After seven years, second-grade students will begin to learn the multiplication table again. A news report in the China Times headlined "Don't overdo it when teaching the multiplication table," was fair. Ever since the implementation of the mathematics curriculum under the Nine-Year Educational Program the public have noticed a decline in students' calculation abilities due to the gap between the old and new programs. What is calculation ability anyway? How can students improve their calculation skills? And how can teachers help their students to learn math well?
What we should pay attention to is a developmental mode of math education in primary and secondary schools: Learning should be developing, teachers should teach with feeling and student learning should be meaningful.
Take the instruction of the 9 x 9 multiplication table for example: One cow has four legs, so how many legs do nine cows have? During almost 30 years of teaching math, we often encountered two problems. First, should students write their answer as 4 x 9 = 36 or 9 x 4 = 36, ie, what number should be the multiplicand, the former or latter? Second, should they memorize the table?
We agree that students need to memorize the multiplication table. The problem is, when and how should they learn it?
When a second-grade math teacher asks that question, those who learned the table in cram schools or from their parents may answer "36" without thinking. "I have learned it already," they may say.
Unfortunately, once a student knows the answer, they just close the window on the learning opportunity, believing that their job is done and that they can stop learning. As a result, they fail to develop their learning motivation, while the teacher does not know how to motivate them again.
According to our professional experience, perhaps the teacher should continue to ask questions, like, "How many legs do 10 cows have? How do you know?" By doing so, he or she can diagnose the students' concept learning skill, and make them feel there is a need to continue learning.
Of course, it is not so simple, as some parents may challenge teachers: "Why do you teach double-digit multiplication when the students are still learning single-digit multiplication?"
The initial instruction on multiplication should not be taught hastily, and distributed learning is recommended. Teachers should consider the range, content and order of the teaching materials. What multiples of what numbers should they teach first? When should they teach terminology and symbols? How can they teach in line with student's levels and experience?
Take the instruction on the multiples of four, for example. Should teachers first ask students what objects always show up as units of four? Situational questions dealing with the single-digit number four could be posed. And should they then propose appropriate questions for single-digit numbers, for example, how many legs do three cows have? Six cows? Students' problem-solving strategies are thus communicated and discussed, and they can connect back to previous problems, repeatedly thinking over various solutions. This makes teaching challenging and joyful.
After this, teachers can introduce terminology and symbols, and pay attention to the need for the symbol "x". Since one cow has four legs, the total number of legs of three cows is written as 4 x 3 = 12 to show that there are three units of four. The quantity per unit is four; the number of units is three. Not teaching students this terminology simplifies teacher communication. Then, at an appropriate time, they should introduce their students to the concept that "multiplicand x multiplier = product." Putting the multiplicand 4 in the front and the multiplier 3 in the back is a convention.
But the Taiwanese convention is the opposite to that of the US. In Taiwan, 3 x 4 means that there are four units of 3, and that is certainly different from 4 x 3, which means that there are three units of four.
After students understand the concept of multiplication, teachers must demand that they familiarize themselves with the multiplication table through games and activities before entering third grade. As for the instruction of multiplication, students must be made aware of what they are actually doing before they familiarize themselves with the calculations.
For example, the straightforward calculation 36 x 27 requires some explanation. The problem for many students when doing long multiplication is that they misalign the resulting second-line sum, 72 -- the result of 36 x 2 -- by placing the "7" in the tens column rather than the hundreds column, and the "2" in the units column rather than the tens column. It is, in fact, not 36 x 2, but rather 36 x 20. Students must be made to understand that the 72 in fact is the sum 720 with the trailing 0 left out.
According to Danish academic Mogens Niss' analysis, structurally, mathematical competencies include two kinds of capabilities: "The first is to ask and answer questions about, within, and by means of mathematics. The second consists of understanding and using mathematical language and tools."
The former includes: one, thinking mathematically; two, posing and solving mathematical problems; three, modelling mathematically and four, reasoning mathematically. The latter includes: one, representing mathematical entities; two, handling mathematical symbols and formalisms; three, communicating in, with and about mathematics; and four, making use of aids and tools.
Students entering the classroom have their own culture and thinking. Teachers must view problems from students' perspective, and communicate with them rationally. They must not be too quick to guide them or press for answers. They must value the accumulation of experience, instead of teaching tricks. They must also emphasize the exploration of methods over results.
The important thing is to build a tacit understanding between teachers and students, and to cultivate a positive and active attitude towards learning math.
Jack Chang is an associate professor in the department of mathematics and information education at National Taipei University of Education. Linda Chou, his wife, is a retired math teacher.
Translated by Eddy Chang
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