What an amazing set of responses to that last post! I’ve been on a much-needed respite from all things educational and I came back to find 14 fantastic comments (plus a shyly emailed one from Mr. B, my dad, a professional engineer). Thank you!
Personally, I’m intrigued by the number of people who actually appear to use, and think with, unconverted fractions. Obviously, I often have to measure with customary units but ever since I learned that an eighth is actually .125, I always convert fractions to decimals for any sort of computation, mental or numerical. I’ve always assumed everyone else did the same. How we teachers think, whether as math users or readers, has so much influence on how we teach, it never ceases to amaze, and frighten, me.
Professionally, I plan to share your many responses with my students, as a means of motivating them to power through our last week of the unit.
I also plan to incorporate your comments in my long-term planning by…
…including more proportionality and conversions. (Diane, Dad) More and more, I believe this is one of the key learnings in math that is simply not well articulated in the standards. Having a keen sense of the relationship between two numbers or two units of measure is a powerful piece of math we all use daily. In 4th and 5th grade, we teach students to recognize that 1,000,000 is bigger than 789,845, but we never really teach them to think about how much bigger it is. Likewise, we generally teach “LCD” and ratios as simply means to an end, not as something that is often used as a tool by itself.
…pushing my students to connect fractions with units of measure. (Kimberly, Eileen, Dawn, Loraine) Because fractions are so difficult and my students’ math backgrounds so barren, I often fear to bring in too much “real-world” math because it will add confusion. But it appears that people are still using cups, ounces and inches, so perhaps my students should be too! I can already imagine how this will allow me to do a wonderful integration of geometry and arithmetic. Thank you for making an upcoming review week fun!
…expanding “arithmetic estimation” to include fraction operations. (Countsepoc, NavyMan, Mr. K) This year, as I looped with my math classes and started with students who were on grade-level, I spent a few weeks at the beginning of the year teaching the sort of mental math that we, educated adults, all do in our heads. (e.g., 5,000 x 300 = 5 x 3 = 15 + five zeroes = 1,500,000) I found that in addition to offering my students much more fluent access to all ensuing topics, it also made my students much more confident and capable when they had to sit down and grind out an arithmetic problem. I hadn’t thought to do the same thing with fractions. A couple days talking about how to estimate what three and a quarter times four and a half sounds very worthwhile now.
Thank you again for all your comments!
Saturday, February 23, 2008
Wednesday, February 13, 2008
Pinnacle of Irrelevance
V---: “Mr. AB, this is boring.”
Me: “Well, V---, I’m glad it’s that easy for you.”
V---: “No, Mr. AB, it’s not that it’s easy, it’s really hard. It’s also just, you know, really boring.”
Me:
---
Dear Readers:
Be honest.
If you were required to add two mixed numbers, say five and one-fourth and three and three-fifths, could you do it? Fluently?
I’m going to guess that unless you teach math, the answer is no. Not because I have low expectations of my readership but because I suspect the last time you did such a problem was when you yourself were in school. It doesn’t matter who you are. As far as I know, not even professional math-people, inclusive of engineers and architects, compute fractions anymore.
Can we even envision a concrete articulation of such a problem? I'll try...
Tom goes to the butcher and buys a three and a third pound roast chicken. His roommate Scott, with inhuman precision, eats two and four ninths pounds. How much is left for Tom, who for inexplicable reasons must know this quantity to the immeasurable and irrelevant accuracy afforded only by unconverted fractions.
Let’s understand why this is worthy of such gripe: in the process of completing this problem, our ten year olds must complete ten constituent steps. This includes deriving the least common denominator, performing two simple and two double-digit multiplication problems, three addition problems, a double-digit long division problem with a remainder and then the isolation of the final answer from that division problem. Depending on the problem, they might also have to do two more division problems to reduce the fraction to lowest terms, bringing us to a full dozen. All must be performed perfectly, as we only give partial credit in high school on up.
Not only is this skill obsolete, it’s not even a prerequisite for ensuing and more important ones. Certainly, I recognize that doing basic operations with fractions is helpful for algebra, but mixed numbers? As far as I know, this skill is at the pinnacle of irrelevance. My students don’t even really need this for sixth grade, let alone later life. Yet I am forced to spend three weeks on it now, because someone, somewhere, decided that understanding how to operate mixed numbers with unlike denominators constitutes “proficiency” in fifth grade math.
To me, the saddest part is this: I can, and I will, teach my fifth graders to master this skill. But why, why, why?!? The list of other, infinitely more valuable aspects of math that I have to leave out (logic, probability, measurement, statistics, just to start…) to make time for this irrelevant crap would make any one who does use math in a meaningful way cry. Imagine the service to humanity if I taught all my fifth graders the difference between mean and median, instead? How many misleading and pointlessly dumb statistics might we be spared? But no…
Now readers, try one and a third subtracted from four and seven-eights… You won't, will you? Because you know it's a waste of your time!
If I am wrong and you are in (or know of) a career where these skills are vital, please correct me. Even with a dream class of budding young mathematicians, teaching this unit is awful. I'd love to provide a reason why it should be more than excruciating tedium. In other words, help me fill in that blank I was stuck in today.
Me: “Well, V---, I’m glad it’s that easy for you.”
V---: “No, Mr. AB, it’s not that it’s easy, it’s really hard. It’s also just, you know, really boring.”
Me:
---
Dear Readers:
Be honest.
If you were required to add two mixed numbers, say five and one-fourth and three and three-fifths, could you do it? Fluently?
I’m going to guess that unless you teach math, the answer is no. Not because I have low expectations of my readership but because I suspect the last time you did such a problem was when you yourself were in school. It doesn’t matter who you are. As far as I know, not even professional math-people, inclusive of engineers and architects, compute fractions anymore.
Can we even envision a concrete articulation of such a problem? I'll try...
Tom goes to the butcher and buys a three and a third pound roast chicken. His roommate Scott, with inhuman precision, eats two and four ninths pounds. How much is left for Tom, who for inexplicable reasons must know this quantity to the immeasurable and irrelevant accuracy afforded only by unconverted fractions.
Let’s understand why this is worthy of such gripe: in the process of completing this problem, our ten year olds must complete ten constituent steps. This includes deriving the least common denominator, performing two simple and two double-digit multiplication problems, three addition problems, a double-digit long division problem with a remainder and then the isolation of the final answer from that division problem. Depending on the problem, they might also have to do two more division problems to reduce the fraction to lowest terms, bringing us to a full dozen. All must be performed perfectly, as we only give partial credit in high school on up.
Not only is this skill obsolete, it’s not even a prerequisite for ensuing and more important ones. Certainly, I recognize that doing basic operations with fractions is helpful for algebra, but mixed numbers? As far as I know, this skill is at the pinnacle of irrelevance. My students don’t even really need this for sixth grade, let alone later life. Yet I am forced to spend three weeks on it now, because someone, somewhere, decided that understanding how to operate mixed numbers with unlike denominators constitutes “proficiency” in fifth grade math.
To me, the saddest part is this: I can, and I will, teach my fifth graders to master this skill. But why, why, why?!? The list of other, infinitely more valuable aspects of math that I have to leave out (logic, probability, measurement, statistics, just to start…) to make time for this irrelevant crap would make any one who does use math in a meaningful way cry. Imagine the service to humanity if I taught all my fifth graders the difference between mean and median, instead? How many misleading and pointlessly dumb statistics might we be spared? But no…
Now readers, try one and a third subtracted from four and seven-eights… You won't, will you? Because you know it's a waste of your time!
If I am wrong and you are in (or know of) a career where these skills are vital, please correct me. Even with a dream class of budding young mathematicians, teaching this unit is awful. I'd love to provide a reason why it should be more than excruciating tedium. In other words, help me fill in that blank I was stuck in today.
Sunday, February 03, 2008
You Win Some, You Lose Some, Some Don't Play
I could write endlessly about the horrors of the pending budget cuts for education, but I don't think it'd get past: You get what you pay for, Governor, you fool.
So let's just sidestep that issue and talk about the kids.
Report card season just passed and gave me the chance to reflect on my students’ progress. In writing the comments, I realized that I have worked with these students for 280 instructional days, and found myself doubly proud of those who I know I have helped and doubly ashamed of those who I have watched decline. But I also realized anew that there are still students who have made tremendous strides and falls completely on their own, wholly outside of my efforts. No single teacher will ever be able to reach every student in a year, or even two, and I wonder how we can design our elementary schools to address that.
Here, for your enjoyment, are two of each kind of story.
S--- was speedily racing down the road to Little Miss Bitch last year, under the influence of a malicious and manipulative friend. But the “friend” left and now S--- is excelling. She earnestly told me at the beginning of this year that she realized she was distracted last year and she was going to change. I had my doubts, but she has erased them. Her hand is in the air constantly. Now, as former friends of hers are starting to turn catty again, she is staying calm and focused on learning. I had nothing to do with this, aside from quietly applauding her good choices along the way. Even her mother doesn’t claim credit, recognizing that her daughter’s change came from within. S--- has forced me to hold out hope for tween girls everywhere.
G--- is one of the few students I know who seems to have regressed in his English Language Development. His speech is less and less comprehensible each week. His writing has become completely undecipherable. I’ve asked him to read it back to me and even he, often, cannot interpret the seeming random combination of letters. He is all heart, very attentive, diligent on all tasks and eager to learn, but the learning, and now apparently even the language, never stays. He worked with me every morning last year on basic English and sight words, he receives two hours a day of small group language instruction this, he practiced furiously on his multiplication last year and is in a small, intensive math class this year. All to no avail, he can hardly parse multi-syllabic words and he still only knows his facts to 5. All of the adults in his life, from his parents to the psychologists, are mystified.
R--- and I are trapped in a vicious cycle. I’m hard on him because he’s a capable kid who doesn’t focus. He doesn’t learn because I’m hard on him and then has trouble focusing because he’s frustrated. He knows the math class is leaving him behind. (The trouble with great self-tracking systems is that they often tell students rather discouraging things.) I’ve tried to praise him; I’ve moved him to the front, back and side of the room; I’ve called and written home. We’re stuck in a loop. I’m going to try and be all smiles on Monday and see if we can start the week well. But it’s the 100th day of school and we’re running out of time.
I’m going to take credit for the success of E---. Shamelessly, I am. In third grade, he was a screaming miscreant who had to be carried into school and bodily held in the seat. In fourth grade, he learned next to nothing because every day was an effort to push boundaries and get attention by any means necessary. He missed a week of school to avoid consequences for his misdeeds. This year, I brought him in before school started to work with me on getting the room set up. I lavished him with attention. I visited him and his Dad at home and we talked about how to work together. For the first weeks, I cut him enough slack so that he didn’t constantly end up knee deep in detention, but held the line firm enough to keep the class moving. I steeped him in praise to the class and to his Dad. Dad was happy, teacher was happy, class was happy and he didn’t want to blow it. Now, I’ve gradually pulled his long tether towards academics until E--- is writing 5-paragraph essays and participating in high-level discussions of our stories. I’ve even started to withdraw the praise because I can tell he doesn’t want his newly acquired “good reader” status to become clear to his too-cool-for-school friends. That's fine, he and I still know the truth.
So let's just sidestep that issue and talk about the kids.
Report card season just passed and gave me the chance to reflect on my students’ progress. In writing the comments, I realized that I have worked with these students for 280 instructional days, and found myself doubly proud of those who I know I have helped and doubly ashamed of those who I have watched decline. But I also realized anew that there are still students who have made tremendous strides and falls completely on their own, wholly outside of my efforts. No single teacher will ever be able to reach every student in a year, or even two, and I wonder how we can design our elementary schools to address that.
Here, for your enjoyment, are two of each kind of story.
S--- was speedily racing down the road to Little Miss Bitch last year, under the influence of a malicious and manipulative friend. But the “friend” left and now S--- is excelling. She earnestly told me at the beginning of this year that she realized she was distracted last year and she was going to change. I had my doubts, but she has erased them. Her hand is in the air constantly. Now, as former friends of hers are starting to turn catty again, she is staying calm and focused on learning. I had nothing to do with this, aside from quietly applauding her good choices along the way. Even her mother doesn’t claim credit, recognizing that her daughter’s change came from within. S--- has forced me to hold out hope for tween girls everywhere.
G--- is one of the few students I know who seems to have regressed in his English Language Development. His speech is less and less comprehensible each week. His writing has become completely undecipherable. I’ve asked him to read it back to me and even he, often, cannot interpret the seeming random combination of letters. He is all heart, very attentive, diligent on all tasks and eager to learn, but the learning, and now apparently even the language, never stays. He worked with me every morning last year on basic English and sight words, he receives two hours a day of small group language instruction this, he practiced furiously on his multiplication last year and is in a small, intensive math class this year. All to no avail, he can hardly parse multi-syllabic words and he still only knows his facts to 5. All of the adults in his life, from his parents to the psychologists, are mystified.
R--- and I are trapped in a vicious cycle. I’m hard on him because he’s a capable kid who doesn’t focus. He doesn’t learn because I’m hard on him and then has trouble focusing because he’s frustrated. He knows the math class is leaving him behind. (The trouble with great self-tracking systems is that they often tell students rather discouraging things.) I’ve tried to praise him; I’ve moved him to the front, back and side of the room; I’ve called and written home. We’re stuck in a loop. I’m going to try and be all smiles on Monday and see if we can start the week well. But it’s the 100th day of school and we’re running out of time.
I’m going to take credit for the success of E---. Shamelessly, I am. In third grade, he was a screaming miscreant who had to be carried into school and bodily held in the seat. In fourth grade, he learned next to nothing because every day was an effort to push boundaries and get attention by any means necessary. He missed a week of school to avoid consequences for his misdeeds. This year, I brought him in before school started to work with me on getting the room set up. I lavished him with attention. I visited him and his Dad at home and we talked about how to work together. For the first weeks, I cut him enough slack so that he didn’t constantly end up knee deep in detention, but held the line firm enough to keep the class moving. I steeped him in praise to the class and to his Dad. Dad was happy, teacher was happy, class was happy and he didn’t want to blow it. Now, I’ve gradually pulled his long tether towards academics until E--- is writing 5-paragraph essays and participating in high-level discussions of our stories. I’ve even started to withdraw the praise because I can tell he doesn’t want his newly acquired “good reader” status to become clear to his too-cool-for-school friends. That's fine, he and I still know the truth.
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