue is that these problems don’t actually train students in that way because they’re so artificial and one-dimensional. what they do is teach students to “translate” from english to math (an important step along the way, i do believe), but not to recognize a situation in which a formula would be helpful or necessary or how powerful it can be.

Two thoughts from a computer science teacher’s point of view:
1) When introducing programming to 15 yrs olds for the first time, we use Python interactive prompt as a calculator first. And the first point is to show the advantages. Variables and funstions (one-liner formulas) simply save work. That is it, that is one of the goals of the whole programming topic anyway. When they do quadratic equations in maths at that time, we are headed that way. And students realize pretty fast: aha, I have to understand how to solve them, but once I do and I describe it properly, I never have to do it by hand anymore. Their understanding of q.e. deepened, they interest in programming increased, and I could naturally introduce a load of important CS concepts on the way. In younger age we do simpler formulas, also like BMI calculation (not only the “area of the circle” kind of stuff, which they, um, do not prefer), but the point is the same: the kids need to see at least some hypothetical benefit for themselves. Having to introduce variables in maths, I would in principle search for a similar approach. (Note: for even younger kids, fun and creativity achieved with Scratch, turtle etc. overweigh the “practical benefits”; but when practicality leads to more fun – win-win!).

2) A good and often forgotten tool between calculation on paper and programming is a spreadsheet. It can store lots of numbers in a structured way and perform basic calculations, what is well understood by kids. And when we want to do anything more complex without getting beards grown, we absolutely need formulas and “variables”. Their advantages are imminent. And the whole time, everything is in plain sight, the level of abstraction is way lower than with programming, making it very accessible for kids. I am of course interested in it “from the other side” – after some decent work in spreadsheets, many more advanced concepts are a step away (for-loop, data type, input-output, function, incremental work on more complicated calculations, debugging etc. etc.). But I believe that thoughtful use of spreadsheets can improve understanding in appropriate topics in maths.