Dear Posnack Parents:
Remember those four-hundred-page heavy math textbooks we carried around school? They could barely fit into our backpacks and they weighed a ton! At Posnack, many of those heavy textbooks have evolved into electronic math books (or e-books). Technology has played a major role in the evolution of the standard math textbook. However, even with the technology changes, the concepts within those “books” are mostly the same as from thirty years ago. In the world of textbook publishing, little has changed in regards to how material is presented (regardless of whether it’s electronic or on paper).
For my letter this week, I want to focus on how those resources (textbooks or e-books) present mathematical concepts to students. I will that the resources we use in classrooms throughout this country continue to reinforce a deeply entrenched cultural phenomenon that actually does a disservice to our children. Let’s take a closer look.
The math textbook (or e-book) is the backbone of any math curriculum. It is the one resource (more than in any other academic discipline) that guides the student’s math learning. There are two competing methods within math textbook design for teaching students how to solve mathematical problems; either the drill-and-practice approach or the cognitive modeling approach whereby material is presented with connecting multiple representations of a step-by-step problem-solving process through worked-out examples. The drill-and-practice approach follows a view of learning as knowledge acquisition and emphasizes the product of problem solving (getting the right answer) In contrast, the cognitive modeling approach follows a view of learning known as knowledge construction, which emphasizes the process of problem solving (meaning the how and why of getting the right answer).
One of the highest scoring nations on international mathematical tests is Japan (the U.S. often falls towards the upper middle of the pack). Interestingly, when you compare the traditional elementary math textbooks of the United States and Japan, you quickly notice that in Japan, the major use of page space within the textbook is to explain mathematical procedures and concepts, words, symbols, and graphics, with an emphasis on worked-out examples and concrete analogies. In the typical U.S. math textbook, the major use of space is to present unexplained exercises in symbolic form for the student to solve on their own. In the Japanese textbook, there is an extensive explanation for answering the why, how, and application questions. The difference in this textbook design supports the Japanese classroom instruction where the emphasis is more on the nuances of the process and less on “finding the right answer” as is emphasized in the U.S. classrooms.
At Posnack, we are aware of this research. We understand that especially during the critical foundation years of mathematical training, the process within the learning is especially important to the developing learner. This approach of emphasizing the conceptual understanding (the how and why to solving a problem) does require a cultural shift on how we teach our children. It does not help that schools traditionally grade students in a manner which reinforces that the only thing that matters is getting the right answer. However, the same feelings of accomplishment for finding all the right answers in the lower grades often leads to frustrations in the upper grades when the “how and why” become critically important to the student’s learning.
In addition, when the Japanese and American students start their mathematical learning in the elementary grades, their skill set of knowledge and level of acquisition of skills is about the same. However, research shows that as the U.S. and Japanese students begin to move through the upper grades, a disparity in their respective learning begins to become apparent. Around the time a student hits middle school and continuing through the high school years, the Japanese student pulls away from the average American student and achieves higher levels of math success. Why? In large part because of the current “finding the right answer” culture which is embedded within our educational system. Students, parents, and educators all want that quick reward and feeling of success even if the skill comprehension of the student is based only on a superficial acquisition of skills.
If we are going to fix this culture in the U.S., it starts with educators and parents reinforcing the idea to our children that understanding the how and why is just as important, if not more important, than getting the right answer (especially in the younger grades). We need teachers to reward students for applying their knowledge and using their conceptual understandings to seek “answers” and not offer points exclusively for only finding the right answer. It is a culture shift. It is a change in the way we view mathematical instruction and learning. In the end, if we teach our children conceptual knowledge along with the skills required for solving the calculations, we will not only teach them to solve for the correct answer but also how and why they are right.
Dr. Richard Cuenca
Head of School
P.S. Make sure you have purchased tickets to our upcoming 3rd Annual Gala and our spectacular Middle School performance of The Little Mermaid.