Module 14

Circumference and Diameter Video:

Describe Ms. Scrivner's techniques for letting students explore the relationship between circumference and diameter. What other techniques could you use?

Ms. Scrivner had students to review their definition's's of the words circumference and diameter as a whole group.  She also had students to compare the world circle to the term circumference as a way of having students to see their similarities.  She had students to realize that both words were measurements that could be used when working with a circle but that the diameter was the distance across a circle where as the circumference was the distance around the outside.
In essence, students in this lesson were learning about the ratio of the circumference to the diameter. Compare how students in this class are learning with how you learned when you were in school.

I was actually surprised to see the ratio of the circumference to the diameter used in this video.  I do not recall learning about the ration between circumference and diameter when I was in school but in learning about in within this video it actually made sense to me.   I remember learning about diameter and circumference by being given a definition, a demonstration, and then exploring on our own.
How did Ms. Scrivner have students develop ownership in the mathematical task in this lesson?
Ms. Scrivner had students to develop ownership in the mathematical task in this lesson by having them to select their own objects that were circular that they had to find the measurements on and then share with the class.  For example: the one group who measured this boys head learned that was incorrect because the head was not in fact circular but in an ellipse shape.  

How can student's understanding be assessed with this task?

Students understanding can be assessed from this task if: 

     -They chose a circular object.

     -If the circumference and diameter are close to a 1:3 ratio.

     -If the students know the difference between circumference and diameter.

For further Discussion:

1.  Pentomino Activities: are something I would use in my own classroom one day.  Although I struggled with using them myself during the lessons I feel like it would be a good learning experience for myself and my students.   Students like hands on activities that teach them and I could see students leaning through the pentomino activities.

2. Nets: I would use nets when teaching about 3-D figures and 2-D figures.  I like the usage of nets when teaching about 2-D and 3-D figures because students are able to see how a 2-D figure can form a 3-D figure and are able to see how a "flat object" can be transformed.

3. Spatial Reasoning Activities from Annenberg: I would use these activities on Spatial Reasoning in my classroom because I feel like the concept of spatial reasoning is taught better through these actual activities than I know how to teach them myself.  I struggled with the concepts of spatial reasoning and I feel like in having an interactive model such as the Annenberg models that the students will gain more knowledge.

4. Quick Images: this is a concept that I saw introduced in this class and one I actually found to be rather interesting.  In showing the picture to the students for a brief amount of time it makes them have to think about what the object is and try and recreate it.   I like this activity also because it allows students to compare the shape(s) they were shown to real life objects.  Students come up with all sorts of explanations as to how they remembered the shape which is quite interesting.

5. Coordinate Grids:  I loved the way coordinate grids were used during module 11.  The website which included all of the different activities to teach about the coordinate grids I really enjoyed and learned a lot from and I could see students doing the same.  Interactive games online I have noticed to be beneficial because students are having fun while learning concepts.

Circles and Pi:

When on Problem A1 I was able to find the diameter and perimeter of the hexagon but I struggled when it came to finding the perimeter of the square as well as the circumference of the circle.

When on Problem A2 and A3 I struggled with pretty hard but then I looked at the solutions and when I read the solutions for both problems the concepts made more sense and I felt rather silly that I did not know how to figure them out at first. 

Throughout the rest of session A I would get parts of the problems correct and then get stumped.  I would then go to the solution and look at it and read through it and they would all make more sense to me than when I tried them originally.  

In session B I found it neat how the circle was able to be cut up and formed into the shape of a rectangle.  I had never seen this before and was shocked to see it now.  I worked through the problems in session B and did good on the first 3 but the last 7 I struggled with.  I enjoy there being solutions to look at because it makes it where if you do not understand at first you will afterwards. 

Module 13

Angles Video and Case Studies:

        In Nadia’s case 14, Martha looked at the triangles that she and thought that when you cut it in half you only came up with two angles so there could only be two angles in the triangle.  Also, she could have not been looking at the fact at where the lines meet on the triangle in the three different places is where an angle is actually formed.  Alana was explaining an angle by the slant of a line.  She understood that lines needed to work at a slant somewhat in order to actually form an angle but she was not completely grasping the concept behind an angle.

     

        In Lucy’s case 15 (line 251), Ron suggests that a certain angle “can be both less than 90˚ and more than 90˚": Ron was explaining how angles can be both 90 degrees or over 90 degrees but when he explained it as a “certain angle” can be both less than or greater than 90 degrees that is where he was incorrect.  He understood you can have angles over and under 90 degrees but did not explain how that changes the name of them.  

     

     In Delores' case 13: the students wrote their different ideas of what the angles were:   

Chad: explained small and large angles in the terms of long and short.  He understood that there were different types of angles that varied in size but I would want for him to actually know the names of the angles as well as a way to actually show them versus the way he explained them.

Cindy: explained angles in a different way to me.  The term slanted was one that was okay with me because when looking at angles they are slanted.  I did not so much like the comment about angles going down.  I feel like angles can be formed in moving in either direction.  I would want Cindy to be able to better clarify what she meant when describing angles.

Nancy: I like the way Nancy described angles because she used the actual terms that go along with angles.  She was able to explain that little angles are acute angles and obtuse angles are larger angles.  The only thing I would want Nancy to further explain would be what makes an acute and obtuse angle.

Crissy: did a great job at explaining angles to me.  She had the names as well as the definitions about what angles truly were.  The only thing that she could have done different was to draw a picture. 

Chelsea: I like how Chelsea included pictures and explained right angles.  The thing I wish she had done different was to draw all kinds of angles.  

  In Sandra’s Case 16: Casey says of the pattern blocks, “They all look the same to me.", Casey believes at first that there is only one kind of angle and that is a 90 degree angle.  As he continues to work with his teacher on the angles he realizes that there are actually other types of angles that are within shapes.

I found the Angle's video to be very interesting in listening to the students different explanations of what the believed an angle to actually be.  I liked watching the one little girl draw the pictures when describing her angles and I also liked how she gave the description's of what she was talking about as she was drawing.  The students in this video for the most part to me seemed to really understand angles and I feel like were more advanced when discussing angles than the students that were in the case studies.  In my classroom I want to allow students to draw images along with giving me an explanation.  Seeing the students give me a visual helps me to better understand if they are grasping the concepts or not.  

Annenberg Angles Module:

The usage of the straw as a protractor in part A was very neat to me.  I had never thought about using a straw before but not I would consider using them in my own classroom because it is a cheaper as well as more flexible device that an actual protractor.  I understand it does not give exact measurements but it still gives a general idea.  Using the straw during session A was a neat way to understand the angles.

In part B I did not have much trouble in answering the questions because angles were always something I enjoyed learning about in school and was pretty good at identifying without actual measurements.  I did have to freshen myself up on a few terms such as vortex but once I got all of my definition's in order I was able to figure out the problems.

In part C  I found the Geo-Logo activity to be fun once I figured out how to do my commands.  I could see students actually enjoying that part of the module because students like interactive computer things and I feel like they also learn better that way.  

Module 12

I apologize for the lateness I just kept running into different road blocks.
Case Studies:
 Barbara’s students in case 12 are under the impression of when referring to measurement you can use the terms, half, bigger, or smaller.  The students would use objects such as the wall to compare to saying it would be half the size of the wall.  Students did not necessarily know the units or distances of the measurements but they do know how to generalize size. The children in Barbara’s class still have to figure out how to take real measurements.  They also need to know correct terms instead of actually using half sized of objects.  

The students in Rosemarie’s case 13 struggled with the number of units in that they did not understand that the larger the unit they use the smaller the number was actually going to be, for example: if you use an inch instead of a centimeter then you would have less inches than centimeters. 

In Delores case 14, students actually were able to recognize the relationship between the sizes and the units.  Students were able to use a visual image to actually grasp the concepts. Everyone didn't measure in kids feet because they did not use kids feet but they used adult feet.  The other students are noticing that not all students feet are the same size because when looking around they made this observation. They are also noticing how most people measured as 60 and that  two students have really high numbers.

In Mabel's case 15, Students are able to recognize that you can in fact use a larger measuring device to measure smaller objects but you have to use the smaller lines that are on the larger measuring instrument.  In case 16 the issue of students choosing not a good tool for their measurements and not starting on the right place on the measuring tape was discussed among the class and solved.  Students worked through where to start when using a measuring tape.  The students understand how to use the measuring devices and realize how to start/where to start when using them. 

Ordering Rectangles Activity:

1. Take the seven rectangles and lay them out in front of you. Look at their

perimeters. Do not do any measuring; just look. What are your first

hunches? Which rectangle do you think has the smallest perimeter? The

largest perimeter? Move the rectangles around until you have ordered

them from the one with the smallest perimeter to the one with the largest

perimeter. Record your order.

When placing my rectangles I came to the idea the rectangles that had the larger lengths and widths would have a greater perimeter and the ones with the smaller widths and lengths would have the smaller perimeters.  The order I chose to use was: E D G A C F B.

2. Now look at the rectangles and consider their areas. What are your first

hunches? Which rectangle has the smallest area? The largest area? Again,

without doing any measuring, order the rectangles from the one with the

smallest area to the one with the largest area. Record your order.

Rectangle G and E with their similar lengths and widths I believe would have bigger areas than the rectangles with different lengths and width like C probably have the smaller areas.  The order I chose to put my rectangles in was: C B F D A E G.

3. Now, by comparing directly or using any available materials (color tiles are

always useful), order the rectangles by perimeter. How did your estimated

order compare with the actual order? What strategy did you use to

compare perimeters?

The perimeters smallest to largest are C D B E  F A G.  My first go around when estimating the order for the perimeters were pretty far off.  The strategy I used to compare orders was to use actual measurement lengths to get a number instead of just eye balling. 

 4. By comparing directly or using any available materials (again…color tiles),

order the rectangles by area. How did your estimated order compare with

the actual order? What strategy did you use to compare areas?

I was way off on my order again on the area.  I came up with the order of E, D, A, B, G, C, F when I did the actual measurements for the area.  When drawing my color tiles and using the area formula I learned that the order I had chosen originally was not correct. 

5. What ideas about perimeter, about area, or about measuring did these

activities help you to see? What questions arose as you did this work? What

have you figured out? What are you still wondering about? Share your

responses to all the questions in 1-5 in your small groups on the discussion

board.

These activities helped me to realize that estimating can give you an idea of measurements but you really need to take the time to do the actual measurements in order to determine the correct measurements.  I am still confused at how I was so off on my estimates though because when looking at the rectangles I thought I was really doing good.

For further Discussion:

As adults we use standard measurements almost without thought. Nonetheless, nonstandard measurements also play a role in speech and behavior. We ask for a “pinch” of salt or a small slice of dessert. We promise to be somewhere shortly or to serve a “dollop” of whipped cream. In some situations we might even prefer nonstandard measurements – for example, in cooking, building, or decorating. Do you or someone you know use nonstandard measurements on a

regular basis or even instead of a standard system? Give some examples and discuss why a nonstandard measurement might be preferred in some cases.

My family and I are all very guilty of using nonstandard measurements on a daily basis especially with direction and cooking instructions.  When asking where somewhere is we will say “right up yonder” or “just up the road.”  People all around where I live say things like that but people who were not around would not understand what we were trying to say.  When cooking my step-dad tells me recipes and he will say “a splash of milk, a pinch of this, a dab of this, and a dash of that”.  When you are trying to cook non standard measurements are hard to follow because you do not want to mess up a recipe.  With directions I do not have a hard time when someone gives me nonstandard measurements because I do not have a good idea of what a mile would be when walking or driving without actually measuring it.  Nonstandard measurements could be good but just depends on the situation.

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Module 11

Coordinate Grid:

Billy Bug was the first one I played because it was the first one on the list and I enjoyed it myself so I know my students would too.  It is a neat way for students to learn their coordinate points and it is interactive which I feel like helps students a lot.  Billy Bug 2 I thought the same of and could be used in collaboration with Billy Bug.  The teacher could use Billy Bug to teach coordinate points and then when teaching about quadrants she could introduce Billy Bug 2 because the students are already familiar with it.  When playing Catch the Fly I enjoyed it as well.  I found it to be like Billy Bug 2 in teaching points and quadrants.  I did not really enjoy the building blocks game because even I got confused on some of the questions it was asking.  I feel like students may have a problem with this but maybe if it is something they have just been introduced to they would be able to do it.  Stock the shelves is a neat game I would like to use in my classroom too when teaching about coordinate planes.  I found it to be fun as well as educational.

I really enjoyed going through and playing with all of the different applets because I like interactive learning tools that are fun.  I advantages of using these computer activities for students are that: students tend to really enjoy them, they are fun, interactive, and usually very helpful in teaching students the content in which you want them to learn.  The only disadvantage I could really think of in using these activities is the amount of computers at hand.  A lot of schools only have enough computers for all students in their computer labs which are generally occupied with other classes.

For further discussion:
*A fellow teacher says that he cannot start to teach any geometry until the students know all the terms and definitions and that his fifth graders just cannot learn them, what misconceptions about teaching geometry does this teacher hold? 

Many teachers believe that to learn materials you need to know all the terms, definitions, and background information before you can teach them a lesson. That is not true.  The teacher in this case is wanting his students to do this as well.  The misconception he is having here is that students can learn as they go.  Many students learn a term better by seeing it first and then being told what it is.  For example: the teacher could show the students a picture of a coordinate plane and then tell them that it is a coordinate plane instead of hitting them with a definition and they have no idea what it will even look like.  Students learn better when they are not being forced to learn a term or definition but yet have something to relate these words to.

Now that you've had some time to explore the world of geometry, how has your view of the 

key ideas of geometry that you want your students to work though changed? 

I can not say that my view has necessarily changed when thinking about they key ideas of geometry but I can say that I feel like I even more about the key ideas of geometry than I did before.  Through this module I have learned from my own experiences that it is easier for the student when you show them something in relation with a term or definition that just making them memorize words.  I also believe that when teaching about shapes it is a good idea to show examples and non-examples.  For example: when teaching about what a circle is show students circular objects and non circular objects and have them to classify which one is a circle and which one is not.  I am sure as I continue to learn more about geometry that my view will probably change on some things.  

Module 10

Pentomino Narrow Passage:

In doing my passages I tried multiple times to try to make mine as large as Dr. Higgins but I was nowhere close to being able to make mine as large.  I was only able to make my passage using 13 pieces.  Each time that I tried to add more in I would wind up having branches coming off.  I am not able to upload a picture at the time because I am using a family members computer and for some reason the picture will not pull into the blog.  I will try an upload a picture on my own computer later to see if it will work. I thought I would be able to form a pentomino narrow passage as Dr. Higgins did because I thought of it as almost like a puzzle (which I am generally good at) and was surprised to see how hard it was to actually use a lot of the pieces.

Tangram Discoveries:

Which polygon has the greatest perimeter?...the least perimeter? How do you know?

When forming my polygons in this activity I had trouble at first with myself in thinking I would not be able to figure out perimeters without a ruler but once I started forming them it made sense to me.  When forming the shapes and looking at the perimeters it appeared to me that the rectangle had the largest perimeter and that the square had the least perimeter.  How do I know?  Well, when looking at the shapes it appeared that the rectangle had the greatest distance around where as the square had the least distance around which is what perimeter is.

Which polygon had the greatest area?...the least area? How do you know?

I believed my rectangle again to have the largest area and my square to have the least area. My reasoning for seeing this (which could be wrong) is in comparing area to perimeter.  If my sides on my rectangle made the greatest perimeter it only made since to me that they would combine to make the greatest area as well.  The same concept I applied to my square.  Also, when looking at my square in comparison to my other shapes the area inside did appear to be the smallest. 

Pentomino Activities:

When going through the PowerPoint and completing the Pentomino activities I had a fun time.  I really enjoyed the first activity of the line symmetry.  I feel like this would be a simple activity to use in the classroom and also one that students would enjoy.  This is a good activity for students to do in order to learn hands on.  The rotational symmetry activity was another hands on activity that I thought would be fun for students to complete.  It is one that I found simple and enjoyed so I feel like my students would as well.  Having students use the pentomino pieces to fine perimeter and area is also another activity I enjoyed.  It gives the students to work with the pieces to find the dimensions instead of having to look at numbers on a paper to configure answers.  The tessellation activity I did not like as much when I had to follow the instructions on the PowerPoint.  I found myself growing increasingly aggravated and not being able to do it.  When using the website I was however able to form the tessellation's and highly enjoyed the activities.

Annenberg Symmetry Module:

When completing Part A of the Annenberg Symmetry Module I found question A1 easy to complete and I completed it with 100% accuracy.  Problem A2 (although asking pretty much the same thing) I had a little more difficulty with.  On the regular pentagon and the regular hexagon I was able to find half the number of lines of symmetry than what really appeared.  I feel the reason I struggled on the was because of the way I was viewing them.  I felt like 5-6 lines of symmetry was just too much so I did not look into finding that many.  Question A3 I did not do well on.  I was confused on the directions so I tried and when I did not feel like I could do it I just looked at the solution and then it began to make more sense to me.

When completing Part B of the Annenberg Symmetry Module I found myself struggling with what the actual degree measurements needed to be.  When I changed the degree measurement each time to rotate the object I was able to see if it was rotationally symmetrical but as for the actual degree measurement and knowing how to determine which ones would work; I need improving on.

For Further Discussion:

I found an article that talked about how the Ancient Greeks used geometry in their art forms.  The Greeks were the first believed to understand "the laws of perspective" which meant they knew how to make a 3-D figure onto a 2-D surface.  Artist of the Renaissance Period also understood the "laws of perspective" and they believed in forming parallel lines in their work as a symbol of forever.  The artist of this time believed that parallel lines went on forever so that's what the parallel lines in their work symbolized. 

Also, I looked at a lot of Mexican paintings just on the Google Images page and it stood out to me that a lot of the paintings that used geometric shapes used a lot of triangles as well as a lot of circles.  In researching I could not find any reasoning behind it but I did find it to be a neat attribute. 

Module 9

Nets Activity and Mathematics Hiding in the Nets Article:

How could you use a similar activity with students in the classroom?  Were you able to complete the activity without too much frustration?  What are some anticipated issues while doing this activity with students?

This activity as well as the activities within the article could all be used in the classroom as a way of teaching students about 2-D and 3-D figures.  The nets could be used to show the students the way that 3-D figures are formed and help them to understand the way the nets look for each of the 3-D shapes.  Students could do activities such as drawing their own nets to form 3-D figures, looking at a sheet of different nets to determine which ones would form the 3-D figures, and students could also use area as a way of finding out which nets would form the figures.  I was able to complete the activities with fairly little trouble.  I remembered using nets in school so I had an idea of which ones would work as well as which ones would not.  I also after reading the article went back to the Nets activity to see how I would do then.  I did do better after reading the article.  An anticipated issue during this activity would students just not grasping how to use the nets.  Some students may not understand the properties that a net must have in order to form a shape.  For example: students may not understand that for a cube you need four connected squares in a row with a square on each side of the row.  If students had trouble understanding that then they could have trouble with ever forming the cube.  Also, some students can get frustrated with trying to form the 3-D figure and mess up when folding the figure, get discouraged, and give up.

Spatial Readings, Annenberg, and Building Plans:

Did you find any of the activities challenging?  If so, what about the activity made it challenging?

I found the first and second activity to be the most challenging.  I had a hard time with the first activity because even after looking at the map the pictures had me very confused.  When I watched the videos of the answers when I was finished I still noticed that I was not good at determining the location of the pictures even with the map.  In the second activity I was just not able to form the figure with the silhouettes.  I realized that my spatial visualization skills are not that great and are something I could definitely improve upon.  The last activity I did good on because I was able to visualize the shapes the cubes shadow would give off.

Why is it important that students become proficient at spatial visualization?

Student's need to be proficient at spatial visualization because it is a big part of working with geometry.  Spatial visualization helps students when learning about their 3-D shapes and helps them to understand characteristics and areas of these objects.  Students need to be able to know that the 3-D shapes can be made up of 2-D shapes and that they can project the shadows of the different 2-D shapes.

At what grade level do you believe students are ready for visual/spatial activities?

I believe 4th grade could be a grade where visual/spatial activities are introduced.  These concepts can be challenging for students in earlier grades because I myself have difficulty with some of the concepts myself.  Students could start by just learning about the 3-D shapes and then move onto the visual/spatial activities.

How can we help students become more proficient in this area?

To help students become proficient in this area we as the teachers need to be proficient in it to start with.  I still struggle in this area somewhat and before teaching it I would actually want to do some more review and try to master it.  Students will have to take notes and do repeated practice activities to really understand visual/spatial concepts.  If students are struggling then the teacher will need to assess what they are struggling with and try and think of a new way to teach them.

For further discussion:

Informal recreational geometry is an important type of geometry in many childhood games and toys.  Visit a you store and make an inventory of early childhood toys and games that use geometric concepts.  Discuss ways these materials might be used to teach the big ideas of early childhood geometry.

One toy I found was the shape block.  The cube had different shapes cut out of it and the child had to find the corresponding shape piece to put through the hole and into the cube.  This can help children to begin to know names of shapes as well as their characteristics.  I also feel like this toy could be used as a tool in geometry in visual and spatial learning because the child will have to determine if the shape will fit in the slot the choose.

The other toy that I found was actually a geometric puzzle that included colors as well.  Students put the puzzle together by connecting lines and colors and different geometric shapes would form throughout the puzzle.  Students could grasp ideas of different shapes in completing one of these puzzles at an early age.  It allows for the child to have fun using large manipulative s while at the same time learning the different geometric shapes that are in the puzzle.

Annenberg Tangrams Module and Creation of Manipulative:

I have never been a fan of tangrams and still am not after completing the activities.  I always had a hard time using them because I would forget the shapes that I was doing.  When completing the Annenberg activities I struggled on all of them.  I had difficulty in understanding what they questions were wanting me to do to and then when I would try and do the questions I would always hit a roadblock.  I am concerned about having to use tangrams in my classroom as well as teach them because I struggled so much with this module.  I feel like it would take me some review time and further instruction to really master using tangrams to the point of being able to teach about them.

Module 8

Quick Images:

Once you've viewed the video, discuss on your blogs what you noticed about their responses. You should also consider how you saw the shape and how your perspective compared to the way the students described what they saw.

I found this video to be very interesting because when I was watching the shape be shown I had the same response that one little boy had in that “he just remembered the shape.”  I watched the shape and did not make a connection to it I just remembered the visual of it and knew what to draw.  Many students made connections to real life objects.  Students gave responses such as: “it looked like a moon”, “it looked like a C”, “it looked like a banana”, “it looked like a boat”, and “it looked like a melon”.  Students were able to look at the shape and then think of an object it reminded them of; even if it was an object they had to rotate.  The teacher would allow the students to show her how they related the picture to the object they named by having them to come to the overhead and maneuver the object to show the rest of the class.  I thought it was neat also that the one little boy named the shape by its actual name which was a “crescent”.  Another thing I found interesting was when he mentioned that there were two points that helped in drawing the picture one at the top and one at the bottom with two curved sides.  I found that to be interesting as his way of looking at the picture.  It was a neat activity to see to understand how students learn because as I mentioned I just remembered a visual image without anything to compare it with or remember it by and was able to draw it but these students had in depth reasons as to how they remembered the shape.  I would like to try this in the classroom just to see the way that my students think about things.

Case Studies: Shapes:

These case studies were very interesting for me to look at to see the way that the children thought about different things.  Looking at Andrea’s case it was a neat way to watch how the students viewed triangles.  Susannah had a hard time grasping that triangles that don’t look like “real triangles” (which I took as what appeared as 3 equal sides) could not possibly be triangles.  After hearing suggestions from different classmates about what a triangle was she finally agreed with one idea suggested by classmate Will that a triangle can have “two slanty sides”.  Evan who was in this case as well seemed to understand what a triangle was the best to me.  He knew that a triangle had points at their ends and were not curvy.  He also suggested a triangle is made up of three sides.  He shared an idea with the class that “no matter how you turn him, he is still Evan.”  This was interesting to me because it was a very valid point.  A triangle is made up of three straight sides that connect and no matter how you maneuver it, it remains a triangle.

Natalie’s case was more to me about definitions and how they need to be exact.  Students were describing squares and rectangles pretty much as the same thing being: “four sides, four corners, and it’s a square.”  That is not true not all shapes that fit that description are a square.  A rectangle fits that description but its parallel sides are the same size.  This case showed me the importance of giving detailed and correct definitions but also made me think that giving visuals would help students to better understand ideas and concepts.

In looking over Dolores’s and Andrea’s cases it was evident that the students were talking about how to make sense of a new idea.  The students in Andrea’s case said something very well when he said “we have to make our eyes and our heads meet.”  He suggests that we have to look outside of just what we already know or what we are seeing and connect the two to have a better understanding of a concept or idea.  In Delores case study a student suggest that “we have to be open to look further” which suggest that sometimes the ideas we have are not always right and we have to keep our minds open to learning new things.

In my classroom I hope to be able to lead my students in a direction that does not get them confused.  I hope to be able to give my students concrete definitions and examples of terms that they can carry with them throughout their years.  I also want to be able to teach my students to have an open mind when learning because things can change and ideas can be elaborated on as well.  I myself as the teacher have to keep an open mind when teaching because I can be introduced to new teaching strategies or new methods of learning that I am unaware of.  I found these case studies to be a lot of help and very interesting.

For further discussion:

If geometry is the mathematics that describes the world we live in, that means geometry is everywhere around us. Use the language and concepts of geometry to describe your own world your home, your workplace, your possessions, your daily commute or other travels. 

As I am laying in bed writing this blog I am going to describe my room using geometrical terms:

• The room is shaped like a SQUARE.
• The mirror is shaped like an OVAL.
• The walls make up 4 RIGHT ANGLES.
• The TV is a RECTANGLE that has 4 RIGHT ANGLES.
• The window has PERPENDICULAR LINE SEGMENTS forming window panes.
• The green chair has INTERSECTING LINE SEGMENTS.
• The desk chair has SPHERE shaped wheels.
• The fan has a 3-D CIRCLE and a 2-D CIRCLE on it.  
• The dresser is a RECTANGLE made of smaller RECTANGLES, SQUARES, and CIRCLES.
• The pin board is one big SQUARE made up of 4 smaller SQUARES.
• The bed is a RECTANGLE.
• I am sure there are many more but I wanted to list the major things that stick out to me.

Geometry really is all around us and you don't think about it until you actually sit there and look around.  I feel like now whenever I am in a new place I am going to be looking around to see the different geometric things.

Module 7

Key Ideas in Geometry:

What are the key ideas of geometry that you want your students to work through during the school year? 

During the school year I would start by having my students recognize all of their shapes by naming them first.  After students were able to visually recognize their shapes I would want for them to be able to draw them, describe them, and label them.  Once the students understood the shapes and their attributes I would want my students to then start to learn the terms of ray, line segment, angles, planes, and all the other geometric terms that they will be exposed to during their time using geometry.  I feel like teaching the students these terms would help them in learning how to to measure angles as well as be able to tell the difference in isosceles, right, and obtuse angles.  The students also need to be familiar that their are 2-D shapes as well as 3-D shapes and that although 3-D shapes favor 2-D shapes the way you measure them will be different.  The process of learning geometry is like any other subject.  You have to start with the basics and build upon them for students to understand the full concepts you are trying to teach them.  

Thinking About Triangles:

How would you structure this lesson for students in an elementary classroom?

I think the way that the lesson was structured in the video was actually pretty to the point and was able to teach a strong lesson.  I enjoyed how Dr. Higgins started out by getting students involved in the lesson by having them to look around the classroom and write down the different objects that they saw that were triangles.  I would then lead a whole class discussion and have them to share the objects that they found and explain how they knew that the objects were triangles.  I would then do the same thing Dr. Higgins did in her PowerPoint of having the students to think of words that started with "TRI" and have them to discuss what those different words had in common.  I would then instead of moving the the geoboard activity as shown in the PowerPoint I would teach/review with students what an equilateral, isosceles, and scalene triangle were.  When students were familiar with those terms and refreshed on them I would move to the geoboard activity and follow the same questions that were addressed in the PowerPoint lesson.  I would then have the students to cut out their shapes they had recorded, find a partner, and group their triangles by similarities.  I would have the students to discuss why they grouped their triangles like they did and also have them to find differences within the groups.  The students may take a while to grasp all of the concepts and as the teacher I may have to alter my lesson to make sure all students understand.  If I felt like the lesson was not going as planned when I introduced it I would stop it, rework it, and teach it again the following day. 

What parts did you have issues with? Did you need to revisit some vocabulary words to 

remind yourself of their meanings? If so, which ones?

I luckily did not have to revisit any vocabulary words because the triangle terms have always stuck with me throughout school...working with triangles was one of my favorite parts of geometry!  I did have an issue tho in forming many right triangles without repeating any.  I was able to only form 4 on my geoboard and put them on paper and I would love to see a visual of where all 14 triangles were to go.  I found it to be difficult to not recreate the same triangles more than once as well.  Maybe if the geoboards were bigger it would have been easier to create more?  

Van Hiele Levels and Polygon Properties Article:

I found the article and the PowerPoint to be very interesting because I was not familiar with the levels until I viewed the PowerPoint and read the article.  I do not think I have mastered what the levels are but I have gained a better understanding for sure.  The PowerPoint talked about how most students that we would be working with would be in the 0-1 level and I can actually see that when working with the students I have in the past year.  

The activity that was in the PowerPoint was one that I enjoyed and one that I would not mind doing with my students (if I was in upper elementary grades).  I feel like the students would really enjoy it and be satisfied with themselves like I was when I was able to get the shapes correct.  Knowing what I know now after looking at the PowerPoint, doing the activity here and above, and reading the article I feel more confident in teaching geometric ideas to my students.  I understand now that it is in fact a process and that students need to learn their basic concepts and build upon them to be successful in geometry as they would in other areas of mathematics.  In my classroom I hope to be able to make learning geometry a fun thing for my students as well as making sure they are grasping all concepts along the way.  

For Further Discussion:

The van Hiele levels can be applied to geometric thinking concepts by concept or to one’s overall 

thinking about geometry. How would you rate your own thinking? Does the level vary or remain fairly 

consistent across the subject matter? For example, are you at different levels working with the 

concepts of two-dimensional or plane geometry than working with three-dimensional or solid 

geometry? Explain. 

I would rate myself at a level 3 of: deduction.  Level 4 of rigor I feel like is someone who is extremely proficient with their geometry skills and I do not feel like that necessarily applies for me.  I feel like my level stays pretty much the same throughout the subject level except if someone was to put an extremely hard formula for me to decipher in relation to a shape.  I may be able to recognize the formula and the shape and know what answer you are asking but not necessarily know how to find the answer.  As long as I am familiar with what is put forth in front of me I would say that I would remain the same.

Module 6

***My formatting looks weird on this one and I am not quite sure why.  I have tried fixing it and can not.  I apologize for the visual appearance***
Dice Toss:

1. Ms. Kincaid wanted the students to make predictions about their experiment on the basis of mathematical probability. Discuss preconceptions that students exhibited about tossing dice even after discussing the mathematical probability.  Discuss the instructional implications of dealing with these preconceptions

The students in this video did struggle with making predictions that were based on mathematical probability. Many students guessed that the most rolls would add up to be 7 because 7 had the most combinations.

Students in this video also based their predictions off of past experiences.  One student mentioned that from previous experiences with dice that she guessed 2 would appear more that 12 because when she rolls dice she gets 1 more than she does 6.  Students in this video even after looking at the chart of combinations and discussing mathematical probability still stuck with their misconceptions.  The teacher needs to let the students know that when working with probability you cant refer to past experiences because probability is more of a "luck" thing and could change from time to time.  

2. Were these students too young to discuss mathematical probability? What evidence did you observe that leads you to believe that students did or did not grasp the difference between mathematical probability and experimental probability? At what age should probability be discussed?

I do not believe that these students were to young to discuss mathematical probability.  The students in the video seemed to grasp the concepts of mathematical probability and understand what it was they just kept wanting to bring up past experience when trying to think mathematically.  When the teacher reviewed mathematical probability and experimental probability with the students they were correct in their explanation and seemed to understand the difference.  I believe that probability being introduced maybe starting in second grade would not be bad.  I do not mean to the full on extent of this exercise but more of a likely/unlikely type of probability exercise.

3. The teacher asked the students, “What can you say about the data we collected as a "group" and “What can you say mathematically?” How did the phrasing of these two questions affect the students’ reasoning? 

When the students were asked about "what can you say about the date collected as a group?" it allowed the students to answer with any sort of answer such as the student who made the analogy about the results looking like a rocket ship.  When asked "what can you say mathematically?" it narrowed down what the students were supposed to talk about and gave the teacher an insight on what the students learned mathematically from their activity.  The students were then able to answer the questions using numbers and not talking about things unrelated to math.

4. Why did Ms. Kincaid require each group of students to roll the dice thirty-six times? What are the advantages and disadvantages of rolling this number of times? 

The dice were rolled 36 times in my opinion because 36 was the number of combinations (sums) that could be made with the two dice.  The groups also consisted of 4 students and using 36 rolls allowed for each student to be able to roll the dice 9 times.  The advantages of rolling the dice 36 times is that it allows a good amount of rolls to collect data off of.  I feel like 36 rolls is a good number and should give a good idea of what number occurs the most.  The only disadvantage I can think of is the students loosing count (despite having a counter) and data being swayed.

5. Comment on the collaboration among the students as they conducted the experiment. Give evidence that students either worked together as a group or worked as individuals. 

The students were able to collaborate well in the video to me.  The students talked about how they wanted to organize their data and then they conducted their experiment in the manner they wanted to.  The groups were able to work well and discuss their ideas.  The students also worked as individuals when doing their own job but their jobs worked collaboratively in making the experiment work well.

6. Why do you think Ms. Kincaid assigned roles to each group member? What effect did this practice have on the students? How does assigning roles facilitate collaboration among the group members? 

I think Ms. Kincaid assigned roles to each member so that everyone had to be involved and so that each person would be responsible for something.  I think this helped the students because although they were doing their individual jobs it allowed for the group to work effectively together.  Assigning roles facilitates collaboration because it makes it where the students have to work together as a group in order for all information to be distributed because they are all doing a different thing. 

7. Describe the types of questions that Ms. Kincaid asked the students in the individual groups.   How did this questioning further student understanding and learning?

Ms. Kincaid did a good job when asking students questions because they were direct and allowed for the students to know what information she was wanting to know.  She also helped guide them through their questions by talking with them and asking them other questions while discussing what was going on.  Ms. Kincaid used a lot of mathematical terminology when speaking with the students as well and wanted them to respond mathematically to questions as well which helped them to understand probability.  

8. Why did Ms. Kincaid let each group decide how to record the data rather than giving groups a recording sheet that was already organized? When would it be appropriate to give students an organized recording sheet? Discuss the advantages and disadvantages of allowing students to create their own recording plans. 

I think Ms. Kincaid let each group to record their data their own way as a way to get them to think on their own as well as a way for her to see the different understandings of data collection that her students have.  Students were able to show what they knew and also able to see which ways of data collection made more sense in certain situations.  I believe that teachers should give students organized recording sheets whenever students are starting out learning about data collection or whenever she wants all students to use one method as a learning tool.  

The advantages of allowing students to create their own recording plans is because it allows them to be creative, really think, and allows the teacher to see what their students know and how they use and sort data. A disadvantage of allowing students to organize their data in their own way is the teacher doesn't really have a certain criteria to grade off of and cannot really tell a student whether their method is the correct one because they allowed the students to use their own method of data collection.

For further consideration...
Knowing what you now know about probability concepts in the elementary school, how will you 
ensure that your students have the background to be successful with these concepts in middle school? 

Knowing what I know now about probability concepts in the elementary school I want to ensure that my students have the background to be successful with these concepts in middle school by making sure that they know what mathematical probability is, experimental probability is, and that they know how to find probability.  I want to do activities like the dice that are shown in the video.  I also want to do many other activities with probability to make sure students know what it is and how to find it.  I would also allow them to record their results in their own ways so I can see how the students are thinking and so that I can work with them if they are having a problem in using the best way of collecting data.  I want to make sure that by students have a solid background when working with probability before middle school so that when they get there they can continue to build on concepts they already know.

A Whale of a Tale:

Impossible                                     Unlikely                                         Likely                             Certain 
1. To run to California in a day.        1. I become a surgeon               1. I become a teacher      1. I am a girl 2. I will be 100 next year                 2. I do not graduate in May        2. I go to bed by 11pm    2. I have                                                                                                                                                       green eyes.