1. ∂ 1 Understanding of Linear Function in Secondary School Students: a Comparative Study Between Shanghai and England Yuqian Wang Patrick Barmby David Bolden

2. ∂ 2 Content Background/Rationale Research Questions Theoretical Framework Results Conclusion Implication

3. ∂ 3 Background: Textbooks policy Commercial / Compulsory Two Levels / Uniform Level School / Home

4. ∂ 4 Rationale England-East Asia Gap in mathematics (Jerrim & Choi, 2013) Textbooks from Top 30% Ranking State Schools Topic: Function to Linear Function Countries Schools Topic

5. ∂ 5 1.Does this topic have similar importance between the two countries' textbooks? 2.What is the background context for the two countries' textbooks respectively? 3.What emphases do the selected textbooks have in terms of the model of understanding function? 4.What approach do the selected textbooks use in terms of application? Research Questions

6. ∂ 6 Theoretical Framework At the macro level (Li, Chen & An, 2009) 1.General arrangement of the topic in terms of percentage of pages allocated; 2.Background context of this topic At the micro level: 1.Understanding levels in examples and exercises on basic knowledge. 2.Application would be analysed based on two aspects: (a) how problems are presented (b) how to resolve them using three main representations

7. ∂ 7 Results at the macro level: Content Placement: TextbooksAverage Pages (%) Foundation Level in English textbooks 1.84% Higher Level in English textbooks 2.53% Shanghai compulsory textbook 2.16% Background Context in linear function: England: straight-line graphs Shanghai: concept of function, proportional function, inverse proportional function

8. ∂ 8 Model for Understanding Function Level 1: Variable perspective Level 2: Dependent relationship Level 3: Translating representations Level 4: Property noticing Level 5: Object analysis Level 6: Inventising min

9. ∂ 9 Results at the Micro Level: Basic Knowledge The Percentage of Example/Exercise at Each Level Level 3: Translating representations Level 4: Property noticing Level 5: Object analysis Level 6: Inventising Higher level 13.6%/32.8%45.4%/41.2%36.4%/16.0%4.5%/10.1% Foundation level 22.2%/ 45.5% 77.8%/ 43.2% 0/ 6.8% 0/4.6% Shanghai18.2%/6.3%9.1%/18.8%63.6%/50.0%9.1%/25.5%

10. ∂ 10 Results at the Micro Level: The Content of Application Types of Examples/Exercises English higher level textbooks: Using graph Shanghai textbook: Word problem Main Type of Solution English higher level textbooks: Equation Shanghai textbook: Mix-method

11. ∂ 11 Real-life situation Graphic approach Algebraic approach English textbooks Definition Algebraic approach combined with graph More abstract algebraic approach combined with graph Real-life situation Shanghai textbook Conclusion How knowledge is presented

12. ∂ 12 Implication For comparative study Student assessments How to borrow For real-field area Teachers: the disadvantage of each approach Schools: teacher’s professional development

13. ∂ 13 Thanks for listening! Any questions? Reference: Jerrim, J., & Choi, Á. (2013). The mathematics skills of school children: how does England compare to the high-performing East Asian jurisdictions? Journal of Education Policy(ahead-of-print), 1-28. Li, Y., Chen, X., & An, S. (2009), Conceptualizing and Organizing Content for Teaching and Learning in Selected Chinese, Japanese and US Mathematics Textbooks: The case of fraction division, ZDM, 41(6), 809- 826