fawn nguyen

Somewhere between Concrete Sequential and Abstract Random

It occurred to be relatively early in during the trimester this past fall that my college algebra students (generally) have no idea what I mean when I say “quadratic function.”  This isn’t because they have never heard it, learned it, or used it.  But that technical of a term simply has not stuck around in their long term memory.

So, similar to linear functions, we start with a pattern:

circle

from youcubed.org and visualpatterns.org

I then had them make posters including how it is growing, what the 10th, 100th, 0th, and -1st cases would look like, table, graph, expression, and relationship to the pattern.  Instead of one large poster, they use 4 smaller pieces of paper and tape them together.  That way each group member can contribute simultaneously.

I noticed:

  • It is difficult for students to describe how an irregular shape is growing.
  • It is even more difficult for them to describe something abstract like the -1 case.
  • Many of them expressed the overall growth as “exponential.”
  • Most could easily see the two rectangles formed and determine the dimensions with respect to “n.”

I wondered:

  • If they could connect the work with patterns to other quadratics.
  • How to have a meaningful discussion around the “exponential growth” issue.

Their homework was to answer similar questions for this pattern from You Cubed’s Week of Inspirational Math:

growing

Spoiler alert: the rule for the pattern is f(n) = (x + 1)^2 or f(n) = x^2 + 2x + 1

So where do we go from here, two days before Winter Break?  My goals are to review some specifics on quadratic functions and simultaneously help the students make connections between different representations.  I know what I must do.  I must channel my inner Triangleman.

[Backstory:  Christopher Danielson and I go way back. At least to 2014. Maybe even 2013.  Seriously though, I strive to organize my college algebra class the way Professor Danielson describes in his blog.  I have picked his brain on more than a few occasions and he is gracious enough to give me advice in certain curricular areas. In short, his philosophy titled “They’ll Need it for Calculus” is the foundation of my College Algebra course. ]

Ok, back to room C118.

Me: Write down everything you know about the function y = (x+1)^2

(Most write down the expanded form, some start to graph, but not many)

Me: What other ways can we represent this function?

Students: Tables! Graphs! Pictures! Words! Patterns! Licorice!

Me: Sweet!  Let’s do all of that, minus the Licorice.

(I give them a few minutes to create a table and a graph.)

Me: NOW, write down everything you know about this function.

I circulate and hand each group a half sheet of paper.

Me: Write down the most important thing on your groups list.

At first I wasn’t really concerned what exactly they wrote down, but how they defended their choice. Then I came across this in all of my classes:

1223151442-1.jpg

We came up with a pleasing list of attributes of a positive parabola that included vertex placement, end behavior, leading coefficients, and rate of change.

Next up for discussion: Parabolas grow exponentially.

Me: Turn to your partner and tell them whether you agree or disagree with this statement and defend your choice.

Students: Yes, words, words, words.

Me: Ok, now the other partner, say how you know something grows exponentially.

Students: Multiplied every time, more words, blah blah blah.

So we agreed that 2, 4, 8, 16, 32, 64… is an example of something that grows exponentially.

Me: Numerically, how can we tell how something is growing?

Students: (eventually) Rate of Change!

We came up with this table and agreed that these two functions were definitely NOT growing in a similar way.

1224151308-1.jpg

Now on to helping them understand what it means for something to grow quadratically…

 

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Sitting in a Circle, Talking about Numbers

“I feel like all we do is sit in a circle and talk about numbers.   It doesn’t even feel like work.”

“This class is more exhausting than my PE class!”

“It’s nice to be confused and then un-confuse ourselves.”

These are words I’ve overheard from my college algebra students this year.  I couldn’t be more pleased with the strides they are making with my problem-solving framework.  I learned the hard way last year that you cannot just throw a problem solving scenario at a student and expect them to immediately persevere, even if they understand the underlying mathematics involved.  Having learned from my mistake, I sequenced the problems this year in a way that has worked to build on their Algebra problem-solving skills.  Furthermore, I’ve put them in groups of 3-4, which has helped tremendously in getting them to talk about their approaches.  Last year, while in pairs, the conversations didn’t occur as naturally as I had hoped.    Here are a few of the problems we’ve tried:

Multiplication Square C thumb (1) thumb

 

Additionally, we’ve used other Nrich problems such as Odds, Evens, and More Evens.

And to add some non-dairy whipped topping to this algebra awesomeness, my students are breezing through visual patterns and having some great conversations about them.  Credit here is due to their fabulous algebra 2 teachers who began visual patterns with them last year and let them struggle with them.  The result has been deeper connections and a more thorough understanding.

 

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Yesterday:  As much as I get frustrated by the attitudes and actions of my 5th hour, much of my resentment stems from the fact that I believe the situation in my class is my fault.  I feel like I’ve conditioned them by accepting disrespectful behavior in order to keep kids in the classroom.  As a result, the entire learning environment has suffered.

Today: So that was the beginning of yesterday’s post. I was concerned going into today’s class. Last Friday of the year and the fact that the school has been a circus compounds the issue. I was expecting chaos, but what I got was mathematical success. The difference was I demanded their attention in a more respectful way. I was firm, but polite, and it payed it’s dividends in student engagement.
We began with a simple math talk that I modeled from Fawn Nguyen’s March 21st math talk:

Today is the 30th day of the month.  Write as many equations you can that equal 30.

I gave them about 5 silent minutes. Then I let them use their calculators to come up with more gems.  At the end, I had them share their favorite or most complicated equation on the whiteboard.

IMG_5388 IMG_5389 IMG_5390

 

Here’s where the real magic happened.

Me:  Look up here and see if there are any equations you disagree with

Lots of discussions ensued about order of operations, square roots, rounding, parentheses, etc.  Overall, the activity lasted 30 minutes, which was about 29 more minutes of math than we did yesterday.

But the fun doesn’t stop there.  To boot, I introduced the Mathalicious Decoder Ring Lesson.  We watched the Christmas Story clip and talked about what a decoder ring does.  What I liked is that most of them were trying to figure out how the decoding worked, rather than just “get the worksheet done.”

Justin Aion stated poetically on his blog today the exact way I feel about this class:

We teach students long-term strategies to accomplish short-term goals and often don’t see any progress.  If we are very lucky, we’ll see the kind of growth we want by the end of the school year, but the growing season on students isn’t as regular as it is for other crops.  Each seed needs its own time to grow.We desperately need to get away from the notion that if it hasn’t sprouted by the beginning of June, then it must be a defective seed.

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Pattern Power

If you have little kids and you’ve been privy to an episode of Team Umizoomi, then perhaps the title of this post evoked a little jingle in your head. You’re welcome; I’m here all day.

My daughter, although she doesn’t choose Umizoomi over Mickey Mouse as often as I’d like, picked up on patterns relatively quickly after watching this show a couple of times.  She’s 3 years old, and she finds patterns all over the place.  Mostly color and shape patterns, but a string of alternating letters can usually get her attention as well.  These observations of hers made me realize that pattern seeking is something that is innate and our built-in desire for order seeks it out.

High school students search patterns out as well.  For example, I put the numbers 4, 4, 5, 5, 5, 6, 4 so that the custodian knew how many desks should be in each row after it was swept.  It drove students absolutely CRAZY trying to figure out what these numbers meant.  I almost didn’t want to tell them what it really was as I knew they’d be disappointed that it lacked any real mathematical structure.

I’m not as familiar with the elementary and middle school math standards as perhaps I should be, but I’m confident that patterns are almost completely absent from most high school curriculum.  Why are most high school math classes completely devoid of something that is so natural for us?

Dan Meyer tossed out some quotes from David Pimm’s Speaking Mathematically for us to ponder.  This one in particular sheds light on this absence of pattern working in high school mathematics:

Premature symbolization is a common feature of mathematics in schools, and has as much to do with questions of status as with those of need or advantage. (pg. 128)

In other words, we jump to an abstract version of mathematical ideas and see patterns as lacking the “sophistication” that higher-level math is known for.  To be completely honest, this mathematical snobbery is one of the reasons I discounted Visual Patterns at first.  Maybe it was Fawn Nguyen’s charisma that drew me back there, but those patterns have allowed for some pretty powerful interactions in my classroom.   I’ve used them in every class I teach, from remedial mathematics up to college algebra because they are so easy to  differentiate.

I think high school kids can gain a more conceptual understanding of algebraic functions with the use of patterns.  For example, this Nrich task asks students to maximize the area of a pen with a given perimeter.   The students were able to use their pattern-seeking skills to generalize the area of the pen much  more easily than if they had jumped right from the problem context to the abstract formula.  

I also notice that the great high school math textbooks include patterns as a foundation for their algebra curriculum.  For example, Discovering Advanced Algebra begins with recursively defined sequences.  IMP also starts with a unit titled Patterns.   I think these programs highlight what a lot of traditional math curriculums too quickly dismiss:  patterns need to be not only elementary noticings of young math learners but  also valued as an integral part of a rich high school classroom.

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