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.