You Can Dance If You Want To, You Can Leave Your Friends Behind...

So here I am, updating my Math blog, even after MAED 314A has finished. I decided that I will keep this updated for some time. This is about my professional growth as a math educator, so why not keep it fresh?

Today, I went to my practicum school to visit my sponsor teachers and hang out in the class. According to my sponsor teacher, I picked a good day.

Sponsor: "So, Paul, did you bring comfortable shoes and comfortable clothes?"

Me: " Uhh, yeah. Why?"

Sponsor: "Because today, in class, we're gonna dance. Didn't you see it in the curriculum?"

She really had me going for a moment. I was pausing to think about how dancing could possibly make its way into a math class, with the exception of some form of interpretive dance (like the short movie we saw in MAED 314A class). Once I figured out what the dance moves were, I thought the idea was brilliant! The "dancing" had four moves, all based on the orientation of polynomial functions on the Cartesian plane.

For odd degree polynomials with positive leading coefficients the dance move was this...
....O
.._|_/
/.. |
..../\
because the function is plotted (from left to right) from quadrant III to quadrant I.

For odd degree polynomials with negative leading coefficients the dance move was this...
....O
\_|_
....|.. \
.../\

For even degree polynomials with positive leading coefficients the dance move was this...
.....O
.\_|_/
.....|
..../\

For even degree polynomials with negative leading coefficients the dance move was this...
.....O
..._|_
./.. |.. \
...../\

The purpose of all of this was to get students to recognize what the general shape of the polynomial is before hitting the [Graph] button on their graphing calculators, so that they know whether what they see on their screens is truly representative of what they should expect to see. All in all, I thought this was a great activity, but only if the students are willing to try it out. This might not work right away with me, unless the students feel comfortable enough with me to act a little foolish. But hey, it's in the name of education!

Unit Plan: Circle Geometry (Principles of Mathematics 11)

http://docs.google.com/Doc?docid=0AXSOUnLC8EogZGc2cDZid25fNDljbXZwZjRkNw&hl=en

Here's the link for my unit plan on Circle Geometry in Principles of Mathematics 11.

Member of the Math (of Least) Resistance

Susan guided our minds in class today to the beautiful coast of some foreign land. I decided to recreate myself as Jacob, the 21 year-old chemistry student (how original!) who, because of the army, was unable to complete his studies. I pined at the loss of my chemical acuity due to inactivity. Remaining unemployed on the coast, I found myself just wasting the days away. That was the case until that fateful night when I was awakened to the harsh knocking on my door. I was brought out to the field in the cloak of night and told that my services were enlisted by a small faction of rebels seeking to overthrow the current army regime. Their request: work with several other math-oriented locals to solve problems vital to the rebel alliance's work. We (Jacobi, Ruth, Alma, Doran, Captain and I) gathered in the cave (also known as the underside of two desks) to discuss two problems: one involving scouting for enemy camps with possible traitors in our midst and one involving distribution of rations. With only a dim candlelight in the cave, we had to work together and solve the math problem.

This has to be one of the most memorable math curriculum/methods lessons I have had so far. The incorporation of drama into getting students to solve simple yet practical problems was excellent. It appealed to my sense of a need for something interesting and it appealed to my sense of getting math done. We worked together on simple problems and it was fun. Overall, I thought it was great. I just hope that I can learn to adapt this method to suit my science courses, as well. (It would be great if I could try this!)

Assignment 3: Math Project (Tessellations, Part 1)

Our group (Erwin, Gigi and Stanley) worked on the Islamic Tiling project. I chose the tiling pattern that consisted of ninja-star-looking units. (This was my first impression of the pattern.) Using a more sophisticated description, the unit cell that I identified had the general shape of an equilateral triangle, with the edges composed of a sine-like curve (1 period). There was an empty space in this unit cell, which contained a Star-of-David-like hexagram. Individually, I was not able to re-create the unit cell--err, pattern. (Forgive me: I keep calling it a unit cell because the chemist in me keeps popping out and remembering first-year chemistry and molecular stacking patterns.) Using straightedge and compass, I created a (more-or-less) similar curvy triangle. The part which gave me the most trouble was creating the empty hexagram with only those two implements. Gigi tried to help me with it and she may have figured it out. (The diagram is on one of the three pages of work.) I did not get through the entire project, but that was the point, wasn't it? To give it a try.

In terms of evaluating the benefits of this project, I think this project caters to the student's artistic flare and mathematical prowess. The first part (drawing the pattern, describing the pattern in words) appeals to those people inclined towards art, not requiring any math skills (except trying to keep things in proportion). The second part (recreating the repeat shape with straightedge and compass, making minor modifications) caters to the mathematician. It requires visual analysis of the shape to look for patterns that can be recreated using the geometry knowledge that students have thus far. Teaching analytical skills will be helpful not just in math but in so many other areas of life. One weakness of the project is that it could be overwhelming for students who cannot see the pattern that can be recreated with the two simple instruments. Tessellations are covered in the Math 8 IRP. Some of these might be difficult. (I will admit: I got a bit frustrated that I could not recreate the hexagram in the middle.) Of course, this problem could be alleviated by choosing simple tiling patterns for the students. I would not modify this too much, but I would pre-select some not-so-challenging tiling patterns or at least have two sets of patterns (medium and hard). One constraint of using this project in my classroom would be time. I might not know how long to prolong this project or how many class sessions to allocate for in-class work. Overall, I could see this being used in my class to get their creative juices flowing.

Liouville Problem, Two-Columned Approach

Together with Gigi and Stanley, we chose to attempt the Liouville problem in the "Thinking Mathematically" book by John Mason, Leone Burton and Kaye Stacey, found on page 179. Here are my thoughts on the problem and my approach. (You will note that in the right column, I wrote "What do I do now?" a lot because it was not clear to me what the question was asking.)

Memorable Moment From My Short Practicum

One of the most memorable moments of my short practicum occurred during my Principles of Math 11 class, on my first full day of teaching. My faculty advisor was present for my first class. After teaching my first class, a review of linear inequalities for the Honours Math group, I was pretty content with what I had planned. Granted, I needed to work on my habits of staying only in one place during the lesson and having my back to the students while writing on the board, but overall I was quite pleased. The next block began and my faculty advisor left and my sponsor teacher was observing me. That's where everything went downhill. I took up homework problems that were considered difficult by the students and during the explanation, I stumbled on my words at several points. I had not looked at that particular question beforehand so I was doing the problem for the first time on the board. I messed up and made mistakes on 3 occasions. This left me wondering, "Why am I here? Do I still want to teach?"

Luckily, things improved for the third block and by that time, I reassured myself and realized that what I needed to do was learn from my mistake and move forward.

Short Practicum Update

Even in the midst of all my lesson planning and UBC assignments that need to be one, I wanted to write a little something about my short practicum so that I do not forget about it. I have been working at St. Thomas Aquinas in North Vancouver for the past week, and it has been a teaching eye-opener. The whole secondary school dynamic is something that I had forgotten about, being almost 6 years out of that environment. My sponsor teachers (2) and faculty advisor have been tremendously helpful in guiding me to be a more effective teacher.

I taught my first lessons today: three Principles of Math 11 classes. I learned some things about my teaching style.

1. No matter how much I try the questions out, I'm bound to forget a negative sign when writing on the whiteboard.
   
2. No matter how well laid-out my lesson plan may be, things will get off-track and I will be racing to finish everything.
3. Timing. Nothing more needs to be said about it.

I hope that I take all my advisors' words to heart and really change my teaching habits. If not for my sake, then for my students' sakes.

Graphing Micro-teaching Lesson Plan

Bridge: Get the students to form pairs in order to work together on small graphing problems.

Teaching Objective: To interpret and comprehend a word problem and to revisit creating data tables from an equation and plotting the data points on a Cartesian plane.

Learning Objective: Students will be able to create lines of best fit for particular graphs and interpolate or extrapolate data points.

Pre-test: Ask if anyone remembers how to graph data points.

Participatory Activity: There are 3 different worksheets. One is about a non-zero y-intercept linear equation (Plumbing Problem), one is about a reciprocal graph (Retirement Gift) and one is about a quadratic function (Carpeting). The students are given 7 to 10 minutes to work on the problems.

Post-test: Representative students who completed one of each of the 3 worksheets will come up to the board, draw their graph and describe how they came to draw that graph. The students then will explain to their fellow students how they answered the corresponding questions.

Summary: Graphing data points on a Cartesian plane is a very useful in Math. It can allow the person to gain insight on mathematical systems that follow patterns. From tracking the trajectory of a ball in the air to predicting the revenue from ticket sales, graphing is important.

Division by Zero

The idea of splitting up a number into smaller parts
Analogy for division, working well in practical ways
Runs into problems with that integer between -1 and 1
Hyperbolas show us some insight into limits of 1/x
Left-hand limit as it gets closer to zero
And right-hand limit too, no agreement, no cooperation
Alas, we are left with inconsistencies
Two trains travelling in opposing directions, never to meet again
I ask my friend to shed some light
"Is it possible to divide something by nothing?"
His response: ERROR.

This poem was created with some inspiration from the 4 minute free-write sessions in class: one on divide and one on zero. After the free-write, I examined all the words written down and wrote this poem. Granted, many of the ideas in the poem were not in my free-write verbatim but I found inspiration in those words. One of the strengths of doing this free-write and poem are that one does not need to be an expert in math to be able to create a poem. Anyone can create the poem from whatever knowledge they have on the topic. Another strength is that it could let the instructor assess how students address a particular subject. If professional terminology is used, then maybe the student understands the importance of that terminology.

Some of the weaknesses of this process is that it is too broad. Students might be like me and write about anything that comes to mind but the content might not even be related to the subject, leading to distract the student from the task-at-hand. Students may conjure up words (like I did) that are completely unrelated and that does not help in understanding some aspect of the topic. Another small weakness in this exercise is that students who like math but not poetry may be thrown off the activity.

Something interesting that I found from the activity were the integration of an extremely academic subject like Math with a subject with openness and fluidity like English. I ended up writing some very un-Math-related things but it was very fun and allowed me the opportunity to write poetry, something that I have not done in a long time. (Aside: Actually, the last time I wrote a poem was for my Materials Chemistry course CHEM 427. I wrote 3 poems about 3 famous Materials Chemists.)

Reflection on Math Micro-teaching

Gigi, Min-Chee and I presented our group math micro-teaching lesson on graphing. It falls under the Math 9 Integrated Resource Package, Prescribed Learning Outcomes B2 (graph linear relations, analyze the graph, and interpolate or extrapolate to solve problems) and B3 (model and solve problems using linear equations of the form a/x = b, x≠0). Some of the positive areas in today's micro-lesson were that we had handouts for the students to use so that they could work through problems in groups of 2, the students successfully displayed their knowledge of graphing and assisted in teaching others, we utilized different types of graphs to introduce students to plotting data points and interpolate and extrapolate from the graph, and we went around the class during the activity and checked in with students that might have problems. The variety of problems, I believe, kept students engaged because the problems were practical and not just theoretically conjured up.

Some of the things which need improvement were our timing, our inability to keep control of explanation and discussion, the clarity of instructions in one of the problem sheets and our team-teaching ability. The timing was not good. The problems probably were not designed for 5 minutes of time to work. They required more time to answer and draw. Specifically, the "Plumbing Problem" had sub-questions that required thinking that would take a long time for a Math 9 student. With the student's presentation and explanation of his solution to the problem, it became very uncontrolled and the student was talking to the class and teaching them for a long period of time. Normally, student-teaching would be a good tool because students will have a better understanding of their fellow students' obstacles. This time, though, the student-teaching took up a large portion of time in our demo. I believe that I should have made the "Plumbing Problem" more explicit in the instructions. I saw a group graphing their points as a step-wise increase in cost. I should have stressed that the cost was gradually increasing from one hour to the next. The final point I have to make is that I spoke very little during the micro-lesson. Even though I did contribute to the preparation, I did not speak much during the micro-lesson and that was a bad move because I should have coordinated better with my team so that we could all give input.

Overall, I think this micro-teaching lesson gave me a lot to think about in terms of student participation and student understanding. I hope that I can come away from this, realizing how to get student involved and engaged so that they can come to appreciate math.

Reflection on "Citizenship Education in the Context of School Mathematics"

After reading Elaine Simmt's "Citizenship Education in the Context of School Mathematics," I have more of an understanding as to the importance of mathematics in everyday life, not only for mathematicians or scientists but everyone. We have become a society of quantifiers, giving numerical values to many things in our lives (e.g. statistics, weather forecasts, hockey performance). Being able to understand a conversation or news report with these data present is becoming increasingly important for people to function in society. By showing real-life connections to math, students will be able to examine how math is presented and how to deal with those essential representations. Presenting problems to students in class would allow me to teach them not only how to solve the numerical problem but those skills applied during the solution can be transferred into their lives and applied as everyday problem-solving skills. Just as with science, I hope to encourage my students to seek answers and to explain themselves clearly. By explaining themselves, they will learn to be succinct and complete in their explanations, learning to interact with others with reasoning and solve the problems to come.

Reference
Simmt, E. (2005) Citizenship Education in the Context of School Mathematics, University of Alberta.

"What-If-Not" Approach

After reading pages 33 to 65 of "The Art of Problem Posing" by Stephen Brown and Marion Walter, I have a better idea of how to plan questions. The authors describe a 5 level process to posing problems.

• Level 0: Choosing a starting point
• Level I: Listing attributes
• Level II: What-If-Not-ing
• Level III: Question asking or problem posing
• Level IV: Analyzing the problem

This process of posing problems can be used by my group for next week's micro-teaching lesson. Our group chose a starting point: we will be talking about graphs, data points, and sets. More specifically, we decided to describe a linear relation. In being more specific, the linear equation will have a non-zero y-intercept and there will be a scenario setting up this word problem. The next step of "What-If-Not" set us up to develop two other graphs. In thinking about how the graph might look like if it wasn't linear, we formulated ideas of problems involving quadratic relations and reciprocal graphs. New questions could be asked in describing these new data points. Of course, analyzing the problems simply is a matter of graphing the data set and looking at the pattern.

I believe some strengths of the "What-If-Not" approach are that you are forced to think about alternatives (e.g. ways your question could go wrong) and you look at all sides of the problem. I believe a weakness of the "What-If-Not" approach is that it can be time-consuming, creating all the attributes and "What-If-Not"s for the situation.

Top 10 Questions/Comments for the Author

In no particular order of importance, here are my 10 questions/comments for the authors:

1. It is very true in math (and even sometimes in the sciences) that given the layout of information, certain questions or problems are assumed (following the x² + y² = z²).
2. Is it our job as future math educators to challenge students' preconceptions and ask them the "unexpected" question?
3. I think the list of possible patterns generated by the class studying the Pythagorean triples shows a higher level of thinking, trying to detect patterns.
4. Something that makes humans unique is our ability to detect patterns in data.
5. A study of the history behind questions and topics can give us background information on the topic and can frame a certain mindset for approaching the question.
6. In discussing the significance of questions and problems, I am reminded of the relational understanding versus instrumental understanding debate. Too often are people lost in plug-and-chug questions that they forget why they were doing them in the first place and lose sight of the meaning.
7. How often do you notice people stuck in internal exploration and don't explore externally?
8. Is it possible to switch gears in children to go from doing questions and accepting everything to learning to challenge the question?
9. "Merely looking for something to challenge will not guarantee that we will find it" (p 18). I find this quote to be very powerful and one of the reasons for doing research in science. If we do research into new areas, we are not sure of the results but that really is the reason for doing research.
10. Why did you choose x² + y² = z² as your first question?

Reference
Brown, S.I.; Walter, M.I. (2005). The Art of Problem Posing, Third Edition, Chapters 1-3, New York, NY, Routledge Taylor & Francis Group.

Letters from the Future

10 Years Teaching Experience

(Letter from student who disliked my teaching) Dear Mr. Siu,

I understand that you are trying to be a great Math teacher. Still, I find that your classes aren't that focused. You don't put an outline of what we're discussing during that class so I don't know what to expect. Also, you spend too much time lecturing during class. If we had more activities to make Math more interesting, or if we spent more time working on problems in-class (as opposed to you telling us the answers) then I think we could learn more. Also, you give way too much homework. My Chemistry teacher gives less homework! I think another way that you could make Math more interesting for us is to get us to do posters or projects on Math topics. That way, we could learn on our own. Also, if you were more available outside of the class hours, then I think I could ask more questions. Don't be discouraged. Thanks for trying to help us.

Kyle



(Letter from student who liked my teaching) Dear Mr. Siu,

I think you are one of the best teachers that I have ever had! You make Math interesting with your puzzles and real-life examples. Making real-life connections with games and toys was amazing! You don't spend the entire class lecturing to us. You give us time to work on problems in-class and you even let us do homework in-class! You're just as cool as my Chemistry teacher. The fact that you're very approachable makes it easy for me to ask questions. Your jokes are sometimes lame, but other times they're hilarious. Thanks for caring.

Sophie



From these two fabricated letters, I was able to verbalize some of my hopes and fears as a teacher. I am afraid of lecturing for too much time during the class or not keeping the students engaged in the class. I hope that I will seem approachable to the students and always devote enough time to my students, looking out for their needs.

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!