## Summer Institute for the Professional Development of Middle School Teachers

### July 8 to July 26, 2013

Deadline for Application: March 31, 2013

Announcement of Acceptance: April 22, 2013

This is a three-week institute on *algebra*for middle school teachers, to be conducted by Hung-Hsi Wu with the assistance of Winnie Gilbert, Stefanie Hassan, and Sunil Koswatta.

The goal of the institute is to provide middle school teachers with the mathematical knowledge for teaching algebra according to the Common Core State Standards in Mathematics (CCSSM). There are basic flaws in the usual algebra curriculum and the CCSSM have corrected most of them. This is especially true with regard to the sequencing of middle school geometry and algebra topics as well as the presentation of basic concepts such as “algebraic expressions”, “variables”, and “solving equations”. In order to successfully implement these changes, we need teachers with the requisite content knowledge. At the moment there seems to be little awareness of this need in the education establishment. This institute will provide teachers with the needed content knowledge.

The twin pillars supporting the learning of algebra are rational numbers and the geometry of similar triangles. Unique among national and state standards, the CCSSM outline the correct sequencing of these topics to properly prepare students for the study of algebra. The Institute of 2012 was devoted to these preparatory materials, and the present Institute will be a direct continuation. However, for those who did not attend the 2012 institute,

*an effort will be made to smooth the transition.*

The proposed schedule of the institute is as follows:

- 2 days: use of symbols

- 4 days: linear equations and simultaneous equations

- 3 days: linear functions and inequalities

- 3 days: rational exponents and exponential functions

- 3 days: quadratic functions and equations

*variable*, promoted so highly in textbooks, is completely unnecessary except as mathematical slang. The heart of the discussion of linear equations in two variables is the theorem that the graph of such an equation is a line, and every line is the graph of such an equation. The proof of this theorem hinges on a correct definition of the slope of a line, which in turn brings in the angle-angle criterion for similar triangles. A brief review of similar triangles will be given before going into the details of the proof. Understanding the proof of this theorem about the graph of a linear equation is critical to students’ ability to handle all problems related to writing down the equation of a line when certain geometric data of the line are given. An important component of the discussion of linear equations is the relationship between the slopes of lines and the parallelism or perpendicularity of the lines, a topic poorly done in textbooks and the source of much confusion.

The next major topic is the introduction of the concept of a

*function*. Why is it necessary? This institute will focus on the bread-and-butter issues of school algebra: linear, quadratic, and exponential functions. It is only when exponential functions are discussed that the laws of exponents are put in the proper context and become learnable. The last part of the institute will be devoted to an analysis of the graphs of quadratic functions (they are

*parabolas*, which will be precisely defined, for a change) and to showing how this knowledge helps make sense of quadratic functions, in much the same way that knowing that the graphs of linear functions are lines helps make sense of the study of linear functions.

In addition to the three weeks from July 8 to July 26, there will be five follow-up Saturday sessions during the 2013-2014 school year (with the precise dates yet to be determined). Each of the 15 weekdays of the institute will begin promptly at 8:30 am and end at 4:30 pm. There will be a total of four to five hours of lectures and seat work (with breaks and lunch); the lectures will be on mathematics (not pedagogical strategies). Material on which the lectures are based will be handed out during the first days; it will be a revised version of the following:

http://math.berkeley.edu/~wu/Algebrasummary.pdf

Teachers will be asked to give short presentations (as if in a school classroom) on topics already discussed in the institute. Each day of the institute will end with small group discussions on pedagogy or on the homework assignment from the day before.

The institute can accommodate 20 teachers, and they will each receive a stipend of $100 for each full day of attendance. Please note, however, that more than two absences during the three-week session may result in being dropped from the institute.

The institute assumes a willingness to work, perhaps even intensely, during the three-week period. With this in mind, we ask that you make a commitment to the following:

- Attend all fifteen days of the institute and the five follow-up Saturday sessions. In particular, no stipend will be given to partial attendance of the three-week institute unless there is a medical reason.
- Do the daily homework assignments.
- Be willing to learn and to participate in discussions.