Module 14

Hello everyone,

 

   We made it! It's hard to believe that it is already the end of the semester but good too know that we have leaned so much. I just wanted to let you all know that I have enjoyed working with you guys and reading your blogs over the past few months. I wish you luck in the coming semesters and in your future teachings!!

 

Circumference and Diameter-Annenberg Video

 

 

• Describe Ms. Scrivner's techniques for letting students explore the relationship between circumference and diameter. What other techniques could you use? 
       I think Ms. Scrivner did a great job of helping her students gain understanding of the concepts of circumference and diameter. After reviewing the terms, she introduced them to the hands on activity they later completed. I like how she let the students sort of help her explain the directions by showing them the charts each group would be using. She focused on helping them understand that each column indicated a different measurement--the circumference and diameter. The way these columns were laid out later helped them see a relationship between the two. The act of measuring and recording data for different items also helped students see connections between the numbers for the different columns. They saw that the numbers for diameter were smaller than that of the numbers for each item's circumference. When sharing these findings with the class, it was great that an item from each group's list was shared on he overhead. While focusing on the measurements of the cork with a circumference of 6 cm and a diameter of 2 cm that one group found, Ms. Scrivner prompted students to discuss a possible connection between the numbers. The class found that multiplying the diameter by 3, you would reach the measurement of the circumference (which implies that dividing the by 3 would result in the circumference). I like how the students were able to come to this conclusion on their own, without being directly told by the teacher. By completing this activity, they were able to gain practice with the idea of division. Moreover I could see how this lesson could be the precursor to a lesson about finding Pi  (π) (3.141592...) which is actually the ratio of the circumference to the diameter. Even though the students did not have decimals in their measurements, you must remember that they rounded their numbers to the nearest centimeter. Just a thought.
       
        Another technique that can be used to help students in this area is by measuring other circular objects with different colored pieces of string or yarn. Each group would cut the string at the length of what they measure. For example, a green string could represent the circumference of the object while a red string represents the diameter. Students can compare the lengths of the string by placing them side by side and observing the difference. Some children might see that it takes a little over three diameter string lengths to match the length of the circumference. I think this would be a good thing to use before introducing students to working with numerical measurements.

• In essence, students in this lesson were learning about the ratio of the circumference to the diameter. Compare how students in this class are learning with how you learned when you were in school. 
      I can't say that I remember learning too much about these concepts in school but I do recall much of the instruction in my middle school and even high school years. The students learned about the circumference and diameter using hands-on activities and group interaction. This is much different from how I learned these concepts because my teachers had us memorize formulas and definitions. As a young student, I would have been able to tell you how to find the radius and measure circumference and diameter but I wouldn't have been able to explain why certain formulas are used and why they work. I think that if my former classmates and I were engaged in inquiry investigation like Ms. Scrivner's students, we would have had a better grasp on the meanings behind the concepts.
• How did Ms. Scrivner have students develop ownership in the mathematical task in this lesson?
       As mentioned earlier, I really like how Ms. Scrivner allowed students to essentially guide the lesson. She asked individual students to look at the measuring data chart and share their thoughts on what they were going to do. I think this really helped them understand the objectives and remember directions. The activity itself gave students the freedom of choosing what they wanted to measure. They were responsible for working with their group members to take their own measurements and then filling out the data chart. I like how the video showed some students even taking ownership of their mistakes. For example, one group was seem measuring the height of the trash can for the diameter while another group decided to measure one classmate's head--which is actually an oval and not a circle. I like how Ms. Scrivner allowed students to work through these issues and make the necessary adjustments.
      The whole-group review time/closing also served as an opportunity for children to take ownership of the lesson because the teacher instructed students to share their findings with the class. In order to determine the relationship between circumference and diameter, the students worked together to share thoughts and ideas. I like how the teacher kept notes on the overhead to guide student thinking rather than just "feeding" them the answers. Essentially, this is how students developed ownership--through guidance and prompting.
• How can student's understanding be assessed with this task?
      I think students can be assessed with this specific task by observation and checking the data charts children fill out. The teacher is able to see which students understand the difference between circumference and diameter by watching what the are measuring. Similarly, it would be easy to spot discrepancies in student work when looking at the measurements on the paper. If their numbers indicate a longer length for the diameter, then this could indicate that they either put the numbers in the wrong box, measured incorrectly, or measured an incorrect part of the item. They could also be assessed on completing the division portion where they use calculators to divide the diameter by the circumference (C/D). If they provide the wrong answer or list an strange number, this could mean they require more time and practice with the activity or concept.

Circles and Pi-Annenberg Module 

 

I'm not going to lie. This was a really tough module! It's been a while since I worked with these concepts so I was easily stumped. Nevertheless, I did the best I could as I worked through each section.

Part AProblem A1: Use the designs to fill in the table below. For the circle, use string to approximate the circumference.

 

 

I don't know about you, but I didn't like this question/activity because it was really hard for me to measure the shapes accurately with the string. I couldn't hold it in place when measuring their circumferences...so I just looked at the solution. I was able to reach the correct answers for the diameter, but that was about it.

 

Problem A2: Look closely at the three designs. What patterns do you see in their measurements?

    I thought this was a bit of a confusing question but just by looking at the 3 shapes, it is clear that the diameter increases by 2 as the shapes get bigger. The sides of the hexagon increase by one. The solution shares more information but these are what stood out the most for me.

Problem A3:

a. For each design, how does the diameter of the circle compare to the perimeters of the square and the hexagon?
   

  In order to answer this question, it helped to write out the numbers and find the pattern. It is clear to see that for each design, the perimeter of the square is 3 times the diameter and the perimeter of the hexagon is 4 times the diameter.

b. For each design, how does the approximate circumference compare to the perimeters of the square and the hexagon?
   This one was a lot harder to solve. I tried to work it out on paper but I easily became confused so I had to look at the solution. After reading it, I did see that the approximate circumference of the circle lies between the perimeter of the square and the perimeter of the hexagon.

c. The circumference of any of these circles is about how many times more than its diameter? If a circle had a diameter of 7 cm, what prediction would you make for the length of its circumference? Why?
   

   As you can see, I was able to determine that the circumference of these circles is about 3.15 or 3.2 cm more than their diameters. Knowing this, I took 3.2 and multiplied it by 7cm to make my prediction. I was correct to say that a circle with a 7cm diameter would have an approximate circumference of 22cm.

 

Problem A4:
   a. For each object, estimate the circumference. Then measure the circumference and the diameter in centimeters to the nearest tenth (e.g., millimeters). Use string or a tape measure. Record your data in the table.

b. Examine the table. What do you notice about the ratio of C to d? Based on these data, what is the relationship between the diameter and circumference of a circle?
      As seen throughout this module, dividing the circumference by the diameter produces the same, or similar values. In this case, all of my values are 3 or very close to 3.

Problem A5:
a. Enter the values from the table for diameter and circumference into a graphing program in your computer or into a table in your graphing calculator to make a scatter plot. Use the horizontal axis (x) for diameter and the vertical axis (y) for circumference. Graph the points. What patterns do you see in the graphical data?

b. What information does a graph of these data provide?

    Even though I have a graphing calculator, I will say that I completely forgot to use this particular feature so I skipped it this problem.

Problem A6: Find the mean of the data in the C/d column. Why find the mean? Does the mean approximate ?

I wasn't so sure why it was necessary to find the mean but I did learn that it is to ensure that no errors were made in previous problems. By finding the mean, I was able to reach a value that was close to Pi.

Problem A7: The symbol r represents the radius of a circle. Explain why C = • 2r is a valid formula for the circumference of a circle.
    Now that I know the reasoning behind Pi, it makes sense why this is an accurate formula to find circumference. We know that we have to multiply Pi by the diameter to find the circumference. 2r is another way to represent diameter so essentially, this would  read C=Pi times d. I hope this makes sense.

Problem A8: An irrational number cannot be written as a quotient of any two whole numbers. Yet we sometimes see written as 22/7 or 3.14. Explain what the reason for this may be.
This question was a little confusing but peeking at the answer helped me a little. It does makes sense that we use these two things in various situations. There are times when you are unable to measure real items using π.

Problem A9: Since π is an irrational number, can both the circumference and the diameter be rational numbers? Can one of them be rational? Explain using examples.
  This question was tough, too. I learned that both of these items cannot be irrational at the same time. It can only be one or the other. The circumference can be irrational which would make the diameter rational. Vice versa, the circumference can be rational while the diameter is irrational.

Problem A10: When mathematicians are asked to determine the circumference of a circle, say with a diameter of 4 cm, they often write the following:

C = π  • d = π • 4

In other words, the circumference of the circle is 4 cm. Why do you think they record the answer in this manner? Why not use the π key on the calculator to find a numerical value for the circumference?
     I would say that just typing these items into the calculator would not yield accurate answers. Mathematicians often look for precise numbers, not approximate numbers

 

Part B

 

 

 

 

 

 

 

 

 

 

 

Problem B1: How does the area of the figure compare with the area of the circle?
    The area is still the same. The only difference is the appearance because the circle has been cut up and arrange in a different way.

Problem B2: The scalloped base of the figure is formed by arcs of the circle. Write an expression relating the length of the base b to the circumference C of the circle.
    I was super confused by this question. I had no idea how to form this as an expression but the solution helped to some degree. I take it that the expression would be C/2 because the top and bottom scallops make up half of the circumference. (??)

Problem B3: Write an expression for the length of the base b in terms of the radius r of the circle.
  Again, I had to look at the answer for this question. It made a little sense that since the new figure is 1/2 the circumference, then you would take 1/2 of the radius. So instead of it being 2πr^2, the new formula would be πr.

Problem B4: If you increase the number of wedges, the figure you create becomes an increasingly improved approximation of a parallelogram with base b and height r. Write an expression for the area of the rectangle in terms of r.

Think about how the activity involving wedges helps explain the area formula of a circle, A = π • r2.
   This question lost me! From reading the solution, I can see why the more number of wedges makes a much straighter line--simply because there are not as many scallops in the top and bottoms. I am just a little confused as to why, that if this is different,  πr is still being used. Nevertheless the correct formula for this problem would be π X r X r.

Problem B 5: Use the circles (PDF document) to work on this problem. For each circle, cut out several copies of the radius square from a separate sheet of centimeter grid paper. Determine the number of radius squares it takes to cover each circle. You may cut the radius squares into parts if you need to. Record your data in the table below.
  I understood this activity but I'm pretty sure that "Circle 1" on the PDF document is wrong. It does not align with the numbers that are given in the solution. Even the sample picture above shows Circle 1 with 6 shaded squares but the PDF only shows 5. For the chart below, I have included the answer that is shown on the solution page. However, I did not understand the part about how many more squares are needed, even though I used the grid paper for his problem.

 

Problem B6:a. What patterns do you observe in your data?
    It is obvious to see that in order to get the radius of the square, you have to square the radius of a circle. In these cases, squaring 6 results in 36, squaring 4 results in 16, and squaring 3 results in 9. I can also make a connection between needed 3 extra radius squares to fill the circle. I connect this to the idea of having a little over 3 diameter lengths to equal the circumference. Or--Pi.

b. If you were to estimate the area of any circle in radius squares, what would you report as the best estimate?
    I would say that since you are going to need a little over 3 squares each time, you would want to use π for the best estimate.

Problem B7: Does the activity of determining the number of radius squares it takes to cover a circle provide any insights into the formula for the area of a circle?

   I'm not sure if I completely understand the connection or even if I'm able to put my thoughts into words but I will give it shot.
   We know that when using grid paper, you can fit a certain number of the given r^2 squares into the given circle. When finding area of a square you are using r^2. So it makes sense to use π x r^2 to find the area of the circle. You would use multiply by π because you need to calculate the 3 or so extra squares.

Problems B8-B10
  I was not able to answer these questions because I was confused by them. I honestly have never worked with giving circles a scale factor of different numbers and even fractions. I feel like I need a lot more exposure and practice with these concepts before being able to solve them.

 

For Further Consideration...

 

We have explored numerous areas throughout this semester. Choose five ideas that you know you will use later in the classroom and write about them on your blog.

     I will say that I have learned a lot of valuable information from this course that I will undoubtedly take with me in my future teaching career.  While it's hard to pick out the most important things, I have chosen the five most interesting things I learned and experienced.

1) Use of manipulatives
   While I have experienced working with certain types of mathematical manipulatives, I have never used some of the ones we worked with from our kits for this course. Things like the pentomino pieces, the tangram puzzle, and that Mira tool (my favorite!) were all very interesting and sometimes helpful items to use. While they helped me in some areas and less in others, I have come to realize that my future students may understand important math concepts when these items are available and taught. I think this is just something to keep in mind.

2) Integration with other subject areas    As a teacher, I know time will be a very precious thing--it will not always be on my side. For this reason, I have learned that it is often necessary to integrate math with other subjects to save some time. One module prompted me to think about integrating geometry with history by having students examine patterns in the items made by different cultures. Many of the NCTM articles described the incorporation of math with science and history. The children's literature project allowed me to think about integrating math with language arts and reading. I think this is super important, not only because it engages students, but because there are many great books that encourage children to learn about math.

3) Data collection  In the first part of this course, it was interesting to learn about how children go through the data collection process. Many of the articles and videos gave some great ideas to use when teaching this concept to them. It was helpful to learn about the different stages children go through when visualizing and representing data. I now know that it is important to give students plenty of opportunities to see data displayed concretely before introducing them to more abstract ways.

4) Use of technology
   It is no secret that we live in such a technology-rich world. I have no doubt that most of the children that enter my classroom will be computer savvy, know how to use the Internet, and will be familiar with using applets on tablets. Many of the modules in this course have shared great examples of mathematical games, applets, and videos children can access to reinforce information they learn in class.

5) Inquiry-Based instruction and learning
    As a visual and hands-on learner, I have always appreciated the fact that some of my teachers used a lot of engaging activities to help me learn and understand information. As a future educator, I hope to give students adequate opportunities to learn in this way. Many of the articles and videos we have watched in this course highlight the use of inquiry-based instruction where students investigate a problem and seek answers using their own ideas and mathematical knowledge. It is clear that using this kind of approach can help make learning math more fun and meaningful for students, especially in the subject of mathematics. I will be sure to remember this.

 

 

Module 13

Hello again everyone,

   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?

    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).

 

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.

 

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.

 

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).

     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.

 

  

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.  

 

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.

  

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.

Module 12

Hello classmates,

   I hope all is well with everyone! It's hard to believe that we have two more modules to complete of this course! I learned so much from module 12 and will trust that all of you have as well. As always, you will find my thoughts and ideas about each section of the module. The required parts are bold red.

Introduction to Measurement

 

This was a helpful PowerPoint and introduction to the concept of using measurement. Before viewing the video, I considered the question that was asked at the beginning:

 

Write down all the ways that you have used numbers in the past 24 hours.

- Tell time on a clock or my phone. Setting my alarm to wake up.

- Entering in my login ID and password to get into Blackboard and my e-mail.

- Checking my grades!

- Helping students with math problems in my field experience classroom
- Reading the speed limit signs on my way to and from various places
- Texting and calling other phone numbers

- Checking the mileage on my car to see if I'm due for an oil change or tire rotation.

- Reading my scores in my Words With Friends games

- Determining how many days are left in the semester (and how much more time I have to complete all of my assignments). Eeek!

- Using my debit card at the store. I had to enter my PIN. Also, I was given a receipt which told me the amount of money I spent

-Checking the current balance in my bank account

- TAXES!

    I am sure that there are many other instances throughout the past 24 hours where I (and you) have used numbers in some kind of way. I think this helps us see that we use numbers for a variety of reasons and that we read them without even think about it or realizing it. Since we need numbers for such a long list of tasks in a day's time, it is evident that teaching children to understand how to read and apply them is crucial. Just a thought.

 

    As prompted, I looked back at my list and thought about which ones involved measuring. Just as Dr. Hargrove explained, it seems as though that about all of my items involve measuring. Whether it's telling time, calculating distances, balancing amounts, or counting money, all of these things require measurement. Very interesting!

 

   It was helpful to learn about the important points of students using measurement. It is clear that in order to successfully use measurement, they must understand the concept and the tools it uses, have measurement sense (familiarity of standard units), and a developing ability to use their sense of measurement.

 

   It was great to have a quick overview of the 3-step measurement process as I really can't remember much from my elementary math days. The PowerPoint did share a useful point--that length is the first unit of measurement children will learn. It was also interesting to learn that younger children should be taught about nonstandard units of measurements but gradually are introduced to standard units like rulers and measuring cups as they enter upper grade levels. It makes sense to do this because their capacity to understand such a concept at a young age is developing. I also did not know about the importance of having students use estimation in the measurement process for it can be beneficial to them in many ways. It's great that this can add motivation and interest as they are challenged in a measuring activity. This is something to keep in mind.

 

    I read the classroom vignette about the student's difficulty with a measuring activity and considered the following questions:

What are some of the issues that the children are having with the task they are asked to complete?

   Its seems as though the children are having trouble placing the ruler at the right spots on the pictures they are measuring. One big problem they have is making mistakes when counting the numbers for measuring on the ruler. Henry demonstrated this when he counted every tick mark on the ruler as an inch, which gave him a very large measurement for his answer. Madison failed to line up the ruler with the picture correctly because she started at the 1 inch mark instead of the 0.

 

What issues have you seen with the students in the classrooms you are working?

   I actually have yet to observe students as they work with measurement concepts with rulers. The class I am working with now has been learning about telling time (a type of measurement) and some still struggle with it. Nevertheless, I'm sure that these students experience similar issues with measuring as the students in the vignette did.

 

What activities/lessons could we do with children to help them with their measurement

understanding?

    I think the best thing to do is to go over the measuring process again. This could mean that you give students another method of measuring the object to compare their ruler measurement. I think children also need to review the appropriate way to measure objects by showing them how to count and read a ruler.

 

Why do you think the students are having difficulty?

   I think they are struggling with this assignment simply because they don't understand how to read the ruler. It seems as though they need more support and assistance with this concept.

 

What misunderstandings are they demonstrating?

   Again, they are misunderstood on where to place the ruler and how to read the marks. They think that they have to place the 1 inch mark at the beginning of the object when they are supposed to start at 0. They are unaware of how to read they marks on the ruler because some of them counted each one to measure the pictures.

 

Have you witnessed any students experiencing some of these difficulties?

  I have yet to see students struggle with measuring objects using standard measurements like rulers but I am sure that I will see this in the Fall semester when I begin student teaching.

 

Annenberg: What Does It Mean To Measure? Parts A, B, & D

 

 

Part A



The first task involved learning about measurement using a rock! Below is a picture of the rock I used in this activity (ruler for scale):

 

 

Problem A1: Make a list of attributes that could be used to describe the rock.

- My rock has jagged edges. It is not smooth.

- My rock has a mixture of black and gray colors

- It is not very large in size as it can fit in the palm of my hand but it is not small like a pebble, either.

- It is very light in weight

 

Problem A2: Some of these attributes might be measurable, and some might not. How do we determine what we can measure?

  I think you have to consider each attribute and think about what unit of measurement you would use to describe it. I would think you would look for standard units of measurement. For example, I would use a ruler or other type of tool to find the dimensions of my rock (inches, centimeters, millimeters, etc.). I would use a scale to measure the weight of my rock (pounds, ounces, etc.). Some attributes like texture and color are difficult to measure, although I'm sure there are ways to do it.

 

Problem A3: If you were to compare different rocks using each of the measurable attributes you listed in Problem A1, what units would you use?

    I think the two biggest attributes I would compare would be length (in inches) and weight (in ounces). Those seem like easiest things to measure.

 

Problem A4: How could you measure these properties?

  I could measure these properties using a standard ruler (inches and centimeters) and a scale (to measure for ounces).

 

Part B

At first, this section was a little confusing for me. Despite my struggles, I tried my best to work thought the activities and questions.

 

Problem B1: How could you use the tinfoil to find the surface area of the rock? Why would you use this technique?

   I think you would simply take a piece of tin foil that is big enough to wrap around the rock. Before wrapping it around, you would have to measure the size of the foil to get an idea how much area you are covering.

 

Problem B2: What unit will you use? Is there more than one choice? Explain.

    Since my rock is not that big but not that small, I decided to use the middle sized unit (.5 cm) to measure the surface area. If the rock was bigger, I would probably consider using the 1 cm measuring unit. If it was smaller, I would use the .25 cm sized unit.

 Note: For the measurement of the rock's surface area and volume, I did not have access to the necessary materials at the time I got to this part (I was away from home). I will revisit this section later.  

The final section of Part B involved measuring the rock's weight:

Problem B7: What information can you gather by using a two-pan balance? Can you determine the weight of your rock with this balance?
If you have access to a two-pan balance, use it to determine the weight of your rock.
   I think the biggest thing you can determine with the two-pan balance is whether or not two different objects have the same weight. If not, you can determine which object has the greater/lesser weight. I would unable to determine the weight of my rock because the balance does not measure the weight. It might give you an estimate but only if you knew how much the other object weighed.
*I am unable to test my object since I don't have access to a two-pan balance.*

Problem B8: How does a three-arm balance scale work? Can you determine the weight of your rock with this balance?
If you have access to a three-arm balance, use it to determine the weight of your rock.
  It looks like this device works much like the kind of scales they use at a doctor's office. You place the object on the scale and slide the small weight to balance the measuring arm. Whatever numbers the small weight slide on is how much the object itself weighs.
*I am unable to test by object since I don't have access to a three-arm balance.*

Problem B9: In science, a distinction is made between mass and weight. What do you know about these two terms?
  Mass refers to the amount of space an object takes up. Weight refers to the heaviness of the object.

Problem B10: How precise are your rock's measurements? What might affect the precision of this measurement?
  I am unable to measure my rock with the instruments. I was not too sure about what would affect the precision of the measurements so I had to look at the solution. Essentially, precision is affected by the pan weights used in the two-pan balance. This make some sense to me.

Problem B11: Now that you've experimented with several different types of measures, which would you use to determine the largest rock in a group of rocks? Should you use a combination of measures?
    Based on my experiences with working with each type of measurement, I think measuring the volume of a set of rocks would help determine which rock is the largest. Then again, I think it really depends on what is meant by "largest". Sometimes people say that and want to know which one is heavier rather than biggest in size. As we have learned, there are ways to test for the volume, surface area, and weight of objects so people can use these methods to answer their own definitions.

Problem B12: There are very interesting relationships among metric measures involving water. One cubic centimeter of water is equivalent to 1 mL of water. In addition, 1 mL of water (or 1 cm³ of water) weighs 1 g. You may then conclude that the amount of water your rock displaced should be equivalent to the weight of your rock. What is faulty about this line of reasoning?
  This was a confusing question so I had to peek at the answer. It now makes sense that this would not be an appropriate method to use because the rock does not have the same density as water.

Ordering Rectangles Activity

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.

  By just looking at all of the rectangles, I think rectangle D has the smallest perimeter simply because it doesn't look like it doesn't it would take as long to go around the outside of the shape. I think A would have the largest. I could be wrong but here is the rest of my lineup (from smallest to largest):

D, E, C, B, F, G, A

 

 

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. 

  By just looking at all of the rectangles, I think rectangle C has the least area simply because it appears to be smaller and take up less space. I think G will have the largest. I could be wrong but here is my guess in rectangle area order (from smallest to largest).

C, D, B, E, A, 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? 

   To find the correct order of the rectangles for perimeter, I used the colored tiles from the manipulative kit. I took the tiles to see how many were used to cover the rectangle. I then counted the sides of each individual tile to find the perimeter. The tiles weren't the exactly the same unit measurements but they helped me determine the ascending order of the rectangles' perimeters. I found that two rectangles shared a perimeter of 14 and three rectangles shared a perimeter of 16. Here is the correct order I found along with each rectangle's perimeter.

 

 C     E     D     B     A    G     F

 10   14    14    16    16   16   18

 

 

    My estimated order was not too far off from the true order of the rectangles. While I did have 3 correctly placed rectangles, I did not have correctly choose the  . I did not suspect that so many rectangles would share the same perimeter. To compare the orders, I used the numbers from the tile method with my ordering to see if I was right.

(Mine) D, E, C, B, F, G, A   

 (Actual) C, E, D, B, A, G, F

 

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? 

    Again, I used the colored tiles to measure the areas of each rectangle. I did this but counting how many tiles it took to cover each rectangle. I discovered that two of the rectangles shared an area of 12. This is the order I found:

 

                                                           C    D     E      B      F     A     G

                                                           8    10    12    12    14    15    16

 

     My estimated order for area was a lot closer than I had expected it to be. I had the correct placement of 3 rectangles (C, D, and G). I just had the others mixed up. I did correctly estimate which rectangles would be the smallest and largest. To compare the orders, I used the numbers from the tile method with my ordering to see if I was right.

 

 (Mine)    C, D, B, E, A, F, G   

(Actual) C, D, E, B, F, A, G

 

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?

     First of all, I learned that you can’t rely on your eyes as an accurate form of measurement. It might look like one rectangle is bigger than another when really, it’s not. It was rather hard to make my own estimates on what order to place the rectangles. As I did this, I questioned my own judgment but figured out how important it is important to have a way to test measurement using a standard device to check the measurements. In this case, I used the colored tiles as the measuring device. The activity also helped me see that it is easy to get the process of finding perimeter and area mixed up (even for me). When counting the perimeter for each rectangle, I started out by accidentally counting all of the tiles, which would lead to the area. I think this is something to keep in mind when teaching children about finding both things. I’m still wondering about the best ways to introduce children to the idea of using measurement and finding things like area and perimeter. In my own experiences, I recall some struggle when learning about these concepts.

 

TCM Article - Measure Up

What ideas will you take from this article into your classroom?

    I think this article presents some interesting information about helping students learn about measurement. I like the idea of allowing students to explore measurement by posing a real problem (having them compare lengths on the wall) and not just questions they complete on a worksheet. Moreover, I like how the article describes students using various instruments to learn about length, volume, weight, and mass. I think it’s important for them to learn and become familiar with devices like rulers, balancing scales, measuring cups/beakers, etc. As the article points out, students need these skills in order to find measurements to compare. In my future classroom, I will strive to give students plenty of opportunities to do this.

Was there anything surprising about what you read that made you change your thinking about children’s understanding of unit or using a ruler?

   Perhaps the most surprising thing I got from this article is that the comparing process seems a bit more complex for first graders than I would have thought. It is interesting to notice a lot of algebraic concepts brought forth in the lesson. For instance, the children learn about comparing objects by assigning them variables and equations (i.e. Length Q, R=Q, etc.). Moreover, it was interesting to read about the notations and statements that were used to describe iterations of units. I had no idea that young children were able to comprehend such concepts. I don’t think I even learned about these things until I reached middle school (this is when letters were added into math equations!). Nevertheless, it was pleasantly surprising to see students in the article be able to understand and successfully complete the assignment.

What possible misconceptions may children have about measurement?

  Aside from the misconceptions described in the module PowerPoint, this article explains that young children first lack the ability to explain how an object is larger or smaller. The students in the article also worked through their misconception that measurement simply involves comparing the length and size of objects. While this is an important aspect, there is so much more involved. I think this is why the MeasureUp program involves the use of symbolic statements and nonspecific quantities. Students don’t really understand that there are relationships among different units.

 

Case Studies-Length

 

1. What ideas about measurement do the children in Barbara’s class (case 12) bring to school before they are taught about it?
    I first noticed that Barbara’s students seem to have a very limited understanding of how to describe the size of an object. Throughout the case, they are observed answering questions using one common word: “big”. I believe that young students like this enter school with the idea that describing objects as “big” or “very big” is perfectly acceptable. I think it’s easy to see that young students are familiar with comparing objects to describe the overall size of something. For example, many students explained that the box in the classroom was big as a part of the carpet or easel while others compared the size of the box to things outside of the classroom—such as a tree and King Kong. While they clearly need more time to develop accuate ways to describe object sizes, I think they had their own ways to communicate their own ideas of “tallness” or “bigness” of the box. Moreover, I think this was a somewhate successful lesson despite the limited language skills the students had.

 

2. Many children struggle with the idea that the larger the unit, the fewer the number of units needed to cover a length. Go through the cases by Rosemarie (case 13) and Dolores (case 14) to identify how different children are making sense of this issue.

     There were many different ideas that came from students in these two cases.  Some had the right idea to say that the less steps you take, the bigger your feet are. Dayna explained her idea that there were different results because some students had different sized feet. It was interesting to read the student’s thoughts on who had the biggest feet. Isaac gave an inccorect answer by saying that Adriana and Kim were tied with the biggest feet because they took 10 ½ steps each. This was interesting considering they both took ½ MORE than Gita which means their feet were a tad bit smaller. I think Courtney make an easy mistake by saying that Miriam had the biggest feet because she took the most number of steps. It is evident that she is associating “bigness” with high numbers—something that children are normally taught.

    Dolores’ students had similar issues. The problem the teacher posed in line 282 about George and Fred generated a lot of different responses. Some students said that George had the larger feet because he took more steps. I really liked the child-adult homework assignment Dolores gave to her students because I think it made the difference in feet size more apparent to the students. It helped many of them reach the conclusion that because their feet were smaller than their parents’, they needed more of their feet to measure the distance of the objects. Les seemed to have the most exciting realization in the lesson. As the teacher used the hands of other students to demonstrate the concept, he quickly realized that if your hand is small, you must use your hand more times to measure the table. He successfully pointed out that big hands and feet will have smaller numbers because they don’t take up as much space—you need more of them. I think he and many other came to understand that in this case, less really is more!

 

3. In Dolores’s case, line 245, Chelsea notices that Tyler and Crissy both measure the width of the basketball court as 62 “kid feet.” Why didn’t everybody measure the width as 62 kid feet? What discrepancy is Chelsea noticing? What is Henry noticing? How are their observations related to the issue that arises in Sandra’s seventh-grade class (case 17)?
      Chelsea and Henry are both noticing that the children who measured the same “kid feet” for one area (basketball court, baseball field) had different measurement for the other areas. I think they understand that these students’ numbers should be the same because they probably have the same foot sizes but are confused by the large variety of answers from different students with these apparent similar sizes. They simply felt that the numbers should be the same all of the time.
     Many students gave ideas of why everybody did not measure 62 kid feet for the width, or why all didn’t measure for the same steps for any area for that matter. They later suggested that maybe the numbers were different because some students lost count of what number of steps they had taken. I think it was great to see some students realize that the numbers were different there were a variety of foot sizes in the class. As this time, they were close to understanding that some students had to take more steps because they had smaller feet.
     Like Dolores’ students, the 7^th graders in Sandra’s casequestioned why they had more/less steps to walk the distance in a similar activity. Students like Deirdre were able to explain that the answers were different because they each had different paces when walking.

 

4. The children in cases by Mabel (case 15) and Josie (case 16) are working out the use of standard tools for measuring length. Specifically, the children in both classes discuss how to place the tool and how to read the number of units. What do the students have to say about these two issues? What do they understand about measuring with accuracy and precision?

      It seems that children in both classes seem to have a lot of different ideas regarding the appropriate uses  measuring tools. Mabel’s students explain the importance of using making sure you use the ruler appriopriately by not holding your place with your finger because it can add more length to your measurement than you need. Other students talked about using the ruler “flipping” strategy to avoid losing your mark when measuring. It was insteresting to read student’s thoughts on what to do when the object you are measuring is shorter than the tool you are using. I think this is where a lot of students demonstrated a number of misconceptions about measurement. Students like Poonam and Maya suggested that if the object ends on or between the “little lines”, then you might decide what number is closer to the middle line. Despite their struggled with this, I do think they are aware that you can gather incorrect measurements when you use measuring devices incorrectly—they just need more time and exposure on how to take accurate measurements.
      I think Josie’s class struggled with similar problems. They simply need more experiences with using and reading the measuring tools. For example, Marcus was observed folding over the tab on the tape measure to collect his data. Other students seemed to use more accurate methods to measure a certain object. Adam used the tape measure to filing cabinet instead of using a ruler multiple times. Later on in the case, I thought it was interesting to read Robby’s comments about “starting with zero” on a ruler when measuring. To support this claim, other students like Deb make great points to say that starting at 1 on the ruler implies that you have already measured one inch—that the end of 1 means the end of 1 inch. It seems like this class still needs some more work on learning to create accurate and precise measuring method. While Adam’s idea to tape meterstick together to measure the height of the wall was clever, he learned that this was not the best way to complete the task since he had a little more length to measure. To me, this suggests the need for more time and practice.

5. By comparing the cases from second, third, fourth, and seventh grades to Barbara’s kindergarten (case 12), can we identify ideas that, by the older grades, are understood by the children and no longer warrant discussion. What are some issues that still lie ahead for Barbara’s students to sort out?

I feel like many of the other classes struggled with similar issues as Barbara’s class, but only to some degree. The classes that did the measuring activities with their hands and feet seemed to be stuck on the idea of “bigness” and comparing sizes as the Kindergarteners did. To me, this indicates that they still need exposure to using standard units of measurement to describe the size of an object. For example, the 1st graders in Rosemarie’s class (case 13) described the foot sizes of certain students by saying things like Gita’s foot was bigger than other students’. It was difficult for these students to see the relationship between someone’s foot size and the number foot-lengths it took them to walk the length of an object. While this class has a better understanding about how to measure “bigness”, they are still developing more accurate ways to compare them to other units of measurements, something that Barbara’s class really wasn’t able to explain.
     Other classes seemed to have a better understanding than Barbara’s students about making precise and accurate measurements with different tools and not random objects like chairs and plastic tubs. These older students also emphasize the importance of appropriately setting up the ruler/tape measure in order to get an accurate reading—something that Barbara’s students will learn later on in their education. While the Kindergarteners and some of the upper graders had little to no knowledge of using measuring tools, the older kids demonstrated knowledge of standard units of measurements. Because they were able to understand that using a range of foot sizes would yield a range of answers. Clearly, Barbara’s students will have to work through the common misconceptions about size and units of measurements as they grow.

Just a note: I think that many of the things brought forth in these case studies highlight important points made throught this module. The PowerPoint and the NCTM article talked a lot about the process in which students learn about measurement. The first step involves giving students something to use to compare to other objects. The case studies clearly show this  because students are using various units to measure objects. Be it feet, hands, rulers, or random items around the room, students used them to compare the size of objects and distances.

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 think it is interesting to think about the many different words we use in our everyday langauge that represent some form of measurement. Once I really think about it, some of the words we (or that my family does) often use aren’t really considered accurate forms or standard forms of measurements. I have providede a short list of the ones that come to mind:

 1)      Sliver—a small slice of something. “Just cut me a sliver of that cake”.

        2)      Hair/tad—a very, very short distance or amount. “Move that picture up a hair/tad to make it level”.
        3)      Bunch—a handful. “Grab a bunch of those paperclips from my desk”.

 I believe that there are some instances where using these nonstandard forms of measuremtns are necessary. Oftentimes, there is no accurate measuremnent to describe how much or how long of something you want. You can’t really be sure of the exact measurement of the piece of cake you want and you can’t really measure right off how mamy millimeters or centimeter you need to move a picture to make it level on the wall. It just seems much easier to use words like these.