Facing Fear

It’s always fascinating to me to watch students step into a new classroom and immediately search for their social comfort zone.  Students aren’t unique in this phenomenon; they are just the group of humans in which I interact the most.  Today being the first day of school, the visible and invisible social boundaries that students draw between one another were clear as I silently observed.

As someone who struggled fitting into a unique social group growing up, I’m most interested in encouraging kids to break away from their cliques. After reading much of what Ilana Horn has written on the subject, I also began to see links between being socially extroverted and status in the mathematics classroom.  For example, kids who are quiet and mostly keep to themselves don’t often have opportunities to display their “smartness,” whereas an outgoing kid willing to contribute voluntarily to class discussion would have their “smartness showcased regularly.  Interestingly enough, when doing the “personality coordinates” activity with my college algebra class today, one group created this graph:  IMG_6508

They defined social achievements as number of friends and academic achievements as GPA.  It allowed us to have a nice discussion about grades and overall intelligence as well as some lovely talk regarding different definitions of social achievement. I look forward to continuing these conversations over the course of the trimester and challenging them to let their popularity guards down.

On a similar note, I tried the Blanket Challenge in my Algebra 2 class.  If you have not read this chapter in Powerful Problem Solving, I’m not sure why you are still sitting here.  Go read it! What impressed me with this group of kids, was they were willing to step out of physical comfort in order to achieve the result they wanted.  IMG_6505 IMG_6506

On the first day of school, in a class that’s tough to adjust to, I can’t begin to express how proud I am of this group of kids for their willingness to work together respectfully and successfully.  I’m hoping to build on the results from this activity in the days to come.

A Speedy Makeover for the Intermediate Value Theorem

As a college algebra teacher, I was not satisfied with the way I presented the intermediate value theorem last trimester.  I felt the lesson was somewhat isolated from other concepts we had studied and definitely was disconnected from the real world.  My approach lacked a hook and was laddened with procedure.   Committed to teaching the concept better this trimester, I recorded the following video while (someone else) was driving:

I know, not a high quality masterpiece, but I think I captured what I needed to illustrate the theorem.

I ask the students to draw a graph of the speed of the car with respect to time.  After playing the video a number of times, I had them share their graphs with their seat partner.  As I circulated the room, I noticed their results fell into one of these three categories:

Graph A

Graph A

 

Graph B

Graph B

 

Graph C

Graph C

After examining the options, I had them choose which graph they felt represented the situation most accurately.  Spoiler Alert:  The overwhelming majority of them chose Graph B.  Their reasoning:  it’s unclear what happened to the speed between seconds 10 and 15 therefore, there should be a space in the graph.  Those vying for Graph C cleverly argued that there was no audible “revving of the engine,” indicating that the car continued to slow.  Others supporting C claimed that even though we could not see the speed, they know how a speedometer works and can make a reasonable assumption about what happened in that time frame.

Enter this student’s graph and the Intermediate Value Theorem (trumpets):

IVT graph

I liked this students “shading” through unknown speed region, so I projected it for everyone to discuss.  They were able to determine the value of the function at ten seconds, f(10), was approximately 45 miles per hour and the value of the function at fifteen seconds, f(15), was approximately 35 miles per hour.  They also knew that the car must have reached 40 miles per hour sometime in between 10 and 15 seconds.  “How do you know that?” I pryed.  Gem response of the day:  “Well, speed is continuous and I can’t go from 45 mph to 35 mph without going through 44, 43, 42, 41, 40 mph, and so on.”  Bingo.  Intermediate Value Theorem.  No boring procedural explanation necessary.

We applied this “new” knowledge to a polynomial function so that they could get a handle on some of the algebra and notation used.   And as a bonus, they also seemed to grasp that this theorem does not only apply to crossing the x-axis, a common misconception students had last trimester.

Moving forward, I’ll definitely work on creating a better video!

Surgery for Function Operations

My college algebra course boasts one of the driest textbooks on the planet. It’s one of those versions that has exercises from 1 to 99 for each section…brutal.   Can you relate?
The topics for college algebra are very standard and cover little more than what students should have encountered recently in their algebra 2 course. I therefore decided that this class would lend itself quite nicely testing out the theory that a high-level, rich question questioning can be facilitated from a traditional, drill-and-kill style textbook.

Previously, I recall that Operations on Functions was a particularly awful topic for both me and my students.  The textbook presents this concept in exactly the way you might think:

f(x) = [expression involving x]  and g(x) = [similar expression involving x]

Find f(x) + g(x), f(x) – g(x), f(g(x), f(x) *g(x), f(x)/g(x)…f(snoozefest)…you get the point.  It’s boring, they’ve done it before, and there’s not much high-level thinking involved.

Fortunately, it’s fixable by asking new questions from the same problems.  For example, have students choose a pair of functions from the book.  We have 99 choices after all!  For example, something quadratic and something linear,  like f(x) = x^2 + 1 and g(x) = 2x+4.

Here come the questions:

  • Which of these function operations are commutative and which are not?  How do you know this?
  • Does this work for all functions, or just the ones that you chose?
  • For what values of x are the non-commutative function operations equal?
  • What do you notice about those values of x for the different operations?
  • Can you prove any of your results?
  • How do the graphs of these new functions compare to the original graphs?

Compositions of functions are the most fun!  Here come some more:

  • For which values of x is f(g(x)) > g(f(x)) for your specific functions?
  •  Do your results hold true if both functions are quadratic?
  • Both linear?
  • How are the graphs of f(g(x)) and g(f(x)) related to both f(x) and g(x)?
  • Don’t forget about f(f(x)) or g(g(x))! How do those relate to our original functions?
  • What about g(g(g(x))) and g(g(g(g(x))))?
  • What do you notice happening each time we compose the function with itself again?
  • Can you generalize your conclusions based on the number of compositions and tell me what g(g(g…g(x)…)) would look like?
  • What do you notice about each of these compositions?
  • What do you notice about their graphs?

A personal favorite of mine is:  If 4x^2 + 16x + 17  =  f(g(x)), what could f(x) and g(x) have been?  This works really well with whiteboards and partners.

I might have students throw out any questions that they find interesting.  In fact, I’ll bet we can come up with at least 99 questions more intriguing than the ones given in the textbook.  Then let them choose which one(s) pique their curiosity.   Now hopefully we’ve taken the time that they would have spend doing 1-99 from a book and turned it into time better spent.

 

 

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.

Full Circle Reflection

It’s almost the end of the trimester already which made today my last official “teaching” day with my Algebra class.  I’ve used a lot of the Math Forum’s Problems of the Week in this class.  Since this is a college algebra class, I use the POWS more as problems of the day rather than the week.  As a member, I have access to the library of problems, which I scour quite frequently to find just the right problem to fit the topic at hand.

Today’s adventure:  Rational Functions

I used a POW in which the first four terms of a patterned sequence of A’s and B’s are shown.  The students are asked to create an expression to represent the number of B’s in the nth term and then create an expression to represent the ratio of B’s to the total number of letters in the nth term.  What I like about this task in particular is that it isn’t a completely obvious fraction-ladened, asymptote-wielding, makes-a-student-want-to-cry rational function.  The students are able to work through most of the problem forgetting that this is in fact THAT type of function.  In fact, since they weren’t immediately scared off with a 1/x or the like, it seemed easier for them to make connections from their solutions to the graph and equation of the function.

What was particularly fantastic about this problem was that the growth of these students in the problem solving process was so evident.  It was clear as I circulated the room that over the course of a trimester, these students’ goals as mathematicians were evolving:  from “fast and correct” to “patient and curious.”

For example, when asked to find the term that results in 35% B’s, I had many students make a table with # of A’s, B’s, Total letters, and ratio of B’s to Total letters.  At the beginning of the trimester, these kids would accept their correct answer, but then reject their method of arriving at the answer because it was not as quick as those able to recall an equation or procedural method.  Now, after 13 weeks, these same kids are able to look at their table and appreciate the extra questions they can now address about this pattern scenario.  Additionally, some students were willing to attempt multiple methods in arriving at the answer.    It was a pretty profound moment for them as problem solvers and me as an algebra teacher.  I don’t know who was more proud, them or me.

Here are some samples of their work:

 

image (2) image (1) photo (2) image

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.