I received this email from Gary Davis with some very points in response to my post on student attention and justification of answers. I think the points are important and I want to pursue them when I have a little more time:
* The teacher (that's you!) said that many of the students showed serious misunderstandings of variables and constants and of fractions and the algebra tiles after he completed a unit. One might say that it comes from many years of rote learning in earlier grades.
* If he is their first teacher who is asking them to think and build their own ideas, it could be that they are in an early cycle of learning these mathematical ideas and need more experiences. We are also not sure by reading this about this particular teacher's expertise in guiding students in developing conceptual knowledge.
* He speaks about putting his foot down and insisting on justification. I'm not sure what that means to him. We have many teachers in our courses that know the buzz words and can talk the talk but when we go in to observe their classes, their actions do not match their words.
* The other thing he speaks about is the students being inattentive so the ways the students are engaged during different moments of his classes and how he handles this may also be factoring into the equation.
* I think it is common for parents not to understand this approach to learning and many teachers receive parent resistance. You remember I did too, when I was a teacher.
Showing posts with label philosophy. Show all posts
Showing posts with label philosophy. Show all posts
Wednesday, December 9, 2009
I am reading Lev Vytgoski
I am also in a moment of deep thought about my belief in constructivism, particularly in math and most particularly in the Middle School age range.
I watch, year after year, how my students superficially work with the algebra models I present them, such as algebra tiles, and then quickly disassociate from these model when they start doing "real algebra" (moving letters and numbers around is "real" to them, and to society as well).
The more I learn about Singapore math, the more intrigued and questioning I am about this model of math education. It isn't constructivist, as I understand, to present the students with exclusive models to think about, but on the other hand, it is through these models that the students often DO "construct" their understanding. Maybe I've been adhering to a very pure understanding of constructivist theory that is divorced from the reality of real learners.
I watch, year after year, how my students superficially work with the algebra models I present them, such as algebra tiles, and then quickly disassociate from these model when they start doing "real algebra" (moving letters and numbers around is "real" to them, and to society as well).
The more I learn about Singapore math, the more intrigued and questioning I am about this model of math education. It isn't constructivist, as I understand, to present the students with exclusive models to think about, but on the other hand, it is through these models that the students often DO "construct" their understanding. Maybe I've been adhering to a very pure understanding of constructivist theory that is divorced from the reality of real learners.
Tuesday, December 8, 2009
Rigorous Math Education
Today I gave my 8th graders a summative exam of the algebra topics we've been covering. This mainly included solving for x as well as solving for y to set up a y=mx+b linear equation.
They have not been terribly attentive, to say the least, to the activities we do in class, so it was with some level of trepidation that I decided to give them the exam. I am not a believer it testing for testing sake, nor for holding grades or exams over my students' heads as threats. But I have to admit that my student culturally go there without me doing anything.
Anyway, many of them showed serious misunderstandings of variables and constants, of fractions and of the algebra tiles we've been using. I saw in front of me the many various misunderstandings of algebra that are interfering with their deeper math education. My admin as well as the parents will quickly tell me that it is my fault: that I am not telling them "how to do it". I will say that it is not the "how" that is holding them up, but the "why". They'll say that I must not be explaining the "why"well enough. I will have to consider this carefully, because every teacher should be humble enough to realize that they are not walking on water when it comes to presenting material. But down inside I know I follow all the best practices associated with conceptual learning, and still my students resist (at times, reject) it.
So it all gets me thinking of the intersection between mass culture and education. If my students and families are hell bent on the "how" (because they learned it that way and because that is the "only" way math should be taught), but I am putting my foot down and insisting on justification and conceptual knowledge, then we find ourselves irreconciably miscommunicating.
It gets me thinking, too, about at what level we, as American educators, can hope to really teaching a rigorously conceptual math program along the level of Singapore Math. This is what I wrote in an earlier post:
When we look at something like math and we don't see what we remember from school, we wonder about its validity. Did it really get "watered down" like all the news reports have been saying. Are modern kids being asked to do anything like the rigorous curriculum we used to work through?
One precept of modern math instruction is that conceptual understanding not only supports math methods, but also should be the precursor much in the way understanding a cat from a holistic standpoint is the important first step before understanding it from a genetic perspective.
I want to believe that I will be able to teach the "real stuff" and not get easily blamed for the "hard" nature of learning math. This evening, though, I can't help but think it might be a fantasy.
When is the next plane to Singapore leaving?
a
They have not been terribly attentive, to say the least, to the activities we do in class, so it was with some level of trepidation that I decided to give them the exam. I am not a believer it testing for testing sake, nor for holding grades or exams over my students' heads as threats. But I have to admit that my student culturally go there without me doing anything.
Anyway, many of them showed serious misunderstandings of variables and constants, of fractions and of the algebra tiles we've been using. I saw in front of me the many various misunderstandings of algebra that are interfering with their deeper math education. My admin as well as the parents will quickly tell me that it is my fault: that I am not telling them "how to do it". I will say that it is not the "how" that is holding them up, but the "why". They'll say that I must not be explaining the "why"well enough. I will have to consider this carefully, because every teacher should be humble enough to realize that they are not walking on water when it comes to presenting material. But down inside I know I follow all the best practices associated with conceptual learning, and still my students resist (at times, reject) it.
So it all gets me thinking of the intersection between mass culture and education. If my students and families are hell bent on the "how" (because they learned it that way and because that is the "only" way math should be taught), but I am putting my foot down and insisting on justification and conceptual knowledge, then we find ourselves irreconciably miscommunicating.
It gets me thinking, too, about at what level we, as American educators, can hope to really teaching a rigorously conceptual math program along the level of Singapore Math. This is what I wrote in an earlier post:
When we look at something like math and we don't see what we remember from school, we wonder about its validity. Did it really get "watered down" like all the news reports have been saying. Are modern kids being asked to do anything like the rigorous curriculum we used to work through?
One precept of modern math instruction is that conceptual understanding not only supports math methods, but also should be the precursor much in the way understanding a cat from a holistic standpoint is the important first step before understanding it from a genetic perspective.
I want to believe that I will be able to teach the "real stuff" and not get easily blamed for the "hard" nature of learning math. This evening, though, I can't help but think it might be a fantasy.
When is the next plane to Singapore leaving?
a
Sunday, December 6, 2009
Math classes are workshops, whether you want it or not.
Students are inquiring, investigating, discussing, and constructing knowledge (just maybe not the exact knowledge we thought). The put forth their ideas in the community of their peers and justify and defend their thinking. In this sense, students are acting like real mathematicians who speak to one another. They ask questions and comment on another's ideas. They defend their ideas to the community, not just to the teacher.
So, let's not try to fix the mathematics, but rather, work with the mathematician. The point is not to get all of the students to agree with your answer, or to fix the mistakes in their products, but to support their development as mathematicians.
a
So, let's not try to fix the mathematics, but rather, work with the mathematician. The point is not to get all of the students to agree with your answer, or to fix the mistakes in their products, but to support their development as mathematicians.
a
Wednesday, November 4, 2009
Taming the Wild Corner, version 4.0
Nature is marvelous and in short supply in San Francisco. My school is bordered by a freeway to the east and many blocks of treeless streets in the other directions. However, as you pass through the front gate, you find a fascinating garden we call the Adventure Playground. I use this garden to consider the nature of school curriculum in general and math instruction more specifically.
David Perkins describes a continuum between tame and wild ideas in school curricula. Tame ideas are closed learning experiences (e.g. five paragraph essays, long division and textbook science) Wild ideas are unpredictable experiences. These can be harder to identify in most classrooms, though examples such as writers’ workshop, project based math and inquiry science certainly are alive and well in many schools.
There is a huge diversity of garden aesthetics around the world, but consider present day California. In irrigated lawn is a tame garden in arid California: little diversity and great predictability. It produces minimal benefit other than the satisfaction of having sculptured a landscape. At the other extreme, the very “wildest” of gardens would be nature left to its own resources. It produces some food which might be difficult to harvest and potentially dangerous to the harvester.
A garden designed to produce food as well as pleasing aesthetics would fall somewhere between these two extremes. This garden represents the sort of sweet spot between wild and tame that many educators aim for. Yet there a wide range of opinions of what appropriate curricula looks like.
Educators work to produce a rich bounty of ideas. They are charged with “taming the wild” so that students can make sense of things. The question that always needs to be asked is: have some ideas been tamed too much? Has the productive garden turned into the irrigated lawn?
Few subjects in schools inspire more “taming of the wild” than mathematics. While the field of mathematics is a vibrantly wild one, it has a long history of tameness in school. Take the classic rhyme to remember how to divide fractions: “Yours is not to question why, simply invert and multiply.” Critical thinking is often weeded from math, It feels neat, defined, and controlled. For many, it feels like the math we learned in school.
But there is a romantic notion about “real world” math that feels entirely wild. Students construct their own understandings and methodologies in math. Some are successful, while many struggle with this approach.
I have seen the effects of overly tame or unduly wild teaching. I have carefully parsed out equations for determining slope of a line without letting them muck around in the patterns. Later, I would observe students freeze when faced with similar problems out of context. They were starving on a flawless lawn.
At the other extreme, I would ask my 4th and 5th graders to “invent” different methods for multiplying multi-digit numbers. Some students did have inventive ways of doing this arithmetic work, but their successes rested more on their home experiences than on their inventive minds in class. These students were starving in a dark and lonely jungle.
We must tread thoughtfully in a zone between the excessively tame and the dangerously wild ideas. I have found a comfortable balance with Problems of the Week (POW’s). These problems are complex, messy, somewhat obscure but not impossible to solve if one persists. One of my favorite POW problems involves a camel crossing a desert:
Camila Camel's harvest consists of 3000 bananas. The market place is 1000 miles away. Camila must walk to the market and can only carry up to 1000 bananas at a time. Being a camel, Camila eats one banana during each and every mile she walks (so Camila can never walk anywhere without bananas).
How many bananas can Camila get to the market?
This problem is wild because it is not solvable by simple algorithms, yet it sufficiently tame so that many people have some entry point to start it. While the necessary math skills are not complex, their application often inspires creativity. It has a best answer but actually there are many good answers that are acceptable approximations.
If we succumb to breaking down math to its bare components, we teach how to take care of a lawn rather than promote diverse gardens. Let’s look at math from an organic gardening perspective. Let’s wild the tame corner!
David Perkins describes a continuum between tame and wild ideas in school curricula. Tame ideas are closed learning experiences (e.g. five paragraph essays, long division and textbook science) Wild ideas are unpredictable experiences. These can be harder to identify in most classrooms, though examples such as writers’ workshop, project based math and inquiry science certainly are alive and well in many schools.
There is a huge diversity of garden aesthetics around the world, but consider present day California. In irrigated lawn is a tame garden in arid California: little diversity and great predictability. It produces minimal benefit other than the satisfaction of having sculptured a landscape. At the other extreme, the very “wildest” of gardens would be nature left to its own resources. It produces some food which might be difficult to harvest and potentially dangerous to the harvester.
A garden designed to produce food as well as pleasing aesthetics would fall somewhere between these two extremes. This garden represents the sort of sweet spot between wild and tame that many educators aim for. Yet there a wide range of opinions of what appropriate curricula looks like.
Educators work to produce a rich bounty of ideas. They are charged with “taming the wild” so that students can make sense of things. The question that always needs to be asked is: have some ideas been tamed too much? Has the productive garden turned into the irrigated lawn?
Few subjects in schools inspire more “taming of the wild” than mathematics. While the field of mathematics is a vibrantly wild one, it has a long history of tameness in school. Take the classic rhyme to remember how to divide fractions: “Yours is not to question why, simply invert and multiply.” Critical thinking is often weeded from math, It feels neat, defined, and controlled. For many, it feels like the math we learned in school.
But there is a romantic notion about “real world” math that feels entirely wild. Students construct their own understandings and methodologies in math. Some are successful, while many struggle with this approach.
I have seen the effects of overly tame or unduly wild teaching. I have carefully parsed out equations for determining slope of a line without letting them muck around in the patterns. Later, I would observe students freeze when faced with similar problems out of context. They were starving on a flawless lawn.
At the other extreme, I would ask my 4th and 5th graders to “invent” different methods for multiplying multi-digit numbers. Some students did have inventive ways of doing this arithmetic work, but their successes rested more on their home experiences than on their inventive minds in class. These students were starving in a dark and lonely jungle.
We must tread thoughtfully in a zone between the excessively tame and the dangerously wild ideas. I have found a comfortable balance with Problems of the Week (POW’s). These problems are complex, messy, somewhat obscure but not impossible to solve if one persists. One of my favorite POW problems involves a camel crossing a desert:
Camila Camel's harvest consists of 3000 bananas. The market place is 1000 miles away. Camila must walk to the market and can only carry up to 1000 bananas at a time. Being a camel, Camila eats one banana during each and every mile she walks (so Camila can never walk anywhere without bananas).
How many bananas can Camila get to the market?
This problem is wild because it is not solvable by simple algorithms, yet it sufficiently tame so that many people have some entry point to start it. While the necessary math skills are not complex, their application often inspires creativity. It has a best answer but actually there are many good answers that are acceptable approximations.
If we succumb to breaking down math to its bare components, we teach how to take care of a lawn rather than promote diverse gardens. Let’s look at math from an organic gardening perspective. Let’s wild the tame corner!
Tuesday, April 24, 2007
What is Algebra????
Learning algebra is a lot like learning to drive a car, or swim, or ride a bike. You must learn concepts at the same time you are developing skills. There are pedals to push, gears to shift, lights to turn on and off, turn signals to manipulate, and a steering wheel to wrestle. Those are skills.
It’s also useful to learn what the brake pedal does, why you need to use your turn signals, what the clutch is good for, and which direction to steer when the car begins to skid. Those are concepts.
Most of the math you have been learning until this year is based in concepts of arithmetic. Arithmetic relates directly to real life. We add apples and oranges, we subtract money, we discover the area of a wall to learn how much paint will cover it. We might assume algebra is going to be like that, too.
But algebra is different. While it can, indeed, be useful in real life, much of it is simply a game that has no direct relationship to anything we can touch or count in our everyday life. Like many games, algebra has game pieces, moves, strategies, goals, and its own vocabulary. Usually, the object of the game is to discover some specific unknown by using available clues. Other times, the object is to translate recurring events into an equation.
Why must we learn this stuff? One answer is because it’s fun. There are many side benefits to many games. Tennis improves your eye-hand coordination. Racquetball is great aerobic exercise. Golf is meditative and social. Chess teaches concentration.
A side benefit of the algebra game is that it may allow you to become a chemist or electrical engineer. It may allow you to calculate interest rates and design cars and computer systems. Many people say that the thought processes they developed by learning algebra were more useful than any direct application. And, of course, if you want or need to learn more advanced math, algebra will be a prerequisite.
So I invite you to join me as we learn the rules and strategies that will make you winners in the game called algebra.
It’s also useful to learn what the brake pedal does, why you need to use your turn signals, what the clutch is good for, and which direction to steer when the car begins to skid. Those are concepts.
Most of the math you have been learning until this year is based in concepts of arithmetic. Arithmetic relates directly to real life. We add apples and oranges, we subtract money, we discover the area of a wall to learn how much paint will cover it. We might assume algebra is going to be like that, too.
But algebra is different. While it can, indeed, be useful in real life, much of it is simply a game that has no direct relationship to anything we can touch or count in our everyday life. Like many games, algebra has game pieces, moves, strategies, goals, and its own vocabulary. Usually, the object of the game is to discover some specific unknown by using available clues. Other times, the object is to translate recurring events into an equation.
Why must we learn this stuff? One answer is because it’s fun. There are many side benefits to many games. Tennis improves your eye-hand coordination. Racquetball is great aerobic exercise. Golf is meditative and social. Chess teaches concentration.
A side benefit of the algebra game is that it may allow you to become a chemist or electrical engineer. It may allow you to calculate interest rates and design cars and computer systems. Many people say that the thought processes they developed by learning algebra were more useful than any direct application. And, of course, if you want or need to learn more advanced math, algebra will be a prerequisite.
So I invite you to join me as we learn the rules and strategies that will make you winners in the game called algebra.
Thursday, April 19, 2007
Making Algebra “Connect” at the SF School
“Though this be madness, yet there is method in it.”
(William Shakespeare)
(William Shakespeare)
Algebra- the word often evokes a strong reaction from people. What do you remember about algebra? What were the "big mathematical ideas" of algebra? What were you really studying?
In case you cannot answer these questions convincingly, try thinking of algebra as the study of patterns and situations in which change occurs. In algebra, we use what we already know to find what we do not yet know. We do this by using patterns and relationships that already exist between numbers. It is a way to see the patterns that are a part of everyday life. These patterns, changes, and relationships can be analyzed and represented in a variety of ways including the use of words, tables, graphs, and symbols.
Traditionally, the goal of algebra instruction has been teaching procedures for manipulating symbols. These procedures are often meaningless to students who try to survive by memorizing and, thus, only retain the ideas for a short time. There is almost no evidence that students develop algebraic and symbolic reasoning from instruction emphasizing teaching procedures for manipulating symbolic expressions. Development of algebraic ideas can and should take place over a long period of time, prior to attempts to deal solely with abstract symbols.
While it is true that SF School 8th graders study what are considered standard algebra topics, they do so within a problem based curriculum. In addition, algebraic concepts such as the use of coordinate graphs, variables, integers and equations are taught in 6th and 7th grades. The ultimate goal is to graduate students with a strong conceptual and procedural understanding of algebra.
At the SF School we are working on the math scope and sequence to support conceptual understanding through concrete and real world examples. This summer the 4th, 5th, 6th and 7th/8th grade teachers will be meeting for three days to examine the math scope and sequence with a particular eye towards solidifying the continuum of concepts and skills that leads towards a successful 8th grade algebra experience. One specific example is how 4th graders work with the concept of fractions by using fraction circles that physically represent the concept of equivalence that later shows up in arithmetic with fractions in 5th and 6th and solving proportions and balancing equations in algebra. SFS students work on Problems of the Week, which are non-routine and often challenging real world applications of math. Evidence of the students’ work blankets the Middle School Math Room’s wall and windows. They do research into the history of math, such as doing Internet research on the life and times of Pythagoras. They have also investigated the mathematical foundations of their cultural surroundings in a project called SCAMP: Story of a Cultural Artifact from a Mathematical Perspective. Student and their families have commented that they had never really perceived all the math that surrounds their everyday world before doing these projects. The combination of mathematical procedural and conceptual knowledge with the historical context allows students to view math not so much as an invention, but rather, as a discovery of pre-existing numerical relationships. We strive to present math as a vibrant area of growth throughout human history and continuing on to the present and beyond.
One advantage of learning algebra in an independent school is that we are not held to unrealistic testing regimens and instead, we can focus on deep understanding as well as procedural fluency. We can base our model on a student-centered approach that moves from concrete materials gradually to algebraic generalizations along these approximate stages:
Build It ... Extend It ... Picture It ... Table It ... Predict It…
… Graph It ... Generalize It ... Formulate a Rule for It.
… Graph It ... Generalize It ... Formulate a Rule for It.
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