SCHEMA-BASED WORD PROBLEM INSTRUCTION AND LANGUAGE SUPPORT IN LOWER PRIMARY MATHEMATICS CLASSROOMS EMILY SUM, PHD
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Schema-based Word Problem Instruction and Language Support in Lower Primary Mathematics Classrooms Emily Sum, PhD
Schema-based Word Problem Instruction for CLD Learners in Lower Primary PART I
What Makes Word Problem Solving Challenging?
玩具店有兔布偶45個、象布偶28個和羊布偶59個。
• 象布偶和羊布偶相差多少個？
• 豬布偶比兔布偶多25個，豬布偶有多少個？
教育出版社有限公司（1下A） HKU 3
Word problems that can
be solved by the same
arithmetic operation but
differ with respect to their
underlying semantic
structure have very
different degrees of
difficultly (Corte &
Verschaffel, 1991, p.119).
教育出版社有限公司（1下A） HKU 4
What Makes Word Problem Solving Challenging?
Numerous steps and
skills are necessary
to solve a word
problem
Comprehension of
the text of the word
problem
Maths embedded
within a linguistic
context
HKU 5
Can We Make Word-
Problem More Explicit?
Students are told how a problem should be
solved and provide them with examples of
good problem-solving strategies.
HKU 6
解答應用題時，應先看清楚
題目中的資料和問題，才決
定用加法或減法來計算。
1下B 單元 7 減法及加減應用題
HKU 7
George Pólya
1945
In conventional word-problem Polya’s Principle:
instruction, students may organize
1. Understand the problem
word-problem information or follow a
mnemonic device to work step-by-step 2. Devise a plan
through the problem.
3. Carry out the plan
4. Look back and check
HKU 8
Identify A Word Problem
As Belonging To A
Problem Type
Warm-up Exercise
HKU 9
Sort Problems into Different Problem Types
1. 妹妹原有貼紙75張，她把21張貼紙送給弟弟後，還有貼紙多少張？
2. 家裏有小保鮮袋18個和大保鮮袋6個。家裏共有保鮮袋多少個？
3. 點心店有蝦餃 79 籠，燒賣比蝦餃少 22 籠。點心店有燒賣 ____ 籠。
4. 書架上有中文圖書 69 本，英文圖書比中文圖書少 5 本。書架上有英文圖書_____ 本。
5. 陳老師買了鉛筆19枝，原子筆30枝和膠水10枝。陳老師共買了筆 _____ 枝。
6. 熱狗比牛角包多25個，豬仔包比牛角包少20個，熱狗和豬仔包相差 ____個。
7. 池塘裏原有魚 38 條，多把 27 條魚放進池塘後，現有魚 _____ 條。
8. 陳先生買了月餅16盒，共有月餅64個。他送給親友11盒月餅後，還有月餅 ____ 盒。
9. 餐廳原有雞蛋85隻，廚師不小心打破了3隻。餐廳現有雞蛋多少隻？
10.家裏有雞蛋12隻和鴨蛋3隻。家裏共有蛋多少隻?
教育出版社有限公司（1下A） HKU 10Problem Type
What is happening in the word-problem narrative.
§ 家裏有雞蛋12隻和鴨蛋3隻。家裏共有 § 妹妹不小心打破了雞蛋3隻。家裏現有
蛋多少隻? 雞蛋多少隻？
HKU 11Problem Type
What is happening in the word-problem narrative.
§ 家裏有雞蛋12隻和鴨蛋3隻。家裏共有 § 妹妹不小心打破了雞蛋3隻。家裏現有
蛋多少隻? 雞蛋多少隻？
§ For Cantonese Learners: § For Cantonese Learners:
• Students may be unfamiliar with the • Act out the application problem by
term 鴨蛋. showing a box of eggs or using cubes
to illustrate the act of taking away.
• Highlight what is expected by a picture
and number sentence (or equation).
HKU 12Problem Type
What is happening in the word-problem narrative.
§ 家裏有雞蛋12隻 。鴨蛋比雞蛋少
9隻。家裏共有鴨蛋多少隻?
If students know a schema for
each type, and understand how to
Once students determine the sort problems into the problem
problem type, they can apply a types and apply the solution
schema (i.e., DIAGRAM, equation, method for each schema, then
or plan) to solve the problems. students should be able to solve
most word problems. (Cooper &
Sweller, 1987).
HKU 13Develop Mathematics Concepts and Vocabulary
家裏有雞蛋12隻 。鴨蛋比雞蛋少9隻。家裏共有鴨蛋多少隻?
§ Although the comparative words, 多(more),
§ To help students in understanding and 少(less, fewer) may be familiar in everyday
interpreting word problems involving context, these phases __ 比 (compare), __
additive/subtractive comparisons of two 多/ 少 are indicating specific mathematical
quantities and to understand the academic quantities which students need to understand.
language terms such as 多(more), 少(less,
fewer), 比(compare), 相差(difference). § Make sure students have a number of
opportunities to communicate and verify their
§ Multiple representations should be used understanding of the language. Practice by
including language, bar diagram/model and saying complete sentences and showing
equations. equations. Writes the sentence frame and
• 雞蛋多幾多? equations on the board:
• 鴨蛋少幾多? • 雞蛋12隻, 鴨蛋3 隻
• 有何不同 (How are they different )？ • 鴨蛋比雞蛋少9隻, 雞蛋比鴨蛋多3 隻
• 相差什麼 (What is the difference )? • 相差9個
HKU 14Problem Type
Bar Model
§ 安琪購物後獲得印花15個， 柏康比安琪多獲印
花9個， 柏康獲得印花多少個?
§ what if … 安琪比柏康多9個?
§ For language learners “instruction should also
focus on how students generalize, abstract,
and describe patterns, rather than only on
how students use individual words”
(Moschkovich, 2007, p. 98)
HKU 15Avoid the KEYWORD Strategy!
The most important approach to solving 樂樂吃了棒棒糖7支
any contextual problem is to analyze
思思吃了棒棒糖5支
and make sense of it using all the
words. The key word approach 二人共吃了棒棒糖多少支?
encourages children to ignore the
meaning and structure of the problem
(Van de Walle, Karp, and Bay-Williams,
2019, p. 178).
HKU 16Schema-Based Word Problem Instruction
Additive Structures
Change (join/add to and Combine (Part-part-
Compare
Separate/take from) whole)
refer to dynamic situations relate to static situations involve two quantities that
in which some event involving quantities, which are compared and the
changes the value of a are considered either difference between them.
quantity, e.g. adding to or separately or in Compare problems are
taking away from a set. combination, i.e. context harder for students to
where everything is learn than either change
present the whole time, problem or part-part-
nothing changes. whole problems.
HKU 17Additive Structures
Change (join/add to and Combine (Part-part-
Compare
Separate/take from) whole)
1.妹妹原有貼紙75張，她把21 2.家裏有小保鮮袋18個和大保 3. 點心店有蝦餃 79
張貼紙送給弟弟後，還有貼紙 鮮袋6個。家裏共有保鮮袋多 籠，燒賣比蝦餃少 22
多少張？ 少個？ 籠。點心店有燒賣 ______ 籠。
7. 池塘裏原有魚 38 條，多把 5.老師買了原子筆12枝、鉛筆 4. 書架上有中文圖書 69
27 條魚放進池塘後，現有魚 15枝和熒光筆10枝。(a)老師 本，英文圖書比中文圖書少 5
_______ 條。 共買了原子筆和熒光筆多少枝？ 本。書架上有英文圖書_____
本。
8.陳先生買了月餅16盒，共有 10.家裏有雞蛋12隻和鴨蛋3隻
月餅64個。他送給親友11盒 。家裏共有蛋多少隻? 6.熱狗比牛角包多25個，豬仔
月餅後，還有月餅 _____ 盒。 包比牛角包少20個，熱狗和豬
仔包相差 ______個。
9.餐廳原有雞蛋85隻，廚師不
小心打破了3隻。餐廳現有雞
蛋多少隻？
HKU 18Van de Walle, Karp, & Bay-Willias (2019) HKU 19
Schema-Based Word-Problem Instruction
A schema is a framework, outline, or plan for solving a problem (Marshall, 1995).
§ Students understand the structure of a word problem is seen as critical to successful problem
solving (Kalyuga, 2007). Students can use schema to
• Organize information from a word problem in ways that represent the underlying structure of a problem
type (Powell, 2011).
• Interpret/ relate the text of a problem situation with schemas, i.e. to derive the semantics out of the
syntax of the word problem, and formulate a solution plan.
• Determine the problem type, they can apply a schema (i.e., diagram, equation, or plan) to assist in solving
the word problem.
HKU 20Approaches in Solving Word Problems
1. Start with contexts that students can relate
Example form PLK to, and make sense of the context through
actions, visual aids and other hands-on
materials. Manipulatives and graphic
representations can minimize language
barriers.
2. Ask students to draw/complete a diagram
that matches the sematic structure of the
problem. This helps students to become
aware of the semantic relation between the
quantities, and use that knowledge for
understanding the problem situation.
3. Represent the situation with a diagram, and
the number sentence, before preforming
computation. Give students ample
opportunities to listen, speak and write
mathematically (Anstrom, 1999).
HKU 21Strategy
To be discussed…
1. Read the problem, oral discussion (scripts contains various prompts)
2. Illustrate the problem by
a. using graphic organizers/ manipulatives
b. drawing pictures/ diagrams/ visual representations
3. Write the number sentences/ equations
4. Solve for the unknown
5. Explain what they were solving for
HKU 22Problem Writing
As tool for determining students’ level of mathematical understanding
§ Use the picture to tell a Maths story. § Uses multiple concrete
representations and visual models to
§ 運用所學的詞語和句型，説出簡短語句:
make the mathematics explicit
三尾魚在水中游泳
§ Student Debrief (匯報) Prompts
students to share their developing
thinking and understanding Provides
opportunities and suggestions for
§ Draw a model to match the story. differentiation
§ Write a number sentence to tell the § Gives formal and informal
story.
opportunities for teachers to check for
§ There are …. understanding
HKU 23From Oracy to Literacy
The role of Cantonese in the learning of Chinese reading and writing (Chan, 2019)
If NCS students can be provided with more intensive training in
their oral skills in Cantonese, involving not just simple
conversations, but different spoken varieties as they progress, so
that their grammatical and pragmatic competencies can be
developed through engaging in different kinds of communicative
tasks, some in relatively more formal settings, it is believed that
their literacy skill and their overall Chinese proficiency can be
gradually enhanced.
HKU 24Lesson Planning
Classroom Discussion
§ What ideas/concepts will I focus on during discussion?
§ What questions will I ask to prompt students to share their thinking and when will
that happen?
§ How will I select which students’ work to be shared? And how will I sequence this
sharing of student representations and/or solution methods to connect and
strengthen all students’ understanding of the content?
§ How can I facilitate student discussion that prompt students to critique the
reasoning of others, justify solutions, ask questions, and learn from other
students’ thinking?
§ What will I connect students’ informal language to precise mathematical
language?
https://achievethecore.org/page/3237/lesson-planning-and-reflection-quick-reference-question-guide HKU 25Lesson Planning
Problem and Exercise
§ Which problem will prompt students to share their thinking and apply their
mathematical language?
§ Which problems will require students to explain and justify their work?
§ What solution methods/representations do I anticipate seeing from students?
§ How will I connect these solutions methods/representations to each other to
strengthen all students’ understanding?
§ How will I address unfinished learning in the context of the on-grade-level work?
§ What extension work will I prepare for students who are ready for deeper
engagement with grade-level content?
https://achievethecore.org/page/3237/lesson-planning-and-reflection-quick-reference-question-guide HKU 26Designing Equitable Learning Environments
https://achievethecore.org/aligned/designing-equitable-learning-environments/ HKU 27References
§ Anstrom, K. (1999). Preparing Secondary Education Teachers to Work with English Language Learners: Social Studies. Resource Collection
Series, 13. http://www.ncela.gwu.edu
§ De Corte, E., and Verschaffel, L. (1991). Some factors influencing the solution of addition and subtraction word problems. In Durkin, K., and
Shire, B. (eds.), Language and Mathematical Education, Open University Press, Milton Keynes, pp. 117–130.
§ Kalyuga, S. (2007). Expertise reversal effect and its implications for learner-tailored instruction. Educational Psychology Review, 19, 509-539.
§ Marshall, S.P. (1995). Schemas in Problem Solving. Cambridge University Press.
§ Matthews, S. & Yip, V. (2011). Cantonese: A comprehensive grammar. Routledge.
§ Polya, G. (1945). How to solve it; a new aspect of mathematical method. Princeton University Press.
§ Tang, S.W. (2002). Guide to LSHK Cantonese Romanization of Chinese Characters, 2nd Ed. Hong Kong: The Linguistic Society of Hong Kong.
§ Van de Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2019) Elementary and Middle School Mathematics: Teaching Developmentally, 10th Ed.
Pearson.
§ Wilkinson, A. (1965). The concept of oracy. Educational Review, 17 (4), 11-55.
HKU 50Do1 Ze6 (M4 Goi1 Saai3)
多謝 (唔該晒)
HKU 51You can also read
