Mainstream Calculus I Enrollments (fall only for 2- & 4-yr colleges & universities) 250 AP Calculus 2004: 225,000 ~150,000 arrive with credit for Calculus I Currently growing at ~12,500/year 200 4-yr colleges & universities 2-yr colleges 150 ~250,000–350,000 retake calculus taken in HS AP Calculus (AB & BC) 100 students (thousands) Estimated # of students taking Calculus in high school: ~ 400,000 50 ~150,000–250,000 will take calculus for first time in college 0 1980–81 1985–86 1990–91 1995–96 academic year 2000–01 Estimated # of students taking Calculus I in college: ~ 500,000 (includes Business Calc)

New Courses Course Number Title Offered Online in 2018? Corresponding Current Courses BSK 1, MTT 1 – 4 Developmental Math Units YES not new courses MTH MTH MTH MTH 4) MTH 6) MTH MTH MTH MTH MTH MTH MTH MTH MTH MTH MTH MTH MTH 9* 111* 133 154 (with MCR MTE Unit Bundle 6-9 Basic Technical Math Mathematics for Health Professions Quantitative Reasoning NO NO YES YES MTH 126 MTH 151, MTH 152, MTH 157 161 (with MCR Precalculus I YES MTH 163 162 167 245 246 261 262 263 264 265 266 267 281 288 Precalculus II Precalculus with Trigonometry Statistics I Statistics II Applied Calculus I Applied Calculus II Calculus I Calculus II Calculus III Linear Algebra Differential Equations Introductory Abstract Algebra Discrete Math YES YES YES NO YES NO YES YES YES YES YES TBD YES MTH MTH MTH MTH MTH MTH MTH MTH MTH MTH MTH MTH MTH *MTH 9 and MTH 111 are not likely to be offered until Spring 2019. 164 166 241 242 271 272 173 174 277 285 291 200 286 • Calculus I, Calculus II, and Discrete Math will all reduce by one credit hour. • Students changing from an old catalog year to 2018-2019 run a risk of losing credits in courses from other disciplines

15.1 About cirquent calculus in general irquent calculus is a new proof-theoretic approach, introduced ently in “Introduction to cirquent calculus and abstract resourc antics”. Its invention was motivated by the needs of computability c, which had stubbornly resisted any axiomatization attempts within framework of the traditional proof-theoretic approaches such as uent calculus or Hilbert-style systems. The main distinguishing feature of cirquent calculus from the known approaches is sharing: it allows us to account for the possibility of shared resources (say, formulas) between different parts of a proof tree. The version of cirquent calculus presented here can be called shallow as it limits cirquents to depth 2. Deep versions of cirquent calculus, with no such limits, are being currently developed.

What is College Readiness? College readiness is a student’s ability to take college-level courses in a particular subject. College readiness at Bergen Community College in English and Math is determined by: *SAT Scores: 450+ Critical Reading, 500+ Math or *ACT Scores: 19+ English, 21+ Math or College courses in English or math taken at another accredited institution *Scores above are for tests taken after March 2016 If you are eligible to be waived from part or all of the Accuplacer Tests, click here to fill out the Waiver Form and upload qualifying test results.** Upon receipt, the Office of Testing Services will enter the information into the database so you can register for your classes. **Students in a degree program requiring the math sequence for pre-calculus and calculus may involve an additional developmental level course. Students can take an additional placement test to potentially waive out of the course, use pre-calculus or calculus CAP courses, AP scores or pre-calculus or calculus taken at another accredited institution (C or higher required). This is required only for students whose degree requires this math sequence. 8

What is College Readiness? College readiness is a student’s ability to take college-level courses in a particular subject. College readiness at Bergen Community College in English and Math is determined by: *SAT Scores: 450+ Critical Reading, 500+ Math or *ACT Scores: 19+ English, 21+ Math or College courses in English or math taken at another accredited institution *Scores above are for tests taken after March 2016 If you are eligible to be waived from part or all of the Accuplacer Tests, click here to fill out the Waiver Form and upload qualifying test results.** Upon receipt, the Office of Testing Services will enter the information into the database so you can register for your classes. **Students in a degree program requiring the math sequence for pre-calculus and calculus may involve an additional developmental level course. Students can take an additional placement test to potentially waive out of the course, use pre-calculus or calculus CAP courses, AP scores or pre-calculus or calculus taken at another accredited institution (C or higher required). This is required only for students whose degree requires this math sequence. 6

Subjects English* Reading, Writing Across the Curriculum, Vocabulary, Grammar, Composition, Fiction, Creative Nonfiction, Nonfiction Writing, Creative Writing, Technical Writing, American Literature, British Literature, World Literature, Poetry, Career Writing, Essay Center, Resume, Cover Letter, Paragraph Submission Math Tutoring Subjects*† Basic Math Skills, College Algebra, Intermediate Algebra, Algebra 1, Algebra 2, Geometry, Trigonometry, Liberal Arts Math, Pre-Calculus, Finite Math Calculus, Single Variable Calculus, Multi Variable Calculus, Applied Calculus, Differential Equations, Advanced Math Topics, Beginning Statistics, Intermediate Statistics, Advanced Statistics, Mathematics for Teachers, Matemáticas en Español * ESOL/bilingual support † master and doctoral level support

Example: Foundational Degree Requirements Communication - Oral: Recognize, send, and respond to communications for varied audiences both as speaker and listener. (LEAP Communication) (State Board Core, Communication) (HS Language Arts - Speech) Complete 3 credits ____ COMM 101 Critical/Creative Thinking and Problem Solving: Engage and demonstrate the ability to analyze and evaluate information and arguments; select or design appropriate frameworks and strategies to solve problems in multiple contexts individually and collaboratively. (LEAP Critical and Creative thinking) (Across the curriculum) (HS Humanities/Fine Arts/Interdisciplinary) Complete 3 credits. ____ ENGL 175 Introduction to Literature 3 ____ ENGL 257 Literature of Western Civilization 3 ____ ENGL 258 Literature of Western Civilization 3 ____ ENGL 267 Survey of English Literature 3 ____ ENGL 268 Survey of English Literature 3 ____ ENGL 271 Introduction to Shakespeare 3 (300 L) ____ ENGL 277 Survey of American Literature 3 (300 L) ____ ENGL 278 Survey of American Literature 3 ____ ENGL 285 American Indian Literature 3 (400 L) ____ ENGL 295 Contemp. U.S. Multicultural Literature 3 ____ FLAN 207 Contemporary World Culture 3 ____ INTR 200 Interdisciplinary Seminar 3 ____ PHIL 201 Logic and Critical Thinking 3 Communication – Written: Recognize, send, and respond to written communications for varied audiences as both writer and reader. (LEAP Communication and LEAP Information Literacy) (State Board Core, English Comp) (HS Language Arts - English) Complete 6 credits ____ ENGL 101 ____ ENGL 102 Mathematical and Symbolic Reasoning: Apply mathematical reasoning to investigate and solve problems. (LEAP Quantitative Literacy). (State Board Core, Mathematics) (HS Mathematics) Complete 3-4 credits ____ MATH 123 Contemporary Mathematics 3 ____ MATH 130 Finite Mathematics 4 ____ MATH 143 College Algebra 3 ____ MATH 144 Analytic Trigonometry 2 ____ MATH 147 Pre-Calculus 5 ____ MATH 160 Survey of Calculus 4 ____ MATH 170 Analytic Geometry & Calculus I 4 ____ MATH 175 Analytic Geometry & Calculus II 4 ____ MATH 187 Discrete Mathematics 4 ____ MATH 253 Principles of Applied Statistics 3 ____ MATH 275 Analytic Geometry & Calculus III 4

Relational Calculus Comes in two flavours: Tuple relational calculus (TRC) and Domain relational calculus (DRC). Calculus has variables, constants, comparison ops, logical connectives and quantifiers. – TRC: Variables range over (i.e., get bound to) tuples. – DRC: Variables range over domain elements (= field values). – Both TRC and DRC are simple subsets of first-order logic. Expressions in the calculus are called formulas. An answer tuple is essentially an assignment of constants to variables that make the formula evaluate to true. Database Management Systems, R. Ramakrishnan

We discuss only DRC Relational Calculus • Calculus has variables, constants, comparison ops, logical connectives, and quantifiers. • Comes in two flavors: Tuple relational calculus (TRC) and Domain relational calculus (DRC) • TRC: Variables range over (i.e., get bound to) tuples. • DRC: Variables range over domain elements (= field values). • Expressions in the calculus are called formulas. An answer tuple is essentially an assignment of constants to variables that make the formula evaluate to true. 03/22/2019 3

The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus is appropriately named because it establishes a connection between the two branches of calculus: differential calculus and integral calculus. It gives the precise inverse relationship between the derivative and the integral. 3

Comparing Physics Teacher Preparation in Buffalo, NY and Helsinki, Finland Dan MacIsaac, Luanna Gomez, Ismo Koponen & Ari Hämäläinen We juxtapose Masters degree programs for physics teacher preparation offered at University of Helsinki and SUNY Buffalo State College. We describe program coursework, candidates and program outcomes. SUNY- Buffalo State College Department of Physics, Helsingin yliopisto Fysiikan laitos Finland has been spectacularly successful in school science student learning according to This poster http://physicsed.buffalostate.edu/pubs/AAPTmtgs/ PISAavailable 2006 (#1 infrom Science; #2 in Reading; #2 in Math) and PISA 2009 (#1 in Science; #2 in Discussion Reading; #1 in Math). OECD (2010), PISA 2009 at a Glance, OECD Publishing. http://dx.doi.org/10.1787/9789264095298-en Helsingin yliopisto and Buffalo State College programs at a glance Finnish physics teacher candidates at university (yliopisto) are supported by the state with free tuition and an independent living allowance for a five year program of study (three years B.S. plus two years M.S.); teachers are a small and highly select population. A considerable proportion of Finnish physics teachers are initially trained as mathematics teachers and add physics as a secondary certification. Finnish high school teachers earn salaries comparable to their NY public school counterparts, well above the general US average. Finnish physics teacher candidates have a varied High School (yhteiskoulu) physics background (comparable to a partial progress through a College algebra-based physics sequence) in Finnish depending upon the individual and their school. All Finnish candidates previously had several years of physics as a separate Middle School subject in Finnish, roughly equivalent to an extended scope conceptual NY Regents physics course (some international schools use Hewitt’s Conceptual Physics text). Finnish HS mathematics is much more rigorous than NY Regents mathematics requirements, with most teacher candidates completing the long form mathematics sequence – basically mathematics every semester with extended conceptual and word problem solving practice ending at about US University Calculus II level. Students completing the yhteiskoulu long form maths sequence are elite students. Hence, Finnish physics teacher candidates may have little or NO university level mathematics course requirements. Buffalo State physics teacher candidates have at least 15cr of mathematics course requirements. NY physics teacher candidates are required to take more advanced mathematics content coursework than their Helsinki counterparts (compared to those Finnish teachers without mathematics certification). Finnish physics teacher candidates spend a great deal of programmatic time conceptually spiraling through the physics material they will teach (only) establishing an enriched conceptual consolidation known as kasitteellinen lujittaminen – which we abbreviate here as lujitta (lou–weet–tyah). Lujitta involves revisiting physics material comparable to the entire canon in US calculus-based texts (e.g. Knight’s Physics for Scientists and Engineers with Modern Physics is the text at the University of Helsinki) at least twice for all candidates, and for some four separate times. Each pass strives for greater conceptual linkage and sophistication augmented with extended practical (phenomenological laboratory) experience and increasing didactical analysis within the same well-defined and delimited scope of physics content. Rich applied conceptual physics understanding builds from a canon first seen in middle school, and is not essentially mathematical in nature. This contrasts sharply with the US bachelors degree practice of presenting (largely by mathematics) and extending new topics beyond the scope of the high school classroom content in upper level undergraduate courses for physics teacher candidates. At Buffalo State, PHY510, 620 and 622 are three courses that are essentially lujitta courses. Buffalo State undergraduate students complete a writing intensive laboratory course, and complete a laboratory report in their capstone laboratory course. Buffalo State masters students complete a master’s project. Helsinki students complete three major written thesis-like projects dedicated to physics instruction (undergraduate teaching report and physics teaching report plus their Gradu master’s dissertation). Buffalo State undergraduates complete intermediate physics courses on many more physics topics than their Finnish counterparts, including theoretical physics, computational physics and a more advanced mathematical course on electricity and magnetism. Many Buffalo State students also complete undergraduate courses on classical (Lagrangian and Hamiltonian) mechanics, and quantum mechanics. Buffalo State physics teachers have typical US laboratory coursework (a three semester calculus based intro sequence with labs, an electronics course with lab, a capstone lab course). Helsinki physics teachers have far more laboratory (practical) coursework – a two year long calculus based physics sequence with lab, and extended laboratory coursework in every year of their five years. Conclusions In the US, a few physics teachers’ graduate courses run by physics departments such as those at ASU (Modeling), Buffalo State , NCSU (Matter and Interactions) and others reprise introductory physics content at a mathematical level no higher than their introductory calculus based sequences, while striving for conceptual consolidation via laboratory and reflective work and adding pedagogical content knowledge (PCK). These courses are a US manifestation of lujitta, though unfortunately most US physics teachers do not take these latter US lujitta physics courses. Such courses should be more widely offered in the US to physics teachers and candidates, E.g. Modeling Physics Workshops. The widespread US claim that “every physics teacher should have an undergraduate major or minor in physics” should be challenged; clearly our highly successful Finnish physics teaching counterparts have a very different scope and sequence of coursework than standard US physics major and minors do. Specialty programs with limited physics content scope and lujitta-like consolidation with PCK and reflection seem remarkably effective elsewhere, and should be considered in the US. A five year bachelor’s plus master’s degree including math and physics certification program might be an interesting Buffalo State experiment, given NYSED teacher licensure. References Aiskenainen, M. & Hirvonen, P. (2010). Finnish cooperating physics teachers’ conceptions of physics teachers’ teacher knowledge. Journal of Science Teacher Education 21; 431-450. Knight, R.K. (2008). 2/e. Physics for scientists and engineers with Modern Physics: A strategic approach. Pearson Addison Wesley: San Francisco. Lavonen, J. (2010). Construction of teacher knowledge in teacher education in Finland. EDUCA 2010 13-15 October 2010, Bangkok, Thailand. Available from the author. Lavonen, J. & Laaksonen, S. (2009). Context of teaching and learning school science in Finland: Reflections on PISA 2006 results. Journal of Research in Science Teaching, 46(8). 922-944. MacIsaac, D.L., Henry, D., Zawicki, J.L. Beery, D. & Falconer, K. (2004). A new model alternative certification program for high school physics teachers: New pathways to physics teacher certification at SUNY-Buffalo State College. Journal of Physics Teacher Education Online, 2(2), 10-16. McKinsey & Company (2009). Closing the talent gap: Attracting and retaining top-third graduates to careers in teaching. An international and market research-based perspective. Accessed Jan 2010 from http://www.mckinsey.com/clientservice/Social_Sector/our_practices/Education/Knowledge_Highlights/Closing_the_talent_gap.aspx. Nivalainen, N., Asikainen, M., Sormunen, K., & Hirvonen, P. (2010). Preservice and inservice teachers’ challenges in the planning of practical work in physics. Journal of Science Teacher Education, 21; 393-409. OECD (2010), PISA 2009 at a Glance, OECD Publishing. http://dx.doi.org/10.1787/9789264095298-en Reinikainen, P. (2007). Sequential explanatory study of factors associated with science achievement in six countries: Finland, England, Hungary, Japan, Latvia and Russia. University of Jyväskylä: Institute for Educational research.

Mainstream Calculus I Enrollments (fall only for 2- & 4-yr colleges & universities) 250 AP Calculus 2004: 225,000 Currently growing at ~12,500/year 200 150 4-yr colleges & universities 2-yr colleges 100 AP Calculus (AB & BC) students (thousands) Estimated # of students taking Calculus in high school: ~ 400,000 50 0 1980–81 1985–86 1990–91 1995–96 academic year 2000–01 Estimated # of students taking Calculus I in college: ~ 500,000 (includes Business Calc)

Implications: 1. The top 25% of students who 20 years ago would have taken Calculus I in college now take it and get credit for it while in high school. They have been replaced by an equal number of students who now make up the bottom 25%. 2. Students who take Calculus I in college either are retaking a course taken in high school or have had to overcome mathematical deficiencies. Calculus I is increasingly taken as a terminal course. 3. Especially at elite institutions but increasingly elsewhere, the traditional Calculus II which presupposes Calculus I at that institution does not serve the needs of the students who take it.

Mainstream Calculus I Enrollments (fall only for 2- & 4-yr colleges & universities) 250 AP Calculus 2005: 240,000 Currently growing at ~13,000/year 200 150 4-yr colleges & universities 2-yr colleges 100 AP Calculus (AB & BC) students (thousands) Estimated # of students taking Calculus in high school: ~ 500,000 50 0 1980–81 1985–86 1990–91 1995–96 academic year 2000–01 Estimated # of students taking Calculus I in college: ~ 500,000 (includes Business Calc)

The Changing Face of Calculus Two-article sequence appeared in Focus in 2004: www.maa.org First-semester calculus has become a high school topic for most of our strongest students. This has several implications: 1.We should ensure that students who take calculus in high school are prepared for the further study of mathematics. 2.We should address the particular needs of those students who arrive in college with credit for calculus. 3.We should recognize that the students who take first-semester calculus in college may need more support and be less likely to continue with further mathematics than those of a generation ago.

Mainstream Calculus I Enrollments (fall only for 2- & 4-yr colleges & universities) 250 AP Calculus 2006: ~255,000 Currently growing at >15,000/year 200 150 4-yr colleges & universities 2-yr colleges 100 AP Calculus (AB & BC) students (thousands) Estimated # of students taking Calculus in high school: ~ 500,000 50 0 1980–81 1985–86 1990–91 1995–96 academic year 2000–01 Estimated # of students taking Calculus I in college: ~ 500,000 (includes Business Calc)

Implications: 1. Students who 20 years ago would have arrived at college ready to take calculus now take it in high school. 2. Students who take Calculus I in college either are retaking a course taken in high school or have had to overcome mathematical deficiencies. Calculus I is increasingly taken as a terminal course. 3. Especially at elite institutions but increasingly elsewhere, the traditional Calculus II which presupposes Calculus I at that institution does not serve the needs of the students who take it.

Into the Future •Pressure to get college-bound students into an AP Calculus class is going to intensify. •The growth in AP Calculus is not about to end. •% increase of BC Calculus will continue to exceed that of AB •% increase in # of students taking BC Calculus before senior year will continue to exceed that of BC generally •More universities will see calculus as a high school course.

Richard Courant, Differential and Integral Calculus (1931), first calculus textbook to state and designate the Fundamental Theorem of Calculus in its present form. FTC in present form does not appear in most commonly used calculus texts until George Thomas’s Calculus in the 1950s.