Ha ha. J/k. There isn’t a picture for polynomials. That’s insane. The question about polynomials comes up, though, especially when we give into the fiction that students can’t enjoy math for its own sake.
Let me highlight two positive externalities of WCYDWT, which is to say, benefits of WCYDWT that don’t limit themselves to the time that we are actually WCYDWT-ing:
- The class understands that non-standard approaches are awesome.
- The class understands that failure is useful, not shameful.
You can capture those benefits using traditional curriculum but you have to work a lot harder at it and if you stop working harder, you capture the negative externalities: students come to understand that math is a right or wrong endeavor in which “wrong” is an destination unto itself rather than just another waystation to “right.”
The last two years of my career I facilitated classes that were often fearless and creative. That meant this: if they were really confident with trinomials like x2 + 7x + 6, I didn’t have to lecture. I’d just write on the board: 2x2 + 7x + 6.
Which would offend them. You know, like, “how dare you bring that weak stuff in here, Meyer? You didn’t see what we just did to the last trinomials?”
Because they were creative and because failure had little stigma attached to it, students would start putting answers down. They’d experiment. In math. Worst case, maybe one of them would throw down (2x + 7)(x + 6) – just banging the numbers from the question together, hoping to see some sparks. She’d call me over and ask if it was correct. I’d tell her to check it. “You know how to multiply binomials.”
She’d see she missed it – 2x2 + 19x + 42 – but we’d notice she nailed the 2x2 – “keep that!” – and ask her to experiment some more. My role in class was to help condense and summarize the findings of student experimentation.
This is how you maintain the spirit of WCYDWT even for concepts that seem to defy the spirit of WCYDWT.
16 Comments
John Scammell
October 20, 2010 - 7:31 am -Kids who can multiply binomials can figure out how to factor (even with leading coefficients other than 1) all on their own. Lots of teacher don’t believe me when I say that. I used to approach it in a similar way to what you describe. Thanks to a colleague with a great take on concrete representations, I am now exploring a more pictorial model, based on polynomial areas (multiplying) and polynomial dimensions (factoring).
Riley
October 20, 2010 - 7:46 am -I think learning happens when you’re comfortable enough with something to experiment. Believe it or not, this polynomial practice reminds me of some of my posts about constructing real environments for kids to explore – they know the rules, they know how everything works, they’re practiced at it, and so now they can figure out _more._
Dropping balls and measuring distance works well because kids are already really good at those things, and they know that they can try again. Your kids had the same comfort with polynomials, so you (and they!) reap the same benefits. Great work!
paul thomas
October 20, 2010 - 7:50 am -I agree with you and John. Driving them to multiply to test is great, but I am also a big fan of algebra tiles. The area metaphor is a powerful one. Here is a random site I just found that illustrates the idea: http://goo.gl/zpKX. The tiles provide more good ways to experiment with factoring.
Christian Bokhove
October 20, 2010 - 8:45 am -The statement -as the poet John Keats knew- on failure is paramount. “The class understands that failure is useful, not shameful.” I use it as one of the pillars in my research concerning mathematics education. The concept has many forms: perturbation (Doll), crisis of learning (van Hiele), disequilibrium (Piaget), impasse (vanLehn), productive failure etc. We all stand on the shoulders of giants. Keep up the great blog!
Phil
October 20, 2010 - 11:26 am -Hi Dan, been reading your blog for a couple of months now, as a mechanical engineer turned about to be maths teacher ( a week form finishing my teachers Qual) I have found your blog seriously inspiring and I have used heaps of your ideas in my classes. The approach you are talking about in this post where students are asked to experiment to “discover” how to tackle the next most complex problem is an approach I used alot… rather than feeding the students the new process they have to think deeply about what is different and how to tackle this new complexity… this improves their mathematical reasoning (for me the ultimate aim of maths education) and also means that they understand the new process on a much depper level once they have made it their own. This approach is often called “inquiry learning” in my experience.. and there is alot of research out there on its use in the maths classroom.
Thanks Dan you have made such a difference to my teaching and have kept me determinedly on the path of making maths interesting and alive, despite swimming against the flow of the vast majority of more traditional (and much more experienced) teachers out there
Nathan Shields
October 20, 2010 - 2:48 pm -Here’s a sweet picture for polynomials:
http://math.ucr.edu/home/baez/week285.html
-N
Daniel Schaben
October 21, 2010 - 6:07 am -I am learning that math needs to be messy for kids. Think finger painting with smocks messy. Guess and check messy.
Example:
Problem: Given a standard 8 ½ inch x 11 inch piece of paper, determine a function which gives the volume of a box (without a lid) made by cutting squares from each of the corners and folding up the sides. Let x be the length of a side of the square, and write the volume as a function of x. What would the dimension of x be to maximize the volume of the box?
I use to show the students the logic behind it by working through the dimensions on a roughly drown picture at the board. Recently. I have noticed that students just didn’t get it after showing them how to do the problem many times. So I have given up on that approach and given them a piece of paper, scissors, and told them each to create their own box. Now I have a pile of paper boxes in the back of my room (messy) Some look like whiskey flasks, some look like short Kleenex boxes. We are right now in the measuring and finding volume stage (collecting the data). We will compile that data and see if we can discover the polynomial function. V(x) = (11-2x)(8.5-2x)x. In creating the boxes we had a great discussion about domain. One student tried to cut a 5 in square out of a corner.
AsteriskCGY
October 21, 2010 - 12:26 pm -(2x+3)(x+2)
Wow, that took me 4 tries to get.
Scott
October 21, 2010 - 12:27 pm -I have been trying to implement problems of this type (the messy ones) over the last few years in college algebra and math for general studies classes. Since we do not meet as often as a high school class, I often struggled with how to “get everything covered” and still do some of the more interesting constructivist/discovery learning type problems. I have been utilizing the concept of the inverted class to accomplish this. Meaning, aspects of the class that were traditionally done in class (e.g. a lecture) are now done outside of (and before) class; things that were normally done outside of class (e.g. the homework problems) are now (mostly) done inside of class. I use Camtasia to create the video lecture where students get the first exposure to the material, thus freeing class time for everyone to “get their hands dirty.”
I am still fine tuning the videos and the in-class activities. And, with common finals, I do still feel obliged to do some drill-skill type problems. So far, I am relatively happy with the results.
Thanks Dan for an incredible resource. I have been reading through some of your “archived” postings. You and your readers are really getting people to think about what it means to “teach” math well.
Scott
Javier
October 21, 2010 - 9:12 pm -No joke. This is what we worked with today. I have a group of seventh graders taking Algebra and they have a bit of that fearless streak in them. We talked a little about binomial multiplication and then told them I was going to show them how to break the resulting trinomials back into the binomial factors.
But first, I wanted them to give it a shot. After the frustration and grumbling started one of the kids says, confidently, “I got it.” She wasn’t asking me, she was telling me. The rest of the class, just in hearing that one of their classmates was able to discover some mathematical secret, dives right back in. A few more students got the first trinomial and one went up and explained it to the rest of the class. After a few cycles, the whole class was taking the trinomials to task and I hadn’t done anything more than hint and nudge. It was magical. The best part is that they owned it by the end of the period. It was all theirs.
Sue VanHattum
October 23, 2010 - 7:55 am -Loving this! Thanks for the lovely polynomial picture site, Nathan.
And Dan, you wrote this right as I entered into the polynomial section of my college course (beginning algebra). I’m delighted.
I would like to head towards the inverted class model eventually. Get some of my favorite topics to lecture about on youtube, and tell them to watch before or after class (depending on whether playing with it first is important).
Elizabeth S
October 24, 2010 - 2:43 pm -I love the discussions like this. Let’s keep taking the gloves off, putting the smocks on the students, and handing them finger paints galore.
Marcia Weinhold
October 24, 2010 - 4:11 pm -There ARE pictures for polynomials. They are called graphs. I started study of polynomials with factors. We built polynomials out of linear functions (do you know what happens when you multiply two lines together?), and played with making the polynomial do what we wanted by placing new zeroes in the polynomial (new linear factors), or making one of the factors have negative slope. We predicted what the new graph should look like by looking at where all the factors were negative or positive, and what sign the product would have (remember sign charts?).
Since Geometer’s Sketchpad started graphing functions, we have used that software to explore because you can make different graphs different colors, and the graphs are a little prettier than a graphing calculator. Adding sliders to the factors, or putting the zeroes as points on the x-axis, adds another dimension to the exploration. To get a “full” picture, you have to be able to add quadratic factors with no real zeroes, e.g. x^2 +2. For more info, see “Designer Functions” Mathematics Teacher, Aug. 2008, pp. 28 to 32.
For some applets to work with the area model (lots of fun), see http://www.fi.uu.nl/wisweb/en/ , choose applets, then geometric algebra 2D. There are four ways to play with these, and you do have to play because the site has no directions.
Thanks for your thoughts, Dan!
Kathy Clark Couey
October 24, 2010 - 6:57 pm -I just figured out why I love reading your blog. Post after post your ‘math is important message so teach it well” message inspires me . . . even though I teach science.
Touzel Hansuvadha
February 27, 2011 - 3:36 pm -Marcia, is it accurate to say that a polynomial is the product of two lines? I feel like a pedant, but isn’t there a slight difference? I mean, ax+b is not a line, but y=ax+b is. And multiplying y=(ax+b) by y=(cx+d) gives y^2=(ax+b)(cx+d)–is that a polynomial?
I feel like such a dick for asking this line of questions…