Big Numbers, Small Numbers

Rationale

My high school promotes a college going culture. In an effort to increase college eligibility, my school offers the opportunity for all students in the junior class to take the SAT. Results indicate that a large proportion of our students struggle on the mathematics portion of the exam. Our students also often struggle with the probability and statistical reasoning section on standardized tests such as the PSAT and SAT. Though these concepts make up a small portion of questions on standardized tests, the gaps in knowledge are obvious and need to be addressed.

Counting and its associated topics are taken for granted in the high school mathematics curriculum my school and district uses. Four years ago, the ESUHSD transitioned from a traditional Algebra 1-Geometry-Algebra 2 course sequence to a three course Integrated Mathematics pathway. 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. 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. There is little emphasis and explanation on how to understand the formulas and why the formulas work.

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. For example, I had students taking AP Calculus BC that were never shown the factorial operator, which is essential in calculus when working with Taylor polynomials and testing for convergence of infinite series. Limited success in higher level mathematics at my school site is due in part to students being unable to access concepts due to gaps in their foundational knowledge.

My students like to see answers that are single and double digits because most of the practice questions that they were exposed to in previous math classes produced those types of results. When counting the number of ways we arrange objects or ways events can occur, numbers grow quickly. I can sense a heightened level of number anxiety and uncertainty when solutions include numbers reaching even the hundreds and thousands place. However, in realistic situations, people deal with numbers of all sizes on a regular basis. Attending to these ideas also might introduce students to take AP Statistics, as topics involving counting, permutations, and combinations which are included in the course. 

Permutation or Combination

The textbook used in Math Analysis gives students the product formulas for permutations and combinations and expects students to use them. In previous years, I noticed students would rely heavily on the formulas for computations without thinking about the problem. Given a word problem, they would scan and identify any two numbers in the problem. The larger of the two numbers was assigned n, and the smaller of the two numbers had to be r since it needed to be smaller. However, they had a very difficult time determining whether a problem was asking for a total number of permutations or combinations. Another point of difficulty for my students is the ability to visualize the large number of permutations and combinations. When we have a set with only a few elements, it does not take much effort to write our all permutations or combinations. However, even with 4 elements, students grumble at the thought of writing our 24 permutations. Once we reach 5 elements, writing all 120 permutations is a tedious and time consuming task. Even a seemingly simple problem, such as finding the number of seating arrangements in a classroom with 32 students results in an answer of 263,130,836,933,693,530,167,218,012,160,000,000, or slightly over 263 decillion outcomes, which can be written as approximately 2.63 × 10³⁵. For comparison, there are only roughly 100 billion trillion, or 10 × 10²³ number of stars in the whole universe (1).