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**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

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**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 **

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*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.

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### 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?”).

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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?”

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**Leung (2009) **

**Four Phases in Problem Posing and ** **Problem Solving **

### Understand (Pose)

### Plan

### Carry Out Look Back

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### 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.

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### 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

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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).

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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].

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**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

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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).

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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].

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**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.

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**3. Method **

**3-1 Case Background **
**3-11 **

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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.

^{18 }

**3. 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)

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• 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).

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**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].

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**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).

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**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.

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**3. Method **

**3.2 Data Sources **

**3.21 Teacher educator’s reflections (TER). **

**3.22 Children’s scripts (CS). **

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**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? *

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**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? **

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**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? **

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**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. **

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**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. **

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**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). **

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**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. **

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**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. **

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**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.

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**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

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**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?” **

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**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?” **

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**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”. **

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**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. **

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**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. **

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**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. **

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**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.

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**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.”

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**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.

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**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?”

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**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.

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**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:

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**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;

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**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;

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**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.

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**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:

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**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].

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**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)

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**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).

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