Beyond Graphing Calculators: Using Technology to Support Discussions in Middle-Secondary Classrooms

Kevin A. Lawrence, Michigan State University kevlawr@msu.edu Twitter: @kalawrence9 & @ipadmath

Nicholas J. Gilbertson, Michigan State Universitygilbe197@msu.edu

About us Kevin Lawrence

PhD student (3rd year) in Mathematics Education (PRIME) at Michigan State University

Taught H.S. math and physics for six years (IN) Interested in technology use by teachers and students, student transition from K-12 to collegiate math

Nic Gilbertson PhD student (5th year) in Mathematics Education (PRIME) at Michigan State University

Taught M.S. & H.S. math for seven years (WI, OR, IL) Interested in technology use, classroom discussions.

Types of Technology Discussed

Would not require one-to-one student access Whole-class inquiry & discussion Focus on conjecturing, justification, and sense making

Includes math specific and non-math specific technologies Math Specific

• GeoGebra• Desmos

Non-Math Specific• Spreadsheets• Screencast Apps

Graphing Calculators

The Good The Bad The Ugly

Graphing Calculators

The Good More available for one-to-one use in some schools (compared to newer technologies)

Graphing Calculators are math-specific technologies

Accepted technologies within the math teaching communities

Graphing Calculators

The Bad Graphing calculators may not be up-to-date.

Easiest (and most common) use is entering in symbolic representation for the production of:• Tables• Graphs

Doesn’t work well in problem-based scenarios that present a representation other than symbolic:• Raw data• Graphs

Representational Translations (Janvier, 1987)

To

From

Situations,

Verbal

Tables Graphs Formula

(Symbolic)

Situations,

Verbal

  Measuring Sketching Modeling

Tables Reading   Plotting Fitting

Graphs Interpretat

ion

Reading off   Curve

Fitting

Formula

(Symbolic)

Parameter

Recognition

Computing Sketching  

Graphing Calculators

The Ugly• Students may use the calculator output as the authority– Unaware of input errors and effect on other representations (tables, graphs, etc.)

– Technology masks the mathematical characteristics that are emphasized outside of calculator use (shapes of graphs of particular function families).

CCSSM Standard for Mathematical Practice CCSS.Math.Practice.MP5  Use appropriate tools strategically. Mathematically proficient students consider the available tools when

solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

We are not here to devalue the use of graphing calculators.

Technology > Graphing Calculators Not all tasks should dictate the same technologies to be used

GeoGebra

GeoGebra

Free download at geogebra.org

Some limitations compared to Geometer’s Sketchpad

Many improvements in the past few years

GeoGebra

A search for a better way to teach the Pythagorean Theorem

Previous approaches Numerically focused Students tended to over-generalize, “this is true for any triangle right?”

Proofs were not very convincing – especially at the middle school level.

Desmos – Online Graphing Calculator Desmos.com and iPad app Similar display on any platform (computer, mobile, app)

Save to Google Doc (not yet on iPad) Pros:

Enter either equation or data (tables) View equation, table, and graph at the same time Sliders for equations provide dynamic change in graphs and tables!

Can import picture to sit behind the coordinate grid.

Desmos

Given raw data as coordinates, find the function that matches Test different types of function families with the use of sliders

Allows the user to test different values See changes that occur when changing the value of parameters (how the graph moves/adjusts)

Desmos – Exponential Growth

Situation/verbal Symbolic Table Graph

A dog of a problem

Aaron and Denise bought a house. After one month of living there, they noticed six prairie dogs in their back yard. After two months, they counted 18 prairie dogs and 54 the month after that.

Find an equation that models the prairie dog population after x months of living in their house.

How many prairie dogs were there the day they purchased their house?

How many prairie dogs would they have in their back yard if the trend continues for one year after they purchased their house?

Spreadsheets

Spreadsheets1. Extend the pattern

2. Write an equation (or expression) that helps describe how to find the number of toothpicks for any shape number

From Variables & Patterns – CMP3, Lappan et al., (2014)

iPad Apps

There are many math apps that are available on iPads and other mobile devices.

There are many apps that are not math specific that can be used for effective communication within the mathematics classroom.

Use of interactive white boards on tablets allows for students to verbalize their thoughts while writing their work in real time by creating a video.

Great for capturing students interpretations of graphs.

Video can be shared in various ways (url links, embed codes, social media, within apps, etc.)

iPad AppsEduCreations ShowMe ScreenChomp

Ask3 Explain Everything TouchCast

Example of a Screencast

http://www.educreations.com/lesson/view/cmpusers14/17503779/?ref=link

Thank you!

Twitter: @connectedmathfacebook.com/connectedmathematicsproject

Kevin A. Lawrence, Michigan State University kevlawr@msu.edu Twitter: @kalawrence9 & @ipadmath

Nicholas J. Gilbertson, Michigan State Universitygilbe197@msu.edu