A 2-for-1 "Soft Skills" Special: The Sit-Stand Paradox and Defective Girls

Written for The Virtual Conference on Soft Skills, July 3 - July 31, 2010

Of all my courses as a pre-service math education major, I think I enjoyed educational psychology the least. When you spend much of every day deciphering the infallibility of mathematics, the "theories" of social science don't hold up well against the scrutiny of a brain hardened by the concept of rigorous proof. I now realize I should have adjusted my perspective in whatever way necessary to ensure I got more out of the class, but even if I had I don't think anything would have fully prepared me for a classroom full of independently-minded students. You just have to jump in there, year after year, class after class.

In my six years of teaching high school math I developed some wonderful relationships with my students. Without overstepping the bounds of a teacher-student relationship, my students became my friends, something I seem to remember being told I should never let happen. But I would look forward to seeing my students each day; I would try to make the most of my time with them, and I would miss them when they were gone. If that doesn't describe "friends," then I apparently don't know what a friend is. I might be in the "ivory towers" of academia now, but I honestly think of my former students from those six years every single day.

That's not to say that there weren't MANY bumps and hiccups along the way, and I regret the lack of effort and deficits in my own character that prevented me from forming stronger relationships with ALL my students. But as I reflect back, two lessons learned (one a realization, the other a piece of advice) helped strengthen that special student-teacher bond.

The Sit-Stand Paradox
Ask any teacher what period of the day is likely to be their least favorite and most will answer, "last period." It doesn't matter if your school has four, six, seven, or eight periods -- there is always a last period. By far my toughest group of kids I ever attempted to teach was a last-period business math class. It's a bad sign when, on the very first day of class, a student who you've just met pulls you aside and tells you, "I don't know who decided to put this mix of students together in the same class, but it's a really, really bad idea." I'd like to think my chances with them would have been much better if I'd seen them before lunch.

So what makes last period so tough? I think the explanation is simple: after a long day at school, students are tired of sitting and teachers are tired of standing. Should you be trying to hide this fact from your students? No! They're people, not circus animals that might attack if they sense fear. If you want a class that works with you, not against you, share your motivations and frustrations. Establish common goals and understandings so you can move forward together. Maybe it's time for an out-of-seat activity, a lesson outside, or a trip to a less familiar room in the school. I know it sounds easier than it really is, but even something as simple as sending your students to the board while you sit at their desk can be just the change in perspective everybody needs. If you're worried that instruction might suffer with a little chaos, think of how much it's already suffering when all the students are watching the clock hoping to be somewhere else.

Don't Treat Boys Like They're Defective Girls
During my third year of teaching I had the privilege to share a classroom with Miss Sandra Miley, a 30+ year educator who took a distinct pleasure in teaching freshman boys' seminar and P.E. If you didn't already know, freshman boys are at that awkward age (which lasts from about 11 to 25, as far as I can tell) that can make them awfully hard to teach. Not so for Miss Miley, who passed on this hard-earned wisdom:

"Do you want to know the secret to teaching boys? It's simple. Don't treat them like they're defective girls."

Ever since I was given that Yoda-like advice, I've been trying to unravel the mysteries contained within. Certainly Miss Miley had a perspective from 30+ years in the classroom that I may never match, but I think I got the point. As teachers, our jobs are made easier (not necessarily more enjoyable or effective) when students sit at attention, take notes, raise their hands, follow rules and instructions, and hang on our every word. If you have students who fit that description, I'd bet dollars to doughnuts that the majority of them are girls. Should you want or expect every student to belong in that category? I sure hope not. So don't punish boys who don't happen to behave like those girls. Such behavior is probably not in their DNA.

If you're not convinced, here's a little anecdote to consider. A highly-respected education researcher shared this hypothesis at a conference last fall (identities have been hidden to protect the unpublished):
"I've never been brave enough to try to publish this, but I've long wondered if boys develop better problem-solving skills because they aren't paying attention in class. Girls who listen carefully to instructions and take notes always know exactly where to start because the teacher told them. Boys who goof off during instructions spend a lot more time and effort sorting out the aspects of a problem for themselves, and that practice pays off in the long run."

We're Not All Math and English Teachers

Yesterday the Des Moines Register published an editorial applauding Colorado for reforming its teacher evaluation laws. The editorial goes on to criticize Iowa's reforms, saying the state is "moving too cautiously." Iowa requires teachers to "provide multiple forms of evidence of student learning and growth," but the Register is disappointed that the inclusion of standardized test scores is not required.

When will the public and policymakers realize that not every subject is covered by a standardized test? I feel like this is one of the most overlooked aspects of this argument. We're not all math and English teachers! Colorado spends 3 hours, per subject, each school year to measure every grade 3-10 student's achievement in math, reading, and writing (plus 3 hours for science in grades 5, 8, and 10). In a state that requires a minimum of 1080 hours of student contact time, this minimum of 9 hours of testing represents less than 1% of a school year. If only it felt like so little!

But that's only for math and English. If you want to mandate teacher evaluations based on standardized tests, you need equally rigorous tests for every subject and every teacher. Can you imagine tests for P.E., music, art, or vocational courses? What if schools could only offer classes that were backed by standardized tests?

Let's also consider the extra time required. Suppose students average 7 Carnegie units (credits) per year. At 3 hours per subject and 7 subjects, we're up to 21 hours of testing per year. If you were to require that standardized tests measure student growth, you'd need to test each student both at the beginning of the year and at the end. Doubling the testing takes us to 42 hours, or almost 4% of the year. It still may not sound like much, but students aren't going to be testing 7 hours a day for 6 straight days. If students tested 2 hours daily, the testing schedule would stretch out to 21 days, or about a month of the school year (half at the beginning, half at the end). You thought finals week was bad? Try two weeks!

Unfortunately, I have yet to work in any school that could carry on with regular learning during standardized test times. We tried everything from one test per day to four, mixed with full-length classes meeting on a rotation to all classes meeting on a shortened schedules. No matter how little testing is done, that testing time affects everything else in the day. Can you imagine a month spent like that?

This little rant has come from me, a guy who has almost always been pro-test. They have their place in the educational system and are part of an effective assessment strategy. But when people want every teacher to be measured by their students' standardized test scores, we have to think about the possible ramifications. So be careful what you wish for, Des Moines Register!

I Love Good Data Visualization. This Isn't It.

Earlier this week Newsweek ran a story titled, "Classrooms or Prison Cells?" Given some of the more recent education coverage from Newsweek I wouldn't have been very surprised if the article came down in favor of prisons.

Thankfully, the article was generally informative and unbiased, and told the story of California's 30-year rise in corrections costs amidst education budget cuts. According to the article, in 1980 California spent 10% of its budget on higher education and 3% on prisons. Now, almost 11% goes to prisons while higher education spending has dropped to 7.5%. If you thought that was a tragedy, check out the graph that accompanied the article:

(Image Source: http://www.newsweek.com/content/newsweek/2010/06/28/classrooms-or-prison-cells/_jcr_content/body/inlineimage.img.png/1277695326254.png)

Do you get the feeling that somebody in the Newsweek graphics department got this assignment at 4:45pm on a Friday afternoon? I would have loved to see the graph try to predict future spending. Given the assumption that these rates are truly linear, you can predict that by the end of this century California will be spending 35% of its budget on prisons and not a single dime on higher education.

Functions of Functions

Mathematical functions are usually introduced formally to students somewhere around the end of Algebra 1 or maybe in Algebra 2. If my Algebra 2 final exam had included the question, "What do you know about functions?" I probably would have said: (a) You use \( f(x) \) instead of \( y \), and (b) if a graph fails the vertical line test, it's not a function. With all due respect to my high school math teacher, this would have been a lousy answer. I might have known some peripheral information, but not the core understanding. Sadly, the mathematical importance of functions is not that difficult conceptually, yet it's crucial to the majority of the content learned in Algebra 1 and 2. Yet students still struggle with what makes a function a function.

My first experience trying to teach the definition in a non-traditional way was using the "Cola Machine" problem in CPM Algebra (Math 1). The problem, several days into Unit 11, describes the following:
The cola machine at your school offers several types of soda. Your favorite drink, Blast!, has two buttons dedicated to it, while the other drinks (Slurp, Lemon Twister, and Diet Slurp) each have one button.

  1. Explain how the soda machine is a relation.
  2. Describe the domain and range of this soda machine.
  3. While buying a soda, Mr. Hagen pushed the button for Lemon Twister and got a can of Lemon Twister. Later he went back to the same machine but this time pushing the Lemon Twister button got him a can of Blast! Is the machine functioning consistently? Why or why not?
  4. When Karen pushed the top button for Blast! She received a can of Blast! Her friend, Miguel, decided to be different and pushed the second button for Blast! He, too, received a can of Blast! Is the machine functioning consistently? Why or why not?
  5. When Loufti pushed a button for Slurp, he received a can of Lemon Twister! Later, Tayeisha also pushed the Slurp button and received a can of Lemon Twister. Still later, Tayeisha noticed that everyone else who pushed the Slurp button received a Lemon Twister. Is the machine functioning consistently? Explain why or why not. (Sallee, et. al., 2002, p. 375)
Most textbooks define functions approximately the same way: a relation is a function if there exists no more than one output for each input. Without a context, however, that definition might not carry much meaning, and relying solely on the vertical line test in a graph may not be helpful enough for many students. The soda machine problem, with its emphasis on consistency, gives both teacher and student a very approachable context within which to discuss what a function is and is not.

Early in my teaching career I thought Algebra 1 basically boiled down to two big ideas: solving equations and graphing lines. If students could do those two things, I felt pretty good about them passing my class. Now I see the big ideas of Algebra 1 differently and basically aligned with Colorado's revised standards for high school mathematics. The six expectations listed for Colorado's high school algebra standard can be summarized as follows:
  1. Functional representations (equations, graphs, and tables)
  2. Function behavior (qualitative)
  3. Function transformations (parameters and parent graphs)
  4. Equivalent expressions, equations, and inequalities
  5. Solving equations, inequalities, and systems of equations
  6. Mathematical modeling using functions
Four of the six expectations explicitly mention functions. The last expectation, mathematical modeling using functions, represents (for many math educators) the ultimate goal for math instruction: to give students the mathematical power to describe and understand the world around them. Instead of just solving and graphing, the big idea of high school algebra is functions, with most linear function work in Algebra 1 and most non-linear function work in Algebra 2.

I think it's a mistake to delay the explanation and definition of functions until late in Algebra 1 or later. Algebra 1 and younger students can understand the cola machine problem or other, similar contexts. Suppose the class plays an "exchange" game. Student A gives the teacher three triangles in exchange for two squares. What should Student B expect to get in exchange for his three triangles? What exchange would represent a function versus a non-function? For something based more in the real-world, this conversation could be set in the context of currency exchange. Also, a helpful model for learning functions might be function machines, such as:

Function machines are helpful models for learning functions, function composition, inverse functions, and even solving equations. They help stress the input-output relationship in a way that words or equations alone might not. Students will also naturally expect a single output for each input. The real challenge for Algebra 1 or younger students is to present a variety of equations that aren't functions, or else risk having students think that every two-variable equation describes a function.

Patterns can also be used to teach functions. If students are given the sequence 2, 4, 8, …, some students are likely to predict 14 as the next term (adding 8 plus 6, the next consecutive even number), while other students might predict 16 (the fourth power of two). Because the fourth term could reasonably be two different values, we can't establish a functional relationship to describe the sequence. This could even be an example worth graphing to discuss the meaning of the vertical line test:

Our calculators also help us distinguish functions from non-functions. If you try to graph a circle on a Texas Instruments graphing calculator, you have to enter two functions: one for the top half of the circle, and one for the bottom half. Therefore, the written forms of the equation of a circle (such as \( x^2 + y^2 = 1\) or \(y= \pm \sqrt{1-x^2} \)) can't be functions. The graphing calculator is also a good tool for discussing the square root function, and why its graph must only be half of a parabola's inverse if we want it to be a function.

There two key obstacles that are likely to remain in a student's way of understanding functions. First, students will continue to focus the vast majority of the time and effort on functions, and there are too few real-world examples of useful non-functions to help distinguish the two. Non-functions are less powerful and less common, but without them we risk having students who casually accept that any equations with an input-output relationship is a function. The second obstacle is notation, and it's not an obstacle we can likely avoid. The change to function notation, such as using \(f(x) = 3x - 2\) instead of \(y = 3x - 2\), is not only difficult to explain as something other than an arbitrary change in symbols, but includes two aspects that are directly contrary to a student's prior knowledge. Now a letter (such as the \(f\) in the preceding example) is no longer a variable, but a name with no numerical value by itself. Function notation also uses parentheses to represent something other than multiplication, adding more work to the already overloaded duties mathematicians place on these two simple arcs. The power of function notation is the preservation of the input (\(x\)) with the output, but it's confusing that the output (\(f(x)\)) reuses the input variable, uses a letter that isn't a variable, and uses symbolism for multiplication of variables that's no longer multiplication. Perhaps a question like this could help students realize the power of the notation:

  1. If \(y=16\), \(y=49\), and \(y=64\), what might be an equation relating an input, \(x\), to the output, \(y\)?
  2. If \(f(4)=16\), \(f(7)=49\), and \(f(-8)=64\), what is \(f(x)\)?
  3. Are the two questions above the same? Which one is easiest to understand? Why?

Sallee, T., Kysh, J., Kasimatis, E., & Hoey, B. (2002). College Preparatory Mathematics 1 (Algebra 1). (L. Dietiker, Ed.) (2nd ed., Vols. 1-2, Vol. 2). Sacramento, CA: CPM Educational Program.

Patterns of Patterns

As a young math student I knew tons of formulas and how to use them, but when it came to counting and generalizing sequences of numbers I often had to resort to brute force, or at least guess-and-check. It frustrated me that I knew there should be an easier way to generalize number patterns, but because I could usually get the right answer (patience and accuracy were on my side, thankfully), I didn't force myself to understand patterns more deeply.

For my first three years of teaching I used the original CPM series, and in their Algebra (Math 1) text they presented students with three key number sequences: the square numbers, the rectangular numbers, and the triangular numbers. These sequences appear so frequently that knowing and understanding their generalizations can be helpful when the sequences appear explicitly or when they are embedded in the foundation of another pattern.

The square number pattern, shown below, is the simplest of the three patterns, although students who are struggling to move beyond a recursive view of the pattern are likely to describe the sequence as "adding the next consecutive odd number." Because of this, the square numbers become a great example for students to see the importance of moving beyond recursive descriptions of patterns and towards expressions that yield the number of tiles for any figure.

The rectangular numbers are the next sequence in the progression of patterns. Instead of the expression (or ) as with square numbers, the rectangular numbers use for one dimension.
The triangular numbers look like a new pattern:

But in fact, the triangular numbers can be seen as half of each rectangular number. This means we can modify the rectangular number expression and represent triangular numbers using .
There are real-life representations of the triangular numbers, for sure (bowling pins, stacks of cans or boxes, etc.), but the real mathematical power behind leading students through this progression is that they see two examples of how to modify a previous pattern and generalization to get a new pattern and generalization.

Our class was presented with a problem known as the "Skeleton Tower," borrowed from http://www.wcer.wisc.edu/archive/nise/Publications/Briefs/Vol_2_No_1/. I've attempted to re-illustrate the tower in the figure below:
Suddenly our figures jump from 2-dimensional to 3-dimensional, but in doing so we've opened up the strategies students might use to generalize the pattern. If a student looks at how the construction of the tower progresses from figure to figure, he or she may think of the tower as a sum of horizontal layers. If a student were only given the table, or the sequence 1, 6, 15, 28,..., he or she would most likely see the mathematics of the sequence the same way.

On the other hand, if a student were familiar with the power of the square, rectangular, and triangular numbers, they may spatially reason the tower to be a single column of blocks of height , surrounded by four "wings," each in the pattern of the triangular numbers. Because the height of each triangular wing is , not , an adjustment to the triangular number expression must be made, giving us a new expression of to describe each of the tower's wings. All together, we can express the number of blocks in a skeleton tower of height as follows:

If a student were to rewrite this expression as , they might visualize the tower as rearranged with two opposite wings removed and each stacked on top of the remaining wings, making a rectangular shape with height and width . Remember, the triangular wings were undersized to begin with; the only reason the width is not is because the center column remains in the middle. Here is the pattern "flattened," which hints at solutions using both the overall dimensions (as just discussed) or using the rectangular numbers plus a center column:

The progression through the square, rectangular, and triangular numbers certainly makes the skeleton tower more approachable. But what if we build a tower that is more complex? What if our three fundamental number sequences are more hidden, and spatial "flattening" strategies are less apparent? Below is one such possible tower, an extension of the skeleton tower.

Unfortunately, after several hours of trying, no generalized expression for the number of blocks in this new tower was found. No amount of spatial reasoning would allow me to rearrange the blocks into an easier shape, but fortunately there were two indications that the generalization would be a cubic.

First, unlike the skeleton tower, this shape is much closer to a true pyramid, so it was difficult to imagine it losing its 3-dimensional character no matter how many blocks were rearranged. (I tried variations of the formula for the volume of a pyramid, , with no success, but that did make me hypothesize that thirds of a cube could be involved in the correct generalization.) My second clue that the generalized expression was cubic was seen in the difference between the terms of the sequence (which I had to extend to a 5th figure):

Because the third differences reach a constant term (four), the generalized expression must be a cubic. This patterning of differences is a very handy tool to have in these cases, but I must admit I don't fully understand why it works, or if it can be used as an aid in determining the expression.

(UPDATE: See update below for ways I could/should have found the generalized form.)

With no real hope of deriving the generalized expression via non-mechanical means, I gave up and used my calculator to find a cubic regression. The generalized expression for this second cube stack is . (The thirds are there, as I suspected, although that was far more luck than intuition.) Even with this expression I still can't imagine a physical restacking of cubes (or fractions of cubes) that would lead a student to this result, nor would knowledge of square, rectangular, and triangular numbers.

I find these five patterns intriguing for a number of reasons. First, we go from very basic to very difficult in a quick but logical way. The skeleton tower is a good choice because it incorporates the triangular and rectangular patterns in its solution, and it adds another dimension (literally and figuratively) that suggests new problem-solving approaches, including 3-dimensional spatial reasoning. As for my pyramid-like tower, I like that it re-establishes a disequilibrium for students (and myself). For Algebra 1 students that aren't ready for quadratic regression, being exposed to such a complex pattern might provide some motivation and could be revisited either later in the course or in Algebra 2. It might also be an example I give to students for a "design-your-own" pattern workshop, where students manipulate different permutations of the square, rectangular, and triangular patterns in 2-dimensional and 3-dimensional shapes.

UPDATE (2010.07.02): With great thanks to two friends/followers, I have two more methods of finding the generalization for my pyramidial stack of blocks.

@msmathaddict sent a tweet a link to the Dr. Math portion of the Math Forum that filled in my missing step from my pattern of differences. Now that I've seen the solution it seems obvious, but we can use a system of equations to devise the correct degree equation we're looking for.

My third differences reached a constant, so I knew the equation describing the equation would be third degree. Third degree (cubic) equations are generalized in this form:

From my original figures and table I had four known pairs: (1,1), (2,6), (3,19), and (4,44). Substituting into the generic cubic form, I can create a system of four equations with four variables:

Simplified, the system becomes:

The system might be most easily solved using matrices (because they are easily entered and manipulated in most graphing calculators):

Solving for , we get , so the general cubic becomes .

The second method for deriving the forumla for my pyramidial stack of blocks was sent to me by my friend Andrew Drenner. He opted to use a summation property to summarize the recursive nature of adding increasingly larger layers. He explains:

In a given horizontal slice of the pyramid there is cubes of height . Thus, the problem of finding the total blocks () in a pyramid tall becomes


Given that the summation sequence for squares is


can be expressed as follows: