These images were created for a high school algebra 1 class. The class in particular has 29 students, many of whom have behavioral issues or learning disabilities. Many of these students failed algebra 1 at the junior high level and are retaking the class. The disabilities fall mostly in the areas of reading, and language comprehension, only 3 students actually have learning disabilities in the area of mathematics. The students are very capable of working, but the majority of the time they would rather goof off and not work or pay attention. Too sum it all up; they are a group of under-motivated students that are in a class they dislike for a second time.
I have created nine images for this unit. The idea behind most of the images was to lend clarity to the formulas, and to give students a path to work through problems, since many of the students will not work in a discovery method for more than a single question. I was also hoping that the students might remember images better since so many of my students have reading and language disabilities. The following will be a justification for each image, followed by a justification for the ordering of the images in the unit.
The first image used in the unit is the slope image. The slope image is a very basic image using a line with some arrows that say rise and run. The purpose of the image was to demonstrate that a line that is diagonal has a constant rise over run, and in the simplest form a student can count the units for rise and run and find the slope. I kept the words to a minimum and tried to make the image all about the shapes and not about reading, since so many students have difficulty reading in my class. The shapes in the image are squares, rectangles, lines, and triangles. I chose these shapes due to their use in educational images. Hansen (1999) stated that squares and rectangles are great for containing information, hence why the labels are enclosed in rectangles and the whole image is enclosed in the square of the coordinate grid.
The next image used is the slope-intercept image which is very similar in construction to the point-slope image. I used the same basic framework for these images which was a thick black line that represents what the student will be drawing that has several text boxes in it with information about each type of equation. Inside the text boxes I color coded the important information so anywhere the slope is mentioned it is in blue, and anywhere the point, or y-intercept is mentioned it is in red. Once again I tried to keep the words to a minimum and let the pictures do the talking. The line gives a reading order that follows natural conventions from top left to bottom right and splits page giving it a symmetrical use of whitespace. This symmetrical creation was two-fold, first Lohr (2008) states that symmetry creates, “A sense of calm and professionalism.” I wanted this sense of calm due to my students’ predisposition towards chaos when over stimulated. The other reason I used a large amount of symmetrical whitespace was to make the amount of information in the image look like it was less than it really was, thus making it more palatable to students that do not like the subject.
The slope, and equation images are followed by the multiple representation image. This image will be used as a handout, and is part of a PowerPoint so there will be two versions one with color and glows and one without. This image depicts the several different ways that the information from an equation, in this case a slope-intercept form equation can be presented. The basic design is a header at the top and then a diamond shape in the middle of the page that contains the equations and lines extending form the vertices of the diamond to the edges of the image so the image is separated into four quadrants. Each quadrant has a different depiction of the information, one quadrant is a table of values, another is a graph, another is a word problem that would create the equation, and the final quadrant is the important information about the equation. Each quadrant is titled with a title about what is in it and surrounded by a reddish glow so that the eye is drawn to it. The information about the y-intercept and slope is color coded so students can link that information to the different forms. I decided to use the color for labeling since that is the first function of four functions of color that Edward Tufte (1990) describes.
The graphic organizer is the next image and by far the most contested image that I created. This image is a traditional mind-map with balloons with text linked together with arrows. To tie it in with the unit I put the map on top of a coordinate grid and I used the coordinate grid numbering system to direct the flow of information. However the flow of information in the coordinate plane is backwards to the normal reading conditions for the United States, thus several people commented that the order seemed confusing. I decided to go with the traditional mind-map as a response to Paas, Renkl, & Sweller’s (2003) cognitive load theory. This theory basically states that a better learning environment can be created by getting rid of extra information. This is why I went with the text instead of using multiple images, since students are familiar with the format and are able to chunk the balloons together to increase retention.
The independent and dependent variable image and the graphing any equation image were both created using the same template. The main ideas was to create a list of information that has the first step as the most important and each subordinate step building off of the work done during the first step. The hierarchy used in the diagram is vertical which Horton (1994) suggests gives the desired outcome of making the first step important. I tried to minimize the number of steps in the hierarchy to increase the ability of students to chunk the information. The images are fairly boring; however they do convey a lot of information in a relativity small space and are appear to be easy to follow. These are the wordiest of the images created but the words selected for the images will have been used in other images and several times in class before this image is given to the students.
The word problem and graphing in two steps image was created to link the important information in word problems to an equation, and then link the equation to the graph. The image is color coded in terms of information to aid in the linking process that I was hoping to create. This image originally had extra background colors so that each section was contained in a different color container, the other students thought it detracted from the image so now the only thing used to create contrast is the box that contains the text and the color of certain textual items. This image is fairly simplistic visually and the number of words is small thus increasing the usability of the visual for several of the students in the class.
The final image on how to use graphing calculators was created using the CARP model which is part of the action step in the PAT model. The CARP model is an acronym for contrast, alignment, repetition, and proximity. The image shows the four steps necessary to graph an equation on a graphing calculator. Each step gives a brief explanation of the buttons to press and what the window will actually look like on the calculator. Step three is the most complex step since the standard graphing window is twenty units wide and twenty units tall, often times students will need to adjust a window to actually see a graph if the numbers are large in the equation, this skill will be explicitly taught through several activities while we work with the calculators. I think the image will work since it breaks the process down to only four steps (only one which takes a little bit of calculation), as well as the real images of the calculator and calculator screens show only the buttons they will need to press and what the screen should look like when they press the button. Contrast was created by using a different font for the title and for the text for each step. I also made the number of each step larger than the actual text so that the steps would be emphasized and students could separate the steps from one another easily. The text for each step was kept to a minimum due to Lohr's instruction that states, "Students learn better when extraneous material is excluded rather than included." Alignment is demonstrated by all of the text being aligned to the left of the page and the pictures are lined up with one another and centered with each other. I thought this made the image readable and easy to follow as everything goes left to right and down the page, since my last image used mathematical properties and the reading went counterclockwise and created some issues for some people. Repetition was demonstrated by following the same process for each of the steps starting with a number and keeping the text in a confined area and the picture is of the same part of the calculator always found to the right of the text description of the step. Lohr states, "Students learn better when corresponding words and pictures are presented near rather than far from each other." With this image I put the pictures of the calculator screens close to the text and lined up the top of the picture with the top of each line of text, thus demonstrating proximity similar to what P. Chandler and J. Sweller(1991) described in Cognition and Instruction.
The unit was put together in a constructivist manner that matches my teaching philosophy. The unit starts simple by looking at slope and creating a discussion based off of the slope image. Since slope is a foundational piece of knowledge in linear functions, as well as something that several of the students have already encountered, it seemed like the best place to start from and build upon. Then the slope is reiterated before each lesson especially in the next lesson that involves creating equations in slope-intercept form. The slope gives the students a starting point and then the only additional information is to find the y-intercept thus lowering the cognitive load dedicated to grasping new material. It is for this reason why I then chose to discuss the point-slope formula, since it firmly based off of slope and can easily be converted into slope intercept form by methods that the students have already learned. The next lesson is all about graphing and trying to link everything they have learned, it begins by giving the students handouts of the graphic organizer and the multiple representation images, these the students can use as reminders and roadmaps as they have to graph problems and find data points. Since real world problems will be used students need to be able to determine what information goes where so they can generate their equations. This information is also highlighted again in the word problem and graphing image. The unit is capped with the inclusion of using technology by graphing using graphing calculators, in this case the TI-84 plus graphing calculator. I feel once students have demonstrated an ability to graph by hand then they can use the technology since they will understand how the calculator created the graph. As mentioned before the unit is very constructivist in nature, that is why the number of images per lesson increases since the constructed base of knowledge can handle more material as the lessons progress.
Reference
Chandler, P. & Sweller, J. (1991). Cognitive load Theory and the format of instruction. Cognition and Instruction, 8, 293-392
Hansen, M. (1999). Visualization tools for thinking, planning, and problem solving. Cambridge, MA: MIT Press.
Horton, W. K. (1990). The icon book: visual symbols for computer systems and documentation. New York: Wiley.
Lohr, L. (2008). Creating graphics for learning and performance: lessons in visual literacy (2nd ed.). Upper Saddle River, N.J.: Pearson/Merrill/Prentice Hall.
Paas, F., Renkl, A., &Sweeler, J. (2003).Cognitive load Theory and instructional design: Recent developments. Educational Psychologist, 38(1), 1-4
Tufte, E. R. (1990). Envisioning Information. Chesire, CT: Graphic Press