Dionna Harvell
CEP 805 Reflections
Spring 2012
Unit Index
Dionna Harvell
CEP 805 - Unit 1 Reflection
Reflection 1
Reading my classmates personal profiles I learned that we all come from different walks of life, but have one underlined common denominator and that is our love for math or the love to understand of technology can be incorporated in math. Reading the Principles and Standards for School Mathematics in achieving solid mathematics curricula, being competent, and having knowledgeable teachers who can integrate instructions with assessments in subjects is something I definitely agree with. I plan on using the same methods in my teaching, while incorporating real life experiences to further help students learn the lesson required. The technology will be used to help students with learning important mathematical ideas with understanding how to incorporate technology. It will be used as a supportive tool to challenge their thinking. “Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning.” Calculators and computers are used to assist the learning practices for visually seeing problems, organizing data, analyzing data, and to give accurate output responses. Incorporating technology in today’s math is very important in getting students attention and involved. It can stimulate areas of the brain that could not be reached with just a pencil and paper.
Moving forward looking at the course syllabus I am most interested in using software tools, such as spreadsheets, calculators, SimCalc, and Geometer's Sketchpad. Although I have done calculations using spreadsheets and calculators I would like to learn other ways of using them that I had not used before. Also, using SimCalc and Geometer's Sketchpad, these 2 soft wares I have heard of but never sat down to test out. As far as the units go I am interested in them all. I feel each unit will open my mind to new and fascinating ways of computing mathematical equations using different forms of technology.
When I initially saw the course learning math with technology I knew this was a class I had to take. What a brilliant idea and combines 2 subjects I love most math and technology. I feel very comfortable in the technological pieces of the course and do not find it problematic at all. The mystery to me is all the reading!!! I am ready to jump into calculations and solving linear algebraic problems using my cell phone, or MatLab.
Dionna Harvell
CEP 805
Reflection 2
My views as a teacher versus when I was a math student are very different when it comes to what I learned in mathematics. When I was a student learning mathematics I learned the rules of how to apply different functions of math. In algebra, geometry, calculus I – IV, and statistics I learned the different rules to calculating each problem to get the correct answer. Once I wrote what I thought to be the correct answer I always used the rule of self check with pencil and paper and then again with some form of mathematical program. As an educator I feel my views of learning mathematics should be broadened. So coming into this class I assumed it would be like my past math classes, but add the use of technology to calculate problems. I am learning more about the usage of math and terminology for the way math is learned like in the Skemp article, understanding instructional versus relational learning. Through this understanding I know that as a student I was definitely a relational learner and I needed to understand the why and how mathematics worked. I did not go on the assumption that 1 + 1 = 2 just because.
Some of the active things I have done in the past when I learned the mathematical content is try to apply it. I wanted to test theories to see if there was a failure or a special case. Every time it would come out the same but it was fun trying and testing. In the future to become a better educator I definitely plan on reading more articles to learn what other educators have done to help their students become better math students.
Active learning can be defined as different things to different people. When I think of active learning and how teachers support the views toward mathematics I think of instructions that students engage in. When stating a new math concept students read the rules, write examples, listen and talk over the steps, and reflect by doing the homework, quizzes, and tests. Active learning can also be an instructional atmosphere where teachers do most of the talking and students listen until it is time to ask questions. Teachers can do some active learning strategies such as individual activities, paired activities, informal small groups, and/or cooperative student projects to get students learning mathematics.
One thing in this unit that really sticks out to me is Benny. I guess because he is so young with such strong ideas on why his wrong answers are correct. I feel there is still time to show him the right way and change his views. Whereas Ma and Pa Kettle are pretty much set in there ways and it would be impossible if at all you could change their views on the way they do their math problems. My view on independent learning has greatly changed. I feel there needs to be some kind of interaction with teacher/student until fundamentals are learned. Once students become adults and began independent learning they are more equipped to handle it because they have the fundamentals needed to grow and develop.
Dionna Harvell
CEP 805
Reflection 3
I agree with the goals represented in the PSSM Technology Principle and Number & Operations Standards because if followed correctly it will give students relational understanding they need for each appropriate grade level.
Understand number – I am looking at how PSSM address goals for 9-12 graders in understanding numbers as “ways of representing numbers, relationships among numbers, and number systems.” There a four expectations and goals for students:
Students must have great understanding in understanding the different ways of doing mathematical equations. Large and small numbers, they must have an understanding of rules of decimals for adding and subtraction it is not the same as multiplying and dividing. Same with fractions the rules are different depending on what sign you use. When taking a test do students understand numbers with more than one symbol and how to get the correct answer? Ex. 1*2-5(3x-3) =? Do your students understand what the value for x will be by looking at this equation?
Vectors and matrices are notational conveniences for dealing with systems of linear equations and inequalities. At this level of learning if given this problem on a test could your student figure this out?
This example used on page 291 “the equation 3x = 1 does not have an integer solution but does have a rational-number solution; the equation x3 = 2 does not have a rational-number solution but does have a real-number solution; and the equation x2 + 4 = 0 does not have a real-number solution but does have a complex-number solution.” To understand how each of the equations work 3x=1, x^3 = 2, and x^2 + 4 = 0, students must first understand the definitions. If they do not understand that an integer solution can be a whole number of negative and positive number and not a fraction then x = 1/3 would not satisfy this and a rational number is satisfied because it stems from ratio which is a/b a fraction and is define as any number that can be made by dividing one integer by another.
Understand meaning - looking at how PSSM address goals for 9-12 graders in understanding meaning “of operations and how they relate to one another.” There are three expectations and goals for students
Student should have a good understanding of the steps to arrive to the correct answer rather than just writing the solution. Once a student have a solution they can explain different rules to why the answer is correct or answer different questions “Listening to students explain their reasoning gives teachers insights into the sophistication of their arguments as well as their conceptual understanding.” (p292) Developing an understanding of properties lets the teacher know students understand the rules of mathematical functions and can apply them to any and every equations where needed. Ex. Applying the property of y = mx+b, this is the equation for a straight line. When teachers teach this property students get and understanding that y is how far up the line goes, m is the slope or how steep the line is and with x being multiplied with it will tell the student how far along in the x direction is the slope going. The b is the y intercept; this is the point where the line crosses the y axis. Knowing this if asked to find m knowing the properties and what they are for you can easily find this by manipulating the equation and know the change in y over the change in x will get the answer.
Compute fluently - 9-12 graders in compute fluently “and make reasonable estimates.” There are two expectations and goals for students. At this level students should be aware of how to figure the answers out using mental calculations, long hand methods, and technology.
Since I am not in a school setting as an instructor, it is hard for me to say how these methods are currently being taught. I can just go by past experiences from my math classes and say that I had very good teachers and in grades 9-12, I feel that my teachers followed a curriculum similar to the way the PSSM structure is, based on my knowledge of what I should know at the 9-12 grade levels in math.
In the Adding it Up reading it states that communication about numbers will require some form of external representation. So PSSM’s technology principal “Technology can help students learn mathematics” goes with this because the external representation can be anything from a mathematical program, to a calculator, or a computer. (P25-PSSM)
I would not change nature of that technology because it aligns with my future goals. Technology is a much needed assisted aide that students should learn to enhance their math, not do their math. Over the last 2 weeks my biggest lesson in math was learning the difference between mathematical and computation. First of all when presented with this question I must admit, I thought these were the same, so I felt the need to investigate more and look up the definition.
- Mathematics – The abstract science of number, quantity, and space.
- Numerical Computation – The study of approximation techniques for numerically solving mathematical problems.
Since I am not yet in a classroom I can only discuss these two terms from my own personal views, looking at the definitions and from sites I have searched I have come to the conclusion that there is a difference between the two. Now when I use the term mathematics I think of concept, reasoning, and theory, when I use the term numerical computation I think of matrices of solving problems.
Dionna Harvell
Cep 805
Reflection 4
My reflection for unit 4 in response to algebra, on how I would teach 9-12 graders, if I have a new understanding of algebra over the past 2 weeks, and how technology enhanced algebra are summarized in 3 separate paragraphs. Beginning with my thoughts on what I would teach in my class, since I am not currently teaching. When I do start teaching I would definitely start with PSSM standards and after gaining some experience deviate if needed to tailor the curricula to my students. Next a new understanding of algebra is very interesting because I never knew the importance of algebra until now. I learned it and caught on to it really well in school, but I never connected it with my understanding of patterns. Patterns and seeing designs manipulating shapes I always thought that was more geometry, but after these 2 weeks I can see the relational understanding that connects everything. Lastly, technology in algebra and how it can enhance student learning.
Key mathematical goals at the grade level I would like to teach for 9-12 grades would be for students to demonstrate technology in problem-solving situations for real numbers, absolute value, and scientific notations using physical materials. Students at this grade level should also be able to test, develop, and explain conjectures about properties of number systems and sets of numbers. Other standards students will use are algebraic methods to describe patterns, explore, model, and identify, describe, analyze, and create patterns in numbers, shapes, and data.
A new understanding I gained about algebra over the past two weeks are the different industries that incorporate algebra. Algebra teaches students to recognize patterns and identify functional relationships. Once students learn the relationships of identifying patterns and functional relationships they can apply it to real life situations. I never realized how important algebra was to learning, but through research I have found that algebra is a gateway for most mathematic communications. It operates at an abstract level and insight in all mathematical subjects (statistics, linear algebra, discrete mathematics, and calculus).
Technology can enhance algebra learning in many ways. Technology can strengthen student learning by making it easier to graph equations and solve problems. It also increases the students speed, they can correct themselves, and act as an aid to make relational connections between mathematical problems. For example, spreadsheets used on a computer can perform various calculations and functions to quickly organize, gather, analyze, and compute data. Once all your data is computed you can plot and interpret the data using a graph or chart. Another tool used to compute algebra systems is Mat Lab. This program is used to assist students in completing complicated math problems, while showing you the process to get to the answers to eliminate mistakes. The most used technology for algebra in schools is the graphing calculators. Graphing calculators’ role for student use is to be a computational tool, transformational tool, data collection and analysis tool, visualizing tool, and checking tool. No matter which form of technology used you must understand the relational understanding of algebra to make sure equations are correctly being used with the right rule. So when you input numbers you get the correct output.
Dionna Harvell
CEP 805
Reflection 5.1
Gathered Information from video:
Doc person needed to save Eagle
Emily helper
Larry pilot down from Cumberland Doctor’s Office
Jasper @ Boone’s Meadow with wounded Eagle
UltraLight total weight = 250 lbs
Can land in the field next to Hilda’s
Payload = 220 lbs
Fuel – 30 lbs, with a 5 gal fuel tank
Flying 30 miles takes up 2 gal of fuel, which is a mile every 2 min
Needs a field 100 miles long for lift off
Pilot (Larry) – 180 lbs
Pilot (Emily) - ? lbs
Not enough information to give an accurate reading on Emily’s weight but I would I am sure she weighs less than Larry based on where the needle was on the scale
Cargo box – 10 lbs
Eagle – 15 lbs
Driving = Speed limit 60 mph
Jasper drives 60 miles to Hilda’s, then hikes 18 miles on foot to reach Boone’s Meadow that takes 5 hours
Doctor is in Cumberland and has to get to the eagle Boone’s Meadow
- Emily = weight is unknown but I am guessing about 120 lbs, she weighs less than Larry who is 180 lbs, based on where the needle is on the scale in the video.
- The cargo box is 10lbs
- The Eagle is 15lbs
Emily, Larry, and the doctor can drive to Hilda’s with the supplies needed to give the Eagle immediate help. The UltraLight can be put in the back of the pickup truck. Emily can fly the Ultra light and hover above Jasper with a rope hanging from the plane to reduce payload, Jasper can secure the Eagle inside his bag and tie it to the rope. Emily can then meet Larry and the doctor in the field to tend to the wounded Eagle. From there Emily and the doctor can drive back to Cumberland and Larry can fly back at his leisure.
- Emily, Larry, and the Doctor’s drive from Cumberland to Hilda’s is 60 miles that will take 60 mins driving 60 mph plus 5 mins to load the plane, 5 minutes to unload the plane, and 5 minutes to take the plane off
- Emily’s flight from Hilda’s to the wounded Eagle over Jasper at Boone’s Meadow is 18 miles at 2 mins per mile 36 mins add 5 mins to load Eagle with the payload (less than 180-pilot, 15-Eagle, 30-fuel)
- Emily’s flight from Boone’s Meadow to the field next to Hilda’s is 18 miles at 2 mins per mile 36 mins, 5 mins to unload Eagle and 5 mins to land plane. At this point the Eagle is getting immediate care from doctor.
- 10 mins for the doctor to evaluate the Eagle and sedate it, Emily, the doctor, and the Eagle drive back to Cumberland 60 miles that will take 60 mins driving 60 mph and Larry can fly his UltraLight back at his leisure
Total mins (1 – 75 mins) + (2 – 41 mins) + (3 – 46 mins) + (4 – 70 mins) =
232 mins
Total minutes to get help for the Eagle is 172 mins
Total minutes to get back to Cumberland is 232 mins
Dionna Harvell
CEP 805
Reflection 5.2
Looking at the 12 Jasper series they all solve a math problem of difficulty, algebra, geometry, statistics, and trip planning (general math). These are problems that I see in everyday life situation. Driving from Lansing to Troy to work I had to figure out the fastest route to get home to avoid the worst of rush hour traffic, going through Flint or Brighton. In a classroom I can see problems arising with students thinking about how much time to get from one class to another if they need to stop by a friend’s locker, or go to the restroom. If they have to stay after class to speak with a teacher can they carry all books needed so they don’t have to make any extra stops to beat the tardy bell or can they relocate a book to a friend’s locker that’s on the way to the next class so that they will not be late. These situations are similar to Jasper series of solving basic math problems to reach a certain goal to beat the bell, or conserve time for the student.
Jasper series relates to the current standards for mathematics education in the NCTM’s Principals and Standards. “Success in learning the mathematics of pre-kindergarten to eighth grade usually meant facility in using the computational procedures of arithmetic, with many educators emphasizing the need for skilled performance and others emphasizing the need for students to learn procedures with understanding.” (Brownell, 1935) Elaborating more on this quote from THE STRANDS OF MATHEMATICAL PROFICIENCY to be this is saying students’ success is measured by what educators teach our pre-kindergarten to eight grade students. If they are taught the skills needed to understand and relate computational procedures of arithmetic to real life situations then goals are achieved based on the NCTM’s Principles and Standards. For example, looking at a slice of pie that’s cut in 4 slices, a pre-k may say there are 4 slices and when one is eaten there are 3 after counting all slices. At a more advanced skill level the four slices become ¼ slices or .25 slices and when one slice is eaten, it becomes ¾ or .75 slices of pie left. Taking it one more level of advancement looking at a whole pie cut in 4 slices with one piece missing relating procedures learned you may have ¼ - x = 1, solve for X to see how much pie is left. Success of students to learn procedures with understanding are based on how well the educator is teaching the subject at that appropriate grade level. The Standards for School Mathematics pre-kindergarten through grade 12 presents 10 standards in the Principles and Standards for School Mathematics presents NCTM's proposal for what should be valued in school mathematics education. Within these 10 standards students should learn numbers, algebra, geometry, measurement, and data analysis and probability at different phases based on grade levels.
Jasper series fall within the 10 standards of learning. The grade level is not clear, but depending on how the problem is solved and the math that is used, I believe it could be easily identified which skill level the student has. When solving Rescue at Boone’s Meadow I tried different sequences to figure out the answer. When you have constraints it makes it a little more difficult to solve and limit options. I based my answers on what was most important and getting the Eagle help was most important so I decided after numerous ways of solving and looking for the quickest time that the below answer was the best way.
Emily, Larry, and the doctor can drive to Hilda’s with the supplies needed to give the Eagle immediate help. The Ultra Light can be put in the back of the pickup truck. Emily can fly the Ultra light and hover above Jasper with a rope hanging from the plane to reduce payload, Jasper can secure the Eagle inside his bag and tie it to the rope. Emily can then meet Larry and the doctor in the field to tend to the wounded Eagle. From there Emily and the doctor can drive back to Cumberland and Larry can fly back at his leisure.
1. Emily, Larry, and the Doctor’s drive from Cumberland to Hilda’s is 60 miles that will take 60 mins driving 60 mph plus 5 mins to load the plane, 5 minutes to unload the plane, and 5 minutes to take the plane off
2. Emily’s flight from Hilda’s to the wounded Eagle over Jasper at Boone’s Meadow is 18 miles at 2 mins per mile 36 mins add 5 mins to load Eagle with the payload (less than 180-pilot, 15-Eagle, 30-fuel) see attached
3. Emily’s flight from Boone’s Meadow to the field next to Hilda’s is 18 miles at 2 mins per mile 36 mins, 5 mins to unload Eagle and 5 mins to land plane. At this point the Eagle is getting immediate care from doctor.
4. 10 mins for the doctor to evaluate the Eagle and sedate it, Emily, the doctor, and the Eagle drive back to Cumberland 60 miles that will take 60 mins driving 60 mph and Larry can fly his Ultra Light back at his leisure
Total mins (1 – 75 mins) + (2 – 41 mins) + (3 – 46 mins) + (4 – 70 mins) =
232 mins
Total minutes to get help for the Eagle is 172 mins
Total minutes to get back to Cumberland is 232 mins