## 1. Visualization

### 1.1. B.442 Gardner (1983) argues that spatial ability is one of the several "relative autonomous human intellectual competences" which he calls "human intelligences."

### 1.2. Relationship between spatial thinking an mathematics B.442

### 1.3. G.5. Children need more than a visual picture

### 1.4. T.276 Duval analyzed the role of visualization in the solution processes of a geometry problem and distinguished several approaches to a diagram in geometry: An immediate perceptual approach that may be an obstacle for the geometric interpretation of the diagram, an operative approach that is used for identifying sub-configurations useful for solving the problem and a discursive approach that is related to the statement describing the givens of the problem.

## 2. Manipulatives and real world objects B 433

### 2.1. Although I feel that taking those objects out of the world and placing them in a false environment e.g. classroom will not be as beneficial than actual real world in the real world.

### 2.2. G. 5. Childrens ideas come from hands, eyes etc, not passive looking.

## 3. B.438 comments about computers helping with the geometric supposer

## 4. E.150 and 151 Lehrer, Jenkins and Osana.

### 4.1. Talk about students not being able to think about angles as turnes etc. just measure the point between the two end points.

## 5. E.164 dynamic geometries allow for exploration of transformations in regard to topology. Cite people.

## 6. G.16 Children learning about spatial attributes through computer activities e.g. logo for directionality, and older children using coordinate based computer games for coordinate locations.

## 7. G.19 Large section on computer benefits for geometry. (Clements).

### 7.1. Provide representations

### 7.2. Advantages over other counterparts - resize, and shape.

### 7.3. Can be saved a retrieved later, and also store configurations. They can play sequences of actions.

### 7.4. Computers can draw symmetrical shapes/anything to what are drawn.

### 7.5. Computers can help children become aware of and mathematize their actions such as moving pieces into place.

## 8. (M. 165-166) Clements, Wilson & Sarama (2004)

### 8.1. Talk about shapes software which is a computer manipulative allowing children to compose and decompose shapes.

## 9. N.276 Logo for learning congruence, transformations, and symmetry

### 9.1. N.283. Logo for map work and coordinates. Although interface must be appropriate and activities well planned.

## 10. P.865 Significant section on learning and teaching geometry. (Battista, 2007)

### 10.1. Early research on the use of computer environments for learning geometry focused primarily on investigating predictions about s student learning made by environment designers. More recent research, informed by initial studies and extensive experience with the environments, has investigated more closely the learning processes that occur within the environments and how the environments affect and shape student learning.

### 10.2. Logo-based computer programs

### 10.3. Geometric Supposers

### 10.4. Dynamic Geometry Environments (DGEs).

### 10.5. In these environments, students are expected to provide explicit specifications so unline paper and pencil, the these computer environments, students cannot make drawings without some level of conceptual and representational explicitness.

### 10.6. P.883 Major comments on DGEs.

## 11. D.98-100. An excellent Battista (2009) summary of technologies for geometry.

## 12. Labordes (1993) Z

### 12.1. Z. 241. Early quote that technology with direct manipulation is opening new possibilities.

12.1.1. Z. 243 User is able to handle objects and relations linked with them.

12.1.2. Z.243 theoretical concepts are reified and can be handled as material entities.

### 12.2. Z.244. Problem with teaching geometry - Space and geometry connection, that it is theory and material objects with perceptual graphical properties.

### 12.3. Z.250 Computer drawings are accurate. Children can change their drawings to make it say something that it should not.

### 12.4. Z.250. Tech, offers the students a chance to experiment, which is good for developing meaning for mathematical phenomena.Also, Z.261

12.4.1. I also think it saves time.

12.4.2. Z.250 connects to visualization

12.4.3. Z.250 They can do repeating trials.

12.4.4. Z. 256 Triggers ideas for other mathematical questions.

12.4.5. Z.257 Helps students develop conjectures.

### 12.5. Models ideas.Z.260

### 12.6. Z. 261. Tasks can be more complex than those given in traditional environments.

## 13. Connect with diagram and pictorial issues from other map.

## 14. X. Clements and Battista (2994)

### 14.1. X.173 Tech easily facilitates the construction of geometrical concepts.

### 14.2. X.173 students score higher using computer environments than non-tech environments.

14.2.1. X.174 especially in coordinate environments

### 14.3. X.174-5. Constructivist learning.

14.3.1. Describing those who advocate for that approach - e.g. van Hiele.

14.3.2. Z.175. Computer environments for learning geometry may make a substantive contribution to such activity-based, conceptual conflict-driven construction of geometrical knowledge.

### 14.4. Logo X.175 onwards.

14.4.1. X.175 Children's representations of space are based on action. The path concept is a good starting point for the study of geometry.

14.4.2. Helps to develop their conceptions of shape. Students are asked to construct rectangles etc which is level 2 descriptive/analytic thinking. Good sentence on external intuitive expectations to become obtrusive and accessible to reflection.

14.4.2.1. Textbooks often fail to do this.

14.4.2.2. X.182. Building autonomy and confidence as the student can think of why something would, or wouldn't work and the computer validates it building confidence.

14.4.2.3. X.178 Past research has shown strongest effects on the meta cognitive ability of monitoring, again this may foster high-level metacognitive awareness that supports memory, especially memory that rebuilds, rather than recalls.

14.4.3. Students work in logo relates to their level of thinking and helps them develop onwards X.175.

14.4.4. X.179 Research supports the hypothesis that work with logo helps students construct more viable knowledge, this is, knowledge schemes in whcih concrete experiences are connected to abstractions with are internally consistent and which can be flexibly applied to a variety of problem situations.

14.4.5. X.183. Connecting with Papert's ideas that turtle geometry are based on personal intuitive knowledge.

14.4.6. X.185. That while students may be taught concepts, they need to make personal connections by doing.

### 14.5. Supposer X.187

14.5.1. Section on this although appears to be for older students. The geometric computer environments can help develop students thinking in geometry. The objects on the computer screen become manipulable representations, a mirror of the students thinking.

### 14.6. X.177 Short comment on being helpful in understanding movement.

### 14.7. Logo environment prompted more discussion. X.178

### 14.8. List of positives to math tech. X. 188 onwards

14.8.1. Elaboration X.188

14.8.1.1. X.188 Students greater elaboration of geometric ideas withing tech, appear to facilitate progressions to higher v.h. levels

14.8.2. Objects as representations of a class X.188

14.8.2.1. X.188 manipulation of screen objects to assist in viewing as a geometric, rather than visual, object.

14.8.3. Viability X.188

14.8.3.1. X.188 Help students construct viable knowledge because students are constantly evaluating a graphical manifestation of their thinking. Also, it is not jsut concept development, but they can apply geometric knowledge to problem solving situations.

14.8.4. Precision X.189

14.8.4.1. Tech demands precision and exactness in contrast to paper and pencil.

14.8.5. Explication X. 189

14.8.5.1. When intuition is translated into a program it becomes more obtrusive and more accessible to reflection.

14.8.6. Personal and Intuitive X. 189

14.8.6.1. Logo's turtle graphics environment encourages students to build geometric knowledge upon personal, intuitive, experiential knowledge.

14.8.7. Mirror thinking X.189

14.8.7.1. Tech environments mirror student thinking. Misconceptions and difficulties can often be masked by other traditional approaches

14.8.8. Ways of thinking X.190

14.8.8.1. Students can test for themselves leading students to move from naive to empirical to logical thinking and encourage students to make and test conjectures. It te4aches children to be mathematicians vs. teaching about mathematics.

14.8.9. Autonomy X.190

14.8.9.1. Aid students autonomy development in learning, rather than seeking authority.

14.8.9.2. New node

### 14.9. Educational implications

14.9.1. X.192 Teacher mediation

14.9.2. X.192 Hands on experiences

14.9.3. X.192 Need time in environments

14.9.4. X.192.Assessment

14.9.5. X.192 Grouping students

14.9.6. X.192 Whole class discussion

14.9.7. X.192 Teacher education and teacher preparation.

## 15. (Y) Goldenberg and Cuoco (1998)

### 15.1. Y.351Describe the term dynamic geometry (DG). Talking about he draggable feature of the program making it different than other geometry software.

15.1.1. Allow to construct and move freely and observe other elements with the altered conditions.

15.1.2. D>G> includes Goemeters Sketchpad, Cabri, Geometry Inventor, and partially the Supersupposer.

### 15.2. Y.354 Interesting point about how we interpret picture. What we choose to see, and what we ignore.

### 15.3. X.357 The D.G. does not provide preselected discrete figures. Instead the students can move the figures and push boundaries of ideas causing disequilibrium which is good.

15.3.1. X.357 the student therefore helps construct definitions which is important -citations.

### 15.4. X.357 The principal contribution of DG to the nature of experimentation is geometry is the dynamic dependency of its display on the position of some movable object.

### 15.5. Y.358 The difference between the Supposer and the DGs. The supposer creates different examples of that figure by creating a triangle with different dimensions. The DG, you can test it on the initial triangle by dragging it into a different shape etc.

### 15.6. Talk about ideas of connecting to theory (obviously older1998) Connects to Honotopy theory.

### 15.7. Y.365 DG is seen as an environment to help students develop the notion of figure attending to underlying relationships rather than to the particulars of a specific drawing.

### 15.8. Y.365 Geometry tech is different than geometry on paper. the process of construction of the drawing involves only action and does not require description.

15.8.1. Good reason to bad drawings Y.365

15.8.2. Y.365 Plastic geometry is better than one that is fixed.

## 16. (T) Laborde, Kynigos, Hollenbrands, Strasser. (2006

### 16.1. T.276 Epistemological perspective

16.1.1. T.276 Both Investigation of axiomatic foundations and also spatial concepts,

### 16.2. T. 276 Historically they did not like diagrams.

### 16.3. T.276 Three cognitive processes involved in geometry

16.3.1. Visualization processes

16.3.2. Construction processes by tools.

16.3.3. Reasoning

### 16.4. T.277 Geometry has two parts, the figural and the conceptual (quote of Fischbein)

### 16.5. T.277 The teaching of geometry is based on theuse of two registers, the register of diagrams, and the register of language.

### 16.6. T.277 The ambiguous role of diagrams is completely implicit in the traditional teaching of geometry in which theoretical properties are assimilated into graphical ones.

16.6.1. T.277 the problem is that the teacher expects theoretical knowledge to construct a diagram where students often stay at the graphical level.

### 16.7. T.277 Researchers stressed the importance of visualization in a geomtery activity.

16.7.1. T.277 It is commonly assumed that the teaching of geometry should contribute to the learning of:-

16.7.1.1. (1) The distinction between spatial graphical relations and theortical geometrical relations.

16.7.1.2. (2) The movement between theoretical objects and their spatial representation.

16.7.1.3. (3) The recognition of geometrical relations in a diagram.

16.7.1.4. (4) The ability to imagine all possible diagrams attached to a geometrical object.

16.7.1.5. These assumptions have lead to a focus on graphical representations provided by computer environments.

### 16.8. T.278Contribution of tech to geo is in the dynamically manipulable interactive graphical representations.

### 16.9. T.278 Papert and constructivist learning and math. Also talking about logo.

16.9.1. T.278 Constructivist perspective - learning is not taken as a simple process of the incorporation of prescribed and given knowledge, but rather as the individuals reconstruction of geometry.

### 16.10. T.278 The past 30 years tech has focued on turtle geometry and DGEs.

16.10.1. Turtle - The main underlying principle was to provide programmability as a means for expressing and exploring mathematical ideas and the joint use of three representational registers: Symbolic programming, graphics, and a notion connection with body movement

16.10.2. T.278 DGEs appeared in the eighties. The main underlying learnig principle was to provide a family of diagrams as representing a set of geometrical objects and relations instead of a single static diagram. To help students see the general apects of a static diagram.

### 16.11. T.279 Noss and Hoyles proposed the Using, Discriminating, Generalizing, and Synthesizing (U.D.G.S) model to describe the conceptualization process of students interacting with technology. Description followed.

### 16.12. T.279 The instrumental perspective developed independently by psychologists in the mid-nineties shares the same idea of the role of the tool on the constructs of the user. Also recently used. Further description followed.

### 16.13. T.280 Developing tech instrument knowledge, may also involve developing mathematical knowledge.

### 16.14. T.280 Two new approaches to geometry and tech. have developed.

16.14.1. 1. Tools and in particular technologies offer opportunities for learning. The subject is faced with constraints imposed by the artifact ad new possibilities of actions to identify to understand and with which to cope. Connecting with the theory of didactic situations.

16.14.2. 2. Following a Vygoskian perspective, operations carried out with technology may be subject to an internalization process with the guidance of the teacher and interpersonal exchanges within the class in the form of collective discussions.

### 16.15. T.280 Teacher role in sharing knowledge culturally.

### 16.16. T. 280 Logo-driven turtle geometry

16.16.1. T.280-281 Parert that while Piaget talked about the shortcomings of children, Papert wanted to look at what children can do when they are offered the right environment to do it in. _ Also look at my paper here for constructionism.

16.16.2. T.281 A good description of what Turtle geometry is.

16.16.3. T. 281 Turtle geo. is based on differential (intrinsic) geometry.

16.16.4. Turtle research is typically focused on the figural products created by the turtle, on the connections students make between the formal programming/mathematical code and the graphical output and how children link experiences of their body movements to the behaviour of the turtle.

16.16.4.1. T.281 Papert named this body connection to the turtle "body-syntonicity"

16.16.5. T.281 There are now over 100 logo type environments.

### 16.17. T.281 Logo based microworlds

16.17.1. T. 281 Definition and the comment about it from Papert - a self contained worlds where students can learn to transfer habits of exploration from their personal lives to the formal domain of scientific construction.

16.17.2. T.282 while Turtle was considered as a microworld within logo, later research was more specific as to what was defined as a microworld.

16.17.2.1. T.282 Gives examples of specific geometrical microworlds.

16.17.3. T. 282 Logo being considered a new type of learning process.

16.17.3.1. T.282 Research on the students learning processes and some aspects of this process emerging from logo geometry environments were recorded. Examples given.

16.17.3.2. T.283 Other research focused on interactions between students and the computer that included the use of: Multiple representations, feedback and editing and constructing./Research on the building of such environments and affordances /Other research using standardized tests or experimental methods -using van Hieles standards.

### 16.18. T.283 Research on geo-tech thinking was influenced by two things - computer access for ordinary people and the constructivist movement. Although they where not a significant focus in math research until the advant of DGE technology.

16.18.1. T.283 Research also focused on the new opportunities for ordinary people (+children) to access computers

16.18.1.1. Leading to a focus on learning strategies not maths.

16.18.1.1.1. led to research on geometry and tech.

16.18.2. T.283 also at that time the constructivist learning movement was also influencing thought at that time. (1980s)

### 16.19. T.283-4 Discussion about formal math and the programs.

### 16.20. T.284 Dynamic Geometry Environments

16.20.1. T.284 Describing DGEs

16.20.1.1. T.284 What is critical about the environments is that they have a quasi-independence from the user once they have been created. The user can drag etc. but geometrical rules have to be followed if the user wants it to or not. - This is different than paper and pencil environments.

16.20.2. T.284 The DGEs link spatial graphical and geometrical aspects together, bridging the gap between theoretical and experimental mathematics.

16.20.3. T.284 There are about 70 DGE environments around the world. - Reported 2006. Although T.285 many are clones and there are otherwise about 10 different ones.

16.20.3.1. T.285 Research on DGEs included Cabri-Geometre, GEOLOG, Geometer's Sketchpad, Geometry inventor, geometric supposer, and Thales.

16.20.4. T.285 DGEs moving from the spatial to the theoretical.

16.20.4.1. Construction tasks

16.20.4.1.1. T.285 Talking about the difficulty in the students connecting between the theoretical and the spatial, or the theoretical /empirical.

16.20.4.2. T.285 The notion of dependency and functional relationships.

16.20.4.2.1. T.285 Talking about the dependency of the fact that the figure when dragged follows geometrical ideas, difficult for students who have not grasped the relationship between the spatial-graphical level and the theoretical level.

16.20.4.3. T.286 the use of drag mode

16.20.4.3.1. A key element of DGEs

16.20.4.3.2. T.286 The mathematical counterpart of drag is variation.

16.20.4.3.3. S.1184 Dragging to test a hypothesis - EXCELLENT CONNECTION TO DIS. about using dragging to figure out angles (acute, obtuse, and right).

16.20.5. T.296 In DG, students actions deal directly with tools producing geometric objects and relations or consist in manipulating dynamic diagrams. Students move from action and visualization to a theoretical analysis of diagrams and possibly to the expression of conjectures and reasoning.

16.20.6. T.288 Constructions

16.20.6.1. T.288 What is done in construction tasks is done by adjusting not only part of the solving process buy they scaffold the path to a definite robust construction.

16.20.6.2. T.288 Healy introduced the distinction soft versus robust constructions to give account of constructions that students could change by dragging in order to satisfy a conditions. More on this..

16.20.7. T.288 Advantages of DGEs

16.20.7.1. T.289 The need in DGE to carry out explicit construction methods based on theoretical properties could lead to consider them as good environments for introducing formal proof.

16.20.7.2. T.289 Students do not often see the need for proofs and the fact that the computer needs it is a good example to show them why.

16.20.7.3. T.289. The explanatory power of proof.

16.20.7.3.1. T.289 Whereas proof is often considered as a means of deciding about the truth of statements, it becomes a means of explanation of phenomena observed on the computer screen that are striking or surprising.

16.20.7.4. T.289 The greater integration of DGE into teaching allows for opportunities to design instructional activities, even sometimes over a long-term period aimed at introducing or fostering deductive reasoning and proof.

16.20.7.5. T.289 Give examples of four studies showing the diverse and novel ways in which proof is provided through DGEs.

### 16.21. T.290 Research trends for tech and geo

16.21.1. T.290 Research has been on ..(listed)

16.21.2. V.40 The visual and spatial interactivity offered by tech means that DGS has perhaps become the best researched area in mathematics education.

16.21.2.1. Personal note - but not with the connection to mobile learning.

16.21.3. T.290 Scarcity of research in loci and use of micro facility.

16.21.3.1. T.290 Technology not fully adopted by teachers - Therefore little research has looked at geometry starting from scratch. Most research investigated the impact of technology on geometry learning for students already introduced to geometrical concepts.

16.21.3.2. T.296 Research should perhaps focus on a novel kind of 3D dynamic and direct manipulation particularly from embodied cognition approach focusing. ---They also go on to talk about the need for new technologies to be focused on.

16.21.4. T.291 Research trends focus on four dimensions (Trouche 2003) 1) Epistemological and semiotic dimension: The nature of geometry mediated by technology 2) Cognitive dimension: Technology supporting learning. 3) Situational dimension: The role of the design of the tasks on learning, and 4) Teacher dimension: The role of the teacher.

16.21.5. T.296 Tech revealed how much the tools shape the matematical activity and lead researchers to revisit the epistemology of geometry.

16.21.6. T.296 Initial research focus was on learner and his/her interactions with tech, then to the design of adequate tasks, then to the role of the teacher. Finally features of software design and how the student manages the tech.

16.21.7. S.1169 The interest in research, new technologies and the impact on learning, teaching and curriculum. S.1172 research also unfolds theories etc and provides a language with which to talk about practice informed by empirical findings and theoretical insights.

16.21.7.1. S.1169 Research informs practice

### 16.22. T.291 The objects offered by technology on the computer or calculator screen are representations of theoretical objects, which behave by following a computerized/mathematical model underlying the software program. There may be some differences between the theoretical and the actual which is called computational transposition.

### 16.23. T.291 Tech and learning of geom. Good section to use.

16.23.1. T.291 Favors learning as students learn by coordinating the reflecting upon the form of their interactions.

16.23.2. T.291 Situated abstraction - the development of the conceptual framework develop by student in such interactions with a computer environment in which they develop. Situated abstractions is related to the exploratory nature of the environments. E.g. Logo TG and DGE allow students to explore and offer a way of accessing the underlying mathematical characteristics of the underlying geometry.

16.23.2.1. T.292 Software tools become extensions of their own thinking. The computational scaffolding contributes to the process of constructing situated abstractions. T.292 more on computational scaffolds.

16.23.2.2. T.292 The concepts of situated abstraction points out the importance of the necessity of a transfer from the computer environment to the world outside the computer.

16.23.3. T.292 The exploratory nature of these environment amplifies the search process of students solving a task and what they actually understand. Noss and Hoyles -It offers a window on the students conceptions and learning.

16.23.3.1. T.292 The tech constrains students actions in novel ways and forces the teacher to notice a students point of view which would not have been noticed with paper and pencil methods.

16.23.4. T.292 Offers a new way of conceptualizing mathematical ideas.

16.23.5. S.1181 Describing student goals - e.g. teacher directed problems connect with the student engaging in exploratory activity and when they figure out what they want to use they are engaging in expressive activity.

### 16.24. T. 292 Design of tasks

16.24.1. T.292 Some researchers also stress that the choice of the tasks in relation to the affordances of the tech geometry environment may be critical of the development of the students understanding.

16.24.1.1. S.1182 It should not be unguided but structured tasks that focus students attention on relevant mathematical notions. Even play should be somewhat structured.

16.24.1.1.1. S.1182 describes the "Play Paradox" where there may be multiple ways of solving the problems, but not in the way the teachers etc would want it to be solved. More examples and descriptions on the play paradox later on this page.

16.24.2. T.292 Relevant combination of tools and problem situations is considered as a good milieu for the emergence of knowledge.

16.24.3. T.292 Arzarello et al 2002 argue that task design and teacher moderation play very important parts in encouraging students to press on beyond perceptual impression and empirical verification in DGE.

16.24.3.1. S.1182 Although off-computer discussions should also be designed to help children explicitly connect with math notions and every day notions.

16.24.4. Pratt and Davison 2003 conclude that IWB and DGS that the visual and kinaesthetic affordances do not help, that they need to be embodied in specific tasks which draw attention to the conceptual aspect.

16.24.5. T. 293 Role of tech in students solving process is multiple. The tools give opportunities that would not be there with paper and pencil.

16.24.6. T.293 Lists four kinds of tasks used by teachers with DGE. USE.

16.24.7. S.1197 mathematical Concordance - Including the point that what students engage in and the objective of the lesson do not always match.

16.24.8. V.41 Tasks should be chosen to be useful, interesting and/or surprising to the students.

16.24.8.1. V.41 It can be helpful if classroom tasks expect students to explain, justify or reason, and be critical of their own and peers explanations.

16.24.8.1.1. V.40 Building of conjectures are best not just at the end of the math activity.

### 16.25. T.293Feedback

16.25.1. T.293 Tech offers feedback to the user. State reasons why feedback is important - check conjectures, look for other solutions, refinement. More on all these.

### 16.26. T.294 Section on teacher use.

16.26.1. T.294 Stress the need for teacher intervention, it cannot happen on its own.

16.26.2. T.294 Talked about teacher attitude.

16.26.2.1. S.1169 limitations on tech due to the old habits and social structures, not being limited by the actual development of technologies.

16.26.3. T.294 Vygotskian approach

16.26.4. T.294 Teacher needs to use all methods not just tech.

16.26.5. S.1191 large section on the role of the teacher. Including roles such as counselor and teacher.

### 16.27. T.295 Logo and DGE belong to the expressive tools category.

16.27.1. T.295Some researchers also consider logo as a tool to forge links between students actions and the corresponding symbolic representations they develop.

### 16.28. T/296 Good close - Tech is still developing at a high pace, DGE and CAS are noteworthy.

### 16.29. T.296 GREAT INTRO or conclusion - studying tech and geo, makes you take into account the complexity of the teaching and learning process. There is a dialectical link between the development of theories and research on the use of technology in geometry and general math ed.. Used available theoretical approaches as well as create new.

## 17. Roblyer book

### 17.1. Great intro about technology on page 319

### 17.2. Graphing calculators 317

### 17.3. Talks about interactive or DGS 322

### 17.4. List of technology integration for mathematics. Not all for geometry.

17.4.1. Virtual Manipulatives 319

17.4.2. Fostering mathematical problem solving 319

17.4.3. Allowing representation of mathematical principals

17.4.4. Implementing data-driven curriculum

17.4.5. Supporting math-related communications

17.4.6. Motavating skill building and practice

## 18. Logo

### 18.1. From Wikipe. Logo is a multi-paradigm computer programming language used in education. It is an adaptation and dialect of the Lisp language. It was originally conceived and written as a functional programming language, and drove a mechanical turtle as an output device. Logo was created in 1967 for educational use, more so for constructivist teaching..... The name is derived from the Greek logos meaning word emphasizing the contrast between itself and other existing programming languages that processed numbers.

## 19. (S) Zbeik, et al. (2007)

### 19.1. S.1170 Two types of mathematical activity: technical and conceptual.

19.1.1. S.1170 Technical dimension of math activity is about taking math actions on math objects or representations of those objects. Procedures can then be built out of sequences of math actions (or out of previously build procedures).

19.1.1.1. S.1170 Tech.math activity is concerned with tasks of mechanical or procedural performance.

19.1.2. S.1170 Conceptual math activity involves understanding communicating and using mathematical connections, structures and relationships. Examples given.

19.1.2.1. S.1170 Conceptual math activity is concerned with tasks of inquiry, articulation and justification.

19.1.3. Affordances of geo tech

19.1.3.1. S.1170 tech. can influence both the technical and conceptual dimensions.

19.1.3.1.1. S.1170 Computer is used for a) gaining insight and intuition, b) discovering new patters and relationships, c) graphing to expose math principles, d) testing and especially falsifying conjectures e) exploring a possible result to see whether it merits formal proof, f) suggesting approaches for formal proof, g) replacing lengthy hand derivations with tool computations, and h) confirming analytically derived results.

19.1.3.2. S.1170 Off-loading, routine computations provides a learning efficiency in terms of compacting and enriching experiences. E.g. computers and calculators providing multiple graphs in a short space of time.

19.1.3.2.1. S.1170 Compacted tech. activity affords an opportunity for enriched conceptual activity. Also, tech that facilitates connectedness and sharing of results heightens those affordances. (I am thinking mobile here.)

19.1.4. S.1170 (Artigue) People often tend to split the two and focus on technichnical activity as entirely mechanical, whereas meaningful learning must be derived from conceptual activity. ABSOLUTELY

19.1.4.1. S.1171 Some think that the tech is it free the student from the technical activity to focus on the conceptual. - But the technical is also important as it is partly conceptual and forms a combination of the two. There is a synergistic relationship where they are both important.

### 19.2. Affordances of geo. tech.

19.2.1. S.1171 Cognitive tools

19.2.1.1. S.1171 tech is different than other math tools (e.g. manipulatives). Tech has special differences and contributions in the activities students undertake, kinds of math activity they engage in, and the math knowledge and understandings they build.

19.2.1.1.1. S.1171 Feedback is important. A cognitive tool must react and respond.

19.2.1.2. S.1171 Pea used the term cognitive technologies. - Those technologies that trenscent the limitations of the mind... in thinking, learning, and problem solving activities.

19.2.1.3. S.1171 Describes what cognitive tools are and that other researchers have categorized them. ALSO, terms are not always used the same to describe the cognitive tools e.g. microworlds and simulations.

19.2.1.3.1. S.1187 Tools designed to show students thinking. Logo makes students thinking explicit...express geo. ideas in the form of a program that represents and "instantiation of students' expression of geometric ideas".

19.2.1.3.2. S.1171 The authors describe cog. tools as both microworlds, simulations, and cognitive toolkits. Description provided of each.

19.2.1.3.3. S.1187 Noss and Hoyles 1196 described programming (logo) as "a tool for expressing and articulating ideas. Illich 1973 described the tools as convivial tools to allow users to express their meaning in action. (Chronological example of people seeing the same value of the tools).

### 19.3. S.1172 Describes what constructs are.

19.3.1. S.1173 Describe three constructs - Externalized representations, Math fidelity, and Cognitive fidelity.

19.3.1.1. S.1173 Externalized representations of math objects provided by or accessed by the cognitive tool and the technical actions on and linkages to those representations available to the user.

19.3.1.1.1. S.1173 students mental ideas externalized, - they can be discussed with others.

19.3.1.1.2. S.1173 Once created, the cognitive tool can act as a "user agent" to perform specific mathematical actions or procedures at their command.

19.3.1.1.3. S.1174 Kaput says that the cognitive tool is best when a dynamic connection is made between 2 representations.

19.3.1.1.4. S.173 the extent to which actions with a physical tool are mathematically meaningful is at the discretion of the user. - It also may be mis-enterpreted by the user.

19.3.1.1.5. S.1174 Also offers rules of engagement for the math activity.

19.3.1.2. S.1173 Mathematical fidelity refers to the faithfulness of the tool in reflecting the mathematical properties, conventions, and behaviors (as would be understood or expected by he mathematical community).

19.3.1.2.1. S.1174 Describe how it must be faithful to the underlying math properties.

19.3.1.2.2. S.1174 Describe how tech may be different than expected math and offers different interpretations (not sure if this is good or bad). A few pages of examples given but appear to be high school. S.1176 Bit of a summary at the end of this section.

19.3.1.3. S.1173 Cognitive fidelity refers to the faithfulness of the tool in reflecting the learners thought process or strategic choices whle engaged in mathematical activity.

19.3.1.3.1. S.1176 The appearance of the drawings etc have to match what the learner is thinking. S.1177 another good summary of why it is important.

### 19.4. S.1178 Section how how the student use the tools.

19.4.1. S.1178 Instrument does not exist in itself, it becomes one when it is used. Not until a relationship is developed does the artifact become a "user-agent".

19.4.1.1. S.1179 Instrumental genesis includes both the user shaping the tool for her or his purposes (instrumentalization) and the users understanding being shaped by the tool (instrumentation).

19.4.2. S.1178-1183 large section not used talking about student use of tech tools and their goals

### 19.5. S.11841187 Large section about students' technology behaviors. Not included in this map as it is not relevant to the comps. Does have a significant sections on positives and also negatives.

### 19.6. S.1192 Tech and curriculum - Tech allows access to content that had not previously been included in the school curriculum.

19.6.1. S.1192 Representational fluency - the interaction between the students and presentation - such as being able to move between representations to draw meaning, and generalize.S.1193 more on representational fluency.

### 19.7. S.1199-1200 Amplifiers and Re-organizers (Pea) Described with examples.

### 19.8. S.1200 Sequencing and emphasis - Talk about a concern with calculator use, then a discussion on micro procedures and macro procedures and studies looked at conceptual and procedural understandings after focusing on different ones.

## 20. (U) Yu, Barrett, & Presmeg (2009)

### 20.1. U.110 Great intro that can be used to describe how geom tech is different and the general affordances it provides. e.g. visualization, manipulation, and discussion.

### 20.2. U.110 Category membership and prototypes - Important curricular and cognitive activity is categorizing geometric shapes according to their properties.

20.2.1. U111 Cognitively this can be difficult. Talk about square rectangle issue. Then the PROTOTYPE EFFECT, that when identifying shapes, some are deemed more representative of that category.

### 20.3. U.111-112etc. Large section on prototypes

20.3.1. U.112 Two different types of prototype. There are prototypes based on a shapes form or properties. The second type of prototype is based on operations enacted on the shape.

### 20.4. U.112 Interactive geometry software has the additional capability of nonisometrically morphing and reshaping certain objects to build relationships with other objects.

### 20.5. Examples of research/activities which influenced:

20.5.1. U.113 Prototype development.

20.5.2. U.119 Processes involved in prototype construction.

### 20.6. Shifting roles of classroom discourse and teachers interactions. U.123

20.6.1. Effective instruction should allow the flexibility and freedom for students to interact and use their own initiative to examine the objects and relations among them.

20.6.1.1. U.123. The noticed that -the Motion-based nature of manipulating the on-screen shapes helped facilitate a more robust, property-based understanding of the geometrical shapes. Their (research students) experiences with the shapes makers expanded their prototypes and mental models for the various quadrilaterals.

20.6.2. U.123. Effective instruction should allow the flexibility and freedom for students to interact with one another and have a common context to share their ideas. Plus more about different types of discussions.

20.6.3. U.124 Effective instruction should include meaningful interactions with the teacher. Teacher guidance is important and should be deliberate and precise while being open to foster studens generation of ideas. Teachers should expose students to proper terminology, make connections between concepts and raise questions to encourage investigations that will help dispel misconceptions that may arise.

20.6.4. U.124. Effective instructional design should be deliberate. Based on the open imagination of the teacher, but also on their experiences, their content goals, and their understanding of how students learn.

## 21. (V) Jones (2011)

### 21.1. V.40 Ubiquitous technologies have been developed which have placed increasing focus on exploiting learners' visual and spatial intuition. This directly connects with geometry so it needs to be considered in regards to these new affordances for teaching and learning.

### 21.2. V.40 While tech has considerable potential in enlivening the teaching and learning fo school geometry, there is much to take account of in terms of enabling this potential to be fully realized.

### 21.3. V.40 It can be a while before people see the benefits of DGS... as they have to learn to use the software and what it can do.

### 21.4. V.41 Mention about tasks - put below.

### 21.5. V.41 Strange thinking about learning 3D on a 2D device. Questions how direct "direct interaction" is.

### 21.6. V.42 Importance of research on suitable teacher professional development.

21.6.1. V.42 Describes the teacher -demonstration approach to teaching students. Where the teacher demonstrates and the students watch and then answer questions.

21.6.2. V.42 Another teaching approach involves the teacher giving previously created interactive files for the students, there the students can experiment with dynamic objects.

21.6.3. V.42-3 Students create their own files for other learners to tackle.

21.6.4. V.42 Jones has a diagram of the different teaching ideas.