Structured Problem Solving

This week in class we discussed structured problem solving in math. We used the example of the growing dots problem. While completing this problem, we were told we must visualize what is happen and draw a diagram. This forced us to solve the problem in a structured way with visuals. Without these instructions many of us may have completed the problem mathematically instead of solving is using diagrams. However even though we were told to use visualizes and diagrams there were still several different approaches to the solution. Two of which can be seen in figures 1 and 2. One is the approach I took of a growing squares and since a square has 4 corners it grows by four dots every minute. The second approach was one that viewed the four corners as orbital’s that grew out by one dot each minute. Both of these approaches come to the same solution but they way the visually problem solved to reach the solution was different. As I’ve said in previous blogs, I think it is very important for teachers to realize that students can approach the same problem in many different ways, and in face teachers should encourage this creativity, as this is what leads to higher order thinking.

figure 1

figure 2

            During this class we also had Diana’s presentation on grade 10 applied math. For her activity we were to solve a problem that was different for each group and can be made based on students’ interests. My group completed a problem about choosing a banquet hall for the school athletic banquet. The solution we came up can be seen in figure 3. This activity is a creative way to show students real life situations when they would need to know how to use substitution or elimination. It was also good that we were not restricted to one method and we could choose if we wanted to use substitution or elimination. We chose substitution as that it was we were all most comfortable with, however we then saw how easy it would have been to use elimination with our two equations, and this may have been the method a grade 10 applied student would have chosen. This is an example of non structured problem solving, which is sometimes more beneficial than structured problem solving as it can lead to more creativity.

figure 3

Using Student's Work As A Mathematical Resource

This week in class we discussed materials and resources in a math classroom. This is something that may be obvious is a school that has access to things such as computers, smartboards and the finances to support a technological math classroom, but this is not always the case. Something that I think many teachers and students don't think about is using themselves as a resource. Students can be used as the primary resource in a classroom through things such as collaboration and presentations of hands on actives with each other. They have the ability to create, draw, write and research and then their work can be used as resources. If you allow students to be creative and come up with their own approach to a problem in mathematics, chances are you will have several different solutions to the same problem. As a teacher you could then use these different solutions to teach how these different approaches work and the math behind them. This allows for collaboration, creativity and higher level thinking in the classroom as well as student centered learning. Using students’ examples to teach would be a great way to use your students as a resource in the classroom, especially if you’re in a school that may not have access to many resources.

            This week I also had my presentation on the grade 10 academic mathematics course. In this presentation we (Ela and I) did an overview of the whole course looking at each major strand: Quadratic Relations, Trigonometry and Analytic Geometry. We then each did an activity that can be used in this course. I did my activity for the Quadratic Relations unit focusing on the transformations of a parabola. In this activity I use a student-centered approach while incorporating technology. In this activity 8 different groups of students would each be given 4 equations representing one a transformation. For example group one would be given 4 equations of a parabola shifted up, group 2 would be given 4 equations representing a parabola shifted down, group 3 would be given 4 equations representing a parabola shifted right etc. The eight groups would be given one of the following transformations: shift up, shift down, shift right, shift left, stretch and compression.  As a teacher you may want to strategically choose these groups to ensure that each group has 1-2 strong students depending on the class size. From here the groups would graph their functions by hand using a table of values and discover the pattern between their graphs. From here each group will present their findings. This is where you can incorporate technology. I used Desmos, which is an online graphing calculator. Using this technology you can either have each group put their graphs onto the same grid so everyone can visually see what each transformation would look like in comparison to a non-transformed parabola. This can be seen in figure one.

Figure one: Desmos Screenshot

 From here students would get into new groups where one student from each of the eight groups form a new group. Each of these new groups would then have an ‘expert’ of each transformation. They will then be given a list of parabolas that contain several transformations of the form y=a(x-d)²+h. They will then come up with a general solution for what each parameter represents and share these findings with the class through a gallery walk and/or class discussion. Desmos also has a parabola transformation software on it that can be used for this part where you adjust each parameter to see how the parabola changes. Other forms of technology can be used throughout this activity such as graphing calculators, a smartboard and gismos. This activity is a student centered lesson which uses the student’s work as resources as well as technology if it is available, but the technology is not required is just there for visual assistance. This activity also allows students to collaborate and explore.

figure two: student graphing their transformation on desmos (student: Ela)

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

Relation vs. Instrumental Teaching in Mathematics

Week 2 Reflection

Is there a right way to teach math? Is there a method that allows students to learn in the best way? Instrumental or relational? Which is more realistic? These are the questions that I was left with after this week’s reading and lecture. 

picture from: http://www.kids-activities-learning-games.com/images/higher-order-thinking-for-math.jpg

            From my experience, math has been taught to me 90% of the time in an instrumental matter, the focus was always on memorizing the information and knowing how to complete the different problems, but never why it is that way or how come that is how you complete the question. But if the focus was on how and why the rules are the rules, I think that this would make mathematics way too complex for students at the elementary and secondary school level. It was not until my last couple years of university that I myself learned by certain things are the way they are. An example I can think of is for physics (my major). All the way through high school science we were always just give the constants and formulas, never told where they come from or why they are that way. Then in higher-level university classes we would spend several lectures on deriving one single formula that was just given to us in high school and even in first year university classes. I specifically remember in my third year quantum physics class taking over two hours to show that Planck’s constant is 6.62607004×10^-34 m²kg/s and that this is a law of the universe not just some made up constant. This derivation and concept was something that I just barely understood in third year university, and would never expect a student in high school to understand or follow along with that. But I do think that a teacher can briefly explain the history and reason behind formulas, rules and constants to simply get students thinking and interested in the topics and also have something to look forward to if they continue with math and/or science after high school. There needs to be a balance, especially in secondary school, between instrumental and relational teaching methods.
            Reflecting back on my learning experiences in high school and even in university, I think I would have a better understanding of many concepts in my math classes if I had known the reason behind things or how they apply to the real world. Now that I am thinking about it, a lot of the materials that I do remember really well are things that I had a deeper meaning of, and knew why and how they were that way. There are many things that I remember being very simple at the time, but because they were just given to my in basic rules and never explained why it is that way or how they connect to the real world, that I now forget. These include simple things such as long division or multiplication with decimals (not using a calculator). Math is a subject that I tutor a lot and it is often these basic things that I have a hard time remembering, but I can usually remember how to do the more complex things such as the application problems.
            I think it is very important for math teachers to have a balance between relation and instrumental ways of teaching. Instrumental is still important because the reality with math is that there are many rules and steps that do need to sometimes be memorized. But students can still be shown or told how and why things are the way they are and also how concepts and material can be applied to real world applications. Students should have the knowledge and be able to apply it and be creative with their solutions. Math should not always be taught in a way that make students think there is only one way to get to the solution. Many problems can be done in multiple different ways and still have the same outcome or result. This is something that I did not realize until the last few years and I believe that if students were aware of this they would be more inclined to think relationally and creatively, which indicates higher-level thinking. I think more students would be successful in their math classes if more teachers establish a balance between instrumental and relational teaching strategies and are more open to explaining the same problem in different ways.

picture from: https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiB6TREQkDDMdc7vDGdgBQ9NJDIDu-KUt5A1FBwa8i79DHnWLbKnP-TBWJ7IcAuKbvaFItRz_Ab9imML9tOHJQJ_P7dc2wA99zaOe1n8EfmaPe2gaeQyNQb4EiY7zwsoXZWefXiplpTmqk/s1600/p-fractions.gif

Learning and Teaching Mathematics

Math has been a strong passion of mine since I was young, I specifically remember in grade 3 realizing how much I enjoyed it. This subject however, often seems to be looked at in a negative way by a lot of people. It is a subject that a lot of people seem to get frustrated with easily. I believe that there is a stigma that some people “just get it” and some people “just don't get it.” I think this is false, and people start to believe that at a young age and then lose hope and confidence in gaining math skills. Through my many years of tutoring, I often whiteness students that come to me and say “I just don’t get math,” it does not take long to realize that most of the time these students are just afraid of making mistakes and need a boost of confidence. I believe that the way a lot of mathematics teacher teach the content only benefits specific types of learners. I would say almost every one of my math teachers growing up taught the same way in which they would go through a note about the topic for that lesson and then give out homework questions to complete about that topic. These lessons would usually be followed by quizzes and a final unit test. There were not many strategies within the math classes that benefited to visual and/or hands on learners.  A lot of times the material was taught in a dry manner, which did not engage students or reflect on student interests, which then lead a lot of students to believing that math is boring. This is definitely not how I feel about math though. I truly do love learning and doing math, and I hope my love to teaching math also grows through out this year and throughout this course. I think a good mathematics student is someone who is willing to make mistakes, is dedicated to learning the required skills and is open minded when it comes to approaching a problem from different angles. I think an excellent math teacher is one whom teaches their lessons in way that engage students and reflect on their interests and learning styles. It also takes someone who is caring and passionate. This is a subject that may require extra time and help for the students.

photo from: http://www.thechalkface.net/teaching/subjectwordle.jpg

 I hope to teach my students that there are many ways to approach a problem and it is more than okay to make mistakes. I have learned throughout my experience as a math student that making mistakes is one of the best ways to learn. This is something I hope I can teach my students. I also hope to show them various ways to approach a problem, such as using graphing, technology and/or mathematic programming etc. I know that math education is very important and I hope to learn new ways to incorporate technology and student’s interests into my classroom in my future as a math and physics teacher.