Rationale
For the last eight years I have taught every level of high school math, from Algebra 1 through AP Calculus BC and AP Statistics. As a result, I gained a unique perspective on the progression of mathematical content and topics in secondary schools. The students I receive in AP Calculus help make me aware of the gaps in mathematical knowledge that accumulate over years. I observed that students who struggle at the pre-calculus and calculus level were successful in the foundational level courses because they memorized formulas and could repeat procedures that closely resembled example problems. However, these students are often unable to explain their reasoning or demonstrate their understanding of the underlying concepts. By my observations, math is not the favorite subject for many of my students. As a consequence, many students take the requisite courses to graduate and simply want to do enough to pass the class. Thus, for various reasons, many students learn superficially through rote memorization and application of formulas and shortcuts, rather than engaging with concepts to gain understanding and build connections between topics within mathematics.
As our school and district transitioned to the Common Core State Standards (CCSS), mathematics courses shifted their focus away from a skills based approach towards the Standards for Mathematical Practice that emphasize both computational fluency and conceptual proficiency.1 With all the changes in the structures of mathematics courses at Overfelt and our district, there was a frequent mismatch between the course objectives and the content provided through the curricula. A major objective, common across all curricula in the Integrated Math 1 course during the first semester, includes the analysis of arithmetic/linear sequences, and geometric/exponential sequences of numbers, and modeling these sequences by writing explicit equations. Afterwards, students would be given “real life” scenarios that follow arithmetic and geometric patterns. Depending on a curriculum’s approach, this topic could span multiple units of study and be extended for weeks to months.
Last year, my district adopted the Big Ideas Integrated Math curriculum last year. In stark contrast to our previous curriculum, which languished through multiple modules about sequences, the current curriculum rushes through sequences by including only two sections about them, each only two to three pages in length. The abrupt presentation of the topics expects students to quickly glance at a sequence, such as 5,11,17,23,… and within the recommended 1 hour for the entire lesson, identify and verify the common difference, write an explicit function that models the situation, and apply their findings to compute later terms of the sequence. Likewise, for geometric/exponential patterns, students are expected to complete the same tasks as for a linear pattern, except they would be asked to identify the common ratio rather than the common difference.
Many students are successful with each individual task regarding a sequence in the specific units of study, but find it difficult to respond to the following question: Is the sequence arithmetic, geometric, or neither? Explain. Most of my students are able to recognize if a pattern is linear, geometric, or neither but have difficulty explaining how they know. If students do recognize sequences as arithmetic or geometric, they have significant difficulty when asked to write an explicit equation. When the two types of sequence are taught as discrete content units in different chapters, the ability to analyze a sequence and make connections between the sequence and its defining characteristics is inhibited. Some students find success at writing explicit equations for arithmetic and geometric sequences, as the general formulas are readily available in our textbook and accompanying resources. However, I notice that many of those same students who can write the equations from a sequence cannot take an equation and explain the relationship between the constants and the features of an arithmetic and geometric sequence.
Quadratic patterns are no longer prescribed in the Integrated Math 1 standards and occur in Math 2. Ironically, however, one of our own district standards for Math 1 expects students to be able to construct and compare linear, quadratic, and exponential sequences and to use them to solve problems. In the pilot year of our current curriculum, my co-teacher and I completed the entire course with enough time to include additional lessons during the year. Thus, there is time in Math 1 to introduce, explore and analyze quadratic functions. Introducing the concept in Math 1 will also provide valuable preparation for the Integrated Math 2 and Math 3 courses, and all of these topics are major components of the first unit in the Math Analysis course.
In my Integrated Math 1 courses, I have many math-phobic students who, due to a lack of success in prior courses, feel as though they are scarred by mathematics. I see many students struggle when they confront the intellectual jump from arithmetic computations to algebra, especially, using variables. For many of my students, it is a major obstacle that elicits allergic reactions to mathematics.
Students who struggle in the higher level math courses often have the opposite problem. They can perform computations and write equations by following steps or replicating the procedures shown in example problems. Where these students struggle is when problems do not look like the examples, or if they do not remember the formulas from the textbooks. One reason for this struggle is that the Integrated Math courses give students quite limited exposure to sequences and functions that that define arithmetic, geometric, or quadratic sequences. I had students taking pre-calculus and calculus level courses who struggled to write and graph linear equations.
One goal of this unit is to provide students with a tool to analyze sequences of numbers. This will be accomplished using difference tables, which provide a structure to organize and assist students as they look for patterns to deduce whether sequences are arithmetic, geometric, quadratic, or none of these. Through examples, this unit shows how to use difference tables to analyze arithmetic sequences with the ultimate goal of writing an explicit equation. For geometric sequences, this unit replaces the difference table with an analogous ratio table, again with the goal of writing an explicit equation. Further extension of the difference table will be applied to quadratic cases. With the quadratic case, a computation based procedure will be shown to enable quadratic sequences to be written as explicit equations.
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