Teaching is Cultural

           This week left me with a similar question as the previous weeks, what is the best way to teach mathematics? This week we learned about the differences between American schools and Japanese schools.  One of the biggest differences that I found between the two was the in the US confusion and making mistakes in mathematics is thought of as a bad thing, but in Japan confusion and making mistakes is thought of as a good thing because this allows students to learn from their mistakes. This is something that I have mentioned in my previous blogs and something I strongly believe, making mistakes is very important especially in math as long as you find out why you have made these mistakes and how to prevent making them when approaching a similar problem. Since teaching is a cultural thing I think that this is a difficult concept for teachers and students to accept, that confusion and mistakes are okay, especially in mathematics.

            Teaching is cultural and teachers often resort back to teaching in the way in which they learned the best when they were a student. Everyone knows how to teach that went to school, it is learned through the culture. Most people have not studied to be teachers but most people have been students in which they have observed for years how teachers teach them. People within a culture share a mental picture of what teaching is like. In a North American mathematics classroom I believe this picture is often of a teacher standing at the front of a classroom teaching a lesson and then providing students with questions to complete on their own followed by a test at the end of a unit. I do believe and/or hope this is beginning to change as students start to become more in control of their own learning and the notion of student centered learning is becoming more popular. This was another characteristic of the Japanese schools that was different from America’s. In Japan teachers would give students problems and allow them to think about it on their own before giving solutions. This allowed for a lot more classroom discussion, creativity and higher-order thinking.

            Giving students time to explore and attempt to solve complex problems will allow them to use this creativity and high-order thinking. Giving them the opportunity to then share and discuss their findings with classmates really allows students to be at the center of the classroom and learn from each other. The teacher can then use their students’ production of work as resources for their teaching and base further lessons on this work. For example in class we discussed the painted cube problem. In partners everyone came up with a solution to the problem and put it onto large chart paper. When these pieces of chart paper were displayed at the end of class, every groups was different. If this was done in a mathematics class in elementary school or high school, the teacher could then make lessons on each of the different methods used to solve the problem. This would allow students to be at the center of their learning, their work to be used as resources and students could see first hand that there is multiple ways to approach a problem. I believe that the Japanese schools have many effective teaching strategies that they use in the classroom and North American schools are definitely starting to bring in some of these strategies. In the pre-service teaching program I have definitely heard about these strategies multiple times, but are teachers who have been teaching for several years open to changing any of their strategies from what they have learned and have been practicing? Will the cultural picture of what teaching looks like every change in Canada?

picture one from:http://www.tidesinc.org/wp-content/uploads/2012/06/1069986-Clipart-Diverse-Stick-Students-Working-On-A-Group-Project-Royalty-Free-Vector-Illustration.jpg
picture two from: https://teachingmathculture.files.wordpress.com/2015/06/part1.jpg

Learning and Teaching How to Problem Solve

Problem solving is a very important aspect of mathematics but it is a process that I believe is difficult to teach. The first question is what is a problem? A problem can be looked at as when there is no apparent path to a solution, because if there were then it would not be a problem in the first place. In order to students to become good problem solvers, and improve at finding the path to different solutions, I think it is important for teachers to not give them step by step solutions but instead just provide them with the knowledge and basics they need, and guide them. A heuristic method should be used in which they are given guidance but not an instruction manual. This allows students to think in a higher-order about problems, as well we be creative.

            The steps to problem solving include, understanding the problem, devising a plan, carrying out the plan and looking back. I think the first three steps are pretty obvious, but looking back is something I found myself often forgetting to do and often see students do this as well. After every problem is complete, students should ask themselves the question “does this answer make sense?” Reflecting back on the problem is a very important step in completing a problem, one that can often lead to students catching a mistake they have made and again thinking at a higher-order.

            I think another important part about teaching students problem solving skills, is letting them know that making mistakes are perfectly okay in fact they are encouraged. I am a huge believer of learning from your mistakes, especially in mathematics. When a student does not first understand a problem, he/she should at least attempt the question before receiving help or guidance on the problem. I have personally experienced this in high school and continue to see it during my placements. At the beginning of class the teacher will check to see if students finished or at least attempted all of their homework, if they have and have any questions then they will take up the problems as a class. This can also lead to students teaching students, in which when one student got to the solution, they can then show and teach their classmates how. It is also important for the teacher to ask if anyone got to the answer using a different approach or provide him or her with alternate solutions. This can often lead to a class discussion as well, which is another important part of teaching mathematics and problem solving.

            Some say mathematicians are the worst communicators (Fulton, n.a). Discussion is something that is looked at as extremely important in other subjects, but if is often neglected in mathematics. This should not be the case, as many students need to discuss with their classmates out loud in order to organize their thoughts and come to a conclusion. I often find myself at first being confused or not understand a problem, but then when I discuss it with a classmate or instructor, just talking about it out loud allows me to realize how to approach it. Mathematics should be encouraged to communicate. This can be done as a class or in groups, both effective and important.

            Problem solving skills are extremely important in order to be successful mathematics students. Teachers should guide students in becoming good problem solvers through a heuristic approach and by encouraging reflection, mistakes and discussion.

picture one from: http://itsfunny.org/wp-content/uploads/2013/07/How-I-solve-math-problems.jpg

picture two from: http://www.groton.k12.ct.us/Page/6359

What mathematical knowledge is needed to teach mathematics?

EDBE 8F83 BLOG THREE

This week’s reading and discussion really reflected on how I have often felt over the last 4 years of being in this program. There have been several occasions when I have thought to myself “why do I need to take this course to teach mathematics or physics at the high school level.” I have become an expert at the subjects not an expert at teaching them, which is what my goal is. I believe that teacher perspective programs should include more classes and experience in teaching. There are many required classes that I believe are definitely not necessary to take in order to be a good math teacher. There needs to be a balance in this program between learning how to each (with experience) and the content needed to do so. This goes for all subjects at the intermediate/senior level, especially in mathematics and science. Ball and Bass (2003) described this as Pedagogical content knowledge, which is a unique kind of knowledge that intertwines content with aspects of teaching and learning. There needs to be a balance of content and how to teach the content. There is no need for us to be taking courses such as linear algebra and differential equations when we are in a program to teach high school math. The only benefit to taking these courses is that you finish with a Bachelor of Science and a Bachelor of Education at the end of this 5 (now 6) year program.

Something that really does not make sense to me is that elementary school teachers teach math but may not have taken a math course since grade 11, while high school teachers take mathematics until 4^th year of university. There needs to be a balance established. I know many elementary teacher candidates who do not know how to compute simple mathematics skills, and these are important skills in which they are ‘qualified’ to teach. I do not think this is the case all of the time, but I have personally seen this on several different occasions.

Another thing that I have learned in the past couple years is that math teachers need to be open to different approaches. Growing up I always believed that to complete a mathematical problem, there is only one way to get to the answer. Through my education course and my in my experience tutoring, I have learned that there is definitely several ways to do a lot of things. The example Ball and Bass (2003) used was multiplying 25 x 35. They showed that there were 3 different ways to do so, and I’m sure there are more ways to compute this as well. I have noticed tutoring too that I will explain a concept and then realize, from the students confusion, that they were taught it a different way. In order to be a good teacher I believe that you need to be open to learning these different approaches from students, and be willing to teach the same concepts in different ways. One approach may make sense to some students and make no sense to others, while another approach would make sense to these students who didn't understand one option and not make sense to the students who did.

I think that a balance between mathematical content and how to teach mathematics needs to be put into perspective teacher programs, especially at the primary/ junior and junior/intermediate level. The content is important in order to have a good understanding of what you’re teaching, and choosing grade and level appropriate material and problems. And the way this content is taught is even more important, in order to maximize students learning of the subject. 

picture one from: https://larrycuban.files.wordpress.com/2013/12/images2.jpg?w=500
picture two from: http://orig12.deviantart.net/6171/f/2013/093/f/a/snoopy_teaching_math_by_sakurapetalwolf-d609pot.jpg