Bringing the Logic Course to High Schools
To test the viability of introducing Stanford's Introduction to Logic into high schools, we have been engaging with high school teachers, administrators, and students. In this section, we report on our successes, lessons learned and future challenges.
Introducing Logic as a Summer Program
We started our efforts by adapting our course as a two-week summer camp to be offered to high school students who will be entering grades 9-12 starting the Fall semester of that year. Our goals for the initial exercise were to (1) assess the enthusiasm of high school students in studying the subject; (2) assess the effectiveness of our course material with the high school students.
The summer camp format closely follows the course outline we use for Stanford offering of the course. We cover propositional logic, relational logic and Herbrand logic. The module on induction is usually left out. The classroom instruction is a mixture of lectures and hands on exercises. The students complete the exercises using the online version of the textbook. The students may work individually or in small groups. Depending on the instructor, the instruction is mixed with on-campus excursions, logic-oriented game activities, and occasional extra-curricular activities.
Since summer of 2016, we have offered seven different summer camps which were attended by a total of 156 students. Two of these summer camps were residential and drew students from all over the world. The remaining five camps were non-residential, and the students came mostly from the local schools of San Francisco Bay area. For three of the five non-residential camps, we selected gifted students, but for the remaining two camps, we selected students coming from under-represented minorities. The students coming from under-represented minorities received a full scholarship while everyone else paid the tuition. The technical course content for the camp for the gifted students vs the course content for the under-represented minorities was identical, except that for the latter, we added short modules on elite college admissions, and cultural awareness. There was no final exam or grade at the end of the course, but we administered a questionnaire to solicit feedback. The feedback (based on 115 responses) is summarized in the table below.
The qualitative feedback from the students revealed that they were drawn to studying logic because they saw a broad applicability of learning logic regardless of their field of study. Some students were drawn by the fact that the course material involves solving lots of puzzles. The students also appreciated the way the course makes connections to STEM topics such as circuit design and troubleshooting.
Based on the above qualitative and quantitative feedback, we can conclude that the material is indeed accessible to high school students and can be made even more challenging. Therefore, we believe that if the course were to be taught during the academic year in a high school, the module on induction could be included without much difficulty. There were no significant quantitative or qualitative differences in feedback between residential and non-residential students, and between gifted students and students coming from under-represented minorities. This provides some initial evidence that the course is effective for a broad cross section of students.Overall, the summer program has provided an initial confirmation of the enthusiasm of high school students in studying the subject and in assessing the effectiveness of our course material with the high school students. In addition, as explained below, it has been a good a good way to train new high school teachers in teaching the material. We are planning to continue this program at Stanford for Summer of 2019. In future, we are planning to expand the summer camp offerings in other cities (e.g., Los Angeles, Seattle, Pittsburgh, etc.)
Introducing Logic in High School Academic Calendar
Academic calendars in High Schools are already crowded, and the immediate questions we needed to address were: Why should this course be introduced in the high schools? If it is to be introduced in High Schools, where and how does it fit in the academic calendar?
In Section 1, we have already explained why we believe logic is important for STEM disciplines as an essential mathematics, and more broadly, for all disciplines, as an essential training for thinking clearly. These claims are backed by existing research [1-3], and there also exist guidelines of the Association of Symbolic Logic on the topics that should be taught at each grade level in the high schools . We have found these arguments to be effective with high school administrators and school teachers. They, however, seek some outside validation of this argument which usually comes from teaching standards.
In California high schools and online schools, any course offered for credit must be approved by University of California (UC) and must appear on the institution's "a-g" course list . The "a-g" courses are to be academically challenging, involving substantial reading, writing, problems and laboratory work (as appropriate), and show serious attention to analytical thinking, factual content and developing students' oral and listening skills. Three California schools have used our Logic curriculum as the basis to propose a "g" course to satisfy the requirement of an elective for mathematics or computer science and have received the UC approval. These schools will be offering the logic course starting the academic year 2018-19. Our long-term goal for California schools is to work towards getting the logic course recognized as a core course for mathematics and computer science. Once the course is recognized as a core course, then there is a natural justification for high schools to find a place for it in their academic calendar.
Another common challenge in introducing our logic course in the academic calendar is that some schools can only offer courses that last the whole academic year. The material in our course is sufficient for a semester long offering but is not sufficient for a full academic year. It remains open for future work to explore different models for a longer offering. One possible design would be to combine a semester long offering of the logic course with a semester of probability and statistics. The combined course could be positioned as a course on critical and/or scientific thinking. In another possible design, a semester of logic instruction could be combined with a semester of instruction on logic programming. The year-long offering could be realized in a way that it satisfies the requirements for the advanced placement course on computer science. Exploring the viability of each of these options remains open for future work.
Outside the context of California schools, we need a better characterization of how the logic course relates to national high school teaching standards. We consider below our initial analysis. A more comprehensive analysis remains open for future work.
In relation to Common Core standards , logic is closely related to two of the standards for mathematical practice. (1) Reason abstractly and quantitatively: The students are able to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents. (2) Construct viable arguments and critique the reasoning of others: Students can build a logical progression of statements to explore the truth of their conjectures. They justify their conclusions, communicate them to others, and respond to the arguments of others. They can distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument, explain what it is.
In relation to Next Generation Science Standards , logic directly supports two of the practices for science and engineering. (1) Constructing explanations and designing solutions: The goal for students is to construct logically coherent explanations of phenomena that incorporate their current understanding of science, or a model that represents it, and are consistent with the available evidence. (2) Engaging in argument from evidence: In science, reasoning and argument are essential for identifying the strengths and weaknesses of a line of reasoning and for finding the best explanation for a natural phenomenon. In engineering, reasoning and argument are essential for finding the best possible solution to a problem. Engineers use systematic methods to compare alternatives, formulate evidence based on test data, make arguments from evidence to defend their conclusions, evaluate critically the ideas of others, and revise their designs in order to achieve the best solution to the problems.
Our long-term goal is to work towards creating an advanced placement course on logic which is at the same level as a course on Calculus. To meet that goal, we are starting to work towards introducing the Logic course in a few states outside California. Before a course can be proposed as an AP course, it must be offered at least 250 schools, be recognized for college-level credit by at least 100 universities and should already have high school student enrollment exceeding 5000 students each year. Achieving these goals requires sustained effort to scale the initial results we have discussed here.
Training Logic Teachers
An essential component to bring logic education to high schools is to train the high school teachers to deliver the material. Towards that end, since 2016, we have been conducting logic teacher professional development (PD) programs.
Our logic teacher PD is a one-week program. Even though the program is open to any high school teacher, but most of the teachers attending our program have been mathematics or computer science teachers. For the first year of the logic teacher PD, there was a significant involvement from us, but after that, it has been led by the high school teachers who have previously taught the course in one of our summer camps.
A key issue in the design of the PD is to determine how much emphasis should be placed on the teaching of the content vs discussing the pedagogy. Most teachers would not have seen our course material before coming to the PD course, and therefore, need some introduction to the material. At the same time, one week is not long enough to master the content that has not been previously seen. Therefore, our approach has been to emphasize the technical content of the course, and along the way, discuss the pedagogical approach. As the PD is taught by a teacher who has previously taught the course at the high-school level, they are able to share the challenges that they faced based on their experience.
By summer of 2018, 24 high school teachers had undergone the logic teacher PD. Three of these teachers have instituted a logic course at their respective schools by seeking the UC approval that mentioned earlier. Several other participating teachers have similar plans.
Our experience with the logic teacher PD has shown us that it has been possible to train a high school teacher to pick up the material in the current form and deliver it effectively. We are planning to further ease the process by providing ready to use lecture notes, and more detailed lesson plans which can be adopted by the teachers with minimal effort.
The teacher professional development program will continue in Summer 2019. Our long-term goal is to find a suitable partnership to train teachers in a scalable manner. For example, one possibility is to team with Stanford's graduate school of education in their effort to develop the course work to prepare teachers for computational thinking. As the high school logic teacher community expands, we will also organize a refresher PD for the teachers who have already gone through the PD once and have been teaching for some time. During the refresher teacher PD, the emphasis will be on improving the pedagogical approach and on addressing the actual challenges the teachers have faced during the classroom instruction.
AcknowledgmentsThis work has been funded by the Infosys Foundation USA, Intuit, and Stanford's Engineering Diversity Program. We thank many of our colleagues for their valuable input including Prof. Dan Schwartz and Dr. Dave Barker-Plummer.
 Nisbett, R. E., Fong, G. T., Lehman, D. R., & Cheng, P W. (1987). Teaching reasoning. Science, 238, 625-631.
 Epp, S. S. (2003). The role of logic in teaching proof. The American Mathematical Monthly, 110(1), 886-899.
 Attridge, N., Aberdein, A., & Inglis, M. (2016). Does studying logic improve logical reasoning? In Csikos, C., Rausch, A., & Szitayi, J. (Eds.). Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, pp. 27-34.
 ASL Committee on Logic and Education. (1995). Guidelines for logic education. The Bulletin for Symbolic Logic, 1(1), 4-7.
 University of California A-G Subject Requirements. https://www.ucop.edu/agguide/a-g-requirements/index.html
 Common Core Standards. http://www.corestandards.org/
 Next Generation Science standards. https://www.nextgenscience.org/