Module 13

Case Studies

This module’s case studies provided a variety of insights on how children perceived shapes and especially angles. I never realized how something that I find so simple like an angle could be such a challenge for a child. In previous modules we spoke about how teaching children without definitions is a better method. However, I believe teaching children the definition of an angle will further help their understanding in this case. Angle is defined as a figure formed as two lines come together at a common point. This definition gives a clear-cut idea of what an angle should look like. Furthermore, in my classroom I would provide students with ample opportunities to see angles work in real life to stray the misconception of a slanting line or object.

adjacent angle

******* Which case study did you find most interesting? How did it help you understand children’s misconceptions?

How wedge you teach the angle concept?

In the very beginning of this article I was encouraged by the idea of inquiry as a mean for teaching angles. I think that this method would be very appropriate to help children steer away from misconceptions. Children often need to see concepts such as angles work hands on to process their meaning. I also think the idea of folding a circle multiple times to create angles is also a way for students to see angles first hand. The article provided a decreasing angle made from a circle to show students that essentially angles are measured from one line meeting another. Building upon children’s prior thoughts is essential to helping them understand angles.

***** What methods do you see best fit for teaching angles?

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Module 12

Case Studies:

From reviewing the case studies, I have learned a few very valuable concepts about teaching children measurement. I learned that children come to the classroom with preconceptions about measurement and ideas about comparing things. In the Kindergarten case study, students were measuring using comparison to other objects. Right away it can be realized that students have pre-knowledge about non-standard measurement. This really helped me tie together ideas from our power points and assignments this week. The power point breaks down the measurement thoughts that we have each day. Students are likely to have the same thoughts and will need to use measurement as well.

Ordering Rectangles:

  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.

My first hunch was that rectangle C appeared to be an inch wide and resembled a ruler like shape. I would say that it would have a fairly large perimeter but not the largest. Rectangle F appears to be double the size of C.

Smallest to largest = A, C, B, E, D, F, G

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.

Area is a bit more of a challenge for this activity. However, I first notice that these rectangles are going to be a close call when comparing areas. I believe that some of the rectangles will have the same areas such as C and D and F and G

Smallest to largest: A, C &D, B, F&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?

Smallest to Largest: D&E, A&B&G, C&F

My estimated was a bit off than what it was when finding actual amounts. The actual orders had some perimeters that were the same, whereas I had paired them to have similar areas. I used small square tiles to compare.

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?

Smallest to Largest: E,C,D,B,F,A,G

My estimate and actual orders were quite different. I realize that my logic was actually around perimeter rather than area making a mix up of things. I used square tiles that were in inch all around as a tool to compare.

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 on your blog. Spend some time discussing your answers with your blog partner.

These activities helped me see the developmental process of measuring . This also taught me to use other things as a mean of measuring such as simply comparing rectangles. My main question was how I could compare these with no tools at all. I found that I was imagining cutting several rectangles in half and seeing if they would fit in another. Although, I did not cut them I tried to work that out in my mind.

** Which of these steps best helped you compare the rectangles?

** Do you think this would be an appropriate assignment to give elementary school children?

TCM Measure up:

Much like this modules case studies, the article introduced us to children’s conversations about measurement. A student stated that the larger the unit the “lesser” times we will need to use it. With that thought in mind, I think it is safe to say this concept will be brought into my classroom as early as possible. This is significant for students to realize because then they will be able to see why Cm measures so much more than Ft. It was also interesting to read about the ideas students had when comparing fractions. One student stated that ¾ is always greater than 2/3. Tying concepts together such as this teacher did with fractions causes for optimal opportunities for student learning.

Further Discussion:

Non standard measurement can be an extremely valuable tool for plenty of field work. I personally use nonstandard measurement with out even thinking. As a wife, most nights I am the one to make dinner for my family. With that being said, I am the cook and I am a cook who refuses to measure anything (unless backing desserts). I often find myself following a recipe but do not use the measuring tools. I often just measure by hand what might be “about a cup” or what a teaspoon/tablespoon might look like in my hand. This helps me move through the cooking faster but still ensures me that the recipes are tasty. I am sure there are other careers out there that provide the need for nonstandard measurement, such as estimators of any sort. If there is anyone like me out there sometimes you must get the job done quick, which is why nonstandard measuring can be significantly helpful.

*** How do you use nonstandard measurement on a regular day?

Module 11

Coordinate Grids

Catch the Fly: http://hotmath.com/hotmath_help/games/ctf/ctf_hotmath.swf

What’s the point? : http://www.funbrain.com/cgi-bin/co.cgi?A1=c&A2=0&A3=0&A4=1&A5=Ov@7tB&A6=[3][4]

Maze Game: http://www.shodor.org/interactivate/activities/MazeGame/

The websites above are games that I explored about coordinate grids. I found them to be very useful and engaging. Students love games and love doing things that do not make them feel as though they are learning. However, there are a variety of tools that can be learned form using these games. Students can practice finding coordinates, placing coordinates and learning which is X & Y. The only disadvantage to using online systems is that sometimes glitches happen when an answer is correct. This happened to me and unfortunately students might look at this as discouraging. I would consider using all three websites within my classroom. Many schools are implementing math rounds as a means for enhancing student learning. With this in mind, these websites would be great for a technology round when working on this unit.

Further Discussion: Misconceptions

Presenting students with definitions to begin a lesson and not go further until they are know is setting students up for failure. Definitions can be wordy and sometimes bring forth a variety of misconceptions for students. One of the largest misconceptions is that students would begin to only visualize geometrical shapes in one way. Therefore, when students see the same shape turned sideways or of different sizes it because a challenge for them to break from this view. Instead rather, this teacher should consider creating definitions with students so that they understand the different varieties of shapes rather than one solid definition. Students need the opportunity to examine shapes and see how they work in different ways.

******How can we avoid these types of misconceptions for our students?

 

How has your view of the key ideas of geometry that you want your students to work though changed?

My view on geometry has changed dramatically after working through these modules. I realize now that I want my students to learn through work rather than definitions. Geometry has a many forms beyond just shapes and I want my students to be able to see that as well. Working with pentominos is a way to see other geometrical shapes put together to create ones that they are familiar with. I also want my students to visualize geometry as a real life concept rather than math only. Using tessellates and patterns can teach students that geometry is everywhere and has been a useful tools for many years.

 

*****How can you implement real world geometry experiences into the classroom?

Module Ten

Tangram Discoveries:

I have discovered a substantial amount of information upon working with the tangram pieces. I have discovered that several shapes can be made from three simple tangram pieces. I also discovered that the big triangles smaller sides are equal to the small triangles large side. With this being said simply by looking and counting the sides of each shape I could determine which shape had the larger perimeter. I found that the square had the smallest perimeter and the triangle shape has the largest perimeter. I also realized after a moment of thought, that the shapes will all have the same area because they were constructed with the same smaller shapes.

Shapess

Pentomino Narrow Passage:

The passageway below is 17 spaces long. There are no branches, and it is closed. The area of my passage is 17 cubic units.

Pentomino******What concepts could you apply narrow path creation?

Pentomino Activities

I will full admit that I was very frustrated when playing the STACKS scholastic game. I did not understand hope I was supposed to possibly make a rectangle with so many different shaped pieces. I found that my first issue was that I was attempting to created a rectangle that was white in it’s center because of the graph behind it. I then realized that I needed to make a rectangle that was filled in. Once I made that connection, I had a better understanding of what I was supposed to do. This would be a great activity to explore with students, and allow them to do these puzzles as a “finished early” activity.

** I stumbled across this website as I looked for more puzzles. It give several other shapes that can be made from pentominoes!

http://puzzler.sourceforge.net/docs/pentominoes.html

Try Stack:  http://www.scholastic.com/blueballiett/games/pentominoes_game.htm

Tessellating t-shirts

Upon reading Tessellating T-shirts, I learned a variety of ideas about tessellation.  I realized that tessellations are a concept that dates back in history and involved the use of geometry in everyday life.  This covers art work, potteries and over styles. To tessellate means creating a pattern with a geometrical shape that fits within the same shape over and over across a plane. I also learned how tessellates can be introduced through items all around us such as tile floors, ceiling tiles, and brick walls. The article demonstrated how T-shirts could be made with the students to express how a tessellate can be used. Here are a few examples that I found of different types of tessellates.

tessimage015

Further Discussion:

1. Mexican Ancient Art: This rug offers an opportunity to observe Mexican art history as well as investigate geometry. When we look closely, shapes such as hexagons, pentagons, quadrilaterals and squares can be found. We could use this to determine the types of shapes we see as well as finding area using the “bead like” stitching as a unit of measure.

Huichol-The-Mexican-Art-07

2. Native American Ancient Art: This bowl is a great example of how geometry was used as a form of art. Students could use the repeating of this pattern to determine where or not it is a tessellate or not. The students could also study how this bowl could be changed to meet tessellate features.

GRCA 13539

3. African American Ancient Art:  This beaded jar has the opportunity for great study within the classroom. Students could use this to determine surface area using nonstandard measurement and visualizing tessellates in real objects. I think this would be an easy way for students to examine patterns as well.

A-Dive-Into-African-Art-HistoryShockey, T. & Snyder, K.  (2007). Engaging pre-service teachers and elementary-age children in transformational geometry: Tessellating t-shirts , Teaching Children Mathematics, 14(2), 82-87.

*****Can you see any other ways this art could be used?

Module 9

Nets with Pentominoes

I could see myself using this activity as an opener or warm-up activity. Using pentominoes seems to be a brain stimulator that could be useful in the beginning of the day. Another fun way to use this activity would be to have students create their own and share them with classmates.

When I was completing the activity, I could do some of the pentominoes with ease and others were confusing. I am not much of a visual person; therefore this was hard for me to think outside the box. However, after reading the article I had a little better understanding. Instead of just looking at them, I could process where I should begin. For my students, the only anticipated issue I would have is the fact that some students may have a hard time trying to “mentally” make a cube. The best way for teaching students about pentomioes would be actually cutting out one and showing them how it would fold into a cube. Some students may struggle with visualizing this activity.

*******How can we make this activity easier for students to visualize?

Spatial Reasoning, Annenburg, and Building Plans

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

I did not find any of these activities terribly challenging. The building plan was a very interesting concept for students to work through. This might become a little confusing for students because they may not realize that the goal is to not just fill into the cubes but must be stacked as well. I also think spatial reasoning can come a little challenging for students who to use there brain as a planning tool.

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

It is important for students to become proficient in spatial reasoning for a few reasons. The first and foremost reason is because students will need this skill for the rest of their life. When doing these activities, I could only think about the time that my husband and I were looking at houses. When I was considering which house to by I was thinking of furniture placement the whole time. This is a concept of spatial reasoning. Also, students will need this skill to plan activities and for some jobs later in their life.

At what grade level do you believe students are ready for visual/spatial activities? How can we help students become more proficient in this area?

I think at any grade level students can start working with visual and spatial activities. Each grade level will have a different level of proficiency that can be mastered. There are less challenging activities that students in grades such as kindergarten can achieve like designing buildings. This will help them become familiar with spatial. We can help students become proficient by providing them with an adequate amount of activities that relate to visual and spatial concepts.

******Were any of these activities a challenge for you?

Further Discussion

Instead of visiting a toy store, I simply took a walk down Wal-Mart’s game isle. I was very amazed to find so many games that were using geometry. In realty, children are using shapes for most games and are moving their players along squares. Students are learning that as they move their player then becomes “blank” squares away from winning. In a sense this is similar to area concepts and how many spaces they are away. Also, in games such as scrabble we are seeing different pentominoes without even realizing it. Games such as this could be used during math rounds or indoor recess.

Annenburg Tangrams

I can truly say that tangrams are a little challenging for those who have to process information. I found that I spent a large amount of time mixing around my tangrams even to create my first square. It is a challenge to use all the pieces as well for making shapes from other shapes. Sometimes I saw a shape with only using some of the tangrams rather than all that were given. I think before teaching with tangrams, students should have the opportunity to explore with them and figure out how they would and what they are used for. Even kindergarteners could make silly pictures from tangrams to begin understanding their use.

tan

Module 8

th

Images

This was an interesting video that made a very valid point. As the students were doing the activity, I tired the activity as well. When drawing the moon like shape, I drew the shape facing the wrong way. I noticed that the students were describing the shape as things that they use everyday. Some students said that it reminded them of a moon, a cantaloupe, or a half of a circle. The students were trying to make sense out of the shape by relating it to personal experience and their world around them. One student had mentioned that he just thought about the shape and remembered it was a crescent. I was impressed with the variety of ways that students visualized this shape.

Considering geometry, what do you think is the purpose of the video’s activity?

Further Discussion

Geometry could truly be considered the mathematics that we live in. Geometry is in fact everywhere we go and all around places we live. For instance, at this very moment I am sitting in my bedroom. To my left I look to the shelf I have hanging on my wall. I can soon tell that there is a repeated pattern on a rectangular prism. Meanwhile, my TV is on and it is a rectangular shape. I can count three square shaped candle shelves hanging on my walls. As well as a jewelry hanger that contains four similar rectangles. The steps for my dog to climb could make triangles if I laid something over it. Our world is full of shapes; we just have to take a moment to investigate for them.

What shape do you notice most in  your bedroom?

Case Study

After reviewing the case studies, I have mainly come to one big realization. That is children will have their own perceptions about shapes before we even teach them geometry. The case studies were based on how children were learning different shapes. Many of the children struggled with definitions and visualizing shapes at a different angle. I found that when teaching geometry, the best way to begin is to learn what the students already know. One child in the case study could not wrap his head around a turned triangle because “it wasn’t the triangles he was used to”. This brings me to the conclusion that students need to see more examples of triangles rather than one standard “this is it”.  Rather than teaching definitions such as ” a three sided shape”, we need to be teaching our students real attributes of triangles. Students need to understand that no matter what size or angle measure a triangle is, then it still is a triangle.  Children also need to learn about attributes to tell the difference between a square and a rectangle. The definitions of these two shapes are the same, however their attributes tell a different story!

How might you clear your students from definition only learning?

Module 7

shapes

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

Depending on the grade level, the key ideas that I would want my students to know would vary. I am currently working with 1st graders for my field experience, so I will focus on this level. I would first want my students to be able to distinguish between different shapes and their attributes to know the shapes. I would want my students to work through constructing shapes. I would lastly want my students to work through recognizing similar shapes.

triangles pernrose borromee

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

When teaching this type of lesson in the classroom, I would first start by getting students thinking about triangles. I would teach students that a triangle is a three-sided figure that is closed. I would remind students that triangles can come in several shapes and sizes and can have different lengths in sides. I would do just as the video said and have students look around the room to find triangles. I would then have students practice making triangles with the geoboard as I call out different types of triangles.

Do you think this activity would be appropriate  for second grade and under?

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 did not have any issues with the geoboard paper. However, I could not figure out why I couldn’t make an equilateral triangle until finishing the PowerPoint. Vocabulary was still fresh in my brain.

How was your vocabulary remembering?

Further Discussion

In terms of my own personal rating, I would say that van Hiele would say I was at the deduction level. With this being said, I am able to classify shapes regarding theirs sides and attributes. It is easy for me to prove why a shape is a shape by explanation. I would say that van Hiele’s  levels would change when considering 2D or 3D shapes. This is because the dimensions will bring forth a different perspective. However, I think that I would remain on the same level personally. This would fluctuate from child to child within the classroom.