Week+4+-+Adventuring+into+space+&+geometry

__**Learning Activity 4.1:** //**Shapes from Squares**// **(Part 1)**__



The shapes that were created with 5x5 piece of paper is:


 * Square
 * Diamond
 * Triangle
 * Rectangle

Achor 17/06/2011

Rectangle

Triangle

Square

Diamond

Right angled triangle

Vicki 20/6/11


 * **Square **
 * **Rectangle **
 * **Triangle **
 * **Diamond **
 * Sarah Wright - 21.6.2011 **

Rectangle, Triangle, Diamond, Square Adam T 22.06

rectangle, square, triangle, quadrilateral, pentagon, hexagon, heptagon - Suzanne W 24/06/2011

__**Learning Activity 4.2:** //**Shapes from Squares**// **(Part 2)**__


 * How are conceptual understanding and language developed in this lesson?

Children's exploration of manipulatives allowed them to form new conceptions about shapes through concrete experiences.

The teacher required the children to write down their thoughts on paper aiding the growth of conceptual knowledge.

The teacher went to the individual children and scaffolded their conceptual understandings on a personal level. For example, when the teacher worked with the child having misconceptions regarding what a side was.

Conceptual understandings were further developed at the end of the lesson when the children were again grouped together but this time to look at what shapes were discovered. The teacher guided the children's knowledge of the various shapes and their sides.

Language in the children was developed when the teacher was instructing the whole class to follow his instructions.

Children were provided an opportunity to work in groups as they explored their manipulatives. While in groups, this encouraged them to talk to each other about the activities they were working through. By doing this, children would create new conceptions and remove old ones that were found not to be useful.


 * How did Mr. Ramirez promote students’ reflection on their own ideas?

The teacher went to the individual children and questioned their thinking in order to promote students' reflection on their own ideas. For example, the child who received individual help by the teacher was engaged in reflection as he realised he needed to dispose of an old way of thinking.

The promotion of student reflection by the teacher was also seen at the end of the lesson where all the children came together to evaluate what they had just been participating in. Through the teacher's considerate choice of questions and the use of models placed in front of the students, the teacher provided the children time to reflect on what they had done and what the teacher wanted them to achieve. - CHORY TYRRELL 15/06

How are conceptual understanding and language developed in this lesson? In this lesson the teacher showed the students that set shapes can be manipulated into other shapes. The student’s concept on shapes changed when they were folding the piece paper and creating their own shapes following the creases. The students can now perceive shapes in a different way now that they know that shapes can be manipulated to form a different shape. How did Mr. Ramirez promote students’ reflection on their own ideas? Reflection in the classroom is often done through questioning. The teacher asked the students individually on their concept of the shapes and how they manipulated the paper to match the shapes on the board. This method allowed the students to share the discoveries so the students that didn’t grab the concept can understand how it was done. It was a very effective method to get students to reflect on what they have learned. Achor 17/06/11

//How are conceptual understanding and language developed in this lesson?//


 * Children can identify how shapes are formed and the relationship between them. Through this exercise children can see the relationship between shapes with straight sides, and can extend this knowledge to see that circles do not have any straight sides. By using shape names, and terms such as sides, folds, corners etc in discussion, students learn the meaning of mathematics language and how it can be applied.

//How did Mr. Ramirez promote students’ reflection on their own ideas?//


 * Challenged individuals to prove ideas and apply learnings to other ideas / concepts (i.e discussion of sides Vs fold). Group discussion, sharing ideas of shapes students found, how they found that shape, what they think it is called and why. Counting number of sides together.

Vicki 20/6/11

**How are conceptual understanding and language developed in this lesson?**

**Students are able to develop their conceptual understanding through the use of manipulative’s which enables them to physically understand the shapes being examined and their relationship to one and another.(Booker, et. al, 2010, p.397) describes that mathematical learning occurs when there is activity with dialogue. This combination is the key to conceptual understanding, a combination of dialogue and the opportunity to view, experiment and create will ignite deeper learning. Being able to have conceptual understanding is very different from memorisation of information as it will allow for recall later on, the information will be able to be applied to new information and help form new ideas and understandings as the lessons become more complex. Language is also developed in this lesson as Mr Ramirez makes his students write in words their findings. This will deepen their learning as they are able to find relationship between language and images, putting names to picture. Mr Ramirez also allows time for questions and class discussion. Students are encouraged to show the class what they have learnt which confirms further their understanding of the lesson.**

** Booker, G., Bond, D., Sparrow, L., & Swan, P. (2010). Teaching primary mathematics. Frenchs Forest, NSW:Pearson Australia. **

**How did Mr Ramirez promote students’ reflection on their own ideas?**

**Students were put into pairs during the exercise to give them the opportunity to discuss and work out with their partner if they are on the right track. Mr Ramirez walked in between pairs and discussed their findings, encouraging them to right their findings down. Finally Mr Ramirez finished the lesson with a group discussion, where students shared their findings and Mr Ramirez went through each shape which was found to clarify to the students the results of their experiments.**

**Sarah Wright 21.6.2011**

**How are conceptual understanding and language developed in this lesson?** Students were able to build on simple shapes to explore and understand the shapes being created and how folds of paper can create more complicated shapes. Students learned how shapes related to one another firstly on an individual basis. Then the students were paired with a buddy to explore, discuss and document their findings to promote their thinking processes and skills. Finally the class sat together as a thinking group and presented their findings about the differnet shapes they created and named.

**How did Mr Ramirez promote students’ reflection on their own ideas?** Mr Ramirez questioned students as to how they came to calculating the number of sides to the shapes and encouraged them to write down the shapes they had discovered. He prompted discussion when a student experienced difficulty visualising sides v's creases. Mr Ramirez used a simple sqaure and ask the student to relfect on how many sides it had and compared the students hand and asked him to indicate where the sides and creases of his hand were. This helped the student understand the differences between sides and creases. Afterwards the class was asked to reflect as a group and discuss their findings. Adam T 22.06

Conceptual understanding and language developed in the lesson by: How Ramirex promoted student reflection on own ideas: Suzanne W 24/06/2011
 * Ramirez providing scaffolding to acknowledge on concept (ie hexagon has six sides) and further build on prior knowledge (that a hexagon is not always a regular hexagon - it is simply a shape with six sides).
 * By getting the children to think outside of the sqaure. Given a piece of paper with a few creases and few instructions to increase their concept of what shapes appear like
 * Scaffolding. Ramirez in talking to the young boy provided scaffolding for the boy to reach his own conclusion through discussion. To build his knowledge on a social level before
 * Group discussion in pairs and also as a class

__**Lea****rning Activity 4.3: Some mathematics learning and teaching challenges**__

//Maths Challenge 1:// Is a square a rectangle? Why or why not? A rectangle is any quadrilateral with four right angles. A square has __four equal sides__ and **four right angles** and so is a rectangle.

//Maths Challenge 2:// Many people use the term ‘diamond’ for certain mathematical shapes. How is this word appropriate or not appropriate in a mathematical context?

Diamond is another name for a rhombus. It is appropriate in a mathematical context because it has a basis. - CHORY TYRRELL 15/06

//Maths Challenge 1:// Is a square a rectangle? Why or why not?

A square is not a rectangle because a square has 4 equal sides and if rotated it can form a diamond where if you rotate a rectangle it will still be a rectangle.

//Maths Challenge 2:// Many people use the term ‘diamond’ for certain mathematical shapes. How is this word appropriate or not appropriate in a mathematical context?

Diamond is a well-known term for the Rhombus shape especially amongst younger students. So yes it is appropriate in that it recognized widely. Achor 17/06/2011

//Maths Challenge 1:// Is a square a rectangle? Why or why not?

Although we think of square and rectangle as different shapes, a square can be classified as a rectangle. A rectangle has four sides, where each angle is a right angle. The square meets this definition.

//Maths Challenge 2:// Many people use the term ‘diamond’ for certain mathematical shapes. How is this word appropriate or not appropriate in a mathematical context?

The term diamond it used to describe the way a shape looks, but mathematical terms for such shapes are often different. Often, a diamond is a rhombus or a quadrilateral. For students to understand geometry, the correct language should be used.

**Learning Activity 4.3**

**Maths Challenge 1: No a square is not a rectangle. A square has 4 equal sides, where as a rectangle has 2 long sides and 2 short sides.** **Maths Challenge 2: Diamond is the term given to a shape also known as a rhombus. Diamond is a familiar term for children and may be used until they learn proper names of shapes.**

<span style="color: #ff5f00; font-family: Arial,Helvetica,sans-serif;">**Sarah Wright**

**Maths Challenge 1: Is a square a rectangle? Why or why not?** A square is not a rectangle as it has four equal sides. A rectangual can fit into a sqaure twice as it has two equal lenths which are long and two equal lengths which are short.

**Maths Challenge 2: Many people use the term ‘diamond’ for certain mathematical shapes. How is this word appropriate or not appropriate in a mathematical context?** I think children are quite capable to remember two different words to describe the same shape eg: diamond and rhombus. For example, my 2 year son, knows that a circle, is a ball, is a bubble etc. Adam T 22.06

Square a rectangle? My brain wants to say no, but yes, it is. A square is a four equally sided object with 90 degree angles. However, a rectangle is a four sided object with 90 degree angles. Technically, a square therefore is a rectangle Diamond an appropriate term? Dependant upon what level of learning the student is at, yes, it is appropriate. Booker, Bond, Sparrow and Swan (2009) discuss how it is necessary for children to use everyday terms that they are familiar with and to introduce correct mathematical terms at a later stage. This would be an instance of everyday language. There are better terms for the shape than diamond, however, if this is a term the student is familiar with, the more accurate term can be introduced with scaffolding at a later stage.


 * __Learning Activity 4.4: An overview of geometry in primary school__**

CHORY TYRRELL 15/06
 * <span style="color: #ff0000; display: block; font-family: 'Times New Roman',serif; text-align: justify;">**Ideas and examples that are** //**familiar**// **to me from my own experiences with mathematics learning and teaching** || <span style="color: #ff0000; display: block; font-family: 'Times New Roman',serif; text-align: justify;">**Ideas and examples related to mathematics learning and teaching that are** //**new**// **to me** ||
 * * The 'geo' in geometry is the Greek word for earth.
 * Geometry has many applications in the real world.
 * Geometry, also known as 'shape and space'. The study of space. || * 'Metron' means a measure in Greek
 * It should be taught practically to children so they can explore the concepts of geometry themselves
 * Geometry like other strands of mathematics, it should be taught with the aim of students obtaining understanding and reasoning so they can problem solve and conceptualise rather than learn in isolation the facts and rules.
 * Visual (informal) geometry and formal geometry
 * The language of geometry can be confusing to children as some terms seem to contradict each other like the naming of polygons and quadrilaterals.
 * Ways of working in geometry: visualising spatial arrangements, communicating orally and in writing, drawing and making models, thinking logically, applying geometrical concepts and knowledge. ||
 * **Questions I have and related things I do not understand from my reading** ||
 * None. ||

**//Achor 17/06/11//** <span style="display: block; height: 1px; left: -10000px; margin-bottom: 0cm; overflow: hidden; position: absolute; top: 0px; width: 1px;">Square - Geometry includes shapes and space. - Discussion is a useful teaching strategy to allow students to construct own learning. - Increased examples of an idea will help children develop a broader view. || - “geometry” is derived from two greek words meaning earth & measure. - Geometry also relates to measurement and spatial sense. - 2 aspects: Visual geometry & formal geometry. - “Language of geometry plays vital role in developing spatial sense” (p. 396). - Polygons = many angled shape - Teachers need to be aware of the terminology used and at first use children’s language before refining it. More important to understand concepts than remember terminology. - “Geometry is not about classifying and naming shapes, must be able to discriminate among shapes, visualise changes and apply geometric ideas to problem solving” (p. 399) - Five ways of working in geometry, which should be emphasised in teaching (p. 400). ||
 * **Ideas and examples that are //familiar// to me from my own experiences with mathematics learning and teaching** || **Ideas and examples related to mathematics learning and teaching that are //new// to me** ||
 * * The earliest concept of geometry was developed by the Greeks.
 * Spatial sense is our own perception on objects, distance and shapes. || * Manipulative are used in class to represent concepts. This can shatter a child’s original perception of things and allow them to construct new knowledge through experimentation.
 * Mathematics help develop process knowledge through interaction with objects, materials and other learning aids. ||
 * **Questions I have and related things I do not understand from my reading**No questions =) ||
 * **Ideas and examples that are //familiar// to me from my own experiences with mathematics learning and teaching** || **Ideas and examples related to mathematics learning and teaching that are //new// to me** ||
 * - Geometry activities should be fun and engaging.
 * **Questions I have and related things I do not understand from my reading** ||

Vicki 20/6/11

<span style="color: #ff5f00; font-family: Arial,Helvetica,sans-serif;">**Learning Activity 4.4** <span style="color: #ff5f00; font-family: Arial,Helvetica,sans-serif;">**Ideas and examples that are familiar to me from my own experiences with mathematics learning and teaching:**
 * <span style="color: #ff5f00; font-family: Arial,Helvetica,sans-serif;">**The use of drawing and model making in geometry to enable the student to see what they had visualised**
 * <span style="color: #ff5f00; font-family: Arial,Helvetica,sans-serif;">**Using various manipulative's such as plastic shapes to join together and create further shapes.**

<span style="color: #ff5f00; font-family: Arial,Helvetica,sans-serif;">**Ideas and examples related to mathematics teaching and learning that are new to me**

<span style="color: #ff5f00; font-family: Arial,Helvetica,sans-serif;">**- Recognition/visualisation** <span style="color: #ff5f00; font-family: Arial,Helvetica,sans-serif;">**- Analysis** <span style="color: #ff5f00; font-family: Arial,Helvetica,sans-serif;">**- Ordering/informal deductions** <span style="color: #ff5f00; font-family: Arial,Helvetica,sans-serif;">**- Formal deduction** <span style="color: #ff5f00; font-family: Arial,Helvetica,sans-serif;">**- Rigour/metamathematical**
 * <span style="color: #ff5f00; font-family: Arial,Helvetica,sans-serif;">**The Van Hiele Theory – while there are different interpretations of these levels, this is a good scaffold.**

<span style="color: #ff5f00; font-family: Arial,Helvetica,sans-serif;">**Sarah Wright**
 * <span style="color: #ff5f00; font-family: Arial,Helvetica,sans-serif;">**The content for geometry can be organised 1) Shape and Structure, 2) Transformation and symmetry, 3) Location and arrangement**

**Ideas and examples that are familiar to me from my own experiences with mathematics learning and teaching:**The drawing of shapes to understand what each one looked like - use of identification.We were also encouraged to shade the shapes in order to show depth and understand the difference between 2D and 3D shapes.Use real examples to understand the relationship between the shape and everyday objects eg: paper towel rolls = cylinder and match boxes = rectangle etc

**Ideas and examples related to mathematics learning and teaching that are new to me:** I nformal (visual) v's formal geometry. Stendents should understand basic fundamentals of geometry before using / learning the correct terminology.Students need to understand and conceptulize geometry rather than memorising facts. They need to record, write, speak, visualise, drew and create geometric objects to gain a wider understanding Adam T 22.06

Ideas/examples familiar to my learning: New learning for me: Suzanne W 24/06/2011
 * Need for scaffolding
 * Need for use of tactile materials
 * Need to encourage students in creating their own learning
 * Geometry is shapes
 * Exactly what is included in geometry
 * That maps are considered to be a part of geometry
 * That it is okay for children to use everyday language rather than correct mathematical terms

// How are conceptual understanding and language developed in this lesson? // <span style="background-color: #ffff00; font-family: 'Calibri','sans-serif'; font-size: 14px;">After watching the complete video //Shapes from squares (# 20// []) discuss with your MathMates the above question. <span style="display: block; height: 1px; left: -10000px; margin-bottom: 0cm; overflow: hidden; position: absolute; top: 0px; width: 1px;">Rectangle If everyone could please have their suggestions in for this week's topic by Friday night, that would be brilliant.I will write it up on Saturday and put it on here for everyone to have a look, make any changes on Sunday and also post to DB on Sunday. Thanks, Suzanne.
 * __FINAL SUBMISSION FOR WEEK 4, COMPILED AND COMPLETED__**

<span style="display: block; height: 1px; left: -10000px; margin-bottom: 0cm; overflow: hidden; position: absolute; top: 0px; width: 1px;">**<span style="color: #ff5f00; font-family: Arial,Helvetica,sans-serif;">Triangle **
 * <span style="color: #ff5f00; font-family: Arial,Helvetica,sans-serif;">Hi Suzanne, **
 * <span style="color: #ff5f00; font-family: Arial,Helvetica,sans-serif;">Here is what I have done so far.. hope its an ok starting point, as I am still a little unsure that I have conveyed how conceptual understanding in particular is developed. Anyway, good luck! Sarah Wright **
 * <span style="color: #ff5f00; font-family: Arial,Helvetica,sans-serif;">How are conceptual understanding and language developed in this lesson? **
 * <span style="color: #ff5f00; font-family: Arial,Helvetica,sans-serif;">Students are able to develop their conceptual understanding through the use of manipulative’s which enables them to physically understand the shapes being examined and their relationship to one and another.(Booker, et. al, 2010, p.397) describes that mathematical learning occurs when there is activity with dialogue. This combination is the key to conceptual understanding, a combination of dialogue and the opportunity to view, experiment and create will ignite deeper learning. Being able to have conceptual understanding is very different from memorisation of information as it will allow for recall later on, the information will be able to be applied to new information and help form new ideas and understandings as the lessons become more complex. Language is also developed in this lesson as Mr Ramirez makes his students write in words their findings. This will deepen their learning as they are able to find relationship between language and images, putting names to picture. Mr Ramirez also allows time for questions and class discussion. Students are encouraged to show the class what they have learnt which confirms further their understanding of the lesson. **


 * <span style="color: #ff5f00; font-family: Arial,Helvetica,sans-serif; font-size: 10pt;">Booker, G., Bond, D., Sparrow, L., & Swan, P. (2010). Teaching primary mathematics. Frenchs Forest, NSW:Pearson Australia. **

Hi Suzanne, here is my contribution ... Children are able to identify how some shapes are formed and the relationship between them. Through Mr Ramirez’s exercise, children can see the relationship between shapes with straight sides, and can extend this knowledge to see that other shapes, such as circles, do not have any straight sides. By using shape names, and terms such as sides, folds, corners etc in discussion, students learn the meaning of mathematics language and how it can be applied. However, Booker, Bond, Sparrow, and Swan (2010) note the importance of allowing students to first use their own mathematics language if needed to help understand concepts. Booker, G., Bond, D., Sparrow, L., & Swan, P. (2010). //Teaching primary mathematics// (4th ed.). Frenchs Forest, NSW: Pearson Australia.

Vicki 23/6/11

Hi Suzanne, here is my contribution to the weekly question.

The children are actively involved in the learning activity. They are given a square piece of paper (a manipulative) to help them understand the concept of shapes and the relationship between the various shapes being made. According to Reys, Lindquist, Lambdin and Smith (2009, p. 28), "learning occurs best when students have a meaningful context for the mathematical knowledge and understand fundamental relationships associated with the knowledge". The children are further encouraged to make sense out of what they are doing by drawing the shapes and naming them on a sheet of paper. The children are also in pairs for this activity. This in accordance with Vygotsky's theory that learning is a social experience and intereaction with others challenges learners to make sense of new ideas (Reys, 2009). The teacher also interacted with the students individually and helps one of the students figure out that a fold is not a side. This student will be more likely to remember this as he figures this out himself. Towards the end of the lesson students were able to share their learning with the class and observe what other students have done. This can help them make sense of what they have learned.

__References__

Reys, R., Lindquist, M., Lambdin, D.V., & Smith, N.I. (2009). Helping children learn mathematics. New York: John Wiley & Sons.

Kerrie - 23/6/11. <span style="color: #808000; display: block; height: 1px; left: -10000px; margin-bottom: 0cm; overflow: hidden; position: absolute; top: 0px; width: 1px;">Diamond

<span style="color: #008000; font-family: Arial,Helvetica,sans-serif;">Conceptual uinderstanding and language was developed through scaffolding of ideas on a individual, paired and finally in a group setting. Students were able to build on simple shapes to explore and understand the shapes being created and how folds of paper can create more complicated shapes. Students learned how shapes related to one another firstly on an individual basis. Then the students were paired with a buddy to explore, discuss and document their findings to promote their thinking processes and skills. Finally the class sat together as a thinking group and presented their findings about the differnet shapes they created and named. Mr Ramirez questioned students as to how they came to calculating the number of sides to the shapes and encouraged them to write down the shapes they had discovered. He prompted discussion when a student experienced difficulty visualising sides v's creases. Mr Ramirez used a simple square and asked the student to relfect on how many sides it had and compared the students hand and asked him to indicate where the sides and creases of his hand were. The students were encourage to record their findings through writing and through verbal discussions. This helped the student understand the differences between sides and creases. Afterwards the class was asked to reflect as a group and discuss their findings. Students were able to adapt their strategies to other childrens thinking providing them with the opportunity to develop and experience further exploration og geometry as part of their on going learnings. Adam T 22.06

(No presenter). (1997). //Shapes from Squares.// Teaching Math. A Video Library, K-4. Annenberg Media. Retrieved June 22 from: http://www.learner.org/vod/vod_window.html?pid=888


 * Allrighty then - this is what I have come up with. Critique away - particularly the reference for the video. I've done it as a web streaming video in the APA guidelines, but I'm not convinced it's right. Suzanne. **

In the lesson, “Shapes from Squares” (Annenberg Foundation, 2011) shown with Mr Ramirez, students have their conceptual understanding and language developed through the use of a manipulative. The manipulative, a square piece of construction paper is used to create shapes without creating new folds other than those already within the piece of paper. This exercise is performed in small groups, followed by a class discussion. In undertaking the exercise in such a manner, Ramirez stretches the students pre-existing ideas of shapes and encourages the use of correct mathematical language.

Booker, Bond, Sparrow and Swan (2010) discuss the benefits of using manipulatives in order for students to have a physical experience to create their own learning experiences. By placing students in small groups to encourage discussion, Ramirez allows students understanding of the concepts of shapes to be further developed. This is in accordance with Vygotsk’y theory that learning is a social experience (Reys, Lindquist, Lambdin & Smith, 2009).

Booker, Bond, Sparrow, and Swan (2010) note the importance of allowing students to first use their own informal language if needed to help understand mathematical concepts. This is seen as students create irregular pentagons and hexagons and not know how to refer to the newly created shapes, other than as five sided or six sided. With the use of scaffolding Ramirez encourages students to question and challenge their existing understanding of what shapes should look like. In asking probing questions such as “What is a five sided shape called?” (Annenberg Foundation, 2011) and “How many sides does the shape you made have?” students can answer their own questions confidently and verbally come to a new understanding of how a pentagon may appear. In doing so Ramirez has taken the students existing conceptual of shapes and further developed their knowledge.

Ramirez further solidifies student’s mathematical language in asking students to create diagrams with the correct names of the shapes made. In talking to different students throughout the lesson, Ramirez does not correct students in their lack of formal mathematical speech, but simply uses the correct name of shapes in his own sentences. In doing so Ramirez is seen furthering the mathematical language development of students without making students feel that their original description was incorrect.

Ramirez then consolidates the learning of students by bringing all the students back together as one class. In the class group setting Ramirez is seen asking students about the shapes they created, the names of the shapes, and how many sides the shapes have. By undertaking this action Ramirez is consolidating for students their learning of what shapes are, both regular and irregular. Ramirez again in this instance uses correct mathematical terms for shapes, which allows students to hear and consolidated earlier learning.

With Ramirez challenging student’s prior conceptual understanding of what makes a particular shape, students conceptual learning is developed. The mathematical language of student’s is also developed through the use of group discussion, both in small and large group settings. Ramirez uses teaching methods that students are comfortable with while stretching pre-existing ideas of what makes a shape.

Reference:

Annenberg Foundation, (2011). //Shapes from Squares,// [Streaming Video]. Retrieved from http://www.learner.org/resources/series32.html

Booker, G., Bond, D., Sparrow, L, & Swan, P., (2010), //Teaching Primary Mathematics,// (4th Ed.). Frenchs Forest, New South Wales : Pearson

Reys, R., Lindquist, M., Lambdin, D., & Smith, N., (2009). //Helping Children Learn Mathematics,// (9th Ed.). Hoboken, New Jersey: John Wiley and Sons

<span style="display: block; height: 1px; left: -10000px; margin-bottom: 0cm; overflow: hidden; position: absolute; text-align: justify; top: 0px; width: 1px;">** What other shapes can be made from this material? **