Previously: [Makeover] 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.
29 Comments
Zach
July 28, 2014 - 11:44 am -Wow. That looks pretty spiffy. I like a lot of the interactivity this offers.
I’m wondering if this is extensible into a real parking lot. (Having multiple rows, a set parking area, etc.) Granted, this is meant to be a basic introduction, but a real parking lot would be an interesting extension.
katenerdypoo
July 28, 2014 - 12:04 pm -fantastic, i love it, and it’s exactly perfect for one of my classes!! can’t wait to try it out in september and i will report back.
one thing: here’s my ongoing request to desmos (and you, since you help design so many fantastic things). i’m really hoping that it’s possible to design all these activities so the teacher can select whether they’d like to work in metric or english system for those of us who want to make use of these things from overseas.
it’s not that my students can’t work in feet, it’s just that they drive me crazy wanting to know what a foot is, about how many meters is it, why don’t they just use meters in america, why is it called a foot, and what about inches and miles and pounds and fahrenheit ad nauseum. normally during the lesson i am more than happy to engage such discussions, but it’s so distracting during a computer lab investigation! :)
Jim Doherty
July 28, 2014 - 12:13 pm -Had the pleasure of working through this at TMC14 when Eli Luberoff was giving his keynote. It is a delightful exercise and one that I will urge my Algebra I teachers to fall in love with. The work at the end where the variable equations need to be constructed is a terrific payoff for the work involved up until that point. My two partners and I were eagerly working through and feeling a touch competitive about it, I must admit. The graphic when you don’t quite get the answer right is fantastic, a gentle recognition that you are not quite down with the task yet. Rather than simply telling the user s/he has not gotten this task right yet. It’s a really wonderful stroke.
Jim Doherty
July 28, 2014 - 12:15 pm -Regarding katenerdypoos request about metric, this came up in a conversation in OK with Nik who made a similar request. This is on the radar of the desmos folks
Grant Wiggins
July 28, 2014 - 12:55 pm -Dan, this is fabulous. This is so much better an introduction to the value of algebra than exists in almost every textbook – all in one nifty little app. And, as noted, the feedback here is fabulous: you did not achieve the desired effect, so it can’t be correct – far better than a red x or some other form of evaluation instead of feedback.
My only suggestion for an improvement would be to ask the student to calculate the number of spaces that could be put into a very large lot so they could see that arithmetic is just not the way to go.
Grant Wiggins
July 28, 2014 - 1:02 pm -OOH – a further idea: what if the spaces are on the diagonal as a later wrinkle, as in many parking areas? (Q: “Why might some towns prefer to use parking places on the diagonal? What are the tradeoffs?”)
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.
Howard Phillips
July 28, 2014 - 1:30 pm -This is brilliant. Now for the keyboard! Or perhaps a violin string might be a gentler intro. And who is stupid enough to count the keys from the bottom, and get stuck with 49!!!
The follow up at some point needs to be off the computer. So far it’s a problem I have faced in shelf building many times. “Bill is making a bookcase. It has to be 6ft high, the top shelf is at a position so that the sides will hold in the books OK. The bottom shelf is 4 inches from the floor. Where is bill going to put the equally spaced 4 other shelves?” – complete with missing information!
Joe Schwartz
July 28, 2014 - 6:37 pm -I can only echo Grant’s comment about the slowing down of the learning process. The very careful build-up, which comes at the learner’s own pace, along with the meaningful feedback, is what makes this activity so powerful. It has tremendous implications for our teaching practice. It also makes me want to get hot wheels and popsicle sticks and build a parking lot to see how it would all work.
Tom
July 29, 2014 - 4:31 am -Very nice. In fact, I just did this in Real Life ™ yesterday when I equally spaced 5 posters on a wall. Love how you extended it to bring in variables.
Mike Lawler
July 29, 2014 - 5:22 am -The videos are long – 15 and nearly 30 minutes, but I walked through this with my 10 and 8 year old last night and they enjoyed it. If you want to see how kids react to this, here you go:
http://mikesmathpage.wordpress.com/2014/07/29/a-review-of-central-park-by-dan-meyer-chris-danielson-and-desmos/
Howard Phillips
July 29, 2014 - 5:40 am -Just a thought:
The first, and second, and third time students meet “variables” is in simple equations such as x+2=5, or slightly more complex ones, 3x+2=14.
In these the x is not a variable, it is an unknown quantity whose value we are asked to guess, find, calculate …
So what are they supposed to do with y=x+2, or x+4y=11, and how is “writing an expression” going to help, when it only has one variable in it. Variables are used to relate two or more quantities together. Equations in one “variable” follow naturally from relations, when one of the variables is nailed down.
The “new” approach is going to make things worse than they already are!!
Laura
July 29, 2014 - 5:46 am -Keyboard 85b: What note, which key, which n? What were the answer choices? Were there any at all? If a students has no musical background then what are the rest of the keys called?
Zach
July 29, 2014 - 6:07 am -@Howard: do you mean by “new method,” Dan’s suggestion? If so, the problems in the activity all include more than one variable. Although not emphasized, you could show students what happens to one variable when another changes by a table, for example. I think that you could extend this parking activity to illustrate y=4x+3, etc.
Howard Phillips
July 29, 2014 - 7:15 am -to Zach
No, I meant the emphasis in the early grades on writing equations to represent simple questions like “what number is 5 more than 12”, and which then shows up on state tests…
I love Dan’s posts, and Central Park.
Regarding Central park I do have a feeling that there is a missing step, which is to formulate the desired calculation with the variables expressed in words, for example
spacing = 55 – 3 X spacerwidth, all divided by 4
Then it is much more straightforward to move on to s = (55 – 3w)/4, and so on, introducing new variables one at a time
Dan Meyer
July 29, 2014 - 8:47 am -katenerdypoo:
Thanks for the feedback, Kate. It’s been heard at the top floor of the Desmos corporate office. We’re on it.
Grant Wiggins:
Agreed. This calls to mind Daniel Willingham’s similar remarks that teachers don’t spend enough time “developing the question.” Print textbooks seem to develop their questions with a heavy foot on the gas pedal, rushing as quickly as possible to these very abstract algebraic expressions, likely owing the cost of paper and ink and distribution. I enjoy developing for a digital medium where I don’t have those same constraints.
Howard Phillips:
No disagreement here, Howard!
Cathy Yenca
July 29, 2014 - 9:06 am -Great to see Central Park materialize after test-driving it with you in June (pun intended). I love how the task does just enough hand-holding along the way, yet appropriately encourages generalizing quite quickly. Can’t wait to try this with students. Well done, Dan and Desmos!
Kyle Pearce
July 29, 2014 - 11:17 am -Awesome stuff! I’m excited to try this out with my students in September. As you mention, students struggle with variables and I think this is where many students make the decision to dislike math. I relate learning variables / algebra or the language of math to learning a spoken language. I found learning French in elementary school very difficult because I just didn’t see how it all worked. I’m sure not having a desire to learn the language didn’t help.
Over this past school year, I have been trying to find ways to make introducing variables visual with animations in Keynote. I would love to spend some time learning how to make these visuals more interactive where students can call the shots on values for the variables, etc. as you have in Desmos with this particular example.
If interested, here is a post that explains one of the visualizations called “Pool Noodles” where we take a boring textbook question and give it some context:
http://tapintoteenminds.com/2014/05/11/pool-noodles/
Since the post is static, here is a video with the animations from Keynote. (Sorry, no sound):
http://youtu.be/nLYfZaUZkJ0
Anyone know if there are tutorials on programming in Desmos to try and do something like the Central Park problem?
Kevin Hall
July 29, 2014 - 11:31 am -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.
Alex
July 29, 2014 - 12:47 pm -Great lesson!
Some specific feedback from (just me) walking through:
– When we were writing expressions with numbers (no variables yet), I wanted to multiply, but the instructions didn’t say how (use an asterisk).
– The “next page” arrows seemed a lot more obvious than the “try it” arrows. I found myself moving to the next page most of the time, and then having to go *back* to test my solution.
– Because there are always three dividers, it’s very easy to pass the initial section of the lesson without really thinking about appropriate lengths. You just place one divider in the middle, and then the other two in the middles of the new spaces. I think you’d make students think more if there was a page with 2 or 4 dividers, so you had to do something more complicated than halving.
– The “let’s see if your equation works for even more lots” moment was supremely cool. That’s where I could feel a “Eureka” bubbling away.
– Why can’t I see everyone else’s titles? That would be a cool learning moment.
– Is there a child safety/language issue with letting everyone see each other’s comments?
– It told me at the end that I’d missed something, but I couldn’t find what. It’s generally a bit of an annoying comment. Can’t the students tell for themselves if they’ve finished?
Overall it was a great innovation. I definitely think it shows of the power of expressions more than most starter lessons (it’s probably something computers are quite well-suited to).
Dan Meyer
July 29, 2014 - 3:24 pm -Thanks for the feedback here, Alex. Lots for our design team to think about.
Harry O'Malley
July 30, 2014 - 3:21 pm -The most promising part of the design is the immediate nature with which the symbolic expression manifests itself physically in the designs as you type. This is makes an immediate link in the mind of the user between the symbolic expression and its spatially tangible counterpart.
One major drawback to the activity is how specific it is. The software cannot be used for any purpose other than learning to “write an algebraic expression that represents the appropriate width of a parking space in terms of the number of desired spaces, the width of the barriers and the total width of the lot”.
In order for this type of applet to be universally useful, one of two things has to be true (or both): 1) it is possible to make A LOT more of these to cover all of the phenomena we wish students to be able to model, or 2) learning how to write the algebraic expression in this applet facilitates a user’s ability to write expressions that represent other phenomena (without the benefit of an immediate link between the symbolic expression and its spatially tangible counterpart, to boot!)
Option 1 doesn’t seem feasible (although it could be!). I question Option 2 as well. I am finding more and more that learning how to write expressions that model phenomena is not a general skill, but instead is highly dependent on the specific phenomena you have learned to model. Even subtle changes in the details of a phenomena are sometimes enough to require more learning before a person can successfully model the new, change context.
Kyle Pearce
July 30, 2014 - 4:33 pm -Harry,
I would agree that it can be difficult to make interactive applets for all math topics, but it is definitely a great start. As you said, if we continue finding ways to make math meaningful for students with tools such as this one, students will be better able to make connections with algebraic representations (I believe, anyway).
When you mention that writing expressions is not a general skill, are you referring to the fact that there isn’t a set procedure or method? I think that is part of the reason why students struggle with it so much. Most of what we traditionally teach in math is procedural and based on a set number of steps whereas writing expressions involves much deeper thinking and understanding.
With an applet like this one, I would hope that students can learn to leverage their use of this applet to help them problem and write expressions in other situations.
Gail C
August 1, 2014 - 11:29 am -Really impressed by this. One small UI suggestion, if you didn’t have a specific reason for not doing it: as a student, I would have liked to see my answer from the previous page shown somewhere on the screen so I could build from it. Much less cumbersome than pressing back to see it (and I didn’t even test whether that worked anyway). It should be exactly as I typed it though, and not a model answer.
Dan Meyer
August 1, 2014 - 1:36 pm -Thanks for the feedback, Gail. We were wondering what to do if the formula was wrong on the first formula screen. Still copy it over?
Simon Gregg
August 1, 2014 - 11:48 pm -Wonderful! Brilliant!
I can see myself using the early pages of this with my Year 4s (3rd grade), because it’s such a clearly set out and satisfying multi-step question.
Even though we do begin some algebraic notation
http://year4atist.blogspot.fr/search/label/algebra
the formula pages are obviously too big a step up.
And I could quite happily leave it at that. But perhaps it’s useful to speculate how it could be even more brilliant…
I agree that the first pages could perhaps not all have the car parks divided into four bays. Three, five. Then there’s more implicit algebraic thinking – it is a variable.
It does feel like there could be a half-way house between the arithmetic and the algebra…
Could there be a page after the arithmetic where you have to say in general what you’re doing. But not type it. Have blocks of text that you can drag around that say “the width of the lot (l)”, “the width of the dividers (d)” along with numbers, brackets and arithmetic symbols. That way, perhaps with the teacher’s help, students could spell out what they were managing to do on the previous pages.
It wold be a formula, but without the big jump to just letters.
It’s one more step then to use the letters instead of the phrases. This would make it more accessible to children in primary/elementary – and maybe also older children too?
One other thing: I agree that it would be nice to have a metric option for those not in the US.
Alex Eckert
August 3, 2014 - 6:03 am -You mentioned in your lead-in post that variable expressions are used by students and programmers. I wonder if you and/or Desmos would be willing to share the variable expressions used in writing the code for Central Park?
Harry O'Malley
August 3, 2014 - 10:12 am -Simon Gregg wrote:
“It does feel like there could be a half-way house between the arithmetic and the algebra…
Could there be a page after the arithmetic where you have to say in general what you’re doing. But not type it. Have blocks of text that you can drag around that say “the width of the lot (l)”, “the width of the dividers (d)” along with numbers, brackets and arithmetic symbols. That way, perhaps with the teacher’s help, students could spell out what they were managing to do on the previous pages.
It wold be a formula, but without the big jump to just letters.”
Good idea.