Guest post by Douglas W. Green, EdD
“The only way to close achievement gaps is to slow down the fast learners.”
Compared to other disciplines, mathematics has a much smaller number of concepts to learn. As a result, the students with high math ability are often needlessly held back. They also run the risk of boredom during math classes.
At the other end, students with poor math skills get dragged on to new concepts before they master current lessons. Since much of math is hierarchical in nature, not mastering early lessons can make later lessons impossible to comprehend. This is less of an issue in other content areas.
While other lessons often rely on background knowledge from previous lessons, students who don’t master previous lessons can still appreciate new material to some degree.
The one-size-fits-all single pace approach that many teachers still use is, therefore, more of a problem in math class than elsewhere. By the time middle school rolls around, most schools sort students into advanced and regular math classes. This is done so that the top students can take some variation of algebra in eighth grade for which they receive high school credit.It allows those that wish to take four more years of math during their high school career for a total of five credits. The final credit is usually some variation of calculus that may include an Advanced Placement class, at the end of which is the national AP exam in May.
Catching the Calculus Train
Students who take calculus in high school are well positioned to major in engineering or physical science in college. Students who miss the eighth grade algebra train most likely won’t be able to take calculus their senior year, and will be at a huge disadvantage if they want to pursue engineering or physical science. This so-called train actually leaves the station for most students at the beginning of seventh grade. In order to take what amounts to math 9 in eighth grade, teachers have to cram math 7 and math 8 into the seventh grade year. This requires schools to identify candidates for the advanced math program near the end of sixth grade.
In my case, my school gave some standardized tests to sixth graders. As fate would have it, I was absent for these tests. Rather than ask the teacher if I was advanced math material, I was dumped into the seventh grade section that was one notch above the class that contained students with major cognitive issues. This meant that I couldn’t take calculus my senior year, but that didn’t stop me from majoring in chemistry. During my calculus I class in my freshman college year, the teacher asked the class if they had calculus in high school. As I looked around I noticed that I was one of only two students who didn’t raise his hand. Ouch!
I struggled, of course, but managed to graduate none the less. My lack of calculus in high school, however, affected not only my math grades but grades in physics and physical chemistry. This meant that my career options did not include a PhD in chemistry. I decided to pick up a masters in education and go into teaching. I guess I can’t complain as I have enjoyed a great deal of success as an educator.
The main point here is if the top students can do two years of math in seventh grade, why can’t they do more than one year of math in every other grade? The answer of course is that they can. It’s just that most schools don’t make it possible. When I was the director of computer services for a small city school district from 1982 to 1993, I hired the smartest students I could find as “student programmers.” I gave them tricky problems to solve and tried to avoid tasks with time pressure deadlines.
At the end of one school year, one of my freshman students who was finishing math 10 asked if he could take math 11 during the summer. At the time, the school’s policy was that summer school was only for students who failed a course. As a member of central office administration, I decided to see what I could do. As an assistant to the superintendent, I was technically above the high school principal so I approached him and asked if my student employee could take math 11 independently during the summer under my supervision.
After a few rounds of that’s not how we do things, I wore him down. When he relented, he told me that I would have to teach the student and that he needed to get an 85% or higher otherwise he would be taking math 11 all over. I accepted the challenge. I picked up a textbook, gave it to the student, and told him to show up for the final in August and be sure to get 85% or better. That was all I did. When I checked the final grades, I was happy to see that he did indeed score better than 85% with no intervention from any adult. I know this is a sample of one, but it does serve to make my point.
Another student programer came to me from a private school that had gotten out of his way so that as an eleventh grader, he had already finished calculus. I was able to break him out of school early so he could take advanced calculus as a junior at a local college. During his senior year he took several more college level math classes. By the time he started college at a place called Harvard, his freshman math course was something called abstract algebra. Raise your have if you have any idea of what that is.
Due to the smallish number of concepts in math, it isn’t unusual for the top students to go beyond their teachers while still in high school. This also happens in music but almost nowhere else as it takes time to develop the knowledge base necessary for subjects like science and history.
While I was computer director, I took part in a knowledge competition involving the high school faculty. This involved a timed test taken as a group. A student team also competed. While the faculty beat the students badly on non math topics, the students routinely clobbered the faculty on the math questions. This was in spite of the fact that the faculty team included several math teachers.
Top Math Students Seldom Go Into Teaching
Students who are good at math have a lot of options. As a result, few of them go into teaching. Since the US educates more teachers than it needs, education is often seen as an easy major on most campuses. This is just the opposite of Finland where you need to be in the top 10% to be accepted in a teacher preparation program and they only prepares as many teachers as they need. People who major in elementary education tend to have even lower math skills and the teacher prep programs do little to address this. If you are wondering why the Common Core math results seem to be stuck at low levels, keep in mind that a few days of staff development will not turn teachers with poor or average math skills into math whizzes.
The other big problem in the math area shows up when students head off to college. This shows up to the greatest degree at the community college level. Community colleges tend to be open enrollment so anyone who wants to go and finds the money can enroll. Since these colleges know that their incoming students aren’t all of the blue chip variety, they make them take placement tests in language arts and math. When I asked the math chair at my local community college how students do on the placement test, he told me that two thirds don’t pass. This means that they have to take what amounts to high school math in college for which the have to pay but they don’t receive college credit. Many of these same students also flunk the language arts test. What this means is that these students have very little chance of picking up a two-year degree in two years. Studies I have seen indicate that my local college is typical of other two-year schools in the US in this regard.
Know When to Flunk PE
So why do so many high school graduates fail the math placement test and what should we do? In most districts, less than four years of math is required. This results in lots of students not taking math their senior year. When they hit the placement test in April or May, they are lost as they are dealing with material they haven’t seen in a year or more.
Fixing this would be easy. Just give all students the placement test their junior year or at the end of their last math course if that happens earlier. Any student who fails can then sign up for math during their senior year, and this could easily be a course designed to prepare them for the placement test. For students who do fail the placement test again their senior year, they should be able to take a summer course or return to high school for a fifth year for more help. The reason this doesn’t happen is that schools are pressured to raise their graduation rates. This causes them to graduate lots of students who are not ready for college math or English. I don’t see how this is ethical.
Students who need to return to high school for math or English help after their senior year could also sign up for some courses at a community college that don’t require the math or English skills they lack. The smart thing for students who fail the placement test as seniors is to avoid graduating. I my school this would be as simple as failing physical education, which is a required course.
All you have to do to fail PE is not show up. When you return for your fifth year to take PE, you could sign up for the type of math and English courses you need for FREE! In New York State you can keep doing this until you turn 21.
Flip It, Flip It Good
During the last decade, there has been an increased interest in flipping classes. This is where students watch direct instructional videos at home or in school. This frees the teacher up to help students individually or in small groups during class time.
An extension of this plan is called flipped-mastery. This where students take unit tests when they think they are ready. If they don’t exhibit mastery, they work with the teacher to learn what they missed and then retake the unit test until they master it. This can allow some students to finish courses in less than a year while others can just keep on going during the summer or the next school year to finish.
This plan replaces the concept of “You Failed Go Back to Square One” with the concept of “You Haven’t Finished Yet so Keep on Plugging Away.” Of all the subjects, I think that math lends itself to this type of instruction very well. Check my summary of Flip Your Class: Reach Every Student in Every Classroom Every Day by Jonathan Bergmann and Arron Sams for more information on flipping at //bit.ly/1atBiOe.
What to Do?
No matter how you do it, do what you can to make math more self-paced for all students. It may not be the most convenient thing for adults, but I firmly believe it’s the best for all students. It rewards hard work and takes failure out of the game. If done right, it will also result in a greater difference in achievement for fast and slow learners but all learners will most likely learn more.
There is also no excuse for not giving high school student the placement tests as juniors so they can get remediation for free while they are still in high school. I also suggest that school leaders grow a backbone and stop booting unprepared students out just to polish their graduation rates.
This stat is totally meaningless to individual students, and in case you haven’t heard, schools are supposed to be in business for the sake of the students, not the adults.
National Center for Education Statistics, First-Year Undergraduate Remedial Course taking: 1999-2000, 2003-04, 2007-08, January, 2013 //1.usa.gov/1gvjWqZ
Ryan Reyna NGA Center for Best Practices Education Division, Common College Completions Metrics, June 2010, //bit.ly/1GfxlZg
Dr. Doug Green is a former teacher of chemistry, physics, and computer science. He has held administrative positions of K-12 science chair, district director of computer services, director of instruction, and elementary principal. He teaches leadership courses for teachers working on administrative certification, and has authored hundreds of articles in computer magazines and educational journals. He retired in 2006 to care for his wife who had Lou Gehrig's disease. After her death in March of 2009, he started his blog at //DrDougGreen.Com to provide free resources and book summaries for busy educators and parents. You can follow him on Twitter @DrDougGreen.
The opinions expressed in Work in Progress are strictly those of the author(s) and do not reflect the opinions or endorsement of Editorial Projects in Education, or any of its publications.