03/30/2013 | Karen Leatherman
common core
Consequently, teachers’ mathematics content knowledge, especially at the elementary and middle school levels, must be addressed both in the preparation programs for future teachers and in professional development opportunities for current teachers. These programs must focus on improving conceptual understanding of mathematics, and in doing so, need to address a key component often slighted or masked by traditional mathematics education. To illustrate this important factor, examine the following task: Find the ybkilt of a nomgan if the berstup is 12. It should be readily apparent that the difficulty has nothing to do with numbers, but language instead. Conceptual understanding of mathematics is founded on a deep knowledge of the language and symbolism, yet the traditional focus is on numbers and procedures. A language focus makes sense because language comprises the intersection of content and instruction — how we define mathematics concepts and how we communicate that meaning to students.
My native language was not English, and as such, I had the dual task of simultaneously learning both the English language and mathematics. These experiences instilled an acute awareness of covert language-based problems in mathematics unnoticed by most math educators. For example, phrases such as similar triangles or a cup of water do not convey the same meaning in standard English than they do in mathematics. It is rare that those differences are addressed in mathematics classrooms.
Language goes far beyond the basic idea of vocabulary. The language and symbolism are an integral part of every aspect of the content and communication of mathematics. Despite the emphasis on language, it is conceptual understanding of mathematics that lies at the core of the publication. Conceptual understanding is difficult to define, but it begins with simple yet deep definitions. With that as the foundation, conceptual understanding of mathematics manifests itself through teachers’ ability to view concepts from different perspectives, to make connections among related concepts, to understand the hidden nuances of mathematical language and visual representation, and to leverage the learning of a new topic with the deep understanding of another.
As noted, building conceptual understanding has deep and thorough, yet simple definitions of key concepts at its core. The paradox is that the shallow definitions often taught in K–12 math classes are, at the same time, often rather complex. Students are usually not exposed to deep yet simple and concise definitions of fundamental mathematical concepts. Although the idea seems contradictory, it is possible both to simplify and deepen math content. Unfortunately, the state accountability exams that are intended to ensure that deep learning occurs can actually serve as false positives regarding students’ level of expertise. As an example, students can know their multiplication facts and illustrate this on a state exam, yet have no clue as to a conceptual definition of multiplication.
The basic concept of multiplication serves as an illustration of the lack of content depth and the missed educational opportunities linked to incomplete or shallow definitions. Most students understand multiplication as repeated addition and have vague notions of division as the opposite of addition. This then leads to only being able to define an average by explaining how to compute it. Defining multiplication as repeated addition is woefully incomplete and substantially handicaps students’ ability to connect to and understand division and average. Contrast that to the surprising simplicity, depth and breadth in connectivity when we add a focus on equal-sized groups to the definition of multiplication as repeated addition. This then provides a smooth transition to division since the typical context involves equal sharing.
At this level, these contexts involve a certain number of groups of the same size which comprise the appropriate total. Students should readily see that the only difference between the operations of multiplication and division lies in which of those three components are known or unknown. This understanding then transitions to a simple yet deep definition of an average as “an equal redistribution.” Multiplication focuses on equal-sized groups, which leverages the focus in division on equal distribution. In turn, a context involving several non-equal groups, an equal redistribution requires the additional step to determine the total, which then transitions to the equal distribution focus of division. And the beauty of this is that all of this power and connectivity is due to a simple adjustment to the basic definition of multiplication.
A focus on language in mathematics has almost unlimited potential and power. An alternative approach to the order of operations as a static set of rules is accomplished by the simple realization that any number has an associated meaning that must involve language. For example, five is not just a five. It is five bears, five computers, five apples, etc. This then connects to the fundamental property in math that only like items can be added or subtracted. Mathematics education has forgotten that an expression such as three plus two carries with it a huge assumption that the three and the two both describe the same things (bears, apples, etc.). This basic tenet eventually leads to the assertion that in an expression such as three plus two times five, the multiplication must be done first to determine how many “things” are represented by two times five, which can then be added to the three. The expression can represent a context such as having three pennies and two nickels. It is necessary to convert the nickels to pennies so that in the end we are adding the same like items, pennies to pennies. Note that this extraction of the order of operations being based on the mathematics itself — rather than a convention — would not be possible without the focus on language and symbolism and illustrates that the order of operations “is rooted in basic mathematics principles and can serve as a powerful model for illustrating the connectivity of mathematics concepts.”
The topic of fractions has often posed challenges for students and teachers alike. This area can be complex and demanding, and it is no coincidence that fractions are also the area in fundamental mathematics most plagued with obstacles in the language and symbolism. This symbolism, which is fraught with assumptions and nuances, is used in many ways with different meanings, easily leading to misunderstanding and misconceptions.
It should be evident that elementary mathematics is not simple even though the term elementary implies that. For substantive improvement in the depth of the mathematics that we teach, mathematics content must be addressed, in particular, the mathematics expertise of the teaching force. Teachers can teach only what they know, and this knowledge is largely dependent on the content provided by the system itself at the K–12 and college levels. Thus, this issue is actually a system-level problem because classroom-level change will not happen without the support of the broader system. But as illustrated by the example of multiplication, division, and average, effective change in the depth of the mathematics we teach can be accomplished with reforms that support and leverage related efforts.