Math mastery comes with balance of ‘why’ with ‘how’
In my Ateneo de Manila University college algebra classes, students are participative and speak out in class, but most make errors in operations that should have been mastered in high school.
Students say they understand the lessons, but during examinations, many cannot answer stock questions. So I urge them to do every single exercise in the book, and to keep on doing so until they succeed.
In mathematics, understanding does not necessarily lead to performance.
In the 1980s, when I was an undergraduate, there was one honors class and one remedial class in math. Today, with the student population more than doubled, while we still have only one honors class, there are now more than ten remedial classes.
National and international tests show that, despite curriculum reforms, tutorial centers, and online information, Filipino students generally have not improved their math performance.
There are bright spots, such as victories in the Math Olympiad. But some years ago, when Pinoy students took the Trends in International Math and Sciences Study (TIMSS), we landed near the bottom.
Various factors may contribute to this dismal performance (poverty, poorly-trained teachers, lack of student motivation, and so on). But now let us focus on the curriculum.
For many months, I have been involved in the K to 12 math curriculum reform led by the Department of Education. Math experts, most of them my friends, debate about competencies and periodically revise drafts. But as I look at my classes; the schools where I am a consultant; parents and students and teachers who ask for help, I know I need more perspective.
New math, spiral approach
I turn to former Ateneo president Fr. Bienvenido Nebres, S.J., now National Scientist, for his lifelong work on math education.
Father Ben first taught math at the Ateneo High School for a couple of years in the 1960s, using the American-style New Math, which was premised on “understanding why.” If pupils understood why something worked, then supposedly they would be able to do the tasks. With a New Math text, Father Ben started discussing sets and systems.
“By the second half of the year, my honors class was not learning high school algebra at the level they would need for college,” Father Ben says. “Too much time was spent on understanding the concepts, and it was not true that understanding would assure mastery of skills. I had to look for another book that would allow me to teach them what they needed.”
Characteristic of New Math was the spiral approach, which had unforeseen consequences. The same sequence of many different topics (for example, whole numbers, fractions, geometry, measurement, statistics, etc.) was followed yearly, only growing in complexity.
Once I asked a college student why he could not deal with rational expressions in algebra, when the operations were similar to what he should have learned for fractions in arithmetic. He said, “All I remember of math is that in Grade One, we added small numbers. In Grade Two, we added numbers a bit bigger. All the way up to Grade Six, we were still adding numbers, but big long ones. The teachers taught us fractions along the way, but I didn’t really get it.”
The student had no choice but to study basic fractions. I got a grade school textbook and helped him with the exercises. Only after weeks of fractions could he finally start on rational expressions.
Father Ben says, “It is well-known that teachers tend to cover the first few topics, such as whole numbers, in the curriculum thoroughly and never have enough time for the final topics. We found students who thought they understood math concepts, but who could not add or subtract fractions.”
In 2001, when then Education Secretary Raul Roco asked Father Ben to help public schools in math, Father Ben’s team decided that instead ofb spiraling through various topics, which was what many schools did, they would use the discipline-based method: Algebra I (only) for first year high school, Algebra II for second year, geometry for third year, algebra and trigonometry for fourth year.
“It is not that the spiral approach is bad,” Father Ben says, “but it demands very well- trained teachers and good classes. For our situation of not-so-well-trained teachers, crowded classrooms, insufficient textbooks, the discipline-based approach is better. On the basis of performance in comparative exams, those following the discipline-based approach have consistently done better.”
The spiral approach in grade school should also be modified to provide more time to topics that are most difficult, such as fractions and measurement.
“Half the year in Grade 4 can be reserved for fractions,” Father Ben says, “and half the year in say, Grade 5, for measurement.”
Learning from the East
In 1988, Father Ben told the International Congress in Math Education, “Balance is needed between methods, which emphasize insight and understanding (understanding why), and mastery of skill (knowing how). Generally, western philosophy emphasizes insight and understanding. If we can get the child to understand why, it will be easy for him or her to learn how. But eastern philosophy places a premium on the right way of doing things. Thus, there is a place for memorization, for group chanting, for methodologies that the West would characterize as learning by rote.”
East Asia (China, Japan, and Korea, as well as Singapore which, though located in Southeast Asia, has a dominant Confucian culture) outperforms the rest of the world in math and science. I have written several times about the Confucian work ethic and other possible reasons for East Asian dominance, but what is relevant now is the way their teachers teach. Instead of centering on the understanding of each concept, teachers design problems that challenge the pupil to think deeply.
Examples are the problems in the TIMSS and in the Singapore Math curriculum, which Father Ben introduced to the Ateneo (and the country).
“The approach that goes from concept formation to skills development (understanding first, then skills) may work fine for talented students,” Father Ben says, “but the approach that goes from doing problems and exercises first and from there to the concepts (skills first, then understanding) works much better for the majority of students.”
Mastery through exercises
Father Ben adds, “Working mathematicians know that the best way to master a subject is by doing lots of problems and exercises, even if they do not always understand the why.”
Heresy? Trust me, we mathematicians experience this many times when doing math.
Research on student learning styles, in the language of the popular Myers-Briggs tests, reveals that “sensing” students need concrete applications first rather than theory, and benefit from a skills-based method.
“Intuitive” learners are the opposite. Studies find that most students are sensing, but most teachers are intuitive, leading to a disconnect between teaching and learning.
The West swings from paradigm to paradigm, and the Philippines follows. After the backlash against New Math, Back to the Basics emerged, then Standards-based Assessment. Now it is Understanding by Design (UBD). Despite all this, Western students do not perform as well in math as East Asians.
The East follows an evolutionary approach, staying with stable curricula and improving them over time. American educator Alan Schoenfeld says, “In China and Japan, curricula change much less frequently and more slowly than in the US. These curricula are carefully conceived, known to be reasonably effective, and are refined on the basis of classroom observations and student performance. Teachers make the curriculum a collaborative object of study, working to find better ways to teach a lesson or improve them. In that way, gradual and sustained improvements are made.”
What should be done? Assess the present curriculum and texts, working with master teachers to help determine how to improve what we have at present, with specific skills to be attained at given levels. Devote time and resources to student mastery of poorly-learned topics, such as fractions.
“The debate on whether emphasis should be on understanding or on skills is not useful overall,” Father Ben says. “Both are needed. We need to balance the two and to develop a curriculum and textbooks that teach both. But we should ensure that the emphasis on understanding does not fall into the same problems of the New Math, which will lead to a lot of wasted years.”
E-mail the author at [email protected].
