The blogs you can read below (all posted in the month April 2009) are part of the work of graduate students at National Institute of Education, Nanyang Technological University who specialize in mathematics education. They have been encouraged ro raise a research agenda on mathematical problem solving based on a problem-solving lesson. Some of their blogs include links to the videos of the lessons.
Saturday, May 2, 2009
Monday, April 27, 2009
Learning how to think and cooperate - Low Kok Soon
Submitted by: Low Kok Soon
In the primary problem-solving lesson video, the setting is discussion in groups, ie, cooperative learning among 3-4 students. I noticed that in the course of the ‘discussion’ pupils were observed to be sharing and justifying their strategies with their peers. However, some seem lost which is not surprising as it happens in a lot of cooperative learning settings. The higher ability ones usually dominate leaving the lower ability ones playing a shadowing role. That set me thinking; perhaps we should arm these lower ability students with certain skills so that they could participate more proactively and will not feel left out.
Kramarski, Mevarech and Arami (2002) cited various studies in their literature review that run a common thread of training students who work in small groups (defined as 4-6 students) to reason mathematically by formulating and answering a series of self-addressed metacognitive questions. Such techniques are called cooperative-metacognitive approach. The rationale is two-fold, supported by theories and research. First, cognitive theories of learning emphasize the important role of elaboration in constructing new knowledge. Second, the well established research on cooperative learning argues that it has the potential for children to improve their mathematical problem solving abilities as it provides them a natural setting for them to supply explanations and elaborate their reasoning. More importantly, the researchers reported positive effects of such instruction on students’ achievement and seem to benefit lower achievers even more. The lower and higher achievers in the cooperative-metacognitive experimental group outperformed their counterparts in the cooperative-only control group , and Effect-Size for lower achievers (ES=1.93) was reported as higher than that for higher achievers (ES=1.06).
With these as a background, my research proposal stimulated by the video would be to investigate whether metacognitive instruction embedded in cooperative mathematics classrooms would exert positive effects on students’ mathematical problem solving ability. The main research question, in particular, relates to the effects of such cooperative-metacognitive instruction on lower and higher achievers. To go about doing this research, I could do anpre-post test experimental-control study on 4 groups of 3 pupils working in groups. Two groups (one higher achiever group, one lower achiever group) will be in experimental groups receiving metacognitive instruction embedded in cooperative learning settings treatment while the other 2 groups (same makeup as above) as control groups will receive only cooperative settings. The experimental groups receive the treatment using methodology such as IMPROVE^ method (Mevarech, et al, 1997) which train students to activate metacognitive processes in small groups and students are taught to formulate and answer four kinds of self-addressed metacognitive questions: comprehension, connection, strategic and reflection questions. The cooperative-only groups will not receive training on metacognitive but employ the usual discuss in groups strategy till a consensus is reached, and asking for teacher help when they encounter diffculties. The details surrounding the statistical tests on their pre-post tessts will be omitted for this discussion.
Such research on cooperative-metacognitive approach will shape instructional practice (perhaps making every student go through metacognitive training as a life-skill and not just in maths?) and further generate useful knowledge for researchers to investigate further for two reasons. First, if we could leverage on such cooperative settings early on in the school days, as they would in the working world anyway, the higher achievers will soar even higher and the lower achievers will benefit more, learnt to be more self-confident and be more motivated to learn (eg the N(T) students?). Second, it benefits the learning environment as it will become more conducive and engender us a step closer to the Thinking Schools Learning Nation vision painted by the ministry.
In conclusion, it may be instructive for the lifelong learner to start learning explicitly how to think, as what Edward de Bono (1993) championed in his Six Thinking Hats theory, one of which is the Green Hat which is thinking about thinking and control of the thinking process.
(number of words excluding title: 644 )
In the primary problem-solving lesson video, the setting is discussion in groups, ie, cooperative learning among 3-4 students. I noticed that in the course of the ‘discussion’ pupils were observed to be sharing and justifying their strategies with their peers. However, some seem lost which is not surprising as it happens in a lot of cooperative learning settings. The higher ability ones usually dominate leaving the lower ability ones playing a shadowing role. That set me thinking; perhaps we should arm these lower ability students with certain skills so that they could participate more proactively and will not feel left out.
Kramarski, Mevarech and Arami (2002) cited various studies in their literature review that run a common thread of training students who work in small groups (defined as 4-6 students) to reason mathematically by formulating and answering a series of self-addressed metacognitive questions. Such techniques are called cooperative-metacognitive approach. The rationale is two-fold, supported by theories and research. First, cognitive theories of learning emphasize the important role of elaboration in constructing new knowledge. Second, the well established research on cooperative learning argues that it has the potential for children to improve their mathematical problem solving abilities as it provides them a natural setting for them to supply explanations and elaborate their reasoning. More importantly, the researchers reported positive effects of such instruction on students’ achievement and seem to benefit lower achievers even more. The lower and higher achievers in the cooperative-metacognitive experimental group outperformed their counterparts in the cooperative-only control group , and Effect-Size for lower achievers (ES=1.93) was reported as higher than that for higher achievers (ES=1.06).
With these as a background, my research proposal stimulated by the video would be to investigate whether metacognitive instruction embedded in cooperative mathematics classrooms would exert positive effects on students’ mathematical problem solving ability. The main research question, in particular, relates to the effects of such cooperative-metacognitive instruction on lower and higher achievers. To go about doing this research, I could do anpre-post test experimental-control study on 4 groups of 3 pupils working in groups. Two groups (one higher achiever group, one lower achiever group) will be in experimental groups receiving metacognitive instruction embedded in cooperative learning settings treatment while the other 2 groups (same makeup as above) as control groups will receive only cooperative settings. The experimental groups receive the treatment using methodology such as IMPROVE^ method (Mevarech, et al, 1997) which train students to activate metacognitive processes in small groups and students are taught to formulate and answer four kinds of self-addressed metacognitive questions: comprehension, connection, strategic and reflection questions. The cooperative-only groups will not receive training on metacognitive but employ the usual discuss in groups strategy till a consensus is reached, and asking for teacher help when they encounter diffculties. The details surrounding the statistical tests on their pre-post tessts will be omitted for this discussion.
Such research on cooperative-metacognitive approach will shape instructional practice (perhaps making every student go through metacognitive training as a life-skill and not just in maths?) and further generate useful knowledge for researchers to investigate further for two reasons. First, if we could leverage on such cooperative settings early on in the school days, as they would in the working world anyway, the higher achievers will soar even higher and the lower achievers will benefit more, learnt to be more self-confident and be more motivated to learn (eg the N(T) students?). Second, it benefits the learning environment as it will become more conducive and engender us a step closer to the Thinking Schools Learning Nation vision painted by the ministry.
In conclusion, it may be instructive for the lifelong learner to start learning explicitly how to think, as what Edward de Bono (1993) championed in his Six Thinking Hats theory, one of which is the Green Hat which is thinking about thinking and control of the thinking process.
(number of words excluding title: 644 )
References
Kramarski, B., Mevarech, Z. R., & Arami, M. (2002). The Effects of Metacognitive Instruction on Solving Mathematical Authentic Tasks. Educational Studies in Mathematics, 49(2), 225-250.
de Bono, E. (1993). Teach your child how to think. London: Penguin.
^ IMPROVE is the acronym consisting of the following teaching steps: Introducing new concepts, Metacognitive questioning, Practising, Reviewing, Obtaining Mastery, Verification, and Enrichment.
Teaching Problem Solving Vs Teaching Via Problem Solving - How do they affect students' performance in higher-order thinking? By Faridah Ahmad
"....think about the problem, then step back
try a different plan of attack
ask for help...ask around
another solution can be found
change directions, replace
if you get stucked, there are other ways
take your time, it will come to you
most problems have answers if you think them through..."
Hahah... written above is actually part of a song lyric that I found in the you tube. Click on the link below to see the whole video clip. When I heard the song, it reminds me of my school days when I was trying to solve mathematic problems. I often get myself very frustrated if I am not able to solve. This song actually gives you another perspective in solving problems. It can be used as a form of encouragement for your students on solving problems not only in mathematics but their everyday lives. Of course, the important message behind is to have the resilience and attitude to keep on looking for a solution and not to give up easily. Definitely, it was something that I could relate to the students in Prof A Takahashi class, who were very determined to figure out the 'mystery' behind the crystal ball.
(Video Clip): http://www.youtube.com/watch?v=XcedSK0NnmI
Based on Prof A Takahashi Lesson Study
Personal reflections/observations from his video:
I was really heartened to see how the students were very engaged in his lesson. It was a very simple lesson to deliver but yet it was made so captivating to the students by simply posing the problem in a very creative manner. The role of the teacher here was more of a facilitator to guide the students in solving the 'mystery' rather than just presenting the solution, which he could do it if he intends to. Indeed, the 'magic crystal ball' was very effective in arousing their attention and stimulating their interest, making them to think how this 'mystery' works. Towards the end of the lesson, it was also observed how the students were intrigued by the 'power of mathematics' behind the mystery, although they looked pretty much exhausted :)
Well, indeed seeing how the students were so engaged by his delivery of lesson, it kind of leads me to think about my own classroom teaching and practices. How much have I engaged my students this while? How much have they actually learnt from me? How much injustice have I done to my students' learning? What could have been better to make them better thinkers? etc...:(
What I feel is productive to research further further into?
Most teachers out there will agree with me that such lesson is certainly very time-consuming. Hey, my lovely classmates, don't you think you agree with me? We need to see the usefulness of such padegogies before teachers are convince of the benefits in the long run. Teachers' beliefs and atittudes are something that it is difficult to change or shape unless they are able to 'see' the results for themselves. Well, not to blame anyone here but rather this is just human nature.
SInce the emphasis have shifted from teaching problem solving to teaching via problem solving (Lester et.al., 1994), many writers have attempted to clarify what is meant by a problem solving approach to teaching mathematics.
So, I am proposing the following research questions:
(1) How the instructions between teaching problem solving and teaching via problem solving differ from each other?
(2) How does the two method of instructions affect students' performance on higher-order thinking questions?
Outline of my research proposal:
-> To conduct literature reviews on teaching problem solving and teaching via problem solving (PS)
-> Based on the reviews, to come up the research methodolgy. What are the operational definitions of teaching PS and teaching via PS? Basically, to come up with a specific type of mathematic instruction that can be measured as teaching PS and teaching via PS. Also, the assessment on higher-order thinking also needs to be operationally defined.
-> To decide on what are things that need to be controlled in this experiment.
Student variables need to be controlled. I need to ensure that the sample of students are of the same capabilites who do not receive any additional help outside curriculum time. Hence, pupil survey on their background is very important. A post test could be given to determine the pool of students in terms of their capabilites. 2 group of students will be selected to undergo the two types of the mathematics instructions.
->Time of lesson and classroom environment needs to also be factored in as this will affect students attitude towards learning. Both instructions need to be conducted during the same time with same typical classroom environment
-> Duration of how much the students will be exposed to the different instructions also needs to be considered. For my research, I am proposing to assess the students after one month of instruction.
Alternatives to my research proposal:
Although the type of instruction has been defined for each method to be used in this research, having two sets of teachers for each group of students may not be fairly accurate since teacher's disposition in class also affect students' learning. Hence, to select the same teacher for both sets of groups will be more reliable in this experiment.
Conclusion - Future Implications to Classroom Practices
Based on the results, teachers will have the knowledge of how well the students will perform in higher-order thinking, depending on the type of math instruction exposed to them. Hence, from this research, time spent for each method of instruction in class should be better justified now. Teachers will realise the importance of instilling thinking skills to prepare them of becoming a better problem solver. Teachers need to plan lessons accordingly also, bearing in mind of the syllabus that needs to be covered given the time that they have in school. Well, teachers may argue about having no time to finish the syllabus but are your students' engaged in their thinking ultimately? To end thhis, let's all of us ponder over this quote by Albert Einstein:
"Education is what remains after one has forgotten everything he has learned in school".
Cooperative Learning and Mathematical Problem Solving (By Woo Ching Nee)
Introduction
In Professor A Takanhanshi’s lesson video, it is observed that he makes use of cooperative learning as one of his teaching strategies. He encouraged students to work in pairs or group when solving the puzzle. He did not present his own work to the students but instead he used students’ work to help them bounce off and generate more ideas from each other. Professor Tankanhanshi gave sole ownership of discovering to the students while he acts a facilitator in their thinking process. From the video, cooperative learning seems to be effective as the students were all engaged and they are making good progress with the problem. According to Vygotsky’s theory, students are able to solve problems cooperatively before they are ready to solve the same problems on their own. Researches have also found that cooperative learning is beneficial towards problem solving (Dees, 1991) and helps to improve higher-level thinking. These probes me into thinking of the role cooperative learning plays in mathematical problem solving.
Research Proposal
In response to the above, I proposed the following research question:
In what ways does cooperative learning enhance students’ problem solving ability?
Possible Research Method: Lesson Observations and Interviews
Observations are to be made of several teachers conducting cooperative learning in a problem solving classroom. The teachers conducting the class must be proficient in cooperative learning strategies. Videos, audios and transcripts are various means to record the lesson and student-work artifacts are also used in the analysis. Interviews are also conducted to capture teachers’ and students’ inputs about the lessons. The data collected are then coded and analyzed to observe patterns that led to students’ development of problem solving skills.
Research Potential
Cooperative learning is in line with Singapore educational initiative of ‘teach less, learn more’ where the students becomes critical an independent thinkers. A better understanding of how cooperative learning enhances problem solving helps teachers to
design more appropriate learning activities
acquire the necessary skills
How does blog promote research in Mathematical problem solving?
In this era where things are highly wired, blog allows us to obtain information with ease in our own comfort zone. Blog allows sharing of information without revealing one’s identity. Thus it is useful in obtaining sensitive data. Blog also help us to communicate to each other at real times and thus can facilitate as a tool for learning circle without the restriction of time and place.
References:
Dees, R.L. (1991). The Role of Cooperative Learning in Increasing Problem-Solving Ability in a College Remedial Course. Journal of Research in Mathematics Education, vol. 22 (pp. 409-421).
Links: http://ruby.fgcu.edu/courses/80337/Lourenco/CoopLearn/
In Professor A Takanhanshi’s lesson video, it is observed that he makes use of cooperative learning as one of his teaching strategies. He encouraged students to work in pairs or group when solving the puzzle. He did not present his own work to the students but instead he used students’ work to help them bounce off and generate more ideas from each other. Professor Tankanhanshi gave sole ownership of discovering to the students while he acts a facilitator in their thinking process. From the video, cooperative learning seems to be effective as the students were all engaged and they are making good progress with the problem. According to Vygotsky’s theory, students are able to solve problems cooperatively before they are ready to solve the same problems on their own. Researches have also found that cooperative learning is beneficial towards problem solving (Dees, 1991) and helps to improve higher-level thinking. These probes me into thinking of the role cooperative learning plays in mathematical problem solving.
Research Proposal
In response to the above, I proposed the following research question:
In what ways does cooperative learning enhance students’ problem solving ability?
Possible Research Method: Lesson Observations and Interviews
Observations are to be made of several teachers conducting cooperative learning in a problem solving classroom. The teachers conducting the class must be proficient in cooperative learning strategies. Videos, audios and transcripts are various means to record the lesson and student-work artifacts are also used in the analysis. Interviews are also conducted to capture teachers’ and students’ inputs about the lessons. The data collected are then coded and analyzed to observe patterns that led to students’ development of problem solving skills.
Research Potential
Cooperative learning is in line with Singapore educational initiative of ‘teach less, learn more’ where the students becomes critical an independent thinkers. A better understanding of how cooperative learning enhances problem solving helps teachers to
design more appropriate learning activities
acquire the necessary skills
How does blog promote research in Mathematical problem solving?
In this era where things are highly wired, blog allows us to obtain information with ease in our own comfort zone. Blog allows sharing of information without revealing one’s identity. Thus it is useful in obtaining sensitive data. Blog also help us to communicate to each other at real times and thus can facilitate as a tool for learning circle without the restriction of time and place.
References:
Dees, R.L. (1991). The Role of Cooperative Learning in Increasing Problem-Solving Ability in a College Remedial Course. Journal of Research in Mathematics Education, vol. 22 (pp. 409-421).
Links: http://ruby.fgcu.edu/courses/80337/Lourenco/CoopLearn/
Role of verbalizations in mathematical problem solving by Lee Wee Hoon
While viewing the video of the primary level problem-solving lesson, I focused my attention on the students and made the following observations. The teacher had given the class a task to do in groups. She wanted them to consider the different sizes of squares in a chess board and find out the total number of squares. A group of three female students was observed working on the task. One student made the following statements:
o "find pattern lah. Like that easier."
o "see got any pattern or not"
o " try 3 first"
o "3 by 3"
o "find pattern lah. Like that easier."
o "see got any pattern or not"
o " try 3 first"
o "3 by 3"
Researchers have established several models of the problem solving processes which describe how problem solvers approach a problem. The most frequently used model is Polya's four-stage problem solving model (http://www.drkhamsi.com/classe/polya.html). According to his model, there are four stages in a problem solving process:
1. Understand the problem
2. Devise a plan
3. Carry out the plan
4. Looking back
1. Understand the problem
2. Devise a plan
3. Carry out the plan
4. Looking back
The above statements made by the student show that she is making a plan to solve the problem. This corresponds to the second stage of Polya's model i.e. devising a plan. Several studies revealed that in most students' solution attempts, self-regulatory activities such as analyzing the problem, monitoring the solution process, and evaluating its outcome are absent. A typical approach used by many students is as follows: the student takes a quick look at the problem, decides the calculations to perform using the numbers given in the problem, and proceeds with the calculations without considering any alternatives even if no progress is made at all.
Through verbalizations, teachers can attempt to identify the processes students use (or do not use) when solving mathematical problems. Teachers can utilize this knowledge to help their students become better problem solvers. Therefore, I propose to do a research to investigate the following:
1. What do the verbalizations of students with different abilities reveal about their problem solving process?
2. To what extent can verbalizations be used in positively affecting students’ problem solving performance?
1. What do the verbalizations of students with different abilities reveal about their problem solving process?
2. To what extent can verbalizations be used in positively affecting students’ problem solving performance?
In this study, verbalizations refer to students' oral descriptions of their problem solving process. Three students with varying mathematical abilities (high, average and low) will be selected to take part in the study. A set of problems of varying levels of difficulty (high, moderate and low) will be given to the students. They will be asked to provide oral descriptions of their problem solving processes while solving these problems. All the students will be videotaped while solving the problems. If the student was silent for twenty seconds, he/she will be prompted “Please tell me what you are thinking.” The videotaped sessions will be categorized according to problem and student. Their verbalizations will be examined and coded according to Polya's four-stage problem solving model. Students will be asked to solve another set of problems with similar levels of difficulty quietly. Their work for both set of problems will be collected and analyzed according to student and problem. The frequency of correct and incorrect answers for all problems will also be recorded.
The findings from this research will allow teachers to have a greater understanding of students' problem solving processes based on their verbalizations. Teachers can then model the problem solving process to students so that they can be more successful in their attempts at problem solving.
The power of blogs lies in the fact that they can reach out to millions of people. Blogs invite people to collaborate and are an excellent way of sharing knowledge with others. They can be used positively to gather feedback as readers are able to respond instantaneously. The use of images, links and videos is also facilitated. Learning becomes more informal and is independent of time and place. Educators from around the world can pool resources together and contribute their knowledge in the area of mathematical problem solving through the use of blogs.
Transfer of Learning in Mathematical Problem Solving by Vasughi d/o Kothandapani
Specific Observations
It is very interesting to observe how the professor brought them through the problem solving process. After asking the students to discover the pattern, the professor made them explain the reason behind the pattern.
Towards the last part of the video, the professor asks the students “why is it that even though there are a variety of 2 digit numbers, the solution is always a multiple of 9?” He questions why the numbers ranging from 80 to 89 always gives 72 as a solution. Using a student’s explanation, he went further to link what the student explained to mathematical expressions using algebra (refer to picture on the right). This shows how he has designed his instructions in such a way to move from concrete examples to an abstract form.
This shows evidence of transfer of learning. Hence, in teaching through problem solving, he leads the students to learning new concepts/skills. He is also able to show through this problem solving exercise the importance of using algebra.
According to teaching model (Ashlock, Johnson, Wilson & Jones, 1983), as shown on the right, there are three phrases in the teaching process namely, Understanding, Consolidating and Transferring.
It is very interesting to observe how the professor brought them through the problem solving process. After asking the students to discover the pattern, the professor made them explain the reason behind the pattern.
Towards the last part of the video, the professor asks the students “why is it that even though there are a variety of 2 digit numbers, the solution is always a multiple of 9?” He questions why the numbers ranging from 80 to 89 always gives 72 as a solution. Using a student’s explanation, he went further to link what the student explained to mathematical expressions using algebra (refer to picture on the right). This shows how he has designed his instructions in such a way to move from concrete examples to an abstract form.
This shows evidence of transfer of learning. Hence, in teaching through problem solving, he leads the students to learning new concepts/skills. He is also able to show through this problem solving exercise the importance of using algebra.
According to teaching model (Ashlock, Johnson, Wilson & Jones, 1983), as shown on the right, there are three phrases in the teaching process namely, Understanding, Consolidating and Transferring.
The first phrase teaches students to understand a concept or skill. The second phase teaches them to consolidate concepts and skills learnt. The third phrase provides opportunities to transfer their understanding to new situations/scenarios. Problem solving requires students to transfer their knowledge of skills and concepts to familiar situations. Cormier (1987) describes transfer of learning as the application of skills and knowledge learned in one context being applied in another context. For more information on learning theories on transfer of learning, one can visit the website http://otec.uoregon.edu/learning_theory.htm
There are three kinds of transfer of learning. They are transfers from prior knowledge to learning, from learning to new learning and from learning to application Transfer of learning helps to make learning more meaningful for students. It builds bridges across different concepts and skills taught and learnt. It also helps the students in making sense of what they are learning. There are evidences to show that less able problem solvers view problems as being related in a non-structural way (Wickelgren, 1974). Hence, in order to develop good problem solvers, it is important for teachers to use transfer of learning in their teaching.
Rather than classifying one problem as a simple and another as complex, Wickelgren (1974) classifies the relationship between problems as
- Unrelated (when problem a and b have no common elements)
- Equivalent (when problem and b have same problem elements)
- Similar (when problem a and b have some common element)
- Special Case (when a is a special case of b, where problem a is included in b)
- Generalisation (when a is a generalization of problem b, where problem b is included in a)
Research Proposal
I would like to research on the effectiveness of the use of transfer of learning through mathematical problem solving. I would carry out a study on an experimental and a control group. In the experimental classroom setting the students will learn a new skill/ concept through the different types of transfer of learning. On the other hand, the control group will learn the same skill/concept through direct teaching. Later, the students in both groups will be assessed on their understanding, meta-cognitive skills and also their affective traits. The assessment will be in the form of paper and pen test, oral interviews and classroom observations.
The finding from this research will generate useful information for teachers. It will convince teachers on using problem solving as a means to carry out meaning learning experiences for the students. Furthermore, it will also highlight the effects of the different types of transfer of learning (equivalent, similar, special case or generalization) on students’ understanding. It will also surface some important teaching strategies to be carried out in order to maximize the benefits of transfer of learning.
Use of Blogs
The use of blogs in teaching and learning is supported by Vygotsky’s theory (Ferdig, 2004). It provides a social platform for rich communication to take place in a collaborative learning environment. It gives people of common interest to the opportunity to interact, surface new ideas and comment/ critic each others’ views. It further enhances one’s learning and helps to scaffolds one’s construction of knowledge.
It also serves as an online repository of information on Mathematical Problem Solving - rich content text, hyperlinks, video clips, audio clips, graphics, pictures etc. Furthermore, it allows one to reflect back on what he/she has written previously (archived blogs) and learn new things from the comments given by others.
Hence the blog on MPS can be a one-stop centre for educators to exchange ideas on problem solving. Teachers can comment on what is their view on problem solving and how they teach problem solving in class. Hence this blog can be used to do the research on transfer of learning. Teachers can post examples of their experiences in using transfer of learning in class and how it has helped the students in their learning.
Hence the blog on MPS can be a one-stop centre for educators to exchange ideas on problem solving. Teachers can comment on what is their view on problem solving and how they teach problem solving in class. Hence this blog can be used to do the research on transfer of learning. Teachers can post examples of their experiences in using transfer of learning in class and how it has helped the students in their learning.
However, one has to be aware that the information in the blog might not be accurate. Hence, the user must use heir discretion in deciding the authenticity of the information posted on the blog.
References
Berinderjeet Kaur & Yeap Ban Har. Mathematical Problem Solving in the Secondary Classroom. National Institute of Education. Singapore.
Gerald Kulm. Information Transfer in Solving Problems. Journal for Research in Mathematics Education. Purdue University.
Yuh Huann Tan, John Ow Eu Gene & Ho Pau Yuen Jeanne Marie. (2005). Weblogs in Education. Educational Technology Division, Ministry of Education.
References
Berinderjeet Kaur & Yeap Ban Har. Mathematical Problem Solving in the Secondary Classroom. National Institute of Education. Singapore.
Gerald Kulm. Information Transfer in Solving Problems. Journal for Research in Mathematics Education. Purdue University.
Yuh Huann Tan, John Ow Eu Gene & Ho Pau Yuen Jeanne Marie. (2005). Weblogs in Education. Educational Technology Division, Ministry of Education.
Sunday, April 26, 2009
What makes a student chooses the heuristic when he/she is solving a mathematics problem? by Chan Wei Liang, Alan
The Video Clips - My Observation
After viewing the video clips of Prof. A Takahashi teaching a class of secondary students on a mind reader puzzle, several questions surfaced from my own reflection of the activity. While viewing the video clips, I took notice on how a student begins to confront or to solve a problem mathematically. I couldn’t help but wonder if there is a set of “fixed” strategies that students would adopt when they encounter mathematical problems, maybe a set of modus operandi that students will follow whenever they take on a math problem. Or perhaps what do students do in order to make sense of the problem presented to them, before solving the problem mathematically.
Based on my observation from the video (reference: “Students Exploring Puzzle”), many students began with the heuristic of guess and check. These students were seen writing down (on their paper) a few examples to work out what are the results when the sum of two digits, (that make up a two digit number) is subtracted from the two digit number. The examples that the students chose to guess and check are random (i.e. 99, 54, 23, 45), with no particular order or sequences to the two digit numbers that were chosen. There was also no limitation as to how many examples that were chosen by the students. Some students tried four or five while others did eight to ten.
Based on the examples, students came up with the conjecture that the results they obtained are “usually multiples of 9” (guess). They were not sure if it is always 9 since they did not cover all the two digit numbers, but they were guessing that it seems the results “tends to be” multiples of 9. When Prof. Takahashi probed further (reference: Professor Sharing Students Work) to whether “everything” that the students work out are multiples of 9 (check), or maybe there is “one” possibility that the result is not a multiple of 9”, students were hesitant in their respond.
Another approach that students adopted to make sense of the problem was looking for patterns. Students were seen listing down a set of two digit numbers (reference: Students Attempting the Puzzle given the Hint) and observe that each set of two digit numbers that begins with the same tens digit (i.e. 23, 24, 25, 26 …, etc) will have the same result (i.e. 18). Students who adopted the heuristic of looking for pattern, were organized and systematic in their approach in making sense of the problem. These students organized and categorized their data to look for patterns or trends that may tell them something about the phenomena that they had observed. They would make conjectures based on the patterns or trends that they had discovered and associate their findings with the mathematical concepts they had learned.
The Research Questions and other leading questions
Based on my observation of these two heuristics employed by students in the video clips, I wanted to find out more about the problem solving strategies used by students. I wanted to know if the types of problem/question posed have a direct influence on the heuristic(s) employed by an individual in mathematics problem solving. What makes a student chooses the heuristic when he/she is solving a mathematics problem?
After viewing the video clips of Prof. A Takahashi teaching a class of secondary students on a mind reader puzzle, several questions surfaced from my own reflection of the activity. While viewing the video clips, I took notice on how a student begins to confront or to solve a problem mathematically. I couldn’t help but wonder if there is a set of “fixed” strategies that students would adopt when they encounter mathematical problems, maybe a set of modus operandi that students will follow whenever they take on a math problem. Or perhaps what do students do in order to make sense of the problem presented to them, before solving the problem mathematically.
Based on my observation from the video (reference: “Students Exploring Puzzle”), many students began with the heuristic of guess and check. These students were seen writing down (on their paper) a few examples to work out what are the results when the sum of two digits, (that make up a two digit number) is subtracted from the two digit number. The examples that the students chose to guess and check are random (i.e. 99, 54, 23, 45), with no particular order or sequences to the two digit numbers that were chosen. There was also no limitation as to how many examples that were chosen by the students. Some students tried four or five while others did eight to ten.
Based on the examples, students came up with the conjecture that the results they obtained are “usually multiples of 9” (guess). They were not sure if it is always 9 since they did not cover all the two digit numbers, but they were guessing that it seems the results “tends to be” multiples of 9. When Prof. Takahashi probed further (reference: Professor Sharing Students Work) to whether “everything” that the students work out are multiples of 9 (check), or maybe there is “one” possibility that the result is not a multiple of 9”, students were hesitant in their respond.Another approach that students adopted to make sense of the problem was looking for patterns. Students were seen listing down a set of two digit numbers (reference: Students Attempting the Puzzle given the Hint) and observe that each set of two digit numbers that begins with the same tens digit (i.e. 23, 24, 25, 26 …, etc) will have the same result (i.e. 18). Students who adopted the heuristic of looking for pattern, were organized and systematic in their approach in making sense of the problem. These students organized and categorized their data to look for patterns or trends that may tell them something about the phenomena that they had observed. They would make conjectures based on the patterns or trends that they had discovered and associate their findings with the mathematical concepts they had learned.
The Research Questions and other leading questions Based on my observation of these two heuristics employed by students in the video clips, I wanted to find out more about the problem solving strategies used by students. I wanted to know if the types of problem/question posed have a direct influence on the heuristic(s) employed by an individual in mathematics problem solving. What makes a student chooses the heuristic when he/she is solving a mathematics problem?
In Polya’s Four Steps to Problem Solving, choosing a heuristic is part of Step 2: Devising a Plan. In our Singapore Mathematics Framework, the component of Metacognition encompasses the selection and use of problem solving strategies by students. Hence, I had narrowed down my research question to “How will the different types of problem influence the use of heuristic(s) by students in mathematics problem solving?”
The Experiment
To conduct the research, I noted that in the research question, the types of problem are the independent variable and this can be operationalised into mathematical problems that are (1) numerical, (2) arithmetic, (3) geometry, (4) statistic (5) probabilistic, and (6) analytical in nature. We want to determine how these variables will influence the dependent variable, the choice of use of heuristics in mathematics problem solving.
With these definitions in mind, I can conduct an experiment to collect data by crafting problems according to the six categories (i.e. numeric, arithmetic, geometrical, statistic, probabilistic and analytical). I will then assess students’ choice of heuristics in answering these problems. Data will be classified into these six categories and the heuristics that students use to solve these problems are noted down. An interview will also be carried out to find out more about students’ approaches and strategies in choosing the heuristic(s) of their choice. The collection of qualitative data through interviews will further consolidate and formalise our understanding in the choice made by students when they are confronted with a mathematics problem.
Useful towards teaching and learning of Mathematics Problem Solving
The answer to the research question will enhance our knowledge into the strategies used by individuals in mathematics problem solving. It will help teachers to be more aware of their students’ mathematics problem solving abilities and provide remedies or interventions to help their students to become better problem solvers.
Teachers will also improve their teaching and review their pedagogical approaches towards students’ learning and understanding of mathematics problem solving. They will become more aware of the various mathematical problem solving heuristic(s) that student will tend to use for a particular type of questions, or that an individual is more prone to using a certain heuristic over the others. Such valuable information will allow teachers to plan their lessons and explore how each heuristic(s) may approach the problem from different mathematical perspectives yet solving the problem.
Website References
http://sc-math.com/math/heuristics.php
http://www.scribd.com/doc/13916762/Problem-Solving-Heuristics-Series-1
http://findarticles.com/p/articles/mi_qa3735/is_200710/ai_n21100613/
The Experiment
To conduct the research, I noted that in the research question, the types of problem are the independent variable and this can be operationalised into mathematical problems that are (1) numerical, (2) arithmetic, (3) geometry, (4) statistic (5) probabilistic, and (6) analytical in nature. We want to determine how these variables will influence the dependent variable, the choice of use of heuristics in mathematics problem solving.
With these definitions in mind, I can conduct an experiment to collect data by crafting problems according to the six categories (i.e. numeric, arithmetic, geometrical, statistic, probabilistic and analytical). I will then assess students’ choice of heuristics in answering these problems. Data will be classified into these six categories and the heuristics that students use to solve these problems are noted down. An interview will also be carried out to find out more about students’ approaches and strategies in choosing the heuristic(s) of their choice. The collection of qualitative data through interviews will further consolidate and formalise our understanding in the choice made by students when they are confronted with a mathematics problem.
Useful towards teaching and learning of Mathematics Problem Solving
The answer to the research question will enhance our knowledge into the strategies used by individuals in mathematics problem solving. It will help teachers to be more aware of their students’ mathematics problem solving abilities and provide remedies or interventions to help their students to become better problem solvers.
Teachers will also improve their teaching and review their pedagogical approaches towards students’ learning and understanding of mathematics problem solving. They will become more aware of the various mathematical problem solving heuristic(s) that student will tend to use for a particular type of questions, or that an individual is more prone to using a certain heuristic over the others. Such valuable information will allow teachers to plan their lessons and explore how each heuristic(s) may approach the problem from different mathematical perspectives yet solving the problem.
Website References
http://sc-math.com/math/heuristics.php
http://www.scribd.com/doc/13916762/Problem-Solving-Heuristics-Series-1
http://findarticles.com/p/articles/mi_qa3735/is_200710/ai_n21100613/
Impact of lesson studies on Problem Solving
My blog entry is based on the secondary 2 problem-solving lesson study.
Overall, I was impressed with how the professor patiently prompts responses to link students’ ideas/observations to mathematical expressions and inspired the students to understand how algebra can be used to communicate clearly and simply. I am interested to know the impact of lesson studies on teacher training and development especially in the area of teaching problem solving to students of all abilities.
In segment one, introducing the puzzle, professor was apt at creating a stimulating environment, encouraging responses. I like the way he arouses curiosity and generates an atmosphere of suspense among the students to work along with him, example, the use of crystal ball and “don’t tell others”. I noted his confident, friendly and patient disposition throughout the lessons. Instructions given were clear and simple. It felt like a privilege to be able to be in the expert teacher classroom. I am curious to know the reflections by the teacher observers, their learning point and how they will teach mathematics problem solving differently after the lesson study. What qualities of teachers are most suited to develop problem solving abilities among the students?
In segment two, exploring the puzzle, professor encourages communications-“talk to your partner”, and giving students ample time to figure out the problem as they explore on the why of the puzzle. At this point, my key concern was how a successful teacher should allocate curriculum time to teach problem solving and complete the mathematics syllabus to accomplish what is set out in the mathematics curriculum? How a teacher can balance the use of curriculum time to helping every student becomes a successful problem solver will be an issue for further research. Professor gives feedback on what the students are currently doing – he tells them that he observes that “some are writing on the note, some are looking very hard”. I noticed that the students were “cautious” and whispering ideas to each other – quite not typical in a boys’ class. Could the crowds of teacher observers affect the learning of some students?
In segment 3 – Exploring giving hints, how likely is it for all students to sit through and persist in the task? What could the teacher do for students who already gave up or were frustrated after a while? In pairing the students, were there special consideration to ensure that the strong is paired with a weaker partner? In lesson studies, are the teacher observers assigned to specific groups of students to observe? It looked to me that the teacher observers were observing professor most of the time. Finally, I was again amazed at how professor walked around and looked for and point out students’ ideas on how they solve the puzzle, without expressing any hint of disappointment/displeasure at their “immature” solution paths. Instead, he saw many interesting ideas and the students needed more time. You didn’t feel like you need to rush through the lessons.
Segment 4, 5, 6 – Professor sharing students’ work and eventually linking the observed patterns by students to mathematical ideas– it strikes me here – mutual respect and helping students to find out why. Professor asked for permission to use students’ ideas and respected the student’s decision that he will not present his solution himself. He patiently led students to see and communicate their solution that truly explains why. That ability to help students draw the link (their solutions to mathematical expression) was so powerful that all students were engaged. How will students’ attitudes and beliefs towards mathematics change by the use of the above teaching approach?
The e-learning has generated a few research questions that have direct implications on how we can train mathematics teachers so that the math curriculum outcomes can be achieved. A possible way to carry out the research: interview expert/master mathematics teachers; observe their lessons in the form of lesson studies, do a protocol analysis, interview students’ to find out about how their beliefs and attitudes have changed. It will shed light and offer possible solutions on the many difficulties faced by classroom teachers to carry out problem solving lessons successfully in the classroom.
Post E-learning Reflection
It’s a good idea and a powerful way to learn. For myself, it was time consuming finding a way to load the video as my laptop admin does not support the downloads of some video configurations. Also, when I finally used another computer to play the video, the video was often disrupted with the need to reload again. The screen view was not clear too. Despite the hiccups, learning was enriching. The blog entries also allow one to view the diversity of other peers views and beliefs and pave way for deeper research work and self improvement.
Reference
Masmi, Max, Yutaka, Takeshi (2007). Japanese Lesson Study in Mathematics. Its Impact, Diversity and Potential for Educational Improvement. World Scientific
It’s a good idea and a powerful way to learn. For myself, it was time consuming finding a way to load the video as my laptop admin does not support the downloads of some video configurations. Also, when I finally used another computer to play the video, the video was often disrupted with the need to reload again. The screen view was not clear too. Despite the hiccups, learning was enriching. The blog entries also allow one to view the diversity of other peers views and beliefs and pave way for deeper research work and self improvement.
Reference
Masmi, Max, Yutaka, Takeshi (2007). Japanese Lesson Study in Mathematics. Its Impact, Diversity and Potential for Educational Improvement. World Scientific
Teacher as Mediator in Problem Solving Instruction by Elizabeth Ee
From watching the video on the secondary problem solving lesson by Prof A. Takashashi at Monfort Secondary SChool, it was observed that the teacher played a crucial role in guiding the classroom discussions, in promoting development of ideas and facilitating the use of students' ideas to eventually solve the problem.
The dynamic role of the teacher as a mediator in guiding classroom discussions in the context of a mathematic problem solving lesson appeared an interesting area to research into.
Research Proposal:
Thus this research seeks to investigate “The Teacher as Mediator in Problem Solving Instruction in Singapore Secondary School Classrooms” and aims to answer the following questions.
1. How frequently is this attribute observed in the natural setting of everyday Singapore mathematics problem solving classroom environments?
2. How this relate to teacher effectiveness in terms of promoting students’ mathematical thinking and students motivation to mathematics?
Research Design:
The sample for this study would be videotaped lessons of teachers teaching problem solving in classroom setting in Singapore secondary schools.
A video rating form of classroom observation inventory could be developed as the instrument used to assess this facet of “Teacher as Mediator” in problem solving instruction. The table below shows an example of the observation instrument.
Table 1: Example of Instrument for “ Teacher as Mediator”
The dynamic role of the teacher as a mediator in guiding classroom discussions in the context of a mathematic problem solving lesson appeared an interesting area to research into.
Research Proposal:
Thus this research seeks to investigate “The Teacher as Mediator in Problem Solving Instruction in Singapore Secondary School Classrooms” and aims to answer the following questions.
1. How frequently is this attribute observed in the natural setting of everyday Singapore mathematics problem solving classroom environments?
2. How this relate to teacher effectiveness in terms of promoting students’ mathematical thinking and students motivation to mathematics?
Research Design:
The sample for this study would be videotaped lessons of teachers teaching problem solving in classroom setting in Singapore secondary schools.
A video rating form of classroom observation inventory could be developed as the instrument used to assess this facet of “Teacher as Mediator” in problem solving instruction. The table below shows an example of the observation instrument.
Table 1: Example of Instrument for “ Teacher as Mediator”
This video rating form could be used by 2 to 3 experienced raters who would each independently rate video lessons on problem solving in the context of a secondary school setting. These raters should be trained through discussion of the rating instrument and an extensive practising phase in which demo lessons were rated. After watching each video, raters would indicate the presence or absence of each item in the observation checklist on a 4- point rating scale as shown in Table 1 above.
Next, to gather information of effectiveness of the problem solving lessons, a students’ perception survey would be crafted to include questions shown below in Table 2.
Table 2: Students’ perception survey of problem solving lesson
The students’ perception surveys should be administered immediately or shortly after each problem solving lesson. The frequency of the attribute of “Teacher as mediator” being observed is then correlated to results from the students’ perception survey.
Research potential
As noted by Grouws & Good (1988) , teachers and indirectly teachers’ behavior is a powerful factor in increasing students’ performance in the domain of Mathematics Problem Solving (MPS). For this reason any findings that would be of assistance in improving the teacher’s instruction would go a long way in ensuring the success of implementing problem solving in the classroom.
In addition, Kliene & Clausen (1999) noted that TIMSS Video Study revealed that Japanese Mathematics teachers systematically use alternative representations and student-developed solution methods to reveal the mathematical problem. They also constantly confront their students with opened-ended mathematics problems. These problems allow the use of alternative solution methods that can lead to correct solutions on different levels of complexity. The teaching approach using open-ended problems has shown to be most effective in promoting students’ mathematical thinking and students’ motivation to mathematics. Hence the potential of having a deeper understanding of the teacher's role in MPS, especially how teachers can effectively act as mediators in problem solving lessons would be a great advantage.
Benefits of Blogging in PS research
New tools open up new possibilities. Blogging would bring sharing of ideas and findings in PS researches among educators and researches in the international community to a new dimension. Its ever availability everywhere and anytime is an attractive alternative to conventional methods of sharing. However, just like any information found on the internet that has not been certified by relevant authority, findings must be applied with caution.
References:
Kliene, E., Clausen, M. (1999). Identifying Facets of Problem Solving in Mathematics Instruction. Paper presented at the Annual Meeting of the American Educational Research Association (Montreal, Quebec, Canada).
Grouws,D. A., Good,T.L.(1988). Teaching of MPS: Consistency and Variation in Students Performance in the Classes of Junior High School Teachers
Biker traversing mountainous terrain during MPS? (Melissa)
I imagine myself as a student in this class. I felt like the biker shown in the picture trying to traverse the mountainous terrain. Though the journey is full of hurdles, when I arrive at the destination, the sense of achievement is “Wow!”. By the end of the lesson, as how the teacher puts it, they were mentally exhausted and were looking forward to the recess break. Though I am just viewing the lesson, I could feel a very high press for thinking. I was mentally challenged throughout the lesson. There was no chance to be passive as there were very interesting questions posed by the teacher and students. There were rich interactions between the teacher and students. I saw in this lesson what Kazemi and Stipek (1997) meant when they talked about “press” students to think conceptually about mathematics.Kazemi and Stipek (1997) did a study that demonstrated what it meant to “press” students to think conceptually about mathematics. He measured “press for learning” by the degree to which teachers emphasized students’ efforts, focused on learning and understanding, supported students’ autonomy and emphasized reasoning more than producing correct answers. His article on “Discourse that promotes conceptual understanding” delineates two important norms in classroom discourse and they are the social and sociomathematical norms. Social norms refer to practices such as explaining thinking, sharing strategies and collaborating. Sociomathematical norms are classified into four categories. Firstly, they are explanations that consisted of mathematical arguments, not simply procedural summaries of the steps taken to solve the problem. Secondly, the errors made by pupils offered opportunities to reconceptualize a problem and explore contradictions and alternative strategies. Thirdly, mathematical thinking involved understanding relations among multiple strategies. Fourthly, collaborative work involved individual accounting and reaching consensus through mathematical argumentation.
My proposed research questions are as follow:
1) What are the practices that establish the social and sociomathematical norms in this lesson?
2) How are these practices associated with the cognitive demand (Bloom’s Taxonomy) of our pupils?
3) Classify the questions generated during teacher to student and student to student interactions according to Bloom’s Taxonomy.
In this lesson, the teacher was providing opportunities for students to be engaged in conceptual understanding. These were some of the interesting questions or points raised by the teacher during facilitation of pupils’ learning:
1) What is the quickest way or shortest way to calculate?
2) I am not interested in the pattern but I want to know how you get to this.
3) How can you get these values in a quicker way. Maybe yours is quicker.
4) Now you are describing to me the pattern which is very interesting way but how to get these numbers?
5) Zi yuan has an idea. Think about his idea.
6) Who else wants to say some more about the method? How can we be sure that it works?
7) Teacher links pupil’s response to concept on area and perimeter.
8) Let’s say if your friend Siu Pah is not in school and you show her your working and she is able to understand.
The teacher does not teach. She uses pupils’ responses to alter her instruction to match pupils’ development of problem-solving strategies. She is not quick to give correct answers but she emphasizes on the process, explanation and presentation of the working. She got them into pairs and group to work out the strategies together. She emphasized on the process of creating a shared understanding.
In 1956, Benjamin Bloom headed a group of educational psychologists who developed a classification of levels of intellectual behavior important in learning. Bloom found that over 95 % of the test questions which students encounter require them to think only at the lowest possible level. Bloom identified six levels within the cognitive domain, from the simple recall or recognition of facts, as the lowest level, through increasingly more complex and abstract mental levels, to the highest order which is classified as evaluation.
Her approach has provided her pupils with opportunities with a high press for learning. They are operating mainly at Bloom’s Synthesis and Evaluation levels. At the Synthesis level, they are always trying to create easier and alternative methods and suggesting solutions during their interactions with the teacher and friends. At the Evaluation level, they listen to their friends’ explanations and provide their opinions and suggestions. There were some differences in opinions and methods and they have to learn to resolve these differences. In this way they learn to develop their own opinions, judgement and decisions on the strategies to use.
What is lacking in the photo posted on the biker is his companions? If only he has a few other companions to encourage him to press on, I am sure his mountainous journey would have been a much smoother and easier one uphill.
References:
Bloom, B. (Ed). (1959). Taxonomy of educational objectives: the classification of educational goals. New York: D. McKay Co.
Confrey, J. (1990). What constructivism implies for teaching (Pt 3, chap.8). Journal for Research in Mathematics Education Monograph, Vol 4, Constructivist views on the teaching and learning of mathematics (pp.107-124).
Kazemi, E. (2002). Discourse that promotes conceptual understanding. Putting
research into practice in the elementary grades: readings from journals of the National Council of Teachers of Mathematics, 44-49. NCTM.
Lane, A. (2007). Comparison of Teacher Educators’ Instructional Methods With the Constructivist Ideal. The Teacher Educator; Winter 2007; 42, 3; ProQuest Education Journals, 157.
Libby K., Bharath S., & Irv J. The Mathematics Educator (2008). A morphology of
Teacher Discourse in the Mathematics Classroom. Association of Mathematics Educators.
Theories and Practices for Teaching and Learning. A Quick Guide. SEED Resource Book. Singapore.
Vygotsky, L.S.: 1978, Mind in Society: The Development of Higher Psychological
Processes, Harvard University Press, Cambridge, MA.
Role of Affective Traits in Problem Solving by Lincoln
Include the specific observations that stimulated your research proposal.
- From the Princess Elizabeth Primary School problem solving video, it was observed that the teacher regularly encouraged student participation and allowed students the freedom of expression. Even when students came up with the wrong concepts or incorrect answers, she did not put them down but offered them other alternatives and various process pathways to generate further thinking.
- The majority of the students displayed good enthusiasm and during the heterogeneous small-group discussions, they appeared to have good constructive interactions among themselves.
- It was also noted that when students did not obtain the correct answers to the questions posed by the teachers, they do not appear to be discouraged and yet showed perseverance to come up with alternative ideas to problem solve.
Your research proposal should include a statement of what you feel is productive to research further into.
For the lesson, it is essential that students have prior cognitive knowledge on the basic mathematical operations. However, as students go about their tasks, it was not easy to evaluate the difficulties and emotions experienced by the individual students and how they eventually react to any stalemates or deadlocks. Other than the cognitive domain, the affective traits could have imposed an influence on the development of the students’ behaviour and thinking. For example, students who were not participative may have believed that if they could not solve a problem in a few minutes, they would never find the solution and thus should not waste time on it. Such beliefs about mathematics shape students’ behaviour and can often produce negative consequences.
Thus, it is imperative that classroom teachers know what the different affective traits are, how they influence the students’ success in mathematical problem solving; and finally, how they can nurture the desired affective traits in the students. It was hoped that from the research, the role of affective traits in mathematical problem solving can be better understood, particularly in the field of proving mathematical statements.
Outline how you would go about to do this research.
Case studies will be done on two students within a Secondary 3 Express class of above-average ability. The class will be tasked to prove a particular mathematical statement and to indicate their proving procedures and the thoughts that accompanied the processes. The goal of this mathematical task is to make the students reflect on what happens when a proof is carried out and to develop skills in analyzing during proving processes. The written protocols of two students will then be examined.
These two case studies aim to illustrate that when it comes to mathematical problem solving, purely cognitive behaviour is extremely rare and that the cognitive and affective domains are intertwined. The expressions and quotes presented by the students will allow us to go deeper into the difficulties faced by the two students in terms of the lack of mathematical knowledge and skills and subsequently, enable us to draw comparisons on the way the students approached their problem from an affective point of view.
You may provide alternatives in your proposal.
Alternative methods of research may take place in the form of questionnaires or surveys where questions are based on mathematical learning and problem-solving, and beliefs about self in relation to mathematics. Other tools such as interviews, could involve questions ascertaining this kind of conviction to paint a more accurate picture of what students believe about the discipline of mathematics.
Conclude with some explanation as to why this research has a potential to generate useful knowledge for classroom practice.
It aims to reinforce that affective variables are vital in influencing the learning environment in a classroom. Students' willingness to work on a variety of mathematics tasks and their persistence in dealing with these tasks might make a difference in their mathematical performance.
- From the Princess Elizabeth Primary School problem solving video, it was observed that the teacher regularly encouraged student participation and allowed students the freedom of expression. Even when students came up with the wrong concepts or incorrect answers, she did not put them down but offered them other alternatives and various process pathways to generate further thinking.
- The majority of the students displayed good enthusiasm and during the heterogeneous small-group discussions, they appeared to have good constructive interactions among themselves.
- It was also noted that when students did not obtain the correct answers to the questions posed by the teachers, they do not appear to be discouraged and yet showed perseverance to come up with alternative ideas to problem solve.
Your research proposal should include a statement of what you feel is productive to research further into.
For the lesson, it is essential that students have prior cognitive knowledge on the basic mathematical operations. However, as students go about their tasks, it was not easy to evaluate the difficulties and emotions experienced by the individual students and how they eventually react to any stalemates or deadlocks. Other than the cognitive domain, the affective traits could have imposed an influence on the development of the students’ behaviour and thinking. For example, students who were not participative may have believed that if they could not solve a problem in a few minutes, they would never find the solution and thus should not waste time on it. Such beliefs about mathematics shape students’ behaviour and can often produce negative consequences. Thus, it is imperative that classroom teachers know what the different affective traits are, how they influence the students’ success in mathematical problem solving; and finally, how they can nurture the desired affective traits in the students. It was hoped that from the research, the role of affective traits in mathematical problem solving can be better understood, particularly in the field of proving mathematical statements.
Outline how you would go about to do this research.
Case studies will be done on two students within a Secondary 3 Express class of above-average ability. The class will be tasked to prove a particular mathematical statement and to indicate their proving procedures and the thoughts that accompanied the processes. The goal of this mathematical task is to make the students reflect on what happens when a proof is carried out and to develop skills in analyzing during proving processes. The written protocols of two students will then be examined.
These two case studies aim to illustrate that when it comes to mathematical problem solving, purely cognitive behaviour is extremely rare and that the cognitive and affective domains are intertwined. The expressions and quotes presented by the students will allow us to go deeper into the difficulties faced by the two students in terms of the lack of mathematical knowledge and skills and subsequently, enable us to draw comparisons on the way the students approached their problem from an affective point of view.
You may provide alternatives in your proposal.
Alternative methods of research may take place in the form of questionnaires or surveys where questions are based on mathematical learning and problem-solving, and beliefs about self in relation to mathematics. Other tools such as interviews, could involve questions ascertaining this kind of conviction to paint a more accurate picture of what students believe about the discipline of mathematics.
Conclude with some explanation as to why this research has a potential to generate useful knowledge for classroom practice.
It aims to reinforce that affective variables are vital in influencing the learning environment in a classroom. Students' willingness to work on a variety of mathematics tasks and their persistence in dealing with these tasks might make a difference in their mathematical performance.
Teachers have a remarkable influence on students’ construction of their beliefs through the ways in which they present their subject matter, the kinds of tasks they set, assessment methods, procedures and criteria. (Pehkonen, 1998; Pehkonen & To¨rner, 1996; To¨rner, 1998). Thus, it is hoped that the research will encourage educators to acquire better ways to deal with affective issues in mathematical problem solving. This will help to motivate the students by stimulating their affective response in various situations and giving them a sense of better understanding of the subject matter. This would then contribute towards assisting students to construct more advanced and productive beliefs about mathematical problem solving.Demonstrate how you can use a tool such as the blog to promote research in mathematical problem solving.
The blog also allows us to interact with one another and to share opinions and feedback on research findings. Blogs offer us the option of enabling a "comments" field after our posts where readers can give us feedback. This not only encourages our fellow teachers to feel more inclined to revisit the blog to view the responses and to keep ourselves updated about any new research findings, but also at our disposal an effective and inexpensive way to get to know one another better.
References
Pehkonen, E., & To¨ rner, G. (1996). Mathematical beliefs and different aspects of their meaning. International Review on Mathematical Education (ZDM), 28, 101–108.
Pehkonen, E. (1998). Teachers’ conceptions on mathematics teaching. In M. Hannula (Ed.), Current state of research on mathematical beliefs V (pp. 58–65). Proceedings of the MAVI-5 workshop Helsinki, Finland: Department of Teacher Education, University of Helsinki.
To¨rner, G. (1998). Mathematical beliefs and their impact on the students’ mathematical performance. Questions raised by the TIMSS results. In M. Hannula (Ed.), Current state of research on mathematical beliefs V (pp. 83–91). Proceedings of the MAVI-5 workshop. Helsinki, Finland: Department of Teacher Education, University of Helsinki.
Pehkonen, E., & To¨ rner, G. (1996). Mathematical beliefs and different aspects of their meaning. International Review on Mathematical Education (ZDM), 28, 101–108.
Pehkonen, E. (1998). Teachers’ conceptions on mathematics teaching. In M. Hannula (Ed.), Current state of research on mathematical beliefs V (pp. 58–65). Proceedings of the MAVI-5 workshop Helsinki, Finland: Department of Teacher Education, University of Helsinki.
To¨rner, G. (1998). Mathematical beliefs and their impact on the students’ mathematical performance. Questions raised by the TIMSS results. In M. Hannula (Ed.), Current state of research on mathematical beliefs V (pp. 83–91). Proceedings of the MAVI-5 workshop. Helsinki, Finland: Department of Teacher Education, University of Helsinki.
What distinguishes an expert problem solver from a novice? By Au Kelly (Based on the research lesson by Prof A Takashashi at Monfort Secondary School
When I viewed the video vignettes, my first thought was, “Hey! I’ve used this in my class before!” A colleague had sent me the link to a website and having figured out how it works, I used it as a class filler after a lesson on Algebra with my secondary 2 students back in 2006. Being the naïve teacher that I was, I was quick to show how Algebra could be used in such a puzzle and did not exploit the potential of the puzzle the way Prof A Takashashi did.
It was interesting to see the way he questioned the students and elicited responses from them to see how they would go about figuring out the puzzle. As I watched the various responses from the students, it occurred to me that there were distinct differences in the way the students tackled the problem, which led me to the research problem: What distinguishes an expert problem solver from a novice?
From the video vignettes, I observed the following:
As it is impossible to open up the heads of expert problem solvers to see what lies inside, the next best thing to do would be to adopt a qualitative approach to the research, as this method will yield richer data. Based on the video vignettes, I would categorise students into three main categories: Novice, Average and Expert problem solvers. I would conduct the research with one student from each category or if time and resources permit, two from each, for better corroboration. Students would be asked to think aloud while attempting to problem solve a specified number of questions and a video recording would be made to capture what the student does and says during the problem solving process.
This method of research would follow quite closely to research that was previously conducted by Muir, Beswick & Williamson (2008) who studied the strategies used by a selection of grade 6 students and detailed three individual cases involving naïve, routine and sophisticated problem solvers, and that of Montague & Applegate (1993) who examined the verbalizations of learning disabled, average achieving and gifted middle school students as they thought aloud while solving three mathematical word problems.
From observations made of these students, results would be able to show some differences between each category of students. For mathematics educators, we can help to leverage the learning of novice problem solvers by teaching them some strategies to employ that were used successfully by expert problem solvers to help them in problem solving.
I think rather than take the stance that some students just have it (problem solving abilities) and some don’t, educators should always try to bridge the gap between novice and expert problem solvers by providing the necessary facilitation and guidance. I believe this research can shed some light on how expert problem solvers think and act and how we educators can help less able problem solvers emulate this.
If I were to conduct such a research, it would be based on convenience sampling, as I would probably use the students I am teaching as research participants. As students in different schools may differ in terms of abilities, instruction received during lessons and strategies exposed to, the use of blogs in researching this problem can extend to a wider reach as other teachers may wish to replicate the study on their students and discussions on the result findings can be made online. The use of blogs as a mode of communication can most certainly be tapped into as it is available 24/7 and is cheap with a wide reach.
References
Montague, M., & Applegate, B. (1993). Middle school students' mathematical problem solving: An analysis of think-aloud protocols. Learning Disability Quarterly, 16, 19-32.
Muir, T., Beswick, K., & Williamson, J. (2008). “I’m not very good at solving problems”: An exploration of students’ problem solving behaviours. The Journal of Mathematical Behavior. 27, 228-241.
It was interesting to see the way he questioned the students and elicited responses from them to see how they would go about figuring out the puzzle. As I watched the various responses from the students, it occurred to me that there were distinct differences in the way the students tackled the problem, which led me to the research problem: What distinguishes an expert problem solver from a novice?
From the video vignettes, I observed the following:
One student only used random numbers and stopped after 3 tries without much attempt at drawing any form of conclusion although he knew the numbers ended up with the same symbol. He also found it difficult to articulate his thoughts and the professor presented the student’s answer on the board and another classmate attempted (to some level of success) to explain it on his behalf.- Some tried more numbers and found that they all ended up with the same symbols when they compared it to the list on the screen but did not progress into much detail.
- Some were able to see the link in that the numbers obtained were multiples
of 9. One boy went into more details, such as observing that numbers in the 90s would yield 81, numbers in the 80s would yield 72, numbers in the 70s would yield 63, right down to numbers in the 20s which would yield 18 and that all these were multiples of 9.
As it is impossible to open up the heads of expert problem solvers to see what lies inside, the next best thing to do would be to adopt a qualitative approach to the research, as this method will yield richer data. Based on the video vignettes, I would categorise students into three main categories: Novice, Average and Expert problem solvers. I would conduct the research with one student from each category or if time and resources permit, two from each, for better corroboration. Students would be asked to think aloud while attempting to problem solve a specified number of questions and a video recording would be made to capture what the student does and says during the problem solving process.
This method of research would follow quite closely to research that was previously conducted by Muir, Beswick & Williamson (2008) who studied the strategies used by a selection of grade 6 students and detailed three individual cases involving naïve, routine and sophisticated problem solvers, and that of Montague & Applegate (1993) who examined the verbalizations of learning disabled, average achieving and gifted middle school students as they thought aloud while solving three mathematical word problems.
From observations made of these students, results would be able to show some differences between each category of students. For mathematics educators, we can help to leverage the learning of novice problem solvers by teaching them some strategies to employ that were used successfully by expert problem solvers to help them in problem solving.
I think rather than take the stance that some students just have it (problem solving abilities) and some don’t, educators should always try to bridge the gap between novice and expert problem solvers by providing the necessary facilitation and guidance. I believe this research can shed some light on how expert problem solvers think and act and how we educators can help less able problem solvers emulate this.
If I were to conduct such a research, it would be based on convenience sampling, as I would probably use the students I am teaching as research participants. As students in different schools may differ in terms of abilities, instruction received during lessons and strategies exposed to, the use of blogs in researching this problem can extend to a wider reach as other teachers may wish to replicate the study on their students and discussions on the result findings can be made online. The use of blogs as a mode of communication can most certainly be tapped into as it is available 24/7 and is cheap with a wide reach.
References
Montague, M., & Applegate, B. (1993). Middle school students' mathematical problem solving: An analysis of think-aloud protocols. Learning Disability Quarterly, 16, 19-32.
Muir, T., Beswick, K., & Williamson, J. (2008). “I’m not very good at solving problems”: An exploration of students’ problem solving behaviours. The Journal of Mathematical Behavior. 27, 228-241.
Saturday, April 25, 2009
Scaffolding & Metacognition by Staphni Sim
In Vygotsky’s concept of the zone of proximal development (ZPD), it was noted that there is a level that every child can independently problem solve and another level that they can potentially develop under adult guidance or in collaboration with more capable peers. ZPD is the distance in between the actual development level and the potential development level.
In the video, the teacher employed the scaffolding teaching strategy that provides individualized support based on the pupil’s ZPD. It is observed that an important aspect of her scaffolding instruction was that the scaffolds were temporary. Once the pupils’ problem solving abilities increased, the scaffolds provided by the teacher were progressively withdrawn.
Observing her lesson, one can classify her scaffolding instruction into the three levels as proposed in the framework of Anghileri (2006). At level 1, learning took place through interaction with artefacts (the chess board) and structured tasks (worksheets). Also, pupils used the templates provided in free play to explore the counting of squares. The classroom was also organised in such a way that learning took place through peer collaboration too. It is also observed that there was no direct interaction between the teacher and the pupils, with only emotive feedback provided.
At level 2, there was direct interaction between the teacher and her pupils. The teacher began this stage by showing and telling the mathematical tasks and recalling the rule. She further developed the pupils’ thinking and own understanding of the mathematical knowledge through reviewing and restructuring as she probed pupil’s ways of getting the answer, getting them to justify their solutions and simplified the tasks.
At the highest level, the 3rd level, the teacher engaged her pupils in a conceptual discourse via extending the problem task. She made connections to the learning activities and developed visual representational tools.
Current paradigm for learning suggests that learning is most effective when there is appropriate scaffolding instruction that allows active construction of knowledge. An interesting area of research could look into the areas of scaffolding and metacognition. Although scaffolding and metacognition differs in the agents that bridge the ZPD, both involve cognition. Teacher scaffolds the pupils’ cognition through the various levels, where metacognition mediate between the pupils and their own cognition.
One possible research question could be:
How can scaffolding and metacognition act together to support pupils’ learning?
An experiment can be conducted to study the pre and post effects of scaffolding and metacognition in the pupils. Both quantitative and qualitative methods can be used to analyze the differences in the quality of their work and their perceptions of which types of scaffolding and metacognition questions best supported their learning. One can adapt the questions used in Holton and Clarke (2006). The data for the study can include pupils’ surveys, pupils’ interviews, related class activities, and transcripts of the teacher's verbal instructions given both at the start and end of the experiment.
It would be interesting to discover how these two cognitive activities can work hand in hand to maximize learning in our pupils. Furthermore, the implication for teaching for such a study is that pupils might also be taught the appropriate questioning techniques to self-scaffold their own learning and metacognition.
References
Anghileri, J. (2006). Scaffolding practices that enhance mathematics learning. Journal of Mathematics Teacher Education, 8(1).
Holton, D., and Clarke, D. (2006). Scaffolding and metacognition. International Journal of Mathematical Eduation in Science and Technology, 37(2), 127-143.
Research Proposal by Tang Shunting
In the primary problem-solving video, these observations were made:
· Teacher got pupils to discuss and solve problems in groups.
· Teacher often got them to talk among themselves to confirm statements made.
· Pupils interacted well with one another and were engaged in ‘mathematical discourse’.
· Pupils were observed to be sharing and justifying their strategies with their peers.
Research proposal:
What is the relationship of peer interaction with mathematical problem solving performances?
Research method:
The study can be carried out by observing 2 different groups of students for their problem solving performances. A pretest is carried out to check for students’ mathematical problem solving performances. After which, both groups will undergo an instruction phase where a series of problem-solving lessons will be conducted by the same teacher using the same materials. During this process, the first group of students will often get to interact with their peers in solving the problems. Teacher will get them to talk to each other about the problem, discuss possible strategies, carry out solution, share different strategies and check on each other’s solutions. The second group of students will not be allowed to interact with their peers. They solve problems individually and have no chance to see and check how their peers solve the same problem. After the instruction phase, a posttest is carried out to check for their mathematical problem solving performances.
A correlational study is then carried out to find out if the presence of peer interaction will have an effect on students’ mathematical performances.
The findings for this research can provide insights for practitioners in planning problem solving instruction and in rethinking the role of group discussions in the mathematics classroom.
Another possible area of research generated from this study can be the investigation role of mathematics discourse in the classroom.
Blogs is a useful tool in communicating and sharing ideas to the targeted audience. In this case, a blog on mathematical problem solving research is able to gather and share insights from other students and researchers in a particular research area. This makes the process of inquiry and research more efficient.
· Teacher got pupils to discuss and solve problems in groups.
· Teacher often got them to talk among themselves to confirm statements made.
· Pupils interacted well with one another and were engaged in ‘mathematical discourse’.
· Pupils were observed to be sharing and justifying their strategies with their peers.
Research proposal:
What is the relationship of peer interaction with mathematical problem solving performances?
Research method:
The study can be carried out by observing 2 different groups of students for their problem solving performances. A pretest is carried out to check for students’ mathematical problem solving performances. After which, both groups will undergo an instruction phase where a series of problem-solving lessons will be conducted by the same teacher using the same materials. During this process, the first group of students will often get to interact with their peers in solving the problems. Teacher will get them to talk to each other about the problem, discuss possible strategies, carry out solution, share different strategies and check on each other’s solutions. The second group of students will not be allowed to interact with their peers. They solve problems individually and have no chance to see and check how their peers solve the same problem. After the instruction phase, a posttest is carried out to check for their mathematical problem solving performances.
A correlational study is then carried out to find out if the presence of peer interaction will have an effect on students’ mathematical performances.
The findings for this research can provide insights for practitioners in planning problem solving instruction and in rethinking the role of group discussions in the mathematics classroom.
Another possible area of research generated from this study can be the investigation role of mathematics discourse in the classroom.
Blogs is a useful tool in communicating and sharing ideas to the targeted audience. In this case, a blog on mathematical problem solving research is able to gather and share insights from other students and researchers in a particular research area. This makes the process of inquiry and research more efficient.
Developing Conceptual Understanding in Mathematical Problem Solving (MPS) Using Games - By Serina Tan
Research lesson by Prof A Takashashi at Monfort Secondary School
Games provide a unique opportunity for integrating the cognitive, affective and social aspects of Mathematics learning. The professor adopted
a technique similar to 5E learning cycle as follows:
1) Engaged students by creating interest and generated curiosity when he called students to follow his instructions and stared hard at the crystal ball with the symbol in their mind
2) Extended curiosity by showing three possible students’ beliefs. Told them today’s task is to find out the trick behind this game based on this belief.
3) Allowed exploration as students started making predictions by observing possible similarities between the numbers and symbols and started devising various problem solving strategies that they could think of to see possible patterns.
4) Got elaboration from students on what their classmate has done. Encouraged students to try several sets of different numbers to see if the same pattern appears.
5) Get students to evaluate their working processes. Showed them the power of coming up with symbols to set the general form of algebraic expressions and get them to clap for their good work during lesson closure.
Research Interest
A possible research area: Developing Conceptual Understanding in Mathematical Problem Solving (MPS) Using Games. Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students will know how to organize their knowledge into a coherent whole, which enables them to learn
new ideas by connecting those ideas to what they already know ( Bransford, Brown and Cocking, 1999).
Participants are from 3 classes of Secondary 1 Express students. Pedagogy adopted is the Multi-Modal Strategy (MMS) where modes of representation are translated into a systematic and practical technique for teaching of MPS. This helps to stress the linkages among different modes of representation, thus deepening understanding (Wong, 2004). Students in the first class will go through the first research lesson where the teacher observed uses a card game, similar to the Ghost Whisperer-crystal ball game which fits the Sec 2 scheme of work i.e. on forming algebraic expressions.
Students’ responses will be recorded down as observations and categorized in an observation form. Entire lesson will be filmed as part of documentation for post discussions where Sec 2 teachers look at the lesson again to identify important misconceptions and come up with areas of improvements. Post discussion will be held and an improved version will be taught to the next class and this entire process is repeated. The next improved lesson will be taught to the last class and a final post discussion was done. To increase inter-coder reliability, a group of sec 2 math teachers from cluster schools can be invited to participate in the lesson study cycle and the coding of results. Measuring variable used would be the improvement in students’ ability in making connections with symbols to form mathematical expressions. End product would be to design a game by applying mathematical concepts and thinking skills that they have learnt during problem solving.
Future research areas
Could be further extended to adopting the 5E Learning Cycle to see the effectiveness of using this cycle in developing conceptual understanding in Mathematical Problem Solving (MPS) using games. It can also be extended from the use of games to authentic performance tasks or alternative assessments which promote mathematical problem solving.
Benefits of the research
Lesson study provides an avenue to the exchange of teaching ideas. It allows teachers to constantly reflect and improve on the lesson within their professional learning committee (PLC). It is seen as a professional development tool and many schools in Singapore have adopted it to improve teacher instruction. A unique characteristic of lesson study keeps students at the heart of the professional learning activity. It emphasizes on process of students’ learning rather than product. An effective lesson study can have a long term effect on students’ learning, since well-prepared classroom practitioners who have a shared strong understanding of the subject content knowledge and pedagogy can produce better results (Takahashi & Yoshida, 2004).
By incorporating games into lesson study, it aroused students’ interest and developed critical thinking, essential in MPS. By exploring the usefulness of games in motivating and engaging students in MPS during lesson study, it allows teachers to work together and refine the lessons, thus helping in their professional development and improving students’ conceptual understanding in MPS, thus increasing their confidence and interest in MPS, the core of Mathematics education.
Usefulness of tools like blogging
Technological advances help in shaping future learning and redefining the skills need to thrive in the 21st century. With internet access been made everywhere, it is convenient and efficient as anyone can view the pages anytime, anywhere. I can initiate a discussion by first creating a platform where visitors can share useful websites on teaching pedagogies or the latest trends in lesson instruction of improving students’ MPS skills in the comments section. For example, I can discuss about my views after attending the Mass Teachers’ conference or the professional development course that I have attended and share with viewers the latest developments. This acts as a trigger to opinions that viewers have and even a healthy discussion on the possible research areas to explore. I can upload certain research articles useful in promoting MPS and get viewers to comment or suggest the viability of using such a measuring instrument in our local context. The use of blogging facilitates independent learning as viewers can comment at their own leisure time. However it is useful only when there is an active exchange of ideas. The user has to constantly update the webpage with information to elicit interest and discussion. Otherwise it will just form another personal blog.
References
Bransford, J., Brown, A. Cokcik, R. (1999). How people learn, Brain, Mind, experience, and school. Washington, DC: National Academy Press.
Takahashi, Akihiko & Yoshida, Makoto (2004), Ideas for Establishing Lesson-Study Communities. Teaching Children Mathematics, vol 10, no 9.
Wong, K.Y. (2004). Using Multi modal Think-Board to Teach Mathematics. A presentation for TSG 14: Innovative Approaches to the Teaching of Mathematics ICME-10, DTU: Technical University of Denmark, Copenhagen, Denmark.
http://faculty.mwsu.edu/west/maryann.coe/coe/inquire/inquiry.htm
Games provide a unique opportunity for integrating the cognitive, affective and social aspects of Mathematics learning. The professor adopted
a technique similar to 5E learning cycle as follows: 
1) Engaged students by creating interest and generated curiosity when he called students to follow his instructions and stared hard at the crystal ball with the symbol in their mind
2) Extended curiosity by showing three possible students’ beliefs. Told them today’s task is to find out the trick behind this game based on this belief.
3) Allowed exploration as students started making predictions by observing possible similarities between the numbers and symbols and started devising various problem solving strategies that they could think of to see possible patterns.
4) Got elaboration from students on what their classmate has done. Encouraged students to try several sets of different numbers to see if the same pattern appears.
5) Get students to evaluate their working processes. Showed them the power of coming up with symbols to set the general form of algebraic expressions and get them to clap for their good work during lesson closure.
Research Interest
A possible research area: Developing Conceptual Understanding in Mathematical Problem Solving (MPS) Using Games. Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students will know how to organize their knowledge into a coherent whole, which enables them to learn
new ideas by connecting those ideas to what they already know ( Bransford, Brown and Cocking, 1999).Participants are from 3 classes of Secondary 1 Express students. Pedagogy adopted is the Multi-Modal Strategy (MMS) where modes of representation are translated into a systematic and practical technique for teaching of MPS. This helps to stress the linkages among different modes of representation, thus deepening understanding (Wong, 2004). Students in the first class will go through the first research lesson where the teacher observed uses a card game, similar to the Ghost Whisperer-crystal ball game which fits the Sec 2 scheme of work i.e. on forming algebraic expressions.
Students’ responses will be recorded down as observations and categorized in an observation form. Entire lesson will be filmed as part of documentation for post discussions where Sec 2 teachers look at the lesson again to identify important misconceptions and come up with areas of improvements. Post discussion will be held and an improved version will be taught to the next class and this entire process is repeated. The next improved lesson will be taught to the last class and a final post discussion was done. To increase inter-coder reliability, a group of sec 2 math teachers from cluster schools can be invited to participate in the lesson study cycle and the coding of results. Measuring variable used would be the improvement in students’ ability in making connections with symbols to form mathematical expressions. End product would be to design a game by applying mathematical concepts and thinking skills that they have learnt during problem solving.
Future research areas
Could be further extended to adopting the 5E Learning Cycle to see the effectiveness of using this cycle in developing conceptual understanding in Mathematical Problem Solving (MPS) using games. It can also be extended from the use of games to authentic performance tasks or alternative assessments which promote mathematical problem solving.
Benefits of the research
Lesson study provides an avenue to the exchange of teaching ideas. It allows teachers to constantly reflect and improve on the lesson within their professional learning committee (PLC). It is seen as a professional development tool and many schools in Singapore have adopted it to improve teacher instruction. A unique characteristic of lesson study keeps students at the heart of the professional learning activity. It emphasizes on process of students’ learning rather than product. An effective lesson study can have a long term effect on students’ learning, since well-prepared classroom practitioners who have a shared strong understanding of the subject content knowledge and pedagogy can produce better results (Takahashi & Yoshida, 2004).
By incorporating games into lesson study, it aroused students’ interest and developed critical thinking, essential in MPS. By exploring the usefulness of games in motivating and engaging students in MPS during lesson study, it allows teachers to work together and refine the lessons, thus helping in their professional development and improving students’ conceptual understanding in MPS, thus increasing their confidence and interest in MPS, the core of Mathematics education.
Usefulness of tools like blogging
Technological advances help in shaping future learning and redefining the skills need to thrive in the 21st century. With internet access been made everywhere, it is convenient and efficient as anyone can view the pages anytime, anywhere. I can initiate a discussion by first creating a platform where visitors can share useful websites on teaching pedagogies or the latest trends in lesson instruction of improving students’ MPS skills in the comments section. For example, I can discuss about my views after attending the Mass Teachers’ conference or the professional development course that I have attended and share with viewers the latest developments. This acts as a trigger to opinions that viewers have and even a healthy discussion on the possible research areas to explore. I can upload certain research articles useful in promoting MPS and get viewers to comment or suggest the viability of using such a measuring instrument in our local context. The use of blogging facilitates independent learning as viewers can comment at their own leisure time. However it is useful only when there is an active exchange of ideas. The user has to constantly update the webpage with information to elicit interest and discussion. Otherwise it will just form another personal blog.
References
Bransford, J., Brown, A. Cokcik, R. (1999). How people learn, Brain, Mind, experience, and school. Washington, DC: National Academy Press.
Takahashi, Akihiko & Yoshida, Makoto (2004), Ideas for Establishing Lesson-Study Communities. Teaching Children Mathematics, vol 10, no 9.
Wong, K.Y. (2004). Using Multi modal Think-Board to Teach Mathematics. A presentation for TSG 14: Innovative Approaches to the Teaching of Mathematics ICME-10, DTU: Technical University of Denmark, Copenhagen, Denmark.
http://faculty.mwsu.edu/west/maryann.coe/coe/inquire/inquiry.htm
Wednesday, April 22, 2009
Can Lesson Study promote Mathematical Problem Solving? by Pauline Neoh
In the video recorded PEPS lesson study (LS) lesson, the teacher posed a series of non-standard tasks to the students. These tasks cannot be solved by simply identifying the correct mathematical operation and performing the subsequent computation. They require the students to consider the context and structure of the problem and engage in the process of problem solving. The students need to understand and then relate the summation of a series of consecutive numbers to the concept of area of a rectangle, and then make connections between the summation of the square of consecutive numbers with the concept of area of rectangle.
Clearly, the teacher was implementing a planned lesson. She assessed the students’ thinking by questioning them and crafted her questions to guide the students to think, reason and progress towards the target solution which the LS group had prepared. The progression in student learning was facilitated with scaffolds in the form of visual prompts e.g. chess board, two-dimensional shapes and manipulative e.g. interlocking cubes.
Clearly, the teacher was implementing a planned lesson. She assessed the students’ thinking by questioning them and crafted her questions to guide the students to think, reason and progress towards the target solution which the LS group had prepared. The progression in student learning was facilitated with scaffolds in the form of visual prompts e.g. chess board, two-dimensional shapes and manipulative e.g. interlocking cubes.
As the students engaged in the problem solving, the teacher, herself, was modeling the problem solving process. She encouraged the students to understand the problem as she explained the tasks and clarified their expectation. She instructed the students to think of how they could solve the problem and think of a related problem. As the students discussed and worked on the problem, she encouraged them to communicate clearly and check their solution. After the students were given time to work on the problem, she demonstrated “looking back” in reviewing the solution process with the class.
Research Question
Do teachers change their instructional practice and focus more on mathematical problem solving as a result of their participation in lesson study? A study can be conducted to illuminate the observation and help define expectations. The research question for a future research study can be:
Do teachers change their instructional practice and focus more on mathematical problem solving as a result of their participation in lesson study? A study can be conducted to illuminate the observation and help define expectations. The research question for a future research study can be:
Are instructional practices in the classroom more focused on mathematical problem solving as a result of teachers participating in lesson study?
Research to examine the effect of LS on instructional practice is timely as this powerful pedagogical tool is increasingly practised in Singapore schools. This mode of professional development is greatly encouraged as it boosts professional sharing and learning and provides teachers the opportunity to implement what they have learnt to benefit student learning. In addition, the research themes of school LS teams tend to be aligned with school needs and educational initiatives and priorities. Lewis (2000)’s study on Japanese school LS teams supported this observation. So, LS is very much relevant nowadays.
Methodology
I propose to do a study on in-school LS groups. Ideally, the study should be on two LS groups so that a comparative study can be made to examine both similarities and differences. Participation in the study should be voluntary to ensure that teachers are comfortable and willing to talk about their own change process. Some Singapore schools are already conducting their LS meetings during curriculum hours and carrying at least two cycles of LS in a semester. Permission can be sought from the school to capitalize on existing school LS programs. It is important that each LS group identify the desired learning outcomes for the teachers in the team and affirm the objective of the LS cycle which is to achieve the identified learning outcomes.
Research to examine the effect of LS on instructional practice is timely as this powerful pedagogical tool is increasingly practised in Singapore schools. This mode of professional development is greatly encouraged as it boosts professional sharing and learning and provides teachers the opportunity to implement what they have learnt to benefit student learning. In addition, the research themes of school LS teams tend to be aligned with school needs and educational initiatives and priorities. Lewis (2000)’s study on Japanese school LS teams supported this observation. So, LS is very much relevant nowadays.
Methodology
I propose to do a study on in-school LS groups. Ideally, the study should be on two LS groups so that a comparative study can be made to examine both similarities and differences. Participation in the study should be voluntary to ensure that teachers are comfortable and willing to talk about their own change process. Some Singapore schools are already conducting their LS meetings during curriculum hours and carrying at least two cycles of LS in a semester. Permission can be sought from the school to capitalize on existing school LS programs. It is important that each LS group identify the desired learning outcomes for the teachers in the team and affirm the objective of the LS cycle which is to achieve the identified learning outcomes.
Each volunteer teacher would be observed at least once teaching a lesson. The lesson would be video recorded, transcribed and coded. Field notes during the LS meetings and the lessons would be made to focus on tasks and teacher actions, especially on how teachers set up and implemented the tasks during lessons. The teachers would be interviewed about the tasks used and the questions they asked during the lesson as these are external representations of teachers’ views about teaching and learning. Each lesson would be segmented into parts that correspond to the main stages in Polya’s problem solving: understanding the problem, making a plan, carrying out a plan and looking back. In each segment, instances when the teachers emphasized the importance of context would be noted and described.
Using the data collected from the lesson observation and interviews, a case was constructed for each teacher to represent the nature of classroom tasks, and the ways these tasks were set up and implemented during the lessons. The 4-1 model described in Yeap and Ho (2009) could be used to measure the degree of achievement of the learning outcomes by examining each type of teacher change with respect to the stated learning outcomes.
Conclusion
The proposed study would be an attempt to collect evidences to concretize teacher learning and monitor teacher growth. It would also be appropriate to use the findings of the study to assess teachers on their instructional practice.
The results could also serve as justification or otherwise for the time and resources invested by the teachers and the school in LS. LS has the potential to impact teaching practice and benefit students learning and the proposed study seeks to examine the actualized benefits of LS as well as provide a means to monitor teacher learning.
Useful Websites
Lederman, E. (2009). Journey into Problem Solving: A Gift from Polya. The Physics Teacher, 47(2), 94-97.
http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PHTEAH000047000002000094000001&idtype=cvips&prog=normal
Lewis, C. (2000). Lesson Study: The Core of Japanese Professional Development. American Educational Research Association Meetings, New Orleans, April 28, 2000.
http://www.csudh.edu/math/syoshinobu/107web/aera2000.pdf
Summary of Problem Solving Process taken from G. Polya, "How to Solve It", 2nd ed., Princeton University Press, 1957.
http://www.math.utah.edu/~pa/math/polya.html
References
Yeap, B. H., & Ho, S. Y. (2009). Teacher Change in an Informal Professional Development Programme: The 4-1 Model. In K. Y. Wong, P. Y. Lee, B. Kaur, P. Y. Foong & S. F. Ng (Eds). Mathematics Education: The Singapore Journey.
Using the data collected from the lesson observation and interviews, a case was constructed for each teacher to represent the nature of classroom tasks, and the ways these tasks were set up and implemented during the lessons. The 4-1 model described in Yeap and Ho (2009) could be used to measure the degree of achievement of the learning outcomes by examining each type of teacher change with respect to the stated learning outcomes.
Conclusion
The proposed study would be an attempt to collect evidences to concretize teacher learning and monitor teacher growth. It would also be appropriate to use the findings of the study to assess teachers on their instructional practice.
The results could also serve as justification or otherwise for the time and resources invested by the teachers and the school in LS. LS has the potential to impact teaching practice and benefit students learning and the proposed study seeks to examine the actualized benefits of LS as well as provide a means to monitor teacher learning.
Useful Websites
Lederman, E. (2009). Journey into Problem Solving: A Gift from Polya. The Physics Teacher, 47(2), 94-97.
http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PHTEAH000047000002000094000001&idtype=cvips&prog=normal
Lewis, C. (2000). Lesson Study: The Core of Japanese Professional Development. American Educational Research Association Meetings, New Orleans, April 28, 2000.
http://www.csudh.edu/math/syoshinobu/107web/aera2000.pdf
Summary of Problem Solving Process taken from G. Polya, "How to Solve It", 2nd ed., Princeton University Press, 1957.
http://www.math.utah.edu/~pa/math/polya.html
References
Yeap, B. H., & Ho, S. Y. (2009). Teacher Change in an Informal Professional Development Programme: The 4-1 Model. In K. Y. Wong, P. Y. Lee, B. Kaur, P. Y. Foong & S. F. Ng (Eds). Mathematics Education: The Singapore Journey.
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