The options are many, the outcomes are uncertain, the needs are tremendous, and the urgency is high. Are you with me? If you follow my blog, and its purpose, perhaps. If not, I speak of the current plight of secondary math education in the U.S. As presently defined, the math competencies of our nation’s youth, on average, are falling off the proverbially cliff; they started on this downward trajectory a few decades ago – why exactly is not known, or if so, not widely shared. Nothing has been able to slow its decline, and it is not clear if it is even possible, regardless of claims from both ends of the educational spectrum.
Conservative stalwarts to progressive reformers, public officials to private foundations, seasoned teachers to fresh-out Teach for America volunteers, all offer their laundry list of “do’s & don’ts” that will reverse the national slide. However, none have shown that their approach is repeatable or scalable, not to mention effective across our great nation with its diverse student populations, many with unique needs that tax even the most patient people we can imagine. What is one to do? I am not sure.
I do know that I plan to do my best to do my part in reversing the slide. My approach comes from a strong desire to reveal the multiple ways math can be approached, that there is no one right way, that no one is incapable of learning how to master math, and giving up on yourself is not an option. I plan to hold students to high standards with high expectations, not for my edification, but for theirs, and if I benefit in the process, so be it, so I can reinvest that improvement back into my students.
Which brings me to my current dilemma. Is it best to show students procedures that are easy to remember in hopes they recall the procedure, apply it correctly, and hopefully, more often than not, respond correctly on the various assessments placed before them? Even if they may not know how to apply what they’ve learned in a problem that deviates slightly in context, content or construction from what they may have grown accustomed to seeing? Is it OK if they simply show rudimentary facility or is comprehensive transference required? What is best for the future of those students and our nation?
And what about those who struggle mightily to recall the simplest facts and procedures? Do we allow them to continue to fail assessments that require proficiency in both facilities? And if we do, what harm, exactly, does that cause our society? Are they less capable of benefiting our local communities because they struggle with algebra, geometry or other “required” mathematical subjects, theorems, rules, etc? Are the plethora of standards truly necessary for all? Is a hybrid approach possible? What does equitable learning look like for students who struggle so? Does our nation have what it takes to get us back to where we were a few decades ago. Or are we doomed to be ranked 12th or worse in the world going forward?
As a specific example of pedagogical dilemma currently rattling around in my mind, here is a simple algebra problem dealing with proportions: 13/6 = 52/z. There are a few ways to approach this problem. The traditional approach taught uses “cross multiplication” to yield (13)(z) = (6)(52), followed by multiplying 6 & 52 yielding 312 which is then divided by 13 to yield 24. This is the popular method to teach struggling learners since it minimizes deeper thinking, by simply following a procedure independent of the numbers themselves, especially if one can use a calculator.
When I teach the same problem, I never multiply the 6 & 52, I leave them and place the 13 in the denominator yielding (6)(52)/13 whereby I then see that 52 divided by 13 is 4 which multiplied by 6 equals 24. This approach is simplest to me, since multiplication of smaller numbers are involved, often just requiring recall of the “tens times tables.” There is no messy multiplication or division that often results in minor mistakes and ultimately incorrect answers; and on multiple choice tests which are so often given at this level, students obtain scores that do not reflect their level of understanding.
So, while I might be limited in revealing different approaches like this when I am a student teacher, I fully intend to explore whether my approach is easier to understand with my future students. Realistically, I may not know which is easier unless I conduct an experiment where randomly selected students are separated into two groups where one receives instruction in one approach and vice versa. However, even this experiment may not reveal whether it is better or not across other classrooms, teachers, etc. So I am left with the dilemma I started this posting with: what is best for the student? our local community? an ethnic group? our city, state, country? If anyone knows, I am all ears.