Different Styles of Teaching (Week 6 Reflection)

The article "Teaching is a Cultural Activity" discusses the different cultural scripts of teachers and their effect on student’s learning. Teachers and students in a school share the same script in their mind around teaching. They know what is expected and what role to play through implicit observation and participation. Furthermore, students seem to be accustomed to similar routines, teaching styles, and day-to-day classroom activities which makes it difficult for students to adjust and perform well when their teacher suddenly implements significant changes to their instructional methods. According to the article, a U.S. fourth grade teacher watched a video of a Japanese lesson and followed that same style of teaching in his next class. Unfortunately, the U.S. teacher’s students failed to respond like the ones in the video. As teachers we should not be repetitive and predictable. Using differentiated instructional strategies early on in the year and consistently will benefit the students and allow them to experience learning using alternate methods. By involving new activities and using different formats to the lesson, students will grow and expand their learning capabilities, resulting in higher order thinking.  

Japanese Classroom

There are different beliefs on how students learn across cultures. U.S. and Japanese teachers have very different views and teaching styles in their classroom. U.S. teachers want their students to learn skills, procedures and follow a systematic approach, whereas Japanese teachers want their students to think about things in a new way and see relationships between mathematical ideas through self-discovery. These goals create a very different classroom environment. U.S. teachers show steps and examples of how to solve a problem by starting with an easy example and then moving on to the difficult ones. They constantly try to keep students focused and attentive during the lesson, otherwise they will get lost if they miss a step. Japanese teachers believe students learn best by first struggling to solve the problem, discussing it with their peers, and understanding many different methods to solve the problem, not just one. They do not need to focus on keeping the student’s attention because they feel the students are already interested in the subject, and they use a chalkboard to have a cumulative record of their class.

U.S. Classroom

Japanese teachers and U.S teachers use completely opposite methods of teaching. I find it interesting how U.S. teachers try to avoid creating confusion, and feel as if they are not doing their job correctly if the students are confused. Whereas Japanese teachers start their lesson by giving the students a challenging question, and have them struggle to solve the problem. I believe it is important to challenge students and allow them to engage in deep thinking because it will expand their problem solving capabilities, thinking capacity, and improve their intelligence. We should allow students to figure out problems on their own instead of always showing them the method and steps to solve every problem they are given.

Throughout my time as a High School student, I have mostly experienced the U.S. methods of teaching where procedures and steps were given through examples and I was required to follow these methods. I enjoyed this method of learning at the time, however when I encountered math problems in University it was a little more difficult to transition since straightforward steps on how to solve each problem were no longer given. As a result of this experience, I will definitely use a combination of both the U.S. and Japanese styles of teaching in my classroom because it will allow students to grow as critical thinkers and prepare them for future success. Students need to learn at an early age how to be critical thinkers and investigate problems without a step-by-step process available to them so that when they get to University they will already be accustomed to the teaching methods they will encounter.

Splurge Diagrams and their use in Mathematics (Week 5 Reflection)

A Splurge diagram is a very useful tool that teachers and students can use to help organize thoughts and ideas about a particular topic. By writing down all of the things that come to mind about a topic, Splurge diagrams inspire new ideas, and help make connections to other topics or concepts. Splurge Diagrams can be very beneficial to teachers because the more these diagrams are drawn, the more teachers can draw connections to other parts of the curriculum and ultimately realize how the curriculum can become more connected. Furthermore, Splurge diagrams can help make it clear what prior knowledge students need before teaching a new lesson, and they can help expand ideas on different teaching strategies.  The book "Adapting and Extending Secondary Mathematics Activities: New Tasks for Old" discusses how the use of Splurge diagrams can have a positive impact on teaching methods and classroom dynamic by offering teachers a structure for considering many other elements that will affect how tasks are designed. Teachers can organize their curriculum expectations, big ideas, learning goals, teaching methods, and a variety of resources through the use of these concept maps, and they can share ideas with their colleagues by collaborating and drawing them together.
              

Students can use Splurge diagrams to their advantage while writing papers. Teachers can teach their students to use these web diagrams to brainstorm and organize their thoughts before diving right into a paper. As a student, I have often used Splurge diagrams before writing an essay to organize my thoughts for the introduction, thesis, body paragraphs, and conclusion. I find it is a great tool to stimulate ideas and to eliminate the struggle of writer’s block. In the book "Adapting and Extending Secondary Mathematics Activities: New Tasks for Old" it is clear that when examining Figure 2.4 A Splurge diagram for Solving Equations, how visible the complexity becomes as the Splurge diagram develops. Although some topics at first may seem rather simple, as you begin to create your splurge diagram and your thoughts start flowing, it can become a very complex structure full of ideas.
                              

Splurge diagrams can also be used in mathematics. The Polya’s Problem Solving process involves Understanding the Problem, Devising a Plan, Carrying out the Plan, and Looking Back. Students can use a splurge diagram to help them understand the problem and devise a Problem Solving plan. Students can organize what information the question provides, brainstorming ways to approach the question, and think of different methods that can be used to solve the problem in their web diagram. This will give students a goal and set them on the right track to solving the problem. Teachers cannot only rely on tools to motivate their students to learn, they must encourage their students and inspire positive attitudes around learning mathematics. A proactive student will be confident to try many strategies, risk making mistakes, have a willingness to persevere when solutions are not immediate, and have the ability to accept frustration that comes from not knowing. Teacher’s need to understand the importance of a positive classroom environment to ensure students are able to develop positive attitudes and feel safe to try or ask anything within your classroom.
        

Enhancing the approach for Teaching Mathematics (Week 4 Reflection)

As a prospective teacher, I have learned various high-level mathematical concepts throughout the many math courses required in my program. However, I have gained little knowledge on how to teach mathematics to students effectively, since very few of these courses were relevant to the grade level I will be teaching.  The article 'Toward a Practice-Base Theory of Mathematical Knowledge for Teaching' argues the notion that learning more advanced mathematics does not necessarily contribute to a prospective teacher’s effectiveness with young students. Therefore, knowing and understanding high-level math does not benefit a teacher’s quality of teaching mathematics. A problem is that prospective teachers do not review or thoroughly study the material that they will actually be teaching in the classroom. Thus, they are not becoming experts on how to teach this material in the most effective way to engage a variety of learners.

 I often hear prospective teachers say that they are worried to teach in their second teachable during their upcoming teaching block because they “haven’t seen some of that material since High School”. It is shocking that even though we have taken many courses related to the subject we will be teaching, we still do not feel prepared or comfortable enough to begin teaching it to students. It seems that inadequate opportunities exist for teachers to learn mathematics in ways that prepare them for work.

            I feel that in order to fix this problem, the concurrent education program should dedicate part of it’s program to courses that review the High School mathematics program and teach us various teaching strategies on how to best teach that material in a variety of ways. We should become experts in the material we will be teaching and be prepared on how to answer any question a student poses during our lessons. Based on the reading 'Toward a Practice-Base Theory of Mathematical Knowledge for Teaching' a teacher successfully showed students how to multiply and divide, however when a child asked for an explanation on why the invert-and-multiply algorithm for dividing fractions works, the teacher could not provide an answer and told the student to just use it as a rule for now.  It is critical that prospective teachers have a strong understanding of the fundamentals of mathematical principles, so that they can properly educate their students on mathematical concepts. Having more mathematical knowledge is useless unless you have the ability to facilitate students learning. It would be most beneficial for prospective teachers to have plenty of practice teaching content to their peers before getting thrown into their teaching blocks. 

            As a student I have not had the opportunity to experience many teachers who were able to properly explain why we use the rules and methods that were being applied. This was very difficult as a learner because I was just memorizing how to do the math instead of realizing when and why certain rules should be used. The teachers that were able to provide explanations had the most positive effect on my learning because remembering the reason why you use a rule has a more lasting impact then simply memorizing when to use it. The understanding of how mathematical principles were derived ensured that I had a deeper and more complete understanding of the material.

Relational Understanding and Instrumental Understanding of Mathematics (Week 3 Reflection)

In the book ‘The Psychology of Learning Mathematics’, Skemp uses many analogies to discuss the two ways a student can understand mathematics, through instrumental understanding, and relational understanding. Instrumental understanding is defined as a student’s ability to use a rule that is given. It is the process of committing these rules to memory and using a step-by-step method to solve a mathematic problem, until it eventually becomes automatic. Relational understanding is defined as knowing both what to do in order to solve a mathematical problem, and why you are doing it. It involves using specialization, generalization, and having the ability to explain your methods logically. Skemp emphasized the importance of incorporating both types of understanding in the classroom since they can be regarded as different kinds of mathematics. This will cater to a variety of learners.

In today’s mathematics classrooms, the majority of lessons are geared towards instrumental understanding. Teachers find Instrumental mathematics easier for students to understand, since it is more efficient while teaching, and the results of student learning are immediate and more apparent. Furthermore, I believe many students will ignore most of the theory and explanations that their teacher provides while learning new math concepts, because they simply focus on the rules they need to follow in order to complete their homework. This can occur when a math class considers relational understanding as a low priority, and students are rarely asked to explain their methods or are even tested for their relational understanding. Most math tests are just drill tests to see if students can achieve the correct answers to the questions, which is why many students do not care about the reason why they are using the methods they are given. This becomes problematic because students are not fully understanding the mathematical concepts that they are applying.  For example, students memorize the formulas they are given, instead of learning how to derive them or understand the logical reason they are constructed. Teachers need to gear their teaching towards relational understandings as well, to ensure students are fully grasping the big picture. If students are only understanding math instrumentally, they can easily make mistakes if they blindly follow the steps they are given, and do not recognize slight adaptations that may need to occur for special cases.

As a future mathematics teacher, I believe in the importance of combining both instrumental and relational understanding throughout my lessons. Instrumental mathematics is highly effective in teaching students to successfully achieve solutions to their math problems, however we should supplement the fundamentals with relational mathematics to provide the highest level of understanding possible. It is sometimes difficult to assess whether a student understands a concept relationally or instrumentally based on what they write to solve a math problem. Therefore, teachers should carry out class/group discussions and ask meaningful questions to guide student learning and become familiar with how they understand. Additionally, it would be beneficial for teachers to ask explanatory questions on quizzes/tests to encourage students to think relationally.

Many teachers would agree that teaching students to understand a math concept relationally is very difficult at times. Through my past experiences as a student, my teachers would try to explain concepts relationally through the use of manipulatives. Manipulatives can be in the form of a physical object or virtual tool. Sometimes it is beneficial to use manipulatives in order to be able to show and explain concepts that seem impossible to illustrate on a 2D chalkboard or to clarify verbally. When we can explain math concepts in a variety of ways, not just through the use of rules and steps, students will begin to understand the deeper meaning of their calculations and problem solving strategies. This will expand the student’s capabilities to understand higher-level concepts, and inspire them to enjoy learning mathematics.

Link to this weeks reading:
The Psychology of Learning Mathematics - Richard R. Skemp

Secondary School Curriculum and the Perception of Mathematics (Week 2 Reflection)

The Mathematics grade 11/12 Ontario Curriculum Document’s Introduction discusses the importance of student’s learning not only mathematical facts and procedures, but achieving a true understanding of the concepts they are learning and applying. I believe teachers should make a point of explaining real world applications and reasons for learning a mathematical concept being taught, so that students can relate mathematics to their daily lives, other subjects and future careers, thus increasing their motivation. Deepening students’ understanding contributes to developing the ability to use learning from one area of mathematics to understand another area of study and make connections. Further, it is important that teachers encourage their students to justify their solutions, communicate them orally or in writing, and reflect on their own solutions, to ensure they solved the problem in the most accurate and efficient way. This will reinforce that students are seeing the full picture of mathematics.

          Another important topic that has surfaced in teaching is differentiated learning. In mathematics, Problem Solving is central, and it is important that students are able to effectively problem solve. I believe that when teaching a class how to solve a problem, teachers must keep in mind the many different learning styles of their students and try to accommodate their learning needs, through differentiated learning techniques. This can be done by adding a visual diagram or using manipulatives for the hands-on and visual learners, structuring the problem in an organized manner that can be easily followed (i.e. Using Step 1,2,3 etc.), working backwards, using technology, allowing students to work/discuss in groups, and showing different methods and approaches to solve the problem. An exceptional teacher would provide many different options to students so that they can become more engaged, willing to learn, and expand their thinking processes. It is important to build new knowledge form prior knowledge. In order to do this, teachers should gage the knowledge their students currently have and then adjust their lesson plans accordingly.

          It is apparent from the article, ‘Hollywood’s Math Problem’, that society has a negative view of mathematics, which is understandable due to the fact that it can be challenging and requires plenty of practice and abstract thinking. The promotion of a negative media opinion towards mathematics may be contributing to students’ disliking math and influencing their poor attitude, which is setting them up to fail before they begin. However, by providing students with a positive learning experience, teachers may be able to change the negative stigma towards math.

          Personally, I have had many positive experiences while learning math. These experiences are what drive my passion for the subject and inspired me to become a math teacher. I want to be able to teach students to enjoy math and hopefully develop the same passion for the subject that I have.  As a student, I enjoyed solving math problems when I understood the material, however like any other student, I quickly became frustrated when I did not understand a math topic. Therefore it is very important that teachers effectively deliver subject matter. However, the onus is not just on the teacher, the students need to be responsible and contribute to their own learning as well. The students will not be successful if they solely rely on the teacher to do everything for them, they need to be actively involved in their learning. A good math student will ask questions to gain a better understanding, attempt each homework question, investigate online problem solving techniques, and come to class with a positive attitude and an open mind. Practice makes perfect, thus students need to put in the effort when learning math.


       

Links to this weeks readings:

Gr.11/12 Ontario Mathematics Curriculum

Hollywood's Math Problem