Dear Posnack Parents and Community:
This week, the “Mathemagician,” Dr. Arthur Benjamin, amazed a standing room only crowd of over four hundred people with incredible mathematical “magic.” Triple digit by triple digit calculations were solved in a matter of seconds. The numerical value for pi was turned into a song. And the day of the week for your birthday was solved regardless of the year. It was an amazing night that showcased the importance our school places on math instruction and the unique approach Posnack takes in teaching math. For my Friday letter this week, I will use the topic of math instruction as my theme.
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 math material is presented (regardless if it’s on paper or electronic).
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, (1) is the drill and practice approach; and (2) the cognitive modeling approach- material is presented with connecting multiple representations of step-by-step problem-solving processes through worked-out examples. The drill-and-practice approach follows a view of learning as knowledge acquisition which 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 to 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). Interesting, 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. This in turn results in classroom instruction in Japan which pays much more attention to the nuances of the process and less emphasis on the U.S. culture of “finding the right answers” being the key.
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 culture 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 feelings of accomplishment for finding all the right answers in the lower grades often leads students to frustrations in the upper grades when the how and why are critically important to the student’s ability to apply their learning.
In addition, when the Japanese and American students start their mathematical learning in the elementary grades the level of acquisition of skills is about the same. They acquire about the same skill set of knowledge in mathematics in the younger elementary grades. However, as the U.S. student and the Japanese student move up through the upper grades, the research shows that on average there begins an apparent separation. Around the time a student hits middle school and continuing through high school, the Japanese student pulls away from the average American student and achieves higher levels of math achievement. Why? In large part because of the current “finding the right answer” culture which is embedded within our educational system. Students, parents, and educators want that quick reward and feelings of success even if the skill comprehension of the student is based only on superficial true 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