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 five 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, recent protests by parents in the Flemington-Raritan school district are raising the same kind of doubts heard elsewhere: kids are unable to do simple math operations in real life, kids are confused by multiple methods of operations and kids are at a disadvantage in later grades when traditional methods are the norm. Readington uses the Everyday Math program, and in our own district there is a low rumble of discontent.
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? Let’s do the math.
Lee Stiff, a past President of 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. He prefers the term “standards” based mathematics, but whatever the term the program is the same. In a defense of these programs Mr. Stiff writes:
“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 aged 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 twelve on paper, two piles of six blocks equals twelve 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 or those children should be allowed to find a method with which they are personally most comfortable is not inconsistent with established early childhood educational norms. 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 elementary school 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 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 and that 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? 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 “Egyption” method are introduced alongside the traditional method. The partial quotients method is introduced for division. A detailed review of these methods can be found here. 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.
However, these methods are not taught as interesting mathematical asides, but as primary methods for finding answers. In fact, classroom tests included with Everyday Math 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 Readington where the program 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. Stories of large numbers of students requiring tutoring in traditional methods in order to participate in higher order math are 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. A November 9, 2005 article in the New York Times detailed the story of constructivist math in the Penfield, NY school district, noting that 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? Traditional math has its own baggage. The derogatory term “drill and kill” came about at least in part due to unending dittos full of math problems divorced from any meaning. Interminable drills in multiplication and other operations may serve to help children memorize math facts but they can also serve to deaden any interest in pursuing the field further. True, a certain amount of memorization is inevitable, but is there no better way to present the lesson? And, what of students who are not strong in memorization skills or who struggle with traditional computational methods? Constructivist math has attempted to answer this problem, but traditional methods offer little guidance.
Traditional programs have a strict focus on accuracy and efficiency—certainly two critical matters when it comes to math! The computational methods stressed in traditional programs are taught because they have been found to be the quickest and most efficient means of getting the correct answer. Imagine a carpenter laying out stair stringers, or a shopper figuring the cost per pound of a product, or a manufacturing clerk taking a quick inventory. Time and accuracy are at a premium to these people and constructivist methods are of little use. To suggest, for example, that adults in real life circumstances would draw a grid, giving each box a diagonal, and then slog through the “lattice” method of multiplication instead of using the traditional method or grabbing a calculator is pure folly. The boss is waiting, time is money, and there are bigger fish to fry. Yet, traditional math curriculum is also legitimately criticized for avoiding the questions of developmental ability, of student differentiation, and of sheer boredom. In traditional programs, one size fits all even when it doesn’t. In traditional programs memorization and mind-numbing repetition are the minimum height for the mathematical carnival ride, even when the student is too short.
Instead of connecting mathematics with real life and offering meaningful reasons for students to do computations, traditional programs often fall back on the unspoken expression “because you have to, that’s why.” It should also be noted that the other areas of curriculum which once helped students understand the value of math skills, such as industrial arts, home economics, and physical education, are cut back or missing from modern schools. Learning math just for the sake of math will only appeal to a small fraction of the student population.
What is our answer, then? If constructivist math programs have laudable goals of reaching out to each student, of providing differentiation and of offering a deeper view of math, it is in the execution where the programs have failed. By ignoring real world circumstances and developmental growth over grade levels, constructivist math programs can bog down students in pointless techniques and processes and stifle chances for later success in math. Parents may find their children going ever sideways and never forward. Constructivist math programs may look good on the drawing board, but they can be slippery where the rubber meets the road. Traditional programs don’t fare much better. By focusing on memorization rather than meaning, and by failing to provide the means for differentiation between students, traditional programs offer achievement for those interested in numbers for the sake of numbers but defeat for many other students.
Teachers in the classroom have offered the closest thing to an answer, and this fact also explains why some communities become bitterly opposed to constructivist programs and other communities tolerate the programs. In many districts teachers simply do not follow the Everyday Math and similar programs as closely as the designers would have preferred. These teachers mix in traditional methods. They leave out or minimize troublesome features. They take it upon themselves to differentiate in their classroom while making certain all the children meet a minimum standard for real world performance. In short, they do their own thing. Such behavior can sometimes drive administrators up a wall, but teachers often are the buffer between children and stupidity. This mix between traditional and constructivist ideas and methods might be the compromise and the solution, except for two problems.
First, in some states and some districts formal or standardized tests used to gauge mathematical achievement necessitate a thorough understanding of constructivist methods for a child to score well. The New York State Regents exam is one example. Children may score well on the tests, but they fail when it comes to real world computation or higher order math classes in later grades. Students win and lose at the same time. Second, administrators have a valid point: fifty teachers doing their own thing means one thousand students with differing standards of mathematical knowledge and achievement. The teachers are making the best of a bad situation, but leadership must eventually unite such efforts.
In the end there must be unification of the constructivist goals of deeper understanding and meaningful connections and the traditionalist goals of accuracy and efficiency. Frankly, neither side has distinguished itself in nationally used programs, but somewhere there must be someone who can solve this equation.
Download File
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? Let’s do the math.
Lee Stiff, a past President of 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. He prefers the term “standards” based mathematics, but whatever the term the program is the same. In a defense of these programs Mr. Stiff writes:
“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 aged 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 twelve on paper, two piles of six blocks equals twelve 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 or those children should be allowed to find a method with which they are personally most comfortable is not inconsistent with established early childhood educational norms. 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 elementary school 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 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 and that 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? 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 “Egyption” method are introduced alongside the traditional method. The partial quotients method is introduced for division. A detailed review of these methods can be found here. 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.
However, these methods are not taught as interesting mathematical asides, but as primary methods for finding answers. In fact, classroom tests included with Everyday Math 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 Readington where the program 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. Stories of large numbers of students requiring tutoring in traditional methods in order to participate in higher order math are 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. A November 9, 2005 article in the New York Times detailed the story of constructivist math in the Penfield, NY school district, noting that 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? Traditional math has its own baggage. The derogatory term “drill and kill” came about at least in part due to unending dittos full of math problems divorced from any meaning. Interminable drills in multiplication and other operations may serve to help children memorize math facts but they can also serve to deaden any interest in pursuing the field further. True, a certain amount of memorization is inevitable, but is there no better way to present the lesson? And, what of students who are not strong in memorization skills or who struggle with traditional computational methods? Constructivist math has attempted to answer this problem, but traditional methods offer little guidance.
Traditional programs have a strict focus on accuracy and efficiency—certainly two critical matters when it comes to math! The computational methods stressed in traditional programs are taught because they have been found to be the quickest and most efficient means of getting the correct answer. Imagine a carpenter laying out stair stringers, or a shopper figuring the cost per pound of a product, or a manufacturing clerk taking a quick inventory. Time and accuracy are at a premium to these people and constructivist methods are of little use. To suggest, for example, that adults in real life circumstances would draw a grid, giving each box a diagonal, and then slog through the “lattice” method of multiplication instead of using the traditional method or grabbing a calculator is pure folly. The boss is waiting, time is money, and there are bigger fish to fry. Yet, traditional math curriculum is also legitimately criticized for avoiding the questions of developmental ability, of student differentiation, and of sheer boredom. In traditional programs, one size fits all even when it doesn’t. In traditional programs memorization and mind-numbing repetition are the minimum height for the mathematical carnival ride, even when the student is too short.
Instead of connecting mathematics with real life and offering meaningful reasons for students to do computations, traditional programs often fall back on the unspoken expression “because you have to, that’s why.” It should also be noted that the other areas of curriculum which once helped students understand the value of math skills, such as industrial arts, home economics, and physical education, are cut back or missing from modern schools. Learning math just for the sake of math will only appeal to a small fraction of the student population.
What is our answer, then? If constructivist math programs have laudable goals of reaching out to each student, of providing differentiation and of offering a deeper view of math, it is in the execution where the programs have failed. By ignoring real world circumstances and developmental growth over grade levels, constructivist math programs can bog down students in pointless techniques and processes and stifle chances for later success in math. Parents may find their children going ever sideways and never forward. Constructivist math programs may look good on the drawing board, but they can be slippery where the rubber meets the road. Traditional programs don’t fare much better. By focusing on memorization rather than meaning, and by failing to provide the means for differentiation between students, traditional programs offer achievement for those interested in numbers for the sake of numbers but defeat for many other students.
Teachers in the classroom have offered the closest thing to an answer, and this fact also explains why some communities become bitterly opposed to constructivist programs and other communities tolerate the programs. In many districts teachers simply do not follow the Everyday Math and similar programs as closely as the designers would have preferred. These teachers mix in traditional methods. They leave out or minimize troublesome features. They take it upon themselves to differentiate in their classroom while making certain all the children meet a minimum standard for real world performance. In short, they do their own thing. Such behavior can sometimes drive administrators up a wall, but teachers often are the buffer between children and stupidity. This mix between traditional and constructivist ideas and methods might be the compromise and the solution, except for two problems.
First, in some states and some districts formal or standardized tests used to gauge mathematical achievement necessitate a thorough understanding of constructivist methods for a child to score well. The New York State Regents exam is one example. Children may score well on the tests, but they fail when it comes to real world computation or higher order math classes in later grades. Students win and lose at the same time. Second, administrators have a valid point: fifty teachers doing their own thing means one thousand students with differing standards of mathematical knowledge and achievement. The teachers are making the best of a bad situation, but leadership must eventually unite such efforts.
In the end there must be unification of the constructivist goals of deeper understanding and meaningful connections and the traditionalist goals of accuracy and efficiency. Frankly, neither side has distinguished itself in nationally used programs, but somewhere there must be someone who can solve this equation.
Download File
