Discrete Math - Day 1

This blog will discuss my course in Discrete Math. My Discrete Math web site explains the orientation of the course, provides summaries of daily lessons and slides for those lessons. This class is an inquiry-based course. The purpose of this is to have students make sense and connections with and between the mathematical concepts presented. Students are encouraged to explore, question, and investigate the mathematics with an eye toward how mathematicians work. This is much different than learning procedures or problem solving techniques that most students experience in their math courses.

The first day is always interesting. I start with the Circle-Name game. It provides an opportunity to learn student names and build classroom community. I then assign students randomly to groups of three or four.

The first task is for students to determine how many names said during the Circle-Name game. I have students work on their own (individual think time based on brain research) that allows students to engage in the mathematics. I then have students discuss their results at their tables and then have a whole class discussion. I don't push for formulas at this time. I am trying to establish class norms on working through problems, sharing results, and discussing methods.

We then explored Pascal's triangle (see below). I like to start with Pascal's triangle because the patterns are observed in so many situations. Depending on what topics you want to cover, Pascal's triangle will show up in combinatorics, number theory, and graph theory. The NCTM's Navigating Through Discrete Mathematics in Grades 6-12 provides several Pascal triangle explorations and is a good resource for other Discrete Math topics.



I gave students five minutes to work on the problems on their own before working in their groups. Not every group came to resolution on the results. We shared partial solutions and results that differed. This gave an opportunity to show how thinking from different groups can build upon each other and how do to go about determining which of many offered solutions is correct. For patterns in the triangle, students made observations about row content, diagonal patterns, and the general structure of the triangle. One student noted that the triangle had a line of symmetry. I was able to illustrate mathematical thinking as one student pointed out that every other row had a middle value. I suggested that given all the patterns in the triangle I wondered if the middle values followed a pattern and whether that pattern connected to some known entities.

After discussing course policies and course topics covered in the semester, we dove back into looking at Pascal's triangle with an eye toward making connections to the number of names said in the circle. This discussion provided some nice connections. One student said she saw the ones on the outside of the triangle as the people standing in the circle. She said she didn't feel it was all that insightful but it was what she saw. Another student made a connection with the third diagonal, noting that these represented the total names said for any given number of students. Students did not have any other connections. As I was listening to the class discussion I thought about the first diagonal representing the individuals and then thought of the second diagonal as representing the number of names an individual would say. The third diagonal represented the total names said. This made a nice connection between the observations and the reality of the situation. I didn't pursue it further but it would be interesting to see if there are any connections between the fourth diagonal and the Circle-Name game.

The class wrap-up was for students to summarize connections between Pascal's triangle and the number of names stated in their own words. This summary is based upon brain research that says that giving students time to think about their learning at the end of class and writing down their thoughts helps to embed the learning in long-term memory.