ELN 122 Lesson 8 Backward Design

Backwards Design provides a way to see if students are actually learning what you want to teach them by looking at assessment results. Overall what I have learned is that I am more concerned with a student being able to think for themselves and be creative than most math teachers, I want students to apply mathematics to solve real life problems not just only math skills, terms and facts. I have always said that there are no answer books in real life. A student has to be able to reason whether or not their  answer is correct.
Other than teaching to national standards, which is a requirement no matter what course you teach there are four important concepts to think about when designing. I found this excellent article on :What is backward design?
The main four points of the article were that backward design helps a teacher select topics that help them teach for understanding.
First, the topic has to have importance and meaning in the student's life. For example, how are patterns used in everyday life. A specific example is a pin number or and id number. How important in a persons life are codes and numbers. You can't get any more important than a social security number, and also it is such an important number in a person's life that codes are developed to protect it.
Second, the topic is one that is essential to the discipline. For example, the difference between numbers used as counting and numbers used as code. There is a big difference when you use a credit card for example between the price you pay for an item, the UPC code of the item, and your credit card number and pin number.
Third, what topic do most students intuitively struggle with? For example, students struggle with the concept of base. They are all familiar with base ten, but students who use other bases everyday do not grasp the mathematical significance. We count money in base 10, time in base 60, eggs in base 12, computers in base 2, and in the US uses a system of measurement based on 12, 3,5280 (horrible!!!).
Fourth, the topic is one that is engaging. For example, the performance assessments I used in the unit on patterns in everyday life.

http://asbmb.org/uploadedFiles/Backward%20design.pdf

Next I wanted to apply what I have learned through the course to the concept of backward design. I found another web site that lets you test your Teaching Goals applying what is important to your course and assessments. There were no right or wrong answers. It inventoried what I thought was important as opposed to most teachers.

"Teaching Goals Inventory. This tool, originally created by Patricia Cross and Thomas Angelo, contains 53 prompts to help instructors identify their goals for a particular course. This on-line version offers rapid self-scoring and data comparisons across goal areas and disciplines"

http://fm.iowa.uiowa.edu/fmi/xsl/tgi/results.xsl

http://fm.iowa.uiowa.edu/fmi/xsl/tgi/data_entry.xsl?-db=tgi_data&-lay=Layout01&-view

https://cft.vanderbilt.edu/guides-sub-pages/course-design/

 

The results of my Teaching Goals Inventory:

1. I rated Higher Order Thinking Skills as my top concern (100%).
2.  Next in importance came Basic Academic Skills (56%0,
3. Then came Discipline, Specific Knowledge and Skills (38%) pretty much tied with Liberal Arts and Academic Values were next (40%),
4. Last were Personal Development(33%) and Work and Career Prep (25%).

I compared it to the results of the general population of teachers taking the inventory:

1. Higher order thinking skills (43%-45%)

2. Discipline Specific (36%-37%)

3. Personal Development ((25%-28%)

4. Work and Career ((21%-26%)

5. Liberal Arts (18%-21%)

6. Basic Skills (18%-22%)

 

 I identified my primary goal as a teacher as "Helping students develop higher-order thinking skills,"

 

I identified the following as skills needed:

• Apply Principals
• Math Skills
• Terms and Facts
• Wise decisions
• Analytic Skills
• Think for Self
• Creativity
• Writing skills
• Openness to ideas
• Problem Solving

Math and Science Teachers overall identified the top skills as:

• Math skills
• Analytic skills
• Problem solving.
• Apply Principles
• Terms and Facts

It seems as though most Math and Science teachers rate skills, facts, and applications more important than reasoning, making wise decisions and creativity..

 

 

 

 

 

 

ELN 122 Lesson 7 eLearning Performance Assessments

 

Define performance Assessment

 A performance assessment is one in which the teacher observes the behavior of the student, or assesses an actual object that the student produced from that performance assessment.

Describe how it can be used in eLearning.

Some of my performance assessment projects (performance assessments) ranged from figuring out how long the Phoenix Sun’s basketball players height/arm ratio were compared to the class arm/ height ratio (linear algebra) to making a sun dial (polar coordinates) to finding the length of a light pole (trig).

I tried a tessellation performance assessment in my face-to-face class. I got very good results. The students enjoyed the performance assessment. In my classroom, on the Whiteriver Apache Indian Reservation, I tried to bring as much hands on performance assessments that I could into the classroom. I think this performance assessment would also work for an online class and student. The student would have to have the capability to take pictures, or do a screen shot of their work, and upload it as a file.

What makes performance assessments truly worthwhile in eLearning?

From what comments I read on the various websites most of the students liked doing the tessellation project. I think something like this project would be good for students who are visual/special, musical, or artistic. Even students who are creative need something to make of their own in a math class whether it s online or face-to-face.

 

The tessellation performance assessment was part of monthly projects I had the students do as a substantial part of their grade. 1/3 was in class work (I didn’t assign homework because it didn’t happen, it was more productive to get them to work hard in class than at home), 1/3 was formative and summative AIMS like tests (so they had practice taking high stakes tests), and 1/3 was projects (performance assessments). This way students who had test anxiety but knew the subject matter could prove that they knew the concepts by applying mathematics to a real life project.

 

This particular tessellation assessment was a unit project. Students had to make intricate tessellations of their own design. They had a choice of using one or more of the regular polygons that tessellate, or they could make an irregular tessellation of their own design. They could embellish the drawing as they wished. They had to explain their reasoning, their process, and the mathematics behind the choices they made in a written paragraph. They would have to take a screen shot of their drawing and paste it into their word document. A few students as usual waited till the last minute while many others created elaborate amazing drawings. I was astounded at the creativity of my wonderful students whom I greatly miss teaching.

 

What can you foresee as the pitfalls and problems with performance assessment in the eLearning environment?

Before any kind of assessment is written by a teacher in Arizona, or most states for that matter, the assessment  must adhere to state and national math standards. That is always in the back of every teachers mind. I will assume that any teacher that does a performance assessment keeps in mind the confines of standards no matter how interesting a concept might be for the class. A teacher must justify the activity or performance assessment. I ran into this quite often when I took my class outdoors for real life math projects. When I did a trig experiment to see how tall a light pole was, I was questioned about taking the students outside on the parking lot during class time even though we were replicating the classic tree shadow math problem. Heaven forbid if I actually took my students to a park.

 

I  found this website for Common Core Symmetry Math Standards:

http://www.corestandards.org/Math/Content/HSG/introduction/

 This one is for Common Core Visual Standards:

http://www.edutopia.org/blog/ccia-10-visual-literacy-strategies-todd-finley

 

 I always tried to make physical performance assessments as cheap as possible. I had over 225 students in my face-to-face classes so I had to keep the cost of supplies down since I paid for supplies out of my own pocket. That's why in the face to face classroom I used markers, colored pencils, drawing paper and 3x5 cards. Most classrooms also have scissors, rulers, protractors scotch tape and 3x5 cards so this project is not expensive to do for students at home or online.

Some of the problems are software related. Many students don't have the money for expensive software programs, and technology. Although most students do have cell phones. In many online classes the minimal requirements are internet, and Microsoft office suite student version. Most printers are able to scan documents , and are fairly inexpensive (the ink costs more than the printer).

I think journal writing or a Microsoft Word document helps students to explain their problem solving abilities along with the finished product.

One of the pitfalls that is common in an online class as any other class is copying and cheating. It is very difficult for an online student to create an exact duplicate of a tessellation. It would be very obvious if they had copied it because they will not be able to describe in detail how the product was made form scratch in their journal writing. 

For example, I had some students copy someone else's homework from the "tree shadow" performance assessment. Basically the performance assessment goes like this:

Suppose you are a light pole repair person, you need to change the light bulb in the light pole. I want you to figure out using the trigonometry of right triangles and how tall the light pole is in feet. You are given a yardstick on a sunny day. Have fun!. First the student would have to use proportions: pole height/pole shadow = your height/your shadow. I also had students be partners since measuring your height might be a problem. I could tell which students copied by the date of their paper and the time of day. Some of the students had to redo the project because they didn't realize that the length of a shadow changes during the time of day.

I would have the student journal the experiment and draw a sketch as they went along, also the time of day when they measured the pole. I would ask for an average of readings over a few days.

I would expect them to notice the change in shadows and comment on this phenomenon. At least this project could get them out in some sunshine.

 

 

 

 

 

 

 

 

 

ELN 122 Lesson 6 Constructed response vs fixed response balance

Blog Entry 6: Describe the differences between constructed-response and fixed-response written assessments. Describe the benefits of using both in eLearning. Finally, describe the necessity for a balance between teacher-graded and computer-graded assessment items.

In my previous blog lesson 5 I described the differences between constructed response and fixed response assessments

The benefits of using constructed response in eLearning are that the learner can communicate their ideas in writing. The problem comes mainly with how to solve math problems. As a teacher I am more interested in how a student attacks a problem, even if they have minor arithmetic errors. I like to see how their math brain works. I once had someone tell me there is only one right answer. In algebra, calculus and higher mathematics there are multiple solutions to a problem. Sometimes it's hard for as student to grasp this concept especially when there are many correct answers for a function. For example, lets just take f(x) = x+4 . There are any number of numerals that can satisfy this equation.10=6+4, 8=4+4, and so on. When you are grading things like graphing, I suppose an online teacher could ask the student to take a photo of the calculator, or graph the equation, or maybe scan the work and upload it.
I would hate to do an entire math course by fixed response items. It forces the student to think the most important thing in mathematics is a simple answer, when sometimes difficult problems have no answers for hundreds of years. Take for example Fermat's last theorem, there were whole branches of mathematics that were invented just to try to solve the problem.
Another example is Pythagorean Theorem, the Greeks ran into trouble when they used the simplest unit on a right triangle and got a number that was irrational. The concept of irrational numbers was as hard to grasp for them as multiple dimensions are for us.
We force students to take summative fixed response tests. Their whole academic career, sometimes even graduation is based on this one test. Does the test tell you how a student thinks? No, it just tells you how many questions a student got right.
When I look at the test scores of my students I am disheartened because I know they can do math, they just have a difficult time taking fixed response tests. I dread to see the upcoming computerized tests. I have seen where students cannot browse through the test and answer what they know first and then go on to more difficult questions. They must answer each question in order. There are students that have test phobia. My granddaughter freezes up on computer tests. She says her mind just goes blank. She can't think of anything. She has an artistic personality and is used to drawing things to solve problems. Now there is no place to draw or write, no sketches. No paper allowed.
The process of solving a problem is more valuable to me than a fixed response. One thing that irritates me as a teacher having worked in the real world outside of academia is that there are no answer books in real life. The answer you give is the answer. My daughter worked for a large company that has massive numeral reports every quarter, She uses huge work books in Excel not just work sheets. The report she gives to her CEO is the answer. Is it right? She has to use problem solving abilities to prove to herself that her report is correct. If a student thinks an essay question is hard, wait till they get into the real world. My other daughter owns her own business, she has to figure profit ,loss, cost of goods sold, predictions about what market trends are coming up. No one can tell her how much to buy, how much to sell. She can't look it up in the back of a math book. She has to make decisions on her mathematics. The mathematics for which there is no answer book, no true/false, no little 4 part multiple choice question. She might have to analyze 20 different scenarios.to make a decision.

ELN 122 Lesson 5 Constructed Response vs Fixed Response Assessments

Blog Entry 5: Describe the differences between constructed-response and fixed-response assessments. When would you use each type of assessment in eLearning? Why?
Constructed Response assessments are essay and completion type formats. The advantage of constructed response is  that the student writes out their response rather than selecting an option provided by the test question as in fixed response assessments such as multiple choice or true false formats. Another advantage of completion constructed response questions are they are relatively easy to construct.  Essay questions can reflect the higher end of Bloom's taxonomy such as create, analyze, compare, prove, defend and contrast. Essay questions reflect  the ability of a student to write clear organized thoughtful answers. A student can explain their logic in solving a problem. I find it fascinating to see how different student's thought processes are even though they all can solve a problem.
The disadvantage of a constructed response completion questions are they are generally limited to the lower end of Bloom's taxonomy in the recall area of information. Sometimes there can be more than one right answer especially if the subject of the question is generic rather than very specific. Spelling and capitalization can make a huge difference. Once when I was helping one of my math students who was extremely frustrated because the student knew their answer was correct but the computer would not accept the answer. In a math problem there are many ways to write a correct answer. Such as  say for example: Factor the number 12:______.
A student could be perfectly correct in answering 12x1, 6x2, 3x4, 12*1, 6*2, 3*4, (12(1), (6)(2),
(3)(4). 3*2*2*1, (3)(2)(2)(1), 3x2x2x1 or 1,12,3,4,6,2 depending on what mathematical symbols you wish to use. Any of these is a perfectly correct answer mathematically. Although once students get into high school they have to stop using x for multiplication, because the symbol x can get mixed up with the variable x. In addition to all these symbols for multiplication and factoring,  I have seen another symbol used for multiplication - a solid large black dot.
The limitations of an essay questions is that it takes longer to grade. Also usually a few essay questions will take the whole test period. Grading an essay question can be subjective unless a rubric is used, and sometimes another teacher's opinion is necessary especially on the borderline answers, and the grey areas. Most of the time it's easy to tell excellent from poor answers, but to tell a high good, medium good, or low good is hard. Even to tell apart a superb excellent from an excellent is difficult.
Just to contrast the difference between constructed response assessments and fixed response questions I as a student am very good at taking multiple choice fixed answer response questions, however I cannot take a true/false test to save my life. I read too much into a true/false test.
When I took the teachers licensing exam I saved 1 hour of the test just for the essay question and it was a doozer! In order to answer the question, I had to know formulas, know geometry, know angles, logic, trigonometry, algebra, and almost every math subject just in order to solve the problem.
With fixed response assessments such as multiple choice and true/false a writer can test many items in a single exam. They can be easily graded on a computer. I used to do an item analysis on every test question I wrote. If practically every student got a question wrong I would not count that question in the grade. I figured either I didn't teach the concept, the students misunderstood what was asked, the question was ambiguous, or the numbers weren't good numbers. I believe these tests should use relatively easy numbers 2, 5, 10, etc not 9, 7 11, weird numbers. If I want to specifically test if students can multiply or divide by higher numbers that should be on a separate test question. Personally I don't give true/false tests. I think they are too susceptible to all or nothing judgments. I can see the exception to every rule.

Constructed Response Assessments: Pros vs Cons

Describe the various types of constructed response assessments. What are the advantages and disadvantages of using these types of assessments? Include pros and cons of making the exam as well as grading and feedback.

One type of constructed response assessment is completion or fill in the blank. The questions are easy to write. They are not multiple choice, only one answer is needed. Next a teacher can include many questions in a test. However the level of knowledge tested is usually on the lower end of Bloom's Taxonomy, such as remembering facts and definitions. Although in mathematics precise definitions of terms is necessary. There is a big difference between the term minus and the term negative. These terms although sometimes represented by a similar symbol are not interchangeable. I liked the example given in the text about sometimes there is more than one answer to a question.
"Who discovered America?" is really too broad a question. Whole archeological dissertations have been written in trying to answer that question. It depends on when, who, where, and other qualifying statements. Also maybe the answer changes depending if you are Norwegian, Italian, Spanish or Native American. One of my student's answer to an essay question about the Pilgrims wrote "They should have stayed home." This from a Native American viewpoint. When I visited Ireland I went to Galway. There is a place there called the Spanish gate. Christopher Columbus came there to Ireland to do research and gather maps before sailing West across the Atlantic. He knew other people had been there before him. He had great PR in my view.
Another question I think is rather humorous is , "There are ______ planets in the solar system."
That's not exactly a one line answer. Depends on when you are asking that question. When I was young it was nine. This is an especially touchy question if you are from Arizona, that question is not exactly popular. Pluto named after Percival Lowell, and discovered by an Arizona astronomer, working in Flagstaff Arizona, is my states' favorite planet. But now it's merely been downgraded to a planetoid. Sigh.
I would not score a fill in the blank with a computer. I think a teacher needs to use judgment when grading an answer.
Essay question are another form of constructed response questions. Especially in mathematics essay questions allow the learner to explain the logic behind their answers. Sometimes essay questions in mathematics involve writing. Written reports are a very common way to express mathematical conclusions. My students used to complain about journal writing and essay questions, and projects in math class. They used to say this isn't English class. However unless a student can articulate what they are doing, why they are doing it and why their answer is correct they don't know how to do math. Even though essay questions are harder to grade I think they give insight into how the student thinks. Although I recently read an article on gender bias in mathematics where teachers were given similar tests to grade with boys names and girls names, although the answers were all correct the teachers gave girls a lower score than boys. The only solution is to have blind grading. Definitely a rubric helps very much to keep the teacher bas out of the grading. Sometimes I used to grade the papers, then sort them into piles regardless of whose name was on the paper. Then I would group like grades with like grades. 1 being 5 being the highest 0 being the lowest. The hardest thing to grade was not an outstanding the difference between an good paper vs. a failure paper but it was in the middle range 4, 3, 2. What was it that made the difference between a 4 and a 3, or a 3 and a 2? Grammar ? Legibility? Logic? ESL ? Diagrams? Arithmetic Errors? I would also allow a day between grading papers and going back for a second look. Sometimes I would even ask a student verbally why they did something. Sometimes they could explain themselves out loud better than on paper.I know this is difficult to do in an online class, but maybe a podcast would help.
Essay questions allow math students to explain their logic. I have told my students that people don't think alike.
How one person solves a problem might not be the same as another person. I like a format such as:
Draw a picture.
Restate the problem in your own words.
Can you write a similar problem using simpler numbers? (1, 2 5, or 10).
What facts are given to you? (Sometimes there are too many facts).
What other background knowledge do you need to solve the problem?(for example when doing a measurement problem you need to know there are 12" in a foot.)
What formulas do you need?
Does the answer make sense?
Check your math!
Can you prove you are correct?

ELN 122 Lesson 3 Best Assessment for eLearners
There are no answer books in real life.

These are some web sites that I found which reflect what I would do if teaching a Freshman level high school math class on-line. I would begin the class by doing some baseline formative assessments to see where my student was in mathematics. I would also want to learn how the student learns. I would want to know the student's reading and writing level. Last being an on-line class, I would want to know the student's technology comfort level.
1) Find out their learning style.
2) Test their math strengths and weaknesses (no one is totally terrible at math, people use it everyday.).
3) Find out what their hobbies are, what do they like to do outside of class. Written simple autobiography. Can they read and write?
4) Just ask what technology they use everyday, this will tell you their comfort level of computer knowledge,after all they will be doing their lessons on-line.

http://www.literacynet.org/mi/intro/index.html

Fun web site to test multiple intelligences. No right or wrong answers just suggestions on how to use your strengths. Mine: Logical Mathematical, Spatial, and Visual. Of course I love math. I see math in colors and pictures in my head. I'm a graphic artist and writer, plus a voracious reader. My lowest score kinesthetic, is that why I hate exercise but I do love to swim! I don't like learning things in a group because I like being alone. I have to force myself into social situations, the best way I learned how to get along with people was working my way through college as a waitress. I also learned a lot about people when I was a sales rep in advertising. So people can learn outside of their comfort zone. Take the test. It's very interesting.

 

https://youtu.be/l2QtSbP4FRg

Actual Howard Gardner interview explaining his philosophy.

 

http://www.math-tests.com/12-Week-Math-Benchmark-Test.html

Eighth grade math test, with pictures and ordinary everyday examples, pizza, stores, tips, etc. Also all grade levels.

http://www.wnyc.org/story/glossary-useful-ed-tech-terms/

Useful discussion blog on at what level is technology comfortable? Also the latest ed tech terms.

http://www.coolmath-games.com/0-fraction-splat
I loved this web site! This is a wonderful example of how to make math fun. Helps to fill in gaps in learning.
https://illuminations.nctm.org/Search.aspx?view=search&gr=9-12
One of the most respected web sites for mathematics education. I used this site all the time when I had my math laptop lab at Alchesay High School.


http://ascendmath.com/index_stndrd.html
Absolutely one of the most horrible examples of on-line teaching I have ever seen, if you want to see a boring "sage on stage" this is the go to website. I cannot believe someone actually is buying this muck for an on-line class.

 

When I became Math department chair at Alchesay High School I had five sections of students who were failing Algebra 1. I knew that this situation was due to the way we had been teaching mathematics. Most of these students were passing in their other classes. So it told me that these students were coming to school, in other words they knew how to play school. They came to school, paid attention in class, did their homework, worked hard, and were successful in school. We were the ones as math teachers who were not reaching the students. "Bless their hearts" as an old saying goes, many of them had taken Algebra 1 two or three times.

I had been reading articles, and a book by Howard Gardner on how people have different learning styles. I challenged fellow members of the math department to start writing lesson plans that included learning mathematics in other ways besides pencil paper. We met once a week to revise our entire curriculum.

I refused to allow anyone to teach as a "sage on stage" The worst way to teach math is to stand with your back to students, mumble the instructions, have a board full of mathematical notations and then say do 100 odds. I would not accept this horrible method of teaching, which I have actually seen as an on-line instruction method. Strangely enough this site is recommended by Rio Salado as a good example of teaching on-line.
When I began to revise the curriculum I tapped into my fellow math teacher's creativity.I challenged each math teacher to bring their area of expertise into our math lessons Each teacher had multiple talents besides math: one gave music lessons after school, another was a basketball coach, one taught a robotics club, one was into hunting and fishing, another taught chess, another had a Japanese animation club, one was an artist, and mine was quilts and graphic arts and computers. Between all of all each of us were doing multiple intelligences in our free time. I wanted to harness that creativity into the math classroom.
The first day of math class I used formative assessments. I used: Learning Styles Test, Eighth grade math test, a simple written biography and I gave the students time on the laptops to observe how expert they were on a computer.
These assessments told me how a student learned so when I put them in groups they would have multiple opportunities to use their strengths in math class. A simple written biography told me what they were interested in outside of school. It also told me if they could read a math word problem, and write sentences. Could I introduce these interests into math class? An eight grade math test gave me an idea of where I needed to focus on filling in the gaps in knowledge. No student is totally terrible at math. Some students just have gaps. I would not put the whole entire class through a review of things they already knew, but I could use the math lab (A+) to focus on gaps in an individual's knowledge. Overall students did very poorly on fractions, so that's where I concentrated on review, but I wanted to do it with realia, manipulatives, and real life examples absolutely not sage on stage.
One of the lines one of my students told me after I retired was " I still like math but it's not as fun as when you taught me." An important idea from Howard Gardner's discussion is that self assessment is not something that is done to you not something that you do for yourself. I encouraged my students to think about their answers. I have told my students, in real life there are no answer books. The answer you give is the answer, whether the bridge falls down, people get sick or you have to pay for a mistake out of you own pocket.
 

 

 

T

ELN 122 Lesson 2 Blog 2 Training vs education

Describe what the difference is between training and education as it pertains to assessment.
Training is task oriented. For example, when studying the unit on symmetry students will be asked to draw and cut out symmetrical shapes. They will be making objects.  When the students make objects they have to have basic hand eye coordination and dexterity skills. Assessing the students would involve questions like: Can they use a scissors, can they use a ruler, a protractor or a compass? Do they know left, right, backwards, up or down? Can they trace a pattern? Can they use a mirror? These are all mathematical tasks that must be accomplished in order to make a symmetrical object. I would walk around and asses how each student is performing the task, guiding where necessary. The mechanics of the project would be good for collaborating in groups.
Sometimes when assessing skills informally in this situation, I have to realize that maybe the student who is awkward with scissors is a left handed person using right handed scissors. Maybe a student who is blind in on eye has no depth perception. Maybe a student only has one arm or hand will take longer. Maybe a student is dyslexic. Maybe a student is color blind. Everyone of these situations has happened to me in my classroom.
Education is a foundation for more learning and problem solving. In order to make a symmetrical object the student learns what symmetry means. They learn orientation is important. They are learning there are types of symmetry:vertical, horizontal, rotational even circular about a point. They learn which objects can tile and those that don't. They make decisions and categorize objects. A summative assessment is given on a high stakes test that is given once a year because symmetry is a state standard. However, students can practice with formative tests at the end of each lesson. As the concepts get more abstract for instance, graphing a shape on graph paper rather than drawing freehand uses conceptual reasoning.
The students can be trained to draw the objects, but they need to be educated on how to interpret a position of the object in a 2-demnsional framework on a graph.