Research on Mathematics Problem Posing for Students, Teachers and Parents: A Personal Journey

Research on Mathematics Problem Posing for Students, Teachers and Parents: A Personal

Journey

Shuk-kwan S. Leung law@mail.nsysu.edu.tw

National Sun Yat-sen University

Department of Mathematics and Information Technology Online Departmental Seminar Dec 15 2021, The Education University of

Thank you

About Myself

School Teacher, Teacher Educator, Parent

About Taiwan

Population is 23,000,000 Size is 36,000 sq km

PME 36, Taipei, 2012

Population Size

SWITZERLAND Maryland, USA AUSTRALIA

Shanghai

About this talk

My selected work as a response to

“manifestations of MPP” by Silver (2013) and

“teaching and learning through MPP” by English (2020).

Sharing results of analyses to data from Taiwan, Hong Kong and US. This research agenda grew over time with participants that were initially students, adding in teachers and finally parents.

Methodological choices I included: variables, construct and intervention (Cai & Hwang, 2020).

Introduction

 Polya (1945) Math in the making

 Freudenthal (1973) Math as a human activity

 Brown & Walter (1983) What-if-not

 Kilpatrick (1987) MPP as goal and as means

MPP Activity

MPP still needs attention

 MPP refers to different activities and its multi-facet nature draws attention to researchers and practitioners internationally (Silver, 1994; Silver, 2013, Singer, Ellerton, Cai, 2015, English, 2020).

 However, it still need attention for learning, teaching and parenting.

 Implementation (Ask what happen, which strategies can meet challenges, and why document designated outcomes) aims to empower involving participants for growth.

Growth in knowledge

 To constitute growth in knowledge one must be curious.

 When curious raises questions (pose) and finds answers (solve).

 After answers ask more questions.

 This cycle goes on and on

 Pose and answer questions in or

擬題、佈題、命題

 擬 題 : Constructing a new problem (Brown and Walter, 1983).

 佈題: Presenting a problem properly (Butts, 1980).

 命 題 : Designing items for examination (Leung, 2006).

Three translations

Problem Posing(擬題)

 Today, the first (擬題) is my major focus

 MPP means the formation of a mathematics problem which invites a solution from an intended problem solver who cares to solve it

 By WHOM and FOR WHAT PURPOSE Silver, 2013,ESM Special Issue

Problem Posing and Solving

 Posing (擬題) is important, related to (yet different from) problem solving (Leung, 2009)

Students’ Learning

 Myself (and team) covered different independent variables and dependent variables (Kilpatrick, 1978).

 Independent variables: tasks, subjects, situation

 Dependent variable: products vs processes;

attitude, beliefs, creativity…etc.



Most frequentiy addressed: tasks

Variables in research

 Posing a story vs posing math structure to match a text.

 Content Specific vs Free Posing

Content: Number, Algebra, Geometry, Statistics

 Posing a goal led to completion of posing with several givens (see Leung, 2013, 6 types)

 In fact, one cannot tell what problem posing is until

MPP Tasks-Nature

MPP Tasks-Openness

Reitman (1965) Silver (1994) Stoyanova &

Ellerton (1996) Leung (2013) Given defined Goal defined

(R1)

Reformulate known

problems (S1) Structured (SE1) 1. Known Problem Given defined Goal

undefined (R2) Formulate new

Problems (S2) Semi-structured (SE2)

2. Algorithm 3. Text 4. Figure/Table

5. Math Topic Given undefined Goal

defined (R3)

Formulate new Problems

(S2)

Semi-structured

(SE2) 6. Answer Given undefined

Goal undefined (R4)

Formulate new Problems

(S2) Free (SE3) WHY NOT

FREE ?

My first piece, Leung, 1993

 Task (X1): Test of Arithmetic Problem Posing (TAPP Version A: with numerical information content NIC, Version B: without NIC). Pose as many problems as you can from a description given in a text.

 Subject (X2): Math Knowledge H/L

 Subject (X3): Creative Thinking H/L

 Posing Performance (Y): TAPP scores as

Leung & Silver (1997)

1.TAPP effectively evaluated arithmetic problem posing

2-1. Problem-posing performance was significantly better when task contained NIC

2-2. Problem-posing performance significantly related to math knowledge

2-3. Problem-posing performance NOT significantly related to creative thinking.

Results on Variables

(X1, X2, X3, Y)

Leung (1994)

 MPP Process: a sequence of problems

After solving, subject used goal to prior posed problem as a given to pose more problem(s).

 A replication: extended TAPP to TGPP

 Addresse relationships between creative thinking and problem posing and problem-posing tasks in mathematics domains.

 Fluency is general in verbal creativity problem posing flexibility is specific in problem posing.

 A general, rather than specific, problem posing competence exists in children and can be measured by the test TGPP.

Leung 1997, ZDM

 “ What problems should I pose? ”

 “ Can you give me examples? ”

 “ Is her posed problem better than mine? ”

 “ Who is going to solve it? “

 “ What if I myself cannot find the answer? “

 “ How difficuit? ”

Strategies Needed: how to teach MPP?

Children’s first posing

Reflections on failure

 When children did not perform well, teach them MPP (Leung, 2009) and show how to perform well.

 We did content specific MPP research and concluded that children CAN performed well only with conditions.

 Provide a variety of rich MPP tasks to children

 Discuss children’s intended or unexpected outcomes

 TRY: teach MPP myself, a math teacher education (MTE)

Attempt of a MTE

 An action researcher who works with teachers;

teach teachers in ways she proposes them to teach (e.g. Math Problem Posing, MPP)

When Where Grade MPP Task Teachers

2001 Hong Kong 5 5x=40 PDGE primary 2009 Kaohsiung 7 Simultaneous Equations In-service middle 2014 Hualin 10 Sine Rule In-service high

5x=40

Grade 5: Leung (in press)

5X=40

Grade 7: ICME (2016)

Grade 10

 Research on students’ learning MPP.

 MTE try first

 Children get used to it gradually

 Use multiple tasks to stimulate children

 Intended vs unxepected outcomes

 Not to be frustrated, work with teachers

What I learn …

 Leung, S. S. (1997). On the Role of Creative Thinking in Problem Posing. ZDM

 Leung S. K., & Silver, E. A.

(1997). The Role of Task Format, Mathematics Knowledge, and Creative Thinking. MERJ

 Silver, E. A., Mamona-Downs, J., Leung. S. S., & Kenny, P.A.

(1996). Posing mathematical problems in a complex task environment JRME

 梁淑坤（1994）。「擬題」的研 究及其在課程的角色。國民小學

數學科新課程概說-低年級臺灣

省國民學校教師研習會編。

 梁淑坤（1999）。從擬題研究提 出數學教學建議。新典範數學。

1. STUDENTS’ LEARNING

Teachers’ Growth

 Problem posing is explicitly mentioned in the part concerning getting students to do inquiry. (Ministry of Education, Taiwan R.O.C., 2008).

 Problem posing can be found in the curriculum guidebook for mathematics. (MOE of Taiwan, 2014)

(108 數學領域課程手冊)

Curriculum Standards revisited

 Why content-wise?

 Addition and Subtraction; Multiplication and Division;

Fractions, Decimals

 Ratio and Rate, Probability and other topics. (For details, see http)

 No need to ask for extra instructional time.

 Teacher understood children’s learning when

Inservice Teachers

 Research on Tasks and on Children’s work

 Content related MPP tasks (n=52)

By strands, by grade, and in six type of posing (unstated goal, Leung 2013)

 Showed teachers how to deal with the 5 categories of children’s work (Leung 1997).

 Used them in teaching.

 All posed problems are useful.

Teachers joined MTE

If I introduce MPP

A new teaching idea to school teachers…

 I must teach MPP myself

 Leung & Wu (1999): Two real classroom examples.

 FIX the ill-posed problems as that it can be solved

 Ratio: Wu as teacher

 Geometry: Leung as teacher

Leung & Wu (1999):

Real classroom examples.

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…

 Do not introduce unless you try and find it great

 Conducted study on MPP during student teaching, received grant (from NCS student research grants, See Leung 2011)

 A course on MPS and MPP (18 weeks, selective)

 Assignments on analyzing children’s posed problems to the 52 MPP tasks (mid-term, Teaching Methods).

Preservice Teachers

Teacher

Educator Shared Teachers Knowledge

Jaworski (2008)

Be Active: Work with Teachers

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].

Intervention Program

No. of Teachers vs participating levels

Year 1 Year 2 Year 3

MPP tasks across grades

Tasks in six types (Leung, 2013)

 an algorithm,

 text,

 a figure/table,

 a math topic,

 an answer,

or a math problem.

MPP tasks across grades

Tasks in six types (Leung, 2013)

 an algorithm,

 text,

 a figure/table,

 a math topic,

 an answer,

 or a math problem.

Three sessions 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).

1. They did problem posing themselves.

2. They analyzed posed problems.

Pose and Analyze workshops

They experienced MPP in pairs as children while I was their “elementary school teacher”.

These “elementary school children” posed problems, and when they asked questions I showed how to answer these questions.

Frequently asked questions were recorded to compare with questions asked by children during the implementation period.

Did MPP themselves

 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

Analyze responses

Teachers’ questionnaires (TQ)

 What type of tasks will teachers consider in MPP?

 What types of problems are posed by children? Can teachers categorize them?

 What are the successes and difficulties of teachers during implementation?

 After half a year…..

 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.

What I learned

Tasks Enactment: How

 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!”

 During the categorization exercise, the teachers were amazed to see real examples of children’s problems in the five different categories.

 At times, they called “Can I use grade five tasks for grade six?”

 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.

Analyze children’s work

 Teachers’ sheared strategies on coding children’s work. “Hard to tell which is insufficient and which is implausible…so… I put them into 2

 Name names: who did try problem posing in teaching, and research on tasks development and enactment. Newsletter by month.

 Building on the findings of this study, 3 teachers from this 60 did action research (Leung 2016) conditions that allow students to perform well on MPP.

Document to empower

teachers

15 graduate thesis: Content wise counting

＋－×÷ graduate decimals Ratio / proportion Equations probability

Curricular Reform

Implemen -tation Teachers

Growth

What I learn: a never ending cycle…

53

 Leung, S. S. (1997). On the Role of Creative Thinking in Problem Posing. ZDM

 Leung S. K., & Silver, E. A.

(1997). The Role of Task Format, Mathematics Knowledge, and Creative Thinking. MERJ

 Silver, E. A., Mamona-Downs, J., Leung. S. S., & Kenny, P.A.

(1996). Posing mathematical problems in a complex task environment JRME

 梁淑坤（1994）。「擬題」的研 究及其在課程的角色。國民小學

數學科新課程概說-低年級臺灣

省國民學校教師研習會編。

 梁淑坤（1999）。從擬題研究提 出數學教學建議。新典範數學。

 Leung, S. K. (1996). Problem posing as assessment. The Mathematics Educator

 Leung, S. S. (2013) 。 Teachers implementing mathematical problem posing in the classroom:

challenges and strategies. ESM.

 Leung, S. S. (2016). Mathematical problem posing: A case of elementary school teachers developing tasks and designing instructions in Taiwan. Springer.

 Leung, S. S., & Wu, R. (1999).

Posing problems: Two classroom examples. MTMG NCTM.

 梁淑坤（1996）。從佈題探討數 學科教科書的評鑑。教師之友。

1. STUDENTS’ LEARNING 2. TEACHERS’ PROFESSIONAL DEVELOPMENT

Parental Involvement

Why Parents’

 Posing at home through diary writing (Leung & Wu, 2000)

 Practical, connect to daily life

 Make math learning continuous

 Promote teacher-child-parent relationship

 I teach in my sons’ classes (1-6)

 I conducted parents workshops (K-6) over time

 Parenting,

 Communicating,

 Volunteering,

 Learning at home,

 Decision making,

 Collaborating with community.

MPP attends to communicating and learning at home.

Epstein, 2000

Leung & Kuo (2011)

Grade 4, MPP at home and in school

 What are the changes in children’s mathematics problem posing after in-class instructions in mathematical problem posing?

 What are the changes in children’s mathematics attitudes after in-class instructions in mathematical problem posing?

 What are the changes in interaction of parent and child

Data Collection

 Problem posing instruction materials

 parent-child posing sheet

 mathematics attitude measure

 students interview

 parents interview

 teachers’ journal

 video tapes

Data coding of posed problems

by Leung & Silver (1997).

Parents’ views on problem posing

 They expressed that normally they read quickly and signed children’s record book.

 They discussed more with their children and became to understand children’s thoughts.

 A long time since I was in elementary school and there are new ways to do math now.

 I learn more on how to supervise the homework after this problem posing exercise.

 I begin to realize the changes in curriculum and from now I know better what how to explain to my child and trap the thinking…

Children’s views on posing 1/2

 Three of them normally did homework with mothers, one with father, one with elder sister and the last one with grand parent.

 “I do homework by myself at home”

 “Dad and Mom are busy, they sign without reading my work”

 After introducing problem posing at home:

“ We spend more time doing homework together, at least 10

Leung & Kuo (2011)

 MPP of children improved over time.(PP, attitude)

 MPP at home increased interactions of family members.

 MPP at home increased parents’ awareness of child’s learning in school

 Problem posing in-class and at home can enhance interactions of parents, children and teacher.

 Views from parents and children are positive towards problem posing instruction and problem posing at home.

 Feasible to use PP, (student-centered)

 Parental involvement assisted young children to perform MPP that extended the activity from school to home, using strategies like reading picture books or diary writing (Leung, 2019).

 Long term project in an-elementary school with walking distance from my school.

 Bi-weekly family math camp

Parental involvement:

An Intervention (K-6)

Data Collection

 Data were collected longitudinally and analyzed using mixed methods (Creswell, 2009).

 Multiple sources: parents’ focus group interviews and questionnaire, videos/photographs, family diaries/worksheets, and teacher educator’s journals.

 Family-math-tasks (family dimension/affective;

Parental involvement: K1

K1: Lengths

Parent and child used items at home and posed a problem for family members to solve by comparing lengths.

Outcome 1: “ Who is taller? ”

Use eyesight: Elder sister is taller than younger sister

Parental involvement: K1

 Outcome 2: Which is longer?

Direct comparison: Length of notebook longer than pen

 Outcome 3: Which is longer?

Overlapping: Napkin box is longer than lunch bag

Parental involvement: K2

 K2 Combine 2 numbers

 Folding a toy. Inside the toys please write numbers you like and draw in corresponding “o”s accordingly.

 Pose a problem

Parental involvement 1: K2

 Chatterbox Surprise

 What is written inside? “2、8

、3、7、6、4、1、5”

 Pose problems

 Numbers: 1 and 5 make 6

 Unexpected:

 Color: Red and white: Pink

Grade 1 and grade 2

Geometric sticks Monopoly

 Posing problems for a math trial

 Parents as Teachers: They made up problems for children to solve.

 Teacher educator took photographs for parents to consider.

 Parents (in pairs) made up problems for peer to review and revise.

Grade 3 and Grade 4

Parents’ posing: Grade 3&4

 Map with 10 checkpoints

Example Tasks

School Entrance

1.

Find the perimeter of the red circular bench

How many sides are there on the surface of this stool?

If you put this into a box find the size of this box

More Examples

6.

Watch the figure on the ground

How many right angles can you find?

Read the shapes:

(1) How many squares?

(2) How many rectangles?

(3) How many triangles?

8.

Please read the figure surrounded in red.

Name the shape.

Try to work with your parent the area of this figure. You can use any units you like.

Walking with parents

 We wash cups of rice but we have guests today and need 2 times as usual. How many cups shall we need today?

 Fair Trade (__weeks, __days)

Grade 5

 Tangrams

Please use 7 pieces given in Tangrams to make the gun below.

Grade 6

What I learn from parents

 Leung, S. S. (1997). On the Role of Creative Thinking in Problem Posing.

ZDM

 Leung S. K., & Silver, E. A. (1997). The Role of Task Format, Mathematics Knowledge, and Creative Thinking. MERJ

 Silver, E. A., Mamona-Downs, J., Leung.

S. S., & Kenny, P.A. (1996). Posing mathematical problems in a complex task environment JRME

 梁淑坤（1994）。「擬題」的研究及其 在課程的角色。國民小學數學科新課程 概說-低年級臺灣省國民學校教師研習會 編。

 梁淑坤（1999）。從擬題研究提出數學 教學建議。新典範數學。

 Leung, S. K. (1996). Problem posing as assessment. The Mathematics Educator

 Leung, S. S. (2013)。Teachers implementing mathematical problem posing in the classroom: challenges and strategies. ESM.

 Leung, S. S. (2016). Mathematical problem posing: A case of elementary school teachers developing tasks and designing instructions in Taiwan. Springer.

 Leung, S. S., & Wu, R. (1999). Posing problems: Two classroom examples. MTMG NCTM.

 梁淑坤（1996）。從佈題探討數學科教科

書的評鑑。教師之友。

 Leung, S. S., & Wu, R. (2000). Sharing Problem Posing at Home Through Diary Writing: Case of Grade One. AMPC, Australia

 Leung, S. S.; Kuo, H. C. (2011). Sharing some results on mathematical problem posing involving children and parents.

Chiayi.

 梁淑坤（ 2008）。教學錦囊01‧逃吧！數

學瞌睡蟲。格子外面文化

 梁淑坤（2015）。晚餐後，幸福的數學時

光。前程出版社。

 梁淑坤（2019）。嗨！我是好玩的數學遊

戲。湖南科學技術出版社

1. STUDENTS’ LEARNING 2. TEACHERS’ PROFESSIONAL

DEVELOPMENT 3. PARENTAL INVOLVEMENT

Conclusion

The investigator is an action researcher herself

 Found that children and prospective teachers were inexperienced in problem posing.

 Developed tasks and systematic assessment.

 Offered a two-credit course for prospective teachers.

 One student received award funding from the National Science Council (Leung, 2001).

 Teacher-training workshops for in-service teachers.

 In working towards a framework on MPP, the contents of this lecture will add to existing literature.

Pose

Existing

knowledge Unknown

(How to teach MPP)

Solve

What did I pose?

Thank you…^0^

Shuk-kwan Susan Leung

QR CODE

http://www2.nsysu.edu.tw/leung/home.html

Research on Mathematics Problem Posing for Students, Teachers and Parents: A Personal

Journey

Shuk-kwan S. Leung law@mail.nsysu.edu.tw

National Sun Yat-sen University