I remember one student who managed to take my Geometry class in 9th grade even though (unbeknownst to me) she had already taken a traditional Geometry class in 8th grade. When her father found out, he was miffed, because she had ruined his plan to hyper-accelerate her. I asked her if she thought she had been mis-placed. Her response: even though she got an A in 8th grade geometry, that course had had no meaning for her; she was happy to finally understand this material.
Yes, that is an anecdote about a single student, but it is representative of a broad social phenomenon. Over the years, there has been a disturbing trend towards hyper-acceleration in math education. It started with a full dose of Algebra 1 in eighth grade, but it keeps getting more extreme, with Algebra 1 in 7th grade, AP Calculus in 10th grade, and college courses in 11th and 12th grade.
The trend is at its most intense in the socio-economic groups that used to have a monopoly on mathematics beyond arithmetic in the days before "algebra for all". In some ways, it can be seen as an attempt to assert that kids from certain families are just better at math, and deserve the various advantages supposedly conferred by being "ahead", such as admission to fancier colleges. Of course, no one says it quite like that, and I'm sure such a thought doesn't even cross anyone's mind. Still, the class context of this trend towards hyper-acceleration is hard to ignore.
In this article, I will put sociology aside, and focus on an educational analysis of acceleration.
In most cases, parents (and others) present this as solving a problem: the child is not being challenged, the child is bored, and hyper-acceleration seems to be the only solution. This is not always a fair assessment of the problem, but even when it is, except in very few cases, hyper-acceleration is not the best response. Watch this short video for an alternate response by Calgary math educator Gord Hamilton, of MathPickle.
In some schools and communities teachers face a tremendous push for hyper-acceleration of certain students. The pressure comes largely from parents, but they are often supported by students and administrators. Many parents believe that hyper-acceleration will help their child's college applications. Many students' self-image is intimately related to being "ahead" in math, as they have received so much praise for this in the past. Many administrators encourage hyper-acceleration in response to parental pressure, and in their attempts to compete with other schools. Such competition is seen as an economic imperative for private schools, and for public schools, it is a way to reduce the hemorrhaging of middle class students into private schools.
None of these motivations have anything to do with high quality math education. In fact, more often than not, hyper-acceleration undermines student learning. Here are some examples of hyper-acceleration, and an attempt at explaining why they are counter-productive.
- While some algebra is a must before high school, moving the entire Algebra 1 course in its 1970's form down the grades is a serious mistake. It shuts the door in the face of many students, and promotes superficial rote learning for the rest. (I wrote about this issue at greater length in my analysis of the Common Core.) I have worked with many, many kids who "did well" in a traditional Algebra 1 in middle school, but in fact learned nothing. When being interviewed to decide on what course they should take when they start high school, they struggle to remember the most elementary techniques, which they once knew by rote. In many schools, rote learning yields good grades, but unfortunately it does not stick.
- Many parents and students want to skip geometry, which they see as slowing down the march to calculus. Or else, almost as bad, they would like to dispose of the topic quickly in a summer course. Yet geometry is a fundamental component of cultural literacy the world over; it is necessary for further work in mathematics, including trigonometry and calculus; it fosters a strong visual sense, which is crucial in many careers, such as architecture, chemistry, and design; it is a part of math where students look at the big picture, literally and figuratively. Note that it is often the students whose strengths lie in the memorization of algorithms, not in problem solving or reasoning, who are most eager to skip or rush through that course. Yet they are the students who need it most!
- While calculus can be a legitimate topic for 12th grade, little is gained by teaching it earlier, and much can be lost. I once visited a BC Calculus class where a 10th grader stated that it was obvious that log(a) + log(b) = log(a+b). Such a misunderstanding is normal in 10th grade, but not so much in BC Calculus. Moreover, schools that allow early calculus have to find something to do with their hyper-accelerated 11th and 12th graders. The solution is typically to offer college-level courses with college textbooks and pacing, thereby making it difficult for students to develop an in-depth understanding of the material since most of them are, in fact, not much more mathematically mature than other kids their age. Colleges quite rightly do not offer credit for those classes, and students end up having to retake them, if they haven't been totally turned off to math by then.
OK, you say, too much acceleration can be bad. But how do I suggest schools deal with students who are not challenged by grade-level math, and are bored?
Deeper, Not Faster
Curiously, there is no pressure on schools to teach Shakespeare or Faulkner to ever younger students. In English, common sense prevails: if the student is exposed to a particular bit of literature at an appropriate age before graduation, that's good enough. In math, however, it is widely believed that younger is better, and that strong students will get bored if they are not accelerated.
Alas, the latter is true if the curriculum is shallow, and emphasizes recall and regurgitation! A certain type of student can handle an enormous amount of this, and since all students reflect the surrounding culture, they can't get enough of it, and get addicted to speed. Combine this with the relatively powerful position of their parents, who are typically professionals who put a fair amount of energy promoting their point of view in the school, and you have a recipe for nearly unstoppable hyper-acceleration.
In actual fact, the better solution for the vast majority of students is for math departments to resist hyper-acceleration. Instead, we should find ways to go for depth of understanding, challenging problem-solving, and enrichment. Of course, this is difficult to do when there is a lot community and administrative pressure going the other way. I propose this four-prong strategy:
- Work to shift the departmental and school culture towards depth rather than speed. Deeper, not faster! One way to do it is to teach core topics with rich activities. Of course, changing the culture is not going to happen overnight, but any motion in that direction is well worth it even in the short run.
- Keep offering problems that are "too difficult" for even your top students, as suggested by Gord Hamilton in the video above. This will benefit these students far more than just feeding them more techniques to memorize. Learning to handle frustration, learning ways to tackle tough problems, learning to work collaboratively as well as independently — all those are more valuable in the long run than rushing through next year's techniques, which in any case will not challenge those students, and will come soon enough. This need not freeze other students out. Deep, curricular, age-appropriate problems work with the whole class. Here is an example of a tough but very accessible problem. It's not difficult to find additional challenging problems on the Web, in professional journals, and in past mathematical competitions.
- This needs to happen in math class, but because of time and other limitations, it should also be pursued in math clubs and teams. Unfortunately, at many schools the latter tend to cater almost exclusively to boys, perhaps because of the overemphasis on competition. I strongly encourage you to look at the approach pioneered by the Julia Robinson Math Festivals which works for a wide spectrum of students. Here is a report on my participation in one of those. The Math Circles movement can help provide worthwhile extracurricular math.
Finally, and paradoxically, a key ingredient in resisting hyper-acceleration is to implement some intelligently-conceived, moderate acceleration which doesn't expand learning gaps and which takes into account students' mathematical maturity and developmental readiness.
A moderate amount of acceleration can indeed be a good thing. Here are some examples:
- Some algebra before 9th grade is an excellent idea. In most countries, including the countries that do well in international comparisons, algebra is integrated into math education starting in middle school. That makes a lot of sense, and is far better than the traditional US approach of rehashing arithmetic in middle school, and then hitting students hard with a gigantic amount of algebra to be learned in a single year. In age-appropriate doses, earlier algebra is a good thing, and I welcome that aspect of the Common Core State Standards for math (CCSSM). However this should be balanced with postponing some of the end-of-book and highly technical Algebra 1 topics, which only become useful in more advanced classes. (For more on this, see my CCSSM analysis.)
- New technology and better pedagogy can make some topics accessible to more students than they once were. For example, electronic graphing and an emphasis on "real world" connections have made the concept of exponential growth and decay accessible to ninth graders, even though it was once an 11th grade topic. Interactive geometry software makes it possible to introduce transformational geometry to eighth graders, when this was once an end-of-book tenth grade topic if it was seen at all. Those are additional positive aspects of the CCSSM.
- Having students of different ages in the same class is one way to reduce tracking and enhance equity. For example, a geometry class with strong 9th graders and not-as-strong 10th graders provides both groups with the same opportunities, at slightly different times in their school career. (This is vastly different from assigning some students to an honors class, and others to a "regular" class, because that entails different expectations, and guarantees different outcomes.) But acceleration by more than one year tends to be counter-productive.
- It is possible to teach some interesting and motivational college-level mathematical ideas in high school. I have enjoyed bringing topics such as different-sized infinities, basic group theory, the fourth dimension, dynamical systems, and fractals to 11th and 12th graders. However to do this well, it needs to be done in a form and at a pace that is appropriate to high school, not by using college textbooks and approaches. (See my Space and Infinity classes for one way to do that.)
The main point is to acknowledge that racing students through topics that they are not ready for leads to a rote mastery which does not stick in the long run. Paying attention to students' developmental readiness, and taking the time to teach for understanding enhances learning and retention. Finally, incorporating very challenging problems at grade level is better in the long run than rushing to more advanced topics, as it teaches perseverance, reflection, and collaboration, all of which are more important than any particular topic.