Transcript Professional Portfolio
Professional Portfolio
by Marcia Stahl For ECOMP6102 Roswell, New Mexico
Table of Contents
Biography
Professional Literature
Goals
Module 2-2
Module 3-1
Module 4-2
Philosophy
Reflections
Module 1-4
Module 2-3
Module 3-2
Module 5-1
“Let’s play school!” “I want to be a teacher when I grow up!” Those are two statements you would have heard very often from me as I was growing up. I knew as a child that I wanted to be a teacher. My first attempt at a six- paragraph essay in 6 th grade entitled “My Life’s Work” said I wanted to be a teacher. That desired never went away. I graduated from college at 20 and started teaching at a small Christian school in Pomona, California. My teaching experience there, however, was not a good one. The school did not provide a curriculum to teach from. Being brand new and just out of college, I needed guidance and received none. I had to figure out for myself what needed to be taught to these very precious children and how to do it. I was not up to the task. I taught there a couple years and quit very disillusioned, saying I would never teach again. I got married, had four children, and stayed home to raise them. My children, their growth and education became a high priority for me. I worked with them at home and volunteered at their school. When they were in preschool, I worked at their preschool. I was the children’s director at church, just to be involved in their growth. Finally, one day someone said, “You need to go back to teaching!” My response was an emphatic – “No!” It took me about a year of mulling it over before I finally said – “Yes”. By that time it had been several years since I had quit teachinh so I needed to go back to college and get recertified. That has been the best decision I’ve made. I’ve now been teaching for 13 years. I can’t say every minute has been wonderful, but I would not
consider doing anything else. Nothing compares with watching a child who has been struggling to suddenly beam with understanding. Teaching math to 7 th and 8 th grades carries with it many challenges which I enjoy immensely. Middle school students are more interested in what their peers think and the opposite sex more than almost anything else. The challenge starts with how to make them want to listen to you and be interested in what you are trying to teach them. They are so much fun. They are just beginning to think like adults and searching for their own personal “way to go”. Presently, I am the department head for the Math Department at Mesa Middle School. I teach Algebra and Pre Algebra. I spend about a half hour a day in the school’s behavior disordered classroom teaching the students there. I teach a class on teen leadership and character choices besides being the building mentor to new teachers to our building. My job at Mesa is not to just teach the students in my classes to solve math problems. I’m concerned about the whole student. I want them to know their multiplication table, sure, but I want much more for them. My dream for them is that they will want to be continual learners and enjoy it.
Philosophy Education should be a process of exploration. That is my basic philosophy in one sentence. I find myself fighting with it. It is so much easier to get the facts I think the students should know and understand, put them all together on a platter, and dish it out to them. I know, though, that unless my student discover for themselves those facts will be meaningless. Knowledge can not be just spoon feed to be remembered and meaningful. One of the most hated statements heard by teachers from students is that “This is boring!” That same statement has given me direction in my approach to teaching. Our students today expect everything to be exciting and stimulating every minute. That is a huge goal for a lesson in math. “How can I make it exciting and stimulating?” I tell my students that math is just solving a puzzle. It is just a game and like all games, there are rules to the game. The rules to the game are basic rules of numerical relationships. If we put the basic rules of the game in our memo ries, the rest is just solving puzzles. Everyone understands that all games have rules that need to be followed. In football, breaking the rules results in a flag on the play. In math, if the basic rules are not followed the answer to the equation is incorrect. I am more interested in my students knowing how to find the answer than the actual answer. Technology is a wonderful tool that helps me make my students more interested and involved with their learning. I have a laptop connected to a projector and s mart board in my classroom. Using the smart board is so much fun for them. Not looking dumb to their peers is so important to them; they try very hard to know how to solve the equation
before they get up there. They all want to be up there, so they learn the basics and have fun solving the equations while using the equipment. The process for grading or assessing my students work has changed dramatically in the last couple of years. If you look at just the answer to a math problem or equation, it is either right or wrong. When you dealing with a problem using the basic operations, such as multiplication or division problems, there is only one right answer. When problem solving or reasoning is involved, the process also needs to be examined. If the process for solving can be followed to a logical conclusion, the answer is not entirely wrong. The process needs to be assessed as well as the answer. The grading process takes longer to complete, but a much clear picture of the students ability and understanding is reached. It is like using a rubric to grade each problem. I also find myself putting much more emphasis on what they do in class. I still have not been able to let go of homework. Math is a subject that I feel requires considerable practice. I have incorporated much more practice into class time, but I still need to know they are practice independently as well. There seems to be a direct correlation between the test grades of those stu dents that do their homework and those that do not. Learning should be a life journey. I want my students to take the right things along on that journey. There are many basic facts they need to have in their luggage. Most of all, they need to know where they are going and what they hope to find there. Their basic knowledge should show them the way. Their journey, research and exploration, should help them find what they are looking.
Reflections on Professional Literature and Research. I.
Multiple Intelligences
by Howard Gardner Since the beginning of my studies in education, the subject of multiple intelligences and learning style had been topic brought up numerous instructors and fellow teachers. I went away knowing that each of us has different interests which contribute to what we enjoy doing and how we learn. The real significance did not become apparent to me until just recently. My final project for the first class in our master’s program was a paper on both multiple intelligences and learning styles. I, first, researched Gardner’s Theory of Multiple intelligences. I used several websites to glean information. I, also, took several different intelligence tests to compare the results of each. Amazingly, the rests of each were pretty much the same. The hardest part was answering the questions honestly and not how I wanted the answers to come out. After researching the intelligences, I became aware of many things about my students that were not nearly as important to me before. Next, I researched the different learning styles. There was a significant correlation between the intelligence of the child and their learning style. The writing of this paper has made a significant change in ho w I teach and my expectations from my students. It has definitely helped me developing appropriate accommodations for corresponding student. II.
The Teaching for Understanding Guide
by Tina Blythe Tina Blythe’s book,
The Teaching for Understanding Guide,
made me look at how I teach and how I should teach. Her book gives a whole framework for
understanding and how to teach to it. The most important thing I think I learned was to let go of my hold on curriculum and the textbook. I needed to know what the overall goals for the course were, decide on how I was going to accomplish it, and finally a ssess it. To accomplish this, I was surprised to realize I had to work backwards – from what I was going to assess forward. After reading and studying Blythe’s book, curriculum mapping made much more sense. I feel a much greater ownership of what I’m teaching and I feel I can defend what I’m teaching much better. My philosophy on assessment changed a great deal as I studied this book. It helped me identify the skills, concepts, and processes that I felt were essential for my students to learn and understand. Now that is what I look for when I grade their work. Before this time, I would grade the papers in math and it was either right or wrong. Now I check for understanding. It is easy to make a simple computation error and get the answer wrong. But if I can tell from examination of their process, that this student understands how to solve the problem, then they do understand and should receive credit for that understanding. The student has demonstrated his understanding and that’s what is important. I.
Student-Involved Assessment for Learning
by Richard Stiggins Our textbook,
Student-Involved Assessment for Learning
by Richard Stiggins, gave me a much broader understanding of assessment and learning. First, the book made me think about what is it that is important for my students to actually know. What little bits of knowledge need to be in every student in my heads that is essential for them to
know? What is the difference between essential facts and skills and those things that can be researched to find the information when needed? What is just a personal bias? A closer look at the standards to which we teach helps us stay on track and teach tho se things that are essential to know. This textbook, also, made me look at reasoning. I realized I have too much of a deductive reasoning approach to my teaching. I teach too often from the top down. I need to teach more from the bottom up, so that my students take greater possession of their learning. There are some things like the rules for exponents and roots that have to be given as notes and accept as fact, but the exploration of those facts to see why this works as it does gives the students opportunity for discovery. Bloom’s taxonomy has never made as much sense as after this class. The charts of verbs and question starters for each level of Bloom’s to lead discussions and writing lesson plans has always been an essential. Now, however, I learned to I need to start with using it to develop my assessments and then my activities to guide my students to the appropriate end. It proves again that we need to know where the end of the journey for the course should be, then plan the road to get there. Stiggins’ book and this class made me look at how I’m to do my job in a much different light. In the past, I have followed the textbook provided and matched the standards to it. I am now looking at the standards as targets to which I am teaching and deciding how best to assess those targets. Planning how I going to get there is coming last in the planning process. I’ve also learned that I was really looking at what the standards said. I have learned to analyze what the standard is actually asking to me to teach. This
class taught to analyze and synthesize what I am doing and gave me direction in how to teach my students to do the same thing. The education community has become as involved in assessment the last few years. We have given standardized test for years, but emphasis on high achievement has never been greater. This course and textbook has helped to better understand the assessments. Just what are they telling us? How can we use this information and how is it shaping education? If as teachers, we know what is going to be assessed, and how, then we can shape the curriculum to meet those needs.
I.
Reflections Reflection on Module 4-2 and 4-3 – Creating a Table of Specifications Creating the table for specifications made me look at the subject I teach in a totally different light. At first I couldn’t figure out how to make anything other than Knowledge based questions. I teach Math – That’s all about definite facts. Then I started to think about how to my students demonstrate their understanding. Knowledge questions only showed basic facts and skills they had retained in the memories. I had to asked my students to demonstrate understanding by applying what they knew. I realize the only way to actually teach my students to use reasoning was to give them something they had to use reasoning to figure out the answer. Consequently, I had to give them problems that used the higher levels to Bloom’s Taxonomy. Being a believer in the task did not make it any easier. I tell my students that Math is just solving puzzles and I looked at creating the problems the same way. I figured out what my targets for learning and assessment were and then developed the problems to fit. I struggled with this assignment but I, also, learned so much. If I were asked to evaluate this assignment, I would say – “It was hard, but a must! I learned so much.” I.
Reflection on Module 2-3 – Breaking Apart a Standard Breaking apart a Standard was an eye opener. I don’t think I had actually analyzed a standard in depth before. I realized I wasn’t thinking about what my students had to know to meet the standard. I wanted the students to just produce something to
show me what they had learned. I was missing the fact that I, first, had to teach them how to produce something. Once they could produce something, they could use that skill to demonstrate their understanding of a particular topic. This module made me analyze what the difference is there between what my students need to know and what kinds of skills they need to have. Just as important though, is what kinds to reasoning powers do they need to have and how I can lead them in the direction to different kinds of reasoning? This particular standard required they evaluate, compare/contrast, and synthesize the technology tools available to them that would best suit their purpose. I realized that I have to synthesize each standard I teach. A more in depth look at each standard is required if I’m going to teach to them adequately and adequately is not enough. I have grown in my understanding of how I am to look at a standard before I even begin to develop a lesson to teach to it. I.
Reflection on Module 5-1 – Reasoning Essay Deductive and Inductive reasoning were just two different kinds of reasoning before starting this paper. Deducing sounded like something a detective did, so I knew it had something to do with investigation. The only thing I was right about was the investigation part. Reasoning does require investigation and that is exactly what I want my students to be doing. In the process of writing my paper on reasoning I learned the difference between deductive and inductive reasoning. An understanding that in inductive reasoning, you
start with valid premises to form a logical conclusion. In deductive reasoning you start with valid theories and prove their validity.
Sparks Notes Study Guide
helped me a great deal in understanding the difference between the two. I realized in deductive reasoning, you started with a hypothesis and tried to prove it. In inductive reasoning, you start with observations to reach a logical conclusion. Also the idea that in deductive you work from the top down and in inductive you work from the bottom up. I realize that in teaching math there are certain rules that have to be followed. My students taking notes and learning those basic rules are the deductive part of my teaching. Inductive reasoning comes in when they can take what they know to be true to disco ver other things that are also true.
Future Goals During of the course of this class, I have become intrigued with how the brain and the three systems work together. Discovering the desire to learn comes from the brain stem made some things very clear to me. I have to instill desire in my students. No matter how well I know my subject matter, I have to stimulate their interest. The use of technology will provide assistance with this. Students enjoy technology so if I can incorporate as much as possible into my teaching, I will capture more o f their attention. I am only intrigued by the different presentation tools for teaching, such as the smart board, but also the assessment technology tools available. I would like to learn more about them, try them out, and hopefully convince the school to purchase one. From what I have observed, it gives the opportunity for immediate feedback to both teacher and student. It helps me to know what needs to be re-taught and the student an understanding of what he needs to investigated further. I find the technologies that are know available to us in the classroom exciting. No matter what the learning style of a particular student is, technology can be used to enhance it. Many of our students with various disabilities would have much more difficulty if it were not for the opportunities provided by technology. Many of our teachers are still in the learning stages themselves when it comes to use of technology in the classroom. I enjoy helping them with learning the use of technology in the classroom and help them over their fear of something new. I tried to incorporate as much as I can into my teaching and hope to help others to do the same. I
want to continue to learn more. I have developed a new curiosity about omething new and interesting to learn about and use in teaching. I enjoy being in the classroom, but if at some point there would be a position available that would give me the opportunity to assist other teachers in learning not only how to use technologies that available to them, but also integrating it into the school curriculum I would be very pleased. I take on many opportunities in that direction now and would like to continue. I think a position that assists teachers with integrating technology into their classroom would be fun.
USERS AND USES FOR ASSESSMENT
Users and Uses – Who uses assessment information about Which Level of Assessment students? What do they use it for?
User Decisions Each User Might Make
does each user depend upon most?
Large-Scale Assessments Classroom Assessments Student Parent Teacher A student might decide to use the assessment to help him/her to know what he needs to be able to do. Students who feel in control of their own chances for success are more likely to care and to strive for excellence. Parents would use assessments as a form of measurement to see what their child is capable of and to know what their child’s targets are for success. Parents would also use assessment information to assess the teacher/program. Teachers use this information to define achievement standards. Teachers use information to meet content standards and validity. Teacher using this information will set clear and appropriate targets. X X X X X X X X X X
Principal School District The principal would be concerned whether the assessment met content standards. The principal would be concerned whether the teachers were highly qualified in the content area. The principal would be concerned whether the curriculum continuously progressed across grade levels or content areas. Districts use assessment information to analyze state and local standards, and compare with their local written curriculum. Districts use assessment to evaluate the local program. Districts would also use this information for professional development. X X X X X X X X X
Module 2.2: What’s Worth Assessing? Kim Kunko, Marcia Stahl, Marabeth Fields Revised by Marcia Stahl 12-4-05 1.
What advice would you give the "critical friends" review team? How would you change the sample test items?
The team should review the state Standards before they begin writing their assessment as this may eliminate much of the personal bias. For example, the Standard might be “how to use reference materials.” The activity to support this Standard would be to explain how/where to find the answer to such specific questions. The specifics of the seal might be an activity that meets the targeted Standard and can be transferred to other learning instead of being the end result as the initial team suggested. It seems that some of the test writers want to have very specific knowledge of the subject. Knowing material also means you know how to find how or where to find the answers. An overall knowledge of the major points would lead the student to find specific information. 2.
What are the curriculum issues that need to be raised before the critical review is complete?
Again, Standards must be the issue and how curriculum supports the Standards being tested. The team needs to take a more global approach to assessment items rather than such specific items. We have a tendency to be too specific. We need to be sure that our curriculum covers relevant broad topics. Then take them apart to be more specific, being careful not to be too nit picking. Leave some information for them to find when needed as in research projects, not in a general knowledge test.
3.
Can you think of an instance - as a teacher, parent, or student - when you took a test that included items you felt weren't worth learning or assessing? How did you react?
Frustration, the sense of wasted time and effort for something that would not benefit real life issues, and desire for more appropriate materials were the emotions felt as a result of poor assessment. I would, first, think that the review team has not done their job. The collaborative team should have made the determination of whether the items were worth
learning or knowing. Were they too specific or too personally biased? Second, I would also wonder if I had to rethink what I thought was worth knowing? Did I go off on a tangent because of my personally bias? I would really think about the material and reassess my understanding of what was important and worth knowing.
1.
In your own teaching, how are decisions made about what is worth learning, teaching and assessing?
The first step we use in our district is for teams of teachers to develop curriculum mapping based on state Standards. We next develop EPSS (Educational Plan for Student Success) which target the deficit areas indicated by state standardized test results. We use these curriculum mapping and EPSS goals to determine what will be assessed.
Module 2-3 Break Apart a Content Standard Dude Burrola, Marcia Stahl, Sabrina Sutherland, Ginger Towle, Lori Hooper 11/5/05
ISTE Standard 5 – grades 6-8 – “apply productivity/multimedia tools and peripherals to support personal productivity, group collabo ration, and learning throughout the curriculum. 1. What do the students need to know to meet this standard? In order for students to apply productivity/multimedia tools and peripherals, they need to comprehend the material they are presenting. They need to know what productivity/ multimedia tools and peripherals are. They need be able how to work in a group and how to collaborate with a group. They need to know how to evaluate themselves. They need to know how to put all this knowledge together in whatever curricular area they are studying to show their own personal growth by the use of these technologies. First, I would have to teach me the use of the different multimedia tools we have available to us at out school. Numerous opportunists for cooperative projects would have to be included in my lesson plans that give them confidence in working with other people and how to work with other people. Since I teach math, I would have to give opportunity for a variety of activities to give multi- curricular experiences in my classroom. W e would have to look at things scientifically, through investigation and reading and writing about the concept. The standard wants me to teach technology in whatever we subject we are teaching. So I would have teach them to present what they have learned through technology. When teaching to a standard, you need to FIRST take a very close look at what the standard says, and then consider what you would have to do to teach and assess that content. What would you need to teach students in order for them to use multimedia and productivity tools? Students also need a competent understanding of the subject matter and how to use appropriate resources. What is the subject matter in this standard? What do students need to know in order to USE PRODUCTIVITY TOOLS or COLLABORATE, or use educational software to learn a subject? Focus here on the Tech knowledge students would need. First students need to be presented with a clear objective or purpose for learning as well as a collaborative rubric, which will be used for their assessment. Next they need to delegate roles to each member of the group to facilitate productive collaboration and quality research. Finally, the students need to evaluate the product towards the rubric and determine the quality of their learning. These look like good teaching strategies, but I am asking you to think about WHAT you will be teaching rather than how you will teach it. This type of teaching tool could enhance
the learning by incorporating multi-sensory strategies so that the learner could confidently express their knowledge with a final product. It is very good to consider learners needs, but that is not part of breaking down a standard into the knowledge, reasoning, skills, products and dispositions you would need to teach in order to say that your students had ACHIEVED ALL THE SKILLS LISTED IN THIS STANDARD. 2 .What patterns of reasoning must students is able to apply? They must be able to use inductive and deductive reasoning skills such as observing, hypothesizing, synthesizing, comparing, contrasting, and analyzing the information in order to create a product that demonstrates their understanding of the knowledge obtained. First the students need to dissect the project into specific tasks. After each student in the group has analyzed the information they acquired, they need to synthesize as a group to determine the validity of their personal findings so that they may realign their hypothesis if necessary. The students need to be able to evaluate the different kinds of technology/peripherals/software available to them. This evaluation process could use comparative reasoning to determine similarities and differences of what is available. They could synthesize or categorize the information to figure out what is best suited for their situation. Complex comparison depends on clear understanding which requires analysis. All of the patterns of reasoning should be used to see the clear picture of what they need to do and use.
Good break down of reasoning skills that students would need to use to well when creating a product using tech tools. However, you should also include the types of reasoning students would need in order to select the most appropriate tech tool for each learning task. For example, students would have to compare and contrast tools in order to choose the right technology tools for the learning task they are doing. What other types of reasoning about technology would they have to do? ALSO what type s of reasoning would have to be done for them to use technology for collaboration? 3. What skills do students need? They need to know basic computer skills such as keyboarding, Office programs, research techniques, Internet usage, operation of multimedi a tools such as Smart Boards, Video cameras, Audio equipment, etc. They also need cooperative learning skills such as delegation of tasks, disposition strategies, listening and participation. Finally they need basic language arts skills. Nice work!
4. What products should they be able to create? They can create many different types of products such as PowerPoint presentations, graphs, video, web pages, spreadsheets, web quests, scavenger hunts, and written reports.
Yes
5. What dispositions will students need to develop to meet this standard? The students need cooperative learning strategies. They also need the ability to pay attention and stay focused, as well as be able to concentrate and to use appropriate study skills
.
Yes
Paul Lessard Steve Nunez Marcia Stahl Dude Burrola Module 3.1 B. Understand and use the
Five Standards of Assessment Quality
below. c. Accurate assessment uses appropriate methods given the context To what extent was this student successful or unsuccessful on this Math Performance Assessment? Please explain why the student was successful or unsuccessful in your opinion.
His answer is incorrect, because he does not give a specific answer or explains his answer properly. He knows the division process with some errors, but he is frustrated with the problem and guesses with an answer. Is the test question written at a level that the student can read?
Can you always tell from a performance assessment what went wrong if the student is not successful? Explain your answer in some detail.
No, we must see all of the student’s work in order to tell what the student was thinking in the process of finding the answer to the math problem. In a performance assessment there could be outside influences that affect his performance.
Would it be of value to use selected response or essay assessments in an situation such as the one presented in this case study? Explain your response in some detail.
The value of selected response will allow the student to explore each option and try to find the answer. Each answer could be used to solve the equation for its validity. A frustrated student however would just guess at a response. The value of an essay would allow the student to be able to explain the answer or tell the teacher how he would have come up with the answer rather than guessing at a selected response. He may be able to figure out the correct answer at his novel approach and be able to explain what he did. This student would have been able to answer in an essay assessment.
How would you change the performance task to ensure that the assessment was valid and reliable? (Validity and reliability are discussed in your text on page 14 and again on page 20).
The question is valid but to increase the reliability an explanation of his thought process needs to be included. The question needs to ask for an explanation of how he arrived at the answer. The assessment may
may not correctly represent his understanding of the concept. He is also deficient in his basic math skills. A possible change could be to make it a 2 part question instead of one. First, portion being multiple choice with answers to explore, and the second portion, an essay question asking for a written explanation or visual showing how the solution was arrived at. An assessment of his reading ability may also show that he was unable to comprehend the problem. Word problems require comprehension, word recognition, and problem solving(reasoning) ability.
Barriers to Using High Quality Classroom Assessments Marcia Stahl, Sabrina Sutherland, Melissa Hartley, Sandra Medina, Lori Hooper 11/5/05 Revision by Marcia Stahl 11/6/05 B. Understand and use the
Five Standards of Assessment Quality
below. e. With the assistance of technology, translate evaluations of student work into records that accurately convey the level of student achievement to students, parents or guardians, and school personnel. Making use of all the information you have learned about assessment might prove difficult unless you identify barriers to using high quality classroom assessments that exist in your classroom and in your school. Using a word processor, create a table like the one below. Then, list all barriers to using high quality assessments in your own teaching situation. Once you have brainstormed all the barriers that exist, try to determine what you might do to eliminate each of the barriers. Barriers Possible Solutions to Barriers 1) Time – Instruction Preparation Administration time for the test and give the teachers Fridays for prep. This 2) Money for necessary, quality, equipment 3) Appropriate facilities, and qualified personnel (especially for will only work if there is data to support this “out of the box” idea. Some schools have convinced their district to allow them to “pilot” this concept and review its success (compared to the other schools in the same district on a traditional schedule) to see if it provides higher success for its student body. To overcome preparation time limitations, assessment could be a collaborative effort. 2) Grant writing, capitol outlay from State special needs learners) . Representatives 3) Community involvement, incentives for highly qualified personnel 4) Vague, unaligned, and ever 4) State adopted curriculum that is pre-aligned to Standards and Benchmarks as well as the
4) Vague, unaligned, and ever changing Standards and Benchmarks in relation to the CRT 5) Teacher’s inability to determine the appropriate assessment type and/or inability to objectively grade the assessment (not valid or reliable) 6) Student’s motivation to perform at their “true” ability, diversity of ability levels and test taking skills 7) Parent’s support of the assessments. 4) State adopted curriculum that is pre-aligned to Standards and Benchmarks as well as the State Assessment. I believe this is the ONLY way to make high stakes assessment valuable, if, in fact, it is to be used to drive curriculum. I would change the order and have the Curriculum Mapping drive the State Standards, which then drives the high stakes assessment. I believe they have it backwards AND unaligned. Alignment, as well as immediate feedback, to allow it to be appropriately used .If assessment is going to be developed before curriculum, we need to have the overarching goals well established. 5) Professional development, district assessment team, 6) A solution to this barrier would be have a lot of test taking practice Have activities that promote success and increase their desire to continue to do well. The students need to understand how the assessments affect them and have incentives in order to “buy into” giving teachers their authentic ability performance. COMMUNICATION and ACCOUNTABILITY. The more the practice resulting in success, the more the student will continue to be motivated to succeed. 7) The same solution as stated for the student’s. It has to be clear how it affects them and why they should support the teacher’s assessment results. COMMUNICATION and ACCOUNTABILITY. The parents need to be involved in the process. More exposure and experience will foster more understanding. With their support, barriers with their children will also diminish.
Module 4-2 Create a Table of Specifications for a Selected Response Assessment Marcia Stahl
Strand: Algebra Standard: Students will understand algebraic concept and applications. 5-8 Benchmark: Analyze changes in various contexts Grade 7 Performance Standard 1. Use variable and appropriate operations to write an expression, and equation, and /or inequality that represent a verbal description involving change. Content Equivalent equation with addition and subtraction Knowledge 2 Compare/Contrast 2 Application 2 Total 6 Equivalent equations with multiplication and division Equivalent equations with fractions and reciprocals 2 4 2 2 2 2 6 8 Total Reflection 8 6 6 20 Students need to understand that both sides of an equation ha ve to remain equal no matter what operation is involved. Two basic rules of beginning algebra say – 1. Do the same thing to both sides of an equation, and 2. Do the opposite of the original. If students understand from the beginning the essential of these two rules, they will save themselves errors in equations in the future. For that reason, proof they know how to solve the equation or
Knowledge
of the process is important. Plenty of practice is essential. It should only take a class period each to teach t he basic operations in simple one step equations. Mastery of them will take several days of continued practice. The fractions and reciprocal portion will take several additional practice times.
Compare/Contrast
will take their understanding of the skill to the level of being able to see and understand the differences and/or similarities between the different operations. Hopefully this will lead them to the understanding the addition and subtraction symbols really represent positive and negative integers. Their understanding will make progressing to more complex equations much simpler.
Students need to be able to transfer their understanding of numbers into the “real” world, thus
Application.
If I can write word problems in accordance with the different operations, that will show the students understanding of this basic beginning step, I have been successful and in turn they are successful. Fractions and reciprocals seem to scare my students. Additional questions in this area will give them additional practice to build confidence. Reciprocals, in particular, cause stress initially. Much more time will be spent in teaching and practice of this concept. The more practice accompanied with success will hopefully reduce that fear. Since fractions and reciprocals seem to be the most difficult, more assessment seems to be necessary. Excellent work – I am impressed with your knowledge and the way you have structured this table. 1. X + 12 = 36, X = __________ 2. If X – 4 = Y+2, does Y = X-6? True or False 3 3x + 6 = 18, is the same as x + 6 = 6 True or Fa lse 4. Write an equation showing the sum of the number and 6 equals 24. 5.
4X = 48 1.) 192 2.) 3.) 4.) 4/48 12 48/4
x
6. How is a. = 24 solved differently than b. 3x = 24? 3 1.) a is divided, while b is multiplied. 2.) a is added, while b is subtracted. 3.) a is multiplied, while b is subtracted. 4.) a is subtracted, while b is added. 5 5 7. Is of what number is 216 similar to 216 is of what number? 8 8 5 5 1.) No, of 216 is not the same as of an Unknown. 8 8 2.) Yes, One problem is just written in reverse order of the other. 3.) Yes, It is impossible to answer either problem. 4.) Yes, Each has a similar answer
8. Matching ____2x + 4 = 12 ____ 2
x
2 4 = 12 ____2 2
x
4 = 12 ____2 2 2
x
4 2 = 12 a. multiply, subtract, and divide b. Multiply, add, and divide c. Subtract and divide d. Multiply, subtract, divide e.
Multiply, add, multiply, subtract, and divide 9. 1 4 1 2 1 8 = 1.) 1 64 2.) 4 4 3.) 1 4.) 8 8 10. Fill in the blank ____ 2 ¼ 1 1 8 = 7 Essay Questions 11.
Write and algebraic expression that tells 3 times the sum of ½ the number and negative 2. After writing the expression, explain why you have written the expression the way you did. 12.
Explain what the two basic rules for solving an algebraic equation are. Explain why it is necessary to use them exactly as they are stated. 13.
Give a complete definition for a reciprocal and why this understanding is important in solving an expression that contains fractions. Explain fully with an example. Be sure to include in your explanation the distributive property.
A Definition and Explanation Of Reasoning And Patterns of Reasoning As children grow up, we want them to be able to make good choices, but before they can do that we have to teach them how. Teaching the various patterns of reasoning should help them become better thinkers, with a broader scope of reasoning powers, which finally should make them competent problem solvers. Reasoning seems to be broken down into two basic categories, but numerous different patterns. Authors seem to differ on how they break the patterns down but all seem to agree that the two basic categories for reasoning are inductive and deductive. Don’t use the word “seem” so much. Jeanne Curran gives this explanation of reasoning, - “The best example I've ever found for remembering inductive and deductive reasoning This is a definition of Inductive and Deductive reasoning, rather than a general definition that could be applied to all the different types of reasoning. is teaching French to little children. If you start with the pieces and move to the whole, you are using inductive reasoning. If you start from the whole, and define it by its parts, you're using deductive reasoning.”(Curran, 2005) Ms. Curran says that if you're in a hurry, deductive reasoning may work the best.
However, if you have the time and understanding, the use of both inductive and deductive reasoning gives the student more opportunity for discovery and greater understanding. The more you use both, the more comfortable you will become. Inductive reasoning means to change our students into processors of information to form their own conclusion. “It is the form or structure of an inductive argument that has little to do with its perceived believability or credibility, apart from making the argument seem more clear or more well-organized. The receiver determines the worth of an inductive argument.” (Inductive and Deductive Reasoning) Inductive logic gives the student some true premises; they use some valid reasoning and reach a sound conclusion, not necessarily true. Inductive reasoning gives our students facts. They then have to sort these facts so they can reach the desired conclusion. The inductive process changes my “job”. First, I have to arouse their curiosity and motivate them to want to figure out the premise. I teach mathematics and that is an exact science, so I have to guide them to an exact conclusion. If they are going in the wrong direction, my job as facilitator is to get them back on track in the right direction by helping them figure out they ar e going the wrong way and get them turned around. Science (math) is also about data collection and organization of that data. The students have to be able to work together, and organize their information together into something meaningful. The whole process from beginning to end is about solving the problem and resulting with a discovery with understanding. My job is to keep them going, no matter how windy the path, to the correct conclusion. Deductive reasoning is so much easier Good Deductive reasoning is NOT easier – it is different, but it does not put as much discovery in the students’ hands. In deductive reasoning, the beginning premise is just as valid as the conclusion because the conclusion
is built in the premise. “It is the form or structure of a deductive argument that determines its validity…. The argument is that if the premises are true, then the conclusion necessarily follows. The conclusion is … contained in the premises.” ((Inductive and Deductive Reasoning) Deductive reasoning is never just partially right. It is either all right or all wrong. Deductive reasoning basically gives the student the instruction manual and tells them step by step how to put the object together. There is much that is taught in our classrooms that has to be taught in just this way. I have to give them the step by step process for finding the surface area of an object. I can take apart a simple object to show them why this process works. Then their job is to take apart a larger object and find its surface area. Some problems, especially in math, seem too complex without a direction to go. Math builds on itself. What I am hoping to illict in my students is that by using the basic facts they have learned and the basic steps in the discovery process I’ve given them, they can discover the bigger idea. The Sparks Notes Study Guide explains deductive reason as “working from the more general to the more specific. Sometimes this is informally called a "top-down" approach. We might begin with thinking up a
theory
about our topic of interest. We then narrow that down into more specific Graphics borrowed from Sparks Notes Study Guide, part 1
hypotheses
that we can test. We narrow down even further when
we collect
observations
to address the hypotheses. This ultimately leads us to be able to test the hypotheses with specific data -- a
confirmation
(or not) of our original theories. Inductive reasoning works the other way, moving from specific observations to broader generalizations and theories. Informally, we sometimes call this a "bottom up" approach In Graphics borrowed from Sparks Notes Study Guide, part 1 inductive reasoning, we begin with specific observations and measures, begin to detect patterns and regularities, formulate some tentative hypotheses that we can explore, and finally end up developing some general conclusions or theories .” (Sparks Notes p. 1) EXCELLENT examples. You have developed a thorough understanding of Deduction and Induction. Stiggins made the statement, “reasoning patterns are rarely used independently of one another.” (Stiggins et al, 2005). I agree with that statement because it seems that no matter with kind of approach I try to have in a lesson, the other seems to creep its way into the students thinking and mine. In analytical reasoning, we are saying that an object is the sum of its parts and figure out what the parts are. We take the object apart and figure out what makes it work. It is like taking a clock apart to see what makes it tick and why. Analytical reasoning requires that we use our prior knowledge to build on. As in finding surface area, we
already know that multiplying the length and the width gives me the number of little squares of a particular size inside that object. In synthesizing a problem, we work in the opposite direction. We know what the parts are and we are trying to figure out what more we can get out of those parts. It is like the small child that took something of Daddy’s apart in the garage just for the pure fun of taking it apart and now taking those exact parts to see if he can make something better or different. If my students can look at a set of operations and computations, and tell me what the original problem was they are showing real understanding. Comparative reasoning is nothing more than giving our students the task of comparing or contrasting. The use of a Venn d iagram is a great tool for our students for this type of task. The characteristics of one go in one circle while the characteristics of the other go in the other circle. Characteristics they both share are put where the two circles overlap. Assessment has proven students have a difficult time finding differences and similarities. In math, an answer for a given problem can be found several different ways. I like to give my students several alternatives, and give them the choice of which would be best to use in a given situation or maybe even easier. Classifying objects gives the student the opportunity to put objects into different categories or groups. To be able to do this, first require the understanding of the requirements for each group and an understanding of the objects into each group. This type of reasoning elicits the ability to analyze the characteristics of the objects and compare them to one another and decide on similarities of objects to be placed in one category together. Obviously, this type of reasoning does not stand alone as a process. If you look at the word “evaluate”, the word itself tells what evaluative reasoning
Graphics borrowed from Sparks Notes Study Guide, part 1 If you look at the word “evaluate”, the word itself tells what evaluative reasoning is. What is the value of this? To determine value, we have to have criteria. Does the work performed follow the given criteria? Before students can provide us with what we want, we have to give clear instructions of what is expected. I like my students to not only solve a problem, but also tell me how they arrived at that answer. This gives them the opportunity to evaluate what they did. This some times shows them they did something wrong or left a step out. By evaluating their own work, they have to demonstrate knowledge of the process and there are critiquing the quality of their own work and how to make it better. Teaching geometry lends itself to using multiple kinds of reasoning to look a one object and answer many questions about. Finding the perimeter, area, surface area, and volume requires comparison and analytical reasoning. Using what we know of corresponding angles and corresponding sides is required to classify the shape and/ or shapes involved. This object is definitely the sum of parts and those parts can be taken apart to construct other whole objects. Finally, the student can evaluate his work and defend his products. I like Stiggins’ bulleted summary on pages 54 and 55. He says all reasoning is related. Synthesis is inductive. Analysis is prior to comparison. Inductive is comparative. Evaluation requires analysis, comparison and criteria. In other words, it is all tied together. Stiggins makes some statements that made me think about how I teach and what changes I need to make. One of those is his explanation of knowing and understanding. Since I teach math, there are many premises that I accept as fact. Much of our math
curriculum expects our students to just learn these premises and accept them as fact. It would make much more sense, according to my understanding of what Stiggins’ is saying, to help my students discover why it is true. Much of this I do, but I’ve decided I don’t do it nearly enough. Another portion of Stiggins’ book, emphasized, for me, that I was assessing my students in the proper manner. “When we reason analytically, we draw inferences about the component parts of something: its ingredients, how they fit together, and how they function as a whole.” (Stiggins et al, 2005) When I “grade” my students’ work on a particular equation, I look at all parts of the work performed. This tells me if they understand the process. Understanding the process and all the components of how to reach the final answer to the equation is much more important than the answer. In my classroom, I want my students to be able to see why “things” work out as they do. An example is finding 3 64 . By prime factoring 64, they need to “think” and “see” the combinations to numbers to “find” the 3 roots of 64. In an article by Vicente Talanquer, called “Pandora’s Box of Misconceptions”, he believes,” Students are experts in common sense problem solving”. (
The Science Teacher, 2005
) They use the knowledge they have and make predictions. This tends to lead to many misconceptions. Talanquer has eight rules or patterns of reasoning. His field is chemistry which seems to apply well to my classes of mathematics. “His first rule is basically cause and effect. He feels his students believe one specific cause will always produce the same effect. Second, his students focus on the most evident variable and base their assumptions on its activity. Third, Students reasoning is focused on observation and not necessarily on research. Fourth, Students
believe what they know to be true. Things are what they are. What’s real is real. Fifth, students see the world in concrete terms not abstract. Sixth, when they do look at the abstract, it is directly correlated to the concrete. This is the same as this. S eventh, Students see laws as laws and no conditions apply. If this, then that is also correct. And eighth, Students don’t see the need to differentiate between concepts. Relationship and discrimination is unimportant.” (
The Science Teacher,
2002) Talanquer realizes his reasoning rules are not complete, but are useful for him to develop his lessons. They help him in designing his lessons, so that his students do not follow a path that leads in the wrong direction. In my estimation, he appears to have a more deductive approach to teaching. He wants to be sure of where his students end up in their discovery. He tries to lead his students in a direction that helps them realize their common sense answers often lead them in the wrong direction. To thinkers of my students, I need to be teaching them by asking them to use all the various patterns of reasoning. In the real world, we have to be able to look at life’s problems in several different ways. We have to teach them all those different kinds of problem solving skills. I have to start with the end result in mind, and teach to that end.
Work Cited Curran, Jeanne, and Takata, Susan R., retrieved Nov.28, 2005, from Dear Habermas web site: http://www.csudh.edu/dearhabermas/beaucoup01.htm Fullerton Community Coll, (n.d.). Inductive and deductive reasoning. Retrieved Nov. 28, 2005, from Fullerton Faculty Web site: http://commfaculty.fullerton.edu/rgass/newpage22.htm. Inductive and deductive reasoning.
Sparks Note, Study Guide
, Retrieved Nov 28, 2005, from http://www.sparknotes.com/math/geometry3/inductiveanddeductivereasoning/su mmary.html. Stiggins, R. (2005).
Student involved assessment for learning
. 5th ed. , NJ: Prentice Hall. Talanquer, V. (2002, Nov ). Minimizing misconceptions.
The Science Teacher
, Retrieved Nov 28, 2005, from www.chem.arizona.edu/tpp/misconst.pdf.