Tuesday, June 21, 2011
Wednesday, June 15, 2011
3 Act Story Problems
I'd never really thought of a story problem as being similar to a three act story. Maybe it's because I'm horrible at writing stories! But after reading examples on blogs such as Mr. Piccini's Mathtabulous Site and A Recursive Process, I think that volume problems in Geometry naturally lead to this type of set-up.
I taught Geometry again this past year, but this was the first year in a long time where I got to teach the second half of the material, which happens to be the half in our curriculum with Volume and Measurement. In fact, I think I've done a few "starter" or "warm-up" problems that could easily be rewritten into 3-act lessons.
For instance, I have done "which is larger?" problems in the past. Here's an example of a problem I wrote and gave to my class as a warm-up earlier this spring. This problem could easily be set up as a 3-act story problem. I'm certain the students would be much more engaged if I gave the set-up first, added the information and back-story, then finally gave the "reveal" at the end. I was intrigued by the idea of adding a "sequel" as well, or letting the students come up with their own math problems.
If I'm teaching Geometry this year, I'll definitely add some problems like these to my curriculum!
I taught Geometry again this past year, but this was the first year in a long time where I got to teach the second half of the material, which happens to be the half in our curriculum with Volume and Measurement. In fact, I think I've done a few "starter" or "warm-up" problems that could easily be rewritten into 3-act lessons.
For instance, I have done "which is larger?" problems in the past. Here's an example of a problem I wrote and gave to my class as a warm-up earlier this spring. This problem could easily be set up as a 3-act story problem. I'm certain the students would be much more engaged if I gave the set-up first, added the information and back-story, then finally gave the "reveal" at the end. I was intrigued by the idea of adding a "sequel" as well, or letting the students come up with their own math problems.
If I'm teaching Geometry this year, I'll definitely add some problems like these to my curriculum!
Monday, June 13, 2011
Textbooks or Not Textbooks?
The secret to being a lazy math teacher? Good textbooks! Yes sir, anyone can be a math teacher if they have a good textbook - just read the material in the next section before class starts, give a few examples of the new material on the whiteboard for the class, assign the even problems to the class (no odds, they're all in the back and all the kids will do is copy!), and sit back in your chair with a good magazine for the rest of the hour while the kids work. The next day, have the kids trade and grade yesterday's papers, and repeat the previous day's process, following the material in your textbook. Give a quiz once or twice a week, a test every 2-3 weeks, and there you go - math class. Right?
Of course I'm attempting humor in the previous paragraph, but the truth is, I'll bet every secondary math teacher in America has tried the formula above once or twice in their career. As long as you have a great textbook, right? The problem is, today's changing math standards don't really follow any given textbook.
I student taught and spent the first decade of my teaching career using the University of Chicago School Mathematics Project (UCSMP) textbooks, and I loved them! The books focus more on reading than previous math texts, they offer a wide variety of story problems, and they encourage project based learning. I attended the UCSMP conferences in downtown Chicago several times early in my teaching career, and I bought right in to what they were selling. I liked having my kids read, and when I got behind and wasn't prepared for the next lesson, I'd read along right with them. In my first years of teaching I taught from everything that UCSMP had to offer: PreAlgebra, Algebra, Geometry, Adv. Algebra, FST, and PDM. You could follow the planning schedule in the book day after day, use the provided tests and quizzes (or accompanying software), throw in the occasional project, and your teaching year was set.
Unfortunately, the State of Michigan switched the standards for HS math in the mid-2000's. Now Statistics were a part of Alg 2, while topics like matrices were not. Suddenly, following the UCSMP books day after day was no longer possible, as we began to introduce curriculum that previously only existed in other courses. My world was turning upside down! To make matters worse, I got hired in a district that didn't use UCSMP at all! In fact, this new district was encouraging us to not use textbooks at all, or use material that could be found or accessed online. By assigning each student a laptop, with access to our "official" texts online, students wouldn't need to lug around texts each day (and our district wouldn't need to spend $ replacing texts).
So now, the question is, what "textbook" do I use, or do I even use one at all? Our Trig and PreCalc books are older than my students, and they don't really even align with the state standards. I can find better material elsewhere, but my district won't buy new texts. So I can try another option .... I can scan a sample textbook that I like, post links to it on my moodle page, and have students use it as a text. The catch is only students with the password can access the material, which keeps me from getting caught for violating copy write laws.
This whole situation would be made easier if I had fewer preps and some consistency of teaching assignment. As I've complained in previous posts, I teach a wide variety of classes, and they switch year to year. If a teacher can focus on just one or two math subjects for consecutive years, I think they'd be a much more effective teacher. Regardless of the textbook.
Of course I'm attempting humor in the previous paragraph, but the truth is, I'll bet every secondary math teacher in America has tried the formula above once or twice in their career. As long as you have a great textbook, right? The problem is, today's changing math standards don't really follow any given textbook.
I student taught and spent the first decade of my teaching career using the University of Chicago School Mathematics Project (UCSMP) textbooks, and I loved them! The books focus more on reading than previous math texts, they offer a wide variety of story problems, and they encourage project based learning. I attended the UCSMP conferences in downtown Chicago several times early in my teaching career, and I bought right in to what they were selling. I liked having my kids read, and when I got behind and wasn't prepared for the next lesson, I'd read along right with them. In my first years of teaching I taught from everything that UCSMP had to offer: PreAlgebra, Algebra, Geometry, Adv. Algebra, FST, and PDM. You could follow the planning schedule in the book day after day, use the provided tests and quizzes (or accompanying software), throw in the occasional project, and your teaching year was set.
Unfortunately, the State of Michigan switched the standards for HS math in the mid-2000's. Now Statistics were a part of Alg 2, while topics like matrices were not. Suddenly, following the UCSMP books day after day was no longer possible, as we began to introduce curriculum that previously only existed in other courses. My world was turning upside down! To make matters worse, I got hired in a district that didn't use UCSMP at all! In fact, this new district was encouraging us to not use textbooks at all, or use material that could be found or accessed online. By assigning each student a laptop, with access to our "official" texts online, students wouldn't need to lug around texts each day (and our district wouldn't need to spend $ replacing texts).
So now, the question is, what "textbook" do I use, or do I even use one at all? Our Trig and PreCalc books are older than my students, and they don't really even align with the state standards. I can find better material elsewhere, but my district won't buy new texts. So I can try another option .... I can scan a sample textbook that I like, post links to it on my moodle page, and have students use it as a text. The catch is only students with the password can access the material, which keeps me from getting caught for violating copy write laws.
This whole situation would be made easier if I had fewer preps and some consistency of teaching assignment. As I've complained in previous posts, I teach a wide variety of classes, and they switch year to year. If a teacher can focus on just one or two math subjects for consecutive years, I think they'd be a much more effective teacher. Regardless of the textbook.
Tuesday, June 7, 2011
Lesson Planning ... ugh!
I knew I was in trouble during my first year of teaching when my principal gave me a lesson plan book. He expected me to turn in my upcoming week's lesson every Monday morning. Plan ahead? Me? I just don't work that way!
I tried as best as I could that year to submit weekly lessons. I was lucky enough to have submitted my lessons the week our staff irritated my principal so much that he entered a negative note into the file of each staffer who hadn't submitted their plans that week! But I can admit it - I'm a seat of my pants type operator. I'm lucky to stay a day ahead, let alone a week or a whole unit!
There was a short time during my first years of teaching that I actually made up monthly calendars! I would print out a schedule for, say, the month of October for Stats class with potential assignments, quizzes, tests, and projects listed for the whole 31 days. And I'd do my best to stick to that schedule! That would force me to plan my units and activities far in advance. I'm not sure when and why I dropped this system. I think it was when I switched schools in 2001 - in a new district, it took me a while to plan ahead for the new subjects I was teaching, and then I got moved from building to building and subject to subject so often that I just gave up.
I started at my current district in 2007, and in 4 short years I've now taught every regular math course our school offers - I'm the "catch all" guy. I've taught Algebra, Geometry, Algebra 2, Trig, PreCalc, Stats (these last 3 are stand-alone 1 semester subjects), and AP Calculus in this time. That's everything my district offers, aside from Tech and Business Math, taught by our Business/Math instructor. So I've taught 7 math subjects - my colleagues have taught 2,3, or 4 different subjects in that time period! My flexibility is good for my administrators and counselors making schedules, but it's tough for me to focus my craft on just a few select subjects.
Currently, I try to plan a chapter or unit at a time. I figure out what assessments I should use, which topics will take the most amount of days, and what special projects or activities I should add into the regular text we use. Sometimes I choose not to use the text at all, if I don't like the question bank for a particular lesson. Then I'll rely on other texts and sources I have.
I love to do special projects, and now that I'm no longer teaching on trimesters, I have more time in the year to assign them. Simple activities like taking a day to make tessellations in Geometry, and doing stats measuring experiments in class I feel encourage students to explore learning math in different ways.
There have been days this year (Mondays?) where I've walked into school 20 minutes before my first class with no clue what I was doing in class that day - and yet since I've taught such a wide variety of topics in my career, I feel very confident on finding different ways to introduce new material to students. But being organized and planning ahead always works so much better!
I tried as best as I could that year to submit weekly lessons. I was lucky enough to have submitted my lessons the week our staff irritated my principal so much that he entered a negative note into the file of each staffer who hadn't submitted their plans that week! But I can admit it - I'm a seat of my pants type operator. I'm lucky to stay a day ahead, let alone a week or a whole unit!
There was a short time during my first years of teaching that I actually made up monthly calendars! I would print out a schedule for, say, the month of October for Stats class with potential assignments, quizzes, tests, and projects listed for the whole 31 days. And I'd do my best to stick to that schedule! That would force me to plan my units and activities far in advance. I'm not sure when and why I dropped this system. I think it was when I switched schools in 2001 - in a new district, it took me a while to plan ahead for the new subjects I was teaching, and then I got moved from building to building and subject to subject so often that I just gave up.
I started at my current district in 2007, and in 4 short years I've now taught every regular math course our school offers - I'm the "catch all" guy. I've taught Algebra, Geometry, Algebra 2, Trig, PreCalc, Stats (these last 3 are stand-alone 1 semester subjects), and AP Calculus in this time. That's everything my district offers, aside from Tech and Business Math, taught by our Business/Math instructor. So I've taught 7 math subjects - my colleagues have taught 2,3, or 4 different subjects in that time period! My flexibility is good for my administrators and counselors making schedules, but it's tough for me to focus my craft on just a few select subjects.
Currently, I try to plan a chapter or unit at a time. I figure out what assessments I should use, which topics will take the most amount of days, and what special projects or activities I should add into the regular text we use. Sometimes I choose not to use the text at all, if I don't like the question bank for a particular lesson. Then I'll rely on other texts and sources I have.
I love to do special projects, and now that I'm no longer teaching on trimesters, I have more time in the year to assign them. Simple activities like taking a day to make tessellations in Geometry, and doing stats measuring experiments in class I feel encourage students to explore learning math in different ways.
There have been days this year (Mondays?) where I've walked into school 20 minutes before my first class with no clue what I was doing in class that day - and yet since I've taught such a wide variety of topics in my career, I feel very confident on finding different ways to introduce new material to students. But being organized and planning ahead always works so much better!
Tuesday, May 31, 2011
Annotated Bibliography
This is a post to keep track of MTH 629 readings. It will be updated
Annotated Bibliography
“Tailoring Tasks to Meet Students’ Needs,” McDuffie, Wohlhunter, Breyfogle, Mathematics Teaching in the Middle School, vol 16, no 9, May 2011.
Four strategies to make curriculum/lessons/instruction fit student needs. Eg. ELL & Special Education students.
1. Switch to a familiar context
2. Supplement foundational gaps
3. Incorporate overarching goals
4. Adjust for reading levels
"Problems That Encourage Proportion Sense," Billings, Esther H.M, Mathematics Teaching in the Middle School, vol 7, no 1, Sept 2001.
Students often fail to consider the reasonableness of answers. The use of non-numeric problems can encourage proportion sense in students.
"Using Assessment for Effective Learning," Lee, Clare Mathematics Teaching, June 2001
Formative Assessment can be shown to clearly raise standards. Some specific methods:
1. Ask Questions worth thinking about - questions without easy answers
2. Questions that last a lesson or two - keep students engaged
3. Students write their own questions - quickly identifies strengths and weaknesses
4. Last 5 minutes - students reflect, tell what they know now that they didn't know
5. Give yourself a score out of 5 - do 5 questions from a lesson, have students score themselves (without seeing the answers first) and then review only the questions that students felt they missed
6. What questions do you have about how to complete a task? - student write the "5 burning questions" highlighting what information is still needed to complete a problem
7. Peer homework correcting, using detailed solutions - students grade each others' work, using answer keys with steps given, and must debate about what indicates showing work correctly.
"Toward an educationally relevant theory of literacy learning: Twenty years of inquiry" Cambourne, Brain, The Reading Teacher, vol 49, no 3, Nov. 1995
A study that quantified the conditions needed for language growth shows that these same conditions could be applied to all types of learning. The conditions were:
1. Immersion
2. Demonstration
3. Engagement
4. Expectations
5. Responsibility
6. Approximations
7. Employment
8. Response
"One Teacher's Story," Collins, Anne M., Mathematics Teaching in the Middle School, vol 16, no 1, August 2010
Account of a middle school math teacher's attempt to focus on teaching problem solving methods. The teacher encouraged the students to use common problem-solving strategies (model it, make a table, etc...) while solving an involved, multi-step problem. Students were expected to clearly communicate their solutions while justifying their steps, and an alternative (1-4pt) grading scale was utilized.
Annotated Bibliography
“Tailoring Tasks to Meet Students’ Needs,” McDuffie, Wohlhunter, Breyfogle, Mathematics Teaching in the Middle School, vol 16, no 9, May 2011.
Four strategies to make curriculum/lessons/instruction fit student needs. Eg. ELL & Special Education students.
1. Switch to a familiar context
2. Supplement foundational gaps
3. Incorporate overarching goals
4. Adjust for reading levels
"Problems That Encourage Proportion Sense," Billings, Esther H.M, Mathematics Teaching in the Middle School, vol 7, no 1, Sept 2001.
Students often fail to consider the reasonableness of answers. The use of non-numeric problems can encourage proportion sense in students.
"Using Assessment for Effective Learning," Lee, Clare Mathematics Teaching, June 2001
Formative Assessment can be shown to clearly raise standards. Some specific methods:
1. Ask Questions worth thinking about - questions without easy answers
2. Questions that last a lesson or two - keep students engaged
3. Students write their own questions - quickly identifies strengths and weaknesses
4. Last 5 minutes - students reflect, tell what they know now that they didn't know
5. Give yourself a score out of 5 - do 5 questions from a lesson, have students score themselves (without seeing the answers first) and then review only the questions that students felt they missed
6. What questions do you have about how to complete a task? - student write the "5 burning questions" highlighting what information is still needed to complete a problem
7. Peer homework correcting, using detailed solutions - students grade each others' work, using answer keys with steps given, and must debate about what indicates showing work correctly.
"Toward an educationally relevant theory of literacy learning: Twenty years of inquiry" Cambourne, Brain, The Reading Teacher, vol 49, no 3, Nov. 1995
A study that quantified the conditions needed for language growth shows that these same conditions could be applied to all types of learning. The conditions were:
1. Immersion
2. Demonstration
3. Engagement
4. Expectations
5. Responsibility
6. Approximations
7. Employment
8. Response
"One Teacher's Story," Collins, Anne M., Mathematics Teaching in the Middle School, vol 16, no 1, August 2010
Account of a middle school math teacher's attempt to focus on teaching problem solving methods. The teacher encouraged the students to use common problem-solving strategies (model it, make a table, etc...) while solving an involved, multi-step problem. Students were expected to clearly communicate their solutions while justifying their steps, and an alternative (1-4pt) grading scale was utilized.
Grading HS Math
So what's the most effective way to grade high school math courses? I've gone through a variety of methods throughout my teaching career, either determined by the math department at whichever school I happened to be employed by at the time, or driven by whatever might be fashionable in current educational development readings.
Here's a breakdown and my critique of some of my current or former grading scales. Most of the schools in which I've taught did the traditional A, A-, B+, etc.. letter scale, though use of this scale could lead to a side-topic of it's own!
At one school the minimum passing grade was 70% - anything less and you failed. After failing an Alg 2 student with a 69% one year, my principal informed me that since I was good with numbers I should adjust scores so that no one in my classes scored between 65 and 69% ever again - they were either above or below! At one time that same school would not allow a student to get a "minus" score: B+ and B were allowed, B- wasn't. So scores or 80-86 were all B's, while 87-89 got B+, and a 90 was an A. No focusing on minuses, which lead to negativity.
I do not like my school's current grade scale - 87-89 is a B+, 84-86 is a B, but 80-83 is a B-. Why does the B- take up 40% of all the B grades, but the B+ and straight B get 30% each? Shouldn't the B+ and B- get 30% each, and the straight B get 40%? So 80-82 would be a B-, and 83-86 would be the B. One of my old districts graded this way. When I suggest this to current math students, they understand the unfairness immediately, but no change has been forthcoming. I think it is due to certain teachers not wanting to award an A for 93% in a class - they want a minimum of 94 or even 95%. Anything to make things tougher for kids.
Back on topic - how to grade. When I first starting teaching I gave category scores - 50% of the final grade was based on tests, 25% on quizzes, 20% on homework, and 5% on participation. That 5% participation grade was held over the heads of the kids that skipped frequently, though whether we should grade on attendance is another story. I immediately found a problem with this scoring system: a student could be failing a class with a 58%, fail a final test with a score of 54%, but suddenly have a passing grade of 60% in the class. If the student's test average was very low (say 45%), the final test of 54% would raise their test average, thus raising their overall class average enough to have the student receive 60% in the class. I didn't like this ironic outcome, so I changed my scoring system to strictly be based on total points.
Using total points is my favorite method of grading - tests are worth 80-100 points, quizzes 10-25 points (usually 20), with homework 2-5 points, usually 3. You get a test score higher than your class average, your overall grade goes up. You get a score lower than your average, your overall grade goes down. The only problem is that the scoring levels are not consistent year to year or even term to term - your final grade could be based anywhere from 45 to 65% on test scores, depending on when tests were given.
My current school technically requires percent scores (final grade is 4% participation, 12% HW, 24% Quizzes, 40% Tests, 20% Final Exam), but now we have a thing called cumulative scoring, which takes the philosophy that "we don't care what you did along the way, we only care about the final results," the final results being what you did on your final exam, even though the final exam can only question roughly 50-60% of all the material covered. Our tests are supposed to be cumulative as well - a Unit 2 test would be 50% unit 1, 50% unit 2, while a unit 5 test would be 20% unit 1, 20% unit 2, ..., 20% unit 5. So theoretically, your final test would cover the same material as your final exam. If a student has 100% (or close) of their homework completed, they could replace all previous tests with the score from their last test, assuming it is their best score. This worked great when we had 75 minute class periods last year, but cumulative testing is nearly impossible in a 55 minute time period (our current class schedule.) Plus, is the final test the only thing? Outcomes only? How about those that participate in the mathematical process throughout the year, but have a bad day during the final exam?
No easy answers, but a lot of questions raised when it comes to scoring.
Here's a breakdown and my critique of some of my current or former grading scales. Most of the schools in which I've taught did the traditional A, A-, B+, etc.. letter scale, though use of this scale could lead to a side-topic of it's own!
At one school the minimum passing grade was 70% - anything less and you failed. After failing an Alg 2 student with a 69% one year, my principal informed me that since I was good with numbers I should adjust scores so that no one in my classes scored between 65 and 69% ever again - they were either above or below! At one time that same school would not allow a student to get a "minus" score: B+ and B were allowed, B- wasn't. So scores or 80-86 were all B's, while 87-89 got B+, and a 90 was an A. No focusing on minuses, which lead to negativity.
I do not like my school's current grade scale - 87-89 is a B+, 84-86 is a B, but 80-83 is a B-. Why does the B- take up 40% of all the B grades, but the B+ and straight B get 30% each? Shouldn't the B+ and B- get 30% each, and the straight B get 40%? So 80-82 would be a B-, and 83-86 would be the B. One of my old districts graded this way. When I suggest this to current math students, they understand the unfairness immediately, but no change has been forthcoming. I think it is due to certain teachers not wanting to award an A for 93% in a class - they want a minimum of 94 or even 95%. Anything to make things tougher for kids.
Back on topic - how to grade. When I first starting teaching I gave category scores - 50% of the final grade was based on tests, 25% on quizzes, 20% on homework, and 5% on participation. That 5% participation grade was held over the heads of the kids that skipped frequently, though whether we should grade on attendance is another story. I immediately found a problem with this scoring system: a student could be failing a class with a 58%, fail a final test with a score of 54%, but suddenly have a passing grade of 60% in the class. If the student's test average was very low (say 45%), the final test of 54% would raise their test average, thus raising their overall class average enough to have the student receive 60% in the class. I didn't like this ironic outcome, so I changed my scoring system to strictly be based on total points.
Using total points is my favorite method of grading - tests are worth 80-100 points, quizzes 10-25 points (usually 20), with homework 2-5 points, usually 3. You get a test score higher than your class average, your overall grade goes up. You get a score lower than your average, your overall grade goes down. The only problem is that the scoring levels are not consistent year to year or even term to term - your final grade could be based anywhere from 45 to 65% on test scores, depending on when tests were given.
My current school technically requires percent scores (final grade is 4% participation, 12% HW, 24% Quizzes, 40% Tests, 20% Final Exam), but now we have a thing called cumulative scoring, which takes the philosophy that "we don't care what you did along the way, we only care about the final results," the final results being what you did on your final exam, even though the final exam can only question roughly 50-60% of all the material covered. Our tests are supposed to be cumulative as well - a Unit 2 test would be 50% unit 1, 50% unit 2, while a unit 5 test would be 20% unit 1, 20% unit 2, ..., 20% unit 5. So theoretically, your final test would cover the same material as your final exam. If a student has 100% (or close) of their homework completed, they could replace all previous tests with the score from their last test, assuming it is their best score. This worked great when we had 75 minute class periods last year, but cumulative testing is nearly impossible in a 55 minute time period (our current class schedule.) Plus, is the final test the only thing? Outcomes only? How about those that participate in the mathematical process throughout the year, but have a bad day during the final exam?
No easy answers, but a lot of questions raised when it comes to scoring.
Thursday, May 19, 2011
Where I am Now
My name is Eric Thuemmel and I'm finishing my 18th (!) year of teaching public school in Michigan. I graduated from MSU in 1992 and started teaching at North Huron High School (Kinde / Port Austin area) in the fall of '93. At North Huron I taught everything from Algebra to PreCalculus. After 8 years at NHHS I got married and moved to the west side of the state. I took a new job with Ludington HS, and I mostly taught Geometry. My years at Ludington were a little rocky, as a budget crunch nearly took my job, but I was lucky to get a spot teaching in their new alternative school for several years before returning to the main high school.
While teaching in Ludington, my family and I lived in Manistee (30 minutes north) where my wife worked. In the summer of 2007 a high school math teaching job opened in Manistee. With this job I'd be able to teach upper level math (Trig, Pre-Calc, Stats- my favorite subjects), as well as have coaching opportunities (track, cross-country, forensics.) In addition to all of these opportunities, I live close enough to the new high school to jog or ride my bike in to work. So I was thrilled to start working in my new hometown.
This year I'm teaching HS Geometry, Stats, and PreCalc/Trig at Manistee High. I have no idea what I'll be teaching next year - with the budget crunch, despite my experience, my job may be on the chopping block. It seems as if everyone and their brother has a math certification in my district.
While teaching in Ludington, my family and I lived in Manistee (30 minutes north) where my wife worked. In the summer of 2007 a high school math teaching job opened in Manistee. With this job I'd be able to teach upper level math (Trig, Pre-Calc, Stats- my favorite subjects), as well as have coaching opportunities (track, cross-country, forensics.) In addition to all of these opportunities, I live close enough to the new high school to jog or ride my bike in to work. So I was thrilled to start working in my new hometown.
This year I'm teaching HS Geometry, Stats, and PreCalc/Trig at Manistee High. I have no idea what I'll be teaching next year - with the budget crunch, despite my experience, my job may be on the chopping block. It seems as if everyone and their brother has a math certification in my district.
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