Many of my students' go-to strategies are on the lower level of different types of problem solving. This is not to say that my students' strategies are less effective than those who use a sophisticated algorithm. I have seen this assumption, or the inference of this assumption, made by some General Education teachers when they make comments like, "I don't have time to pass out manipulatives" or "Letting them use the blocks takes so much time out of my instructional time". It's difficult not to sympathize with these teachers because they make some valid points; I struggle even offering wait time to 5 students in my 30 minute lessons. Teaching time is a precious gift, but fundamentally I can't deny a student their right to think the way they prefer. We have to stretch our minds to overcome the difficulties we face and differentiate our methods for our students. They are what truly matter, not our timetables.
I have a student who offers quite a different take on addition and subtraction-or does she? When I first met her I assumed she was just confused about which direction to start her computations. I observed her teachers shut down all discussion or communication about this apparent "mistake". However, I got to thinking...perhaps my student was using an inventive strategy to solve the problem. When I look back at her Brigance pre-test I notice that on a few of the problems that require difficult regrouping my student attempts to start the problem on the left and work right. Perhaps she was attempting to use mental math and known facts to work around the difficult regrouping? For example, given a problem like 32-19, the student could use her knowledge of 32-20=?, and then added 1 to even out the equation. I will never be sure of the truth because the talk moves exhibited by her teachers (which should be nicknamed "extinguish") have ceased all discussion. I feel as though I should ask the student to explain her thinking in my small group instruction, however I worry that by doing so my student will be singled out by her teacher as doing it "wrong" again. I suppose it comes down to feeling as though I lack the authority to so drastically change the model of instruction in another's classroom. Perhaps one day this lesson will prove fruitful for me and my class.
My student in the Problem Solving Interview did not have any invented strategies, but relied upon direct modeling or (for the easy numbers) recalled facts. I wish there was more relevant information I could write about my interview, but sadly there isn't.
Monday, October 18, 2010
Tuesday, September 28, 2010
Math Talk Moves
the idea of a place value chart as a strategy. I thought this was a
sweet way of putting the spotlight on this student's
cleverness, and disguising what makes them different from others in the classroom. I also heard the teacher asking students to use the revoicing strategy, and the restating someone else's strategy. The latter was used primarily to call out students who were off-task and not listening. Though, occasionally she asked a student who is struggling to repeat their neighbor's strategy, in the hopes of them picking up some new tips. In the cases where it was a behavior management tool, I still see it as beneficial for those who are paying attention to hear it explained again, by someone else and in different words. 
Tuesday, September 21, 2010
Stepping Into Teaching
So, here's (1st picture) one of my students working on a place value game in their General Education Math class. The game involves taking turns rolling a 9-sided die to fill up a certain amount of place value blanks. First, the group of students (4 kids at their desks which are already grouped together) decides how many place value blanks they want to work towards. Example: this group here decided they wanted to work towards the hundredths place since they had to include a decimal in their game for today. They take turns rolling the die, and place the number they rolled into one of the blanks; it's their choice where to place it.
Here (2nd picture), my student placed an 8 in the ones place. As the teachers and I walk around we ask the students to identify the place values of the numbers they've rolled. This is an excellent way to informally assess students' understanding of place value. It's also a prime opportunity for students to teach each other and/or model their strategies.
Afterward (3rd picture), the students line up in order from least to greatest, or vice-versa. Here the teacher has partnered with two struggling students as they play. She later left them with the group at the next table once she felt they had a sufficient understanding of the concept. This group then had 6 students in it which made for a challenging team collaboration, but also a very rewarding challenge for the group in the end.
Here's (4th picture) the first group's finished work. Afterward, they expanded their problem to the thousands place (before the decimal) and the thousandths place( after the decimal). It also became a contest as to who could write the largest number; the nine on the 9-sided die became a coveted roll. Note to class, this was not a teacher-planned competition. It appears that the "greatest" amount meant more than place value to my students!
Finally (5th picture), once the game had ended the students returned to their seats to correct their quizzes for partial credit. They're allowed to use their books to help solve the problems, and they work alone or with a teacher. One decision I greatly appreciate is my teacher's choice to give full credit on homework if the students correct their work (with a red pen) while they go over it the day it's due. As long as the student tried and is willing to learn from their mistakes, she says there's no reason not to give them a 100%. If they didn't even attempt it, they have to make it up at a later time without full credit.
Shown here and below (6th picture until last picture) are the accommodations my students with IEPs use for every task. They're laminated folders with several different charts, mnemonic devices, and facts pasted onto them. the students may use them during any activity; quizzes, tests, warm ups, group activities-anything! I think they're really wonderful tools for these students who are working towards accountability and self-monitoring.
So, it's obvious that I don't have pictures from the halls, walls, or facade of the building...truth is I didn't feel comfortable taking these sorts of pictures. There's always a class in the hallways and their artwork lines the walls; there wasn't a good picture to take if I had to avoid faces and names! Regardless, my school is very laid back, friendly, and proud of their students. Everywhere you look there's a student's name posted for a chivalrous deed (the school's mascot is a knight, so they earn notoriety/respect- their name is announced on the morning announcements if they're caught performing a chivalrous deed), examples of the class' projects, and educational posters (ex: 3 Science questions for the month; students can turn in their answers to the front office for a raffle held at the end of the month.) It's a warm place; one I consider applying to even if Steve and I move to Central Austin. We shall see. :)
Thursday, September 2, 2010
Response to Readings Class 9/2/10
1) Problem solving involves exploration and discovery. While there is a need for some prior knowledge, it's not necessary to teach the skills to solve the problems first. Teaching occurs during the trials, mistakes, and final understandings after the explorations.
2) My personal feelings towards math will impact my teaching skills and the way my students view me as a competent, comfortable teacher. If it's clear in my actions or expressions that I'm not confident in my skills, this is easily passed on to my students. They in turn may become nervous, unconfident learners.
3) Students, given the proper guidance and problems to solve, will not have to reinvent the wheel. Much of the responsibility falls on the teacher's shoulders to keep the discussion moving at a decent pace, keep the discussion on topic, and keeping the students motivated. It can be done!
4) When a student has struggled with a concept for a long period of time, a good idea may be to have a peer offer to help, or to ask probing questions which might lead the student onto the right track. Sometimes a simple hint, leading question, or statement can guide someone to new insights and possibilities.
5) I read this article so long ago that I can't remember specific examples. I know that the researchers used modeling to assist students' learning; more specifically, they used manipulatives. They also allowed for student discussions in small groups during the problem solving activities.
2) My personal feelings towards math will impact my teaching skills and the way my students view me as a competent, comfortable teacher. If it's clear in my actions or expressions that I'm not confident in my skills, this is easily passed on to my students. They in turn may become nervous, unconfident learners.
3) Students, given the proper guidance and problems to solve, will not have to reinvent the wheel. Much of the responsibility falls on the teacher's shoulders to keep the discussion moving at a decent pace, keep the discussion on topic, and keeping the students motivated. It can be done!
4) When a student has struggled with a concept for a long period of time, a good idea may be to have a peer offer to help, or to ask probing questions which might lead the student onto the right track. Sometimes a simple hint, leading question, or statement can guide someone to new insights and possibilities.
5) I read this article so long ago that I can't remember specific examples. I know that the researchers used modeling to assist students' learning; more specifically, they used manipulatives. They also allowed for student discussions in small groups during the problem solving activities.
Tuesday, August 31, 2010
My Math Life Story
Peak Experience ~
In my early years of schooling I found math to be easy; just another part of the day when I learned fun, interesting facts. I excelled in the games we played and in the practical expressions of mathematics (counting money, for example). Looking back now I find that almost every time I grasped a concept in those classes, manipulatives were heavily involved (or just provided). Actually seeing and visualizing how an object may represent a numerical value was crucial for this kinesthetic learner. I need something that grounds theory to real-life; something I can hold in my hands. I also had a series of teachers who were kind, supportive, and encouraging. I was a pretty sensitive, anxious child, so to look back at myself and see a confident mathematician is a huge compliment to my teachers! Just goes to show you that getting to know your students' personalities thoroughly will come back to hug you in the end. Sadly, this peak experience was quite brief...
Nadir Experience ~
Oh where to begin....I suppose the big switch came when the very basics of Algebra were introduced in 4th or 5th grade. I have a very vivid memory of the teacher handing out laminated papers with scales drawn on them, and lots of chess pieces (pawns and bishops). I got excited for a (fleeting) moment since my best friend had just taught me to play chess and I wanted to impress my teacher with my mad skills (HA!). However, I'm not sure if the teacher neglected to introduce the manipulatives and their respective values thoroughly, but I had NO CLUE what they stood for (Can you explain to me how/why a pawn stands for -1???). It stayed that way for the entire unit because, try as I might, I couldn't 'balance' my stupid scale correctly with those manipulatives. These units yielded the worst grades I'd ever received in math thus far....little did I know that they would continue to drop.
Then (dun-dun-duuuunnn!), my Dad tried to help me with my homework when my grades suffered and it went horribly wrong...lets just say it's a miracle we still speak to each other. ("Why can't you get this?! It's EASY!" is not something you ask your child while they're crying over their homework.)
Later on, when I was failing 7th grade math, I was told by my female math teacher that, "It's ok you're not good at math, you're a girl." This statement, needless to say, did NOT improve my outlook on the subject. I figured I was screwed by my genetic imprint, so why set myself up for failure? I took regulars classes from then on, ending high school with Algebra II under my belt (barely). In college I attempted to take college level Algebra 3 TIMES before I succeeded. When I realized my colleagues excelled and surpassed me after 2/3 of the course had passed, I dropped the course. At that point my comprehension of the topics was reduced to almost nothing.
Turning Point ~
In college I made 2 discoveries about myself as a mathematician: First, I need someone who is removed from my situation to tutor me. They don't judge you outwardly, they're more patient, and they usually have experience explaining these sorts of topics to people. I went frequently! Secondly, I need a teacher who doesn't drone on and on about the same topic forever. Nothing will make my mind wander faster than a subject that bores me (sorry Teddy!) and a professor who can't move on to a new topic when it's been beaten to death already (it's dead, give it up!). I used to find my myself trying to catch up halfway through a new topic because I'd lost interest in the previous topic. My college Algebra professor, Steven Alwin, kept a perky pace (as did my mind), an enthusiastic attitude, and a supportive demeanor. I passed with an +B (and he made us really work for it!)!
Another turning point occurred when I took a course that focused on teaching mathematical strategies for elementary-aged children. I hadn't thought about the reasons why we perform the steps we do when we solve an equation; this course made me think past the "memorization", "don't ask questions", "this is THE way to solve these problems" methods I've always hated/resented I really appreciated a course where we were encouraged to explore different ways of reaching a solution, and then to accept all of them.
Other Experiences ~
Here's a funny one: I will NEVER forget my freshman year regular Algebra professor...wherever he is, I hope he knows I still feel pity for him. It was his first year of teaching, and they WALKED ALL OVER HIM! I sat in the front row, center, headphones on (to tune out my rude classmates), and lip-read my way through the course. I didn't do very well in that class either; it wasn't that he was a bad teacher, he just never stood a chance against 25 rowdy 14 year-olds. Poor guy. However, it was in this classroom that I made an important self discovery that wasn't strictly attached to mathematics: I will fight for my education (come on Cohort F, you saw what happened in our Reading class!). One day I'd had enough of the boy in the desk behind me screwing around, shouting at the top of his lungs. So, before I knew it I was on my feet, red-faced and shouting right back at him, "Sit down and shut the hell up!" Now, I'd never even spoken without raising my hand in that class -which probably added to my mystique and the internal struggle going on in this boy's mind (how crazy will this chick go on me if I don't shut up?)- because once he realized it was me shouting at him he actually shut up! It was the quietest class we'd ever had, and after that my professor would threaten to sick me on students who acted up. I don't think I knew how much I cared about my education until that moment, but it was a relief to know I'm a passionate fighter when it's threatened.
My Greatest Challenge ~
I believe conquering the phrase, "It's ok you're not good at math, you're a girl" was a huge challenge. It doesn't help that our society is so wiling to accept, even embrace, the "I'm no good at math" mentality. It took me 22 years to discover that I could do math MY WAY (I do what I want!); in a way that makes sense to me! I also see this as a challenge in the future because I do not believe that this mentality has diminished all these years later. It will fall on my shoulders and on the shoulders of my colleagues to stamp it out completely. I can only hope that our influence on students will be stronger than decades of excuses, billions of parents shrugging away low grades, and millions of students' wills that are crumbling in the face of mathematics. My resolve is strong; I'm a fighter, after all.
As a Special Education Teacher ~
I feel it's my responsibility as a human being to treat others equally, which really explains my attraction to Special Education. I am a ready and willing advocate for those who are not able to be one for themselves. I found that my will to receive my education extends to those around me. I have witnessed and encountered many "professionals" who have denied services to those who (legally) require them to succeed, and I have seen those individuals fail. My heart is pained by the memory of those experiences, so I set out to do it right. No one is perfect, but I believe I am receiving one of the best educations in the field of Special Education and I couldn't ask to be better prepared for what I will face. My knowledge will become others' power, and I find that extremely rewarding.
In regards to mathematics, I have seen too many children become discouraged because they cannot relate their knowledge to the various problem solving methods. It is exciting to think of passing on the tools they will need to succeed, to help them solve problems as they see fit.
It feels as though it's coming full circle for me; the struggling learner I was, the confident learner I've become, and the students who are struggling and are just waiting for a teacher like me (US!) to assist them on their journeys through mathematics.
In my early years of schooling I found math to be easy; just another part of the day when I learned fun, interesting facts. I excelled in the games we played and in the practical expressions of mathematics (counting money, for example). Looking back now I find that almost every time I grasped a concept in those classes, manipulatives were heavily involved (or just provided). Actually seeing and visualizing how an object may represent a numerical value was crucial for this kinesthetic learner. I need something that grounds theory to real-life; something I can hold in my hands. I also had a series of teachers who were kind, supportive, and encouraging. I was a pretty sensitive, anxious child, so to look back at myself and see a confident mathematician is a huge compliment to my teachers! Just goes to show you that getting to know your students' personalities thoroughly will come back to hug you in the end. Sadly, this peak experience was quite brief...
Nadir Experience ~
Oh where to begin....I suppose the big switch came when the very basics of Algebra were introduced in 4th or 5th grade. I have a very vivid memory of the teacher handing out laminated papers with scales drawn on them, and lots of chess pieces (pawns and bishops). I got excited for a (fleeting) moment since my best friend had just taught me to play chess and I wanted to impress my teacher with my mad skills (HA!). However, I'm not sure if the teacher neglected to introduce the manipulatives and their respective values thoroughly, but I had NO CLUE what they stood for (Can you explain to me how/why a pawn stands for -1???). It stayed that way for the entire unit because, try as I might, I couldn't 'balance' my stupid scale correctly with those manipulatives. These units yielded the worst grades I'd ever received in math thus far....little did I know that they would continue to drop.
Then (dun-dun-duuuunnn!), my Dad tried to help me with my homework when my grades suffered and it went horribly wrong...lets just say it's a miracle we still speak to each other. ("Why can't you get this?! It's EASY!" is not something you ask your child while they're crying over their homework.)
Later on, when I was failing 7th grade math, I was told by my female math teacher that, "It's ok you're not good at math, you're a girl." This statement, needless to say, did NOT improve my outlook on the subject. I figured I was screwed by my genetic imprint, so why set myself up for failure? I took regulars classes from then on, ending high school with Algebra II under my belt (barely). In college I attempted to take college level Algebra 3 TIMES before I succeeded. When I realized my colleagues excelled and surpassed me after 2/3 of the course had passed, I dropped the course. At that point my comprehension of the topics was reduced to almost nothing.
Turning Point ~
In college I made 2 discoveries about myself as a mathematician: First, I need someone who is removed from my situation to tutor me. They don't judge you outwardly, they're more patient, and they usually have experience explaining these sorts of topics to people. I went frequently! Secondly, I need a teacher who doesn't drone on and on about the same topic forever. Nothing will make my mind wander faster than a subject that bores me (sorry Teddy!) and a professor who can't move on to a new topic when it's been beaten to death already (it's dead, give it up!). I used to find my myself trying to catch up halfway through a new topic because I'd lost interest in the previous topic. My college Algebra professor, Steven Alwin, kept a perky pace (as did my mind), an enthusiastic attitude, and a supportive demeanor. I passed with an +B (and he made us really work for it!)!
Another turning point occurred when I took a course that focused on teaching mathematical strategies for elementary-aged children. I hadn't thought about the reasons why we perform the steps we do when we solve an equation; this course made me think past the "memorization", "don't ask questions", "this is THE way to solve these problems" methods I've always hated/resented I really appreciated a course where we were encouraged to explore different ways of reaching a solution, and then to accept all of them.
Other Experiences ~
Here's a funny one: I will NEVER forget my freshman year regular Algebra professor...wherever he is, I hope he knows I still feel pity for him. It was his first year of teaching, and they WALKED ALL OVER HIM! I sat in the front row, center, headphones on (to tune out my rude classmates), and lip-read my way through the course. I didn't do very well in that class either; it wasn't that he was a bad teacher, he just never stood a chance against 25 rowdy 14 year-olds. Poor guy. However, it was in this classroom that I made an important self discovery that wasn't strictly attached to mathematics: I will fight for my education (come on Cohort F, you saw what happened in our Reading class!). One day I'd had enough of the boy in the desk behind me screwing around, shouting at the top of his lungs. So, before I knew it I was on my feet, red-faced and shouting right back at him, "Sit down and shut the hell up!" Now, I'd never even spoken without raising my hand in that class -which probably added to my mystique and the internal struggle going on in this boy's mind (how crazy will this chick go on me if I don't shut up?)- because once he realized it was me shouting at him he actually shut up! It was the quietest class we'd ever had, and after that my professor would threaten to sick me on students who acted up. I don't think I knew how much I cared about my education until that moment, but it was a relief to know I'm a passionate fighter when it's threatened.
My Greatest Challenge ~
I believe conquering the phrase, "It's ok you're not good at math, you're a girl" was a huge challenge. It doesn't help that our society is so wiling to accept, even embrace, the "I'm no good at math" mentality. It took me 22 years to discover that I could do math MY WAY (I do what I want!); in a way that makes sense to me! I also see this as a challenge in the future because I do not believe that this mentality has diminished all these years later. It will fall on my shoulders and on the shoulders of my colleagues to stamp it out completely. I can only hope that our influence on students will be stronger than decades of excuses, billions of parents shrugging away low grades, and millions of students' wills that are crumbling in the face of mathematics. My resolve is strong; I'm a fighter, after all.
As a Special Education Teacher ~
I feel it's my responsibility as a human being to treat others equally, which really explains my attraction to Special Education. I am a ready and willing advocate for those who are not able to be one for themselves. I found that my will to receive my education extends to those around me. I have witnessed and encountered many "professionals" who have denied services to those who (legally) require them to succeed, and I have seen those individuals fail. My heart is pained by the memory of those experiences, so I set out to do it right. No one is perfect, but I believe I am receiving one of the best educations in the field of Special Education and I couldn't ask to be better prepared for what I will face. My knowledge will become others' power, and I find that extremely rewarding.
In regards to mathematics, I have seen too many children become discouraged because they cannot relate their knowledge to the various problem solving methods. It is exciting to think of passing on the tools they will need to succeed, to help them solve problems as they see fit.
It feels as though it's coming full circle for me; the struggling learner I was, the confident learner I've become, and the students who are struggling and are just waiting for a teacher like me (US!) to assist them on their journeys through mathematics.
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