Measurement Misconceptions
Why do you think the students are having difficulty?
I think the students are having difficulty reading the ruler. They do not understand that the 0 is the start of the measurement. When counting, students start with the number one and they are trying to do the same with the ruler.
What misunderstandings are they demonstrating?
They are misunderstanding how to compare the units with the attributes of the object being measured. Like the student who measure 41 inches. He knew that was wrong because he "thought the pencil was shorter".
Have you witnessed any students experiencing some of these same difficulties?
I have not witnessed this yet with students.
What types of activities could you implement that would help these children?
If I had students that were struggling with this concept, I would start the day with a Measuring Morning work. Give the student a chance to get familiar and perfect measuring one day at a time.
Have you seen a student struggle with measuring? If so, in what way? Were you able to work with teh child?
Angles Video and Case Studies
Discuss how the children in the video view angles. What ideas make sense and what ideas need further development?
One child described angles as "when two lines meet and it is the space between them". The students were able to show an angle using their arms. One student mentioned how the angle is at the opening of a cross, but an angle is considered at the small end as well as the larger opening. Another student mentioned that when you put your two hands together, an angle is created. She then uses her classmate as an example. The pushes his hands closer together and explains that moving these hands together creates a smaller space and that is a smaller angle as well.
One thing the students need to have a further discussion about is what makes an angle. The student drew on a board a curved line and a straight line creating a point. She said this creates an angle. Her classmates also agreed. I think the students need to go through and further discuss why this is not considered an angle and what can create an angle.
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?
I think Nadia is showing one angle which uses the bottom and one side. Instead of doing the same for the top angle, the student split that angle to distinguish the making of the second angles.
Have you come across a student that had a similar understanding of 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?
This one is a little difficult. A line represents 180 degrees, but I think the student is more referring to the slant line as one of the lines that form an angle.
3. In Lucy’s case 15 (line 251), Ron suggests that a certain angle “can be both less than 90˚ and more than 90˚.” Explain what he is thinking.
I think that Ron is showing an understanding that during the action, the skater started at one angle, then moves to another. On paper, if we showed the action on paper, the skater could show both 120 and 60 degrees next to each other on a line.
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 mentions how you have a small angle with two pencils together and a larger angle when you put your arms together. I think the thing that chad needs to consider is that the lines that are making the angle are different sizes, but the angles could be the same.
Cindy mentions how angles can be in any direction, but doesn't explain what an angle is. I would want Cindy needs to learn more about what an angle is, and how it they are formed. She doesn't show much understanding what an angle is.
Nancy shows a great understanding with little words. She understands that there are a variety of angles, obtuse and acute. I would move on with the student and show the student various angles that looks like the are different in size.
Crissy also has a great understanding of angles. She mentions acute, right and obtuse angles. She doesn't quite have a great description of an acute angle.
5. In Sandra’s case 16 (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?
Based on the images provided in the case study, I am wondering if the student is confusing the linear line as a 90 degree angle since that's whats in common of the three shapes.
Casey made the connection that a 90 degree angle is still a 90 degree angle not matter the size of the object creating it.
What do you think causes students to have confusion about angles when comparing the same sized angles from different objects or shapes?
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?
When the students began performing the activity, it amazed me that the students were able to use prior knowledge of angle measurement to help determine an angle. For example, one student said “The whole circle is 360 degrees; so if you cut it in half, that straight line would be 180 degrees. If you cut it into fourths, … each angle would be 90 degrees.” This is great use of their prior knowledge. I remember as a child, one thing that I did not understand was that a circle was an angle (360 degrees). That was a hard concept for me to grasp because I assumed angles were created by two straight lines that made a point. This could be something stdents struggle with when learning about angles.
Exploring Angles with Pattern Blocks
Post your responses to the questions posed within this task. 
Was this activity difficult for you? If so, which part?
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 grandmother to this day uses nonstandard measurements when cooking. My grandmother prefers this because she always said they never made appropriate measuring cups for the amounts she needed. So she always said measurements like spoonful, handful, a pinch, a tad bit, or even a "good" amount of something. It was measurements that she was comfortable and familiar with using.
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