342 unit plan

1. Integrated and in-depth

    How does that make sense to students?

    How to apply the basic concept in life? 

    Does this application make sense/interest to students?

    What happened in the history?

2. In a very practical way! Think as a teacher!

3. Realistic task, activity, question, prompt, and time allocation

Everything we learn in math comes from things people inquiry in lives

Dan Meyer math 3 act

Hook them clearly and visually

Drop in resources and tops?

Resolutions?

Inquiry questions from Dan website

Lesson flow

Time do the class/activity time management

Connection

Goal/objectives: at the beginning or not, but keep in mind

School culture appropriate

Routine, rule, flexibility/adaptibility

Short activities that connected with one another

Things to interrupt the flow

Adjustment of the flow

Teamshape app......: Tean Shake

Welcome students warmly as a group

Factor appropriateness

Safety issue

1145 Henry Angus 254 Finding the right fit

Pauline O'Brien, director of educational recruitment services

Al Scott, school support & evaluation officer

Finding the Right Fit

CIS (council of international schools) community of educators

690 schools, 527 universities and colleges, representing 111 countries

Quality schools internationally

Vision:

To inspire the development of global citizens through high quality international education: connecting ideas, cultures and educators from every corner of the world

Edu recruitment services

International university admissions & guidance services

School support & evaluation services

Promoting the IB education certificate cohort to meet international standards

Provide services, language learning, operational systems

Why it is a good experience to work internationally

Collaborative teaching environment

Attractive Salary and benefits

Professional growth

Global travel opportunities

Curricular and technology resources

School as a community center

Multi-national students

Profile of a successful international educator

Demonstrated success

Language acquisition training

Differentiated instruction

Collaborative teaching style and methodology

Integrating technology into instruction

Skills offered outside the classroom

Ability to adapt new environment

Recruitment starts in October

Peak: December to February

Year around activity

Use of online interviews to fill positions

Recruiting fairs still fill 50% of teaching vacacies

Recruit strategies

Value-added to school community as international schools tend to be isolated communities

Technology around curriculum and community

Adapt new environment

Education philosophy

Leadership

Enthusiasm and confidence

Languages

Communication

Collaboration

Experience

School atmosphere

Community

Salary and benefits, compensation

Career growth plan, promotion opportunities

School mission and programme,

School website

Research country and community

    Cultural and political

    Environment and security

Misconceptions identified quickly

Citizenship restriction

Need for certification

Housing for dependent a

VISA restriction (age, nationality, etc)

Training partners

Pets

Tax situation

Www.cois.org

CV

Motivational statement

Philosophy statement

Certificate

Three referneces (current)

Collegial Anglo Colombiano: www. anglocolombiano.co

Amsterdam international community school: aics.espritscholen.nl

Raffles international school Dubai: www.rafflesis.com

360

Digital/online/Internet literacy

Critical consumer of Internet-based information

ELL website

Newsela: exchange big/tough words to simple words so that students can understand

Wiki has a Basic English feature that change hard words to simple words

1107 20+ apps

Bitly: shorten rul addresses

Plan board: teacher app for teachers to plan daily schedules

Remind/celly: send msg to students' cellphones using PC (try not to identify students)

Screenleap: run a small program on teacher's PC and share it to students' screens (good for distance class)

Classdojo: good for group work?

Edpuzzle: flip learning: use videos found online, pause the video and ask questions to students

Instagrok: tells all the related definitions and knowledge about the input words......good search engine

Google cultural institute: collected museums around the world

Pear deck: student response system: using connected devices to check students' understandings

Kahoot.it

Quiziz

Socrative: student response system

Plickers: very powerful and efficient to test students' answers of a multiple-choice question

Quiver: making 2d objects to 3d moving objects

Element 4d: science, maybe math, especially 3d geometry

Google translate

Arloon

Photomath

Augmented reality & creation

Lucid chart: mind mapping flow chart outline 

Thinklink: take a picture and describe it 

Explain everything: ......

Screencast-o-matic

Creation tools

342

Integers operations:

All about examples and actions.

From very easy to hard, let student catch up

Then student stand up and try the same activity

Trust students are good learners and teachers as themselves

Much better than the internet addition I tried in Churchill

Functions and equations

I'm thinking of a number...... (Write x)

What's next? ...... Great amount of waiting time

Whistle when showing the division

Students talking and discussion

Repetitions and emphasizing

Moving around the class

Teacher gestures: match actions, notations, and mathematical expression 

Waiting for students' reply by doing the action......

490/450

Does collecting data in school improve student performances?

Information, quantitative, qualitative

Feedback, reflection, mapping records, measurable, control

Scores, family background, attendance, medical information,

CEM Centre for Evaluation and Monitoring

ALIS 

Ability than achievement

Fluency than knowledge

ALIS test score sample analysis

 

Nov 4-5

I've discussed my first lesson plan with Ms Lin and Ms Knott. Even though Ms Knott will not be my SA (b/s she has had an SFU TC already) and is substituted by Ms Leslie, I will still teach three classes in her class.

Tomorrow I'll teach the mandarin class about lyrics. It is a song in a famous TV episodes in China: 何以笙箫默. 

When a teacher stands in front of the class, all the teacher needs to think about is to communicate with the class. Eye contacts, gestures, ask for confusing, etc, are commonly used as ways of accessment. Also, be strong and be prepared to stop any inappropriate behaviors.

Nov 2-3

On Nov 2 I was in Ms Lin's class. Ms Knott was not there as she only comes on day 2. There is an old style light projector in each classroom and an Epson projector with VGB input port. Normally there are less than 30 students in a class.

Ms Lin teaches mandarin and math. She is kind and friendly. I was invited to a mandarin teaching training on Nov 14. 

On Tuesday Ms. Knott was there and we chatted about my teaching practice in her class. She will give me three lessons to teach: math 9 on multiplying polynomials (in three classes). I like the staff.

490 & 450

Reflexction every two days on blog and sent to email

Resilience and thrive

Churchill!

Take responsibility on my documents

Haircut, dress, school website information (VSB, principal, whether it is a match)

Self intro, occ, ministry guideline, 

First day at 8, but every day if possible

School office sign in everyday

VP explanation of the school

Form: 

secondary TC form

Yellow: SA-math & mandarin

Green: FA

Rules/regulations

Daily schedule (bell schedule, block rotation, breaks)

Safety procedures

Admin office

Equipments, staff rooms, classrooms, names of school teachers, principals

School clubs & activities

Special program

Teacher blogs

Observe the detail how they do things

Understand your students as learners

342

Math 8;

Math 9; locally developed adapted math 9;

Foundations of math 10 and precalculus 10; apprenticeship & workplace math 10

Foundations of math 11 (required for graduate); precalculus 11; apprenticeship & workplace 11;

Foundation 12; precalculus 12; A&W 12;

Calculus 12

AP math; IB math

PD day 1 with Jim

First day with Jim (retired math teacher in Churchill)

teaching mathematics is not about teaching math i learnt b/c what i learnt may not be the math they need in the future

make sure you engage your kids
the success depends on how i engage the kids

Textbook: Heise 

Oct 20 Mulgrave visit

I was the driver. Neil, Jessica, Breanna, Joyce were the passengers.

We arrived at 710. The meeting started at 800.

After the group meeting, our DP math teacher candidates went to Rm 180 and had a short conversation with 3 DP math teachers (Kayle, Micheal, Prior) and 1 MYP physics/math teacher. We were told about the big jump of difficulty levels from G10 extended to G11 math HL. It is also true that parents may be a factor of pushing students to study HL math. The class pace is fast for HL while the SL is ... Also fast...... We were shown a student's G12 SL term project using chi-square analysis and a lot of statistics knowledge. It was really astonishing when we told that all the statistics knowledge was taught in class. The teacher also mentioned that they have to teach extra-curriculum knowledge to students within insufficient amount of time.

At 910 we were in Rm 182, Dr. Frewin's math 12 HL class. The topic was derivatives. Students were asked to play a card game to match on-card information to different functions. By the way, the classroom was clean and tidy with several math artifacts.students were invited to play the card game on teacher's laptop, then to explain their work on the whiteboards on the two side walls. Frequently, the teacher recalled definitions of some math concepts as well as derivatives and average velocities. One can told that physics contents were introduced inthe class (and of course the students are taking physics 12 this year). Students were encouraged to express their thoughts and understandings at anytime of the class.

We were invited to watch a performance after this class.

The at 1100 we went back the Rm 182 and attended a math 10 class. Students were engaging but the class pace was slow. The topic was about tangent and started with slope. Graphical calculators were widely used and large amount of time was spent to explain how to use them. Kids have little knowledge about numbers as I was invited to explain why a decimal number cannot be a perfect root square of an integer.

Then we had a short break and attended a ToK class taught by the Vice Principal. Topic was about how knowledge was acknowledged through eight aspects: sense perceptions, 

Then it was the lunch time. Students were arranged to have lunches at different times.

At 1405 I had my last class visit in Rm 165 with Mrs. Chen. It was such a wonderful feeling to speak Chinese legally in classroom. Unfortunately there were only two students who took Mandarin as language B SL. They were struggling in this course as G10 Mandarin emphasized too much on listening and speaking, with limited skills in reading and writing. However, students can understand what I was talking (in mandarin) and respond in English. It is truly amazing that this communicating process improves their perceptions of Chinese culture.

Mulgrave is such a wonderful school that is only available to some families. Even though, it is wonderful, wordless to say.

490450

Certificate must be finished today

E-folio

Study skills

1) asking questions and wait for answers 

2) engagement: teaching is supposed to be a game

    don't embarrass students and talk about the issue after class

3) teach integrity

4) study sheet: summary (brain map, chart, diagram, etc), 

Delivery method: observe and record the teaching strategies used

Techniques: what strategies were used to check learning?

Talk 5 minutes to make notes

Search: record chart, Cornell note taking/Cornell margin

What to observe in the visit/practicum 

401

Training program: first aid, etc

Government funding

Freedom of implementing policies among schools

490&450

In the news

What is Inquiry-based Learning?

Reflection #2 is to be a comment on your readings about Inquiry and your thoughts on the process as a teaching strategy. Because this is a  reflection about your perceptions of the topic, please discuss how you believe it will be manifested in your classroom or simply ask important questions that you need to be answered before you are able to understand and apply the process.

490 & 450

To do: 

Reflection

Goal file

Tool kit

Group feedback

Orgnized and time control

Preparation, role identification

Overview of the whole project

Rubric

A table

  Group work-roles.    Speed dating.    Read share.      Tool kit.           Teaching strategies.      

  Presentations-from group.                 Rubric.              Goal setting

  Interview.                  Blog.                 Reflections.       Projects.         Collaboration

  Skills.                        Challenges.       Ethos. 

Kids need to be listened to rather than to be told.  

Volkswagen Das sth...... Just be fined by the U.S. Government.

401

401

ZOTERO

www.zotero.org in Firefox

  Education policy information

    What kind of policy

342

MathThatMatters

By David Stocker

Object: Angles > 360 exist

Knowledge to teach: what is an angle, principle arm, terminal arm, degree, reference angle

Kinesthetic activity: find ways of showing a 90 angle

Tools: clock second hands and minute hands

Monday class notes

401:

Halloween preparation

Courageous conversation

    Stay engaged

    Experience discomfort

    Speak your truth

    

Policies:

Warning-principle office-further punishment

0 tolerance but do NOT over-react

What if the blind person is your close friend? Will you still be happy to laugh at him/her?

You may disagree with the law, but plz realize the importance of the existence of the rul/law

Opinion: ++/+/?/-/--

Discuss with same opinion

Discuss with one degree away opinions

.....................two........

Stand in a shape of pi

Discussion in a whole group

Change opinions

Conclusion

Social justice issues in classroom

Antiracism

LGBTQ

antipoverty

Environmental justice

Status of women

Peace and global education

Disabilities

The 3 "S"s 

Social Service

Social Responsibility

Social Justice

308

ADHD drug addiction and long term side effects

Migrant crisis map

Stanford-Binet Intelligence Scale

The dark side of IQ testing

Nazi murder people based on IQ score

U.S. Sterilized low-IQ people

Teach kids how to deal with FAILURE ( in math test)

Team score is determined by group members' individual tests average (it transmits teacher's responsibility to group LEADERs).

490

First IBDP cohort meeting on Tuesday

Brain map: how does (math) influence the beliefs on individuals

Use note margin to review so that only 5% words are read but everything is gone through.

Facilitating sometimes means PUSHING!

342 class today 1

zSkemp:         relational ---------- instrumental

David Pimm: understanding ----------- fluency

                     In-depth understanding ------------ procedural algorithm

Go straight 100 meters, turn right and walk 200 meters, then .....

Go to book store, go to the village, go to the bank, etc.

Square x, square y, plus them together, then take the square root, then that is the length of the hypothenuse of a right triangle.

Consider the right triangle, use Pythagorean theorem.

Multi step algorithm....... Need explanation

    Students get the first few rules, but the rest......

    Teachers can't fully explain why to to these steps......

    

Some cases are instrumental friendly as long as students can SEE the reasons.

Basically connect what is learnt to things students are familiar with

Ken Robinson's TED talk in class & instructor's late to be in class

Right now is 10:30 am. I am just off of my 401 class.

Bathseba was late about half an hour today due to the traffic jam. Good to learn the feelings of being in a non-teacher class.

We watched a TED video "Do schools kill creativity?" addressed by Ken Robinson. Here is the link to the speak transcript: http://www.ted.com/talks/ken_robinson_says_schools_kill_creativity/transcript?language=en. In this video there are one thing I want to talk about.

Ken mentions the "academic hierarchy" in the current school system: language and math > humanity > arts. He summarizes this hierarchy as the result of industrialization. In class we also discussed it as the result of "the desires of the riches". Well, from my own perspective, it is the result of society selection. Constrainted by limited resources, it is not reasonable to attribute resources equally to subjects like arts and math. However, this situation may be changed in future.

It is worth saying that read the assigned readings prior to class is good for learning. I wonder if I can do something to my math class.

Learning notes 3

Wonderful lectures today even though the gold price is still dropping!

Isobel made a good example of arrange group discussion. Each group member should have a role. This role settlement pushes students to focus more on group work and to engage in the discussion.

Isobel also arranged a "no-talking" game in the 450 class. In the game, all the DP students needed to stand in alphabet order of our first name/last name. Moreover, we needed to play the game according to our birth dates, and even the distance from birth places to Vancouver general hospital.

The purpose of the game is inspiring: (a) in order to play the game, students need to communicate through body languages (remember, talking is not allowed), which in some extent creates a "fair" environment that is regardless of language, skin color, or any other factors. Also, (b) it challenges students' ability of analyzing problems. In this case, students need to break down the problem through creating comparable categories. For example, in the birth place case, students need to compare the birth places by country, then province, then city, and then communicate through a silent method. Many of us use distance (in km) to demonstrate birth places (but it is imaginable that at the beginning the class is totally a chaos as people communicate differently). Some even communicated by airplane flight hours. But all in all, this is inspiring.

This game is all about showing what an inquiry-learning class is. I will summarize the inquiry-learning materials in this blog on Saturday and Sunday.

In the afternoon 490 class, we were informed that all the IB candidate teachers must participate in some CAS activities (basicly volunteering) for two hours per week. Neil later gave me a wonderful suggestion that to be a basketball (or soccer) coach in Richmond. Maybe I will join him.

We also did a claim/counter claim exercise in 490 class. The topic is "You can't be a well-rounded, knowledgable person unless you take math (this is my teaching subject)". One thing to be noticed is that all the subjects have sufficient reasons to be known, but students may choose some subjects that s/he may be interested in for further development. Rejection of any subject(s) makes a student grow up imbalancely.

Learning notes 2

Today I had two classes. EDUC 401 by Bathseba Opini, and 342 by Susan Gerofsky.

Bathseba asked students to discuss a problem with the student next to him/her, and then to share the discussion results in class. This is a helpful teaching strategy for classes with discussion activities (but not that appropriate to classes with more lecture activities).

Susan gave us a very impressive lecture today as she could almost tell everyone's name in the first class. This is an extremely helpful method as: (a) it first builds relationships other than teacher-student relationship with students, (b) it shows how much efforts you have made to help students, and (c) it reminds students the high costs of absence from class.

In 401 class there are lots of tech-aid students. In 342 class Susan emphasized that it is quite reasonable to adjust group settlement in class as long as the adjustment is beneficial. Also, moving around in the class is also important.

Nothing else to mention for today. Sunny, Ella and I had a wonderful dinner together after school.

Learning notes 1

Inquiry learning.
    Frankly, what we did in the teacher education class can be considered as inquiry learning. In class we (teacher candidates) broadly discuss issues that may rise from teaching practice, and look for solutions.
    It requires students' collaboration as the learning process may fail if students feel disengaged. This means that a teacher cannot simply use a topic for class discussion.

If everyone cheats, then cheating is not cheating. Agree or disagree?
(a) define cheating: (i) breaking rules; (ii) unfairness; etc.
(b) statements VS counter-statements

    John Yamamoto used a very interesting way to demonstrate the practicum policy. Before class started, John gave 20 sealed envelopes to randomly-selected students. In each envolope, there was a prepared question that covered a small part of the class content. The questions were framed in the format "A + B + C", where:
                "A" is a self-introduction (example: "my name is......", or "my parents named me ......", or "I prefer you call me ......");
                "B" is a self-praising statement (example: "though this wonderful lecture ...". or "I really like your informative class", or "Thank you, John! You really helped me a lot");
                "C" is the new question part.

    I like this activity because:
    (a) It get everyone's attention through reading the question. Because the questions are new and selected by the teacher, everyone knows they are important;
    (b) "B" is a smart design. The self-praising statement----or self-joking statement depending how you see it----relaxes everyone, and it plays the role of class games that make students enjoy the class;
    (c) Even though the questions are not the students' questions, they are asked by students and "sounded like" students'. This design makes students concentrated.

hw 1 sep 2-3 ibdp

Sep 3

Reflections on discussions of education

What/so what/now what

On September 1, all 60 IBDP/MYP students participated in a group discussion activity. Each of us was given a small rectangular piece of paper whose color was chosen from 12 different colors. Then according to colors we grouped together. Thus there were 5 members in each group. In each group, members were assigned with numbers from 1 to 5; based on the number, members were assigned with different roles. For example, all students with number 1 became the group leader who was responsible to manage the discussion; number 2 students were the recorders who were in charge of recording what was discussed; for number 3 students, observation was the role (maybe to observe each members’ behavior? The problem here was that this role was not clearly defined, at least from my perspective); number students were the presenters of groups who would present the discussion results at the end of the discussion; number 5 students were given the role to summarize and submit the results in words one day later. One interesting design of the discussion was that members in each group were not fixed. For instance, after a discussion, number 3 and 4 of group white should go to, group yellow for example, and number 3 and 4 of group yellow would go to ANOTHER group. Because of time limitation we did not change members thoroughly.

From my perspective, these arrangements-----group color, numbering the members, and switch team members----are smart arrangements for students to break through the feelings of unfamiliarity so that they are FORCED to communicate with people they used to communicate with. Let’s see some other group designs: (a) teachers let students freely group together; or (b) teachers assign each student to a planned group; or (c) a mixed design of (a) and (b) that after (a) teachers may adjust the group members. It may be clear, compare to (b) and (c), that students may not reject the grouping method we used in class as the teacher creates a protocol that everyone must obey; this method is also better than (a) as it creates communication opportunities for students who are not familiar with; meanwhile, it can also be seen as a small game in class that can make students enthusiastic about the class; what’s more, when students are switched among groups, they actually are sharing different opinions or perspectives on one topic.

This discussion design is good for 

September 2, 2015. Third day in IB training program.

Sorry that the first blog of IB training is actually from the third day. In the first two days, the whole IB DP/MYP/PYP programs learned something about some facts of IB's current situation, and experience the CAS (creativity, action, and service) that is actually a series of volunteering activities. There will be a lot to write about and I will gradually show you what we have done for the first two days.

Now is the third day of being in the IBDP program. It was a bad experience for me because I got cold last night. For the today's lecture from 9 am to 3:30 pm, I was in a headache mode and had to push myself almost to the limit. But I have to say today's lecture is wonderful.

We first review the IB's current situation (which has been demonstrated by Gary Little in the second day), and then Isabel showed us the requirements of receiving the final IB diploma. The DP program starts from Grade 11 and finishes in Grade 12. It contains groups of subjects: Group 1: first language (English, French, Spanish); Group 2: second language (a language that is different from the first language. For example, mandarin, german, etc.); Group 3: Humanity (history, geography, business management, economics, etc.); Group 4: science (biology, chemistry, physics, environment system, etc.); Group 5: math (math studies, math I, math II, further math); Group 6: arts (dance, painting, musics, etc.). To achieve the final diploma, a student must have 3 HL (higher level) courses and 3 SL (standard level) courses.

According to the requirement I create my own IB plan: English (SL), Mandarin (HL), economics (HL), physics (SL), math (HL), and chemistry (SL). For the sixth course I select chemistry but not a course from Group 6 arts.

Also, we did a inquiry design for a unit in math. We chose the Pythagorean Theorem as the topic.

Some thoughts about learning/teaching math history

It may worth to specify to whom the math history is written. I assume the history is written to students no higher than secondary school, and it may be true that the younger the reader, the better the effect, i.e., the understanding of the math knowledge, some concepts, and of course, some stories.

However, it is also true that unlike stories of other findings, stories of math findings are usually boring and hidden under other findings. I received a gift from my grandpa in my 12th birthday. It is a book recording stories of most major math and scientific (mainly physics, chemistry, and biology) findings. I still remember the feelings when I read those math stories: boring. The first math story was about Pi. At that time I have learnt Pi in school, but why should I be excited when an ancient spent all his life finding some values of Pi? I mean, there are more vivid stories like Archimedes' principle of buoyancy, the stories of a number are something not as interesting as the others.

However, the story of using wheat to fill up a chess board is intersting (2^0, 2^1, 2^2, ... 2^64). It demonstrates how fast exponents grow as well as how stupid the king was. And so does the story of square root 2, in which a man was killed by being thrown into the ocean. Similar math stories are rare; but if there is one, it must be the story rather than the math finding itself that attracts my attention.

At this moment I carefully examine my feelings of reading this book. I notice that I can clearly remember some math findings, but all of them are story-related. Here is the most important feeling I believe: math is everywhere in this book, but my feelings of math is only that it is powerful. I feel that it is the stories that touches me rather than the math. Right now I am trying to imagine the feelings if I encounter a chapter full of mathematical proofs. I would just skip the chapter in that age.

Then about the teaching the math history. No doubt this will help students understand math. In fact no matter what we teach, as long as they are willing to learn, they can always make improvements. I would suggest if we can combine the math history with some non-math components, like science findings and real applications, and less proofs and calculations. As long as we implement a feeling that math is powerful in students' heart, and even more, if we let them think the power of math outweighs the struggles of learning math, we succeed.

Technologies in math education

This week I read the second article written by Smith, King, and Hoyte, Learning Angles through Movement: Critical Actions for Developing Understanding in an Embodied Activity. The Kinect design is awesome, which is similar to a project that helps student visualize the relationships among line, line segment, and their 2D equations.

But this is not the only thing I thought during the reading. When I saw the graph (Fig 3, p. 101) showing the improvements in percentage correct rate of the test, I could not stop but asking myself two questions: (1) is it possible that these students may perform similar or better in the test after taking a normal lecture about angle? And (2) by visualizing these math concepts, are we helping our students prepare for their future?

I understand the second question is a bit strange. It comes from a recent discussion between my cousin and I. I try to convince him that he needs to read more books instead of playing games and watching videos, while he tries to convince me that since watching videos and seeing things are more efficient ways of learning, it is unnecessary for him to read books at this time. And by the way, he is in Grade Six right now.

It seems impossible in the short run to develop technologies that can directly input information into human brains, so the fact that reading is the main learning method may not be changed in the short run. However, the daily-updated technologies do provide approaches for students to learn things “faster, more efficiently, and more accurately” (by Kevin, LOL). Then the mismatches here inevitably occur.

So far, learning through doing/acting is still a non-main-stream learning approach. Not only because of the developing technologies, but also its internally conflicted nature to the main-stream learning style. Well, we can also say that the main-stream style is limited by technology, but overly using technologies in learning may not be a good solution to teach students. However, it also depends on the things that students learn. It is also true that students may benefit from learning the technology used in the learning process. 

Impression of FLM 1-1

This is a good opportunity for me to think about a journal as a whole, especially the very first volume of its kind.

In July 1980, the first volume of FLM, For the Learning of Mathematics, was published. The cover page is simple, only the title of the journal and a symbolic graph containing iterated tiles and triangles. I cannot see the color as in pdf it is black and white, but I tend to believe the color may be yellow or green. Meanwhile, the back page is empty, with short statements telling that this is a space for future improvements.

Then I start to read the journal. In total there are 54 pages excluding the cover and the bottom pages. The goal of the journal is printed on the second page as:

The journal aims to stimulate reflection on and study of the practices and theories of mathematics education at all levels; to generate productive discussion; to encourage enquiry and research; to promote criticism and evaluation of ideas and procedures current in the field. It is intended for the mathematics educator who is aware that the learning and the teaching of mathematics are complex enterprises about which much remains to be revealed and understood.

This journal is published three times in a year, July, November and March. The price of subscription is also printed on the second year, along with the staff members of the journal. The instruction of writing articles to FLM is stated in detail at the end of the journal.

Three out of the nine articles published in this volume talk about geometry, with the rests talk about different topics. The lengths of these articles vary a lot. The “Two Cubes” is a one-page article with most of the page occupied by two reversed cubic boxes. The longest articles are “About Geometry” and “Alternative Research Metaphors and the Social Context of Mathematics Teaching and Learning”: 9 pages. The former contains two graphs, while the latter illustrates 15 graphs and tables. It is clear that these articles are of different writing styles.

Dick Tahta_Is there a Geometric Imperative

I have to say, reading Dick Tahta’s Is there a Geometric Imperative is not as straight-forward as the other articles I have read. Tahta (2010) summarizes the ways of framing a “coherent geometry course” (p.1) as imagining, construing, and figuring.

Imaging, as it is defined, happens when people try to visualize things that are not presented in front of them, or “seeing what is said” (Tahta, 2010, p.1). The key concept here is that no one can guarantee that the describer and the audience have the same images in their mind----and indeed, it is almost impossible that they visualize the same thing. But how can we see the similar things when we hear the descriptive words? I got the idea of vocabulary and the perspective, but that is not all what Tahta is trying to say. I will explain this in the following article.

Construing, which Tahta (2010) interpreted as “seeing what is drawn, and saying what is seen” (p. 1), contains two parts. Seeing what is drawn is the input, while saying what is seen is the output. Both are built based on an assumption that the listeners who are listening to the describer have the same vocabulary as the describer. And because what is described is an image, the perspective is with the same significant level as the words used to describe the image. It is true that an image must be different when viewers see it through different perspective.

And finally, figuring tells us to “[draw] what is seen” (Tahta, 2010, p.1). What I understood is that geometry is a subject about images. It is of great importance to build a bridge between image and descriptions. We human spend hundreds of years trying to figure out how to express what we see on some intermediates.

I really like the imaging, construing, and figuring ideas. It is not only the way used in geometry; rather, it is the way we used to describe the nature. We use abstract intermediates, such as words, definitions, and images, to symbolize and explain our surroundings.

Thinking mathematically by John Mason

John Mason in his book Thinking Mathematically demonstrated what we should do when encountering a conjecture (it is not necessarily a mathematical conjecture): (a) to convince ourselves; (b) to convince our friend; and (c) to convince our enemies. What he means by these is that we should describe the conjecture (as part (b) and explain it using formal language as proper as possible), and then see every statement as a conjecture and try to disprove it. These steps are significant in learning a mathematical concept.

But the below is what I feel valuable:

“Each version tends to become more abstract and formal, trying to be precise and to avoid hidden assumptions of informal language, but incidentally causing the reader to have to work harder at decoding the original insight and the sense of what is going on.” (p. 98)

This is why the math concepts are hard to read: the insights are covered under the intricate math signs. It might be helpful to demonstrate the true meanings of math concepts by showing students their origin. 

Thoughts of mathematical proofs

            The topics for this week’s reading are somehow out of my expectation, although they are interesting as well. I have thought some relevant topics about mathematical proof may be to explore why the reasoning process is a relatively boring subject for students, but Hanna and Barbeau’s article discusses the potential achievements students can make beyond the wordy mathematical proofs, while Zack showed his understanding and improved teaching skills through a series of classes dealing students’ counterarguments of a given plausible but wrong solution.

            Indeed, proof is a powerful and helpful tool for students to understand mathematical concepts. Zack’s improvement and understanding of children’s math thoughts strongly supports Hanna and Barbeau’s argument that teachers and students can improve their understandings through proofs. But proof itself may be scary for first-time learners if his/her instructor focuses more on the format of writing a mathematical proof. The situation may be more challenge if a student has a below average math skills because there is (almost) no formula for finding a proof. My own experience tells me that math mainly contains two parts: reasoning (proofs) and application. Things (do math problems) can be easy if my job is to simply use some rules; but it is always a challenge task to understand a proof because of its mathematical writing styles and the relationships between each step. Things can be out of control if I am asked to write a mathematical proof. That means I have to construct a persuasive reason and convey it into mathematical language.

            I believe this is the challenge part for some students, as conveying the reason into proper mathematical language always involve a certain level understanding of necessary mathematical knowledge. This requirement is hard to meet because (a) an average students may not master all the previous knowledge well, and (b) there are (is) limited (not as open as proofs should be in a class/test setting) ways for students to choose when conducting a mathematical proof because of their understandings and knowledge background. And thus a proof is somehow boring if students cannot figure out solutions by themselves.

            Anyway, what interest me in proof are the different-than-usual proofs of some familiar proofs. I am looking forward to explore more on this area.

Ps: I wonder where are the additional readings for this week as I am absent from the off-campus meeting.

The Marchall Island Mapping Models--------really drove me crazy....

Marcia Ascher showed me a series of elusive but beautiful findings of the Marchall Island maps. In the article Models and Maps from the Marshall Island: A Case in Ethnomathematics, the author demonstrated the mathematical thoughts in Marchall Island sailing tools. Local Marchall residents use mattings, rebbelith, and meddo to save secrets of their geological knowledge and to train their navigators.

Frankly, I found the matting part difficult to understand. The Marchall mattings are usually symmetric, and contain lots of triangles, sectors, angles, arcs, etc.. Local people use them to position in the ocean. However, even though I know that the arcs and swells are somehow similar to the shapes of waves in the Marchall sea area, I am still REALLY confused about how the tools work. The other two tools are somehow easy to understand: local Marchall people use intersects to identify islands so that whenever they see two islands, they can locate all the other islands.

I think this article is a good example of showing ethnomathematics, as it is difficult for me to understand the explanations in an off-background situation. I can understand every individual word; but when relating them together, I just lose my mind. I guess this is how Marchall people keep their secrets…….. 

Some thoughts of reading the article "problem posing in mathematics education" by Stephen and Marion

In Stephen and Marion’s article “Problem Posing in Mathematics Education”, five topics are mentioned: (a) some math questions are posed abstractly so that they do not make sense to students; (b) besides solving a question, there may be some other educational works that students might engage in, i.e., create questions using newly learnt math concepts; (c) a question with detailed descriptions may be reasoned similar to other questions, which means these questions reflect different sides of a coin. Students may benefit from exploring these questions; (d) based on part (c), students may learn more if they can construct new problems using the given conditions. This may broaden their understandings of the knowledge; (e) knowledge is constructed on social context, and so should be the math problems.

What we are doing as math teachers is to compress math thoughts of hundreds of years on students’ brains so that they can pass exams. The object here is so clear that solving the question is much more important than experience the process of learning. For those who can understand math concepts easily, they more or less follow this thinking process, and we believe that they can relate math concepts to their living experience. However, for those who cannot do math problems well, can teachers really “teach” the living experience they need? Even if we can teach the experience, are students willing to learn? Or they prefer to learn the “fast, accurate, and efficient” ways that can help them pass exams?

I spent hours writing the above paragraph, and I realize the situation here may be not because of the way we teach, but the object we teach and students learn. If the issue here is that math teaching and learning are exam-driven, students will no doubt focus more on methods of solving problems instead of methodologies used in mathematics (correct me if I use inappropriate words). The math education in Finland may be an example of showing how math can be taught in an exam-free educational system. 

Victory Over Maths????????

Dick Tahta in his Victoire sur les Maths (victory over maths) demonstrates me several examples of children with limited ability in performing math problems because of their psychological issues.

Most of the examples are about Lusiane Weyl-Kailey, a therapist worked in a Paris clinic. All the children mentioned in the article have some kinds of family issues that psychologically prevent them from understanding mathematical concepts.

OK, up to this point I was totally curious about Dick’s idea: how can family issues influence children’s understanding of mathematics?

Well, family issue does, more or less, influence children’s learning abilities. Dick believes if the family issue is somehow relevant to some mathematical concepts, and because a child always tries to AVOID unhappy events in his/her youth, s/he may PRETEND/ALTER the math concepts as a way of changing the reality s/he is facing. (In this article all the children are male.)  

There are things I need to mention here:

      1) Most treatment mentioned in the article took a year. Children are normally put away from math works, and are treated patiently by the therapist.  

    2) Despite the exact treatment plans are not mentioned in the article, I feel that the treatments improved the children’s math work gradually. Meanwhile, the treatments also changed the children’s feelings toward their family members.

     3) It is in Paris, and it takes a year, so the treatment must be expensive. 

     4) Math concepts were relevant to family issues. This implies that math is taught from unconsciousness, especially in early year math education.

Sorry guys for the late post. 

Summary and Thoughts about the article

The author’s demonstration starts with the language used in a US mathematics textbook.

Voice:

operationalize the notion of voice by focusing on the construction of relationships and roles for the author and reader (teacher and students) by examining particular linguistic features of the textbook.

Language choices can construct the reader as a thinker and in other cases as a scribbler.

Interpersonal function (the speaker’s meaning potential as an intruder into the context of the situation) is central as it focuses on personal relationships established in a text, and focuses attention on power and authority.

Inclusive imperative (“consider”, “define”, “prove”) demands that the speaker and hearer institute and inhabit a common world or that they share some specific argued conviction about an item in such a world, and so they construct the reader as a thinker.

Exclusive imperatives (“build”, “bring”, “create”) requires a reader to carry out a specific activity, which construct the reader as a scribbler who performs actions.

In order to do mathematics, people need to both scribble and think.

Ideational function includes (a) who is involved in doing what kinds of processes; and (b) the depiction or suppression of agency.

Textual function can be examined through investigating the ways in which the text maintains consistency.

The material researched here is a 64-page student edition of Thinking with Mathematical Models (TMM, not CMP). It is the only one that both the teacher and the student read (implying students do not read textbooks?).

Findings:

            Sometimes when the authors said they were going to ask questions, they actually used imperatives, i.e., “These questions will help you …”, and more than half the reflections were actually imperatives, which were instructions to direct actions. This confusing of imperative sentences with questions is a common feature of all mathematics textbooks.

Within the imperatives, more than 2/3 are exclusives, emphasizing the reader’s role as a scribbler. In turn, these exclusive imperatives highlight the authors’ authoritative voice as they were in a position of telling the students what to do and how to do.

Meanwhile, first-person pronouns were entirely absent from the student edition, which obscure the presence of human beings in the text and affects “not only the picture of the nature of mathematical activity but also distances the author from the reader, setting up a formal relationship between them”. On the contrary, “you” was widely used in the textbook. Statistics show that the “You + verb” form is the most common form (165 times our of 263 times) in the textbook, normally appears when the authors defined what they thought the reader was doing. Also, the authors used “you may find …” to control the sense that a reader made of something and were defining what s/he should have taken away from an activity. This means that the authors were attempting to define and control the common knowledge in the classroom.

It is true that math textbook in this style keeps students away. I do not want to read a book like this until I really realize that I need to read it. This may explain why students refuse to read textbooks, lol. But what should we do? 

W1 reading response

    This week’s reading response is difficult, as we must predict what the article is about when we read the first three sentences. The first three sentences of William Higginson’s “On the Foundations of Mathematics Education” are the following:

    “In the fall of 1726 a book was published in London with the title ‘Travels into Several Remote Nations of the World’. The author, described as, “first a Surgeon, and then a Captain of several Ships”, was reputed to be one Lemuel Gulliver. Behind Lemuel and his fictitious journeys there was, of course, the brilliant mind and savage wit of Jonathan Swift.”

    OK, I totally could not understand what this article is about. Apparently Lemuel Gulliver is a person in Jonathan Swift’s classic 1725 tale.  Well, I confess that I could not understand the two names until I searched on Google. However, how does Gulliver’s journey reveal some foundations of math education?

    Higginson constructed a MAPS model (M-mathematics, A-philosophy, P-psychology, S-sociology) that explains the four dimensions of our math education. If delivering knowledge to students is similar to telling a new story to audience, a teacher needs to explain “what, when, who, where, why”, and “how” to students, with “what” in mathematical dimension, “why” the philosophy, “who and where” the sociology, and “when and how” the psychology. If students find it hard to understand the math, is it possible that we focus tooooooooo much on “what”, and perhaps “how”, while somehow ignoring the rest, so that our students cannot understand the whole story? I personally believe that it is necessary for us to define explicitly what are “what, where, when, who, why, and how” under the MAPS setting before we go any deeper. 

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