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.
Tuesday, October 27, 2015
Friday, October 23, 2015
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?
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?
Monday, October 19, 2015
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.
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.
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.
Actual Howard Gardner interview explaining his philosophy.
Eighth grade math test, with pictures and ordinary everyday examples, pizza, stores, tips, etc. Also all grade levels.
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.
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.
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
Tuesday, October 6, 2015
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.
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.
Friday, October 2, 2015
Lesson 1 learning objectives vs performance objectives. Their roles in e-learning assessment.
ELN 122 37714 Lesson 1
Martha Knox
Describe the difference between learning outcomes and performance objectives. Include the different categories of learning outcomes and types of performance objectives. What are their roles in assessment?
The categories of Learning Outcomes are: declarative knowledge, procedural knowledge, and problem solving. Declarative knowledge is knowing, recalling and verbalizing specific facts. Procedural knowledge is knowing how to identify, demonstrate and classify the knowledge. Problem solving is knowing how to strategically apply known procedures to solve unknowns in a problem.
Performance objectives asks what observable events can the student demonstrate by using specific behaviors to prove they have learned the indicated knowledge. What specifically can the teacher see the student after they have been taught the concept. They include: capability, behavior, situation, and special conditions.
Problem solving is my favorite method of assessing knowledge. I always tell my math students there are no answer books in real life. The answer you give is the answer, whether the bridge falls down, someone gets too much medicine, or you have to pay for the wrong change out of your own pocket as a cashier. The answer you find is the answer. The most important thing is can you analyze your answer in terms of what you know to prove to yourself that your answer is correct?
One of the most common math problems from ancient times to modern day cray computers is to find prime numbers in a random list of numbers.
It relies on declarative knowledge: multiplication tables, procedural knowledge identifying classes of numbers, and problem solving: generating a solution from the knowledge they have about the characteristics and relationships of numbers. (Solving this classical problem has resulted in many new theories about how numbers are related.)
Let's try a list of numbers , not in any particular order: 56, 34, 73, 29, 81, 22, 55, 69, 2, 15, 1010, and a big one 70600809789664532.
Declarative Knowledge: First we or an explorer of math has to know multiples. We have to know for example multiples of numbers from at least 1 through 10, (although 11, 12, and 13 would be great).
Procedural Knowledge: Then we have to classify the list of numbers, maybe using the knowledge of prime numbers. 2 is the only even prime number. So if a number is more than a googleplex , it still is not prime, if it is even. This means any number no matter how many infinite digits is not prime if it is even; so we have eliminated half of all the possible integers.
Then we need to classify numbers as odd or even, since even numbers are multiples of 2 thus 56, 34, 22, 1010 could be eliminated, also the biggest one.
So the numbers 73,81,55,69, 15 are left. Let's work on the odd numbers. We reason that 5 would be a great number to try. Maybe the other fairly easy numbers could be tried for example: If a number ends in 5 then it is not prime. So now we're left with 73,81,and 69. So we reason that 81 is a multiple of 9x9, and 69 is a multiple of 3x23, thus neither are prime.
Problem Solving: so we are left with 73. Now comes the problem solving part. We need to prove 73 is prime. We reason that 73 could be a multiple of any prime number between 2 and 37 since anything equal to or above 37 as a factor would result in a fraction.
73 is not even, dividing by 3 is a fraction, it doesn't end in 5, it's not a double digit 77 so that means 11 is out of the running, 13 is out because 13 numbers in a deck of cards and 52 cards in a deck plus 26 is 78, 37 is out because 37 x 2 is 74 which is more than 73, the upper limit, so we only have to guess and check: 17, 19,23,29,and 31; of which none are multiples.
The prime numbers between 2 and 36 are 2,3,5,7,11,13,17,19,23,29,31 and 37 none of these are factors of 73 so 73 is prime. Fun. Like a puzzle.
So how can we assess learners? Information, concept, and rule
First, find out what information they already know.
Second, ask learners to classify different examples of the same concept.
Third, give learners a brand new example so they can apply the rule to a brand new example.
Martha Knox
Describe the difference between learning outcomes and performance objectives. Include the different categories of learning outcomes and types of performance objectives. What are their roles in assessment?
The categories of Learning Outcomes are: declarative knowledge, procedural knowledge, and problem solving. Declarative knowledge is knowing, recalling and verbalizing specific facts. Procedural knowledge is knowing how to identify, demonstrate and classify the knowledge. Problem solving is knowing how to strategically apply known procedures to solve unknowns in a problem.
Performance objectives asks what observable events can the student demonstrate by using specific behaviors to prove they have learned the indicated knowledge. What specifically can the teacher see the student after they have been taught the concept. They include: capability, behavior, situation, and special conditions.
Problem solving is my favorite method of assessing knowledge. I always tell my math students there are no answer books in real life. The answer you give is the answer, whether the bridge falls down, someone gets too much medicine, or you have to pay for the wrong change out of your own pocket as a cashier. The answer you find is the answer. The most important thing is can you analyze your answer in terms of what you know to prove to yourself that your answer is correct?
One of the most common math problems from ancient times to modern day cray computers is to find prime numbers in a random list of numbers.
It relies on declarative knowledge: multiplication tables, procedural knowledge identifying classes of numbers, and problem solving: generating a solution from the knowledge they have about the characteristics and relationships of numbers. (Solving this classical problem has resulted in many new theories about how numbers are related.)
Let's try a list of numbers , not in any particular order: 56, 34, 73, 29, 81, 22, 55, 69, 2, 15, 1010, and a big one 70600809789664532.
Declarative Knowledge: First we or an explorer of math has to know multiples. We have to know for example multiples of numbers from at least 1 through 10, (although 11, 12, and 13 would be great).
Procedural Knowledge: Then we have to classify the list of numbers, maybe using the knowledge of prime numbers. 2 is the only even prime number. So if a number is more than a googleplex , it still is not prime, if it is even. This means any number no matter how many infinite digits is not prime if it is even; so we have eliminated half of all the possible integers.
Then we need to classify numbers as odd or even, since even numbers are multiples of 2 thus 56, 34, 22, 1010 could be eliminated, also the biggest one.
So the numbers 73,81,55,69, 15 are left. Let's work on the odd numbers. We reason that 5 would be a great number to try. Maybe the other fairly easy numbers could be tried for example: If a number ends in 5 then it is not prime. So now we're left with 73,81,and 69. So we reason that 81 is a multiple of 9x9, and 69 is a multiple of 3x23, thus neither are prime.
Problem Solving: so we are left with 73. Now comes the problem solving part. We need to prove 73 is prime. We reason that 73 could be a multiple of any prime number between 2 and 37 since anything equal to or above 37 as a factor would result in a fraction.
73 is not even, dividing by 3 is a fraction, it doesn't end in 5, it's not a double digit 77 so that means 11 is out of the running, 13 is out because 13 numbers in a deck of cards and 52 cards in a deck plus 26 is 78, 37 is out because 37 x 2 is 74 which is more than 73, the upper limit, so we only have to guess and check: 17, 19,23,29,and 31; of which none are multiples.
The prime numbers between 2 and 36 are 2,3,5,7,11,13,17,19,23,29,31 and 37 none of these are factors of 73 so 73 is prime. Fun. Like a puzzle.
So how can we assess learners? Information, concept, and rule
First, find out what information they already know.
Second, ask learners to classify different examples of the same concept.
Third, give learners a brand new example so they can apply the rule to a brand new example.
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