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# Category Archives: math

## what’s so important about algebra?

Every student in the US has to learn algebra. If this statement is an exaggeration, it’s not much of one. Almost all students take at least two years of algebra before graduating from high school and millions take it again in college. In addition, algebra skills are required in most science, engineering, and other course. But as technology evolves and what it means to be an educated person changes, I think it’s time that we think about why we teach algebra and the way we use it in education. In particular, I think it’s time we stop making algebra skills a barrier to success in college.

Now don’t get me wrong – I love algebra. Really. It’s a beautiful achievement, solving problems that challenged humanity for centuries. It’s also fun, and, as a math teacher at a community college, I enjoy supporting people as they learn algebra’s intricacies. I hope algebra is always available for those students who want to study it. However, if we’re honest about the knowledge and skills needed by 21^{st}-century graduates, workers, and citizens, algebra does not rank high on the list. Even in the technical fields, I seriously question how often algebraic skills are actually required.

The issue is especially relevant in the community college setting because large percentages of incoming students are placed into developmental algebra courses, or below. These are the same courses most of us took in high school, but students have trouble retaining the algebraic skills they learned, especially if those skills aren’t related to their majors. As a result, many students struggle to learn algebraic content that, if they’re not going on to calculus, they don’t need for their next courses – topics like factoring polynomials and solving rational equations with variables in the denominator and synthetic division. The data reveal that students who place into algebra or below are very unlikely to ever pass college level math. And because first-generation college students and students of color are placed disproportionately into low-level math courses, the algebra barrier perpetuates educational and economic inequities.

For all these reasons, in 2010 I partnered with a colleague to develop a new course designed to prepare students who were going on to take college-level statistics. The fact is that relatively little algebra is needed to learn statistics and we thought we could help students succeed in statistics using a different kind of course, a course containing only the algebra students would need for statistics. We hoped to help the majority of students who aren’t heading toward calculus and who need statistics to complete their associate degrees and transfer to four-year colleges.

Fortunately, we were not the only ones working on this idea and we learned a lot from professors at other community colleges already trying this approach. (Learn more about the “pre-stat” community at: http://accelerationproject.org/.) With their help we were able to create our course, called Preparation for Statistics, and piloted it in Fall 2011. In the course, we asked students to engage with real data, using statistical ideas in an interactive and constructive teaching and learning style. We even helped them create their own surveys, collect data, analyze the data, and present it to their classmates. It was work to teach this way, but it was also the most fun I’d ever had in class.

Most important, it worked. Data from our college, combined with other colleges teaching similar courses, show that students from pre-statistics courses are successful in college-level statistics and that they are much more likely to complete their math requirements than students that who took the traditional algebra sequence. The evidence also suggests that the courses helped close achievement gaps for underrepresented students. (http://rpgroup.org/system/files/CAP_Report_Final_June2014.pdf) At our college, the evidence was strong enough to expand beyond the pilot stage. Each year we were helping hundreds of students reach and succeed in statistics.

If taking algebra in college is not necessary for success in statistics, what about other math courses? What about science courses? Isn’t algebra the mathematical foundation of modern science?

Questions like these got me thinking about mathematical prerequisites for general education science courses. These are the science courses that non-science majors usually take to satisfy the science requirement for their degrees, things like astronomy, biology, geology, geography, and basic chemistry and physics. I looked for studies of math prerequisites in courses like these, but have yet to find one (if you have one, I’d like to see it). The marked lack of statistical evidence that either supports or refutes the need for math prerequisites in science courses (or any courses, for that matter) is telling. At my college, most of these courses do not have math prerequisites, precisely because they want to attract non-technical majors to the courses (some of the courses advise completion of algebra, but don’t require it).

I did find some unpublished data, collected at my college and two other California community colleges that offer pre-statistics courses. Aggregating the data from all three colleges, students who took pre-stats courses before statistics were almost exactly as successful in their general education science courses as students who took the traditional algebra preparation for statistics (84% vs. 83%). Even disaggregated, the difference between the success of students at each college was never greater than 10 percentage points and the college (my own) with the lowest success rate for pre-stats students in GE science courses was still 72%, compared to 78% success for their traditionally algebra-prepared peers.

These results beg the question of how students without as much algebra are doing so well in general education science courses. One answer, suggested and bemoaned by some, is that instructors of those courses are reducing the mathematical content of the courses to accommodate students who haven’t had algebra since high school. Another potential answer is that, since almost all students took algebra in high school, a little reminding and prompting enables students to use algebra to the extent that they need to solve the problems.

While both of these are possible, I have yet to see any data that support those answers or any other. In the absence of evidence, I think it much more likely that the real skills needed to do well in general education science courses are things like numerical literacy, critical thinking, the ability to connect evidence to an idea, and academic skills like going to class, reading your book, taking good notes, turning in your homework on time, and, perhaps most important, belief in your ability to succeed. All these skills are taught in both algebra and pre-statistics courses; my experience is that more attention is paid to them in pre-statistics courses than in algebra.

But, what if it were true that science instructors have reduced the algebra content of their classes? Would this be a problem? I say, no. From my perspective, science classes exist to teach science concepts, not to test students’ algebraic knowledge. If, indeed, science teachers are making science concepts more understandable for students with less algebra experience, that would be a testament to the quality of their teaching ability. As I like to say, it’s easy to make an idea complicated and hard to understand; the difficult task is to make ideas simple and clear.

We have been making most science and math courses harder to understand by forcing algebra into them, even though it’s not needed or needed only minimally. For example, in a physics class the height of an object thrown in the air can be modeled quite well by a quadratic equation. Understanding of the scientific principle is demonstrated by setting up the equation. Solving the equation is purely algebraic, but most of the time these “physics” problems aren’t correct until the equation has been solved. In a science class, the science concept should be the primary goal. Solving the equation by hand should be less important, especially when computers with powerful solving algorithms are so readily available.

Here’s another example, from a geometry course:

The geometric concept being reinforced is that the sum of the angles in a triangle is always 180**°**. But, in order to solve the problem, you have to perform some algebra. We don’t need algebra to understand the geometric idea, but if a student can’t do the algebra they won’t get the problem right.

We force students to do similar (and often more complicated) algebraic manipulations in chemistry, biology, oceanography, geography, economics, trigonometry, calculus, statistics and many others. In my experience it is algebra that trips up most students in these courses, not the non-algebra content. Limits, differentials and integrals are challenging ideas in calculus courses, but factoring from beginning algebra is frequently the biggest barrier to completing a calculus problem.

Of course, reinforcing algebraic skills throughout the math and science curriculum is not necessarily a bad thing, but I think too often we do it because that’s the way we were taught, not because of any considered pedagogical reasons. The cost of this decision is high because algebra courses and algebra’s continued use throughout the curriculum is, as I mentioned earlier, so often a barrier preventing students’ success.

And, while algebra can teach attention to detail, mastery of algorithms, symbol manipulation, logic, critical thinking, problem solving, teamwork, numerical literacy, and more, there are other ways to teach those same skills. My experience teaching pre-statistics suggests that we can teach those skills as well or better outside of the abstract context of algebra.

Higher education is changing at an unprecedented pace. These changes are driven partly by increases in the percentage of the population who go to college, partly by pressures from the federal and state governments for more return on their education dollar, partly by employers’ demands for well-prepared, 21^{st}-century graduates, and partly by huge technological advances. In mathematics, the traditional algebra and geometry sequence, familiar to most of us from our own mathematical careers, is being questioned. The algebra sequence, after all, is designed to prepare students for calculus and beyond. But in a world where the most students are not seeking science, technology, and engineering degrees, do we really need to prepare all students for calculus? I don’t think so and I’m not alone. According to the 2015 report *Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation*, “Alternatives emphasizing statistics, modeling, computer science, and quantitative reasoning that are cropping up in high schools and colleges are beginning to challenge the dominance of the familiar math sequence.” (http://edpolicyinca.org/publications/degrees-freedom-diversifying-math-requirements-college-readiness-and-graduation-report-1-3-part-series) These alternatives are emerging because the knowledge and skills needed by informed citizens of the 21^{st} century can be taught as well or better in other ways and because the cost of continuing to insist on algebra is too high.

I’m open to being persuaded that algebra is as important for college students as we have made it. But, to change my mind, you’re going to need to show that the benefits of algebra are algebra’s alone and that they outweigh the costs of forcing everyone to do it.

## It’s on math teachers to change society’s attitude toward math

The two most common responses when I tell people I meet that I’m a math teacher are:

- People tell me how much they admire and even like math;
- People tell me how bad they are at math, squirm uncomfortably, and kind of sidle away.

In the second, much more common option, I seem to represent all the inadequacies they have felt about math for much of their lives. They try and fail to put on a polite face, instead looking at me as they would a particularly disgusting plate of food. It’s an occupational hazard to have these awkward moments at parties.

Both responses more or less make an assumption that I love numbers and all things numeric. But, as much as I enjoy learning about and teaching math, I don’t love numbers. I love people and ideas. For me teaching is pretty close to a perfect combining of these two: I get to talk to people about ideas. Teaching statistics, in particular, is about helping people sort through all the individual stories they know, looking for a trend or trends; students learn to organize and communicate about data, helping them make decisions in the world.

And as a teacher, the stories my students tell me about struggling with math and with education in general are often heartbreaking. Everyone has a math autobiography, too often filled with a teacher or relative who said something like “math is hard and it’s not for everyone” — a backhanded way of saying telling the student he or she doesn’t cut it. Sometimes it’s more direct (“Math is not your subject” or even “You’re not smart enough to do math.”) and usually these messages are delivered to children at ages 8, 9, or 10.

Another common narrative is of the young person who enjoys math till a certain class or till a certain teacher — frequently around 9^{th} or 10^{th} grade — where the student gets the message that they have “reached their math level” and anything beyond it is impenetrable for their meager talents.

There are, of course, many other stories (and I invite you to share yours in the comments) and this kind of introduction often leads people who are pro-math-and-science to exhort our society to change its attitude toward math, moving to a more math-positive message for children or to calls for increased time and emphasis on math in school. And while that would be nice, I don’t think it’s the job of society to change its attitude toward math and science. I think it’s the job of teachers to change society’s attitude.

Clearly, many people have been trying for generations to do just that, so I’m not suggesting I have all the answers or that I know what everyone should do. But I would like to share some of my experience co-creating a math course using the principles of backward design, just-in-time remediation, attention to the affective domain, and the assumptions that students are capable of high-quality work and that context is important for learning.

Three years ago, a colleague of mine and I set out to create a course that would prepare students to take college-level statistics using these principles. Unlike myself, my colleague was an experienced statistics instructor and as we talked about what to put in the preparatory course and explored the curriculum together, I constantly asked “why do we cover that topic?” and “what’s the purpose of that skill?” To her credit, she never responded that we do it that way because we always have. She never said “trust me.” She always gave me a good reason — or we tossed out that topic. The result is a course in which, unlike every other math course I have ever taken or taught, there are no extra topics; that is, in our course (“Preparation for Statistics”) every topic and every activity and every assignment are directly relevant to preparing students for the next course.

And while statistics is in general easier to contextualize than algebra, if a student does ask why we are studying a particular topic, the answer is always, in addition to any other uses, that it will be used again next semester.

This intentionality about everything we do in the class creates more buy in for students. Combine it with the assumption that students are capable of doing the work and the practice of appropriate support and we have a course that alters students’ perceptions of math (toward being more useful to them) and of themselves (toward being capable of understanding and using math).

For generations math teachers have debated amongst themselves and with others about the best ways to justify and explain the importance of mathematical education, with more and less success. Arguments about the development of problem solving and reasoning skills assume a privileged place for math that is disciplinarily arrogant and willfully ignorant (or even insulting) of the intellectual rigors in every other discipline. Discussions of the utility of skills such as factoring polynomials, solving inequalities, and calculating the volume of a frustum (much as I enjoy these topics) are unconvincing and potentially disingenuous.

It is our job to do better. And from my experience, when we do a better job of connecting what we’re doing in class to something the students want to know, the students respond with curiosity and engagement — the kind of engagement that leads to empowerment, learning, and a new attitude toward math.

And when that happens, people no longer have feel that sinking feeling inside when they see some numbers in an article they’re reading, they no longer have to cringe when they meet a math teacher, and my social life gets a little less awkward.

(*Big ups to the work of the California Acceleration Project and its founders Myra Snell and Katie Hern – my work and the assumptions it’s based on could not have happened without their leadership, intelligence, and support. Snell and Hern are among the finest educators I know.)

## the signal and the noise

Nate Silver has made a name for himself in recent years, largely as the founder of Five Thirty Eight, a blog that uses statistics to discuss and predict the outcome of elections and other political issues. His book, *The Signal and the Noise: Why So Many Predictions Fail — But Some Don’t*, is an extended discussion in context of the ideas behind his methods. Whether exploring the statistics of gambling, sports, the recent housing bubble, the stock market, weather reports, hurricanes, disease, or anything else, Silver’s thoroughly researched writing is almost always approachable and compelling, more narrative than demonstration of technique.

But I think the real point of the book is to suggest that Bayesian probability is an important, perhaps the best, way to understand the world. The main Bayesian idea is to start with some assumption of how likely an event is and then, as new information is acquired, modify the chance of the event as often as necessary, coming closer and closer to the truth. This explicitly probabilistic view of the world expects you to make predictions and to test them against what happens. If you refuse to do this, you are either dishonest, don’t recognize the biases you bring to the way you see the world, don’t believe in your own assessment of the likelihood of an event, can’t or won’t see the world probabilistically, or some combination of these. One proof of Silver’s methods is that he correctly predicted the outcome for every state in the nation in the last two presidential elections.

Lest you think Silver is bombastic or trying to force an ideology on the reader, let me assure you: on the contrary, the writing is almost humble in its willingness to question itself and tries hard to present the evidence and let you decide what seems right to you — an especially good example of this is the chapter on global warming, in which Silver, who appears to believe that global warming exists and is a problem, acknowledges the strength of the skeptical arguments and responds to them respectfully.

As a math teacher, I appreciated the wealth of examples and the deep conversation about probability, statistics, assumptions, models, uncertainty, and heuristics. Any reader would enjoy the book for its careful and clear handling of complex topics.

## Uri Treisman rocks

Uri Treisman, already well-known for his work improving the success of Calculus students, continues to impress me. (And—I had the chance to meet him last summer—he’s a nice guy.) In this talk at the WestEd Board of Directors’ 2010 Forum, Treisman talks about the work Carnegie is doing on developmental math at the college level. He makes many smart points, often backed by research and data. One of my favorite parts of his talk is that he frequently refers to actual student feedback—a radical notion, by definition.

## the importance of listening to students

Good teaching is as difficult to define as other arts and the debate over how teachers should be evaluated and what it should mean is raging all over the country. While reading a paper on teacher evaluation put out by Accomplished California Teachers, I realized that, though the study is useful and the recommendations good, it misses a fundamental issue. Too often, in the discussion among professionals about teaching and learning, we neglect the voice of students.

That’s one of the reasons I like the draft study done by James W. Stigler, Karen B. Givvin, and Belinda J. Thompson, “What Community College Developmental Mathematics Students Understand About Mathematics.” In it, they try to eplore what students get wrong and what they don’t and why. They listen carefully and respectfully to students, thoughtfully writing about what they find.

One of the most profound questions that students pose when asked to solve a problem during the interview is, “Am I supposed to do it the math way, or just do what makes sense?” The question reveals a fundamental disconnect between what students experience in their lives and what they experience in the classroom. Not a revelation: the disconnect is completely consistent with my experience listening to community college students in developmental math classes. Any teacher paying attention is aware of it. However, as I read this question and the rest of the study, I began asking a series of different questions:

- Is the math we teach connected to students’ lives?
- Is the math we teach connected to our own lives?
- Are we, as math teachers, so indoctrinated into a mathematical perspective that we force the connections between math and our lives?
- Would it be beneficial to math students for teachers to call out the cultural framing that we are bringing to the subject and that we are trying to help them assume?

Clearly, I’m not going to answer the first three questions here. People make variously good and bad arguments about math’s “utility” that are usually circular, starting from the assumption that math applies to most, if not all, the natural world. Rather, I think we must continue to ask them of ourselves and of our curriculum. The question of perspective and acculturation is complex and probably unanswerable. Philosophers of science, much smarter and more capable than I (e.g., Karl Popper, Imre Lakatos, Thomas Kuhn, and Paul Feyerabend) have been arguing about it for years without full resolution.

But the last question is easier for me. Cultures around the world do math, so math seems to be a fundamentally human activity. However, that math is not usually what we’re teaching. As such, I firmly believe it is helpful for students to see that the math we teach in our classrooms is a cultural construct and not necessarily “natural”; in fact, the math in our modern textbooks is a carefully contrived version of math. It is made to appear smooth, a straight line of development from numeration, to fractions, to factoring, to graphing, to functions, to differentiation, to integration, and beyond. If students don’t see how smooth and “obvious” it all is, then it is their fault. And when the story isn’t quite so smooth, we just pretend it is — “don’t you see?”

Acknowledging the culture of math and its interplay with the other parts of our culture is an important step to demystifying math and to being intellectually honest, toward having students realize that they can bring all their intuitions, experience, and knowledge to bear on problems, both in and out of math class. At the same time, it helps remind us, as teachers, to listen to students, because their experience of math is part of what math is in our classrooms. More, their experience of math will survive us, long after we’re retired, helping to create the culture of math in the world to come.

As teachers, we have spent years mastering our content and working to be better teachers. Yet, students still sometime disparage our work and/or our chosen field of study. Working with as many students as we do, it is often hard to see what we can learn from the next batch. Truly listening to our students takes effort and focus. I frequently fail to do it well, but every time I do, I am rewarded with a better connection and a better class. Listening to our students is part of the art of teaching. We fail to listen at our own, our profession’s, and our culture’s peril.

## from fish to infinity

In a *New York Times* opinion piece, Steven Strogatz does a great job of articulating both the power and abstraction of numbers. It’s something I try to talk about in my classes — probably with less success than Strogatz. I often use the example of shepherds keeping track of how many sheep they have by creating piles of stones. The usefulness of carrying around a number, say “24,” rather than 24 stones, to express how many sheep one has is pretty self-evident. At the same time 24 applies equally well to sheep, bombs, dollars, stones, people, and more, which presents a potentially dangerous abstraction; that is, we don’t really want to treat 24 dollars the same as 24 people.

Strogatz writes about this issue and more in the article. His goal is to discuss “the elements of mathematics, from pre-school to grad school, for anyone out there who’d like to have a second chance at the subject — but this time from an adult perspective. It’s not intended to be remedial. The goal is to give you a better feeling for what math is all about and why it’s so enthralling to those who get it.”

If future installments are as well done as the first, the project will be useful for the math-interested and not.

(Thanks to Alisa for the heads up.)

## the eyeballing game

For the geometry/technology geek in us, there’s the eyeballing game. So far I haven’t made it to the good side of the distribution on it. (You’ll know what I’m talking about if you play.)