Mixed Mathematics Class

This week we completed activities that would be completed in the grade 12 mixed math class. The first activity was about domain and range of a function. First four stations were set up around the class: inequalities, words, number line and list of numbers. Then each student was given 3 pieces of paper that represented one of the stations. After this, the students from each station then collaborated to match there equivalent statements that were represented in the different ways. This allowed for collaboration and for students to actively see that a range of numbers can be represented in four different ways. At the end we were given a hand out with graphs and students would state the domain and range using two different notations. This then allowed students to apply what they just demonstrated in the beginning of the activity.

The next activity we did was about investigating exponential functions. To begin the activity an example was shown using an email chain. The first person sent the email to one person the email to two people, and then those two people sent to two more people and then all of those people sent the email to two more people etc. Everyone who got an email stood at the front of the room in a line for each group of emails, It was shown that in the first group there was one person, then two, then 4 and then 8 (figure 1). This was then graphed using Desmos to show that it is an exponential function of y=2^x. This was a great way to visually show how exponential growth works as well as get the students up and moving. Then a handout was given and students were asked to use their personal devises to access Desmos and graph three groups of exponential functions. One group exponential growth, the other was exponential decay and the third was 1^x (a horizontal line of y=1). This was a great way to visually see the graphs and see how the variable a in y=a^x affects the graph. This was another great and engaging activity that incorporates technology. 

Engaging Activities

This week we completed 3 activates in class from the different presentations. The first activity was for a grade 10 applied math class and the activity dealt with measurement. In this activity we were first asked to estimate the conversions between the metric system and the imperial system. Then we were asked to measure various body parts with a partner using a meter stick. This was great for the students, especially applied level, to get them up and moving and actually physically measuring and seeing the measurements.

Next we completed an activity about triangles and using trig laws along with sine and cosine law. First we worked in small groups to complete a few simple problems using soh cah toa and cosine and sine law. After each question was complete we were given a pile of letters and once we competed all of the questions we had to unscramble a word that completed a sentence. This was a great way to motivate students to complete each question quickly because they would be excited to complete the word and phrase. Lastly we were given a real world problem that incorporated the students’ names and interests. My group’s problem was about measuring the distances of ships to a dock, to see which one was closer, only being given the distance between the boats and two angles. Since it was not a right angle triangle we used sine law to solve for the two unknown side lengths and discovered that the red ship was close to the dock than the blue ship. We then shared our solution with the class using chart paper, which can be seen in figure four. Using students interests and names in problems is a great way to get the students involved in a math class, this makes the students more excited about the problems and I think intrinsically motivates them to complete it.

The next activity was completed using graphing calculator and motion detectors. In this activity we were to recreate a circle graph using the detector. At first my group thought we simply just turn in a circle but then realized that the motion detector needs to stay fixed on one thing. We then realized that we can just move the motion detector in a circle by holding it close to you and then going around so your arm is stretched out in front of you and then back close you to you, all in a circular motion. Graphing calculators are definitely a great way to incorporate technology into the math classroom but I personally find them difficult to use, which is probably just due to my lack of experience. I think computerized graphing technologies such as desmos and/or gizmos seem to be easier to understand and are more user friendly. It is definitely important to some sort of graphing technology into the classroom, if it is available, as I think it really helps the visual learns as well as prepares students for post-secondary school. I remember getting to University and having to use graphing software in many of my math and physics classes and it was a huge learning curve since I wasn’t really exposed to a lot of it in high school.  

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