![]() ![]() In my previous years of teaching, I noticed that many students lacked basic understanding of the counting principles that forms the foundation for a variety of topics in higher level math courses. There is little emphasis and explanation on how to understand the formulas and why the formulas work. Our district includes probability computations as a major standard in the curriculum pathway, but designates permutation and combinations to a supporting role, which is often interpreted as meaning “optional content.” The combinatorial topics that remained in the Integrated Math II curriculum emphasize counting permutations and combinations using prescribed formulas. ![]() In this transition, topics related to counting have taken a back seat, or were dumped, in favor of more emphasis on algebra skills, graphing of functions, and geometric concepts. Four years ago, the ESUHSD transitioned from a traditional Algebra 1-Geometry-Algebra 2 course sequence to a three course Integrated Mathematics pathway. Though these concepts make up a small portion of questions on standardized tests, the gaps in knowledge are obvious and need to be addressed.Ĭounting and its associated topics are taken for granted in the high school mathematics curriculum my school and district uses. ![]() Our students also often struggle with the probability and statistical reasoning section on standardized tests such as the PSAT and SAT. Results indicate that a large proportion of our students struggle on the mathematics portion of the exam. In an effort to increase college eligibility, my school offers the opportunity for all students in the junior class to take the SAT. My high school promotes a college going culture. This is in stark contrast to even some of the other high schools in our school district, such as one school located just 3 miles away with an enrollment of approximately 2500 students but only 17% of students being socioeconomically disadvantaged. Over 80% of students at Overfelt qualify for federal free or reduced lunch, and over 50% of students’ parents are not high school graduates. The East Side is its own unique, vibrant, and strong community, and is a different world from the rest of San Jose. There are seven elementary partner districts that feed directly to ESUHSD. Overfelt High School is one of eleven high schools in the East Side Union High School District (ESUHSD), a high school only district serving approximately 24,500 students in the East Side of San Jose. On the East Side of San Jose, there are approximately 100,000 people, with a majority Hispanic/Latino population followed by Asians, including Filipinos. San Jose, California, is a city of over one million people. Furthermore the connection between the number of combinations and the coefficients of the binomial theorem will be demonstrated through examples. This unit emphasizes the relationships between the three key content concepts and gives students the opportunity to understand the general formulas and properties through relatable examples. The connections between these three topics is often not explained or emphasized, though the structures and formulas for computation are similar. ![]() In high school, permutations and combinations are emphasized in Integrated Math II (or Algebra II) and the Math Analysis (precalculus) courses. The Fundamental Counting Principle is introduced in elementary and middle school and forms the foundation for enumerating quantities given varying choices. Counting allows us to enumerate all possible ways an event can occur, and from these counts we can make inferences that will inform our decision making. People use basic principles of combinatorics in many everyday situations. Events are the set of all possible outcomes resulting from a particular situation, experiment, or process. When was the last time you were in a situation where you thought to yourself, “There are too many options, I can’t decide what I want!” or “What are the chances?” Nearly all situations can be quantified using methods or concepts in combinatorics, the branch of mathematics that includes selections and arrangements of objects with prescribed conditions and configurations. Other decisions are more complicated and may involve breaking a situation down into smaller choices. Some of these decisions are only two options, being ‘yes’ or ‘no’. It has been estimated that the average adult makes 35,000 decisions per day (7). Enumerating Daily Life with Counting Principles, Permutations, and Combinations by Lawrence E. ![]()
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