VIEWS: 9 PAGES: 10 POSTED ON: 5/23/2012 Public Domain
Irma Crespo EDD 514 October 2, 2009 Posted by Irma Crespo on Tuesday, September 29th, 2009 12:02 PM Wow! There's so much information out already. I just checked yesterday and there weren't much. I was worried about being redundant from our first discussion but I wrote something about how mathematics came to be. I'll research more on the classroom perspective and peruse more on the other pieces suggested for reading. What is mathematics? In order to understand the obvious impact of modern technology on mathematics education, it is worthwhile to step back in time and find out how far mathematics has gone. As Grandgenett said, posing the question on what the study of mathematics is "is not a simple question to answer." (2008:149) Before the term mathematics was conceived,"mathematics originated with the practical problems of counting and recording numbers." From the ancient period, there were no verbal or written words for numbers. They were intangible ideas relevant to few, enough, many, or too much. Hence, counting was vague until it became concrete from simply knowing one and two to counting with fingers and other objects for tracking. However, in the course of exchanging commodities, the tasks became tedious at a point when goods were innumerable. Representations were needed, thus, tallying was the response to this necessity, which evolved from scratches on posts, knot-tying, to stick drawings. (Burton. 2007:1-2) Fast forward, mathematics literally meant "any subject of instruction or study" from the Greek word mathemata. (Burton. 2007:1) But this denotation does not apply in today's sense anymore. Students in all levels define it according to topics from arithmetic, algebra, geometry, to higher level branches such as calculus. The assigned reading chapter quoted that it is the "science of space and number…the science of patterns." (Grandgenett. 2008:150) With Koehler and Misha's wicked problem, these patterns have evolved to a juncture where mathematics should be realistic and feasible with the exponentially growing digitization. Next segments: The Mathematics TPCK Teacher and What Makes TPCK Work in Mathematics Posted by Irma Crespo on Tuesday, September 29th, 2009 12:50 PM Hi, Midori: You always bring up very interesting points. You made me realize that another view of incorporating technology with pedagogy and content is engagement. Since children already have preconceived ideas that mathematics is difficult upon entering the classroom, why not use technology to stimulate their interest? In EDD 569 (Reading Course), I thought, it's useful to make a lesson plan that will show the concept of Algebra in another perspective after teaching them variables and constants so they can find the relevance of Algebra in their own lives. I chose to integrate reading and music with mathematics while using technology to show them that there is Algebra in the rhythm of music. Whether or not they play instruments, at least, they will appreciate the fact that variables are not only x's and y's, they can be represented in many different ways. I thought, it is TPCK combined with Integration in other subject areas (reading, music, and kinesthetic), which caters to multiple intelligences. If anybody's interested, I have my complete lesson plan with addendums for example that also includes the PowerPoint presentation and labyrinth game I created for interactive participation. 1 Irma Crespo EDD 514 October 2, 2009 Posted by Irma Crespo on Tuesday, September 29th, 2009 1:13 PM Spreadsheets are indispensable productivity tools in the workplace. In our reading, Grandgenett gave an example of a veteran teacher who had her students utilize this application. From this model, I became aware that TPCK can also be a venue to prepare students for the workplace. If I were to follow her approach, I can use spreadsheets in the unit on Formulas and Representations. Given varied shapes and their corresponding formulas: triangles, squares, rectangles, rhombus, cylinders, and circles, among others, they can create rows and columns for the measurements and apply the formulas on the "formula bar," so they can get either the parameters or areas of the shapes. Other instances are word problems where they can key in given information and create their own formulas to generate the answers on the spreadsheet. With probability and statistics, they can create their own graphs or charts to make predictions or determine a pattern. Posted by Irma Crespo on Tuesday, September 29th, 2009 1:25 PM Hi, Cynthia: In the textile business, fractals are already being used to create prints for clothing designs. There are fashion design softwares now that use fractals. What's amazing about these softwares is that the user can choose just about any object and the program will miniaturize it in so many ways repetitively since fractals are fractional geometry. These can be as simple as lines to the elaborate patterns of the peacock feathers. Just an interesting fact to share:) Posted by Irma Crespo on Wednesday, September 30th, 2009 11:54 AM Hi, Midori: I just sent you, Jennifer, and the Math group a zip file of the Algebra Labyrinth Lesson Plan with addendums. Thanks, Irma Posted by Irma Crespo on Wednesday, September 30th, 2009 12:14 PM Hi, Cynthia: There's more about fractals from PBS. I actually watched one of Nova's episodes entitled Hunting the Hidden Dimension. It's available online for viewing: http://www.pbs.org/wgbh/nova/fractals/program.html I hope this will pique your interest further. 2 Irma Crespo EDD 514 October 2, 2009 Posted by Irma Crespo on Wednesday, September 30th, 2009 1:07 PM Why TPACK Works for Mathematics - It can combine with other methods or approaches in teaching and learning (adaptability is TPACK’s strongest point). - It can adjust to multiple learning abilities because its perspective of technology refers to both analog and digital besides being aware of ill structures and wicked problems. - The implementation is not tedious given the proper training and professional workshops. - The duration at which the success of teaching and learning becomes apparent is reasonable. A formative assessment during classes will give the teacher an inkling of how effective his/her TPACK approach is. - It is flexible in preparing students for standardized testing. The curriculum can be modified to fit state or national requirements. - Communicating the mathematics concepts, verbally and in writing, can be adjusted to specific learning needs. For example, technology can provide the sights and sound for the visual and hearing impaired. These are all for now. Do you have other strengths to add on this list? Can you think of any limitations or constraints with TPACK? If so, how do you fix these? Posted by Irma Crespo on Wednesday, September 30th, 2009 2:41 PM Check on this article, which sums up the ideas on integrating technology with content and pedagogy. I believe, this is what TPACK is all about. Promoting Appropriate Uses of Technology in Mathematics Teacher Preparation by Joe Garofalo , Hollylynne Stohl Drier , Suzanne Harper , and Maria A. Timmerman (University of Virginia), Tod Shockey (University of Wisconsin—Stevens Point) http://www.citejournal.org/vol1/iss1/currentissues/mathematics/article1.htm Dr. Duran, I added the pictures on this document. Thank you. Posted by Irma Crespo on Wednesday, September 30th, 2009 4:39 PM For those who do not have time to read the article I recommended: Promoting Appropriate Uses of Technology in Mathematics Teacher Preparation, the following is a summary. I tried to explain each example as simple as possible for those who are not in mathematics. Guidlines on the appropriate use of technology in mathematics: - introduce technology in context - address worthwhile mathematics with appropriate pedagogy - take advantage of technology - connect mathematics topics - incorporate multiple representations” (2) 3 Irma Crespo EDD 514 October 2, 2009 Technology in Context Example: Simulating Freefall With Parametric Quadratic Equations “Derive an expression for the height of an object dropped from 500m above the surface of the Earth, as a function of time.” Tool: Calculator 2 Quadratic expression: y = 500 – 4.9x Student interpretation: outward downward, instead of straight down. 2 Equation set to Parametric: 500 – 4.9T Students clearly recognize the freefall with parametric – straight down. Benefit: With proper use of the calculator, students are able to visualize what the problem asks them to simulate or re-create. Worthwhile Mathematics with Appropriate Pedagogy Example: Exploring the Pythagorean Theorem 2 2 2 Requires students to realize that a, b, and c of a + b = c are side lengths. With the use of the software, Sketchpad, where a right triangle is created, regardless of side 2 lengths, which the students can manipulate, (a x a) + (b x b) = (c x c) = c . The visual aid in this case is the squares created from each side of the triangle that clearly show the lengths of the sides, which are also labeled with their calculated measurements. Benefit: The students are encouraged to manipulate the side lengths of a right triangle and discover themselves that the Pythagorean Theorem is valid. There is exploration through technology for informal proof. Taking Advantage of Technology Example: Exploring Sierpinski Polygons 4 Irma Crespo EDD 514 October 2, 2009 Introduce fractals or fractional geometry to students. The students maneuver/play around with the parameters of the already existing and well-known Sierpinski Polygons, specifically the ratios. Starting from a triangle to the seemingly complex Sierpinski Polygons, which was simply created by repetitively creating triangles using the interior midpoints of each triangle created. The Sierpinski Polygon With the Manipulated Ratio of the Sierpinski Polygon Set at 1.5 Benefit: Technology afforded the students the opportunity to learn functions, ratios, and similar triangles. They are able to imagine and to visualize what happens when shapes are reduced to miniature sizes repetitively: another concept to consider - iteration. Connect Mathematics Topics Example: Connecting Infinite Series and Geometric Constructions Making students understand Infinite Series geometrically, which are supposedly two different topics in mathematics. The task necessitated students to visualize the series: With Sketchpad, an equilateral triangle is first divided into four equal triangles using the internal midpoints while rotating the color scheme clockwise as the students create another set of four equal triangles within a triangle. 5 Irma Crespo EDD 514 October 2, 2009 Benefit: The students become aware of the fractional segments of the series visually and are able 1 to generalize that the series is equal to /3. Certainly, ¼ gets lesser and lesser in the series because the triangles get smaller and smaller. Geometry helped in the understanding of Infinite Series. The other example in this aspect was connecting mathematics with Social Studies wherein the students learn about margin of errors from Presidential Election results that were gathered from the newspapers. A calculator was used to understand this. Incorporate Multiple Representations Example: Exploring Maximum Area “Explore multiple representations involves a popular problem in which students find the maximum area for a rectangular pigpen given a fixed amount of fencing.” Using Microsoft Excel (a spreadsheet), the students key in the parameter given by the problem and play around with the length of a side to find out the maximum. Benefit: The students are able to represent the problem and solve it in multitude ways with technology. They can tweak the parameter or the length of x while still satisfying the requirements of the problem. While there are more examples in the reading, I just took one application for each guideline to exhibit how technology should be used based on these guidelines. 6 Irma Crespo EDD 514 October 2, 2009 Also, I have pictures on my Word document. I tried to copy and paste them so you can see what I'm talking about but it's not working. However, you can check the article and look at them since I shortened and simplified the examples already. Posted by Irma Crespo on Thursday, October 1st, 2009 5:46 PM Hi, Midori: I checked on the sites you provided. Unfortunately, some of the simulators required downloads. My computer is in need of repair. At this time, I am using my sister's laptop. I can't take up so much memory. I don't want her computer to slow down, either. I like Learning by Seeing: Fun Visualization Tools That Educate.It shows what happens when manipulating the different parts of a given formula (ex. What is needed to increase lift or to maximize velocity?) The EggMath is appealing for geometry especially for ellipses while the Famous Curves Applets (this is where I wasn't able to run the applets) is an excellent way to manipulate variables for visualization. I think, this latter application is catered more to students in AP classes or in math undergrad. Most of them were covered, at least, at Calculus 2 level and higher Geometry such as Non-Euclidean. Posted by Irma Crespo on Thursday, October 1st, 2009 6:56 PM Yeah, I was intimidated by the Applet Curves. I asked myself if I really have to go back and review those! When I took them up, I literally "burned the midnight oil!" When I looked back at the MI Content Expectations, the four years of math may include Geometry but not that high of a level. At this stage, which is usually in the junior year, they are still learning how to make proofs. I agree, as teachers, we have to be careful at choosing our sources. We both learned our lessons there. Posted by Irma Crespo on Thursday, October 1st, 2009 6:18 PM Following Dr. Duran's suggested format: Technology in Context Example Content: Quadratic Equations and Parametric Equations Pedagogy: Comparing the Equations by Graphing Technology: Calculator Worthwhile Mathematics with Appropriate Pedagogy Example Content: Pythagorean Theorem Pedagogy: Making Informal Proofs Technology: Sketchpad Taking Advantage of Technology Example 7 Irma Crespo EDD 514 October 2, 2009 Content: Fractals Pedagogy: Re-creating and manipulating the parameters of the Sierpinski Polygon Technology: MicroWorlds Connect Mathematics Topics Example Content: Infinite Series Pedagogy: Geometry Integration - Visualizing Number Series with Geometry Technology: Sketchpad Incorporate Multiple Representations Example Content: Maximum Areas Pedagogy: Representations Technology: Spreadsheet Posted by Irma Crespo on Thursday, October 1st, 2009 6:47 PM Hi, Erica: Integrating technology in mathematics is not only for demonstration and reinforcement. There are numerous reasons to incorporate technology. One of these is keeping the students at par with the changing generation. Certainly, we wouldn't want them using a typewriter when state-of-the-art word processors are widely utilized and are readily available. Should we keep them away from learning if they are unable to attend school when there's video-conferencing at hand? They can be physically absent but virtually present with other students in class using this technology. As math teachers, do we painstakingly ask the students to solve large numbers or graph infinite series by hand when there are calculators or softwares such as Sketchpad? These all boil down to the basic role of technology - make tasks more effective and more efficient. In teaching, it's to make learning more meaningful. Posted by Irma Crespo on Thursday, October 1st, 2009 7:26 PM Hi, Marguerite: Just as Midori and Jennifer indicated there no content standards for integrating other subject areas but there are studies out there on "subject area integration." The plus side about it is preparing the students to think globally. The downside of it is that teachers specialize in certain subject areas and incorporating other curriculum subjects means understanding them, too. For instance, a math teacher who is introducing the concept of functions and relations with the science of mass and gravity also has to get acquainted with the concept of the latter in order to make connections. It's another learning hurdle the teacher has to make. I made a lesson plan on incorporating literature and technology with mathemtics. This is using the Harry Potter series in developing problem solving skills since the books offer a lot of opportunities 8 Irma Crespo EDD 514 October 2, 2009 to think and do mathematics. There's the concept of currency exchange with Harry Potter's galleons, knuts, and sicles or the ideas on proportions with Harry Potter potions, or the geometry of angles in analyzing Harry Potter's Quidditch games, and many more. With these, the students are encouraged to use varied forms of technology to demonstrate the concept of the word problems and their corresponding solutions. Besides the multitude mathematical scenarios found in Harry Potter books, the students can readily connect since it's a best seller and kids love to read them. In my rationale, students think the math, represent the math, communicate the math, apply the math in new situations, and choose the proper technology for the math. My grading tool here is a rubric. Posted by Irma Crespo on Thursday, October 1st, 2009 7:33 PM Correction: "Just as Midori and Jennifer indicated there are no content standards..." TPACK bridged the gap between the literature, the mathematics, and the learning. Posted by Irma Crespo on Thursday, October 1st, 2009 11:22 PM Hi, Midori, Jennifer, and the Math Group: If you're wondering why the fractions are not simplified on the Algebra of Music, it's because simplifying them changes the measures, thus, the tempo. Musicians/composers need to know beats per measure - this determines how fast or slow the music is. Thus, the mood it conveys. This is where students apply Algebra in the proper context. Besides, the main purpose of the lesson is not really the fraction but the representation of variables in Algebra. This is where Koehler and Misha's wicked problem comes in - when content is taught in varied contexts, which provide a more open mind and understanding of applications. Posted by Irma Crespo on Friday, October 2nd, 2009 11:09 AM Also, since the lesson plan is geared for high school students, the Algebra in context can be a tool for critical thinking. What happens to the beats per measure when the fractions are simplified? Why do you think this change occurs? These are older kids and they'll probably find out even before these questions are raised. Posted by Irma Crespo on Friday, October 2nd, 2009 9:06 AM Marguerite. Thanks. It's sad, though. I have so many lesson plans in my portfolio but have not implemented them yet. So, I haven't seen those faces light up (or the opposite) yet. We read all the Harry Potter books. Infact, there's so much potential in literature that there are multitude of opportunities for teachers to create their own word problems. I have also read all the books in the following series: Twilight, Inkheart, Narnia, and Rodin's. There are more to read! Posted by Irma Crespo on Friday, October 2nd, 2009 10:12 AM Here's the link for High School Math Standards: http://www.michigan.gov/documents/Math11-14-open1_142202_7.pdf 9 Irma Crespo EDD 514 October 2, 2009 Participation in Online Discussions Rubric CATEGORY 4 3 2 1 Active Participates actively Participates actively Sometimes Seldom participates participation in discussion. in discussion. participates actively actively in Responds to others' Responds to others' in discussion. discussion. ideas and questions. ideas and questions. Responds to others' Responds to others' Poses thoughtful and Sometimes poses ideas and questions. ideas and questions relevant new questions. Enters Enters discussion in a quick and questions. Enters discussion mid-week late-(Thursday). superficial manner. discussion early. (Wednesday) Enters discussion late (Friday). Makes Connections Makes relevant and Sometimes makes Rarely makes Does not make insightful connections between connections between connections between connections between the text and own the text and own the text and own the text and own current or previous current or previous current or previous current or previous experience, other experience, other experience, other experience, other texts, current events. texts, current events. texts, current events. texts, current events. Asks Questions Poses thoughtful Poses questions. Poses questions. Rarely poses questions. Grapples Tries to answer Answers to thoughtful questions. with these questions. these questions. questions are Answers to Offers an Sometimes offers an superficial. questions are interpretation or interpretation or superficial. analysis of the analysis of the selection. selection. Uses Examples Returns to the text Returns to the text Rarely returns to the Does not return to from Text often to cite specific sometimes to cite text sometimes to the text to cite examples. Provides specific examples. cite specific specific examples. specific evidence Demonstrates examples. Does not that demonstrates familiarity and Demonstrates some demonstrate familiarity and understanding of the familiarity and familiarity and understanding of the assigned reading understanding of the understanding of the assigned reading selection. assigned reading assigned reading selection. selection. selection. Reflects on Practice It is evident that the It is sometimes There is weak There is no evidence student is reflecting evident that the evidence that the that the student is thoughtfully on their student is reflecting student is reflecting reflecting beliefs and teaching thoughtfully on their thoughtfully on their thoughtfully on their practice. Growth is beliefs and teaching beliefs and teaching beliefs and teaching evident. practice. practice. practice. Adapted from http://rubistar.4teachers.org/index.php?screen=ShowRubric&rubric_id=1193320& 10