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/
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