# Standards Based Grading in an AP Calculus AB Classroom

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Standards Based Grading in an AP Calculus AB Classroom
Taylor Gibson - North Carolina School of Science and Mathematics

Presentation Overview
Overview of Standards Based Grading What we’re doing at NCSSM Questions

A Story: Part I 1912: Starch and Elliot
147 English Teachers grade two English papers Paper 1: Scores range from 64 to 98 Paper 2: Scores range from 50 to 97, 15% failing, 12% “A” Starch, D., & Elliott, E. C. (1912). Reliability of the grading of high school work in English. School Review, 20,442–457

A Story: Part II 1913: Starch and Elliot
128 Math Teachers grade Geometry papers Scores range from 28 to 95 Starch, D., & Elliott, E. C. (1913). Reliability of the grading of high school work in mathematics. School Review, 21,254–259

A Story: Part III 2012: Hunter Brimi
73 High School Teachers grade the same student paper 20 hours of training in writing assessment Scores ranged from 50 to 96 Brimi, H. M. (2011). Reliability of grading high school work in English. Practical Assessment, Research and Evaluation, 16(17), 1–12.

A Story: Part IV 1918: Johnson and Rugg
Move towards scales with few categories Excellent, Average, and Poor Excellent, Good, Average, Poor and Failing (A, B, C, D, F) Johnson, R. H. (1918). Educational research and statistics: The coefficient marking system. School and Society, 7(181), 714–716 Rugg, H. O. (1918). Teachers’ marks and the reconstruction of the marking system. Elementary School Journal, 18(9), 701–719.

Tenets 1&2 of Standards Based Grading
Grades represent only student achievement on learning standards Percentage based grading and averaging are poor measurement tools to describe student learning

To communicate, to all stakeholders, student achievement towards a set of learning goals at a certain point in time

Tenets 1&2 of Standards Based Grading What Does Not Go into a Grade
Student Behavior Late work penalties Cheating Attendance Bonus Points Relative Grading Zeros for missing assignments

Tenets 3&4 of Standards Based Grading
No (or less focus on) summative or omnibus grades Grades should engage students in the learning process

Sample Mathematics Report Card (Middle School)
Marzano, Robert J, and Tammy Heflebower, Grades That Show What Students Know, Educational Leadership 69-3 (2011)

Tenets 5&6 of Standards Based Grading
A students grade can change on a standard through reassessment The most recent evidence of learning counts the most when determining mastery on a standard

AP Calculus AB

The Standards AP Calculus AB

First Trimester Wrote our own standards
Grouped learning objectives into 3 major types: C-level: Skills based standards B-level: Content specific conceptual understanding A-level: Overarching Mathematical Skills

First Trimester Struggled with: How many standards?
How to word learning objectives? How to align assessments with these objectives?

AP Calculus Curriculum Framework

Our Updated Standards

Limits Students will understand that:
The concept of a limit can be used to understand the behavior of functions Continuity is a key property of functions that is defined using limits

Derivatives Students will understand that:
The derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies. A function’s derivative, which is itself a function, can be used to understand the behavior of the function. The derivative has multiple representations and applications including those that involve instantaneous rates of change.

Integrals and the FTC Students will understand that:
Antidifferentiation is the inverse process of differentiation. The definite integral of a function over an interval is the limit of a Riemann sum over that interval and can be calculated using a variety of strategies. The Fundamental Theorem of Calculus, which has two distinct formulations, connects differentiation and integration. The definite integral of a function over an interval is a mathematical tool with many interpretations and applications involving accumulation. Antidifferentation is an underlying concept involved in solving separable differential equations. Solving separable differential equations involves determine a function or relation given its rate of change.

Derivatives: C-level Deriv.C.3 Calculate explicit derivatives LO2.1C
Students will know that… Direct application of the definition of the derivative can be used to find the derivative for selected functions, including polynomial, power, sine, cosine, exponential, and logarithmic functions. Specific rules can be used to calculate derivatives for classes of functions, including polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse trigonometric. Sums differences products, and quotients of functions can be differentiated using derivative rules. The chain rule provides a way to differentiate composite functions

Limits: B-level Lim.B.1 Analyze functions for intervals of continuity or points of discontinuity LO1.2A Students will know that… A function 𝑓 is continuous at 𝑥=𝑐 provided that 𝑓 𝑐 exists, lim 𝑥→𝑐 𝑓 𝑥 exists, and 𝑓 𝑐 = lim 𝑥→𝑐 𝑓 𝑥 . Polynomial, rational, power, exponential, logarithmic, and trigonometric functions are continuous at all points in their domains. Types of discontinuities include removable discontinuities, jump discontinuities, and discontinuities due to vertical asymptotes.

Integrals: C-level Int.C.# Approximate a definite integral LO3.2B
Students will know that… Definite integrals can be approximated for functions that are represented graphically, numerically, algebraically, and verbally. Definite integrals can be approximated using a left Riemann sum, a right Riemann sum, a midpoint Riemann sum, or a trapezoidal sum; approximations can be computed using either uniform or nonuniform partitions.

Assessing the Standards

Proficiency Scale 0: No Evidence of Learning 1: Beginning 2: Developing 3: Proficient 4: Advanced Adapted from Frank Noschese

Sample Question #1 Deriv.B.1
Use derivatives to analyze properties of a function

Sample Question #2 The number of jobs in North Carolina, in thousands, is modeled by the function 𝐸 𝑡 , where 𝑡 is the number of months that have passed in the year Interpret the following mathematical statements in context using correct units. 𝐸 3 =4373 and 𝐸 ′ 3 =14.4 𝐸 ′′ 3 =−4 Deriv.B.3 Interpret the meaning of a derivative within a problem

Reassessment

Reassessment Students may be reassessed on previous content
Teacher or Student Initiated If student initiated, must demonstrate improvement before reassessment Most recent assessment counts 60% of score

Reporting the Standards

Reporting the Standards