There are basically two goals in teaching and learning mathematics: one is to learn the methods required to arrive at correct solutions, and the other is to develop an understanding of the meaning behind the mathematics. Both are important to teach, but a deep meaning of mathematics is longer lasting and more likely to translate to other areas of life. Memorized formulas and algorithms fade quickly and do not contribute to a student’s numeracy or mathematical reasoning. Unfortunately, meaning is harder to measure than method, leading schools to place a higher emphasis on teaching and testing on algorithms and memorized knowledge.
In my secondary education, I was mainly taught from a methods standpoint, even though my school was not subject to standardized tests. I remember simply studying “how to apply a formula”, rather than coming home with a better understanding of where the formula came from, why it is useful, and how it connects to my life and prior knowledge. This stayed the same in my undergraduate lower-division courses, where I gained little conceptual understanding. It was not until my upper division classes where I was tested on conceptual understanding. Whether I was being tested on a proving or programming, it was near impossible to do without understanding and being able to discuss the underlying math.
As a teacher, I will strive to teach my students meaning over method. I will not forgo method, as it is an important part of demonstrating knowledge on standardized tests and in college. However, I will attempt to ask questions, structure activities, promote discussions and assign work that builds my students’ conceptual understanding. In my opinion, conceptual understanding is built through connecting the mathematics to student’s prior knowledge and giving them the opportunity to work on their own intuitions of the way the world works.
As teachers, it is our job to make these connections and guide students along this path; correcting, adjusting and challenging along the way. Once this foundation has been built, I think the methods we are teaching will be better received and internalized.
For example, when teaching linear equations, it is important to begin with real life examples, rather than y=mx+b. There are so many examples that students understand and can connect with, but they do not yet have the ability to talk about these in mathematical terms. For example, students might understand how to find miles per hour by dividing on how far a car drove by the time it was driving, but not even realize they are finding the “average rate of change”, or “change in distance over change time”, or the “slope of a linear equation”. Making this connection throughout a unit on linear equations is much more effective than memorizing that “m = slope” and “m= y2-ya/x2 -x1”, when these variables and symbols hold no meaning for the students.
Fortunately, the new Common Core Standards also seem to place a higher emphasis on meaning over method. Hopefully, this means that standards tests will be formatted to test students’ understanding through problems where students are able to make their thinking transparent and defend their work. If this were in place, math teachers would be able to spend more time on teaching meaning, rather than focusing on a myriad of methods students need to know.
In closing, I once heard that there were four steps in using math to solve a problem. The first is figuring out what problem you need to solve and what information you need to solve it. The second is choosing the best mathematical method to solve the problem. The third is carrying out the method, and the fourth is verifying your answer or checking to see if it agrees with reality. The only step that can be easily done by a calculator or computer is the third step, but this is the main focus of most math classrooms. All the other steps require mathematical reasoning and understanding, skills that can be applied to all other content areas. As a teacher, I wish to teach my students how to carry out all four steps, with a particular emphasis on steps one, two and four. By doing this, I believe the students will become problem-solvers that are prepared to face whatever challenges they encounter, whether in the workforce or academia.
In my secondary education, I was mainly taught from a methods standpoint, even though my school was not subject to standardized tests. I remember simply studying “how to apply a formula”, rather than coming home with a better understanding of where the formula came from, why it is useful, and how it connects to my life and prior knowledge. This stayed the same in my undergraduate lower-division courses, where I gained little conceptual understanding. It was not until my upper division classes where I was tested on conceptual understanding. Whether I was being tested on a proving or programming, it was near impossible to do without understanding and being able to discuss the underlying math.
As a teacher, I will strive to teach my students meaning over method. I will not forgo method, as it is an important part of demonstrating knowledge on standardized tests and in college. However, I will attempt to ask questions, structure activities, promote discussions and assign work that builds my students’ conceptual understanding. In my opinion, conceptual understanding is built through connecting the mathematics to student’s prior knowledge and giving them the opportunity to work on their own intuitions of the way the world works.
As teachers, it is our job to make these connections and guide students along this path; correcting, adjusting and challenging along the way. Once this foundation has been built, I think the methods we are teaching will be better received and internalized.
For example, when teaching linear equations, it is important to begin with real life examples, rather than y=mx+b. There are so many examples that students understand and can connect with, but they do not yet have the ability to talk about these in mathematical terms. For example, students might understand how to find miles per hour by dividing on how far a car drove by the time it was driving, but not even realize they are finding the “average rate of change”, or “change in distance over change time”, or the “slope of a linear equation”. Making this connection throughout a unit on linear equations is much more effective than memorizing that “m = slope” and “m= y2-ya/x2 -x1”, when these variables and symbols hold no meaning for the students.
Fortunately, the new Common Core Standards also seem to place a higher emphasis on meaning over method. Hopefully, this means that standards tests will be formatted to test students’ understanding through problems where students are able to make their thinking transparent and defend their work. If this were in place, math teachers would be able to spend more time on teaching meaning, rather than focusing on a myriad of methods students need to know.
In closing, I once heard that there were four steps in using math to solve a problem. The first is figuring out what problem you need to solve and what information you need to solve it. The second is choosing the best mathematical method to solve the problem. The third is carrying out the method, and the fourth is verifying your answer or checking to see if it agrees with reality. The only step that can be easily done by a calculator or computer is the third step, but this is the main focus of most math classrooms. All the other steps require mathematical reasoning and understanding, skills that can be applied to all other content areas. As a teacher, I wish to teach my students how to carry out all four steps, with a particular emphasis on steps one, two and four. By doing this, I believe the students will become problem-solvers that are prepared to face whatever challenges they encounter, whether in the workforce or academia.
RSS Feed