**Discrete Math Project Collaborative **

## Summer Institute for Discrete Mathematics

Dates: July 5 - 14, 2023

Participating teachers: 9

Facilitators: Anne Marie Almaraz, Melody Morris

Participating teachers: 9

Facilitators: Anne Marie Almaraz, Melody Morris

*Fig. 10. This summer’s DMPC Teacher Professional Development program served new and current discrete mathematics teachers from Sweetwater Union High School District (Eastlake and Bonita Vista High Schools) and San Diego Unified School District (Clairemont and Lincoln High Schools and Logan Memorial Education Campus).*

After completing graduation-required math classes, thousands of California high school students are only too happy to opt out of the mathematics pathway, yet this severely limits their chances for college admission and closes many potential career pathways. The Discrete Math Project Collaborative's (DMPC) main goals are to promote a college and career readiness culture and help traditionally disenfranchised students qualify for and succeed in college-level mathematics courses, ultimately broadening the base of student populations in STEM-related college majors. Studies show that students who miss the opportunity to take any mathematics course in their senior year are less likely to complete a four-year degree program. The DMPC has been helping to solve this problem since 2017.

To help avert this possible academic misstep, a growing number of San Diego County school districts are offering discrete mathematics, an approved A-G (‘C’ - mathematics), fourth-year high school math class taught in California, Iowa, South Carolina, and some New York City schools. Discrete mathematics expands the mathematical options available for junior and senior high school students to complete three or four years of rigorous math in high school, especially those students in low-income/high-need schools.

As of October 2022, 60 teachers from 29 high schools throughout the state and from around the U.S. have participated in DMPC’s discrete mathematics professional development.

To help avert this possible academic misstep, a growing number of San Diego County school districts are offering discrete mathematics, an approved A-G (‘C’ - mathematics), fourth-year high school math class taught in California, Iowa, South Carolina, and some New York City schools. Discrete mathematics expands the mathematical options available for junior and senior high school students to complete three or four years of rigorous math in high school, especially those students in low-income/high-need schools.

As of October 2022, 60 teachers from 29 high schools throughout the state and from around the U.S. have participated in DMPC’s discrete mathematics professional development.

**What is discrete mathematics?**

While discrete mathematics incorporates a broad range of topics, the typical content choices for schools include combinatorics, number theory, graph theory, iteration and recursion, sequences and series, cryptography, and discrete puzzles and games. Discrete math can be taught for several reasons including for computer science or as an introduction to proof course for mathematics majors. The DMPC course focuses on advancing students’ problem-solving, argumentation, communication, and modeling abilities to prepare them for college-level mathematics.

Why study discrete mathematics?

Focus groups including students, teachers, and counselors indicate that students take the DMPC course for a variety of reasons. Aside from satisfying institutional requirements, students want to try something new, interesting, and different. They may want their own voices to be heard in mathematics classes rather than be told how to solve problems.

In the hands of a skilled teacher, the study of discrete mathematics can offer a rich and engaging opportunity to develop mathematical problem-solving skills and ways of thinking called for in the Common Core State Standards.

“In discrete mathematics, students experience math as a ‘sense-making’ endeavor. They solve puzzles, identify patterns and explore counting theories, a striking contrast to traditional high school math classes that emphasize algorithms and formulas,” said DMPC Director Dr. Osvaldo “Ovie” Soto.

Focus groups including students, teachers, and counselors indicate that students take the DMPC course for a variety of reasons. Aside from satisfying institutional requirements, students want to try something new, interesting, and different. They may want their own voices to be heard in mathematics classes rather than be told how to solve problems.

In the hands of a skilled teacher, the study of discrete mathematics can offer a rich and engaging opportunity to develop mathematical problem-solving skills and ways of thinking called for in the Common Core State Standards.

“In discrete mathematics, students experience math as a ‘sense-making’ endeavor. They solve puzzles, identify patterns and explore counting theories, a striking contrast to traditional high school math classes that emphasize algorithms and formulas,” said DMPC Director Dr. Osvaldo “Ovie” Soto.

Background: DMPC

The Discrete Mathematics Project Collaborative was initially funded in 2016 by a grant from the California Department of Education’s “California Mathematics Readiness Challenge Initiative” to San Diego State University’s Center for Research in Mathematics and Science Education (CRMSE) and the Sweetwater Union High School District. DMPC is currently housed at SDSU’s Department of Mathematics & Statistics.

To introduce and prepare teachers to teach discrete mathematics, San Diego State University’s Discrete Math Project Collaborative (DMPC) offered a two-week professional development program this summer called “Discrete Math Pre-Collegiate Summer Professional Development for Teachers,” held July 5-14, 2023 at UC San Diego’s Center for Research on Educational Equity, Assessment & Teaching Excellence (CREATE). Nine current and future discrete mathematics teachers from San Diego Unified School District and Sweetwater Union High School District attended the training.

Funding for this year’s DMPC Summer Institute for teachers was provided by CSU’s Center for the Advancement of Instruction in Quantitative Reasoning (housed in the CSU Chancellor’s Office). Math for America San Diego (MfA SD), a teacher professional development program, also provided support. MfA SD was founded in 2008 by a consortium of three public universities (CSU San Marcos, San Diego State University, and UC San Diego), and in partnership with five school districts (Escondido Union High School District, Grossmont Union High School District, San Diego Unified School District, Oceanside Unified School District, and Vista Unified School District). This year additional support for DMPC came from the UC San Diego Mathematics Project. In the past, the project received funding from the U.S. Department of Defense STEM ( DoD STEM ) and College Futures Foundation.

The Discrete Mathematics Project Collaborative was initially funded in 2016 by a grant from the California Department of Education’s “California Mathematics Readiness Challenge Initiative” to San Diego State University’s Center for Research in Mathematics and Science Education (CRMSE) and the Sweetwater Union High School District. DMPC is currently housed at SDSU’s Department of Mathematics & Statistics.

To introduce and prepare teachers to teach discrete mathematics, San Diego State University’s Discrete Math Project Collaborative (DMPC) offered a two-week professional development program this summer called “Discrete Math Pre-Collegiate Summer Professional Development for Teachers,” held July 5-14, 2023 at UC San Diego’s Center for Research on Educational Equity, Assessment & Teaching Excellence (CREATE). Nine current and future discrete mathematics teachers from San Diego Unified School District and Sweetwater Union High School District attended the training.

Funding for this year’s DMPC Summer Institute for teachers was provided by CSU’s Center for the Advancement of Instruction in Quantitative Reasoning (housed in the CSU Chancellor’s Office). Math for America San Diego (MfA SD), a teacher professional development program, also provided support. MfA SD was founded in 2008 by a consortium of three public universities (CSU San Marcos, San Diego State University, and UC San Diego), and in partnership with five school districts (Escondido Union High School District, Grossmont Union High School District, San Diego Unified School District, Oceanside Unified School District, and Vista Unified School District). This year additional support for DMPC came from the UC San Diego Mathematics Project. In the past, the project received funding from the U.S. Department of Defense STEM ( DoD STEM ) and College Futures Foundation.

“By fall 2024, nearly every high school in the Sweetwater Union High School District will offer discrete mathematics classes,” Soto said. “DMCP, as the Discrete Math curriculum developer and provider, is excited to conduct professional development for the 24-plus SUHSD teachers who will teach the course next year.

“And while Lincoln High School is currently the only San Diego Unified School District high school to offer discrete mathematics, there are 24 SDUSD high schools on deck with plans to implement the class in the next few years,” he added.

“It’s gratifying to support our valued MfA SD consortium partners--Grossmont and San Diego Unified school districts--with implementation of discrete mathematics classes in their schools,” Soto said. “We look forward to additional consortium partners [Escondido Union High School District; Oceanside and Vista Unified School Districts] joining us in this work.”

“And while Lincoln High School is currently the only San Diego Unified School District high school to offer discrete mathematics, there are 24 SDUSD high schools on deck with plans to implement the class in the next few years,” he added.

“It’s gratifying to support our valued MfA SD consortium partners--Grossmont and San Diego Unified school districts--with implementation of discrete mathematics classes in their schools,” Soto said. “We look forward to additional consortium partners [Escondido Union High School District; Oceanside and Vista Unified School Districts] joining us in this work.”

*SUHSD mathematics teachers Melody Morris and Anne Marie Almaraz have forged a successful partnership as DMPC Teacher Leaders responsible for training and supporting current and new DM instructors.*

**Participants**

This year’s two week discrete mathematics Summer Institute for Teachers marks the seventh year of DMPC’s professional development efforts. To serve teachers’ needs, the summer institute changes locations yearly. Early in the life of the project, it was held at San Diego State University before moving to school districts and now to UC San Diego.

This year’s DMPC Summer Institute was led by SUHSD’s Anne Marie Almaraz, mathematics teacher at Otay Ranch High School and Melody Morris, mathematics teacher at Olympian High School. Almaraz and Morris are members of the DMPC Leadership Team responsible for providing professional development for discrete mathematics teachers.

Teachers attending the program included SUHSD mathematics teachers Arielle Farrell, Eastlake High School; Mariana Flores, Bonita Vista High School; SDUSD mathematics teachers Lisbeth Mehregani, Clairemont High School; Andrew Ojeda, La Jolla High School; Aurmon Harchegani and Derek Mesman, Lincoln High School; and Vianey Rodriguez and Lila Das, Logan Memorial Educational Campus. Also attending virtually, Pui Ho Wilson Tsang, a retired mathematics teacher from Santa Clara, CA.

Content Focus

Professional development sessions immersed the participants in DMPC’s Discrete Math (DM) content and curriculum. While ‘teaching the teachers’, Almaraz and Morris modeled a research-based pedagogy called DNR. DNR is an instructional framework for teaching mathematics developed by UC San Diego’s Dr. Guershon Harel, a distinguished professor in the Department of Mathematics.

“The curriculum offers an assortment of highly engaging and accessible math tasks with multiple entry points and approaches for solving to students of all mathematics levels,” Almaraz said. “In this class, students learn the reasoning and creativity required in understanding discrete mathematics has tangible applications in the real world.”

“By teachers ‘doing’ discrete math problems, they are able to uncover methods to advance students’ ways of thinking, outlined in the CCSS Standards for Mathematical Practice,” Soto said. “They also discover how to spark students’ mathematical curiosity and the skills needed to problem-solve for this particular area of mathematics.”

Professional development sessions immersed the participants in DMPC’s Discrete Math (DM) content and curriculum. While ‘teaching the teachers’, Almaraz and Morris modeled a research-based pedagogy called DNR. DNR is an instructional framework for teaching mathematics developed by UC San Diego’s Dr. Guershon Harel, a distinguished professor in the Department of Mathematics.

“The curriculum offers an assortment of highly engaging and accessible math tasks with multiple entry points and approaches for solving to students of all mathematics levels,” Almaraz said. “In this class, students learn the reasoning and creativity required in understanding discrete mathematics has tangible applications in the real world.”

“By teachers ‘doing’ discrete math problems, they are able to uncover methods to advance students’ ways of thinking, outlined in the CCSS Standards for Mathematical Practice,” Soto said. “They also discover how to spark students’ mathematical curiosity and the skills needed to problem-solve for this particular area of mathematics.”

The discrete mathematics high school curriculum features Game Theory, Graph Theory (Connectivity and Traceability), Combinatorics, Cryptography, and the Study of Sequences and Series. For this PD session, teachers sampled and solved an array of lessons from three units: Cryptography, Games, and Connectivity and Traceability, a solid base for nearly 75 percent of the year's content.

Cryptography

*“Getting the text message is not the goal; getting the mathematics is the final message.” ~ Trang Vu, DMPC assistant director and mathematics teacher.*

Ms. Almaraz kicked off a series of DMPC cryptography lessons starting with “Investigating Keypad Encryption.” Like all problems explored by the group, Almaraz presented the lesson content and modeled the pedagogy she practices in her discrete math classrooms. She began with a movie clip from “The Imitation Game,” featuring Benedict Cumberbatch as Alan Turing, the British mathematician and game cryptanalyst who led efforts in code-breaking the German Enigma machine. Teachers were asked what they noticed and wondered in the film clip as a way to engage and prepare them to work on a range of encryption problems.

Next, Almaraz presented an older cell phone, the kind used when text messages could only be generated using keypad buttons. Teachers were asked to decipher a series of ‘text messages’ using a cellphone keypad to decipher for following problem:

Next, Almaraz presented an older cell phone, the kind used when text messages could only be generated using keypad buttons. Teachers were asked to decipher a series of ‘text messages’ using a cellphone keypad to decipher for following problem:

*Alice wants to communicate a message to her friend, Blake. Alice makes use of the phone’s keypad and writes the encrypted message. Question: Can you decrypt Alice’s message? Briefly describe her encryption process.*

Answer: Want to surf this Saturday?

Although seemingly simple, cryptography problems like these require students to use a range of mathematical reasoning, such as functional thinking (the ability to find and state rules that clarify a mathematical relationship between two or more quantities). When studying cryptography, students are also introduced to a few basics of modular arithmetic (e.g. a clock with 12 hours uses only 12 numbers). Modular arithmetic is often used to disguise messages by assigning each letter in an alphabet to a number and manipulating the values mathematically. They also necessitate the use of counting strategies, including fundamental counting principles (counting the total number of possible outcomes), permutations, and combinations.

The follow-up problem proved more difficult for the group:

Some teachers had calculators out. “What were you doing with the calculator?” Almaraz asked. “If we needed to write a program so we could input this message and have it decrypted, could we make an algorithm? Could it decrypt words, letters, expressions, and equations? Could we automate it?”

This is an example of how the teacher leaders attempted to advance teachers’ own functional thinking and point out that advancing students’ functional thinking is the goal of the unit.

The follow-up problem proved more difficult for the group:

*Blake quickly figures out Alice’s method and responds. How did you decrypt Blake’s message? Describe your thinking method.*Some teachers had calculators out. “What were you doing with the calculator?” Almaraz asked. “If we needed to write a program so we could input this message and have it decrypted, could we make an algorithm? Could it decrypt words, letters, expressions, and equations? Could we automate it?”

This is an example of how the teacher leaders attempted to advance teachers’ own functional thinking and point out that advancing students’ functional thinking is the goal of the unit.

*“If you like to be perturbed by a problem, this one’s great.” ~ Anne Marie Almaraz*

Connectivity and Traceability: The Handshake Problem

The Dinner Party lesson, a.k.a. The Handshake Problem, offers a deep-dive into graphs, connectivity, and traceability. Problems like this necessitate the use of graphs to represent relationships and connections. The actual name for this mathematical concept is called the handshaking lemma, part of graph theory.

Ms. Morris led teachers through the content and pedagogical elements of the following problem:

Jack and his spouse, Juan, invited three other married couples to shake hands with the same person more than once.

After all the handshakes were over Jack asked each guest and their spouse (but of course didn’t ask himself) how many hands they had each shaken. Surprisingly, each person gave a different answer. Make a prediction.

The Dinner Party lesson, a.k.a. The Handshake Problem, offers a deep-dive into graphs, connectivity, and traceability. Problems like this necessitate the use of graphs to represent relationships and connections. The actual name for this mathematical concept is called the handshaking lemma, part of graph theory.

Ms. Morris led teachers through the content and pedagogical elements of the following problem:

Jack and his spouse, Juan, invited three other married couples to shake hands with the same person more than once.

After all the handshakes were over Jack asked each guest and their spouse (but of course didn’t ask himself) how many hands they had each shaken. Surprisingly, each person gave a different answer. Make a prediction.

- How many handshakes took place?
- How many hands did Jack shake?
- How many hands did Juan shake?

Teachers worked on the problem by themselves and then shared their estimated predictions with the group on the number of handshakes that took place at the party. Morris polled the class on possible estimates of handshakes. (See Dr. Soto’s paper for more information: “Promoting a set-oriented way of thinking in a U.S. High School discrete mathematics class: a case study”.)

Then the class acted out a similar handshaking scenario. Morris and another teacher took the roles of Jack and his spouse Juan. Some of the teachers shook all hands, some shook less; and all kept in mind the rules: 1) Spouses didn’t shake each other’s hand; 2) one couldn’t shake their own hand; and 3) no one could shake another’s hand more than once. Teachers determined how many hands they shook, how they counted a handshake, the difference between a handshake versus a handshake ‘event,’ and their ability to keep track of their own handshakes.

*Aurmon Harchegani explains his reasoning for the Dinner Party handshaking problem (watch the 2:04 video here).*

“We forget what it feels like not to know how to solve a problem,” Harchegani said. “We haven’t been stumped by an authentic problem for a while and it’s a nice feeling. I haven’t had this feeling since college.

“These kinds of problems are best for gaining empathy for students. They let us ‘sit’ in the issues of our students,” he added.

“These kinds of problems are best for gaining empathy for students. They let us ‘sit’ in the issues of our students,” he added.

The class also explored an additional problem from the DMPC curriculum (Nice to Meet You) in depth. For this activity, teachers were restricted to a different set of rules for solving the problem: 1) Only one handshake is allowed between any two people; 2) no one may shake hands with themselves; 3) you don't need to shake hands with anyone if you don’t want to; and 4) all handshakes are between two people at a time.

Teachers presented their representations to the group on how they were solving the problem. “When we saw a certain representation isn’t working, we had to confirm it with another representation,” Morris noted. “It’s so lovely to see all the different representations.

“I notice students will stick with the way they started,” she said. “I stop students after five minutes and show different attempts on the document camera so students can see each other’s work. I ask them to make a list of the other students’ conjectures and list them out. Get them all out in the room so students can see and pick up on each other.”

Teachers presented their representations to the group on how they were solving the problem. “When we saw a certain representation isn’t working, we had to confirm it with another representation,” Morris noted. “It’s so lovely to see all the different representations.

“I notice students will stick with the way they started,” she said. “I stop students after five minutes and show different attempts on the document camera so students can see each other’s work. I ask them to make a list of the other students’ conjectures and list them out. Get them all out in the room so students can see and pick up on each other.”

“If a group of people shake hands, the sum of all the hands shaken must be even. Always true or sometimes true?” asked Morris. “Never true is not correct. Rotate (change groups) three times to justify your arguments with your colleagues.”

“This reminds me of my college math—set theory and combinatorics,” said Derek Mesman (above). “It really gets away from algebra, which is good for high school students because it touches on the types of mathematics you run across in college.”

“This reminds me of my college math—set theory and combinatorics,” said Derek Mesman (above). “It really gets away from algebra, which is good for high school students because it touches on the types of mathematics you run across in college.”

Takeaways

Virtual participant Pui Ho Wilson Tsang’s takeaways from the DMPC professional development were emblematic of other participants’ comments:

“The summer PD for DMPC ended earlier today. I was able to participate virtually in all 8 days, thanks to the accommodations provided by the DMPC team. It was an amazing experience. I have summarized my thoughts as follows:

All in all, it was time well spent for me.”

For more information on discrete mathematics teacher professional development, visit the DMPC website.

Virtual participant Pui Ho Wilson Tsang’s takeaways from the DMPC professional development were emblematic of other participants’ comments:

“The summer PD for DMPC ended earlier today. I was able to participate virtually in all 8 days, thanks to the accommodations provided by the DMPC team. It was an amazing experience. I have summarized my thoughts as follows:

- The low-floor-high-ceiling characteristics of many activities in the DMPC curriculum help promote logical thinking and sense-making.
- Teachers should experience the curriculum as learners before attempting to teach it.
- Since the topics in the DMPC curriculum are not part of the Common Core Math Curriculum, DMPC might provide a more "level playing field" for students who are marginalized in traditional math classrooms.
- Through this PD, I have gained a deeper appreciation for discrete math as "the mathematics for computer science." Many of the puzzles we tackled in class can readily be converted into programming exercises.

All in all, it was time well spent for me.”

For more information on discrete mathematics teacher professional development, visit the DMPC website.