Have We Settled on What to Teach and How to Teach Mathematics?
by Mahesh Sharma ©2008
When a mathematics idea, theorem or result is discovered, within three to four years, a scientist will get hold of that idea to explain a scientific phenomenon. Within three to five years after that, an applied scientist or engineer will use that scientific explanation to find a way to solve a practical problem or design something new to achieve better results. Within three to four years of that, there will be a product, process, or service in the market for the public to use. This is the cycle of scientific progress that propels a country to prosperity. History has demonstrated it for centuries.
For this process to happen, we need a large pool of mathematicians-a group of people who can and want to devote their time to this endeavor. As long as America had access to this pool, American science, engineering, research, innovation, and technology flourished; we prospered. This pool is dwindling, as the National Mathematics Advisory Panel has aptly pointed out in their report.
We need to do everything possible to turn this dangerous trend. If we do not, the national security, the economy, and the standard and ways of living that Americans have come to expect is at stake.
The National Mathematics Advisory Panel’s (NMAP) report has made some very sensible recommendations about mathematics teaching in our schools.
There is a definite emphasis, in this report, on competence, fluency, and proficiency. The mathematics curricula for a long time have emphasized exposure rather than mastery. The report focuses on teaching key concepts at each grade level with mastery rather than just exposure to concepts, processes, and skills. By the eighth grade, children should master with proficiency.
The NMAP also debunks “claims based on theories that children of particular ages cannot learn certain content because they are ‘too young,’ ‘not in the appropriate stage,’ or ‘not ready’ and their ‘brains are insufficiently developed’ when they are actually ready.
Another key recommendation: raise curriculum standards and streamline delivery. American standards are low and fragmented from grade to grade. It is important to note that our lower standards begin in Kindergarten. The K numeracy standards do not match the literacy standards. Most children leave Kindergarten with mastery of letter/sound correspondence in upper and lower case. Yet most children can acquire considerable knowledge of numbers and other mathematical ideas before and during kindergarten. Establishing strong numeracy skills in kindergarten and first grade is an essential prerequisite for mathematics achievement over the academic lifespan.
The NMAP’s emphasis that we teach algebra in the eighth grade is also very important. This will strengthen the curriculum in lower grades. For this to happen, the curriculum must focus on developing mathematics language, conceptual understanding, computational fluency, and problem-solving skills.
The NMAP states that “debates regarding the relative importance of these aspects of mathematical knowledge are misguided. These capabilities are mutually supportive, each facilitating learning of the others.” Taken together, linguistic and conceptual understanding of mathematical operations, fluent execution of procedures, and fast access to number combinations jointly support effective and efficient problem solving.
The panel also supports the idea that effort in mathematics is very important. Some of the lack in achievement is ‘rooted in the erroneous idea that success is largely a matter of inherent talent or ability, not effort.’ Children of high achieving countries believe that effort is responsible for their success in mathematics compared to American children’s belief that it depends on ability. Children’s goals and beliefs about learning are related to their mathematics performance. We need to change the belief from a focus on ability to a focus on effort. This change in belief can do wonders in the mathematics classroom. It will increase engagement in mathematics learning, which in turn improves mathematics outcomes: When children believe that their efforts to learn make them “smarter,” they show greater persistence in mathematics learning.
Every student has a right to be taught by an effective teacher. Effective teachers are those who have three major characteristics: they know their content, they know their children, and they know how to connect the content with their children. They produce results. We need to identify and support such teachers. The impact on students’ mathematics learning is compounded if students have a series of these more effective teachers.
In the last thirty years, a debate has existed amongst educators about student-centered and teacher-led classrooms. ‘High-quality research does not support the exclusive use of either approach.’ Teachers need to teach to the whole class and ‘know, acknowledge, and meet the needs of the students.’ Quality instruction is the judicious use of the two.
The panel’s recognition of the needs of gifted children is very timely. Promoting the mathematically gifted students is in the best interest of the country. We need to pay more attention to them than we pay to top athletes and performers. Mathematically gifted and talented students should be provided with excellent teaching, extra resources, and acceleration.
American textbooks are a joke-they are too long, sometimes incorrect, and not focused. The Panel should have gone farther than just recommending making them shorter. We should not adopt a textbook with extraneous and incorrect information.
We can solve most of the problems of school mathematics and teacher training if we can have a national curriculum that is meaningful, rigorous, and comparable with other industrialized nations. Then we can have quality textbooks, meaningful and uniform national assessments, standard teacher preparation programs, and reliable national achievement studies.
The most important message from the report clearly is: stop mathematics curriculum wars. For the last fifty years, there have been several reform movements in mathematics education. At present, lines divide different groups of mathematics educators on: learning basic skills vs.conceptualization of mathematics ideas; use of calculators in elementary school; concrete models or symbolic manipulators. These are artificial and dangerous dividing lines.
It is counter-productive to emphasize one or the other aspect of mathematics. In order to receive, conceptualize, and communicate mathematical ideas, a student has to master language, concepts, and procedures of mathematics with fluency and confidence. Similarly, to master a mathematical idea, one needs know it at: intuitive, concrete, pictorial/representational, abstract/symbolic, applications, and communications levels.Leaving a child at the concrete level of mastery is not enough and is problematic. They should be able to solve problems and communicate at all levels.
The arguments both for and against calculators are also artificial. Children should be allowed to use a calculator for a procedure, when they can demonstrate the skills: 1) good number sense and mastery of arithmetic facts, 2) being good at estimation, and, 3) knowledge of the concepts. In the absence of these skills, the calculator, rather than being useful, becomes an impediment to mathematics learning. By allowing calculators in elementary and middle schools, we are producing a generation of innumerates. The calculator can be a very powerful and useful tool for learning and applying in algebra, trigonometry, calculus, probability, and statistics. Schools should ban the use of calculators before the seventh grade.
The important areas that the panel did not devote much time are: How much classroom time should be devoted to mathematics teaching per week? At present, it is not enough. What does a results oriented classroom look like? At present, we do not have any guidelines for such a classroom. What should principals and supervisors know about mathematics so that they can provide meaningful leadership in observing, supervising, and giving feedback on mathematics lessons? At present, they do not have any training in this area. What should we do with children and adults experiencing mathematics anxiety and demonstrating symptoms of specific learning disabilities in mathematics, such as dyscalculia? What training should teachers get in these areas? At present, very few teachers have this training.
The National Mathematics Advisory Panel has provided very good recommendations, which we should take seriously. We have a lot of work ahead of us; however, it can be done.
©2008 Mahesh C. Sharma