Integrating Technology into Mathematics (6-12) MELT 2015 Appalachian State University Kayla Chandler DAY 1

Agenda TimeActivity 8:30 – 10:00 Introductions Area Under a Curve 10:00 – 10:15Break 10:15 – 11:45 The Lottery Problem Sequence Generator 11:45 – 1:00Lunch 1:00 – 2:30 Things to Consider When Using Technology in Mathematics Evaluating Technology Tools 2:30 – 2:45Break 2:45 – 4:15 Cognitive Demand of Tasks The Cell Phone Task Thinking Through a Lesson Plan Protocol 4:15 – 4:30Wrap up

SESSION ONE

Goals for the Week Overview of math specific technologies Mathematical Action Technologies GSP, GeoGebra Survey

Google Forms What are some benefits and drawbacks of using Google forms versus traditional paper surveys? How might you use Google forms in your teaching of mathematics?

Google Forms and Grading Flubaroo – grading tool for Google forms Sample data:

Getting to Know You Visit the link below and introduce yourself. Double click in the blank space and input your name, a picture, and an embarrassing story about an experience you’ve had with technology.

Using Padlet How might you use Padlet in your mathematics classroom to encourage collaboration? Are there particular tasks or activities for which it might be useful to use Padlet?

Area Under a Curve Without using technology, find the area under the curve for the following function: f(x) = -0.26x 4 – 0.85x x x + 4

Helpful Formulas Left-hand sum Right-hand sum Midpoint sum Trapezoidal sum

Try it Using Technology On the MELT wikispace for today, you will find three sketches. Explore the sketches and think about the following questions: 1. What is happening in the sketches? 2. What do you like/dislike about the sketches? 3. How might you use the sketches to help students conceptually understand Riemann sums?

SESSION TWO

Dynamic Mathematics Environments (DMEs) Connected, multiple representations (dynamically linked) Use dragging to explore relationships Facilitate generalization Examples: The Geometer’s Sketchpad, TinkerPlots, Fathom, GeoGebra, TI-Nspire

GeoGebra Open GeoGebra and take a few minutes to explore to see what types of things you can do! Can you construct a line segment? Can you graph a function? Can you find how to view the spreadsheet?

The Lottery Problem Imagine you won the lottery and have to choose between 2 options for the distribution of your winnings: 1) you could take $100,000 each day for 1 month, or 2) you could take a penny the first day and then twice as much each day after for the duration of a month. Which option would you choose and why? Solve using the spreadsheet in GeoGebra.

Questions to Consider How did you solve the problem using GeoGebra? What other representations that might be helpful when making the decision between the two different payout options? In what ways might building the sequence and cumulative sequence in a spreadsheet to represent the lottery payout options help or hinder students’ understanding? What’s the difference between an arithmetic sequence and a geometric sequence? Which type of sequence is each payout option? How are these sequences represented graphically? What are some potential questions you could ask students about this task?

Sequence Generator With your partner, create your sequence generator and answer the questions provided.

Homework: Lesson Plan Generate a lesson plan that appropriately utilizes technology to teach mathematics. Any tool and topic Any format – any teacher should be able to follow your lesson Planning sessions Presentations on Friday

SESSION 3

Ways technology can be used in mathematics Illustrating mathematical ideas, Posing mathematical problems, Opening opportunities for students to engage in mathematical sense making and reasoning, or Eliciting evidence of students’ mathematical thinking (Dick & Hollebrands, 2011, p. xi)

Questions to Consider Interactive activity on graphing trig functions: How do a, b, c, and d affect the graph of Was the technology in this task used as a servant or used to develop mathematical understanding, or both? Explain. What advantages does this technology provide to the teacher?

Evaluating Technology Tools: Fidelity Mathematics Pedagogy Cognition

Mathematical Fidelity Is what is represented in the tool an accurate representation of the mathematics?

Pedagogical Fidelity Is the tool intuitive? Does it allow for actions and provide appropriate feedback? Would students think the tool is confusing or does how it should be used seem clear? Consider this Algebra Tiles applet:

Cognitive Fidelity Is the feedback provided by the tool consistent with the students’ mathematical thinking? In other words, is the way in which the tool operates mathematically consistent with the way a student would think mathematically about the task? Solve

Cognitive Fidelity Go to wolframalpha.com. Type the equation in the input line as 5x2 + 2x – 4 = 6x.wolframalpha.com To what extent is the solution provided similar to or different from how you solved it? Compare and contrast the cognitive fidelity of the tool(s) you used to solve the equation and Wolfram Alpha. To what extent did they reflect and support your mathematical thinking?

Evaluating Technology Tools: Design

Questions to Consider For each of the applets below, use the design principles to critique the exploration questions and the technology. Which design principles are followed? Which are not? What changes would you make? Why? Function Flyer: Fish Farm: zzle=40#Open%20Java zzle=40#Open%20Java

SESSION 4

Cognitive Demand of Tasks

Questions to Consider How would you classify the lottery problem? How would you classify the cognitive demand of this task? Jeremy has been on his parents’ cell phone plan, but now that he has graduated college, he has to get his own plan. He has a particular smartphone that only works with one company and the company has many options Jeremy can choose from. Jeremy has narrowed down the selection to two monthly options: Plan #1: no flat fee, $0.16 per minute OR Plan #2: flat fee of $20 + $0.10 per minute. Write an equation and generate a graph to represent each cell phone plan. Use your equations and graphs to determine which plan is the better deal if he typically talks 45 minutes per month.

Choosing a Cell Phone Plan Jeremy has been on his parents’ cell phone plan, but now that he has graduated college, he has to get his own plan. He has a particular smartphone that only works with one company and the company has many options Jeremy can choose from. Jeremy has narrowed down the selection to two monthly options: Plan #1: no flat fee, $0.16 per minute OR Plan #2: flat fee of $20 + $0.10 per minute. Which is the better deal? How do you know?

Questions to Consider How would you classify the cognitive demand of this revised cell phone task? Solve the problem using GeoGebra. Which is the better deal? How do you know? What representations are most helpful for considering this question? Why? What mathematical concepts must students draw upon in order to answer this question? Describe several different ways students might approach this problem. Think of or find a task that is of low cognitive demand. Rewrite the task to make it high-cognitive demand. What strategies can you use to change the cognitive demand of a task?

Thinking Through a Lesson Plan Protocol Stage 1: Selecting and setting up a mathematical task The purpose of this initial stage is to identify your mathematical goals for the lesson, identify tasks that will help you meet those goals, and consider the ways that students will draw on their previous knowledge to engage with the task(s) to meet the learning goals. This includes identifying: the different ways that students might solve the task; what misconceptions they might have; and what errors they might make.

Thinking Through a Lesson Plan Protocol Stage 2: Supporting students’ exploration of the task The purpose of this stage is to identify how you will support students’ learning as they work on the task. This includes planning the questions that you might ask to: help a group get started; focus students’ thinking on the key mathematical ideas in the task; assess students’ understanding of key mathematical ideas; and encourage all students to share their thinking with others or to assess their understanding of their peers’ ideas.

Thinking Through a Lesson Plan Protocol Stage 3: Sharing and discussing the task The purpose of this stage is to consider how you will have students share their ideas so that you accomplish your mathematical goals. Planning the questions that you might ask so that students will: make sense of the mathematical ideas you want them to learn; expand on, debate, and question solutions being shared; make connections among the different strategies presented; look for patterns; and begin to form generalizations.

Questions to Consider What could be the mathematical goal of the task? Jeremy found that there were other cell phone plans that include a certain number of minutes, or unlimited minutes, within the flat fee. He wants to compare these plans to the flat fee plus per min plans he found earlier. Cell phone plan #3: $ $0.45 per min over 450 min Cell phone plan #4: $ $0.40 per min over 900 min Cell phone plan #5: $69.00 unlimited min

Questions to Consider Of the new plans, which is the best deal for Jeremy? Describe the circumstances under which you would suggest that Jeremy choose each of the new plans. Explain your reasoning. Do plans 3 and 4 ever cost the same amount? How do you know?

Questions to Consider Assume students have not yet learned about piece-wise functions. What other mathematical concepts and skills would students need to know prior to engaging with this task? What strategies would students use if they had not yet learned piece-wise functions? What difficulties would you expect students to encounter?

Questions to Consider What questions or prompts might you use to help students transition from the linear cell phone plans in the initial task to the piece-wise plans in the extension? Describe how you might facilitate a discussion allow students to share their thinking on the task and their informal ideas about piece-wise functions. Also describe how you would transition to formally defining and discussing piece-wise functions. Provide as much detail as possible.