Unit Plan for Math 10 Systems of Linear Equations

Unit Plan

Response - John Mason: Questioning in Mathematics Education

1) Do Mason's ideas connect with inquiry-based learning in secondary school mathematics? (And why or why not?)
2) How might Mason's ideas about questions in math class be incorporated into your unit planning for your long practicum?

After reading John Mason's article, I think his ideas connect with inquiry-based learning in the secondary math classroom. The majority of the questions that I had often heard from my high school teachers and ask students myself are in the category of "asking as telling" which directs student attention to easier and easier answers. This type of questioning is not very helpful for student to genuinely learn to think mathematically. I was impressed by how Mason articulates the form of "teaching by listening" in that the teacher is not listening for an expected answer but listening to student's thought process and how they might justify their answers. Eventually, students internalize the metacognitive-type of questions about their own solutions so that they are capable of thinking mathematically without the teacher's verbal prompts.

In my planning for long practicum, I will try to incorporate time to interacting with students during lessons. I will write in questions such as, "How do you know...?" and "Will that always be the case?" I want to encourage students to articulate what they understand rather than being fearful of responding incorrectly to my questions. For a formative or summative assessment task, I will have students construct and solve their own problems, even ones that will challenge their peers. As a teacher, I will model the mathematical thinking myself by not immediately asking funnel-oriented questions or supplying answers but asking, "What did you do yesterday when you were stuck?"

Reflection on group micro-teaching

We did well in terms of time management and organization of the lesson parts. I think this group micro-teaching lesson shows much room for improvement. The beginning of the lesson and the hook were not demonstrated clearly. I would bring in some visuals such as ads from magazines that show percentages to relate the lesson to everyday life. The lesson development or examples of problems were good but I feel they could be more student-centered. Perhaps, the calculations can be some certain characteristics taken from the group of students so they will feel more involved. I think the activity went pretty well as students were interested to find out predictions of their lifespan. Also, we were able to generate further follow-up questions for discussion to make up the remaining time. See pictures of peer- and self-evaluation forms here.

Group micro-teaching lesson plan

Teachers: Jessica, Mandeep, Simran

Topic	Intro to Percentages
Grade level	8
PLO	A3 demonstrate an understanding of percents greater than or equal to 0%
Objective	Students will be able to use the given percentage to find the required information. Students will be able to find percentage of two given whole numbers. Students will have the understanding of the percentage sign.
Materials	White board and marker. Lifespan cards.
Prior Knowledge	Students know division and multiplication.
Intro/Hook	4 mins	Percentage sign %, no calculations here. Discuss how many pennies make up a dollar. (nickels, dimes, quarters). Each group still represents one dollar. On the board, draw out a circle representing a dollar and then fill in 25% to represent 1 quarter.
Development	4 mins	Finding a percent of any number with respect to another number (comparative number) Here percent represent the fraction with denominator 100,While the number represent the amount. Example1 -  if i have 40  halloween  candies ,i gave  10 % of candies to my son ,then what is the number of  candies i gave to my son? Solution- Here, 40 is comparative number    10% of 40 candies = 10/100* 40 =  4 candies to my son Example 2-   If i have 40 halloween candies ,i gave 10 to my daughter,then what is the percentage of the candies i gave to my daughter? Solution-       10/40*100 =25 % of  the total number of candies to my daughter
Activity	3 mins	-Give each student a “Lifespan” card - Ask them to fill out the blanks on the card, their age etc - Explain the question on the card and tell them to do their calculations on the back of the card - let the students work in pairs of two for 2 minutes - go around and make sure everyone's clear on the activity
Closure	1 min	go over the idea of percentage and how it can be used in daily life. For ex: at the shopping mall, calculating the time you have left before you have to get ready for school or dividing your food into portions and eat a certain percentage at a time. It becomes very easy to picture what's gone and what's left if you convert things into percentage.

2-column solution to "Thinking Mathematically" puzzle

Exit slip: video of Dave Hewitt's math classes

Watching the videos of two of Dave Hewitt's classroom caused me to think about my own style of teaching, what my goals of each math lesson should focus on, and students' response to a lesson. I like how Mr. Hewitt utilized the entire perimeter of the classroom; the shift in orientation with regards to each student benefits student attention. Also, I was impressed at how extensively repetitions were used to emphasize the main points. Another area of interest was the choral response from the students since most of the time, I imagine a math classroom to be mostly quiet with either the teacher or one student speaking. I think the choral response is generally a good approach but I would like to check the answers at some point. A student may have said a different answer than his peers but maintain that his was correct. The second lesson in which Mr. Hewitt introduced the idea of variables and different operations was quite interesting as well. I think how he subtly wrote down "x" while saying "I'm thinking of a number" was a great way to express these equivalent relations to students.

Response - Dave Hewitt: Arbitrary and Necessary

Arbitrary is something that cannot be worked out. It is in the realm of memory where students must be informed of this information. A square can just as well be named a "syervel" or any other name. There does not seem to be a definite reason why it is named a "square". Such knowledge is arbitrary.

Necessary is something that can be worked out. Students can use their awareness or prior understanding to figure out a mathematical fact. For example, they can determine the position of a quarter-turn and a half-turn in reference to some point of origin.

For a math lesson, the larger part of the time allotment should focus on working out the necessary mathematics. Less time should be spent on practicing the arbitrary. This influences my lesson plans in that the teacher should have a shorter amount of time talking about what is arbitrary: the names, symbols, notations, and conventions, and the students would have the guidance to work out what is necessary: the properties and relationships in mathematics.

Exit Slip: Math Fair at the Museum of Anthropology

For this afternoon's class, we participated in the math fair presented by grade 6 students from West Point Grey Academy. A few weeks ago, the students were given math problems to work on in pairs. After they have figured out the solution, they visited the Museum of Anthropology and modified their problems to incorporate an artifact of interest to them. Then, they prepared the presentation for the day of the math fair, making display boards with a description of the problem, hints, and solutions along with an interesting back story and manipulatives.

During the math fair, we were encouraged to go around in groups of two or three to visit each project. In a group with two others, I was particularly impressed by three presentations. For the first one, the group made two sets of the game board with numbered sea shells for two people to use as aids to solve their problem. In addition, they provided an extra white board for the third person in our group to write on. I think it was quite thoughtful of them to be so prepared and they also gave us each a little present after we successfully solved their math problem. The second group’s model of their problem was not entirely workable so they directed us to pointing our fingers on a drawing on their display. What they lacked in physical manipulatives they made up in their oral presentation. One of the little girls explained the conditions of the problem and guided us through solving it with a high level of clarity and confidence. They had figured out that we were training to be math teachers because we were over-thinking the problem! As we exited, I spied another project that caught my attention so I went back by myself to talk to this group. It was a problem on inverting a triangle made out of multiple blocks. The girl led me to solve several similar problems using less number of blocks before tackling their challenge problem. I think that scaffolding really helped to build my confidence as the number of blocks increased. Then without me mentioning (for a previous group, we asked the presenters, “What will happen if you started with a larger number?”), the girl told us that we can use the same method to obtain a solution for a triangle with n blocks. She also pointed out how the problem is related to the Fibonacci sequence which I had not thought of either.

Overall, I saw that many of the parents came to support their children’s work. I am also encouraged by the level of effort and professionalism displayed during this math fair.

Math Fairs

I have volunteered for several math fairs that took place on campus with the UBC Math Department and PIMS. However, I did not fully understand the rationale behind these math fairs. After reading through the introduction in this booklet, of which I have seen many times at these events, I feel that this type of problem-based fair provides one of the most beneficial approaches to mathematics for people of all ages. The participants ranged from elementary children to mathematical education professionals. Everyone was actively involved and shared the joy in solving mathematical problems, sometimes not even conscious that they were using mathematics. Hence, I would love to run a math fair at my practicum high school if possible. I would first ask the students if they have any problems of interest in mind before giving them one. For the math fair problems, I would allow some class time over two weeks for groups/pairs to work together both on solving and preparing the presentations.

Reflection on micro-teaching

For my micro-teaching lesson, I taught my peers to make fruit/vegetable sculptures based on Saxton Freymann's picture books. Initially, I had wanted to make giraffes out of bananas but decided against it since it was too complicated to finish in 10 minutes. Hence, I simplified the lesson to using bell peppers and adding lively features such as eyes and a smile. Overall, I think the lesson went well and everyone enjoyed this fun activity.

In my first micro-teaching lesson in another class, I ran out of time so I wanted to be extra careful not to plan too much. But in this case, since the activity had been simplified and everyone was cooperative, we were left with a few minutes to spare. I expounded on the picture books I showed them in the beginning. In hindsight, I would have elaborated more on the sculptures themselves, such as adding other distinctive elements on each bell pepper or ask the students to brainstorm other extension ideas. Moreover, I intended the "take a photo of your final product" to be the assessment part but it may not have come across clear enough because some feedback indicated a lack of thorough assessment.

Response - Battleground Schools reading

For this blog response, please comment on the fraught history of mathematics education in North America and the ways that you think this might affect your own situation as a math teacher.

I was surprised to discover so much contention in the history of mathematics education in North America. Progressivists had thought of and Dewey attempted to implement inquiry-based learning in the last century with little reform in the general scene of math education. However, the university-level math curriculum gained ground in response to the space race between the former Soviet Union and the United States. I started to consider where I fit into this history that unfolded in the US. As an elementary student in California, I remembered learning math in a basic, traditionalist way. When I moved back to Taiwan, I experienced an even more "conservative" approach to teaching mathematics but at a higher level. The international ranking of different countries eighth graders' mathematical ability heightened the emphasis on math education in North America and thereby influencing NCTM Standards.

For my future career as a math teacher, I will certainly encounter parents and other teachers who are math-phobic. In addition, there will be ongoing "math wars" and revisions to the current math curriculum. I believe that the students should come out of the math classroom feeling accomplished about the sense-making and practical applications of mathematics, not merely the drill and skill of accurate computation.

Lesson plan for micro-teaching: Food for thought

Teacher: Jessica Chen
Topic: Food for thought

Objective	Teach students that you can play with your food, using fruits and vegetables to make interesting, everyday objects.
Materials	Bell peppers, beans, exacto knives.
Prior Knowledge	Safe use of exacto knives.
Intro/Hook	Materials prepared in front of each student. Ask “What can we make out of this?” Show the picture books by Saxton Freymann.	2-3 min.
Development	1. Teacher demonstration.       2. Individual student work.	5-6 min.
Assessment	Students take photos of their creations.	1 min.
Closure	Remarks about ways you can have fun with your food.	1 min.

The giant soup can puzzle

I think everyone will have different points of measure so the answers we come up with will be different. I also noticed that the camera angle of this photo, the way the bike is positioned, and the partially buried portion of the water tank can alter the results. With that said, I estimate from this picture that the water tank's diameter is 2 times the bike's height. 

From a simple Google search result, "The dimensions of a typical bicycle are a handlebar height of 0.75 - 1.10 m". I will estimate that Susan's bike is 1 m, or 100 cm. Therefore, the diameter of the water tank is 100 cm × 2 = 200 cm, and the radius is 100 cm.

According to the other picture of the Campbell soup can and my own measurement of one at home, the ratio of the height of the soup can to the diameter is 1.51. For the water tank, the length can be determined by 200 cm × 1.51 = 302 cm.
The volume of the water tank is estimated to be (pi)×(radius^2)×(height)=pi×(100 cm)^2× (302 cm)=9488000 cm^3 = 9488000 ml = 9488 L.

(Imaginary) letters from future students

Dear Ms. Chen,
I was in your math class in grade 10. To this day, I always tell other people that I love learning math and you are my favorite math teacher. Why? It's because you really care about the students' learning. You would make the effort to explain a concept several times by showing different examples. I enjoyed doing the fun in-class activities that engaged me in mathematical thinking and working with my peers helped me understand better too. Whenever I felt like I needed extra help, I know that your classroom is a warm door for me even after school. Thank you for inspiring me in such a unique subject as math and genuinely caring for us students.

Dear Ms. Chen,
You might not remember me in one of your grade 11 math class. I sat in the back of the classroom where you probably don't notice much. Sometimes I had trouble hearing your voice, especially if you were speaking too fast. I didn't like my partner when we worked in group activities because he always got the answer right away so I didn't feel like trying to solve the problem. Doing the math project was fun but that's because I prefer anything other than quizzes and tests. I am sure I learned quite a bit in your class but math is still a difficult subject for me.

Reflection:
I hope to implement interesting activities in my math classroom so that it is not just pencil and paper. Also, caring for students in every aspect is one of my main goals. I will make myself available and approachable for extra help after school.
I am afraid that I may not be aware of speaking too fast or too soft in class. I hope to walk around the classroom so that I'm not always in the front talking. In addition, I would like to make different types of assessment in math such as a math project. Another thing that I'm worried about is how to pair up students, those with similar or differing levels of math together, and how to carry out group work.

Math/Art project: 60-card polyhedron

For the math/art project, my group created a deltoidal hexecontrahedron out of 60 playing cards using the template provided by George Hart. I have always been fascinated by the art seen in geometrical shapes and solids but using playing cards to construct polyhedra is a new idea to me. The slits on each card must be cut accurately since the construction is made by interlocking the cards together without tape or glue. The most difficult part was to figure out the 3-fold lock which required a little bit of bending to create the equilateral triangle with each point above the next. This task was easy for only one group member who is a very visual learner. It took the rest of us about 30 minutes individually to learn to do the 3-fold construction.

Rather than putting together 60 cards randomly, we arranged the cards in a specific pattern. Each layer is the same suit of cards in ascending order and the layers are alternating between red and black suits. This required more careful planning when we were constructing the cards but our final product looked awesome. We can explore shapes and patterns in this 60-card polyhedron. For example, we can ask students, "How many pentagons are there? How many vertices are formed by 3-fold locks?" These questions can lead to lessons in geometry about platonic solids and various other kinds of 3-dimensional models.

The fire alarm incident during our presentation gave us an opportunity to experience "teaching math outside the classroom." I think our classmates tried their best at the activity but it was more difficult than we had planned. Getting everyone's attention outside with many distractions around was not an easy task. I believe it is important for teachers to adapt to changing environments. Sometimes, unpredictable circumstances will require us to relate the material to the surrounding or give insight into other perspectives. I will continue to examine math in an artistic way, endeavor to discover/create art in math and bring these new points of view to my students.

Dishes puzzle

We know that every 2 guests shared a dish of rice, every 3 shared a dish of broth and every 4 shared a dish of meat. What is the smallest number of guests that can fit this scenario?

If we only take into account the first two conditions, that is, 1) every 2 guests shared a dish of rice and 2) every 3 guests shared a dish of broth, then we need at least 6 people for this to work (since 6 can be divided into both 2 and 3). Now, for the multiples of 2, 3, and 4, let's see where they share a common multiple!

Listing out the multiples:
2, 4, 6, 8, 10, 12, ...
  3,   6,    9,     12, ...
    4,     8,        12, ...

So 12 is our friend here. Let's consider the case where there are 12 guests. How many dishes of each type would there be?

12 ÷2 = 6 dishes of rice
12 ÷3 = 4 dishes of broth
12 ÷4 = 3 dishes of meat

6 + 4 + 3 = 13 dishes in total
The question has 65 dishes. (65 ÷ 13 = 5)
Since we multiply 13 dishes by 5 to obtain 65 dishes, we also multiply 12 guests by 5 to obtain our final answer: 60 guests in total.

I think cultural context certainly would have an effect on the problem or the attitude of students towards solving this problem. If put into a Chinese context, a student with a Chinese cultural background may be more likely to tackle the problem without much thought. Another student without this background may not be familiar to the idea of so many people sharing multiple dishes, therefore be a bit confused when solving this problem. I found this same question on the NCTM facebook page using Mexican food as the cultural context. This shows that math puzzles can be tailored to meet the specific needs of certain students/community.

Mathematics for social justice

After reading the excerpts from David Stocker's Maththatmatters, I have gained a whole new perspective on the teaching of mathematics in school. Through thoughtful planning with the goal of engaging students' interest, the teacher can transform ordinary math lessons to topics that address social and environmental issues in the world. I was impressed by how often we take a "real-life" problem to do math operations when instead it may be more beneficial to apply math skills on issues that matter to the society. In this context, Stocker's topics would not necessarily be considered as "neutral" because he does incorporate his own point of view. This kind of approach tends to raise concerns over the bias of a topic but I believe it is also our duty to encourage students to think critically.

Even though Stocker's book focuses on middle school mathematics, I am positive that these topics can be tailored to teach secondary mathematics as well. Secondary students may be able to take a stand strongly on these issues or going further, to develop some topics into their own research projects. This will definitely be more time-consuming but ultimately the students will learn math in more depth, consider social justice matters and eventually take actions to make a brighter future.

Influential math teachers

My fifth grade teacher in California was the most memorable first teacher who made math an interesting and fun subject for me. I came out of that school year telling everyone that I loved math. At the beginning of each class day, we worked on the exercise called "Star Voyager" which consists of small math problems and one challenge question. Each student accumulated points based on the number of problems solved. I don't remember exactly but it somehow added to our imaginary adventure into outer space. I hope to bring fun and challenging problems that would further interest students in the subject.

I think a not very exciting math teacher would be my junior high math teacher. She excelled at solving problems and answering our questions. I learned a lot from following her examples but the format of every lecture was the same. In my teaching, I will plan to have a diverse range of activities to approach different concepts in problem solving. 

In my Pre-Calculus class in high school, the teacher was enthusiastic about the subject. This really made me desire the same love for math and teaching math as I had a role model there. I found that helping my peers with the concepts and assignments in math was very enjoyable and rewarding. The teacher provided a space for student pair/group discussions which I think is crucial in learning math.

Reflection on Teaching Perspectives Inventory(TPI) results

This graph depicts my results from taking the TPI test. My scores were pretty consistent in the first four perspectives, transmission, apprenticeship, developmental, and nurturing, around 34-36 with the highest score in apprenticeship. Due to the similar scores and the fact that none is exactly above the top line, I would not say that a specific perspective is dominant. However, I scored the lowest in social reform, 25, which is 9 points less from the nearest neighbor so social reform is a recessive perspective. On average, I received a score of 32.6 with a standard deviation of 3.88. 

From the definition of the apprenticeship teaching perspective, this test shows that I strongly value good teachers being skilled practitioners of what they teach. As a teacher, I am there to demonstrate to students the knowledge through performing examples and to guide them in different levels of learning. For the other three in which I scored equally high, I think what the test reveals is true. Of course, the transmission of content is important but so are understanding each learner's way of thinking and nurturing the students to have confidence in themselves. I really echo that part in the definition about caring for the students by way of encouraging their efforts and challenging them while supporting their goals.

What surprised me was that my score for the social reform teaching perspective appeared so low on the scale. After much thought, yes, mathematics is not a social studies class and we may not think much about incorporating messages about societal change into the math classroom but there are ways to direct the students to think critically. For example, students can work on projects on a social issue of their interest and apply the appropriate mathematical skills in their presentation and analysis of data. Other lessons could incorporate optimization and decision-making for instance, in education funding or low-income housing options. I would try to be more creative in this area of lesson planning. Regardless of the various activities, the goal is for students to associate their learning with the environment we live in and how their actions would impact the future world.

Provincial Pro-D Day

On October 23rd and 24th, I will be attending the IB-DP Mathematics workshop.

Number of squares in a chessboard

How many squares are in an 8-by-8 chessboard?

The first thought that pops into the mind is that 8*8 yields 64 squares. However, these only account for the 1-by-1 size squares. We quickly realize that squares can be seen in different sizes on a chessboard.

Let us consider how many 2-by-2 squares are possible.
I'd like to think that there are 7 horizontal positions for a 2-by-2 square since a square cannot move outside the board. Likewise, there are 7 vertical positions for squares of this size. 7 times 7 gives 49 squares here.

Similarly, for 3-by-3 squares, there are 6 horizontal positions and 6 vertical positions. =>36 squares.
Also, for 4-by-4 squares, there are 5 horizontal positions and 5 vertical positions. =>25 squares.

We see a pattern!
Taking it all the way, we get 1 square for 8-by-8 size.

The total number of squares is the sum of the squares: 64+49+36+25+16+9+4+1= 204 squares.

Ways we can extend this puzzle:
Q: How many 3-by-2 rectangles are in a chessboard? (The student must think about positioning these rectangles on the board.)

Q: How many triangles are there below?

Reflection on integrating instrumental and relational learning

After reading Skemp's article, I fully stood on the side of relational understanding of mathematics and hence was surprised to discover a good number of the class in support of the instrumental approach. Throughout the discussion, some stressed teaching students instrumentally first so that they would gain confidence in learning math; however, others contended that it is more important to present students with the relational understanding before showing them the shortcuts/formulas. 

For example, on teaching angles in polygons, one could easily give out the formulas for computing the sum of the interior angles (180 degrees*(n-2)) and calculating each interior angle of a regular polygon (180 degrees*(n-2)/n). Nevertheless, it is much more helpful to have students start by looking at the sum of interior angles in a triangle, which is 180 degrees. Then, we see that a square/rectangle is composed of two triangles and that a pentagon can be divided into three triangles, etc. From this point, the students can come up with the formula themselves about the sum of the interior angles in a polygon. Furthermore, they can also figure out each interior angle in a regular polygon. 

I now feel that both instrumental and relational learning are crucial to mastering mathematics and that it is the teacher's responsibility to discern how to integrate both approaches with regards to specific lessons. 

Response - Richard Skemp's article

I stopped to think about the course of my education in mathematics, how I was taught, and my attitude toward the learning of the subject. Were they instrumental or relational ways of understanding? One of the things that struck me in the article were the clear analogies differentiating instrumental mathematics and relational mathematics. If we see mathematics in this light, the approaches to teaching and learning the subject would be revolutionized. I was also surprised by the fact that once concepts were learned relationally, these carry on a long-term, even organic(!) effect in the student's future learning. In conclusion, I believe that relational mathematics is far more beneficial to teachers and students alike than instrumental mathematics. We would love to grade a 100% exam but we would be doing learners more justice by instructing the ways to navigate and construct their mathematical knowledge.

Welcome to my blog on teaching secondary mathematics

Hello, it's wonderful to see you on my first blog ever!