# 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.

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:

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.)

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:

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.

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

# 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:

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.

# 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.

# A Visual Comeback

Please excuse me while I geek out for a few minutes about Visual Patterns.  My love affair with this versatile website has made the transition from autumn to winter as I engage in select patterns with my Algebra classes.  I didn’t start using these until a unit on quadratics last trimester, so I was very pleased that a linear pattern could create just as much conversation and mathematical excitement.

For example, this is a replica of pattern #114 that we looked at in class today:

The equation y = 3x + 4 was not terribly difficult for these kids to decipher. But the fun began, as usual, when I asked them to relate their equation back to the figure.  Here are some of their findings:

1.  Students used the idea of slope and recognized that the slope is the change in the number of squares divided by the change in the step.  The y-intercept is the value when the “zero” step is determined.

2.  There are always 4 squares in the corner and each “branch” off of that square has a length of x.

3.  SImilarly, there is one square in the corner and each branch from that one square has a length of x+1

4.  There are always x “sets” of three squares, and four squares left over.

5.  The arithmetic sequence formula works nicely here, common difference of 3 and first term of 7.

The final observation deserves its own paragraph, as I was completely blown away by the thought process.  The student noticed that if we made each step in the pattern a square, then the formula would be (x+2)^2.  He then noticed that the portions that were missing were two sections, each consisting of a triangular number.  Recalling the formula we worked out last week (by accident) for the triangular numbers, (.5x^2 + .5x) he took (x+2)^2 -2(.5x^2+.5x) and simplified it.  The result is, you guessed it, 3x + 4.  Below is a photo of this amazing insight:

What I like most about these visual patterns this time around is that it helps the kids get comfortable having a mathematical conversation.  Students build on each other’s thinking and discover new insights by listening to their classmates.  This was difficult to do last trimester with a similar group of kids.  I think that by starting with a linear patterns, rather than quadratic, the students have acclimated themselves to different ways of approaching the patterns.

# Oh, UNIfix cubes! I get it!

I’ve done a lot of professing my new found love for Visual Patterns lately, and today will be no exception.  If I haven’t convinced you of the flexibility and differentiation available in these seemingly simple patterns, let me have one more stab at it.

Today, my College Algebra class looked at pattern # 28.

I took out the unifix cubes for those who wanted to actually have the three dimensional shape in front of them.  This was helpful for some, however, I realized the limitations of the cubes…the fact that they only will “fix” to one other cube (hence the name UNIfix). This may not be mind blowing information to many of you, but I just put those two things together in my brain today.  Because of the one fixture, they were hard to take apart in usable “chunks” without the whole figure falling apart.

Anyway, back to the pattern. I wanted to workout ahead of time all of the possibilities that students would come up with so that I could more effectively use the 5 Practices of Orchestrating a Mathematical Discussion and anticipate their responses.  I’ll tell you what, I played around with those expressions so many times, and thought for sure I had came up with at least the majority of responses I would encounter. They were all quadratic. Then, out of left field, the students threw me for a loop. The majority of students came up with n^3 – (n-1)^3!

Now, you might be thinking, duh! It’s 3 dimensional AND a portion of a cube.  However, my algebraically trained brain started with quadratic expressions and stuck with them since I saw from the difference of differences table that this pattern was in fact, quadratic.  Yes, the n^3 terms get cancelled out when the expression is simplified and the simplified expression becomes quadratic but this opened up a whole new avenue of discussion with my class. We were now able to talk about the misconceptions of expanding something like (n – 1)^3, because if they found the expression for the nth step another way, they could use that as a check for simplifying their answer.

What was eye opening for some of the students that chose an algebraic method (such as using a table of differences and then setting up a system of equations) was that the “c” value in ax^2 + bx + c was hard to conceptualize.  It was very difficult for students to grasp that the first term and the non-existent “zero term” had the same number of cubes.

Finding the surface area formula for step n was even more awesome, because it was in this portion of the pattern that I was able to see real growth in my students’ willingness to attempt a more conceptual method.  There are certain students whose default method is to set up a system of equations using the table of values for the pattern steps. These students are noticing more that they encounter errors much more often than those who have a conceptual understanding of how the pattern is built.  I found this time around, less students relied on the algebraic method (about 7/35) whereas last week, probably 15/35 of them were starting algebraically.  As we are covering more and more concepts in this course, the students are realizing that they do not remember specifics about formulas and procedures from their previous algebra courses.  They remember “learning” the topics, but they usually can’t quite nail down the specifics of each method.  I really feel that we are making some good headway toward solidifying their conceptual understanding of the algebra as I see more and more students break away from the procedural methods toward a more conceptual one.

We talked about this pattern for an entire 60 minute class period.  You know it’s a good day when kids look at the clock and say, “whoa! Class is over already?”

# Visual patterns with a side of awesome sauce

Regular old Wednesday turned amazing today when I posed pattern #2 to my math recovery class, a remedial math class for kids to recover credit from a previously failed course. It may not need mentioning, but just to be clear, these kids hate math and think they’re no good at it. In pattern #2, the kids need to find how many cubes are in step 43 and the surface area of step 43. Side note:  My kids wondered, why 43, Mrs.Nguyen?

Anyway, finding the surface area was where the magic started to happen. I had 4 or 5 kids out of this class of about 15 get seriously invested in finding out the answer. They were drawing pictures, explaining their thinking to one another, figuring out different ways to think about the problem. It was inspiring and motivating for both them and me.

As if that wasn’t enough to make it a great day, I decided to pose the problem to my College Algebra class as a starter and try my hand at the 5 Practices for Orchestrating Productive Mathematics Discussions. My expectation was that they found the number of cubes and surface area of step ‘n.’ What was gorgeous about this problem was not necessarily the answer, but the numerous ways they came up with to arrive at the nth step. Here are a few:

n + n + (n-1) + (n-1) + n + (n-1) + n + (n-1) + 2

4(n-1) + 4n +2

4(2n – 1) + 2

6 + 8(n-1)

4[n+(n-3)] +10

6(n-1) + 2n + 4

8n – 2

What was even more powerful was, as Ben Blum-Smith calls, an effing game changer.  He’s right, and this was beautiful.  I used the tactic he lays out in his blogpost where students are asked to summarize the ideas of someone else.  I had a few try to slyly summarize their own ideas, but alas, I would have none of it.  As a result, I had more engagement, more involvement, and more buy-in that this problem solving process is helping them to understand the mathematics more deeply.

Here is an exchange between two students (T and C) that is worth highlighting.  T is the student who came up with 6 + 8(n-1) as the surface area for step n:

C: Oh, I see.  T just used the arithmetic sequence formula.  The first term is 6 and it goes up by 8.

T:  Actually, that’s not what I was thinking.  I thought that there were 8 sides of the figure that had ‘n-1’ squares and then 6 squares left over, two on the caps and 4 in the corner.  OH, you’re right, it is the formula.

Then the lights came on.  This girl who had probably only known mathematics and algebra to be a long list of rules, procedures, formulas, and practice was able to experience that developing a conceptual understanding of this pattern help her to create the arithmetic sequence formula.  It was the bottom-up approach that I’d been talking about all trimester where developing conceptual foundations are where real math learning happens.