We are getting closer to the end! I don't know about you but I learned a lot about angles this. I've always thought that this is a harder concept for many children to grasp so it was great to read more about the methods and strategies to use. As always, you will find my thoughts and ideas about each section of the module.
Angles-First Grade Classroom Video
Describe how the children were talking in the video.
I thought it was really interesting to watch this 1st grade class give their thoughts about angles and what they are. Many of the students had some creative ideas but really seemed to struggled putting their thoughts into words that made sense to the teacher and their classmates. I heard a variety of responses such as that 1) angles are the point where 2 lines meet and 2) are things that have 2 lines that are exactly straight. a I noticed that some students shared some of the same ideas. These children were set on the idea of angles having a "space" in between two straight lines and explained that angles have to have an area that is open from the points they meet. While it was hard for them to explain, I think the kids did a good job of sharing their ideas even if they were somewhat inaccurate. It was interesting to watch as one of the students (name is Leila? at 2:48 in video) talk about and demonstrate her idea of an angle. At first, she described an angle having 2 straight lines but in the second clip, she said that they could be made up curvy lines, too. I couldn't help but notice the her differing explanations.
What are your ideas will you take with you to your classroom?
I thought it was really interesting to watch this 1st grade class give their thoughts about angles and what they are. Many of the students had some creative ideas but really seemed to struggled putting their thoughts into words that made sense to the teacher and their classmates. I heard a variety of responses such as that 1) angles are the point where 2 lines meet and 2) are things that have 2 lines that are exactly straight. a I noticed that some students shared some of the same ideas. These children were set on the idea of angles having a "space" in between two straight lines and explained that angles have to have an area that is open from the points they meet. While it was hard for them to explain, I think the kids did a good job of sharing their ideas even if they were somewhat inaccurate. It was interesting to watch as one of the students (name is Leila? at 2:48 in video) talk about and demonstrate her idea of an angle. At first, she described an angle having 2 straight lines but in the second clip, she said that they could be made up curvy lines, too. I couldn't help but notice the her differing explanations.
What are your ideas will you take with you to your classroom?
This short video was very informative in that it helped me see some of the ideas and misconceptions children have about angles. Moreover, I think the video shows the teacher using certain instructional strategies in the lesson. I like how she has the students seated in a group and engages them in a discussion where they feel comfortable to share their input. I also like that regardless of what the students say--right or wrong, she let them share their ideas. She also encouraged them to use a variety of methods to demonstrate their thoughts. In addition to verbal explanation, the students were seen using their hands and arms and also drawing angles on the big paper. These are things to keep in mind for future teaching.
Case Studies-Angles
1. In Nadia’s case 14 (lines 151-158), Martha talks about a triangle as having two angles. What might she be thinking?
I thought Martha made a very interesting observation. She seems to think that triangles only have two angles because they are made by adding two separate angles together. Going by the pictures she drew, she knows points A and C are both angles but does not see point B (angle ABC) as an angle. This was a little surprising considering her grade level. I imagine her looking at this triangle as an addition problem. In Martha’s eyes, perhaps taking 1 angle and adding 1 angle results in 2 angles (1+1=2).
I thought Martha made a very interesting observation. She seems to think that triangles only have two angles because they are made by adding two separate angles together. Going by the pictures she drew, she knows points A and C are both angles but does not see point B (angle ABC) as an angle. This was a little surprising considering her grade level. I imagine her looking at this triangle as an addition problem. In Martha’s eyes, perhaps taking 1 angle and adding 1 angle results in 2 angles (1+1=2).
2. Also in Nadia’s case (lines 159-161), Alana talks about slanted lines as being “at an angle.” What is the connection between Alana’s comments and the mathematical idea of angle?
It appears that Nadia knows that angles are drawn to look “tilted” or “slanted” but is leaving out a second line in which to connect it. Her underdeveloped idea of angle connects with a definition from mathisfun.com that says an angle refers to "the amount of turn between two straight lines that have a common end point (the vertex)". I think her ideas somewhat connect to this definition because her angles are “turned” or are placed “at an angle”. She seems to know a little about the position of lines but needs to learn more about angles having more than one line.
It appears that Nadia knows that angles are drawn to look “tilted” or “slanted” but is leaving out a second line in which to connect it. Her underdeveloped idea of angle connects with a definition from mathisfun.com that says an angle refers to "the amount of turn between two straight lines that have a common end point (the vertex)". I think her ideas somewhat connect to this definition because her angles are “turned” or are placed “at an angle”. She seems to know a little about the position of lines but needs to learn more about angles having more than one line.
3. In Lucy’s case 15 (line251), Ron suggests that a certain angle “can be both less than 90˚ and more than 90˚.” Explain what he is thinking.
I don't know about you, but it took me a while to really understand the students' thoughts and ideas about the picture with the skater. In saying that the angle in question can be both less than and more than 90 degrees, he is considering both perspectives—the degrees before reaching the skater’s position and the degrees after reaching the position. I think he realizes that all angles have two parts in terms of how one angle came to be. I think Sarah's comments helped him make this realization. She essentially said that it really depends on the way you are looking at the angle.
I don't know about you, but it took me a while to really understand the students' thoughts and ideas about the picture with the skater. In saying that the angle in question can be both less than and more than 90 degrees, he is considering both perspectives—the degrees before reaching the skater’s position and the degrees after reaching the position. I think he realizes that all angles have two parts in terms of how one angle came to be. I think Sarah's comments helped him make this realization. She essentially said that it really depends on the way you are looking at the angle.
4. In case 13, Dolores has included the journal writing of Chad, Cindy, Nancy, Crissy, and Chelsea. Consider the children one at a time, explaining what you see in their writing about angles. Determine both what each child understands about angle and what ideas you would want that child to consider next.
Chad: Chad seems to focus on the length of the sides in an angle but I think he might be overlooking the sizes of actual angles. I think what is confusing him is the fact that he is comparing the smaller angles (the tip of his pencil) with larger angles he sees (angles he makes with his arms). Clearly, comparing these angles would result in the appearance of different sized angles. As his teacher, I would want to question Chad on what characteristics he is focusing on--the side lengths or the angle in itself. I would also want him to explore the idea that angles can have certain lengths but still make a variety angles (I hope that makes sense).
Chad: Chad seems to focus on the length of the sides in an angle but I think he might be overlooking the sizes of actual angles. I think what is confusing him is the fact that he is comparing the smaller angles (the tip of his pencil) with larger angles he sees (angles he makes with his arms). Clearly, comparing these angles would result in the appearance of different sized angles. As his teacher, I would want to question Chad on what characteristics he is focusing on--the side lengths or the angle in itself. I would also want him to explore the idea that angles can have certain lengths but still make a variety angles (I hope that makes sense).
Cindy: Her mention of “slanted sides” might mean her understanding that angles appear to have points that come up and/or come down (like a ramp at an incline). However, she does not demonstrate knowledge that angles can take on different sizes, she just seems to focus on an angle's position in space. Perhaps I would have her consider other angles that have different sized "slants" to make them look different.
Nancy: Interestingly, Nancy also made a similar comparison as Chad about the tip of a pencil being a small angle. . However, she may be thinking in terms of the overall size or length of the angle. I think she understands that certain angles take on different characteristics but needs more help determining what these are. I would have her explore the types she may not be familiar with.
Crissy: Crissy seems to have a pretty solid understanding about angles. She knows the certain characteristics of the different angle types. She knows what certain types of angles look like and that they have different "degrees". However, we can't be too sure what she understands by using this term. This would be something to build on in future instruction for her.
Chelsea: I think Chelsea has some basic knowledge and understanding about angles but she needs clarification on certain rules and characteristics angles possess. Her drawings shows a misunderstanding that right triangles can have more than one right angle. I would want to work with her on reviewing and learning more about triangles. Doing this might help her understand characteristics of angles themselves--when they are not included within a triangle. Perhaps I would even go over other types of angles she did not mention in her writing.
5. In Sandra’s case16 (line 318), Casey says of the pattern blocks, “They all look the same to me.” What is he thinking? What is it that Casey figures out as the case continues?
I think Casey first considers the patterns blocks to be the same because he is only familiar with 90° angles. The pictures helped me see that he is not focused on making the block fit, he is simply trying to make the edge or corner of the square block meet with the edge or corner of the other shapes. Later in the case study, Casey learns that in order to visualize the 120 degree angle he could not understand, he had to add angles. It seems like he was unaware that it was possible to add angles to make new ones.
I think Casey first considers the patterns blocks to be the same because he is only familiar with 90° angles. The pictures helped me see that he is not focused on making the block fit, he is simply trying to make the edge or corner of the square block meet with the edge or corner of the other shapes. Later in the case study, Casey learns that in order to visualize the 120 degree angle he could not understand, he had to add angles. It seems like he was unaware that it was possible to add angles to make new ones.
How Wedge You Teach? TCM Article
I will be completely honest and say that it was really hard or me to follow along with some of the information presented in this article. The description of the students' demonstrations were really confusing to understand since I couldn't see what they were doing or really understand parts of the images they were pointing to. I thought the "wedge" activity itself was interesting even though I was a little lost at some parts. Nevertheless, I'm pretty sure I have come to understand the main ideas the authors were making.
What ideas will you take from this article into your classroom? Perhaps the most important thing I will take away from this article is the idea of using inquiry-based instruction in the classroom. I have always heard about the positive effects of using inquiry lessons but reading this class' experience with exploring angles with makeshift "protractors" really put things into perspective for me. The children seemed to learn so much from engaging in a particular problem that they had to work through for themselves (but in groups). Moreover, they each seemed to really hone in on their peers' ideas and offer their take on them.
Another thing I will take from this article is the idea of not rushing into teaching students the use of protractors. The most important thing is to make sure they understand the meaning of a unit of measurement before focusing on the use of this tool.
Lastly, it was really helpful to read this article and learn about the dilemma of whether or not to use traditional instruction before inquiry or vice versa. I found it interesting how experts recommend using the former approach because focusing on superficial characteristics of a mathematical problem can hinder students' ability to develop specific concepts. This was an interesting thing to consider because I am sure there are many teachers that are struggling to find an appropriate balance in their traditional and inquiry-based instructions.
Another thing I will take from this article is the idea of not rushing into teaching students the use of protractors. The most important thing is to make sure they understand the meaning of a unit of measurement before focusing on the use of this tool.
Lastly, it was really helpful to read this article and learn about the dilemma of whether or not to use traditional instruction before inquiry or vice versa. I found it interesting how experts recommend using the former approach because focusing on superficial characteristics of a mathematical problem can hinder students' ability to develop specific concepts. This was an interesting thing to consider because I am sure there are many teachers that are struggling to find an appropriate balance in their traditional and inquiry-based instructions.
Was there anything surprising about what you read that made you change your thinking about children’s understanding of calculating an angle?
As the students discussed their ideas about completing their group activities, I thought some of them gave some really interesting responses that demonstrated solid understanding of complex ideas for their grade levels. I was surprised by this and have learned that we adults don't really give children as much credit as they really deserve when it comes to learning about these things. It was only a little surprising to learn that other students lack the understanding of what units of degrees are when it comes to measuring angles. Moreover, it was surprising that many students at this age can't provide a reasoning behind why an angle is considered acute, obtuse, or right. Once I think about this, I really shouldn't be surprised considering that many people my age cannot provide an explanation, either. It was also interesting to learn that students first begin learning obtuse and acute angles before learning about right angles. They then use their previous knowledge of these angles to form accurate ideas about others.
As the students discussed their ideas about completing their group activities, I thought some of them gave some really interesting responses that demonstrated solid understanding of complex ideas for their grade levels. I was surprised by this and have learned that we adults don't really give children as much credit as they really deserve when it comes to learning about these things. It was only a little surprising to learn that other students lack the understanding of what units of degrees are when it comes to measuring angles. Moreover, it was surprising that many students at this age can't provide a reasoning behind why an angle is considered acute, obtuse, or right. Once I think about this, I really shouldn't be surprised considering that many people my age cannot provide an explanation, either. It was also interesting to learn that students first begin learning obtuse and acute angles before learning about right angles. They then use their previous knowledge of these angles to form accurate ideas about others.
What possible misconceptions might children have about angles and what misconceptions did you have about angles?
I learned a lot from this article regarding the extent of certain mathematical abilities students have. As mentioned earlier, many students are unable to explain characteristics of different named angles. I learned that children really don't even understand the concept of a degree of measurement when referring to these items. They simply think of the word "degree" as a label attached to a number of space and not as a specific unit.
For me, the student's discussion about same-sized circles having different number of wedges and angles was really interesting. I was wrong to assume that the size of a circle would change the number of wedges it could hold. It appears that I need further instruction on the idea that there is a a connection between an angle-measure unit and a fraction of a circle. Moreover, it was surprising to learn that before really understanding what a unit angle is, children must be able to identify the number of units in a circle. While this idea is still a little fuzzy, it has given me something to think about.
For me, the student's discussion about same-sized circles having different number of wedges and angles was really interesting. I was wrong to assume that the size of a circle would change the number of wedges it could hold. It appears that I need further instruction on the idea that there is a a connection between an angle-measure unit and a fraction of a circle. Moreover, it was surprising to learn that before really understanding what a unit angle is, children must be able to identify the number of units in a circle. While this idea is still a little fuzzy, it has given me something to think about.
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