False Dichotomies in Education
We have heard the debate on content versus higher order thinking. Pause for a moment and ask if a person can really exhibit thinking at any level without content. This is really a false dichotomy. Thinking and content go hand in hand. This brings me back to a classic paper written by Hung-Hsi Wu, professor emeritus of mathematics at the University of California in Berkeley. Wu's paper is entitled "Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education". Here are excerpts from this paper:
Looking back at a previous post in this blog, "Making Sense Out of Numbers: Math Common Core", one can appreciate better the following:
The Mathematics Common Core provides specific content and practices, yet it allows for teachers to do the final finishing touch so that it will be effectively implemented inside the classroom. Is this for real? Barry Garelick wrote in the Atlantic back in November of last year the following article, "A New Kind of Problem: The Common Core Math Standards". First, here are his comments for the first 97 pages of the Math Common Core Standards:
Let's look first at the 97 pages of what are called "Content Standards." Many of these standards require that students to be able to explain why a particular procedure works. It's not enough for a student to be able to divide one fraction by another. He or she must also "use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9, because 3/4 of 8/9 is 2/3."
It's an odd pedagogical agenda, based on a belief that conceptual understanding must come before practical skills can be mastered. As this thinking goes, students must be able to explain the "why" of a procedure. Otherwise, solving a math problem becomes a "mere calculation" and the student is viewed as not having true understanding.
This approach not only complicates the simplest of math problems; it also leads to delays. Under the Common Core Standards, students will not learn traditional methods of adding and subtracting double and triple digit numbers until fourth grade. (Currently, most schools teach these skills two years earlier.) The standard method for two and three digit multiplication is delayed until fifth grade; the standard method for long division until sixth. In the meantime, the students learn alternative strategies that are far less efficient, but that presumably help them "understand" the conceptual underpinnings.
I guess the point of providing the student multiplication to help appreciate division by fractions is to overcome the limitations of manipulatives, of things we could actually hold, in explaining what happens when we divide something by a fraction. A child can easily relate to dividing by a number greater than 1. A pizza can be divided into four parts, each slice being a quarter of the whole pizza. But how does one divide a pizza by 3/4? Why is the answer 4/3? It is bigger than before. Making the student perform the reverse process brings the equation to something much easier to relate. 3/4 of 4/3 is one. This can be visualized.
Garelick does end the article with a hopeful view:
Garelick does end the article with a hopeful view:
As the Common Core makes its way into real-life classrooms, I hope teachers are able to adjust its guidelines as they fit. I hope, for instance, that teachers will still be allowed to introduce the standard method for addition and subtraction in second grade rather than waiting until fourth. I also hope that teachers who favor direct instruction over an inquiry-based approach will be given this freedom.
At the end, for any curriculum, teachers provide the determining step. I do hope that teachers do make sense out of this curriculum.
In General Chemistry, students struggle with calculations of pH for aqueous solutions that contain the conjugate base of a polyprotic acid, such as phosphoric acid. How does one calculate the pH of a 0.1 M aqueous solution of sodium dihydrogen phosphate? The pH for this solution is halfway between the first and second pKa's of phosphoric acid. Arriving at this quick solution requires likewise a deeper understanding, in this case, of simultaneous equilibria conditions.
These are algorithms that are proven and we should take advantage of these procedures. This is how we stand on the shoulders of intellectual giants who came before us. These algorithms are less cumbersome and can be generally applied. For an entire classroom, waiting for each student to come up with his or her own own way of solving each problem will simply lead to mistakes and gross inefficiency. There is really no need to reinvent the wheel all the time.
On the other hand, passing to students these proven algorithms and recipes without the underlying principles and explanation leads to rote learning. It is how we teach these algorithms that make the difference. It is only when we fully understand what these shortcuts really are can we teach them properly. And yes, these need to be taught.
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