**~ Sweetwater Union High School District ~**

**Summer Math Academies Teacher Professional Development**

Dates: June 5, June 12, 2023

Number of teachers attending: 8

Number of SUHSD high schools represented: 4

Number of teachers attending: 8

Number of SUHSD high schools represented: 4

To prepare for this year’s Summer Math Academies in the Sweetwater Union High School District (SUHSD), UC San Diego’s Dr. Guershon Harel, a distinguished professor in the Department of Mathematics, led a series of DNR professional development (PD) sessions for eight SUHSD mathematics teachers. The sessions were held June 6 and June 12 at Otay Ranch High School.

The PD program, often described by teachers as a mathematics retreat, provides teachers with a rare opportunity to consider their own knowledge of mathematics, identify and discuss how students learn, and take time to think and reflect deeply on their own instructional practices.

The PD program, often described by teachers as a mathematics retreat, provides teachers with a rare opportunity to consider their own knowledge of mathematics, identify and discuss how students learn, and take time to think and reflect deeply on their own instructional practices.

*Fig. 2. SUHSD is a low-income/high-need secondary school district in San Diego County, with more than half of its student population identified as socioeconomically disadvantaged.*

The PD program, often described by teachers as a mathematics retreat, provides teachers with a rare opportunity to consider their own knowledge of mathematics, identify and discuss how students learn, and take time to think and reflect deeply on their own instructional practices.

Participants

SUHSD teachers attending the summer PD program were introduced to a theoretical framework for teaching mathematics developed by Prof. Harel called DNR (Duality, Necessity and Repetition). The main components of DNR include 1) Teachers’ knowledge of mathematics, which includes the importance of questioning their own understanding of the math they teach; whether their understanding goes beyond the level they teach; and what is a desirable level of understanding; 2) Knowledge of cognition; how students learn; and 3) Knowledge of pedagogy; what to teach and how to teach it.

Participating teachers included mathematics and special education teachers Anne Marie Almaraz and Ricardo Oyorzabal, Otay Ranch High School; Hunter Alzate and Marco Amaral, Castle Park High School; Evelyn Castaldi and Amy Lee, Hilltop High School; Melody Morris and Jose Alberto Vega, Olympian High School.

SUHSD teachers attending the summer PD program were introduced to a theoretical framework for teaching mathematics developed by Prof. Harel called DNR (Duality, Necessity and Repetition). The main components of DNR include 1) Teachers’ knowledge of mathematics, which includes the importance of questioning their own understanding of the math they teach; whether their understanding goes beyond the level they teach; and what is a desirable level of understanding; 2) Knowledge of cognition; how students learn; and 3) Knowledge of pedagogy; what to teach and how to teach it.

Participating teachers included mathematics and special education teachers Anne Marie Almaraz and Ricardo Oyorzabal, Otay Ranch High School; Hunter Alzate and Marco Amaral, Castle Park High School; Evelyn Castaldi and Amy Lee, Hilltop High School; Melody Morris and Jose Alberto Vega, Olympian High School.

“At the Summer Institute, teachers ‘do mathematics,’ and consider research-based learning processes that account for what students do when given similar problems,” said Dr. Osvaldo “Ovie” Soto, executive director of Math for America San Diego. “Teachers think deeply about what math should be taught, how to teach it, and how students think about it,” he said. “DNR puts the integration of content taught and students' intellectual needs at the center of the instructional effort."

*Teachers work on a range of math tasks and experience what it’s like “as a student” to solve problems.*

Taking a DNR Approach to Student-Centered Math Content and Curriculum

To experience how DNR helps guide teachers’ instructional goals, Prof. Harel presented two versions of a problem:

What general principles can you draw out from these two problems?” Harel asked.

Teacher responses included: “In [the traditional version], I made a table first, not an equation—as directed; [the DNR version] allowed us to be creative.”

Prof. Harel stated the DNR version offered the element of choice in how to solve the problem, emphasizing its lack of cues about conceptual tools to use; i.e.; the traditional version asks students to solve with a table, graph, and an equation.

- Version:
*Our family has a small pool for relaxing in the summer that holds 1500 gallons of water. I decided to fill the pool for the summer. When I had 5 gallons of water in the pool, I decided that I didn’t want to stand outside and watch the pool fill, so I had to figure out how long it would take so I could leave and come back to turn off the water at the right time. I checked the flow on the hose and found that it was filling the pool at the rate of 2 gal/min.*- Use a table, a graph, and an equation to create a mathematical model of the number of gallons.

- Use a table, a graph, and an equation to create a mathematical model of the number of gallons.

- DNR-based Version:
*Our family has a small pool for relaxing in the summer that holds 1500 gallons of water. I decided to fill the pool for the summer. At 12:30, when the pool’s dial showed 442 gallons of water in the pool, I decided that I didn’t want to stand outside and watch the pool fill. I knew the hose fills the pool at a rate of 2 gal/min.*- At what time should I come back to turn off the water when the pool is filled?
- My brother came home at 2:30, he looked at the pool’s dial. What number did he see?
- My brother entered the pool when the dial showed 1020 gallons. What was the time then?
- There is a second dial that shows a graph with the amount of water in the pool at any given moment. How does the graph look when the pool is full?

What general principles can you draw out from these two problems?” Harel asked.

Teacher responses included: “In [the traditional version], I made a table first, not an equation—as directed; [the DNR version] allowed us to be creative.”

Prof. Harel stated the DNR version offered the element of choice in how to solve the problem, emphasizing its lack of cues about conceptual tools to use; i.e.; the traditional version asks students to solve with a table, graph, and an equation.

“

“You can reason with the conceptual tools available to you at any age. DNR guides teachers in planning lessons that teach students the importance of knowing ‘how and why’ mathematics works, in addition to finding an answer. Teachers must embrace and encourage their students to use their existing conceptual tools and necessitate the use of the next conceptual tool,” he added.

*DNR calls these kinds of problems holistic problems.*Holistic problems are those that every student, regardless of their level of math knowledge, can at least start. This advances the process of finding multiple ways to a solution,” Harel said. “Teaching with these types of problems instills a wonderful sense of confidence in students. If they are able to master this way of solving problems, it will serve them well in future mathematics learning.“You can reason with the conceptual tools available to you at any age. DNR guides teachers in planning lessons that teach students the importance of knowing ‘how and why’ mathematics works, in addition to finding an answer. Teachers must embrace and encourage their students to use their existing conceptual tools and necessitate the use of the next conceptual tool,” he added.

**Next, read how teachers applied their new knowledge at the Summer Math Academies at Otay Ranch High School and Hilltop Senior High School.**