Math "Indeduction"
Working at a brand new school offers many, many outlets to being creative. Because nothing has been developed curriculum-wise, we can go in any direction that we see fit to help our students succeed. For the past week, I have been engaged in a professional development that is being called a "curriculum institute". Throughout the "institute", many of my previous beliefs have finally been able to be listened to. Many of these pertain to questions about why certain topics are covered in math curriculums without much relevance to the real world (i.e. factoring polynomials, solving for contrived quadratics, etc.).
In order for me to make these ideas work in my head (if they don't, I find myself endlessly reflecting), I reflected about my approach to the teaching and learning of math. The following is the result of my reflection.
I would rather not force students to focus on learning about each concept, but to actually put them right in the mix.
For whatever reasons, there are many that think that math is a huge set of facts and processes (to be explored later). They think that these facts are simply understood by someone (teacher) and then passed down to someone else (student).
For obvious reasons, students find this both boring and not involving their life at all. However, why is it that I NEVER felt this way? Why did I love math so much? Was it because I had a knack for it and was only enjoying because that's where I found success? Is that all it takes for someone to like a subject? I reflected some more...
I firmly believe that math is the exact preciseness of pure logic. There are no units in pure math... no objects... just quantities, or better put, ideas of quantities.
This preciseness is stimulating, beautiful, and ever-changing.
Thinking logically involves persistence, eagerness, question-pondering, sense-making, analyzing, synthesizing, reanalyzing, pattern seeking, prediction-making, and problem-forming.
This seems a lot like daily life to me. My math teaching approach involves creating an environment in which this sort of thinking is what we do.
Pure logic does not have a huge place in today's multivariable, multifaceted, multi-perspective, "connected at the speed of light" way of life. Today's society are getting tasks done and leaving the "basic" things for computers to work out. Back in the day, the "basic" tasks used to be arithmetic (+, - *, /). Nowadays, we have wonderful tools like WolframAlpha that are able to solve entire multi-stepped math problems with one click. Much like when people stopped memorizing things so much when information was at their fingertips, people are now stopping the understanding of certain processes. Why would anyone make sure they know the Pythagorean theorem when there are online calculators designed to tackle that specific problem? The brain used to be a rememberer and multistep processor. This is now replaced by an "extension" of the brain (aka the computer).
Where does this leave the human brain's responsibilities?
Don't get me wrong. I'm not saying that pure mathematics does not have its place as a course. I feel that there needs to be a different model for the non-mathematician bound students that do not love math for the sake of math. There needs to be a model for students to match the applications of math to real life without getting bogged down with the details.
Most students aren't interested in "ideas of quantities". They want something real.
So if you have to teach math, why not just teach it as applied math? Well, rather than apply knowledge/processes to find other knowledge, why not have the students discover it for themselves?
To me, the best context to "presenting" math content is one that allows students to create their own predictions, see their own patterns, and create their own generalizations to these patterns to solve other problems. This will help create a "real-world" problem for the students where they come up with a "real world" solution.
The reason some students hate math is that it doesn't make sense to them because it is processes and "tricks". Students say that they don't see the practical applications and hate math because of it. I'm reminded of a great xkcd comic that proves a point much better than me saying it.
What I am proposing is having students discover (with the help of a math teacher, of course) math principles, ideas, theorems, etcetera. This will be called "deduction". Students will go through an open inquiry activity in which students will work collaboratively to deduce that week's lesson. Guided inquiry will be on an as needed basis for each group although what I am hoping is that students will work together to discover the lesson goal(s).
After students' discoveries, I envision other days being more about "induction" in which students use their discovered principles and apply them to novel situations. These involve analyzing a problem and breaking it down in order to solve the problem.
As an algebra teacher, this was not as complicated, but I am looking forward to the challenge of trying this with my precalculus classes next year.
I'll keep you posted... :)
In order for me to make these ideas work in my head (if they don't, I find myself endlessly reflecting), I reflected about my approach to the teaching and learning of math. The following is the result of my reflection.
I would rather not force students to focus on learning about each concept, but to actually put them right in the mix.
For whatever reasons, there are many that think that math is a huge set of facts and processes (to be explored later). They think that these facts are simply understood by someone (teacher) and then passed down to someone else (student).
For obvious reasons, students find this both boring and not involving their life at all. However, why is it that I NEVER felt this way? Why did I love math so much? Was it because I had a knack for it and was only enjoying because that's where I found success? Is that all it takes for someone to like a subject? I reflected some more...
I firmly believe that math is the exact preciseness of pure logic. There are no units in pure math... no objects... just quantities, or better put, ideas of quantities.
This preciseness is stimulating, beautiful, and ever-changing.
Thinking logically involves persistence, eagerness, question-pondering, sense-making, analyzing, synthesizing, reanalyzing, pattern seeking, prediction-making, and problem-forming.
This seems a lot like daily life to me. My math teaching approach involves creating an environment in which this sort of thinking is what we do.
Pure logic does not have a huge place in today's multivariable, multifaceted, multi-perspective, "connected at the speed of light" way of life. Today's society are getting tasks done and leaving the "basic" things for computers to work out. Back in the day, the "basic" tasks used to be arithmetic (+, - *, /). Nowadays, we have wonderful tools like WolframAlpha that are able to solve entire multi-stepped math problems with one click. Much like when people stopped memorizing things so much when information was at their fingertips, people are now stopping the understanding of certain processes. Why would anyone make sure they know the Pythagorean theorem when there are online calculators designed to tackle that specific problem? The brain used to be a rememberer and multistep processor. This is now replaced by an "extension" of the brain (aka the computer).
Where does this leave the human brain's responsibilities?
Don't get me wrong. I'm not saying that pure mathematics does not have its place as a course. I feel that there needs to be a different model for the non-mathematician bound students that do not love math for the sake of math. There needs to be a model for students to match the applications of math to real life without getting bogged down with the details.
Most students aren't interested in "ideas of quantities". They want something real.
So if you have to teach math, why not just teach it as applied math? Well, rather than apply knowledge/processes to find other knowledge, why not have the students discover it for themselves?
To me, the best context to "presenting" math content is one that allows students to create their own predictions, see their own patterns, and create their own generalizations to these patterns to solve other problems. This will help create a "real-world" problem for the students where they come up with a "real world" solution.
The reason some students hate math is that it doesn't make sense to them because it is processes and "tricks". Students say that they don't see the practical applications and hate math because of it. I'm reminded of a great xkcd comic that proves a point much better than me saying it.
What I am proposing is having students discover (with the help of a math teacher, of course) math principles, ideas, theorems, etcetera. This will be called "deduction". Students will go through an open inquiry activity in which students will work collaboratively to deduce that week's lesson. Guided inquiry will be on an as needed basis for each group although what I am hoping is that students will work together to discover the lesson goal(s).
After students' discoveries, I envision other days being more about "induction" in which students use their discovered principles and apply them to novel situations. These involve analyzing a problem and breaking it down in order to solve the problem.
As an algebra teacher, this was not as complicated, but I am looking forward to the challenge of trying this with my precalculus classes next year.
I'll keep you posted... :)