Tools, Tasks, and Strategies: Higher-Order Thinking, GIMP, Pick's Theorem, and Pixel Counts

1. Tools, Tasks, and Strategies:Higher-Order Thinking, GIMP, Pick's Theorem, and Pixel Counts Brian H. Giza, Ph.D. & Olga Kosheleva, Ph.D. The University of Texas at El Paso NCTM National Conference April 27, 2012

2. Navigation • The Tools, Tasks, and Strategies (TTS) Framework • Open source software and software alternatives in the context of TTS • Mathematics and TTS: Pick’s Theorem (1) • The Gnu Image Manipulation Program (GIMP) an open-source graphics tool • Mathematics and TTS: Pick’s Theorem with the GIMP • Questions ? • Obtaining the GIMP (and other free tools) • NCTM & Common Core Standards • Contact information • References • Presentation website

3. Tool-Task Dependence vs.Tool-Task Independence • Teachers are often provided with technology tools that range from the very simple to the very sophisticated. And when teachers receive technology professional development it is often targeted at the use of a particular technology tool, usually how to use it in only one context. • Teachers are essentially taught to function at the application level (Tool-Task Dependence). Teachers are rarely encouraged to adapt that tool to support additional learning activities. • Tool-Task dependence operates at the lower, application level of Bloom’s cognitive taxonomy.

4. Tool-Task Dependence vs.Tool-Task Independence (2) • In order for teachers to learn to use technology tool to support learning goals that diverge from the original design, they must • (1) master the tool, • (2) master their instructional environment, and • (3)be able to design new and creative ways of achieving a learning goal by adapting a tool to novel uses. • This higher level of tool mastery is termed by the author as “Tool-Task Independence“, in which the user operates at the higher, creative Bloom’s level. The Tool, Task, and Strategies Framework is the authors’ approach for helping teachers achieve tool-task independence.

5. Tool 1: What is a hammer for?

6. Tools 1 revisited: What is a hammer for? • A hammer is a tool for learning about the physics of cantilevers. Activity and image source: http://pbskids.org/dragonflytv/superdoit/balancingact.html

7. The hammer balance in terms of Tools, Tasks, and Strategies • The Tools: A hammer, a rubber band, a ruler, and a table edge • The Task: Demonstrate an understanding of the physics of a cantilever • The Strategy: provide pairs of students with the tools and coach them to think through how the principles of the fulcrum and lever, and center of mass can be used to demonstrate a principle used in architecture. Back to the main navigation slide

8. A sample Task and Strategy (1) • To teach the elements of plot a language arts teacher wants pupils to collaborate in small groups in writing short stories (Mott & Klomes, 2001; McLellan, 1992). Each story must include a description of some characters, their motivations, and how they interact. It must have a beginning, a middle and an end. • The task the teacher is setting the pupils to do is in learning certain elements of plot. • The strategy the teacher is using is collaborative work. In this situation the task is a learning task, and it can be described in terms of a learning objective. The strategy (or pedagogical approach) has its own issues of course - effective collaborative learning requires certain elements that we shall pre-suppose are present. • There are a variety of tools that can be used to accomplish the desired task while using the collaborative learning strategy.

9. Examples in which the task and strategy remain relatively constant but the tools change: (1) Paper & pencil • The teacher gives each pupil in the group a set of index cards and pencils and tells them that they must each create a character, discuss and come to a consensus as to how the character will interact with each of the other group-member's characters. Then they will each write one or two paragraphs of the story and then pass their card to another group-member who will write the next section and so-on until the story is complete.

10. Examples in which the task and strategy remain relatively constant but the tools change: (2) PowerPoint and one computer: • The teacher gives the group access to the 'group computer'. They are given PowerPoint and shown how to enter text in a text-field as well as create new slides. Each pupil in the group will take turns at the computer. To start they must each create a character, discuss and come to a consensus as to how the character will interact with each of the other group-member's characters. Then they will each write one or two paragraphs of the story in a PowerPoint slide and then give another group-member access who will write the next section and so-on until the story is complete.

11. Examples in which the task and strategy remain relatively constant but the tools change: (3) Internet-based collaborative writing environment (WIKI and Chat): • The teacher provides WIKI access to each pupil in the group (they are all working from home). the instructions for the activity are on the WIKI lesson home page. A chat room is set up in which the members of the group meet, discuss characters, and use to come to a consensus as to how the characters will interact . Then, using a WIKI page for the story, each person writes one or two paragraphs of the story and then turns writing access over to another group-member who will write the next section and so-on until the story is complete.

12. As may be seen in examples 1, 2, & 3 above, the tool can be considered separately from the task and the strategy. Even though the actual design of the learning situation may vary considerably, the commonalities of the task and strategy are evident. This is an important consideration because the TTS Framework has as its underlying paradigm that: "a good tool used with an inappropriate strategy will not accomplish the desired learning task".

13. FLOSS tools and TTS • To master technology tools , especially software, an educator must use it on a regular basis in order to achieve the needed level of proficiency. Making these tools available to K-12 pupils for their use in classroom projects is even more important. Although schools may provide the tools on site, pupils who can afford the tools at home have a distinct advantage over pupils who cannot afford productivity and multimedia production tools. • One group, the “haves” can explore and experiment at leisure, increasing their proficiency with the tools, while the other group of pupils, the “have-nots” are limited to school time, with resultant restrictions on time for learning the tools, etc. • Free (Libre) and Open Source Software (FLOSS) provides an answer for this, with licensing that provides for students' home use.

14. Summing up Part 1: the TTS Framework & FLOSS Software • The Tools, Tasks, and Strategies framework helps teachers accomplish teaching/learning tasks by using tools in new ways, and of adapting effective teaching strategies to new tools or tasks. • Free (Libre) and Open Source Software (FLOSS), as well as open standards for software file types helps teachers adapt technological tools to new uses. Back to the main navigation slide

15. Part 2: Introducing Pick’s theorem • Dr. Olga Kosheleva: Exploring Polygons • Dr. Brian Giza: Applying Pick’s Theorem to real-world situations using an open-source graphics tool Back to the main navigation slide

16. How does one compute the area of polygon when the shape is irregular? • What do you think children will do? Do you think knowing the formula in this case will help? Let’s discover a formula that may help.

17. Polygon 1 Polygon 2 Polygon 3 Let’s explore! • What is common in all these polygons? What did you notice?

18. Can you show on your geoboards another shape of the same type that has the same area? How many boundary points does it have?

19. Polygon 4 Polygon 5 Polygon 6 Let’s explore! • What is common in all these polygons? What did you notice? ______________________________________ ________________________________________________________ ________________________________________________________

20. Can you show on your geoboards another shape of the same type that has the same area? How many boundary points does it have?

21. Polygon 7 Polygon 8 Let’s explore! • What is common in all these polygons? Can you show on your geoboards another shape that has the same area? How many boundary points does it have?_________________________ Is there a relationship between area of a polygon and number of pegs? Can you describe it? _______________________________________________________________

22. Can you show on your geoboards another shape of the same type that has the same area? How many boundary points does it have?

23. More explorations! An Alternative way to find an area of a triangle.

24. Polygon 9 What happens if we have interior pegs? • What is common in all these polygons? Can you show on your geoboards another shape that has the same number of boundary pegs and interior pegs? Can you find its area?______________________________________ What is your conclusion?____________________________________ ________________________________________________________

25. Can you show on your geoboards another shape that has the same number of boundary pegs and interior pegs? How many boundary points does it have?

26. Polygon 9 What happens if we have interior pegs? Can you show on your geoboards another shape that has the same number of boundary pegs and interior pegs? Can you find its area? What is your conclusion?

27. Polygon 10 Polygon 11 Possible solutions: What are the areas of these polygons? What is your conclusion?

28. Let’s recall: Polygon 1 Polygon 2 Polygon 3 What was the change in the area when we got one interior point?

29. Polygon 12 What happens if we have two interior pegs? Can you show on your geoboards another shape that has the same number of boundary pegs and interior pegs? Do they all have the same area 3.5? What was the effect of adding one more interior point?

30. Teachers can continue with more examples until your students are ready to see the connection between Area and B and I. Suggestion to students: Write your hypothesis: Area =____________________________ Pick’s formula: George Alexander Pick. Born: 10 Aug 1859 in Vienna , Austria Pick's theorem which appeared in his eight page paper of 1899 Geometrisches zur Zahlenlehre published in Prague in Sitzungber. Lotos, Naturwissen Zeitschrift. The result did not receive much attention after Pick published it, but in 1969 Steinhaus included it in his famous book Mathematical Snapshots. From that time on Pick's theorem has attracted much attention and admiration for its simplicity and elegance. Pick died in a Nazi concentration camp at the age of 82 (1942). Back to the main navigation slide

31. Applying Pick’s theorem in the classroom using a real world example and a free graphics tool • Five E model • In order to guide students toward higher order thinking with student-led explorations, we shall be using an adaptation of the well-known Five-E learning cycle model in this tutorial (Bybee et al, 1989; Bybee et al, 2006). • The Five Es are: Engage, Explore, Explain, Evaluate, and Extend. In this tutorial we shall emphasize the Explore and Explain components, combining them together as the Explore step-by-step activities. The Evaluation is simple: Can the user perform a measurement of the change in beach area in two satellite images, before and after a tsunami using Pick's Theorem and a graphics program? Back to the main navigation slide

32. Pixel counts versus Pick’s theorem • There are already some excellent resources for analyzing satellite imagery with free tools. LuAnn Dahlman of the Carleton University’s Center for Science Teaching and Learning has an excellent lesson using the free ImageJ tool and pixel counts* – but we use the GIMP because it is widely supported for image editing, has a large number of ‘how-to’ tutorials at locations on the web, including YouTube, and because it is available (as is ImageJ) in portable versions • Plus we want to do an exercise with Pick’s theorem! • * http://serc.carleton.edu/eet/measure_sat2/index.html

33. GIMP: A Free and Portable Graphics Program Pick's theorem provides a simple method for finding the area of an irregular shape. It relates area to the number of points on a grid found inside or on the perimeter of an irregular shape. To use it we need a way of superimposing a grid on our irregular shape (the oil slick on the satellite image), as well as a way to calibrate the size of our grid. We also need to have a tool that lets us draw the perimeter on the image so that we can count the points on it or inside of it. The Gnu Image Manipulation Program (GIMP) is a powerful graphics editor that is available as a 'portableapps' version that can run from a USB drive in modern versions of Microsoft Windows: "GIMPPortable." (There is also a portable version for Macintosh OSX users.) The portable version does not require any administrator rights or privileges for a user to use it - it is not actually installed, just running from the USB drive - so it is very convenient for use in school settings. (As many teachers have found out, trying to install software can be a headache on school computers.). Teachers can even give it away or share copies of it as much as they wish, since that is the expectation of the user license under which the program was developed.

34. General information about Pick's Theorem in terms of the GIMP • Pick's theorem concerns the area of so-called "lattice polygons". These are simply polygons whose vertices all lie at points whose coordinates are integers. Such points are called "lattice points". Here it is: • Let P be a lattice polygon. Let b be the number of lattice points that lie on the edges of the polygon, and i be the number of lattice points inside the polygon. Then the area of the polygon is exactly b/2 + i - 1. • The GIMP provides tools for overlaying images with lattice polygons, allowing users to apply Pick’s Theorem to measures of image areas.

35. Exploring Pick’s Theorem with the GIMP

36. Studying tsunami-induced beach erosion using the GIMP, satellite images, and Pick’s theorem Open the GIMP graphics program and load the first (before) image of the beach, prior to the tsunami. This is an edited version of the GeoEye image of a location on the Japanese coast before and after the tsunami of March 2012. Although GeoEye is a commercial site, there are many other images through sites such as NASA or NOAA Image source: GeoEye (2012). Japan Earthquake and Tsunami. GeoEye. Online. Accessed March 16, 2011 from http://www.geoeye.com/CorpSite/promotions/Image_Slider.aspx.

37. Step by step instructions with the GIMP • Step 1. Import image 1 (3-15-img4-cropped-before.jpg) • Step 2. Once you have loaded the image in the GIMP, add a new layer using the Layer: New layer command. We do this so that our work does not alter the original image

38. Step 3: Use the GIMP's View: Show grid command to lay a grid on top of our image. We can configure the grid to make it a lattice of points suitable for use with Pick's theorem.

39. Step 4: Use the GIMP's Image: Configure Grid command to define your grid. We want to set the Line style to Intersections (crosshairs) and the Foreground color to red. This will make our grid of points more visible as a layer on top of our original image. Once we select the OK button our grid will be added to the image as a new layer.

40. Step 5: Use the pencil tool and an easy-to-see color such as green, draw a perimeter around the beach. Draw your perimeter carefully - if you need to, you may use GIMP's View: Zoom in menu sequence to enlarge your view. A view of a crudely completed perimeter drawing is available here. The most difficult decision that you must make is which grid points appear to be on the boundary and which are in the interior (come to a consensus on what will be included for each - it is suggested that if a cross-hair touches a boundary, it can be considered to be on it). Just don't count a point twice!

41. Do the same thing with the second image. Issues to consider: the images must be precisely correlated. You can ensure this by copying and pasting one image onto another as a separate layer and then using the GIMP’s Layer properties slider to adjust the transparency to ensure that the images are vertically aligned (superimposed properly). We shall actually use each image separately, drawing a perimeter and counting points similarly on both to apply Pick’s theorem to derive our beach areas, before and after the tsunami.

42. Step 6: Count the lattice points on the boundary (perimeter) and inside of it so that we may apply Pick's theorem. (we shall do the same thing with the second image later). Step 7: Copy the first, ‘before’ image from to the clipboard and import it as a new layer and reduce its’ opacity (increase its transparency) with the Layer slider controls so that you can see where your original beach was. Outline the post-tsunami beach with your pencil tool and count the points on and inside of the boundary so that you may apply Pick’s theorem).

43. Compare the two (superimposed) images by eye. It is apparent that there is change in the beach in the before and after images…but how much? (What is the proportional change in the beach between image 1 and 2?)

44. Evaluate: Using Pick's theorem, calculate the area of the change in units. For each image (or the perimeters on the superimposed images) Count the perimeter and interior points and apply Pick's Theorem. How many interior points (i) did you get? Divide this number by 2. How many perimeter points (b) did you get? Add the number of perimeter points to the ‘number of interior points divided by two’. That is the number of units in terms of units of area common to both images. Compare the two areas. What is the proportional change between the before and after images of the beach? What are some limitations associated with this technique? Area of ‘before’ perimeter points (b) was= 117 interior points (i) was: 282 Pick's area calculation for area was 282 + (117/2 - 1) or 282 + 57.5, giving an area of 339.5 Area of ‘after’ perimeter points (b) was= 135 interior points (i) was: 111 Pick's area calculation for area was 111 + (135/2 - 1) or 111 + 66.5, giving an area of 177.5 Estimated Ratio is: 339.5 to 177.5 or 177.5/339.5 = 0.52 (meaning that roughly half of the beach is gone).

45. Questions? Contact Information Dr. Olga Kosheleva: olgak@utep.edu Dr. Brian Giza: bhgiza@utep.edu Intrigued? Also visit the 2011 NCTM Pick’s theorem application to the Gulf Oil Spill Activity online at: http://www.educationtechnologies.com/modules/picks1 Back to the main navigation slide

46. Locations for obtaining Portable FLOSS software • Portable Apps for Windows (including GIMP Portable) online at http://portableapps.com. • Portable Apps for Macintosh OS X (including GIMP Portable) http://www.freesmug.org/portableapps/ • The GIMP home website: http://www.gimp.org/

47. Standards • NCTM Standards • Measurement • Instructional programs from prekindergarten through grade 12 should enable all students to: • understand measurable attributes of objects and the units, systems, and processes of measurement; • apply appropriate techniques, tools, and formulas to determine measurements. • Common Core Standards • Mathematics (6th grade)* • 6.RP.1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. • 6.RP.2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. • 6.RP.3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. • http://www.corestandards.org/the-standards/mathematics/grade-6/ratios-and-proportional-relationships/ Back to the main navigation slide

48. Contact information • Brian H. Giza • Associate Professor of Science and Technology Education • The University of Texas at El Paso • bhgiza@utep.edu • http://www.educationtechnologies.com • Olga Kosheleva, Ph.D. • Associate Professor of Mathematics Education • The University of Texas at El Paso • olgak@utep.edu We would like to express our thanks to EAI Education (Booth 925-926) for their kind Geoboard support Back to the main navigation slide

49. References (1) • Ben-Chaim, D., Fey, J., Fitzgerald, W., Benedetto, C., & Miller, J. (1998). Proportional Reasoning among 7th Grade Students with Different Curricular Experiences. Educational Studies in Mathematics, 36(3), 247-273. • Bybee, R.W. et al. (1989). Science and technology education for the elementary years: Frameworks for curriculum and instruction. Washington, D.C.: The National Center for Improving Instruction. • Bybee, R. W. et al. (2006). The BSCS 5E Instructional Model: Origins and Effectiveness. A Report Prepared for the Office of Science Education, National Institutes of Health (Appendix D). Available online at: http://science.education.nih.gov/houseofreps.nsf/b82d55fa138783c2852572c9004f5566/$FILE/Appendix%20D.pdf • Comeaux, P., & Huber, R. (2001). Students as Scientists: using interactive technologies and collaborative inquiry in an environmental science project for teachers and their students. Journal of Science Teacher Education, 12(4), 235-52. • Culp, K.M., Honey, M., & Mandinach, E. (2003). A retrospective on twenty years of education technology policy. Washington, DC: U.S. Department of Education, Office of Educational Technology. Retrieved April 24, 2012, from http://www2.ed.gov/rschstat/eval/tech/20years.pdf • Dahlman, L. (2011). Measuring Distance and Area in Satellite Images. Earth Exploration Toolbook: Step by step guides for investigating Earth system data. The Science Education Resource Center at Carleton College, Carleton, MN. Online. Accessed April 23, 2012 from http://serc.carleton.edu/eet/measure_sat2/index.html. • Daviddarling.com (n.d.). Pick's theorem. The Internet Encyclopedia of Science: Geometry Polygons. Daviddarling.com Available online at http://www.daviddarling.info/encyclopedia/P/Picks_theorem.html. Accessed July 9, 2010.

50. References (2) • Donovan, M. S., & Bransford, J. D. (2005). How Students Learn: Science in the Classroom. Board on Behavioral, Cognitive, and Sensory Sciences and Education (BCSSE), National Research Council, National Academies Press, Washington, DC. • GeoEye (2012). Japan Earthquake and Tsunami. GeoEye. Online. Accesed March 16, 2011 from http://www.geoeye.com/CorpSite/promotions/Image_Slider.aspx. • Giza, B. H. (2011). Science Technology and Young Children. Chapter in Blake, Winsor, & Allen (Eds) Technology and Young Children: Bridging the Communication-Generation Gap. IGI Global, Hershey, PA. 2011. • Giza, B. H. (2010). Using open-source graphics, animation, and video tools in STEM education. Chapter In Maddux, Gibson, & Dodge. (Eds.) Research highlights in technology and teacher education 2010. pp. 243-250. Society for Information Technology & Teacher Education (SITE). Chesapeake, VA: AACE. Available http://www.editlib.org/p/35314. 2010. • Giza, B. h. & Kosheleva, O. (2011). Proportional Reasoning, Pick's Theorem and an Open Source Graphics tool. Abstracts of the Annual Meeting of the National Council of Teachers of Mathematics NCTM 2011, Indianapolis, IN,, March 2011. • Haigh, G. (1980). A 'Natural' Approach to Pick's Theorem. The Mathematical Gazette, Vol. 64, No. 429 (Oct., 1980), pp. 173-177. • Kosheleva, O. (2009). Geometry and Algebra Make Good Bedfellows! Explorations of Area on Geoboards. Abstracts of the Annual Meeting of the National Council of Teachers of Mathematics NCTM'09, Washington, DC, April 22-25, 2009, p.125.