1
Mathematical Problem Posing in School and at Home: Case of Tasks Use by School
Teachers and Parents in Taiwan
Shuk-kwan S. Leung law@mail.nsysu.edu.tw
National Sun Yat-sen University, Taiwan
2014. 03. 04 彰師大科教所
A simple study
• About one particular study I completed
• TRY:
seeing the general from one particular
• Compulsory Education
Part I. The current educational system
1968 9-year program (6 years of
elementary, 3 years of junior high)
12-year program
(elementary, junior high and senior high or
vocational education)
2014
3
• Higher Studies
Graduate Institutes in Mathematics Education – Master degrees
(NTNU: 1995; NKNU: 1992; NCUE: 1987);
– Doctoral degrees
(NTNU: 1986; NKNU: 1998; NCUE: 1993).
4
5
Curricular Reform
Implemen -tation Teachers
Growth
Is it a never ending cycle?
Students learning day and night
Cram schools go year round
and on school holidays
A decision to make…
On Math Problem Posing
• What would you do? (Zimmermann)
• What are the considerations they might have in re-formulation? (Kilpatrick)
• Would you start with posing one problem with a similar structure? (Carrillo & Cruz)
• What would they consider as similar
(Mason)? In what sense?
IMAGINE:
How can we enhance children’s learning by:
a) working with teachers (Leung, 2013a)
b) working with parents (Leung 2013 b)
Working with teachers/parents to enhance children’s math learning
10
Why and how do teachers enact problem- posing task materials in an elementary mathematics curriculum?”
I explore, analyze, and discuss my own learning and that of teachers by recruiting active teachers to connect research to practice.
11
Making MPP happen
Leung (1994) discussed 4 characteristics MPP
1.MPP can be idiosyncratic. When one is considering some givens and poses a problem, one is trying to connect the various givens to a goal. For example, suppose the given information in a problem is, “There are 10 boys and 20 girls in a class.” A person may pose, “How many children are there altogether?” but another person may pose, “What is the ratio of boys to girls?”
2.Problem posing involves plausible reasoning (e.g. “Consider the change in this ratio if the total number of children is the same but the number of boys is 11 instead?”).
12
3. Problem posing can happen before, during, or after problem solving. For example, after posing and solving for the total number of children, one may ask, “What is the percentage of boys in the class?”
4. A posed problem which is not yet solved can be insufficiently specified, impossible or even impossible, such as “If 25 bottles were distributed to 10 boys and 20 girls and each one gets one, how many bottles were left?”
13
Leung (2009)
Four Phases in Problem Posing and Problem Solving
Understand (Pose)
Plan
Carry Out Look Back
14
Leung (2009)
• Leung, S. S. (2009). Research efforts on probing students'
conceptions in mathematics and reality: Structuring problems,
solving problems, and justifying solutions. In Lieven Verschaffel, Brian Greer, Wim Van Dooren, &
Swapna Mukhopadhyay (Eds.).
Words and worlds: Modelling
verbal descriptions of situations.
Sense Publishers, 213-225.
TASKS as a variable
Kilpatrick’s (1978) classified research on problem solving by task and subject variables.
In practice, teachers need to know which tasks to use for which student.
16
Part I : working with teachers (introduce MPP tasks)
• Leung, S. S. (2013a). Teachers
implementing mathematical problem
posing in the classroom: challenges and strategies. Educational Studies in
Mathematics, 83(1), 103-116. [SSCI]
Including teachers
• From year one to year three
A teacher may stay (1-3 yrs, depends)
• Taiwan in-service teachers training system
• National Level
• State Level
• School Level
• District supervisors
If I ever wanted to introduce
a new teaching idea to school teachers…
• I must understand well that idea
• I must study how that can be integrated in the curriculum
(without adding instructional time, as the teaching progress sheet is on every
school’s website, parents will read)
Would you try it (ever)?
If you introduce a new book to student…
If you introducing a new car to your customer…
If you feed a baby new food..
Leung (2013a) Teachers Professional
Development: Participation in Curriculum Reform
Regard student teachers, teachers, and teacher educators as teachers.” (Krainer, 2008, p. 1) . Additionally, they can work together to solve problems arising in the new teaching methods, thus acting as co-learners (see Jaworski, 1999).
Jaworski (2008) reflected on her work as editor of the Journal of Mathematics Teacher Education (JMTE) and commented that JMTE’s work mostly falls into the category of a mathematics teacher educator (MTE) passing knowledge to teachers. She drew a Venn diagram to represent how working
together can share knowledge. 22
One circle carries MTEs’ knowledge of research and theory while the other circle holds teachers’ knowledge of students and schools. The sharing of knowledge is represented by the intersection of the two circles, which means that the passing of knowledge is bi-directional. That is to say, teachers also pass knowledge to MTEs.
Teacher
Educator Teachers Shared
Knowledge
23
Henson (1996) discussed teacher educators working with teachers and included three levels of teachers’
involvement:
teacher as helper (Level 1),
teacher as junior partner (Level 2), and
teacher as lone researcher or collaborator (equal partner, Level 3).
24
Leung (2013) Educational Studies in Mathematics.
Level One: Teachers assist in developing task by data collection. [Helpers; I only].
Level Two: Teachers decide on when to use which tasks, use coding scheme then suggest how to revise task and coding.
They also suggest ways to use children’s work in teaching.
[Junior partners; II, III only].
Level Three: Teachers conduct action research. [Equal partners;
I, II, III].
25
Leung (2013a) Teachers Professional
Development: Participation in Curriculum Reform
Regard student teachers, teachers, and teacher educators as teachers.” (Krainer, 2008, p. 1) . Additionally, they can work together to solve problems arising in the new teaching methods, thus acting as co-learners (see Jaworski, 1999).
Jaworski (2008) reflected on her work as editor of the Journal of Mathematics Teacher Education (JMTE) and commented that JMTE’s work mostly falls into the category of a mathematics teacher educator (MTE) passing knowledge to teachers. She drew a Venn diagram to represent how working
together can share knowledge. 26
One circle carries MTEs’ knowledge of research and theory while the other circle holds teachers’ knowledge of students and schools. The sharing of knowledge is represented by the intersection of the two circles, which means that the passing of knowledge is bi-directional. That is to say, teachers also pass knowledge to MTEs.
Teacher
Educator Teachers Shared
Knowledge
27
Henson (1996) discussed teacher educators working with teachers and included three levels of teachers’
involvement:
teacher as helper (Level 1),
teacher as junior partner (Level 2), and
teacher as lone researcher or collaborator (equal partner, Level 3).
28
Level One: Teachers assist in developing task by data collection. [Helpers; I only].
Level Two: Teachers decide on when to use which tasks, use coding scheme then suggest how to revise task and coding.
They also suggest ways to use children’s work in teaching.
[Junior partners; II, III only].
Level Three: Teachers conduct action research. [Equal partners;
I, II, III].
29
2. Literature Review
2.1 Mathematical Problem Posing
2.12 These six types (Tsubota, 1985):
• an algorithm,
• text,
• a figure/table,
• a math topic,
• an answer,
• or a math problem.
7
3. Method
3-1 Case Background 3-11
17
Case
Research Team a teacher education
two research assistants
ten schools 60 teachers
L(g 1, 2) M(3,4) H(5,6) (20) (20) (20)
3. Method
3.1 Case Background
Year I: Development MPP tasks by grade, content strand, and type
according to Tsubota (1987);
52 tasks were resulted.
183. Method
3.1 Case Background
3.12
Year 2: 60 selected teachers
participated and used the tasks to teach mathematics.
There were 10 teachers from each grade (grades 1 through 6).
Sent back real examples of children’s work (52 tasks)
19
• The temperature of the cup of tea is 50℃.
NOT-A-PROBLEM
• Is there soup or tea in this cup? NON-MATH
• The temperature of the cup is 45℃; the volume is 450 cc., how many altogether?
IMPLAUSIBLE
• The temperature of the cup of tea is 50℃, it is cooled down in 2 hours, what is the final
temperature? INSUFFICIENT
• If the cup of tea is 49℃ and heated to 5 degrees warmer, what is the final
temperature? SUFFICIENT
3. Method
3.1 Case Background
3.12
In the final year, three teachers conducted
action research and
continued their journeys (reported separately).
20
3. Method
3.1 Case Background
3.12 Note that the number of teachers participating in this phase was
controlled by me.
I chose to allow only a smaller number of teachers as the level of involvement (Hensen, 1996) increased over time [105, 60, 3].
21
3. Method
3.1 Case Background
3.12 Three sessions of seminar one were
conducted on the first three Wednesdays of the term with 20 teachers each from the
lower (grade 1-2), middle (grade 3-4), and upper grades (grade 5-6).
22
3. Method
3.1 Case Background
3.12 They experienced MPP in pairs as children while I was their elementary school teacher.
These “elementary school children” posed problems, and when they did not
understand, they asked questions and the
“teacher” showed how to answer these questions.
Frequently asked questions were recorded to compare with questions asked by children
during the implementation period. 23
3. Method
3.1 Case Background
3.12 Categorizing children’s given problems.
This exercise in coding consisted of
problems posed by teachers during the
seminar as well as those posed by children during the first year of the study.
I introduced categorization scheme by Leung &
Silver (1997) for this coding component.
24
3. Method
3.2 Data Sources
3.21 Teacher educator’s reflections (TER).
3.22 Children’s scripts (CS).
25
3. Method
3.2 Data Sources
3.23 Teachers’ questionnaires (TQ).
• 1. What type of tasks will teachers consider in MPP?
• When deciding on a task, what concern(s) do you have?
• In implementing the task, how did you get your children to do MPP?
26
3. Method
3.2 Data Sources
3.23 Teachers’ questionnaires (TQ).
• 2. What types of problems are produced by children? Can teachers categorize them?
• Is there any difficulty in using the categorization scheme?
• Is there any recommendation for revising the scheme?
27
3. Method
3.2 Data Sources
3.23 Teachers’ questionnaires (TQ).
• 3. What are the successes and difficulties of teachers during implementation?
• Did your children understand the task? Did they ask questions? What did they ask?
Please describe the feelings of the children.
Please describe how the teacher felt when seeing children pose problems. Write about any fun or pain about implementing posing
problems. 28
3. Method
3.2 Data Sources
3.23 Teachers’ questionnaires (TQ).
• 4. Are you interested in doing action research in the next school year?
29
3. Method
3.3 Data Analysis
• The investigator and graduate research
assistants double checked that children’s work on the scripts was placed by the 60 teachers into the correct categories (Leung & Silver, 1997).
• Descriptive statistics showed counts for each of the five categories from tasks 1 to 52.
30
3. Method
3.3 Data Analysis
• I began with a particular analytic framework with expected categories of responses (Leung
& Silver, 1997) and examined how teachers used the framework.
• For teachers’ journals and teacher
questionnaires were analyzed qualitatively, in Creswell (2009); a more inductive, grounded- theory kind of approach.
31
4. Results and Discussion
• Teachers were at involvement level two (Henson, 1996), where both teacher
educators and teachers function as learners in a mutual reciprocal influence of knowledge
(Jaworski, 2008); where both are teachers in a broad sense by Krainer and Wood (2008).
32
4. Results and Discussion
• I used the three data sources (TER, CS, TQ) to improve my knowledge of how my teachers a) considered and used tasks, b) used the
coded children’s work and results in teaching, and c) responded to matters arising in
teaching.
• In each case, I used at least two of the three data sources to inform my understanding.
33
4. Results and Discussion
• What did I learn from working closely with teachers?
• I consider why and how teachers enact research-based tasks,
• how teachers use the coding scheme to analyze children’s posed problems,
• and the techniques, challenges and strategies they employed in enacting MPP.
34
4. Results and Discussion
4.1 Enactment of Research-based Tasks 4.11 MTE shared knowledge of research-
based tasks.
As mentioned in detail above, seminar one consisted of a briefing on problem posing
tasks. Teachers were divided into groups of two and were asked to pose problems as
elementary school children.
I purposely asked teachers to watch how I responded to questions and led the
discussion when the teachers orally
presented the problems they posed 35
4. Results and Discussion
4.1 Enactment of Research-based Tasks 4.11 MTE shared knowledge of research-
based tasks.
When the learners asked for a problem-posing example, I did not give an example but said,
“No, no. Try to think it over. If I give you an example, you will pose a problem similar to my example!” All the problems posed by teachers during seminar one belonged to the sufficient category.
36
4. Results and Discussion
4.1 Enactment of Research-based Tasks 4.12 Teachers shared knowledge from
enacting MPP tasks.
I made notes on how teachers posed problems, raised questions. In my reflections on
curriculum integration, I realized that
(1). They wondered if they could prompt children to pose problems that were suitable for
specific topics
37
4. Results and Discussion
4.1 Enactment of Research-based Tasks 4.12 Teachers shared knowledge from
enacting MPP tasks.
(2). e.g.
• “What if children pose a problem that does not belong to that chapter?”
• “What if the problems are too difficult for them to solve?”
38
4. Results and Discussion
4.1 Enactment of Research-based Tasks 4.12 Teachers shared knowledge from
enacting MPP tasks.
(3). The teachers also asked questions
about the format of students’ answers to the tasks.
• e.g. “Can children draw or read out loud instead of writing?”, “Do I need to give examples?”
39
4. Results and Discussion
4.1 Enactment of Research-based Tasks 4.12 Teachers shared knowledge from
enacting MPP tasks.
(4). teachers made suggestions on the wording and format of the tasks.
• e.g. “Algorithm” is a difficult word for lower grades. Question number “30.” and the
beginning of item “15” should be separated to avoid children reading it as a decimal
“30.15”.
40
4. Results and Discussion
4.2 Analyzing Children’s Work Using the Coding Method
4.21 MTE shared knowledge on the coding method.
In seminar one, the second part of the session focused on coding student work
samples. In order to facilitate the transition of roles from experiencing posing as
children to categorizing children’s problems as teachers, I announced, “Let us read
children’s posed problems.
41
4. Results and Discussion
4.2 Analyzing Children’s Work Using the Coding Method
4.21 MTE shared knowledge on the coding method.
Remember that you are teachers now, not children.” During the categorization
exercise, the teachers were amazed to see real examples of children’s problems in the five different categories.
42
4. Results and Discussion
4.2 Analyzing Children’s Work Using the Coding Method
4.21 MTE shared knowledge on the coding method.
Teachers asked me to clarify the distinction between the Not a Problem and Non Math categories as well as the distinction
between Impossible and Insufficient.
43
4. Results and Discussion
4.2 Analyzing Children’s Work Using the Coding Method
4.22 Teachers’ reports on coding children’s work.
At times, they called me and asked
questions like “Can I use grade five tasks for grade six?” Some of them collected
children’s diaries on an irregular basis and kept journals themselves.
44
4. Results and Discussion
4.2 Analyzing Children’s Work Using the Coding Method
4.22 Teachers’ reports on coding children’s work.
When students posed problems that fell under the Not a Problem or Non Math categories, teachers directed children’s attention to the instructions and asked them to “make up a problem” instead of using the word “pose.”
45
4. Results and Discussion
4.2 Analyzing Children’s Work Using the Coding Method
4.23 Sharing knowledge by reviewing children’s scripts.
• all 52 tasks were implemented; there was no task that the sixty teachers did not
attempt to use in instruction.
• 84% of all children posed problems were plausible mathematics problems (75% with sufficient information, 9% without).
• 11 tasks contained children’s work in all
five categories. 46
4. Results and Discussion
4.2 Analyzing Children’s Work Using the Coding Method
4.23 Sharing knowledge by reviewing children’s scripts.
• For example, in one item there is a picture of a mug of hot liquid next to a thermometer
showing 49 degrees Celsius.
47
4. Results and Discussion
4.2 Analyzing Children’s Work Using the Coding Method
4.23 Sharing knowledge by reviewing children’s scripts.
• “If my brother had a fever, and his
temperature in the morning was 25 degrees and also 25 in the afternoon, what is his total temperature for the day?”
48
4. Results and Discussion
4.2 Analyzing Children’s Work Using the Coding Method
4.23 Sharing knowledge by reviewing children’s scripts.
• In addition to learning about students’
understandings and misunderstandings, the exercise of classifying responses enabled the teacher educator to see if teachers
understood the categorization scheme.
49
4. Results and Discussion
4.3 Responding to Matters Arising During Teaching MPP
• teachers discussed the challenge they came across when children asked questions about problem posing.
• teachers were also concerned with children’s ability to understand the process of MPP
since it was their new way of learning.
• in addition to the above concern on
understanding of MPP, teachers were also
attending to children’s feelings toward MPP. 50
4. Results and Discussion
4.3 Responding to Matters Arising During Teaching MPP
• I captured teachers’ feelings, 33 were positive towards this implementation. Some of the
positive comments made by teachers are listed here:
51
4. Results and Discussion
4.3 Responding to Matters Arising During Teaching MPP
* I look forward to the day when I can do posing;
* I do not differentiate high and low achieving children… in posing.
* Unlike problem solving, children do problem posing without pressure;
52
4. Results and Discussion
4.3 Responding to Matters Arising During Teaching MPP
* I realized that some of my children have hidden potential;
* now I see their levels and can adjust my
teaching to give appropriate challenge at their levels;
53
4. Results and Discussion
4.3 Responding to Matters Arising During Teaching MPP
• the MTE made sure that teachers’ effort were recognized and shared in group settings.
54
5. Conclusion
• Given that the three parts of this MPP
research project were developing research- based tasks [I], enacting tasks [II], and
classifying children’s work and use results in teaching [III]), then the three levels of teacher participation in research map onto the three parts as follows:
55
5. Conclusion
• Level One: Teachers assist in developing task by data collection. [Helpers; I only].
• Level Two: Teachers decide on when to use which tasks, use coding scheme then suggest how to revise task and coding. They also
suggest ways to use children’s work in teaching. [Junior partners; II, III only].
56
5. Conclusion
• This book of tasks, with real sample work from children for the five possible posed problem categories (see Leung & Silver,
1997) and a section of tips for teachers who try problem posing in teaching, will allow
more depth for teacher training sessions and research on teacher development. An
example can be found in Parke et al. (2003)
57
5. Conclusion
• Building on the findings of this study, future research is needed on how to implement
problem posing and the conditions that allow students to perform well on MPP, including teachers’ inexperience with implementation (Leung, 2009).
58
