CALCULUS I Expectations & Teaching Strategies Mitch Anderson, - - PowerPoint PPT Presentation

CALCULUS I

Expectations & Teaching Strategies Mitch Anderson, UH Hilo Erica Pultar, UH Maui College Amir Amiraslani, UH Maui College

Student Expectations for entering Calculus I

1. Algebra Fluency

• Be able to show the domain of an expression using interval notation and when necessary find

the intersection of intervals

• Be able to find the common denominator of symbolic expressions with fractions
• Be able to factor expressions using basic identities such as binomial identities
• Be able to rationalize expressions including radicals when necessary
• Be able to simplify radical expressions and use absolute value when necessary

Student Expectations for entering Calculus I

• 2. Function Fluency
• Families of Functions
• Recognize polynomial, rational, power (square/cube root), exponential, log/ln, and trig functions,

both from symbolic and graphical representations

• Recognize and be able to apply the types of algebraic manipulations you would expect to be

able to perform for each, and know when a computer is more appropriate

• Interpret function notation in conceptual context
• E.g. By properly interpreting the numerator of [f(x + h) - f(x)]/h, conclude this represents a secant

line slope, or average rate of change of f over the interval [x, x + h]

• Interpret functions defined recursively: f(x + 1) = c f(x) and f(x + 1) = f(x) + c simply say respectively

that the “next” output is a constant multiplied by the “current” output, and the “next” output is the “current” output plus a constant, which are the primary definitions of exponential and linear functions

Student Expectations for entering Calculus I

• 3. Conceptual Fluency
• Intellectual maturity and sophistication appropriate to the level of the course
• Function fluency assists this process
• Ability to go beyond following an algorithmic process: e.g. being able to get information off a

graph helps students gain a deeper understanding of the symbolic representations

• Ability to intellectualize higher order concepts

Teaching Strategies

The Problem Solving Rubric ➔ What is the problem to be solved?

◆ e.g. Define instantaneous velocity

➔ What high school formula would you normally use to solve this type of problem?

◆ r =d/t

➔ Why can’t you use this formula directly?

◆ Change in time cannot equal zero (can’t divide by zero)

➔ What is your strategy for overcoming this difficulty?

◆ Approximate ◆ Take better and better approximations and look for convergence

Teaching Strategies - The Limit Concept

➔ Not a discrete topic: Limits permeate most of Calculus 1 and 2 ➔ Problem: How to train students to think deeply about limits

◆ Hands on: they need to visualize the limit process ◆ Appropriate Tools: Graphing calculators and computers

➔ Graphing Calculator Example: Slope at x = 3, begin with h = .1 and -.1

h = .1 h = -.1 6.1 5.9 6.05 5.95 6.025 5.975 6.0125 5.9875 6.00625 5.99375

The Definite Integral Arc Length Euler’s Method x^ln(x) from 1 to 3 sin(x) from 0 to 1 dy/dx = xy, y(0) = 1 3.590778541 1.311442498 1.648721271

Teaching Strategies - Precise Conditioning Program

n L(n) 10 3.364598787 100 3.567427475 1000 3.588436089 10000 3.590544223 100000 3.590755109 n L(n) 10 1.311382031 100 1.311441892 1000 1.311442492 10000 1.311442498 n L(n) 10 1.547110398 100 1.637820458 1000 1.647623038 10000 1.648611365

How do we know they truly understand the concepts? Answer: Group Projects 1. The Derivative: Sin Gun - How fast is the bullet traveling when it hits? (Intuitive Parametric Function) 2. Integration as a Process: The River Skipper - How long does it take the boat to arrive, if the current is determined by position (as opposed to time)? ODE 3. Integration as a Process: George the Slug - Where is George 5 minutes later, if his direction and speed are determined by his position? 2-dim Differential Equation. Note: We don’t cover differential equations until AFTER the two group projects. Students independently discover Euler’s Method in 2-dimensions.

Teaching Strategies - Assessment

Teaching Strategies - Group Labs

My goals for group labs:

• Offer a hands on activity besides lecture
• Incorporate technology
• Schedule some lecture time in a computer lab
• I’ve been utilizing free online graphing calculator www.desmos.com
• Want students to make visual connections to topics
• Want students to experience very light computer programming
• Get students to talk to each other.
• Make conjectures, challenge each other, etc.
• Get students to support and justify their hypotheses using precise language

Teaching Strategies - Group Labs Version 1

In this version, students are building the graphs they need to answer questions. Plan these activities before the topics are formally introduced in lecture

Example 1: Zooming to find Derivatives

Teaching Strategies - Group Labs Version 1

Example 2: Function & Derivative Relationship

Teaching Strategies - Group Labs Version 2

In this version, I give students pre-made graphs/programs to use and analyze.

Example 3: Epsilon-Delta Relationship

Teaching Strategies - Group Labs Version 2

Example 4: Newton’s Method

Thoughts

• Want students to feel comfortable in this part of class
• Writing thought-provoking questions/prompts is hard
• Planning is important

Teaching Strategies- Adding a group project component

• Projects about real-life applications of one or more major topics covered in the course assigned

within the first four weeks of class (10% of the total grade)

• A presentation rubric and a scientific report rubric provided
• Group presentations and reports at the end of the semester
• Could lead to a problem-based learning approach to calculus in the future

Examples:

• Can the sun become a black hole?
• Optimally designing a tent

Common Themes

• Technology
• Group work
• Making deeper connections with material, not just memorizing rules
• Using both precise math and English to explain or justify answers

Q & A