Reflection of Dave Hewitt's Classroom

Watching the video of Dave Hewitt's classroom gave me the opportunity to witness a new style of teaching. The sudden striking of the board with a ruler and getting students to count confused me because I couldn't see any numbers where he was striking. His technique of getting the students to verbalize their answers was a good way to get them to understand the patterns. Jumping back and forth and getting them to understand counting with positive integers and negative integers was a good way to segue into discussion of linear equations. The repetition in the older classroom of the pattern of solving a linear equation was a good way to get the students to understand why they perform certain operations to solve linear equations. As discussed in class, one might perceive his constant repetition of problems as over-the-line but if you examine the students in class, they do not seem to mind the repetition. In fact, they seem to be paying attention to him. Overall, I think the levels of interactivity and engagement of the class in the lesson were good and are a model for me possibly to integrate into my teaching style.

Interview with Math People (Group Report)

We interviewed a grade 8 mathematics teacher from a middle school in the Coquitlam school district via e-mail. He had expressed the biggest challenge was trying to teach the basic operations such as addition, subtraction, multiplication, and division to the struggling students such that the students would not give up. Consequently, he would also have to keep the top students from being bored. For the top students, he allows them to be peer tutors and provides math challenges or puzzles for them. The math class schedules goes as: class game of Bus Driver, presentation of the math challenge, lecture (25 minutes) and assigned questions (30 minutes). Bus driver is a game with multiplication flash cards. These cards are never shuffled. The bus driver is the winner from the last game. They start off facing off another student in the class. Whoever is the fastest at answering the next flipped up card is the winner for that round. If it's a tie, the teacher keeps flipping the cards (sometimes many at a time) to see who is the fastest. The winner is the bus driver and goes to the next student. The game ends when everyone has a turn and is proclaimed the bus driver and has their name written on the board as the bus driver. Even the struggling students love participating and sometimes may guess the answer ahead of time to win.

By interviewing two students in different grades, we were able to observe some interesting similarities and differences in their responses. First half of the interview questions dealt with what the students like and not like about mathematics. When we asked them what their favourite parts in mathematics were, the grade 9 student responded that he likes to work with integers because they are straight-forward and it is easy to remember the rules. The grade 11 student responded that he likes algebra for a similar reason, but he also likes riddles and logic puzzles. It is interesting how the grade 9 student likes the straightforward and easy concept, while the grade 11 student likes the challenging puzzles and riddles that allow him to think beyond the simple rules and concept.

We then asked them to share some of the challenges they encounter in mathematic classes. The grade 9 student responded that one thing he finds really difficult is translating the word problems into the equations. Also, he is confused when the same symbols are used to represent different things. Similarly, the grade 11 student had a problem with understanding the idea behind the concepts and rules. For example, he has difficulties with understanding the differences between the inverse function and the reciprocal function. Some of these areas of difficulty in math reminded us of the articles that we have read by Skemp (1976) and Robinson (2006) stressing the idea of relational understanding instead of instrumental understanding. We thought that the students are having difficulties in those areas, because they lack the relational understanding of the concepts.

They also expressed an interest in class when the teacher used different media for explanations (that is, anything but the chalkboard). The use of geometric shape blocks on the projector depicting larger shapes and ideas was found as interesting. Presenting students with “challenge-of-the-week” (COW) puzzles and problems got the students interested in math beyond the daily lessons. With the grade 11 student, humour made the classroom more relaxed and the math lessons more interesting–but we can’t all be comedians! In the students’ opinions, logic puzzles would be an interesting addition to math classes. Real-life applications of math would also make the classes more practical. When asked about group work, the reactions were mixed. The grade 9 student did not prefer group work too much because of the added distractions from getting the work done. The grade 11 student has had math projects, such as designing a water slide using cubic functions, and enjoyed the idea of working with others, finding value in comparing answers and thought-processes with others.

Interview with Math People (Personal Reflection)

Communicating with math teachers and students provided me with great insight into the current status of math in schools. Contacting math teachers allowed me, as a teacher candidate, to ask five questions about the teaching profession in order that I might learn some handy hints. Interviewing math students allowed me to ask five questions of my future “audience” and to understand what challenges I will have to face in order to help my students.

Many of the teacher’s responses coincide with what I envision doing in my own classroom. According to the teacher interviewed, one of the biggest challenges faced by him is keeping students interested when going over basics and keeping struggling kids from giving up. I agree with this response from the teacher. It is also one of my fears as a teacher candidate: “Will I be able to teach to the whole class and not lose anyone?” Students who understand basic concepts are bored during initial review and students who do not understand have immense frustration learning them. I hope that I will be able to compile my own repertoire of games suitable in high school math classes. In conjunction with what I have learned in my science education class, the teacher suggested a class format, whereby the beginning of the class consists of a game, followed by lecture and assigned questions (with the time lecturing and working on assigned questions are equally weighted).

Some of the areas of difficulty in math for the students reminded me of the articles that we have read by Skemp (1976) and Robinson (2006) stressing the idea of relational understanding instead of instrumental understanding. When the students expressed a dislike of word problems and graphing inverse and reciprocal functions, I realized that these types of questions are disliked because of their higher-level assessment. Word problems require one to synthesize the question into a practical question. The difficulty with inverse and reciprocal functions comes from not knowing why the graphs look like how they do. As a teacher candidate, I hope to incorporate “buzz groups” and other forms of group discussion into my math classes because the students may be more open to asking for help from peers than from the teacher.

In summary, this interviewing assignment has helped me see both sides of the class and, from an external perspective, assess how I might react in those situations. It also has helped me to give me realistic expectations of a classroom setting. I hope that I can utilize the information gained from this assignment and become an effective teacher.

Teacher’s Questions

1. What is the biggest challenge in teaching students that learn at different paces?
2. What is the most difficult section/topic to teach?
3. What techniques/methods do you use to engage the class in active learning?
4. How often do you give the students a small break from lecturing? (Anecdotes or math questions.)
5. What is some advice you have for math teacher candidates such as ourselves?
   

Student’s Questions

1. What is your favourite part in math and why do you like it?
2. What do you think is a challenge for math students like you?
3. Can you think of a time where your math teacher got you really interested in math during class? What did he/she do?
4. Is there something you wish you could do in math class to make it more interesting?
5. Do you think there should be more group work in math classes answering problems?

Battleground Schools Summary

After reading “Mathematics Education” by my very own math education instructor, Susan Gerofsky, I developed a better feeling of how “political” math can be. The article reads like an historical timeline, with the description of the Progressivist Reform, the New Math era and the National Council of Teachers of Mathematics (NCTM) Math Wars. In the beginning of the Progressivist Reform, there was a desire to teach more of a relational understanding. Besides being able to do math, Progressivist proponents wanted students to learn how to know math. By encouraging active engagement of students in class with activities and both individual and group work, they believed this would be most beneficial for students. I find it interesting that the ideals of the Progressivist Reform are what I am hearing and learning about now, as a teacher candidate.

The New Math era seems to me like an example of children growing up too fast. The abstract ideas and concepts that the New Math curriculum was teaching were too sophisticated. Teachers with regular math backgrounds (in elementary or secondary math) were now teaching topics as hard to comprehend as “set theory, abstract algebra, linear algebra [and] calculus.” The New Math method was ideal for preparing “future elite scientists” but fell short of practicality for everyday people.

The Math Wars was an era where standards were developed for math (and also other curricular subjects). Establishing a set of standards was difficult when all sides were pushing for their input. It seems to me that for all of these eras, in trying to decide on standards, everything became “political” and polar. I suppose that wherever there are decisions to be made, trying to reach a balanced consensus is difficult. As long as both sides remember to keep the students’ best interests in mind, whatever decision is made will be suitable.

Reference
Gerofsky, S. Mathematics Education, Battleground Schools, 2008, Vol. 2, 391-400.

NOTE: I couldn't resist typing this out, but I thought a cool alternative to the title of the book "Battleground Schools" would be "Battlestar Scholastica" but I'm not sure if that infringes on any copyrights or is too close to--well, you know :)

Constant Assessment of Teaching

In response to Heather J. Robinson’s article on “Using Research to Analyze, Reform, and Assess Changes in Instruction” in the book Teachers Engaged in Research: Inquiry Into Mathematics Classrooms, Grade 9-12 (2006) I find the content easy to read about but practically harder to implement. As a teacher candidate, I hope that I will be capable of the same level of self-assessment to realize my shortcomings in a classroom setting. In a course such as mathematics where the norm in schools is lecturing, throwing in a little variety into the classroom with activities such as group problem solving and student teaching are excellent ways to encourage students to discuss with their peers. The idea of restricting lecture time in a class period would help the teacher assess his or her lesson plan and whittle everything down only to the most salient and most essential points, while at the same time ensuring enough class time for the other activities which will engage students in active learning. By re-designing class activities and old knowledge tests to ensure that they require more “critical thinking skills [and] require deeper thought” the teacher can lead the student away from being a machine that makes only one product to a product engineer who designs what the machine can make. The importance of Think-Pair-Share (TPS) in classrooms should not be discounted. Even in my science education courses, we have learned about the utility of TPS. By having students discuss their thoughts (and thought processes) on a problem, the group of students can gain the insight of other students and perhaps gain a new perspective.

When looking at specific strategies for teaching math (science, and any other subject area) it all boils down to this: teachers need to be in a state of constant self-assessment. “Do the students really understand what I’m teaching? Can they apply what I’ve taught them and take it to the next level?” I hope that as a teacher, I do not lose sight of the students’ best interests, which is what teaching is all about.

Reference
Robinson, H.J. Using Research to Analyze, Inform, and Assess Changes in Instruction. Chapter 4 in Teachers Engaged in Research: Inquiry Into Mathematics Classrooms, Grade 9-12, 2006.

Most Memorable Math Teachers

One of my most memorable math teachers is Ms. Lui (Grade 9). She is memorable because, being a young teacher, she had an immense desire to make math interesting. Not only were outlines for each unit given to us (assigned homework, due dates, tests, quizzes) but there were also optional questions worth bonus marks that were challenging and were incentive to do extra work.

The other memorable math teacher is Mrs. Wong (Grade 11). Most of her teaching style was focused on lecturing (as is the case with almost all math teachers and who can blame them--it's math!) but she also let everyone teach the class one topic in the geometry section. This gave us an opportunity to ensure we were confident with a specific concept, enough so that we could explain and teach the class. One way for someone to know that they "know" is by teaching.

My other math teachers were also amazing people, but these two stuck out in my mind. Their approaches to engaging the students in the learning process are examples of some of the methods that I want to inherit as a future math educator. Reflecting on the teaching styles of these two math educators, I realize that I can draw on their methods and integrate them into my own style. The use of a good outline can help the students keep on-task and give them goals. Bonus questions are good ways to encourage students to challenge their knowledge. Having students teach and present a certain topic helps the students cement their knowledge on the subject. My goal as an educator is to integrate all the positive math experiences that I have had with the suggested methods that I am learning in my classes.

Microteaching Reflection

The microteaching lesson in class went smoothly. I taught my group of 3 (in a sense I re-taught 1 person) how to play Old Maid. Using the 10 minutes, as written in my BOOPPPS lesson plan, I introduced the game, relayed the rules, enacted a small simulation and finally engaged the entire group in playing together. According to the group feedback, the objectives for my lesson were clearly outlined: they would be learning to play Old Maid. The participatory activity and the post-test were very easy to implement, as I could assess whether the group understood the rules by playing Old Maid. The best to learn something (when it is safe to do so) is through the hands-on approach. This is the basis of co-operative education, apprenticeships/internships, and most certainly this applies to playing games. The group found the lesson captivating, interactive and just plain fun, which are all good characteristics for an effective lesson. The only aspects on which I could improve, in their opinion, are a slower explanation of rules and a clearer outline as to what we would be doing.

As is often the case, people are more critical of themselves than others. I believed the engaging activity and introduction were well done, but I could have spent more time planning all possible outcomes. When faced with a possible situation (a "what if" question) I stumbled on my own words because I was not entirely sure about the actual outcome. As a teacher, I will need to prepare for the worst-case scenario as best as I can.

Overall, I believe this microteaching lesson was very effective, not only in teaching others about something that I know, but also in assessing my strengths and weaknesses, how good I am at preparing a lesson and carrying out that lesson plan.

How to Play Old Maid

Bridge: What card game has a history of being popular as a drinking game in bar halls, despite its calm title? Technically speaking, any card game could be played in a bar, but I'm talking about Old Maid.

Teaching Objective: To teach the group to play Old Maid, to get the group to play together, and to address any misunderstandings about the rules.

• Introduction - 1 minute
• Rules - 3 minutes
• Interactive Game-play - 5 minutes

Learning Objective: Students will be able to play Old Maid following all the rules of the game.

Pre-test: Ask if anyone knows how to play Old Maid.

Participatory Activity: Following the explanation of rules, students will engage in a round of Old Maid together. The rules are as follows:

1. One queen card is removed from a standard deck of 52 cards.
2. A dealer deals out all remaining 51 cards to all players, regardless of how many cards each player has.
3. After all cards have been dealt, all players discard pairs of cards (i.e. two "3" cards or two "J" cards) in the centre.
4. The dealer presents his/her cards face-down to the player on his/her left. The player selects one card and removes it from the dealer's hand. If the card completes a pair, discard the pair in the centre pile. If this card doesn't complete a pair, continue around the circle.
5. The game continues until one person holds the unpaired queen. The person holding this "Old Maid" is the loser and the game is over.

Post-test: Assess during game-play whether players made any mistakes. If game-play goes smoothly, then all players have learned the rules and can play Old Maid. Asking questions about clarity of rules during the game is a good way to assess understanding.

Summary: Old Maid is one of many card games that can be played by a group of people, whether you're drinking or not. While the game can be competitive, the best way to play it is to just have fun.

Relational and Instrumental Understanding

After reading Richard R. Skemp’s paper on “Relational Understanding and Instrumental Understanding” (1976) I find myself assessing what type of mathematics teacher I intend to be. His arguments fostering the easy-to-teach instrumental understanding in students pose valid concerns. The teaching with mnemonic devices without sufficient explanation to answer why allows “rewards [that are] more immediate [and] more apparent” for students. Students respond well to positive outcomes and are motivated to continue. From a professional perspective, not giving detailed reasoning can be detrimental. Without explanation, the memorization of directions to get the answer becomes like “a multiplicity of rules rather than fewer principles of more general application.” Students sacrifice understanding concepts for the sake of passing important examinations, “answer[ing] correctly a sufficient number of questions.” One clashing of intentions rises when “pupils whose goal is to understand instrumentally [encounter] a teacher who wants them to understand relationally.” I like Skemp’s map analogy, comparing getting lost in using the right equations to making a wrong turn in a series of directions. As a teacher candidate, I want to stress understanding of concepts so that they can be applied to the corresponding problems seamlessly, helping their intellectual growth in the long run, while at the same time not forgetting to balance both relational and instrumental understanding.

Welcome to My Blog

Hi there! You've somehow made your way to my MAED 314A blog. I hope that you enjoy reading my thoughts and opinions on mathematics. (That being said, if you occasionally spot a molecule or chemical structure, don't be frightened. You are still viewing my blog, I just have a fondness for chemistry--being a chemist does that to me.) Enjoy!