Tectonic Instructional Shifts in Mathematics
Catherine A. Hilario
A Mile Wide and an Inch Deep
Li Ping Ma’s seminal work, Teaching and Knowing Elementary Mathematics, has had profound influence on the teaching of mathematics in the United States.TheStanford-educatedresearcher,inherbookbasedonherdissertation, illustrates the importance of profound understanding of mathematical concepts starting in elementary school. Her study shows two big differences between the teaching of math in the United States and China lie in the teachers’understanding of concepts and the number of topics in the curricula of the two countries, explaining in part the difference in the achievement of students from both countries. She claims that in the United States, teachers teach too many topics to the detriment of a deep understanding of mathematical concepts, hence the expression “a mile wide and an inch deep” curriculum.
Mathematical Instructional
Procedural Understanding
The study has some profound implications on the teaching of mathematics and has contributed to the research and instructional shift on what works in teaching mathematics: from the emphasis on procedural math to conceptual understanding. How much time do teachers spend precious instructional time in helping students understand concepts compared to understanding the concepts?Consideringhowourstudentsfareininternationaltests,probably very little. A cursory examination of textbooks would reveal mostly formulas which are hardly illustrated conceptually and computations of abstract notations, which mean very little to students.
Conceptual Understanding 1: The Problem with Math is English
One important consideration is language. Math is a language in itself. As Filipino students grapple with English, they have to grapple with the language of mathematics, which has its own meanings. Li Ping Ma explains that Chinese students have at least a year head start in conceptual understanding of the base ten concept which we use in mathematics. Instead of saying ten, eleven, twelve, etc., Chinese students learn to say ten and 1, ten and 2 ten and 3; 20 and one, 20 and 2, etc., emphasizing the concept of base ten mathematics.
Other examples include notational symbols for multiplication: x, 4(2), *, ab (position as multiplication). These are just a few of the numerous examples. Indeed, learning math language becomes more complicated as students move up in grade level. One important strategy is to pre-teach or front load the language students need to understand to perform procedural tasks. Conceptual Understanding 2: Concrete and Visual before Abstract
One essential component of conceptual understanding is introducing concepts in concrete and visual ways, preferably in that order. Consider understanding the concept of linear and exponential. One way students can see it is to manipulate objects such as rods or any type of material to show how in real-life prices increase gradually or how viruses grow exponentially by not just explaining it but by students manipulating objects to show the concepts before it is shown graphically. There are various videos on-line which illustrate the real-life representations of these concepts. The world-wide success of the Singaporean model lies in the use of graphical rods in their textbook. I suggest starting with hands-on manipulation and transition to visual representation before going into the abstract notation in teaching these concepts.
These two important approaches in helping students grapple with the complexities of mathematical concepts which precede the development of development of procedural skills. It is important that math teachers examine and question their teaching practices to help our students move up in the totem pole of math proficiency. It is of paramount urgency that we shift our instructional practices.
Shift:
Equal focus on both Conceptual and
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The author is SST – III at Angeles City National High School