On Algebra

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There have been a whole flock of article recently addressing the question, “Should we teach algebra to all high school students?” It started (I think, anyway) with Andrew Hacker’s Op-Ed post, “Is Algebra Necessary?” in a recent issue of the New York Times. His conclusion: No, algebra isn’t necessary. A careful reader ought to question, “What does Hacker mean by algebra?” and “Do we want an emeritus political science professor to make decisions about the mathematical education of the masses, given that there are so many people whose entire careers are dedicated to mathematics, the research of mathematics teaching and learning, and being mathematics educators?” But today I wasn’t planning on addressing those questions or Hacker’s article. (See Daniel Willingham’s response, “Yes, algebra is necessary” if you’re interested.)

Instead, there are several important issues that I think are worth pondering whenever anyone starts talking about the necessity of algebra.

1. What do they mean by algebra?

Out of curiosity, I asked WolframAlpha to tell me about “algebra”. It gave a lot of responses (but no definition). It did, however, provide a clear distinction that algebra is something more than equation solving. Yet in Schank’s first paragraph, he seems to conflate all of algebra with the quadratic formula:

“Whenever I meet anyone who wants to talk about education, I immediately ask them to tell me the quadratic equation. Almost no one ever can. (Even the former chairman of the College Board doesn’t know it). Yet, we all seem to believe that everyone must learn algebra.”

I’ll skip over a discussion pointing out that he means the quadratic formula but wrote “the quadratic equation” (as if there’s only one).

In any case, very few people have given the word “algebra” a good-enough definition, from my viewpoint. When I put on my research mathematician hat, I like thinking about universal algebra, which is somehow even more broad and amorphous (and beautiful) than the above definition could convey.  (I am even a published, theorem-proving universal algebraist.) I define algebra like this:

Algebra. Noun. The branch of mathematics that deals with the study of structure.

Yes, that includes studying equations. But it also includes wilder animals like  finite fields or nonfinitely axiomatizable equational theories and the varieties they generate!

2. Who is the “they” and who is the “us”?

Schank also asks,

“Are mathematicians the best thinkers you know? I know plenty of them who can’t handle their own lives very well.”

It turns out that, despite lots of evidence to the contrary, mathematicians are people. As in, real people with real lives and real feelings and real kids and real cats and real hobbies. And, sometimes, real problems. I don’t know why this is news. Surely no one would suggest that we ought not listen to music (or teach music in schools) because some musicians have had difficult lives. The people behind the subject is what makes it compelling. If only robots did math, I’d probably be less interested in math. [And more interested in robots.]

How do mathematicians think? Wow, that’s a fabulous question. Look, no mathematician I know claims that we think “the best.” That’s a “they versus us” distinction if I ever heard one. However, many of us do claim that we think differently.

To believe anything, a mathematician requires a proof.

This is different from every other academic discipline. Mathematicians have a very strict code for how we think about problems. Intuition is never enough.

3. Who taught these people mathematics?!

Moving toward his conclusion, Schank writes,

“You can live a productive and happy life without knowing anything about macroeconomics or trigonometry but you can’t function very well at all if you can’t make an accurate prediction or describe situations, or diagnose a problem, or evaluate a situation, person or object.”

(It humors me that the end of the article talks about making “sensible political choices,” but here it turns out we don’t need to understand any macroeconomics. Say what?)

I wonder who taught these people algebra! None of my students will survive my courses unless they are able to demonstrate that they can use the mathematical content knowledge from my class and apply it to real-world problems about situations that involve optimization or diagnostics or evaluations or predictions. Who are these [potentially imagined] math teachers who teach nothing but endless factoring of polynomials without any motivation?

4. What’s a better question to ask?

Schank and I agree that “[t]he ability to reason from evidence really matters in life.” He thinks algebra doesn’t help develop the skills to do this — I disagree hugely. Algebra can help develop this skill. Does every math educator teach it perfectly? No. Could all of us math educators do a better job? Yes.

My goal as an educator of mathematics is to converge asymptotically on being amazing at my job. Could teaching algebra help students understand logic, reason, and critical thinking? Absolutely. Does it always? Maybe not. But that is not a reason to say, “Don’t teach algebra.” We should say, “We need to teach algebra better.”

And, I promise, I am trying to do just that — along with thousands of my colleagues around the world.

First Day Activity

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I really enjoyed today’s “First Day” activity in Precalculus. I found the idea on Becky Lyon’s blog; you can also find her on Twitter: @rhlyon.

I had the students find someone to work with and told them one member of the pair would be the Explainer and the other would be the Grapher. The Explainer was supposed to sit facing the projector screen, while the Grapher was supposed to sit facing the door (i.e., away from the projector where they could not see it at all).

The idea of the activity is this:

  1. Display a picture or graph on the projector screen for about one minute.
  2. The Explainer has to describe the graph only using words — no hand gestures allowed!
  3. The Grapher tries to re-create the picture or graph from the description.

The pictures I used started out easy (a giant smiley face) and got progressively more difficult. To give you an idea, I uploaded the exact graphs I used to my public Dropbox space: It’s http://dl.dropbox.com/u/59433434/111-Day1.pdf. After we were done, we went through the graphs together and talked about what descriptions had been given and what people could have said to make it easier.

This gave us a great opportunity to review vocabulary like “degree” and “vertex” and “parabola” and “quadratic” and “intercept” and “slope” and “local maximum [versus global maximum]”. It also gave me feedback as to what the “groupthink” occurred and at what level my students are starting. (For instance, some of them volunteered the idea that an even-degree root of a polynomial behaves differently on a graph than an odd-degree root!)

They seemed to enjoy the activity. It helped cement for them that I will expect them to do things in class, not just be. And, happily, it gave them the opportunity to practice my Friendship Policy.

My three favorite comments from today include:

  • “This will be the most FUN class!” –a student said to her friend, at the end of class
  • An e-mail I received after class said, “What a great first day of class! Super exciting and thank you for your approach!“
  • A fantastic Tweet (admittedly from a calculus, not precalculus, student):

     

Friendship Policy

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I have my first course meetings this morning. Right now I’m enjoying a one-hour break between classes in what will become my Office Hours once students figure out what Office Hours are for. I thought I’d take the time to write about an important topic I covered during today’s PreCalculus class.

A Very Important Course Policy:

One of the notable policies I have on my syllabus is called my Friendship Policy: Students in my courses are required to make two friends from class. For those of you who, like me, haven’t been a college student in a number of years, this policy may seem very silly and totally unnecessary! However, the policy has an important function at fixing a “problem” I noticed a few semesters ago.

Before class, I would find students sitting on benches in the hallway for several minutes waiting for the previous class to end. There would be, say, ten or twelve students all from the same course, standing in the same hallway, and it was library silent. No person was talking to any other person! Instead, every single one of them was texting someone on their phone, checking Facebook on their iPad, playing a game on their laptop, etc. Eventually they would all enter the same classroom and continue their technologically dependent anti-social activities.

When I pointed this out to my students, they had never noticed this phenomenon and they didn’t understand why I thought it was weird!

“Back in my day,” says the professor…

There were no cell phones. In order to fill the awkward silence, students in my classes would talk to each other, real-time, face-to-face. Sure, we would talk about course-related things like homework or exam studying, but we would also talk about social activities or sporting events or movies or whatever. This is how we made new friends.

I realize that students in my class have lots of friends. (Otherwise, who would they be constantly texting?) But I still have not figured out how they make new friends. Hence the birth of my Friendship Policy:

Friendship Policy:

You are required to make friends with students in this class. If you are absent from class, your friends will be very happy to lend you their notes to copy! In fact, I think cooperative learning is so important I am going to leave blank space on this syllabus for you to write down the names of two of your class friends and their contact information.

After explaining all this to the students, they usually look at me with confused faces until I say something along the lines of, “Friendship Time: Commence!” and then stare at my wristwatch expectantly. Within seconds, the room explodes in conversation. Occasionally, I have to nudge some of the shy students in the right direction.

Results and Analysis

After several classes over several semesters, this policy seems to make a big difference. First, no one sits before class in techno-quiet. They talk to each other, get to know each other, and occasionally I have caught them teaching each other how to do math problems. Second, I no longer get e-mails asking, “What did you cover in class yesterday?” Third, I learn a lot from my students by participating in before class conversations. For example, in this morning’s class, one student is here on a golf scholarship from Sweden! (How awesome is that!)

I still have two more classes this morning. We’ll see how those groups take to forced friendship-making time.

Project Based Learning

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Our classes for the Fall 2012 semester start today. Thankfully, my teaching schedule doesn’t include Tuesdays, so I don’t start until tomorrow! I’m hoping to use Tuesdays this semester to work on several other projects, including adding more blog postings. Wish me luck.

This semester I’ll be teaching two sections (Section 05 and 17) of our Pre-Calculus class (Math 111) and one section (Section 05) of our Calculus I class (Math 120). Each class meets for 50-minutes per day on Mondays, Wednesdays, and Fridays, and an additional 75-minutes on Thursdays. The longer meetings on Thursdays will be useful in my current quest to incorporate Project Based Learning (“PBL”) into my classes.

I’ve begun the task of designing “Lab assignments” for students to work on, in small groups, during our Thursday meetings. Ideally they would be assignments that require no pre-lecture and ask the students to draw from their course content knowledge to form connections between ideas. By working together in a group, the students could collaborate (hopefully allowing for some peer instruction), ask questions, have a discussion, and digest what we’ve talked about during our other class meetings. According to my calendar, the students will have ten lab assignments over the course of the semester.

Yesterday I began working on the third lab assignment for Calculus. The topics covered earlier that week will be limits at infinity; asymptotic behavior; and continuity. I found an activity called “Carousel Game” from the NCTM‘s Illuminations series and modified it for my class. Here’s a brief overview of this lab:

  • Topic: Graphing rational functions
  • Goal: To correctly determine the equation that corresponds to the problem situation or graph
  • Technology Required: None allowed!
  • Warm-up: Vocabulary assessment, including: asymptote,  rational function, exponential function, end behavior, domain, range
  • Activity: Students will use a description or a graph to find the equation for twelve functions
  • Assessment: After finding the functions, students will find domain, range, vertical asymptotes, horizontal asymptotes, and all intercepts. This will be turned in and graded.

I also uploaded a copy of the lab instructions to my public Dropbox. If you are interested in seeing the entire lab, check it out here: http://dl.dropbox.com/u/59433434/120-Lab2.pdf. (Notice that it’s 120-Lab2, even though I mentioned before it is really our third lab — I start numbering things with zero.)

I’m hoping to reuse this activity in Pre-Calculus later in the semester, once we cover material about rational functions.

Wordle

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Earlier today, Derek Bruff (@derekbruff) tweeted a link to a Wordle done by graduate student Jessica Riviere. Jessica blogged about her Wordle, so check out this link for what she had to say. Her Wordle contained data from her teaching evaluations and what her students had commented. This was clever and fun and it inspired me to make one as well.

I used my course evaluations done by College of Charleston students during the last academic year (Fall 2011 through Summer 2012). Altogether I have data from eight courses (covering several sections of Elementary Statistics, Pre-Calculus, and Linear Algebra) for a total of 114 evaluations. To make the data collection easier, I restricted my focus just to the “Comments on Instructor” and “Comments on Teaching” prompts. This meant ignoring data from sections called comments on “Organization,” “Assignments,” “Grading,” “Learning,” and “Course.”

The most frequently used words were: and, the, to, I, is, she, a, class, was, of, her, Owens, with, Dr. Several of these were removed by Wordle since I had chosen to “Remove common English words.”  I also removed my first name and corrected some misspellings (ex: “explaiend” to “explained”). I enjoyed the following word counts: awesome, 6; funny, 5; humor, 5; and enthusiastic, 9.

Wordle: Eval Cloud

E-Seminar on “Mathematics Teaching and Learning”

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In a previous post, I wrote about finding an E-Seminar from the NCTM (National Council of Teachers of Mathematics). A full list of available topics can be found here. One of them I mentioned before is called “Mathematics Teaching and Student Learning: What Does the Research Say?” Check out the description on their webpage. Today was our first day in my summer SMFT course. Since the students are all in-service math teachers I thought they would benefit from watching the seminar. I hope that they got something out of it, especially considering that it took up around 75-minutes of our limited class time. Here are the top three take-home messages I got from the re-watch:

  1. The idea that teaching is a cultural activity. In other words, we all learn how to teach in a process of cultural immersion during our school years. We get some ideas about what a classroom is “supposed” to look like, what a teacher is “supposed” to be doing during class, and what students are expected to do. Many educators are not taught effective teaching methods and it is easy to revert to teaching how we were taught instead of how we would like to teach (or, how the research says we ought to teach).
  2. The idea that effective teaching is learned. It is not an “innate talent” and it requires a lot of “hard, relentless work.” This is a freeing idea since it allows us to ask questions like, “How do I learn to become a better teacher?” and “What is effective teaching, anyway?”
  3. The idea that improving teaching is a process instead of a goal. Instead of focusing on a large (unattainable?) goal of becoming an effective teacher, instead we can aim for a concrete, step-by-step process of making tiny changes in our classrooms over a long period of time. The seminar suggests to begin “by designing a few lessons with great care” — maybe even just one or two — and after implementation, then gather evidence on the lesson’s effectiveness. A lot of important work should take place after the lesson is introduced when we can consider how to improve it next time.
With these things in mind, one of the major assignments in my SMFT course this summer is for my students to engage in this third item: They are each required to create two lessons for use in their own classrooms. Although they teach for hundreds of hours per year, by focusing a lot of energy and attention on just one or two lessons I hope that they begin to make those small changes. In the meantime, I hope to change the culture of our classroom and move away from being the “Lecturing Professor” character.

Microsoft Mathematics Add-In

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Apologia
I have a confession to make: All of my teaching documents are created in MS Word. Among professional mathematicians, this is heresy. Don’t get me wrong, I know and love and appreciate all the features of LaTeX. In fact, in graduate school, I took my laptop with me to classes and took lecture notes “real-time” in LaTeX, keeping a running, self-updating Index, Table of Contents, and Bibliography.

I even really like writing in LaTeX, I like coding graphics and figures in TikZ, and for a while my favorite hobby was writing the LaTeX code for a great out-of-print book called Algebras, Lattices, and Varieties:Vol I (by McKenzie, McNulty, Taylor, ISBN 0534076513). In fact, you can see the PDF output of my efforts on Ralph Freese‘s course homepage for his universal algebra class. So for stuff I want to look really “pretty” (like the paper I published or my PhD dissertation), I’m down with all the LaTeX fans.

The problem is that I generate a lot of teaching documents. I provide my students with complete lecture notes for my courses, and as they will happily complain, they end up with a three-inch binder of printed materials. So I need something that I can quickly create and edit from a variety of places. Getting WinEdt installed with all the LaTeX packages I use, on machines that I don’t own or Administrate, it is beyond my threshold for acceptable frustration. So, hello Microsoft Word, my dear old friend!

Mathematics in Microsoft Word
If you haven’t used the built-in Equation Editor in Microsoft Word in a while, you might be happily surprised with what it can do now. First, I can input an equation easily using [Alt]-[=], and they are WYSIWYG. No compile/view/re-compile process! Second, it has gotten a lot easier to save a Word document as a PDF file. (I should say that I’ve had some difficulty getting the PDF producer to “play nice” with parentheses, but in that case I can always revert to CutePDF.)

Third, and most important, the Equation Editor has learned some LaTeX. It knows the stuff you use most often: Things like “\ldots” and “\delta” and “\Int_0^2” all do exactly what you think they should. It even has some {align} or {eqnarray} environment functionality, where you can align a series of equations at an equals sign.

But none of this is as cool as the Mathematics Add-In.

Microsoft Mathematics Add-In
It’s a computational engine that will display graphs, solve equations, and do lots of things your favorite graphing calculator can do, too. It’s available free at http://www.microsoft.com/en-us/download/details.aspx?displaylang=en&id=17786.  

It will generate awesome graphs of multivariable functions easily:

Did I mention that it is free?!?

If you want some quick documentation on how to use the Add-In, check out these Dropbox files: docx format or pdf format.

If you want some longer documentation, Microsoft has a support webpage with even more information. I found out about the Mathematics Add-In from a brief article I read, I think in The Mathematics Teacher, that I can’t find now! It made me scream, “How come no one told me about this sooner?! It’s awesome!” So check it out.

Statistics Group Projects: In Progress

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The Elementary Statistics students in my course are wandering the streets of downtown Charleston gathering data for their group projects. There are five groups and here are the questions they are asking (along with their best guess as to what they will find):

  • Do you own a bike? Best guess: At least 30% “Yes” response rate
  • Do you have a passport? At most 40%
  • Are you on vacation? About 30%
  • Do you consume alcohol? At least 80%
  • Have you ever had a fake ID? Around 50%

I look forward to seeing their data!

Statistics Group Project

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Project Motivation
There are two class meetings left in my “Elementary Statistics” summer course. This class time will be devoted to students working together on a group project. Last semester when I taught this course for the first time I really wanted to implement some type of end-of-term project. I wanted the project to be collaborative in nature since both my own experiences and recent research in education have shown that students explaining concepts to each other is as important to their learning process as hearing their instructor’s explanations. I also wanted the project to be somewhat self-designed by the groups themselves. It was my hope that giving them some freedom in their projects would increase their interest level in what they were doing.

The topics we finished covering at the end of the course were about creating confidence intervals and performing hypothesis tests (sometimes called tests of significance). Because we discussed this material so recently, it seemed appropriate to have this be the jumping-off point for the projects.

Project Introduction
I wanted the students to have experience going out into the “real world” to gather data, so the project asks them to conduct interviews with people they find around campus. Since it’s only a week-long project (instead of over an entire semester), to make things easier each group has to agree on a single”Yes” or “No” question to ask their random sample. There are three rules for the question.

  • First, each member of the group must agree with the group’s decision on the question. They have to discuss different ideas, vote on them, and eventually reach consensus.
  • Second, the question must be “interesting.” This is hard to define, but basically I want them to avoid boring questions like “Are you a human being?” or “Have you ever been to Mars?” that will result in boring data.
  • Third, the question must be “appropriate” — it has to be something each group member would feel comfortable asking a perfect stranger or their grandmother or their kid brother. (Hopefully they would know to avoid offensive or disrespectful or inappropriately personal questions, but who knows?)

Once they have chosen their question, each individual is asked to guess (to the nearest 10%) what proportion of interviewees will answer “Yes” to the question. After reaching an individual conclusion, the groups discuss what they expect as a group. I wrote a handout describing the “What” and “How” of prior probability distributions and each group works on creating [a very basic] one before they are allowed to leave to gather data.

Project Report
The groups have the rest of the class time to gather data together. I tried to avoid giving them much direction on who they should interview, or where they should find the people, or what types of people to ask. (For instance, do they want to focus on College of Charleston undergrads, or are tourists okay too?) I suggested to them that they need to keep in mind a lot of the ideas we discussed in the class, like:

  • What’s an appropriate sample size?
  • What sampling method should we use? (Convenience, cluster, stratified, systematic, etc.)
  • Should we expect bias in our data? If so, what types? (Sampling bias, response bias, nonresponse bias, etc.)
  • Can we do anything to eliminate bias?

Eventually the groups must produce a typed project report, outlining their process from how they decided on a question and constructed their prior to where they conducted their interviews. They must use the methods of inferential statistics that we learned in our class to create a confidence interval for the proportion of subjects who said “Yes” and give a correct interpretation of the confidence interval. They also have to perform a one-proportion hypothesis test. They are expected to use their prior probability distribution to formulate a claim to test. They are graded on both their data analysis and interpretation of results.

Project Grading
I created a grading rubric for the project. It’s available as a public Dropbox file: 104-project-rubric.pdf A colleague looked over it in the copy room and commented, “You sure are overly detailed with that thing!” This is probably a fair criticism, but mostly I was trying to avoid hearing lots of student questions that boiled down to, “What is the least I have to do in order to get an A?”

Something that comes up a lot in discussions about graded collaborative assignments is the “slacker problem”, i.e., How do you keep students from getting by doing zero work? I don’t have a good answer for this. I know that when I was a student, I was annoyed by free-loaders, so I have empathy for students who feel the same way. One of the categories on my grading rubric is “Teamwork Assessment.” Each student must individually send me a confidential e-mail discussing how their group functioned as a team and how they contributed to the overall project. They are asked to give their team a grade of how well they worked together. It was my hope that telling them on the project rubric that (a) they are responsible for their group functioning as a team and (b) they are also responsible for ensuring they contribute to the group that it would create a cultural pressure toward equal collaboration. I can’t say for sure how successful this was last semester, however I was happy that out of nearly 100 students, I only had one or two complain about slackers in their groups.

Looking Forward
This summer’s class has a strong group mentality, I think partly because we have been spending ten hours a week together in class. I hope that this will contribute to great collaborative effort toward these projects. I am also excited to see what questions they will ask and what their data will show. I’ll end this post with a few questions I remember from the class projects last spring:

  • Do you have a fake ID?
  • Do you own an iPhone?
  • Will you vote to re-elect Barack Obama?
  • Did you drink alcohol last weekend?
  • Do you have blue eyes?
  • Do you have a car on campus?

Science and Math for Teachers

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This week is the last week of our “Summer I” term and so my “Elementary Statistics” course is coming to an end. My next course begins on July 9th. It is part of a graduate here at CofC that offers a Master in Education in Science and Math for Teachers. My students will be participating in a professional development program called the Mathematics & Science Partnership.

The class itself is called “Applications of Algebra for Teachers.” The prefix for the course is “SMFT” since it’s part of the “Science and Math for Teachers” program. This is a new course, both under the SMFT prefix and in the summer Partnership program itself. I’ve been working on course development since late March, starting with the course description:

Applications of Algebra for Teachers (SMFT 697) – A course designed for middle-level and secondary teachers to investigate applications of algebra in science and technology. Topics will include numeration systems and number theory; linear, quadratic, exponential, and logarithmic functions; and matrix algebra with linear programming. Investigative labs, collaborative learning, and active learning approaches will be fundamental to the course structure.

(Other course descriptions for this summer’s program can be found here.) Our official textbook is “Reason and Sense Making in Algebra“, published by the NCTM. I have also used “Real-World Math with Vernier” for inspiration, since I hope at least some of our labs will use their LabQuest devices. We will also be using current and back issues of The Mathematics Teacher.