Recently, there has been a lot of pushback from parents and students alike about the use of models in the math classroom. I believe that much of this stems from a misconception that a model must be in the form of a picture or diagram.
There are times when a model is in the form of a diagram. The array model is commonly used for multiplication in many contexts and is helpful in illustrating the distributive property. However, the model exists to help students visualize the math, not solve a multiplication problem and then draw what they did. Creating problems at a challenging level helps students use the model for understanding instead of document what they already know how to do.
In other cases, a model may not be a picture. For example, when determining the prime factorization of a number, students are often shown how to create a factor tree. It is less often that they are explicitly told that this is a model. One student came up with a prime factorization of 22 x 32 and then drew a picture of two groups of two and three groups of three with a multiplication symbol in between the groupings. This picture may be representative of the values used, but does not help show the student’s thinking and does not model prime factorization. The student first determined a solution and then created an arbitrary way to try to show it. With this interpretation of models, students are left thinking that a model is extra work and using one is more of a hindrance than a helpful tool.
Another common example is when working with fractions. In lower grades, students are shown many models of fractions, such as the one below.
These models can be useful when examining the meaning or worth of a given fraction, when comparing fractions, or when determining if fractions are equivalent, but they break down when students begin to add, subtract, multiply, and divide fractions. When trying to model fraction operations, students often draw representations of both fractions and then put an operation symbol between them. This often leaves students feeling unsure of what to do next. In this case, students are not modeling the problem or the mathematics within it- they are simply drawing a picture that they think matches the values in the problem (see below for an example of student work).
We need to be helping students see that models help them use mathematics to think about and describe the world around them. The models are not arbitrary and they should be helpful in SOLVING the problem, not just something that needs to be added on as an afterthought to “show work”. The more frequently students are exposed to models (both diagram and non-diagram) and can use both types of models as a way to make sense, the more they are likely to use them in problem-solving.
Elizabeth Masalsky is a 6th, 7th, and 8th grade mathematics teacher at Battery Park City School in Manhattan. She has a post-baccalaureate in mathematics from Brandeis University and her master’s in secondary math education from Bard College. Elizabeth is a Math for America Master Teacher and continues her professional development through workshops with Math for America, Metamorphosis and Math in the City.