**~ Sweetwater Union High School District ~**

## Hilltop Senior High School Summer Bridge Math for incoming 9th graders

- Dates: June 13 - 16, 2023
- Number of students: 11 incoming ninth graders
- Teachers: Marco Amaral, Evelyn Castaldi, Ricardo Oyorzabal, Hunter Alzate, Amy Lee, Jose Alberto Vega

*Fig. 4. Close to 50% of Hilltop Senior High School students are from socioeconomically disadvantaged families and*

**nearly 80%**are Hispanic.Participants

The six SUHSD teachers who observed instruction at Otay Ranch High School’s Summer Bridge Academy were ready to co-teach the same curriculum to 11 incoming ninth graders at Hilltop Senior High School the following week (June 13-16).

Academy days 1 and 2 were co-taught by SUHSD teachers Marco Amaral, special education fundamentals teacher, at Castle Park High School; Evelyn Castaldi, mathematics teacher, at Hilltop High School; and Ricardo Oyorzabal, mathematics teacher, at Otay Ranch High School. Academy days 3 and 4 were co-taught by Hunter Alzate, mathematics teacher, at Castle Park High School, Amy Lee, mathematics teacher, at Hilltop High School; and Jose Alberto Vega, special education, mathematics teacher, at Olympian High School. On the days they didn’t co-teach, teachers observed their colleagues and participated in planning and the debriefs.

The six SUHSD teachers who observed instruction at Otay Ranch High School’s Summer Bridge Academy were ready to co-teach the same curriculum to 11 incoming ninth graders at Hilltop Senior High School the following week (June 13-16).

Academy days 1 and 2 were co-taught by SUHSD teachers Marco Amaral, special education fundamentals teacher, at Castle Park High School; Evelyn Castaldi, mathematics teacher, at Hilltop High School; and Ricardo Oyorzabal, mathematics teacher, at Otay Ranch High School. Academy days 3 and 4 were co-taught by Hunter Alzate, mathematics teacher, at Castle Park High School, Amy Lee, mathematics teacher, at Hilltop High School; and Jose Alberto Vega, special education, mathematics teacher, at Olympian High School. On the days they didn’t co-teach, teachers observed their colleagues and participated in planning and the debriefs.

Problem: The Grand Canyon Train Ride

The Grand Canyon Train Ride mathematics problem provided students with an outstanding opportunity to explore two different patterns of growth—one linear and the other quadratic—depending on the patterns the students observed while working on the problem. This produced a varied set of representations, solutions, and arguments from students, each valid relative to the student’s goals.

A train with 100 seats picks up its first two passengers at First Street. The train is headed to the Grand Canyon. When it arrives at its destination it is carrying 100 passengers. How many stops will it make before it reaches its final destination?

Clues about the situation:

A train with 100 seats picks up its first two passengers at First Street. The train is headed to the Grand Canyon. When it arrives at its destination it is carrying 100 passengers. How many stops will it make before it reaches its final destination?

Clues about the situation:

- When the train leaves Second Street it is carrying seven passengers.
- When the train pulls up to Fifth Street the number of people who board the train is twice the number of people who were on the train when it pulled into Third Street.
- The number of passengers boarding at each stop is increasing at a constant rate.
- If the number of passengers boarding at each stop is increasing at a constant rate, then the increase in passengers boarding from one stop to the next is always the same number.
- There are 15 people on the bus when it leaves Third Street.

Students in groups used a central whiteboard for reasoning collaboratively on the problem. By using a shared middle space, students have the opportunity to travel to the mind of another student to co-create problem-solving approaches, conjectures, and arguments. Understanding the reasoning of others is essential in DNR both for teachers and students. In this learning configuration, it is more natural for students to engage in argumentation and to reflect on and critique their own reasoning and the reasoning of others.

*Alberto Vega and Hunter Alzate work closely with students to help them with any reasoning roadblocks as they work out the train problem.*

After the groups finished their shared work, they presented their train ride solutions on the classroom whiteboards. The students were instructed to do a gallery walk and use a clipboard to write down questions they had about the other groups’ work (e.g.; What does this variable mean?, How does this equation represent the situation?, etc.). They also had to write a positive comment specific and applicable only to a particular group’s representation of the solution.

*Students’ work included questions and answers. Growth patterns were represented using tables, functions, and words.*

*During the academy, students helped each other by comparing answers and pointing out where errors occurred.*

**Daily Debrief**

At the debrief, the Grand Canyon Train Ride problem sparked an engaging discussion among teachers. They were especially pleased with student thinking and how well the group responded to the problem.

- Students were using equations today; there was literacy with math, e.g.; “boarding the train.” There were nuances that students picked up on to help solve the problem.
- The train problem allowed students to feel that they could do it this way, or do it that way and gave them permission to solve it any way they could. There was a sense of “I feel I could do this, now that I can approach it from different directions.”
- Each group presented the solution in different ways - some were very detailed, used a table, expressed the solution in an equation, and/or provided an illustration of each train stop.

*Danny Saldivar (center, at the 2023 Oceanside High School Summer Academy) studies mathematics at MiraCosta College. She provided near-peer support to incoming high school students at the SUHSD and OUSD Summer Math Academies.*

**Why a good mathematics teacher matters**

Summer intern Danny Saldivar was also present at all debriefs. Saldivar assisted with classroom logistics for the two-week SUHSD and the three-week Oceanside Unified School District Summer Math Academies. Saldivar, a 2023 UC San Diego STEMULATE intern, is currently a mathematics student at MiraCosta Community College with plans to become a mathematics professor.

At the debrief, Saldivar was asked what it was like for her to be in a room with eight math teachers and two groups of incoming ninth grade students for an extended period of time.

“Observing the pedagogy and how teachers worked with students moved me on a personal level,” she said. “I know from experience that classroom interactions can mark students for life. I was overwhelmed by the students’ positivity and energy. Just hearing them say, ‘I want to be confident and try,’ I knew the students felt safe to make mistakes. They understood that teachers were here to really help them. It was 180 degrees from what I experienced in my own high school math classes. This is the way mathematics should be taught,” she added.

At the debrief, Saldivar was asked what it was like for her to be in a room with eight math teachers and two groups of incoming ninth grade students for an extended period of time.

“Observing the pedagogy and how teachers worked with students moved me on a personal level,” she said. “I know from experience that classroom interactions can mark students for life. I was overwhelmed by the students’ positivity and energy. Just hearing them say, ‘I want to be confident and try,’ I knew the students felt safe to make mistakes. They understood that teachers were here to really help them. It was 180 degrees from what I experienced in my own high school math classes. This is the way mathematics should be taught,” she added.

**Takeaways**

At the final SUHSD debrief,

**teachers shared what they found most valuable**about their Summer Math Academy experience.- Being able to watch students and support their own methods over these past two weeks and encourage them with the process. The gallery walk in today’s class was a great way to focus on process vs equations.
- I really appreciated that students realize there are many ways to solve a problem.
- I liked the opportunity to work with these kinds of problems. I don't give these problems to my students and see that they liked these kinds of problems.
- I'm leaving with so many tools. I’m excited to apply the methods, conversations, and goals in freshman classes.
- Thinking about open-ended problems, I think about them differently. Before, I thought I was teaching “open-ended” problems, which meant making a table, making a graph. I learned I can provide better questions. I learned that without any structure, students can still do it.

- I liked the pleasure of working with great colleagues and learning different things from other teachers. It makes my teaching more exciting–for me and the students.