Research Lesson on Graphs of Linear Functions

A group of teachers in Kranji Secondary School, as part of Ministry of Education Partner Schools Programme, are exploring ways to make the 21st Century Competencies a reality in the classroom.

Yesterday, I had the good fortune to facilitate a post-lesson discussion for a mathematics lesson on graphs of linear functions.

The lesson was to use technology to help students learn how a linear graph changes as the values of m and of c in y = mx + c change.

I learn something through my observations. I would like to share two of them here. I also learn many things by listening to teachers who shared their observations. Again, I will share two of them here.

Observation 1: Language. The six boys I observed used word like "goes upwards" and "tilt" to describe the graph. Three of them described the graph as "moves upwards". Even the more careful student wrote "when looked from left to right, slants upwards". I learn that students tend to be at a decriptive level in the learning of graphs. If they stay at this level, the main purpose of learning graphs is lost. We learn graphs so that we have a good way to represent relationships between quantities - so that we are able to make predictions or generalizations, among other things. Why didn't any of these students use language that are indicative of their appreciation that it is about relationships? I wonder if any students said or wrote when x increases, y increases when they describe a graph of y = mx + c when m > 0?

Reflecting on the I See.I Think. I Wonder. tool, the use of language indicates that the six students are at I See. Can a teacher move them to the next level by asking: Why do you think the shape of the graph is such?

I wonder how the graph looks like if the value of c increases / decreases / remain unchanged but the value of m becomes positive.

Note to self: Try to learn in oter research lessons if consistent use of a thinking routine such as I See. I Think. I Wonder. will affect students' use of language which is reflective of their level of understanding.

Observation 2: Technology. I was surprised that 14-year olds who are so adept at figuring out all things technological took some time to figure out a fairly straight forward software such as the graphing software they were using.

I See students struggling to figure out a relatively simple software. I Think this is unusual given that I have seen many kids and definitely teenagers capable of figuring out things on a cell phone or on a laptop screen. I Think too much instructions on how to use the software can be not a good thing. I Wonder if it is sufficient have the students open the software and tell them, please show the graph of y = 2x + 1 using the software; go figure out.

What I Learn by Listening to Other Observers (1) The class was entire engaged but even when some student is interested to pursue a point further, the partner was not entirely interested. Even when two students had contradicting findings, they were happy to let it go without questioning their findings. I learnt after listening to three observations that as a teacher I need to distinguish between superficial engagement and deep engagement.

(2) One observer was wondering about what if we use a "more progressive" worksheet. Combining what she shared and my own observation, I learn the limitation of the use of a standard worksheets in a 21st CC classroom.

I am interested what she envision a "more progressive" worksheet would look like.

What would I do if I were to teach the same lesson? I would start by asking the students to log on to Ace Learning and go to I.Tools. I would then say go figure out with the help of your partner how to show the graph of y = x + 2. Now, what if you change the value of the coefficient of x (the number in front of x) to other positive values? What do you see? Why? I wonder what happens if the coefficient of x is negative. What do you see? Why is the value of y decreasing when the value of x goes up?

Now work with your friends to (1) see what happens when the value of the constant changes (2) think of an explanation for what you see.

To conclude the lesson, I wonder why the graphs are always straight lines in this lesson. I wonder how the values of y changes when the value of x increases if it is not y = x + 1 but instead it is y = x^2 + 1 that is what if the x is now the square of x? I wonder what will happen? Is it right? Someone says the value of y increases more for the same increase in x. Can you show using the graph.

I wonder what happen if the equation is y = 2^x + 1?

Kranji Secondary School

Kranji Secondary School is one of Ministry of Education Partner Schools on implementing the 21st Century Competencies initiative. Together with Dunman Secondary School, they are helping teachers develop the knowledge necessary for such transformational learning. Both schools are in Marshall Cavendish Institute's newly-crafted programme that combines workshop-style learning (familiar to Singapore teachers) with authentic setting professional learning (fairly new in Singapore). If we want teachers to be facilitators, they themselves should be familiar with this style of learning. Most of us, as students, would have rare opportunities to learn with teachers who are facilitators. I am glad these schools have sort of feel that lesson study - where teachers learn through facilitation - should be blended with workshop-style professional learning.

The research theme, or learning goal, for the teachers is to develop confident learners with high achievement. The first meeting in which teachers studied two lesson plans in English Language and Mathematics for Secondary 2 written by their colleagues.

The goal is to prepare teachers to be able to have an image of desired behaviours in such classrooms. What do I hope teachers gained from the session? I hope they re-think about what is commonly practised now and how we can enhance student learning. We spoke about the use of an anchor task to get to deep thinking- too may little tasks tend to get students to be engaged at superficial levels. We spoke about teaching at metacognitive level as opposed to at cognitive level - teach to make good thinking a habit.

The English lesson is on persuasive language and the mathematics lesson is on linear graphs.

Students will learn to identify use of language in different ways as they explore videos of advertisements. Students should learn how advertisement persuade us by playing on our emotions, how it prey on our beliefs and how it appeal to our logic. The plan calls for using numbered heads. I wonder if thinking routines can be used in a more substantial way.

In mathematics, students learn to see how the coefficient of x and the constant affect the appearance of the graphs. Students explore the graphs of different linear functions on a software. The plan did not explicitly identify a thinking routine. I wonder if see-think-wonder can be used here and regularly in other mathematics lesson.

Both lessons have great potential - one main task, collaboration, tehnology. I wondered aloud if the teachers may unwittingly introduced 'contaminants' by planning or doing too much in a way that students will be distracting from deep thinking and busy themselves completing what is expected. Both lessons have one main tasks, but I have seen too many such great plans fell through because we are not aware the need to keep the task lean so that students can get to the depths of thinking about the core goal (persuasive language elements, two main factors influencing a linear function).

I also wondered aloud if both lessons have been planned and later will be delivered with the main aim of having students develop enduring qualities i.e.. those beyond content.

Hiroshima International Conference

Hiroshima International Conference 26th – 28th January 2012, Hiroshima Japan

Organized by Graduate School of Education, Hiroshima University and funded by Ministry of Education in Japan. We would like to thank Ministry of Education japan for funding the trip for our six Singapore educators which is led by a teacher trainer, Peggy Foo (from Marshall Cavendish Institute). Team members include science and mathematics teachers from government schools and one special school.

Chair: Koyama Masataka (Professor of mathematics education)
Vice-chair: Isozaki Tetsuo (Professor of science education)


Conference Theme
Lesson Study in Science and Mathematics
Four East Asian Nations International Conference on Improving Mathematics and Science Teachers’ Teaching Competency by Lesson Study

Objective and Aim
The objective of this conference is to share and discuss ideas for improving the teaching and learning of mathematics and science at lower secondary school level (age 13 – 15 in Japan) in order to develop students’ thinking and representing ability in mathematics and science. At the same time, this conference aims to raise awareness among the mathematics and science educators and researchers on the role of Lesson Study for teacher professional development in their teaching career.
We want to share the state-of-the-art of Lesson Study of mathematics and science at lower secondary school level in four Asian countries of China, Korea, Singapore and Japan, and to discuss issues and challenges related to the teacher professional development through Lesson Study of mathematics and science in these countries.

Invitees
China: Wang Xiaoqin, Pan Sudong, Jing Zejin, Qui Yecheng, Xu Ping, Liu Da,
Shi Jun, and Zhang Renli
Korea: Kim Beom-ki, Lew Hee-chan, Kim Tae Sun, Lee Jaechon, Park Jae Hwan,
Kim Won Jung, Kim Jung Hee, and Cho Young Chang
Singapore: Peggy Foo Pei Chie, Lim Eik Kheng, Soh Poh Suan, Poh Yong Beng,
Seow Jun Jie, and Vanessa Neranjani d/o Muhundan
Japan: Ohtaka Izumi and Fujii Toshiakira

PROGRAM
Day 1 Thursday 26 JAN: Registration, Opening Meeting, School Visit,Sightseeing, and Welcome Dinner

10:00～11:00　Registration
11:00～12:00　Opening Meeting
12:00～13:00　Lunch
13:00～13:30　Bus from Mielparque to Hiroshima Higashihara Junior School

Hiroshima Higashihara Junior School
13:30～15:00 School Visit to observe mathematics/science lesson followed by short discussion

15:40～15:50　Boat from Miyajima-guchi to Miyajima Island

Miyajima Island
15:50～17:30　Sightseeing in Miyajima Island
17:30～18:00　Boat from Miyajima Island to Aki Grand Hotel

Aki Grand Hotel
18:00～20:00 Welcome Dinner

Day 2
Friday 27 JAN: School Visit and Workshop

8:00～9:30 Bus from Mielparque to H. U. Attached Fukuyama Junior School

Hiroshima University Attached Fukuyama Junior School
9:30～11:00 School Visit to observe mathematics and science lessons
9:30～10:00　Pre-session
10:00～11:00　Lesson Observation

11:00～11:30 Bus from the school to Fukuyama New Castle Hotel

Fukuyama New Castle Hotel
11:30～12:30　Lunch (Restaurant)

13:00～16:30　Workshop in Two Groups Mathematics Group Moderator: Koyama Masataka Science Group Moderator: Isozaki Tetsuo
13:00～14:30　Lessons from China, Korea, and Singapore
Mathematics
Shi Jun (China)
Kim Jung Hee (Korea)
Lim Eik Kheng (Singapore)
Science
Zhang Renli (China)
Kim Won Jung (Korea)
Vanessa Neranjani d/o Muhundan (Singapore)

14:40～16:00　Discussion and Sharing Ideas for Improving Lessons
16:00～16:30　Summary

Day 3
Saturday 28 JAN: Symposium on Lesson Study and Reception

7:00～7:45 Breakfast (Mielparque)
8:00～8:30 Walk from Mielparque Hiroshima International Conference Center

Hiroshima International Conference Center (HICC)
9:00～17:00　Symposium on Lesson Study

9:00～91:00 Opening Session
9:10～10:00　Plenary Lecture
Kiyohara Yoich (Japan Ministry of Education)
Strategies for mathematics and science education in Japan
10:15～13:15　 Symposium on Theme 1 (Science Education)
Improving Science Teachers’ Teaching Competency by Lesson Study: A critical analysis of lessons for improving students’ PISA Science Literacy
Moderator: Isozaki Tetsuo (Japan)
Speakers:
Ohtaka Izumi (Japan)
Pan Sudong (China)
Kim Beom-ki (Korea)
Poh Yong Beng (Singapore)

14:00～17:00　 Symposium on Theme 2 (Mathematics Education)
Improving Mathematics Teachers’ Teaching Competency by Lesson Study: A critical analysis of lessons for improving students’ PISA Mathematics Literacy
Moderator: Koyama Masataka (Japan)
Speakers:
Fujii Toshiakira (Japan)
Wang Xiaoqin (China)
Lew Hee-chan (Korea)
Peggy Foo Pei Chie (Singapore)

Lesson Study Meeting at Assumption English School

The school employs the gallery method where all the other learning teams observe the mathematics learning team do their lesson study meeting. In the MCI Lesson Study Porgramme, our lecturers model the facilitation process for critical meeting. This is the firts critical meeting when the learning team have met to craft their research theme and are about to embark on selecting the unit / lesson to be studied.

In such a meeting, facilitators should help team members build mental models of their object of study. For example, the learning team has identified their research theme to be "engaging students in mathematics learning" which is based on the school's focus on engaged learning. In the meeting, the facilitator asked various team members to share what they expect engagement to look like. In the end, the various mental models are either a process-type image of student engagement (e.g. to be able to see where is the starting point to learn algebra and where the ending pint is - perhaps academically weak students are not engaged because their starting pint is so removed from where teachers typically start) or prodict-type image (e.g. engagement as reflected by achievement scores.

In this meeting, facilitators should also help team members prepare for subsequent lesson planning meetings. For example, there are at least a few directions the team may take. (1) using real-life situations to engage students (2) using model drawing to engage students (the textbook also suggests this approach) (3) using the CPA vApproach that Singapore tecahers are familiar with (4) using counter-examples to engage students.

A team member also alerted the team to the PETALS Framework that Singapore teachers already have when thinking about engaged learning.

Introduction to Lesson Study

For the relevant readings, please see the previous blog entry.

LSP101 Introduction to Lesson Study

View more presentations from Jimmy Keng

Tampinese Junior College and Nanyang Primary School.