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August 3, 2010

Doing the “Math!”

Nick Diaz

Typically it is the “lattice” method of multiplication that pushes parents over the edge. This method taught to elementary school students under the Everyday Mathematics program, one of several national programs collectively labeled “constructivist” or “Chicago” math, is so jarring to those raised in a traditional math program that it ends up being the last straw.


For the last 10 years or so, parents and some educators across the country have raised doubts about “constructivist” math, sometimes generating enough protest to have the program thrown out of their school district. Even locally, protests in recent years by Frederick County parents have been raising the same kind of doubts heard elsewhere:


Students are unable to do simple math operations in real life.


Students are confused by multiple methods of operations.


Students are at a disadvantage in later grades when traditional methods are the norm.


Many public and private schools and school systems use the Everyday Math program. In our own Frederick County Public Schools there is a low rumble of discontent over “Investigations,” popularly known locally as “TERC math.”


Is there really a problem? Is this a case of parents stuck in their ways, unable to see beyond their own childhood experience? Do constructivist math programs like Everyday Math offer innovative strategies for modern students, or do they simply confuse students with pointless computational methods removed from the real world? Is traditional math instruction any better?


Well, let’s do the math.


The National Council of Teachers of Mathematics rejects the label “constructivist” math. The term was coined because these programs aim to have students construct their own knowledge through their own process of reasoning. This math teachers group prefers the term “standards-based mathematics,” but whatever the term, the program is the same. The council defends its preference thus:


“Reform-minded teachers pose problems and encourage students to think deeply about possible solutions. They promote making connections to other ideas within mathematics and other disciplines. They ask students to furnish proof or explanations for their work. They use different representations of mathematical ideas to foster students' greater understanding. These teachers ask students to explain the mathematics.


“Their students are expected to solve problems, apply mathematics to real-world situations, and expand on what they already know. Sometimes they work with other students. Sometimes they work alone. Sometimes they use calculators. Sometimes they use only paper and pencil.”


It is hard to argue with a statement like that. It sounds reasonable enough. Who would disagree that students should not have a deeper understanding of math?


It might be that some of the roots of constructivist math are in the field of early childhood education where preschool and kindergarten children have long been encouraged to understand mathematical concepts in multiple physical and intuitive ways.


Maria Montessori pioneered the use of what modern teachers call “manipulatives.” These physical teaching aids, which might be a simple as blocks, help young minds grasp the nature of mathematical concepts through their senses. Just as two times six equals 12 on paper, two piles of six blocks equals 12 on the classroom floor.


Such techniques are long recognized as useful and necessary to promote developmental growth. A variety of available physical outlets for understanding mathematical concepts means that young children will be able to develop a comfortable relationship with numbers on their own.


That same sort of philosophy is part of the constructivist math program. The idea that children could have different methods for reaching the same answer is not inconsistent with established early childhood educational norms. Children should be allowed to find a method with which they are personally most comfortable.


Yet, there is one key difference with constructivist math programs: Now we are much further along on the developmental scale. Everyday Math and similar national programs are used, not in preschool but in the intermediate grades such as 4th and 5th grades, and on up to sixth or even eighth grade.


In writing curriculum, “invented” spelling is allowed in lower grades so as not to stifle creativity for the sake of accuracy. In later grades, though, spelling should, (and sometimes is), examined and corrected, and eventually accurate spelling is required. This principle does not seem to have a corollary in constructivist math. The disparaging term “fuzzy math” is a reference to this fact. Constructivist math programs do not make the kind of distinctions for developmental growth found in other curriculum areas; this means that, in later grades, there is not a particular emphasis on efficiency or accuracy, at least compared to traditional programs.


What exactly is taught in Everyday Math and other constructivist programs? Algorithms for addition include the partial sums method and the column addition method, plus the traditional method most adults use. In subtraction the “trade-first” and “left to right” methods are introduced. In multiplication the “lattice” method, partial products method and the “Egyptian” method are introduced alongside the traditional method. The partial quotients method is introduced for division. Some of these methods, while not traditional, do approximate what many adults would do in their head to come up with an answer without pencil and paper or without calculator.


These methods, however, are not taught as interesting mathematical asides, but as primary methods for finding answers. In fact, classroom tests included with constructivist programs require students to do problems using more than one method. Many critics have also noted that the program is inconsistent over grade levels. Certain methods are required in early grades, perhaps encouraging a particular student to rely on a favorite method for multiplication calculations. Then, on tests in later grades, that favorite method may be disallowed on tests or a different method now unfamiliar may be required.


What is more, in districts like our school system, where “Investigations” (TERC) is not utilized throughout the child’s whole educational career, there may be a sudden harsh adjustment when the switch to traditional math occurs. Calculators are introduced in very early grades in constructivist programs, leading some to wonder if they are a quick path to a permanent crutch. Critics nationwide have also pointed out the difficulty children in constructivist math programs have moving on to algebra and other higher order mathematics curriculum where a thorough knowledge of traditional math methods is expected. Anecdotal evidence of large numbers of students requiring tutoring in traditional methods, in order to enhance their probability of success in higher order math, is common.


In 2003, the Minnesota legislature removed constructivist math from its state curriculum. The director of Undergraduate Mathematics Education at the University of Minnesota, Dr. Lawrence Gray, said near this time that constructivist or “reform” math was depriving Minnesota students of a good math education because:


“1. University students who had taken reform math in K-12 were at a huge disadvantage in being able to succeed in university-level mathematics.


2. Students taking reform math were not learning enough algebra to prepare them for college math.


3. Many university students who took reform math were dropping out of math classes when they discovered they would have to take remedial math to succeed at the university.


4. High school students taking reform math were one to two years below grade-level in their math skills.”


These are common sentiments. In the Penfield, NY, schools, the district itself is now offering remedial classes in traditional methods to some 300 students to help answer the demands of angry parents.


Is switching back to traditional math curriculum the simple answer, then?


To be continued…

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