Module 13
Hey Jess! I hope you're having a great week!
Measurement Misconceptions
Why do you think the students are having difficulty?
The students are not using the ruler properly causing confusion. They are not realizing that the ruler shows different units of measurement.
What misunderstandings are they demonstrating?
While they understood where to place the ruler to begin measuring, they were thrown off by different lines leading to different units of measurement. Madison wanted to start at zero rather than one. The students seem to be misunderstanding the general layout of a ruler and could benefit from a review of what everything on the ruler means. As teachers, it is easy to assume that students know these things, but we need to be careful to look out for small, simple misunderstandings like this.
Have you witnessed any students experiencing some of these same difficulties?
Unfortunately I have not observed any units involving measurement in my field work.
What types of activities could you implement that would help these children?
As I mentioned above, I think the students could benefit from a review of the tools they are going to use before they use them. It may be a little time consuming to talk about the parts of a ruler and what it means, and it could be redundant to some students, but it would clear up confusion before they begin measuring. Sometimes taking the time for a quick review like that can help avoid problems later on. Do you think taking the time to review the use of a ruler would be beneficial, or would it be better for them to learn through their mistakes?
TCM Article – Rulers
What ideas will you take from this article into your classroom?
Was there anything surprising about what you read that made you change your thinking about children’s understanding of unit or using a ruler?
Discuss the possible misconceptions many children have about measurement?
This whole activity is definitely something I would use in my classroom! Using different units, even nontraditional units, can help students understand that units differ and how measuring with a larger unit will lead to a smaller number answer, and vice versa. As I discussed above, teaching these basic concepts may seem obvious to us as teachers, but it is so important to break it down for students who have never used these tools before. I have not really had any experience teaching measurement yet, but this did reinforce that simple things still need to be taught in-depth to build a foundation of understanding. A few of the biggest misconceptions I have seen in this class when children are learning about measurement is that they do not know what to do if they are measuring something that exceeds the length of the ruler, and that they to not understand why using a bigger unit won't give them a bigger number. These are pretty easy things to demonstrate and help them with if we take the time to do so.
Angles Video and Case Studies
Discuss how the children in the video view angles. What ideas make sense and what ideas need further development?
Respond to the following questions after you read the case studies:
1. In Nadia’s case 14 (lines 151-158), Martha talks about a triangle as having two angles. What might she be thinking?
Martha didn't seem to understand that ABC was also an angle. She was only looking at the two equal angles.
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?
Alana is making the connection that an angle has at least one slanted line, but she is using the term "angle" more as terminology and not really in its true definition. However, this will help her to understand that parallel lines will never form an angle and at least one will have to be slanted for them to intersect and form an actual angle.
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 think Ron is thinking of lines that intersect. If we have two lines making an X shape, two of the angles can be less than 90 degrees and two can be more. I am not sure if I'm clearly explaining my thought process on this, so let me know if I'm not making sense and I can try to word it better!
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.
I believe that Chad understood that the lengths could be different sizes, but I was a little unclear about what he meant when he said they could be long or short. I think he is trying to say that angles can be large or small, but I'm not positive. I would work with him on terminology next. Cindy understands that an angle can be rotated or flipped and still makes the same angle. She demonstrates that she knows that, but does not really demonstrate knowledge about angles having different sizes. I think the next step for her would be associating angles with different degrees and discussing them in different sizes. Nancy understands that angles are pointed and that small angles are called acute and big angles are obtuse. I do not think, however, that she has any connection to what degrees create which angles, and I would focus on that next. Crissy, however, does demonstrate knowledge that angles associate with different degrees and that obtuse is larger than 90 and acute is smaller. As her teacher, I would want to ensure that she know that 90 degrees is a right angle and she knew what that looked like, to make sure that she could recognize these things rather than just memorize them. Finally, Chelsea seems to know the terminology, but she does not have the correct angles associated with that terminology in her mind. She seems to have memorized things like right, acute, and obtuse without really knowing what they mean or being able to recognize them. I would want to work with her as well on associating these with actual degrees and being able to recognize what these angles look like.
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?
Casey is only looking at the square, so he sees that all of the angles making up the square are equivalent. He finally realized that the 90 degree angle and the 30 degree angle make the 120 degrees that his peers got.
How Wedge you Teach?
What ideas will you take from this article into your classroom? Was there anything surprising about what you read that made you change your thinking about children’s understanding of calculating an angle? What possible misconceptions might children have about angles and what misconceptions did you have about angles?
I really enjoyed reading about the wedge activity and I think the fifth graders I worked with this semester would have loved it! I love the idea of using an inquiry approach with mathematics and I think that allows the students to get more out of the lesson. I hope to use similar approaches in my classroom when applicable. I feel like I have been working with angles for so long that a lot of this article was surprising to me. I think that we will find that a lot in our first few years of teaching. Students will always think through things in a way that surprises us, or have misconceptions we would not have even considered prior. That is one of the reasons I enjoy articles like this so much, because it helps to prepare me for what may come in my future classroom and it reminds me to keep an open mind in regards to student understanding. Even in the beginning of this article, it was surprising to me to think that while students may recognize a right angle, acute angle, or obtuse angle, they may have no connection as to what degree those angles may be. They may have no idea that 90 degrees is equivalent to a right angle. This is one of those times where it may be even more challenging for the students to learn the vocabulary terms first, because while they recognize the angles based on the definition, they do not correlate them with certain degrees. Do you think knowing the definition of a right angle first without associating it to 90 degrees is harmful or helpful when teaching angles?
Exploring Angles with Pattern Blocks
Green: This is an equilateral triangle with the sum of the angles being 180 degrees, making each angle 60 degrees,
Blue Rhombus and Red Trapezoid: The acute angles are 60 degrees and the obtuse angles are 120 degrees.
Tan Rhombus: The acute angles are 30 degrees and the obtuse angles are 150 degrees.
Yellow Hexagon: The angles are all 120 degrees, making the total 720.
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.
I can't think of anyone who uses strictly nonstandard measurements rather than standard, but I can think of plenty examples of people using them as mentioned in the checklist! Using nonstandard measurements is much more convenient in casual conversation or when estimates are acceptable. For example, when pertaining to food, some recipes will call for an exact amount of something, but things like salt can be to taste. It all depends on the circumstances. People also round a lot when discussing measurement. Rather than taking the time to pull out a tape measurer, they will say that they saw a deer not 10 feet from the road. Or they will say its a quarter till two, when really it's 1:47. In the classroom when teaching math, most things need to be exact in order to measure correctly and find the right answer, but I think this is part of the reason we teach rounding to students. Rounding teaches us how to estimate closely. When we are using nonstandard measurements, while we do not need the exact amount, we need to be speaking in a way that gets our point across. Can you think of times where nonstandard measurement is used in the classroom, or specifically math, rather than standard measurement?
Measurement Misconceptions
Why do you think the students are having difficulty?
The students are not using the ruler properly causing confusion. They are not realizing that the ruler shows different units of measurement.
What misunderstandings are they demonstrating?
While they understood where to place the ruler to begin measuring, they were thrown off by different lines leading to different units of measurement. Madison wanted to start at zero rather than one. The students seem to be misunderstanding the general layout of a ruler and could benefit from a review of what everything on the ruler means. As teachers, it is easy to assume that students know these things, but we need to be careful to look out for small, simple misunderstandings like this.
Have you witnessed any students experiencing some of these same difficulties?
Unfortunately I have not observed any units involving measurement in my field work.
What types of activities could you implement that would help these children?
As I mentioned above, I think the students could benefit from a review of the tools they are going to use before they use them. It may be a little time consuming to talk about the parts of a ruler and what it means, and it could be redundant to some students, but it would clear up confusion before they begin measuring. Sometimes taking the time for a quick review like that can help avoid problems later on. Do you think taking the time to review the use of a ruler would be beneficial, or would it be better for them to learn through their mistakes?
TCM Article – Rulers
What ideas will you take from this article into your classroom?
Was there anything surprising about what you read that made you change your thinking about children’s understanding of unit or using a ruler?
Discuss the possible misconceptions many children have about measurement?
This whole activity is definitely something I would use in my classroom! Using different units, even nontraditional units, can help students understand that units differ and how measuring with a larger unit will lead to a smaller number answer, and vice versa. As I discussed above, teaching these basic concepts may seem obvious to us as teachers, but it is so important to break it down for students who have never used these tools before. I have not really had any experience teaching measurement yet, but this did reinforce that simple things still need to be taught in-depth to build a foundation of understanding. A few of the biggest misconceptions I have seen in this class when children are learning about measurement is that they do not know what to do if they are measuring something that exceeds the length of the ruler, and that they to not understand why using a bigger unit won't give them a bigger number. These are pretty easy things to demonstrate and help them with if we take the time to do so.
Angles Video and Case Studies
Discuss how the children in the video view angles. What ideas make sense and what ideas need further development?
Respond to the following questions after you read the case studies:
1. In Nadia’s case 14 (lines 151-158), Martha talks about a triangle as having two angles. What might she be thinking?
Martha didn't seem to understand that ABC was also an angle. She was only looking at the two equal angles.
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?
Alana is making the connection that an angle has at least one slanted line, but she is using the term "angle" more as terminology and not really in its true definition. However, this will help her to understand that parallel lines will never form an angle and at least one will have to be slanted for them to intersect and form an actual angle.
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 think Ron is thinking of lines that intersect. If we have two lines making an X shape, two of the angles can be less than 90 degrees and two can be more. I am not sure if I'm clearly explaining my thought process on this, so let me know if I'm not making sense and I can try to word it better!
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.
I believe that Chad understood that the lengths could be different sizes, but I was a little unclear about what he meant when he said they could be long or short. I think he is trying to say that angles can be large or small, but I'm not positive. I would work with him on terminology next. Cindy understands that an angle can be rotated or flipped and still makes the same angle. She demonstrates that she knows that, but does not really demonstrate knowledge about angles having different sizes. I think the next step for her would be associating angles with different degrees and discussing them in different sizes. Nancy understands that angles are pointed and that small angles are called acute and big angles are obtuse. I do not think, however, that she has any connection to what degrees create which angles, and I would focus on that next. Crissy, however, does demonstrate knowledge that angles associate with different degrees and that obtuse is larger than 90 and acute is smaller. As her teacher, I would want to ensure that she know that 90 degrees is a right angle and she knew what that looked like, to make sure that she could recognize these things rather than just memorize them. Finally, Chelsea seems to know the terminology, but she does not have the correct angles associated with that terminology in her mind. She seems to have memorized things like right, acute, and obtuse without really knowing what they mean or being able to recognize them. I would want to work with her as well on associating these with actual degrees and being able to recognize what these angles look like.
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?
Casey is only looking at the square, so he sees that all of the angles making up the square are equivalent. He finally realized that the 90 degree angle and the 30 degree angle make the 120 degrees that his peers got.
How Wedge you Teach?
What ideas will you take from this article into your classroom? Was there anything surprising about what you read that made you change your thinking about children’s understanding of calculating an angle? What possible misconceptions might children have about angles and what misconceptions did you have about angles?
I really enjoyed reading about the wedge activity and I think the fifth graders I worked with this semester would have loved it! I love the idea of using an inquiry approach with mathematics and I think that allows the students to get more out of the lesson. I hope to use similar approaches in my classroom when applicable. I feel like I have been working with angles for so long that a lot of this article was surprising to me. I think that we will find that a lot in our first few years of teaching. Students will always think through things in a way that surprises us, or have misconceptions we would not have even considered prior. That is one of the reasons I enjoy articles like this so much, because it helps to prepare me for what may come in my future classroom and it reminds me to keep an open mind in regards to student understanding. Even in the beginning of this article, it was surprising to me to think that while students may recognize a right angle, acute angle, or obtuse angle, they may have no connection as to what degree those angles may be. They may have no idea that 90 degrees is equivalent to a right angle. This is one of those times where it may be even more challenging for the students to learn the vocabulary terms first, because while they recognize the angles based on the definition, they do not correlate them with certain degrees. Do you think knowing the definition of a right angle first without associating it to 90 degrees is harmful or helpful when teaching angles?
Exploring Angles with Pattern Blocks
Green: This is an equilateral triangle with the sum of the angles being 180 degrees, making each angle 60 degrees,
Blue Rhombus and Red Trapezoid: The acute angles are 60 degrees and the obtuse angles are 120 degrees.
Tan Rhombus: The acute angles are 30 degrees and the obtuse angles are 150 degrees.
Yellow Hexagon: The angles are all 120 degrees, making the total 720.
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.
I can't think of anyone who uses strictly nonstandard measurements rather than standard, but I can think of plenty examples of people using them as mentioned in the checklist! Using nonstandard measurements is much more convenient in casual conversation or when estimates are acceptable. For example, when pertaining to food, some recipes will call for an exact amount of something, but things like salt can be to taste. It all depends on the circumstances. People also round a lot when discussing measurement. Rather than taking the time to pull out a tape measurer, they will say that they saw a deer not 10 feet from the road. Or they will say its a quarter till two, when really it's 1:47. In the classroom when teaching math, most things need to be exact in order to measure correctly and find the right answer, but I think this is part of the reason we teach rounding to students. Rounding teaches us how to estimate closely. When we are using nonstandard measurements, while we do not need the exact amount, we need to be speaking in a way that gets our point across. Can you think of times where nonstandard measurement is used in the classroom, or specifically math, rather than standard measurement?
Hey Megan!
ReplyDeleteI think it would beneficial to review rulers before because either way there is still going to be mistakes made. Then, when students make a mistake they can still learn from it and their peers. Reviewing first may tie up some loose ends and eliminate some possible mistakes. I think it is always important to continue to encourage our students through their mistakes and help them learn from it, and to never be embarrassed. This makes me think of when my teachers would make mistakes and it would put me at ease, knowing they make mistakes too and that it is okay as long as we grow and learn from them. Your next questions is interesting and I have really been debating sides in my head. As we saw, angles are a complex portion of math education so I think it may differ student to student if knowing the definition and associating it with 90 degrees is more beneficial or not. Personally, I probably would briefly give a definition but also associate it with 90 degrees, giving students more to visualize and more knowledge to connect together. I like the examples you gave and agree with the importance of being more exact in the classroom. My mom always used to say its a quarter to, quarter of, quarter after, and I would have to think about it for a second, especially if I saw on a clock it was not exactly that time. I think it never hurts to be exact in the classroom, we could point these non-standard units of measure out to our students to make them aware in case they hear them, but I think I would stick to more exact or standard units most of the time. Especially since their exams and such will more than likely have standard units. Great post, Megan!