1. How digital environment could contribute to geometry education
Mingyu SHAO
Supervisors
Main topic of my PH.D project, but it is a large domain
Luc TROUCHE
Ecole Normale Supérieure de Lyon, France
Jana TRGALOVA
Université de Lyon 1, France
Jiansheng BAO
East China Normal University, China

2. Outlines The main question of Ph.D. research
Retrospect of my master thesis—HLT for analytic geometry
A preliminary comparison between Chinese and French curricula
General structure, influencing factors, detailed comparison in geometry, two local didactical theories
How could the comparison inform the Ph.D. research

3. Under which conditions and by what means the Dynamic Geometric Software (DGS) could support students’ growth along the Hypothetical Learning Trajectory(HLT) of geometric concepts and reasoning
Besides the opportunities offered, does the integration of DGS use bring any challenges or disadvantages?
Complex mechanic techniques--utilization scheme on the digital artefact and the conceptual meaning behind this should be constructed (instrumental approach, Verillon & Rabardel, 1995; documentational approach, Gueudet & Trouche, 2009)
Cognitive load caused by the informative dynamic digital environment--student attention should be carefully directed to the essential mathematical part (Mayer, 2009; Sweller et al., 2011)
The key idea is to investigate
Based on this objective, we can ask some subquestions as…
One problem might be, so I will refer to …to deal with this aspect

4. Under which conditions and by what means the dynamic geometric software (DGS) could support students’ growth along the hypothetical learning trajectory(HLT) of geometric concepts and reasoning
Schema of the HLT (for a particular concept, to be focused) (Clement & Sarama, 2004)
Level of understanding expected
type of reasoning expected
student mental activity
Student behavior
The role of DGS
Mathematics history, epistemology, conceptual analysis of the targeted content,
Usual teaching / approaches to this content and their effects, didactical theories and their application
Student conceptions, difficulties and obstacles in this field，theories of cognitive development stages in geometry
The opportunities that DGS offers for geometry (or some more specific concept) education

5. Retrospect of my master thesis—HLT for analytic geometry
Four concepts as representatives
Construction of rectangular coordinate system; Equations of lines
Conic sections and their equations; Algebraic representations of geometric transformation
Content analysis method
confront
with
the curriculum standards of 7 countries
the framework which has been identified from theories of cognitive psychology, interview with experts, literature of empirical researches

6. Retrospect of my master thesis—HLT for equations of lines
Level 1 geometrical determinants of 2-D line → Cartesian equations and other algebraic representations of 2-D lines
Level 2 use these algebraic expressions to solve geometric problems related to lines within plan
Level 3 Cartesian equations of 3-D lines (constitute of two plan equations )；
vector equations of 2-D/ 3-D lines (equations of the two relevant plans can be conducted)
Level 4 definition of polar coordinate system，polar equations of lines

7. approach based on Locating, Spatial Orientation Models, Navigating
Some results of master thesis—HLT for rectangular coordinate system
approach based on Locating, Spatial Orientation Models, Navigating
approach based on Number and Operation on the Number Axis …… …… …… ……
C1 2-D rectangular coordinate system defined as orthogonal number axes/defined as a reference system for locating
distance ： ( 𝑥 1 − 𝑥 2 ) 2 + ( 𝑦 1 − 𝑦 2 ) 2 ； position relationship：dual order between 𝑥 1 and 𝑥 2 , 𝑦 1 and 𝑦 2
approach A starts from “locating, spatial orientation model and navigating” in concrete 3-D backgrounds, seeing Cartesian coordinate system as a reference system to locate a point.
While the other approach D originates from “point, number and operation on the number axis” in one-dimensional world, seeing the coordinate system as a means to connect figural and geometric world.
The two approaches Perpendicular models emerge, The composition of vector and… in c2 and c3 could help students better understand Decomposition of distance and the order relationship
C2 plane vector and plane structure
Composition and decomposition of vector
base vectors and dimensions of space
C3 3-D rectangular coordinate system, space vector and spatial structure

8. Some results of master thesis—conception progress of “locus”
Level 1 locus as a set of points having a certain property which could be expressed by
an equation
Level 2 locus as the image (under a mapping or transformation) of an moving object
Level 3 Locus as a necessary and sufficient condition of its equation
completeness and purity of locus well understood
Those results, especially those concerning the vectors and structures of the space, could kind of serve as a reference when I go into the spetial geometry

9. Back to the Ph.D. project -- didactical engineering approach
Session 2 in China
How the differences and commonalities between the institutional contexts in France and China could influence the teaching practice in the two countries ?
successive lessons
successive lessons
successive lessons
Session 1 in France
Session 3 in France
1 Preliminary analysis-
2 Instruction design and a priori analysis
- HLT schema, affordance and pitfalls of DGS
Negotiate the HLT with the teachers, and the possible conflicts between HLT and their initial teaching plan /national curriculum, and the collection of instructional tasks and principles needed to promote growth.
𝑑𝑖𝑑𝑎𝑐𝑡𝑖𝑐𝑎𝑙 𝑠𝑖𝑡𝑢𝑎𝑡𝑖𝑜𝑛, 𝑡𝑎𝑠𝑘𝑠,𝐷𝐺𝑆−𝑎𝑖𝑑𝑒𝑑 𝑐𝑜𝑢𝑟𝑠𝑒𝑤𝑎𝑟𝑒 homework exercises/tests
role of teacher ?
role of researcher ?
3 Teaching experiment
In-lesson and after lesson data
4 A posteriori analysis
HLT, instrumental/documentational genesis

10. Outlines The problematic of Ph.D. research
Retrospect of my master thesis—HLT for analytic geometry
A fundamental comparison between Chinese and French curricula
General structure, influencing factors, detailed comparison in geometry, two local didactical theories
The presentation up to now is just a conceiving of how the research would be organized, to proceed. As I have situated it in two different context, a comparison of the two countries curricula might be a more fundamental work before all these, this is also what I have done up to now, next I will introduce this preliminary comparison with respect to several aspects

11. General Structure Chinese standards (Jan. 2018)
French Program (Sep. 2016)
no yearly division
for grades 1-9 : 3-year cycles, yearly division in programs for high school
Three modules corresponding to different assessment requirements
Three options within each of the three streams
General Vocational Technological
General Vocational
1 Compulsory (graduation, GAO KAO)
2 Selectively compulsory (GAO KAO)
from second year
WE can mainly see three streams of French high schools. Within each streams there are three options
In china we also have such a distinction of streams, but a little different is Chinese standards arranged three modules according to three different assessment requirements, which are compulsory, selective and optional. The options only happens in the final year of high school, and because it is mainly oriented toward the independent enrollment by universities
3 Optional (independent recuitment)
sciences humanities
sciences and economics
Math
physics and technology;
sociology and economics
language and history
sports and arts
math in daily life and local culture
no entrance exams for universities, like GAOKAO
Baccalauréat as a hidden criteria

12. General Structure Chinese standards (Jan. 2018)
French Program (Sep. 2016)
Four main line in the compulsory part
Functions
Geometry and Algebra
Probability and Statistics
Mathematics modelling and inquiry activities
Three different topics across three years
Functions/Analysis
Geometry
Probability and Statistics
two transversal topics
Algorithms, Reasoning and Logic
The new topic “modelling and inquiry activities” besides three traditional topics
Without distinction between knowledge and skill, but underline their connection with core competencies
For each topic, program table comprises several lines like

13. Factors influencing mathematics curriculum policies – France
Reform of “Modern Maths ” in the 1970s, grounding mathematics teaching on very formal approaches (Trouche, 2016)
CREM report in 2002 (Gueudet et al. 2017)
connecting rigor and imagination；
experimental aspect of mathematics；
Encompassing the mathematical practices especially in computer science；
importance of reasoning and proof
Create a new image of experimental aspect of mathematics
Common core (Bodin, 2008): a set of knowledge, skills, and attitudes in French curriculum
Another milestone is the crem report
Tools for visualization and representation, numerical or formal calculation, simulation and programing have been largely suggested by program;
Would be considered as another important factor, it has contributed to the development of individualized…
Results of PISA assessment and the European common framework of “key competencies” (OECD, 2006) (Gueudet et al. 2017)
Introduce common core in 2005
individualized teaching practices and personal support
inquiry-based mathematics learning

14. Factors influencing mathematics curriculum policies – China
Tradition of “Two Basis” in Chinese obligatory education (from 1950s)
“basic” knowledge (memorizing concept, understanding proposition) firmly constructed
“basic” skills (of proving and operating) fluently performed
Curriculum reform, the first version of curriculum standards (beginning of 21 century) and its revision (2011, 2017) .
From result oriented to process oriented objectives；Two basis+ basic thought and basic activity experiences
Ten key words ：sense of number, consciousness of symbols, spatial vision, geometric intuition, perspective of data analysis, operation competence, reasoning competence, Idea of modelling, consciousness of application and innovation
Believe in “practice makes perfect”, “memorization promotes understanding”；Appreciate the well, direct class control
As for Chinese curriculum policies, I think it takes its root in the tradition of two basis, that means the basic knowledge…here they don’t emphasize the mechanic practice and memorization.
Then the reform in the beginning of 21, witness a transform from result…
Basic ideas and activity experiences were added to the already existing two basis…
Curriculum Revision in 2011 for primary and junior high school, experts have draw up a ten key words, the six parts inderlined were later developed into 6 key competencies in senior high school, consistency between junior and senior high school
Finally it’s worth noting that Chinese curriculum also fall under the influence of international policies and theories
Reports by ministry of education, 2014《教育部关于全面深化课程改革落实立德树人根本任务的 意见》
To construct the “core competency system” for each scholar periods (“教育部将组织研究提出各学段学生发展核心素养 体系“）
Definition given by research group of Peking Normal University, 2016
Framework of “key competencies” (OECD, 1997, 2006) (Shi, 2018)

15. China France Mathematics core competencies Six key competences
mathematical abstraction
intuitive imagination
logical reasoning
mathematical operation
mathematical modeling
data analysis
Six key competences
representing，communicating，reasoning，
computing，modelling，searching
General objectives
criticism, treatment of information, changes in the register
Design, realize and analyze algorithmic
Now let’s see the key competencies listed and general objectives listed in both curricula
« L’acquisition de techniques est indispensable, mais doit être au service de la pratique du raisonnement qui est la base de l’activité mathématique des  élèves »
Tools for visualization and representation, numerical or formal calculation, simulation and programing have been largely suggested by program;
design and realize algorithms to solve problems and the analysis of algorithms
,changes in the register (graphic, numerical, algebraic, geometric) take priority

16. China France Mathematics core competencies Six key competences
mathematical abstraction
intuitive imagination
logic reasoning
mathematical operation
mathematical modeling
data analysis
Six key competences
representing，communicating，reasoning，
computing，modelling，searching
General objectives
criticism, treatment of information, changes in the register
Design, realize and analyze algorithmic
Three levels for each core competencies to assess student’s progression--formative evaluation
Create proper situations (daily life, other disciplines, modern science and technology, culture) for introducing concepts
« L’acquisition de techniques est indispensable, mais doit être au service de la pratique du raisonnement qui est la base de l’activité mathématique des  élèves »
Tools for visualization and representation, numerical or formal calculation, simulation and programing have been largely suggested by program;
design and realize algorithms to solve problems and the analysis of algorithms
,changes in the register (graphic, numerical, algebraic, geometric) take priority
Relative relation between techniques and reasoning

17. China France dialectic relation between proof and observation
emphasize reasoning expressed by mathematics language (special notation/vocabulary)
problem type VS knowledge/competences behind the solution
process of concept development, key events and main figures in mathematics history
respect students’ autonomy and initiative in the resolution of problems while encourage communication and collaboration in student activities

18. Detailed comparison into the domain of geometry —commonalities
Geometry in the space ：
autonomic use by students software for visualization and construction
Vectors：transition from plane to space
Analytic geometry：
slope, director vector and cartesian equation of lines in the plane
canonical equation of a circle
GEOMETRY: appear early in the curriculum
Complex number
algebraic forms and geometric representations，operations
trigonometric forms and exponential notations

19. Detailed comparison into the domain of geometry —differences
Trigonometry in grade 11
link of vectors with equations
Director/normal vector and Cartesian equation of lines in the plane
Normal vector and Cartesian equation of planes in the space
Parametric representations of lines/planes in the space through non-collinear vectors
Non-orthogonal coordinate system
Solutions in C of a second-degree equation with real coefficients.
Physic background of the notion of vector
Concept of projection vector
Equations of conic sections
France
Optional
vector space and subspace, outer product
Parametric representations of lines/planes in the space through non-collinear vectors
systems of linear equations in three variables
Third-order matrix and determinant
Spatial isometric transformation and its matrix representation, group of transformation matrix
China
GEOMETRY: appear early in the curriculum

20. QINGPU experiment (Gu, 1997, 2007)
Two classical didactical theories growing from local communities
Theory of didactical situation (Brousseau, 1986, 1998)
QINGPU experiment (Gu, 1997, 2007)
activity-internalizing E S
conceptions
attitudes
behaviors
languages
externalizing
Notion of “intermediary of activity”(活动中介): human activities, which can be distinguished into cognitive aspect (conceptions, emotions, attitudes) and behavior ones
（体现中国传统特色的学习理论）是一种以人的活 动为中介的学说，这种活动可区分为内部的心理活 动--认知,情意--和外部的行为活动—行为语言—两类
Notion of “milieu”: the adversary system of the student, all that will bring out positive or negative feedbacks to the student’s actions
contradictions, difficulties allows the students to adjust this action, to accept or reject a hypothesis, to evolve from one solution to another, les rétroactions du milieu permettent des adaptations
It is the student who will act on the milieu and reconstruct the equilibrium which has been lost after the he/she received the feedback of contradictions, difficulties from the milieu.

21. QINGPU experiment (Gu, 1997, 2007)
Two classical didactical theories growing from local communities
Theory of didactical situation (Brousseau, 1986, 1998)
QINGPU experiment (Gu, 1997, 2007)
activity-internalizing E S
conceptions
attitudes
behaviors
languages
externalizing
milieu
A gap between behaviors and conceptions
researcher/teachers need make sure that the expected behaviors, will result well from the conceptions intended;
Students‘ external behaviors (including language) can be internalized into internal cognitive activities. The two not only are genetically homologous, but also develop in an isomorphic and synchronous manner （Gu, 2007a, p.30）
GEOMETRY: appear early in the curriculum
If the student does not commit a mistake, it may be because the (digital) system does not allow it (IT transposition, Balacheff)

22. Two classical didactical theories growing from local communities
A theoretical tradition based on constructivism, genetic psychology (Piaget, 1975)
Didactical theories rooted in ancient educational philosophies（学而时习之，知行合一）
combining receive-based learning with inquiry-based learning (活动式与接受式)；the teacher need present successively the knowledge structures with different complexities, like a set of boxes (套箱结构，序进原理)
students have to establish the solutions to the problem autonomously without being told by teachers, the only recourse is their previous conceptions
Tradition does not engage the necessary (mathematical) knowledge and can not appropriate them
Teacher’s work：make sure that subject is able to identify the feedback and recognize the existence of a gap not acceptable with respect to her intention, and, on the other hand, that the milieu can provide identiable feedback.
Lorsque le milieu permet ceci sans aucune intervention de l’enseignant relative au savoir, et sans tenir compte des attentes de l’enseignant, on dit que l’élève est dans une situation adidactique From conception to knowledge: a procedure of institutionalization
the organic combination of accepting based learning (directly obtain the knowledge and experiences well developed) with inquiry based learning (put the knowledge into practice and communicating 学而时习之)
(observation,, experiment, conjecture, reasoning and conclusion)
so that the one in junior grades could be compatible with the more complex one in senior grades
discern essential features or connotation of the concept by exploring varying embodiments of the concept or of other concepts / understand the formulation process of the concept by using different semiotic registers, improve problem solving abilities by transforming the new, complex problem into easier, already solved problems
Didactical variables: the pertinent values of which will change the hierarchy or the best resolution strategies to the problem, thus allowing students to evolve the necessary conceptions
Theory of variation: conceptual variances / procedural variations to highlight something unchanged (essential features or connotation of the concept) or to deepen one’s understanding of knowledge

23. Outlines The problematic of Ph.D. research
How DGS could support student along HLT ?
Retrospect of my master thesis—HLT for analytic geometry
A preliminary comparison between Chinese and French curricula
General structure, influencing factors, detailed comparison in geometry, two local didactical theories
How could the comparison inform the Ph.D. research

24. Back to the Ph.D. project…
How the differences and commonalities between the institutional contexts in France and China could influence the teaching practice in the two countries ?
Session 2 in China
successive lessons
successive lessons
successive lessons
Session 1 in France
Session 3 in France
1 Preliminary analysis-
2 Instruction design and a priori analysis
3 Teaching experiment
4 A posteriori analysis
- HLT schema, affordance and pitfalls of DGS
Negotiate the HLT with the teachers, and the possible conflicts between HLT and their initial teaching plan /national curriculum, and the collection of instructional tasks and principles needed to promote growth.
decisions and intentions will be captured and motivated, to keep track of the development of the designers’ (both teacher and researcher) insights
what can be at stake of this situation for the student (action, choice, decision, control and validation
𝑑𝑖𝑑𝑎𝑐𝑡𝑖𝑐𝑎𝑙 𝑠𝑖𝑡𝑢𝑎𝑡𝑖𝑜𝑛, 𝑡𝑎𝑠𝑘𝑠,𝐷𝐺𝑆−𝑎𝑖𝑑𝑒𝑑 𝑐𝑜𝑢𝑟𝑠𝑒𝑤𝑎𝑟𝑒 homework exercises/tests
role of teacher ?
role of researcher ?
In-lesson and after lesson
HLT, documentational genesis

25. How could the comparison inform the Preliminary analysis and instruction design
Confine ourselves to a content more focused…3-D geometry and logic reasoning
within the commonalities between Chinese and French geometric curricula,
represent some typical difficulties of geometry and convenient to be facilitated in dynamic geometry environment
Different approaches to design tasks
varying the values of the didactical variable: calculate →→ calculate 14+59
varying embodiments of the concept to highlight the connotation of the concept : three examples of non-coplanar lines——
When digital technology get involved…
affordance, pitfalls, utility, utilisability and acceptabilité, instrumental/documentational genesis

26. Workshop

27. Try to solve the following problems
An example in sésamath e-textbooks concerning patrons et perspectives cavalière des solids
General description of sésamath outils.sesamath.net/manuel_numerique/index.php?ouvrage=ms2_2014&page_gauche=170
(An example given in Chinese curriculum standards) when we cut a cube by a plane, what kind of shape might be obtained as the cross section ?
how many different types of triangles can be obtained
how many different types of quadrilaterals can be obtained
could we get some other polygons ?
Is it possible that the cross section is a regular pentagon? right-angled triangle? or the polygons with more than 6 sides? Why?
which type of the triangle-shaped intersections would have the largest area?
you can recur to the geogebra page
Describe the context where the dynamic website is located
Sésamath is an association of mathematics teachers in France, most of whom teach at secondary school. the Sésamath website is just for these teachers to designing publishing and use free resources. Sésamath e-textbook for grade 10. It is freely accessible online when browsing the page with the mouse, some “complements” windows appear, offering different tools: animated helps, dynamic figures, etc. This window also offers the source file of the page: the teachers can download it to make all the modifications they wish to introduce

28. Consider the questions below
What are the targeted knowledge/competences behind this problems (in relation with each curriculum)
What manipulations can we do with these objects and what can we not do?
Does the digital dynamic tools here really help you better master the knowledge/competences compared to the paper-pencil approach？
With respect to the targeted learning, how could we design a suitable didactical situation (series of tasks with some variation)? do you have some suggestions for improving the digital dynamic page so that it could be better integrated into the teaching practice?

29. References
Clements, D.H., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking & Learning, 6(2),
European Parliament. (2006). Key competencies for lifelong learning. European Reference Framework. Official Journal of the European Union. Retrieved from europa.eu/LexUriServ/LexUriServ.do?uri=OJ:L:2006:394:0010:0018:en:PDF
Gueudet, G., & Trouche, L. (2009). Towards new documentation systems for mathematics teachers? Educational Studies in Mathematics, 71(3),
Gueudet, G., Bueno-Ravel, L., Modeste, S., & Trouche, L. (2017). Curriculum in France: A national frame in transition. In D. Thompson, M. A. Huntley, & C. Suurtamm (Eds.). International perspectives on mathematics curriculum (pp. 41–70). Charlotte, NC: International Age Publishing.
Gu, L. (1997a). 青浦实验──一个基于中国当代水平的数学教育改革报告(上).[Qingpu experiment: a report on mathematics education reform based on level of contemporary china (1st).] 课程.教材.教法(1),
Gu, L. (1997b). 青浦实验──一个基于中国当代水平的数学教育改革报告(下). [Qingpu experiment: a report on mathematics education reform based on level of contemporary china (2nd).]  课程.教材.教法(2),

30. References
Gu, L. et al. (2007). 变式教学：促进有效的数学学习的中国方式. [TEACHING and learning with variations: a Chinese way to promote effective mathematics learning.] 云南教育：中学教师(3),
Shi, N. (2018). 中小学数学课程改革——从“双基”到数学核心素养. [Maths curriculum reform in middle and primary schools: from “double bases” to “mathematics core literacies”.] 新课标教师培训讲座 [The report of “new curriculum” for teacher training]
Trouche, L. (2016). Didactics of mathematics: Concepts, roots, interactions and dynamics from France. In J. Monaghan, L. Trouche, & J. Borwein (Eds.), Mathematics and tools, instruments for learning (pp ). New York, NY: Springer.
Verillon, P., & Rabardel, P. (1995). Cognition and artifacts: a contribution to the study of though in relation to instrumented activity. European Journal of Psychology of Education, 10(1),