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.