July 11-14, 2022

Instructor: Paul Gruber

Paul Gruber Headshot

After 30+ years in education, Paul Gruber still has a love for teaching and has a blast with his students, His classroom is mixture of constant work, lots of movement, and kids always laughing. To that end, he enjoys doing APSI workshops in Calculus where he travels and meets many wonderful people while experiencing different parts of the country and sharing his knowledge along with his passion in the greatest profession in the world.

For his entire career, because of his belief in equity, he has taught in urban districts. While teaching at Elizabeth HS in Elizabeth, NJ, after the previous AP teacher retired, thirteen years ago, no one signed up for the class – the principal was bothered by this and asked the top students what would it take for them to take AP Calc. They said, “If Mr. G teaches it, we’ll take it”. The first year, twelve students signed up. In ten years, the program has grown to 150+ students taking AP Calculus AB as well as BC with three different teachers teaching the classes.

As for his APSI presentation, he does not only present Calculus, but to create a better learning environment, he integrates what he knows about brain-based learning through the proper use of transfer, closure, primacy-recency theory and more.

Aside from being a teacher, he is also a grader of AP Calculus exams for nine years and an APSI facilitator for six, As for Paul’s education, he earned a BBA from Baruch College, an MA in Math Education from CCNY-NY and was a EdD Candidate at Columbia University / Teachers College in Math Education. Paul is married for 38 years, has three children and (finally) two grandchildren. Paul also has passion for baseball where he used to coach Varsity baseball and has visited 24 of the stadiums! Another fun fact, between his freshman and sophomore year in college (in the summer of 1976), he traveled and worked with the elephants of Ringling Brothers and Barnum and Bailey Circus.

Course Description

The primary goal of this class is for the teacher to acquire the techniques necessary to successfully teach Advanced Placement AB Calculus at the secondary level. This course will focus on classroom strategies that encourage teachers not only to enjoy teaching calculus, but also to learn how to creatively engage students in mathematical investigations that enable the students to “discover” the major concepts. We will examine a variety of strategies and skills that can be used to explore the fascinating world of teaching AP Calculus. This course is intended for to provide appropriate training for the AP Calculus teacher by providing opportunities to expand, refresh, and solidify your theoretical understanding of the calculus, become familiar with the level of knowledge required for student success on the AB AP Calculus Examination, gain expertise in the creation and use of appropriate assessment vehicles, including technology, and connect pedagogical theory to practice in the AP classroom.

Calculus is magic!   In this course, aside from covering many of the standard topics of Limits, derivatives and integrals with and without calculator usage, we will explore how many of these topics could be taught whereby students get a better understanding of what they are learning.  This is done by using transfer of connecting old knowledge with new knowledge.   We will also explore the AP Calc CED, the AP Classroom as well as understanding the end of year reports and look at how to answer Free Response Questions as well as Multiple Choice questions.  Most importantly, we will have fun doing it!

Course Agenda Overview


  • Graphing Calculator
  • Laptop
  • Please ensure College Board registration is complete

Day 1:

  • Introductions to the CED
  • Grading the AP – what it looks like.  Speaker’s Experience and the process.
  •       Looking at limits
    • Limit as X approaches K
    • Limit as X approaches 0
    • Limit as x approaches infinity – discussion of transfer. – the book
    • Limit with trig functions – developing UBD
    • Limit with L’Hopital’s Rule
    • Looking at AP type questions using limits
  • Course Time Sequence – Creating a calendar for the school year, before the school year begins!  The game plan.
  • Access and Equity.
  • Establishing Your Course.  Changes in the Curriculum… Developing Understanding by Design

Day 2:

  • Intro to AP Classroom
  • Discussion of Key Takeaways, as it pertains to pacing the curriculum.
  • Review of homework example and set up of first 1-1.25 months of curriculum
  • Looking at the derivative
    • The definition of derivative. Review sample MC question.
    • Proof of product rule
    • Look at quotient rule and chain rule
    • Develop understanding of implicit differentiation
    • Looking at derivative of Ln, working with e, trig functions.
    • Developing a pacing guide for the derivative portion. More UBD
    • Reviewing finding derivatives using the rules with an equation, graphically and a tabular approach. Look at question 6, 2017
  • Brain Based Learning and the TIMSS (to me, this is the MOST important piece that I can do with teachers).  Inculcating Transfer, Primacy-Recency Theory and more
  • Writing in the Curriculum. Looking at how to answer an AP question using Theorems, and connections between f(x), f’(x) and f”(x)
    • Looking closely at “justification”
  • Looking at resources on line —- manipulating the College Board web site / audit/other. Looking at other resources i.e. Teachers Pay Teachers.

Day 3:

  • Review of Homework
  • Related rates – a hands on approach.
  • Using the Calculator in the Classroom
    • Sto Key
    • Basic Calculator functions
    • Graphing
    • Derivative
    • Integral
    • Numerical Solver
  • Understanding the grading process
  • Continue to develop the curriculum – rework days already done to include calculator use.
  • Intro to Integrals. Making connection between velocity graph and distance traveled
  • Riemann Sums
  • Connections between Riemann Sum and working with Limits and Summation of area under the curve (the definition) . AP MC Sample questions.
  • Looking at Fundamental Theorem of Calculus Part II, Final position = initial position + displacement.
  • Working with Mean Value Theorem of Integration – m&ms
  • Introduction to Volume – Intro to discs
  • Looking at the Calendar and developing the game plan for the school year
  • Looking at a variety of AP questions
  • Grading of an AP FRQ

Day 4:

  • Review of Homework
  • Review of Volume
  • More Hands on with Volume, using the disc method.
  • Looking at AP question(s) using volume.
  • Looking at College Board Web Site as well as other sites that are recommended.
  • Developing Cross Sectional understanding of volume.
  • Looking at an FRQ AP question.
  • Looking at several AP Multiple Choice questions
  • Filling out the evaluation


Message to participants:  While this is my tentative agenda, please email me at P_Gruber@yahoo.com, if there is anything in particular that you want to better understand.    Please note that I hope to constantly be spiraling the entire curriculum and include topics such as differential equations and lope fields, somewhere in the presentation