To me, it seems as though our school system is of two minds.
On the one hand, we seek to prepare our youth to have a facility of mathematics. It has been shown that mathematical literacy is a necessity in industry, in finance, and in business. In an economy that finds itself more and more technological, mathematical literacy has become more and more essential. Therefore, it behooves our society to produce workers with increasing facilities of mathematics. To do this, we must discover how people learn best, develop methodologies which will enable us to implement these discoveries, and use them in the classroom to increase facility of mathematics.
On the other hand, the teaching of mathematics is seen by most as synonymous with traditional teaching styles: teacher-centric classrooms that employ the banking method of teaching and learning, memorization, and drill-and-kill homework assignments. There are other teaching methods, true, but those styles are “ineffective”: They produce students who are over-reliant on technologies; these students are button-pushers, not thinkers. They lack the facility of basic skills required of students of mathematics.
The great irony, here, is self-evident: That in a system that wishes to see its students succeed, both sides work against each other, allowing for the inevitable failure of our students. To me, this is why when so many math students enter college—no matter their ethnicity—they find themselves unprepared.
In order to cease the continual failure of our students, changes need to be made. Traditionalists, of course, would deny that the system needs amelioration: After all, not all students are math students. I refute this belief. We no longer live in a world where the least the students must do is read contracts and sign their names. Students must do more. They must be able to compute, negotiate math texts (graphs, equations, spreadsheets), and problem-solve. They must be able to analyze a situation, model it abstractly, use critical thinking to determine a solution, and then determine if that solution has applications in reality. To me, the notion that all students are not math students is antiquated: just as all students are readers and writers, so are all students math students.
Given this notion that all students are math students, I feel that the teaching of math needs to change, not only in high school, but especially in college. But let me be clear just as to what I mean. I am not arguing for an integrated approach to mathematics: I like the Algebra/Geometry/Algebra II course progression, even though I feel that each is at best a misnomer of a three-part high school mathematics sequence. I am, instead, arguing for a revision in approach. I am asking that teaching be more dynamic, less the static instance of teaching found in the college lecture hall.
Allow me to embellish this point: During our NSF group projects, some of my team members felt that the ARISE curriculum would not adequately prepare students for college mathematics in that the curriculum was student-centric. They argued that college mathematics is a singular, non-collective activity. It is all lecture, study, problem sets, and test. There is no group-function. And as such, ARISE would not prepare students for the passivity of college mathematics.
Let me reiterate: “ARISE would not prepare students for the passivity of college mathematics.” The passivity of college mathematics—Is this truly the goal of college mathematics? No. College mathematics aims to develop thinkers—thinkers—who will become captains of industry and pioneers working at the frontiers of mathematics. And yet, how can this aim truly be met if its students are docile, quiet, and passive?
Another problem is inherent: Our society needs not only thinkers, but creativity. It requires vibrant creators of thought who can analyze a problem, interact with others in a group capacity, and together synthesize new and different solutions. But if all thinking is taught using one, uniform and singular methodology—a methodology understood by only by those who think in said way—can originality truly develop? And further, if learning is developed alone—and not collectively—how can we expect our thinkers to work together to further the development of ideas once graduated from college mathematics? Truly, if all these aims are to be met, college mathematics must broaden its approach beyond that of the lecture hall.
At the same time, school mathematics must not be ignored. Unlike college mathematics, there have been efforts to reform it. There needs to be more. If college mathematics’s aim is to produce thinkers, school mathematics’s goals should be similar. Clearly, it needs to develop skills and procedures so that when students reach college, they are second nature. But it needs to do so much more. Skills and procedures are nice, but they mean nothing without context. It is like much like Plato’s metaphorical cave. Teaching skills and procedures are like looking at shadows on a cave wall—they mean nothing without a context. School mathematics should provide that context. It should make the math real.
And maybe if it were real, more students would understand it. Or try to.
Wednesday, April 15, 2009
Wednesday, April 1, 2009
NSF-Curricula
Somehow I let this blog get away from me. It must be that time of year. I apologize for that.
Anyways, as many others in the class have noticed, with some differences, each of the curricula presented last Thursday shared many themes. On a whole, they all relied on the constructivist model of learning—student-centric classrooms with the teacher as a guide, cooperative group work, and open-ended contextual activities to motivate learning. This was unsurprising as each curriculum grew out of the NCTM standards which through their process standards urge for this approach to learning math.
What I found most interesting, however, was the array of differing curricula that the publishers were able to build based on the NCTM standards, and the degree to which each publisher applied constructivist ideologies. For example, I presented ARISE. Probably the first thing that struck me about that program was the degree of text on the page, versus the amount found on a traditional mathematics textbook. A cursory flip through the ARISE books begs the question: Is this still a math text? Digging further shows that it is, but those who look to math as a bastion devoid of reading and writing will be sorely disappointed. On the other hand, Prentice Hall’s four-color Connected Math, while certainly more open-ended and non-routine than a traditional math text, struck me as being quite conventional, almost standards-based-lite, or constructivist math for the newly-converted.
Something that further occurred to me while reviewing my NSF-curricula and viewing each group’s presentation was how different mathematics teaching and learning could be from what I got, should these and other approaches (e.g., The Algebra Project) be implemented for present and future generations. Let me clarify this. I’m not referring so much to the banking method versus the construction of knowledge, as I learned in both ways in school. Nor am I thinking so much of contextual versus theoretical approaches to learning math, as I also learned math in both ways. Rather, I’m speaking to the amount of reading, writing, and critical thinking inherent in all of these curricula, which I certainly did not get from math in school.
We worked in groups and alone. We constructed knowledge in groups and with the class. We learned from our peers and from our teacher. Together, both groups scaffolded informal learning with formal knowledge. However, the text from which we read was short, writing was minimal (if even present), and critical thinking was narrowly developed. Standards-based curricula, however, requires math teachers to heed the mantra of literacy educators everywhere: Every teacher a literacy teacher. Using standards-based curricula, math educators are required to buttress students’ reading skills with guides—how else will struggling and intermediate readers be able to negotiate and thus best understand the text? Math educators are further required to provide students with writing prompts taking the form of exercises, math journals, or blogs—how else will students hone their critical-thinking abilities, or be capable to writing elegant, even convincing, proofs in their future math courses? Simply put, mathematics exercises will not be enough. These types of curricula will force students to be knowledgeable, not only of mathematics, but also of reading and writing.
To me, this is a fascinating concept for it accomplishes that which I as a future math teacher hope to accomplish—it allows for the development of students with greater, more critical comprehension of math and of their world.
Anyways, as many others in the class have noticed, with some differences, each of the curricula presented last Thursday shared many themes. On a whole, they all relied on the constructivist model of learning—student-centric classrooms with the teacher as a guide, cooperative group work, and open-ended contextual activities to motivate learning. This was unsurprising as each curriculum grew out of the NCTM standards which through their process standards urge for this approach to learning math.
What I found most interesting, however, was the array of differing curricula that the publishers were able to build based on the NCTM standards, and the degree to which each publisher applied constructivist ideologies. For example, I presented ARISE. Probably the first thing that struck me about that program was the degree of text on the page, versus the amount found on a traditional mathematics textbook. A cursory flip through the ARISE books begs the question: Is this still a math text? Digging further shows that it is, but those who look to math as a bastion devoid of reading and writing will be sorely disappointed. On the other hand, Prentice Hall’s four-color Connected Math, while certainly more open-ended and non-routine than a traditional math text, struck me as being quite conventional, almost standards-based-lite, or constructivist math for the newly-converted.
Something that further occurred to me while reviewing my NSF-curricula and viewing each group’s presentation was how different mathematics teaching and learning could be from what I got, should these and other approaches (e.g., The Algebra Project) be implemented for present and future generations. Let me clarify this. I’m not referring so much to the banking method versus the construction of knowledge, as I learned in both ways in school. Nor am I thinking so much of contextual versus theoretical approaches to learning math, as I also learned math in both ways. Rather, I’m speaking to the amount of reading, writing, and critical thinking inherent in all of these curricula, which I certainly did not get from math in school.
We worked in groups and alone. We constructed knowledge in groups and with the class. We learned from our peers and from our teacher. Together, both groups scaffolded informal learning with formal knowledge. However, the text from which we read was short, writing was minimal (if even present), and critical thinking was narrowly developed. Standards-based curricula, however, requires math teachers to heed the mantra of literacy educators everywhere: Every teacher a literacy teacher. Using standards-based curricula, math educators are required to buttress students’ reading skills with guides—how else will struggling and intermediate readers be able to negotiate and thus best understand the text? Math educators are further required to provide students with writing prompts taking the form of exercises, math journals, or blogs—how else will students hone their critical-thinking abilities, or be capable to writing elegant, even convincing, proofs in their future math courses? Simply put, mathematics exercises will not be enough. These types of curricula will force students to be knowledgeable, not only of mathematics, but also of reading and writing.
To me, this is a fascinating concept for it accomplishes that which I as a future math teacher hope to accomplish—it allows for the development of students with greater, more critical comprehension of math and of their world.
Wednesday, February 25, 2009
Linear Programming
My general feeling with math is that as teachers we make the argument to students that math has applications in real life, but rarely do we provide them with situations in high school where this math could be used. In fact, it is my feeling that the years between Algebra I and Calc tend to be a wasteland of math barren of practical applications (but full of forced ones). Therefore, if I were to teach a class during that time, I would make it a priority to teach linear programming, probably after covering systems of linear inequalities. (Ironically, I did not learn of linear programming until sometime during undergrad, even though it’s a rather simple topic and has very real applications. I believe that it’s in some Algebra II textbooks, but I suspect that even then it’s rarely taught.)
I would introduce the topic using a rather standard problem like the following:
| Dee works at a doughnut factory that specializes in making chocolate doughnuts and frosted doughnuts.
If Dee makes a profit of $8 for each tray of chocolate doughnuts and $10 for each tray of frosted doughnuts, how many doughnuts of each kind should she make to maximize her profit? How much money will she make? |
After allowing my students to try to solve this problem (some might make the connection from before to graph inequalities, but probably wouldn’t know what to make the feasibility region or the how to calculate the maximum profits, let alone the amount of money Dee will make). I would then teach them about linear programming using the example as a foil; explaining ideas like the feasibility region, the constraints, and how to calculate the maximum (or minimum) for each subject.
For homework, I would then give them a series of word problems that would require them to apply what they learned using the concept of linear programming. I’d start them off with items that asked less than the problem presented in class, and then progress in difficulty to same level as the problem in class. As a challenge, I’d also provide questions that required the use of more than two variables; however, I would note to the students that these problems might require innovations of their own to be solved. I would also try to make the problems a little more relevant to their lives, possibly using a social justice math slant.
Sunday, February 8, 2009
Why is it important for High School students to learn Math?
Why Math?
I’m sure all of us are familiar with the age-old quips: “Why do we need to learn math?” or better “When am I going to use this in my life?” In elementary school, teachers argue Math is important because it provides you with the tools in order to tell time, to make change, to measure distances/areas/volumes, and in cooking. While this is a solid argument, it isn’t enough. You can’t tell a 15-year-old that they need to learn math so that they can tell time, make change, or to cook: By that point, most are rather articulate in those skills. In fact, this clarifies the question. It should be “Why is it important for students to learn math?” but rather “Why is it important for students to learn math beyond arithmetic?”
Some teachers will argue it develops students’ critical thinking and problem-solving skills. While there’s no arguing that it can, at the same time one might still argue that a student can develop those same skills using a less symbolic form of thinking. English, Social Studies, and most of the humanities require some form of critical thinking, usually taking the form of research papers and projects. At the same time, Shop projects also rely on students’ problem-solving abilities, and these also seem terribly useful in students’ day-to-day lives. Modern education, in fact, is centered on the fostering of critical thinking and problem-solving skills in its youth. So this begs the question: If all of education can develop this form of thinking, is it really necessary for students to take math beyond arithmetic?
So where does that leave math? Some might argue that math is needed to do science. For students who intend to pursue careers in science, this might be enough. However, for those students who see science as yet another kind of math —i.e., one of those useless things that they teach you in school— arguing that math is useful in science is hardly a solid argument.
Those who know of Gardener’s theories of multiple intelligences might argue that while some students might not understand mathematics well, other will understand it better. For that reason, students should take mathematics. Even if this so, the opposite makes a stronger argument: If some students understand math better, let them take math; all others can be exempt.
Thus, this returns us to same problem: “Why is it important for students to take math beyond arithmetic?”
The reason is simple: It is a Literacy that humans have developed over the course of the past 5,000 years that enables humans with the ability to model and predict real life with great precision. Naturally, math is a useful tool to have if you go in one of the hard sciences, or economics, or computer science. It is of equal value to those who pursue degrees in the softer sciences: For them, statistical analysis will be a mainstay of their education, if not of their professional careers. Learning the mathematics that prepares students in performing these analyses will free them time to focus on learning theories and ideologies of greater importance to their studies.
But what of the students who intend to go into construction? What of those students who go into sales? Or of those who decide to run their own businesses? What about the students who drop out of school, who end-up working as janitors, or end-up in vice selling drugs, or running with a gang, or worse? Do they need math beyond arithmetic?
Even these students need math, and here’s why: construction, sales, hourly-jobs, even vice runs on the basic principles of business. And for any business to remain competitive, it requires the most up-to-date tools of the trade. In specific, there needs to be an understanding of how to maintain supply, what fuels the demand, how to balance budgets, and how to predict the market. To accomplish each need easily and successfully, at least some proficiency in algebra, basic geometry, data analysis, statistics, and even calculus is required. Any student that finds himself in a business with opportunities for advancement will soon discover those who understand the math of business, are more likely to succeed than those who do not.
I’m sure all of us are familiar with the age-old quips: “Why do we need to learn math?” or better “When am I going to use this in my life?” In elementary school, teachers argue Math is important because it provides you with the tools in order to tell time, to make change, to measure distances/areas/volumes, and in cooking. While this is a solid argument, it isn’t enough. You can’t tell a 15-year-old that they need to learn math so that they can tell time, make change, or to cook: By that point, most are rather articulate in those skills. In fact, this clarifies the question. It should be “Why is it important for students to learn math?” but rather “Why is it important for students to learn math beyond arithmetic?”
Some teachers will argue it develops students’ critical thinking and problem-solving skills. While there’s no arguing that it can, at the same time one might still argue that a student can develop those same skills using a less symbolic form of thinking. English, Social Studies, and most of the humanities require some form of critical thinking, usually taking the form of research papers and projects. At the same time, Shop projects also rely on students’ problem-solving abilities, and these also seem terribly useful in students’ day-to-day lives. Modern education, in fact, is centered on the fostering of critical thinking and problem-solving skills in its youth. So this begs the question: If all of education can develop this form of thinking, is it really necessary for students to take math beyond arithmetic?
So where does that leave math? Some might argue that math is needed to do science. For students who intend to pursue careers in science, this might be enough. However, for those students who see science as yet another kind of math —i.e., one of those useless things that they teach you in school— arguing that math is useful in science is hardly a solid argument.
Those who know of Gardener’s theories of multiple intelligences might argue that while some students might not understand mathematics well, other will understand it better. For that reason, students should take mathematics. Even if this so, the opposite makes a stronger argument: If some students understand math better, let them take math; all others can be exempt.
Thus, this returns us to same problem: “Why is it important for students to take math beyond arithmetic?”
The reason is simple: It is a Literacy that humans have developed over the course of the past 5,000 years that enables humans with the ability to model and predict real life with great precision. Naturally, math is a useful tool to have if you go in one of the hard sciences, or economics, or computer science. It is of equal value to those who pursue degrees in the softer sciences: For them, statistical analysis will be a mainstay of their education, if not of their professional careers. Learning the mathematics that prepares students in performing these analyses will free them time to focus on learning theories and ideologies of greater importance to their studies.
But what of the students who intend to go into construction? What of those students who go into sales? Or of those who decide to run their own businesses? What about the students who drop out of school, who end-up working as janitors, or end-up in vice selling drugs, or running with a gang, or worse? Do they need math beyond arithmetic?
Even these students need math, and here’s why: construction, sales, hourly-jobs, even vice runs on the basic principles of business. And for any business to remain competitive, it requires the most up-to-date tools of the trade. In specific, there needs to be an understanding of how to maintain supply, what fuels the demand, how to balance budgets, and how to predict the market. To accomplish each need easily and successfully, at least some proficiency in algebra, basic geometry, data analysis, statistics, and even calculus is required. Any student that finds himself in a business with opportunities for advancement will soon discover those who understand the math of business, are more likely to succeed than those who do not.
Wednesday, January 21, 2009
Introductions
googol [goo-gawl, -gol, guhl]
–noun
a number that is equal to a 1 followed by 100 zeroes and expressed as 10^100.
googolplex
–noun
a number that is equal to a 1 followed by a googol zeroes and expressed as 10^(10^100).
When I was in second grade we had to give a report on a book that we found in the library. I found a book on numbers. I can’t tell you the name of the book today, nor its content. What I can say is that it had a black cover with red, blue, and white convex polygons on the cover, and that it introduced my verdant mind to the imaginative possibilities that mathematics can offer; in this case, of the numbers googol and googolplex.
It’s from that enigmatic book that I account for my love of numbers and mathematics. Or at least, that’s the earliest case I can recall that links mathematics with my adoration of the field. Much to my regret, however, I never really met a teacher with a similar love of mathematics, or in the very least, a contagious interest in the field. Certainly there were a number of *good* mathematics educators (teachers and academics) who introduced me to interesting activities, or whom I felt could relate the concepts better than most. Meditating on these educators brings me back to Mr. Crosby’s Trig and Calc classroom: In it, I relive his jealousy of his Galois, his proposal to his wife on a bridge near the University of Minnesota, even his tricks for remembering trig function values (so long as I have a hand and can remember “All Students Take Crack,” I shall never go wrong).
Mr. Crosby aside, I never felt like I learned under an educator who could spark my imagination like those whom I had in other fields, and even then, I’m not sure that Mr. Crosby sparked anything that wasn’t already abundantly present. You might say that as I had lacked experiencing a mathematics educator with the uncanny ability to spark the math-bug in others, that I would like to be that teacher for students. I genuinely enjoy mathematics, am continually fascinated by whatever new gems of math that I can learn, and try to share this excitement with others, both by helping others and by discovering answers when engaged in problem solving. To me, professionals of this nature must continually challenge others, and especially themselves; they are duty-bound to push the limits of understanding so that all might better understand those limits.
To accomplish these ends, I applied and enrolled at Montclair State’s MAT program in mathematics. Having a Bachelor’s in Mathematics and five years as a Mathematics Content Editor, I knew I was well-prepared to teach either Middle or High School, but lacked a formal education. I have been at MSU for one semester, and in that time I have learned scores of education theories, all of which will be a boon to my future career. Regardless, I eagerly look forward to what I will learn in Math 579, as I feel that where education classes help prepare students to be teachers, methods courses provide students with tools they can practically apply to their careers as teachers. I see this class as a vehicle for those tools, and eagerly await the challenges that await me. Through their engagement, I know I will be a step closer to my goals and my future.
–noun
a number that is equal to a 1 followed by 100 zeroes and expressed as 10^100.
googolplex
–noun
a number that is equal to a 1 followed by a googol zeroes and expressed as 10^(10^100).
When I was in second grade we had to give a report on a book that we found in the library. I found a book on numbers. I can’t tell you the name of the book today, nor its content. What I can say is that it had a black cover with red, blue, and white convex polygons on the cover, and that it introduced my verdant mind to the imaginative possibilities that mathematics can offer; in this case, of the numbers googol and googolplex.
It’s from that enigmatic book that I account for my love of numbers and mathematics. Or at least, that’s the earliest case I can recall that links mathematics with my adoration of the field. Much to my regret, however, I never really met a teacher with a similar love of mathematics, or in the very least, a contagious interest in the field. Certainly there were a number of *good* mathematics educators (teachers and academics) who introduced me to interesting activities, or whom I felt could relate the concepts better than most. Meditating on these educators brings me back to Mr. Crosby’s Trig and Calc classroom: In it, I relive his jealousy of his Galois, his proposal to his wife on a bridge near the University of Minnesota, even his tricks for remembering trig function values (so long as I have a hand and can remember “All Students Take Crack,” I shall never go wrong).
Mr. Crosby aside, I never felt like I learned under an educator who could spark my imagination like those whom I had in other fields, and even then, I’m not sure that Mr. Crosby sparked anything that wasn’t already abundantly present. You might say that as I had lacked experiencing a mathematics educator with the uncanny ability to spark the math-bug in others, that I would like to be that teacher for students. I genuinely enjoy mathematics, am continually fascinated by whatever new gems of math that I can learn, and try to share this excitement with others, both by helping others and by discovering answers when engaged in problem solving. To me, professionals of this nature must continually challenge others, and especially themselves; they are duty-bound to push the limits of understanding so that all might better understand those limits.
To accomplish these ends, I applied and enrolled at Montclair State’s MAT program in mathematics. Having a Bachelor’s in Mathematics and five years as a Mathematics Content Editor, I knew I was well-prepared to teach either Middle or High School, but lacked a formal education. I have been at MSU for one semester, and in that time I have learned scores of education theories, all of which will be a boon to my future career. Regardless, I eagerly look forward to what I will learn in Math 579, as I feel that where education classes help prepare students to be teachers, methods courses provide students with tools they can practically apply to their careers as teachers. I see this class as a vehicle for those tools, and eagerly await the challenges that await me. Through their engagement, I know I will be a step closer to my goals and my future.
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