Ricchio Week 13

The student work I examined for this week's blog was a homework assignment my students completed. The homework is on "Change Number Stories" where students use a change diagram to represent problems in which a starting quantity is increased or decreased. The question gave a number story that asked, "Becky ate 11 grapes. Later in the day she ate 7 more grapes. How many grapes did she eat in all?" The student I examined said the total was 4 grapes and wrote 11+7=4 as her number model. She was the only one with this answer and I found it interesting since she answered the other number stories correctly. What I think this student was doing was she used the numbers given (7 and 11) and found the difference giving her the answer 4. What interested me most was that she wrote her number model as 11+7=4.

One way to advance this student’s thinking is to ask her again if 11+7=4 and see if her response is the same. A future math task I could give her is to count backwards and find the solution to a problem where she actually needs to find the difference. Then I would have her do a problem with finding how many there are of something all together by adding. This will help this student pay attention to the wording of problems and understand when to use addition or subtraction. 

King Student Work

This week students were working on divisibility tests. For example, how to determine if a large number, such as 33,992, is divisible by 7, 11, or 13. The steps for these tests were provided to students in writing with a sample number to test on a worksheet. Then students were asked to practice these tests with another large number on the worksheet.

One student who I helped complete this worksheet struggled to understand the steps for each divisibility test and, in turn, struggled to apply the steps to another large number. I walked through the steps with the student, showed the student what each step looked like, and made sure that the words that were used to in the explanation of the steps was clear to the students. The student still acted confused in an attempt to get me to give her the answer, but she eventually realized that I was not just going to give her the answer and she was able to figure out the answers with my support.

This interaction tells me that this student is beginning to understand the concept of testing divisibility, but does not have the confidence to do it on her own. It also tells me that she is used to being able to "play stupid" and then being given the answer by the teacher or a classroom, which is a gap in many students' general math understanding in my classroom.

One way to advance this student's mathematical understanding is to continue to push her to figure out answers on her own. I can also advance her mathematical understanding by supporting her in times when she is struggling and help her to become more comfortable in situations where she is unsure of herself or her understanding.

Student Work- Robb

This week my students worked on measuring skyscrapers, which they constructed from paper. This was not part of my own unit but part of the Math Trailblazers series. Because my students had had experience measuring with non-standard units during my unit, I was interested in seeing how the students would fair with this activity.

I took down several notes when conversing with students about their procedures for measuring and their thinking, sort of like an informal post-assessment. The following are quotes that I found the most inspiring.

I went to one of my students who is generally not that strong in Math, first. I wanted to see if this particular student had a good understanding of how to measure with non-standard units because he did struggle during my unit.
"Can you tell me how you measured?", I said.

 "Yeah. Ummm, like, well you have to start at zero? Right?" the student said.

"I think so, but can you show me?"

"Well, I was thinking *this* is zero, like the bottom." he said this as he pointed to the bottom of his skyscraper.

"Good!" I said. "So then what do you do?"

"Then I go, 1-2-3-4-5 and I make the links go next to the tower." He placed the links end-to-end, right up next to the tower.

"So then, how tall is it?" I asked.

"It's 5, 5 things tall." He said smiliing.

This student placed his links end-to-end, he did not connect them. Another student I worked withhad a different approach.

When working with one of my female ELL students, I asked her to go through the process of measuring with "links" from step one. She began by saying, "First, you get your links. And then you can put them together if you want."

"You should make the links taller than your skyscraper", she said as she built a link chain quite tall. She then moved the bottom of the chain to line up with the bottom of her paper. "It should go here, if you don't put it here you aren't looking at the right spot." Then, looking at her full length chain, she took off the excess chain at the spot where her building ended.

"What do you do next?" I asked.

"I taked it away here because this is how big it is. Then I count the links and I get 6."

I think this is an essential thing to do, you have to let your students think about their thinking and you have to accept their answers, no matter what variations they come in. If I had told either student that they were "wrong" or told them to measure any other way, I would have defeated them. I would have shot their confidence. I did not see how connecting or, not connecting, the links made either answer more or less valid. Students working from their own ideas are students who are exploring and making the connections that build true mathematical thinkers. When we tell our students to only do things one way, we aren't truly doing our students, or mathematics as a field of study, any justice.

Week 14 Bode

          My first graders were working on telling time to the half hour earlier this week. The picture I have attached is a student’s work from their Everyday Math journals. For the most part this child is doing an average job on the assignment and in the classroom. I know this child and know that sometimes he doesn’t listen to directions so he can get confused or need the directions repeated when we go back to work at the desks. As I reviewed his work I noticed that he may know how to count by 2s (in problem number 3 page 38) however he didn’t follow the directions that you had to start at 12, 14, 16, and continue. He would have gotten the whole problem wrong if we were grading this. So it’s great that he was able to count by 2s but he has to follow directions and see where the number pattern is beginning. Telling time on the clock is his biggest struggle I’d say. He is normally good, as you can see, with telling time to the hour. He can see where the hour hand is pointing and also making the clock that needs to have that time. However, when it comes to telling time to the half hour this is where he has his difficulties. It seems that he gets the hour hand and minute hand confused. He isn’t sure which hand shows the “half-past” and which hand shows the hour. In the first clock on the top left you can see that he says its half-past 5. He got confused which hand was which and thought that the minute hand was the hour and so he felt that was close to 5. You see in the middle clock on the right the child was supposed to write the “half-past” part in and realize it wasn’t 8 o’clock but instead it was half-past 8 o’clock. This child saw that the hour hand was close to the 9 so anticipate that that’s why they wrote it was 9 o’clock but didn’t see the minute hand to check the half hour.

            I would say this child needs some help with telling time and realizing which hand means what and maybe more clarification as to what the word “half-past” means as well. One strategy I would use to help this child advance in their understanding would be to look at the model clock is go over the concept of how many minutes are in an hour (60) and that when you look at the clock doesn’t it look like circle (yes). So when you cut the circle in half straight up and down, what two numbers do they cut? (12 and 6). The 12 is what we point to when we say o’clock and it means that it is a whole hour. The 6 is what we point to when we say “half-past” because the 6 is half of 12 like we can see in this cut in half circle of the clock. The next thing I’d want to do is make the child try to remember the hour and minute hand by color-coding them. Maybe red would be the hour hand and blue could be the minute hand. By doing this the child will then be able to visualize how the hour and minute hand would and that they are not the same hand but in fact they mean something very different to each other. Hopefully these two ideas will help this student be able to visualize and understand the wording we use when referring to telling time.

Maxwell-Week 13

Last week, my students started a new unit in math that focuses on measurement.  Although my students were able to understand the importance of using standard units when measuring, they held several misconceptions about measurement.  During one of our lessons this week, students were asked to measure a line segment in their student math workbooks.  The students were provided with 12 inch rulers. These rulers have both centimeters and inches written on them.  My MT announced to the class that the workbook is asking for a measurement of the line segments in inches, not centimeters, so it was important that they make sure their ruler was on the "right" side. 
Nearly all of my students were able to recognize the difference between centimeters and inches, and they quickly flipped their rulers around to begin measuring.  While watching one particular student, I noticed that he had measured several line segments starting with the "12 inch" end of the ruler.  I asked the student how he was measuring his lines.  He replied, "I am measuring in inches".  "Great job", I said.  Then, I asked why he was starting his measurements from the 12 inch end.  The student stated that since 12 was as big as the ruler got, it had to be a smaller size than that, so he was "measuring down".  His explanation made sense.  He could recognize that the line segment was smaller than 12 inches, therefore, the number of the measurement would be smaller than 12.  Measuring from zero made it seem like the line was getting bigger and thus had the possibilitity of getting a measurement above 12 inches, which my student knew was not a logical answer. I explained that we start from zero, because it will give us a clear answer to our measurement.  If the line ends at 5, then it is five inches long.  However, if we were to measure from the 12 side and the line stopped at 7, we wouldd have to perform a subtraction problem (12-7) to find our measurement in inches.

McCormick Week 13

The piece of math work that I examined was of a student's pre-assessment for the upcoming unit. In particular, I looked at one question that asked the child to fill in the missing numbers in the frames-and-arrows model. The rule that was given to the student was "count back by 2s". The first frame was filled in and showed the number 32. The student was required to fill in the next 5 frames using the rule. The student labeled the arrows above each frame to show -2. She filled in "31, 30, 49, 48, 47" in the missing frames. This student clearly knows how to analyze the rule and put it in symbol form. She knows how to label each arrow to demonstrate what she will do for each frame. She does not yet understand how to fill in the frames appropriately to show what the rule is asking for. She counted back by 1s from 32 to 30 and then did the same again from 49 to 47. She did not show that she knew how to count back by 2s.

One way I may advance this student's thinking is to ask her to count back by 2s from 50 and I would record each number she says. Another way to advance her thinking is for her to circle all of the numbers she says as she counts back by 2s on the number grid. This would help the student identify what numbers she is actually counting. One question I may ask her is "What made you count back 32, 31, 30 and then 49, 48, 47?" A future math task that may be beneficial for this student is work with other skip counts and count them backwards. Eventually, she will recognize that she is saying the numbers that the frames-and-arrows task is asking for.

Montague

Yesterday our students worked more with addition and subtraction problems. We started the lesson off by doing a quick warm up with cards on the board. We started with 5 cards and put 3 more up. We asked them how many we had all together. The students counted them all and gave the answer. As they were doing this, my MT went over how to write the problem with numbers and draw dots next to it to match. Then we followed by reading a book called 16 runaway pumpkins, which gave us an introduction to the subtraction problems. After the book, I modeled different subtraction and addition problems that were drawn on the board. I then had students come up and do it with me and followed by having students come up and work together to do it. Then they were sent back to their seats to work on the worksheet on their own.

I looked over all of the students’ papers and for the most part, the majority of the class performed rather poorly on this activity. I think that they must not have had enough time before going off on their own or did not listen to directions. The paper that I am going to focus on is one where a student who got all of the subtraction problems correct but missed all of the addition problems. This artifact obviously reveals that he has a good grasp on subtraction but not necessarily for addition. I do not believe it means he does not understand addition, I think it could mean that or it could be that he was in a hurry and miscounted those. I thought that this student’s worksheet was actually quite interesting because I have typically considered addition to be the ‘simpler’ of the two operations therefore to see this student do so well on the subtraction and so poorly on the addition it was rather surprising.

One way that I might advance this student’s mathematical understanding given my analysis of this work is to give this same worksheet again, with more clear directions. I would also like to see if more time would help. Then if it still shows the same results, I would be more concerned that he does not understand addition. However, I do not think I would be worried. The fact that he can understand the subtraction problems makes me feel confident that he will be able to catch up with the addition but just needs a little more practice and instruction. Therefore, another way I might advance this student’s mathematical understanding is give more opportunities for him to explore with addition problems. I would give him manipulatives to work with and even some centers in which he can practice with addition problems. I think that a good high level or open ended addition problem may be good for this student so that I can see where his understanding lies in terms of addition. 

Cosmas Student Work

This piece of student work was an in-class assignment done by a second grade student.  The first part of the page deals with money equivalencies which my MT has been working with students on a lot.  The bottom portion of the page deals with using three digits to make the smallest and largest number possible.  First, this student does something interesting on the top portion of the page.  Number 2 asks for how many pennies are equal to 1 quarter (25).  This student puts 23.  At first glance this leads me to believe she has some misconception about how much a quarter is worth OR the 1 to 1 correspondence that pennies represent.  However, what’s interesting is that in numbers 3 and 5 she proves me wrong.  In number 3 she correctly identifies that the equivalency of 2 quarter is 10 nickels.  Here she demonstrates that she understands 2 quarters equals 50 cents and that 10 nickels is another way to arrive at 50 cents.  Then, in number 5 she correctly identifies that it takes 75 pennies to equal 3 quarters.  Taking numbers 3 and 5 into account makes it difficult to support my initial claims about her performance on number 2.  This student may struggle with the 1 to 1 correspondence of a penny, but she was able to conduct it easily when the question was about 3 quarters.  This student also may struggle with how much a quarter is worth, but she answers correctly when asked to put this in nickels so that leads me to believe my assumption would be false again.  This makes me think that her response for number 2 may have just been a careless mistake and needs revisiting. 

              I also noticed some mistakes on the bottom portion of this assignment.  The first two examples she completes correctly with the smallest and largest numbers, but the next two she answers incorrectly.  One example has the digits 5, 3, 8 and she writes the smallest number as 538 and the largest as 583.  This is puzzling to me because my MT has been working on place value with these students extensively lately.  In the two previous examples though, she starts with the smallest number in the hundreds and the largest number in the hundreds when appropriate.  This leaves me wondering why she used the 5 as the digit in the hundreds place for both the smallest and largest number.  She does make a smaller and larger number in the 500’s but not the largest number she could’ve made.   The student does something similar in the next example as well.  The digits are 7, 6, 9 and she creates the smallest number correctly 679, but as the largest number she writes 796 failing to recognize that the largest number would begin with 9 in the hundreds place not 7.  Though she made a larger number than 679, it is not the largest.  This is interesting because in the second example, the largest number requires a 9 in the hundreds place and she does so correctly.  So why did she not recognize that here?  Her performance on these two examples leads me to wonder if: 1. She didn’t understand the task was to create the largest number possible, or 2. If she wasn’t sure how to construct the largest number beginning with the hundreds place.  I would assume that she understood the assignment due to her performance in-class as well as on the first two examples, but I can’t be sure.  From my observations I believe that she understands which digits are larger but did not fully comprehend how to construct them in a manner that would make the largest number.  In the last example, since 7 and 9 are only one number apart, she most likely chose 7 to use first and then made the largest number in the 700’s that she could. 

Student Work Blog: The Past Few Weeks in GLT

          Although my summative assessment of my unit plan was a reflection of where my students excel and where they still need work, I am able to see my third grade students’ growth throughout my unit (through formative and summative assessments). For example, I see many students understand what place value means and an example of place value. This is an important question because through this, I am able to see student thinking and understand how the students can explain what place value is in their own words. Not only do I understand that my students know how to explain place value, but I can see throughout my summative assessment they use their knowledge of what place value means in order to answer questions related to place value and think about how to solve a problem on a deeper level. 

            After evaluating my summative assessment of my unit plan, I am able to identify where my students need more support. My students still need work on rounding numbers and estimating sums and differences. During this part of my unit, I reflected that students did not understand the mathematical terminology sum and difference. Throughout the unit, I made a conscious effort to use these mathematical terms and reiterate and restate what they mean when using them. For example, on my student’s summative assessment, I noticed many of my third grade students were still unsure of what difference means (and in many cases, to subtract the larger number first). 

            On day three of my unit, I gave my students an exit slip as a formative assessment. While reading through exit slips and my summative assessment, I found student work that showed me my students started to master addition with regrouping. Through student work, I am able to better understand that not only do the students know how to solve an addition with regrouping problem numerically, but they understand how to physically represent the value (to further show they understand what “carrying” the one really means and how it is physically represented). Although many of my students have shown understanding for this concept, I would like to ask my students further questions such as: What does the one mean when you "carry" it? How can we represent this physically? 

Markert-Student Work Blog

This week my students have been working on addition and subtraction. We have been working on the idea of “hopping up” and “hopping back.” The students have been using the number line to demonstrate answering word problems and number models. The students have been using the number line to also show their thinking and demonstrate their personal understanding of addition and subtraction.  We have been doing a combination of all class examples and also individual work. The students have overall demonstrated a strong understanding for this concept. They are able to show the hops on the number and also write the number model most of the time accurately. One area that they have been struggling with in terms of the number model is the addition or subtraction sign. Sometimes the students mix up which one to use even if they demonstrated “hopping up” or “hopping back” correctly. I believe this is due to a lack of understanding of what the sign actually means. This may come with time and practice to familiarize themselves with these symbols.

Even though the students have been working with the number line, counting by 5’s and counting by 2’s there still is some confusion on how to show this. Today I noticed while walking around the room that the students were having a difficult time demonstrating hops on a number line to show counting by 2’s, counting 5’s and counting by 10’s. Many of the students did not know where to start or what the question entailed. Even after explaining to the students what the question is asking for many students were just making hops on the number line. I believe that maybe the students are not use to showing counts by 2’s, 5’s, and 10’s. If you ask any student in my class they are able to tell you counts of these numbers but something about showing their understanding of this proves difficult. One student I sat down beside him and asked “ Sam*, can you count by 2’s for me?” He replied counting by 2’s all the way to 20 (number line ended at 20.) When I asked him to then show the hops he immediately became confused. I am wondering if they were just not use to showing their understanding of this or if consistent hops made things confusing for them. I believe one way this could become a higher-level task is by asking the students to explain why they made hops the way they did. This would allow the teacher to see what they are thinking and where the confusion lies. Since this was just a workbook page even having the student verbally explain their thinking about the problem would be beneficial and informative. To me, it seemed that the students just did not understand what to do but maybe they had thinking behind it. It could also help me explain to them where an error may occur. I also think it would be beneficial to do a few examples of these types of questions on the whiteboard prior to the worksheet to help any misconceptions that may occur simply by not understanding the problem. 

Week 13 Bode

          This is a piece of work from one of my low level students. We were working on patterns and I was trying to work with her to help her complete this worksheet but not help too much because I wanted to see what she could do on her own. This first grader in my class often needs a lot of support and normally goes to the reading interventionist who also works with students on math. The students were asked to use the shape templates to draw what comes next in these patterns. Even using the pattern shape templates she had a hard time tracing and figuring out which way the triangle would go for the first couple problems. I don't know if it was the template that made her confused as to which was the triangle should go but maybe this is a sign that the template hinders some children's understanding of what comes next in a pattern because they need to visualize it without a template. Once it was explained to her by me that we need to look at the pattern and right what comes next like "the triangle points up, then down, then up, then down, what comes next?" I think her hearing the pattern helped her recognize it. It was the same for the next two problems, I read the patterns out loud again so that she could hear the pattern and continue it. In problem where it says make up your own she seems to start off knowing how to do a pattern but tries to add a new shape which tells me she's not quite sure how to make her own pattern. She does a triangle, parallelogram, a rhombus then another parallelogram when it is supposed to be a triangle. so it looks as though she was trying to make a pattern with the three shapes but confused the order of the pattern.
          One solution would be to have her read the patterns and tell it to me out loud instead of me reading it to her. This way she is engaging herself in the work and understanding the pattern as she reads it herself. If she can read out loud what the pattern looks like then I think that will be able to help her stay focused on the task because she's explaining it to someone and also hearing the pattern and if there is a mistake present. Another solution that I think will be helpful to her is giving her multiple examples of patterns that are true patterns and then ones that aren't and she has to identify what they are. By doing this she can start to see and explain to herself why a pattern would work and then why it wouldn't. After having her see completed patterns, have her add on to them so that she can get practice in finishing the patterns. Once she has practiced finishing the patterns and said them out loud to correct herself I want her to try to make one up again. Then have another student help her determine if it's a true pattern or not so that the little girl can correct her work. A lot of work should be done to help this little girl understand and create her own patterns!

Behrman_Student work

This example of student work was a homework sheet where students were directed to break apart the number they were subtracting into tens and ones. Since some students had struggled with a similar activity previously, I hand-wrote an example of how students were to break the number down at the top for clarification. However, this student's sheet shows two possible areas of difficulty that resulted in her poor performance on this homework task. One possibility is that she was confused by the directions and would have benefited from additional examples, including an additional modeled example prior to passing out the homework.

Another possibility is that she is confused about strategies for breaking down numbers. Previously we have talked in class about breaking down numbers into making friendly numbers (when the subtrahend is a single digit number). Now that we are using two-digit numbers, the focus is on breaking them into tens and ones in order to more quickly use mental math (or number lines or hundreds charts) to solve the problem. Even if she was focused on breaking down only the number in the ones place, many of these numbers are not accurate ways to break the number down.

One next step for me would be to talk with this student and have her discuss her strategy breaking down numbers. Sometimes she attempted to break the two-digit number down and sometimes only the number in the ones place, so I would like to know how she decided which number to break down. A second step would be for her work with this same type of activity but use coins to show me how she would break this down. By using this visual representation, I think that she would be able to more accurately and confidently break down a number in tens and ones. Also, it would give her a real context to apply this concept to and a chance to continue practicing it beyond this one assignment.

Ricchio Student Work

In math, my students are working on frames and arrows. I have been teaching them how to apply different rules to number sequences. For homework, they were given a set of number sequences, some with rules given and some with rules missing. Its a good way for students to practice applying addition and subtraction with a pattern. The students work I chose to look at has a "plus 5" rule. The first two boxes are blank, the middle box has 17, and the last two boxes are blank. The correct sequence should be 7, 12, 17, 22, 27, however this student put 27, 22, 17, 12, 7. Therefore, her numbers were correct, however their placement was not. I could see how she could be confused since she didn't know whether the numbers were going up or down on either side, but the arrows were going forward so she should have been adding 5 to left and subtracting 5 on the right. It was also interesting that she was the only student who put the numbers in that order. I would like to ask her about why she chose to put the numbers in certain positions to get see her reasoning. I would also give her a similar question and see if she did it again.  I could provide her with more patterns to further her learning and explain that the direction of the arrows is an important thing to notice.

McCormick Student Work

This week students worked on telling time to the half hour. I taught students about telling time to the half hour by showing them the clock and reviewing times to the hour. We worked with half hours on the large demonstration clock, on small clocks that students had, and in writing on the whiteboard. Most students appeared to understand the concept of telling time to the half hour. Students worked in their math journals and were supposed to write the time such as "half-past 4 o'clock" under the clock that showed 4:30. The particular student work that I examined was of one boy. Below the clock that showed 2:30, he wrote "half-past 6". He wrote similar answers for other problems.

This student currently understands that the time on the clock does not show a time to the hour. He understands that when the minute hand points at 12, the clock shows whichever hour the hour hand is pointing to. He understands that this particular clock problem does not show 6 o'clock. The gap in his understanding is that he does not understand how to read the clock hands to find out that the time is half-past 2. He may think that when the minute hand points to 6, that the time is showing half-past 6. Another gap in his thinking is that he did not take into account the hour hand at all. He does not yet understand that the minute hand that used to always be pointing to the 12 is now pointing to the 6. This may be a confusing concept for him since we have only worked with times to the hour up to this point.

A way to advance his thinking would be to have him compare two clocks that show different times using the same hour. He could compare a clock that shows 2:30 with one that shows 2:00. This would help him to notice the placement of the minute hand. A further way to advance his thinking would be to have him compare two clocks that do not have the same hour and have him state the times. These two times could be 4:00 and 9:30. This may help him to see more examples of times and how to read the times. One question I may ask him if I had the opportunity would be to explain his thinking behind writing "half-past 6" on his journal page. I want to know which hands he is looking at while he is reading the time. One future math task that he would benefit from is hearing a time said to him and showing the time on his own clock. This would help him to maneuver the clock hands in a way that helped him remember where each hand belongs.

Cosmas Student Work

Over the past two days in my classroom we have been working on “change to more”/ “change to less” problems and their appropriate number models.  As anticipated, this was a mildly difficult task for me to teach and for students to understand.  We started with change to more and using the specific language of “add 5” and “plus 5” to introduce the + and = symbol in creating number models.  We went through this process step by step by having a start number, a change of “add X” and an end number to create the number model.  Students used their number grids to start at a number, and manipulate the “hops” or added amount to arrive at their ending number.  This was fairly easy for students to grasp because manipulating the number grid and getting bigger is something they are very familiar with.  However, once we moved to change to less, students had difficulty hopping backwards specifically doing a backwards sweep. 

              The piece of student work I will be looking at is an example of the difficulty encountered when doing change to less problems and counting back on the number grid.  This sheet was given during independent work time and at first the student did every example as a change to more and added on completely ignoring the “minus” clue and all the examples we had done in guided practice together.  Once I explained again that minus means take away, and when we take away we move back on the number grid the student started over.  I sat with him for the first problem which required him to start at 15, and “minus 4”.  He accurately ended at 11 and wrote the appropriate number model with a subtraction and equal sign, so I moved on to other students with the intention to return in a moment.  When I came back I noticed something strange.  He had some correct responses where he didn’t have to complete a reverse sweep (ex: from 11 up a row to 10) and others were about 10 off.  I asked him to show me how he completed these problems.  One example was start at 22 and “minus 6”.  He began correctly at 22 and his fingers followed this route, “22, 21, 30, 29, 28, 27, 26”.  As I watched him do this I realized that as he needed to make a sweep from 21 to 20 he instead swept to 30, a bigger number.  When I asked him if it made sense to go from 21 to 30 he agreed it didn’t but wasn’t sure where he should’ve gone instead.  We went over the process of counting back together until he showed me independently he understood the concept.  The mistake he was making leads me to believe he simply had a misconception not about how to count back but which way to go about manipulating the number grid.

Estimating Inches- Kayleigh Robb

In my measurement unit, we have worked on making estimations about height in inches an feet during several lessons. This Tuesday, I asked my students to act as architects and to design the buildings they would want to create for our little-giant city. I then asked them to estimate how many inches their buildings would be and watched intently as they began their first designs.

            One student really took me aback when I witnessed her making her estimation. She took her thumb and index finger and began counting how many of those measures tall the building she had drawn would be. I asked her to do it again when I witnessed it because I was extremely intrigued by her thinking. I had not explicitly modeled that skill in particular, I had just given counting up as a short example of how you could estimate.

            On her own thinking she said, “I take my fingers, and this is like, an one inches, and then I make them go up like with the candies and then I get nine. So nine inches.” I smiled so big right then, I could not help it.

“I love how you thought about that!”, I said, “Do you think you could show everyone? I think that is a great way to estimate!”

            I had the student show how she made her estimation to the whole class. I was excited when some students, who had already completed their assignment, went back to ask for their math journals to try again. This student had really inspired her classmates.

            I love to see this kind of thing happening. The “empty vessel” model of education is, as we know, is so outdated, so “old hat”. The way that my student was measuring was something of her own design, and it is a conclusion she came to on her own. It not only benefitted her, but benefitted some of her classmates as well.

            I was unsure, when I came to kindergarten, how much my students could learn from one another, but I can say with confidence, that my students learn exponentially from one another everyday.  (Pictures soon!)

Markert-- Student Work

This week my students are working with even and odd numbers. As part of this lesson the students continued working with counting by 2’s. To begin this lesson the students sat on the carpet for a mini lesson on what it means to be an even or an odd number. To introduce this my MT had the students count by 2’s starting at 2 then from 20 then from 40. This gave the students practice counting by 2’s but also a way to introduce what kind of number “2” is. Then my MT had two students stand up and explain that these two have a pair then had a third stand up and explain that this student is the “odd one out.” We did this has a group until we had 6 students standing. Then we began discussing what numbers are even and why as well as what makes an odd number. We went over that all even numbers end in a 0,2,4,6, or 8 and all odd numbers end in the remaining numbers (showing the students on the board.) The students really seemed to grasp the idea of an even versus odd number throughout the mini lesson. We did a few examples of writing a number on the board and asking the student if they are even or odd. The students all called out the answer and seemed relatively confident in their response. The student’s were then asked to go back to their seats and get a personal white board, marker and eraser.

Once back at their seats the student’s were asked to write the numbers 1-6 down their boards. My MT then explained to the students that she is going to say a number and they are to write that number down and then write an “O” or an “E” next to the number if they are odd or even. I decided to circulate the room and see what the students were doing and see how they were thinking. I got to one student for number three who wrote the number “16” with a “E” next to it. This is correct, however the number they were asked to write was number 19. This student is one of the lower students who is going to RTI for math. I believe that she wrote the number 16 thinking that she had down 19 and made a relatively common mistake of flipping her number. I then realized she looked up at the board and saw that the number “6” is one of the numbers that is always even. I am glad that I saw her use her resources to answer the question and she did do that correctly however she wrote the wrong number, which would have been an odd number. The importance of knowing your numbers is crucial at this point because it is hard to build knowledge when the framework is not secure. To make this task a high level task I believe it would have been beneficial to have the students explain why the number that they have written is even or odd. Another way that students could demonstrate understanding of even or odd would be to have the student draw a picture or use manipulatives to show that a number is even or odd. Having the student explain the reasoning would be helpful because then I would also know how the student is thinking about even and odd. Did she know that 6 is even because they each have a pair or because it is on the board and the teacher said so? Did she truly mix up her number or was she just not really listening? There are so many questions that could be answered if the students were asked to explain their reasoning. This was a quick lesson due to a field trip and other time constraints so I will be interested in what this particular student does next. I also will like to see how the students grow in their understanding of even and odd numbers. 

Student work week 11

This week the whole fourth grade (my class and Ms. Lewis's (Marie) class) went to Camp Duncan. This camp was all about team building, working together and overall just suppose to be a great adventurous and unique opportunity for the campers to be apart of.

Several students in my class have difficulties working with others when they are in a large group. A lot of them are leaders and prefer students who will just listen to their ideas and say what they want however, that does not work out so well when every student in the group is like that. They all do not listen to each other and they all got mad. This has affected several students from finishing their in class assignments which results in a zero for their grade.

This two day, overnight field trip really helped those students come together and work well. It was not easy getting them to this process and making sure that they were listening to each other and agreeing more instead of disagreeing but eventually they got there. It was very nice to see my students in all the activities today work well, help one another out and make sure that every one was included. No one purposely excluded someone else, or tried to lead the whole group they all realized that by working together and participating they could really accomplish a lot more.

Student work week 10

This week was the last week of my student teaching for math. My students took to the multiples of 2, 5, and 9 really well as well as some properties after explicit instruction however, they were not doing so well with distributive property.

Immediately after going over problems students were sighing and saying "This is so easy, I got this." So I decided instead of doing the rest of the examples with them, I would allow them to do them on their own in their notebook and I would walk around and monitor them.

Now while most students got the steps of the property they had a few misconceptions on what exactly they were suppose to do for two of the steps. The first common problem I saw was students were breaking down the factor into two numbers which is correct however, instead of having those two numbers add up to the larger number they thought it had to multiply. Example 7 was broken down into 7 and 1 instead of 6 and 1.

The second problem I noticed was that students were not using the other factor that they didn't break down as their first factor in both parenthesis. They would take the second factor that they broke down and put it as the first factor in the second parenthesis. Example 6 x 7 break apart 7 into 6 and 1 then they would set up the problem as, (6 x 6) + (7 x 1) so they were using the 7 rather than the 6 in both cases.

Once I saw these misconceptions I was able to get the whole class together for a class discussion and make sure to explicitly model and think aloud each process and explain further why I was doing every step. This really helped the students and it definitely showed on their final assessment.

King Week 11

Nothing very worthwhile in terms of worrisome or overly exceptional mathematics happened this past week. On Friday, the class was given the opportunity to play a math game called Angle Tangle in which the students paired up and took turns drawing angles then estimating and measuring their partner's angle. Students then got points for the difference between their estimate and the actual measure of the angle. They play five rounds and whichever student has the fewest points wins. For example, say student A drew an angle and student B estimated the measure of that angle to be 60 degrees, student A then measures the angle and finds that it measures 50 degrees, student B would get 10 points and then the two students would switch roles. One round consisted of both student A and student B drawing an angle and estimating the measure of their partner's angle.

I required students to play at least two whole games if they had time to play more than one game. The two most interesting outcomes of this game was my ESL student no only participating in the game, but the fact that he understood how to play, which is a statement to the growth in his understanding of English, and one student who drew a reflex angle and as I was walking around monitoring game play, he was able to tell me that he had drawn a reflex angle when I asked him about it.

One thing I could have done to further student thinking for those students who understood the game and were able to progress through the 10 rounds(2 games of 5 rounds each) quickly would be to have them play a game using only reflex angles. This would further their thinking by allowing them to get more practice using their full circle protractor, more practice drawing reflex angles, and more exposure to what various angle measures look like.

Bode Week 11

           I do not have a picture of the student work because it was the written assessment for my math unit and my teacher needed the tests to plug in the grades for the students report cards. On this piece of work, the assessment, the student answered that 2 nickels and 3 pennies was 25 cents and on the second questions she answered that 1 nickel and 5 pennies was 30 cents. When it asked her to circle the largest amount of coins shown she circled the 25 cent answer.

            This student work piece shows me that this student is not recognizing the difference in coin amount when counting combinations of coins. She sees 5 coins as nickels rather than 2 nickels and 3 pennies. She counted the 5 coins by 5s and that’s why she answered 25, this is the same reasoning for the next question. She answered that 6 coins shown made 30 rather than seeing that there was 1 nickel and 5 pennies. She doesn’t understand the counting combinations of nickels and pennies, instead she’s only counting like they were all nickels. As for the second part of the question that asked her to circle the largest amount I do not know the thinking behind this answer. What I believe may have happened was that she looked at another students paper and saw that that space on the test was circled and so she circled hers as well without looking at the amounts and working it out on her own. There are gaps in her learning as far as realizing the difference between counting nickels and pennies when they are in a combination and also understanding the coin amount that is more. It concerned me to see these answers she gave because we did much practice in my unit with counting coin combinations.

            In order to advance her thinking I have at least one solution I know will help this student visualize. I would not want to just tell this student that a nickel is larger than a penny because then when they learn dimes they will be very confused. Instead I want to help her identify the coins with the use of our class coin chart. Helping her see and asking her to identify coin faces will help her to be able to distinguish the faces and values faster on the test or any coin combination problem she’s given. Another solution to advancing her thinking is having her practice writing the coin values above the coin itself so that she can count what she wrote down instead of trying to remember if she identified the coin as a nickel or penny. I’ve seen students do this where they write the value above the coins and it has helped them count the combination every time. This could be an excellent trick for someone like this student who has a hard time distinguishing the coin values and then counting all the coins together. 

Cosmas Student Work

The piece of student work I chose for this week comes from the summative assessment I gave at the close of my unit.  One of the topics covered on this assessment was exchanging pennies and nickels.  I taught students a grouping strategy to help them see this concept of “exchanging” more clearly.  They would have a row of change that looked like this (with circles around each):

 N   P    P    P    P    P

Students would first need to count the amount of money and write it (using cent notation) on the line given.  Then the directions would say to “Show this amount of money with less coins”.  Students were taught to start with the nickels, draw a line down, and redraw the nickel as is however many times appropriate.  Then, students were taught to draw a bracket around 5 pennies, draw a line down, and draw 1 nickel instead.  To show these pennies were “exchanged” they would cross off all 5 to show they were gone and to double-check their grouping of 5.  This would continue until no more groups of 5 pennies can be made. 

On this piece of student work, the student did something interesting with his counting and exchanging.  The assessment had 3 problems like this and on all of them he first counted how much money was there incorrectly.  He was always 1 cent too high. However, then when he would start his grouping and exchanging the end amount he showed would be equal to the start amount showing that he understood the process correctly.  The interesting part is he would write (for example) 13 ₵ when the example was 12₵, but then in his exchange would show 2 nickels and 2 pennies equaling the correct amount of 12 cents.  This tells me that the student did not double-check his work otherwise he would’ve found that the two amounts he was dictating were not the same.  Since his counting response is always 1 off this leads me to think that the student has some type of misunderstanding when counting up the coins.  I assume that he understands that a nickel is worth 5 and starts counting there (because of the way he exchanges 5 pennies for 1 nickel and because his answer is only 1 cent off, not 5).   However, I assume also that he does not understand to start counting by 1’s right away once there are not more nickels.  He may have started to do something like this, for example: 5, 10, 12, 13, 14.  He may not understand the idea of counting on by 1’s from 5 or 10 or whatever interval of 5 he leaves off with.  Also, he may just not be taking his time and accurately counting the amount of pennies.  One thing that would help this student would be to slow down and make some type of marking after each coin he counts to actively keep track of his thinking.  I would also like to ask this student to do an example with me or to look back at their work and explain.  During the unit the student seemed to have no trouble with this, however when the assessment task was fully individual he showed this possible misunderstanding of counting and exchanging pennies and nickels.

Maria Ricchio

This week I gave students dominoes to use to make math problems from. I modeled it and then asked them to come up as many problems as they could. Some students did really well with this concept and were able to take the two numbers to make an addition or subtraction problem. However, many students also used completely different numbers than the ones on their domino. If the domino had 2 dots on top and 3 dots on the bottom, they could write 3+2; 2+3, 5-2, or 5-3, but for some reason I was seeing a couple students write 10-7, 9-3, and so on. I'm not sure where those numbers came from but it is a possibility they were just focusing on making the different math problems and not on using the correct corresponding numbers.

Behrman-Student Work Blog Wk 10

To introduce my unit about subtraction, I worked with students to break down numbers and use their knowledge of addition to aid in this transition. Some of my students challenged themselves and turned these single-digit problems into two, three, or four-digit numbers. However, this student's work shows that when he was breaking down numbers into two groups, he put zero in one of the "groups" on more than one occasion.

This reveals to me that this student does not understand conceptually that although he can write a zero in one of the boxes, in reality that means that that second group does not exist. I think a helpful next step for him would be to practice this type of activity again with bingo chips and a sheet of paper with three squares drawn on it (a larger version of the above worksheet's format). I would have this student show me the problems using bingo chips and have him put all the chips in the top box, then decide how many to move into one of the connected boxes, and then move the remaining chips into the other box. By explaining that both of the connected boxes need to have something in them in order break down the number, this student would be able to see that he can break a number down into zero and itself. Additionally, I would have him practice breaking down the same number in different ways so that he understand the existence of different possible combinations as a result of breaking down numbers.

Week 10 Andrea Delise

For this week's student work, I want to talk about the class discussions that have been going on during math. When I ask the students "why" or "how do you know?" they love being able to explain something to me, instead of vice versa. The responses I get have been showing me exactly what the students are thinking and what they know. For example, I asked the students, "If I paid for an apple that was 7 cents with one dime, how much change do I get back?" One student said three pennies because, "I know that a dime is ten cents and ten minus 7 is 3, and one penny is one cent, so three pennies is three cents." This tells me that he understands the values of the coins, and that he uses subtraction to make change. I also see students counting up with their fingers, or making "hops" on their number grid. This lets me know how students are thinking and what they use to come to the answer. Today during my elapsed time activity, students were working collaboratively on an activity. They choose an activity card with various activities that either my students ALWAYS talk about, or are interested in, such as Obama's house and how ALL of them have met him. So, I made that an activity. Anyways, they roll a dice that tells them a time (3:00, 6:00, etc.) Then, they roll a dice with an elapsed time ( 45 minutes, an hour, etc.) They are to write these on a recording sheet, and find out the end time. Students were really into in and when I went around to the tables, I had them tell me how they knew. I found it interesting that students weren't just counting up on their clocks, or counting by fives. For example, the start time was 4:00 and the elapsed time was 2 hours 30 minutes. One student said, "Four plus two equals 6, plus 30 minutes, 6:30." This lets me know that this students knows that you are adding two hours and 30 extra minutes. Overall, asking students how and why are super important, and have really been giving me a good insight to what they know and what they need to continue to work on!

McCormick Student Work

When I was teaching math this week, students were learning about patterns. Student homework required students to wear patterned clothing to school the next day. When I asked students about the patterns they were wearing, most were able to say the color pattern on their shirts and socks. One child raised her hand to tell me a pattern. She said, "pink, white, yellow" referring to the colors she saw on her shoes. I asked her if it was a pattern and she said yes. I reviewed what a pattern was after she said this.

At this point, the student does not understand the repetition part of a pattern. She simply thinks that stating colors is a pattern. She may think this because many other students said color patterns, such as red, blue, red, blue, etc. The gap in her understanding is that she is not actually stating patterns. One way to advance her thinking would be to show her multiple types of patterns visually so that she can start to see why we call it a pattern. Although patterns were introduced to her, she could benefit from a stronger introduction with visual, oral, and musical patterns. Another way to advance her thinking would be to show her multiple types of patterns including AB, ABC, and AAB patterns. This would show her that not all patterns take the same form and that not all patterns are color patterns. A question that I might ask her is how she would make the color of her shoes into a pattern. I would ask her to draw the pattern and verbally say the pattern so that I could hear her thinking. A future math task that would be beneficial to her would be to describe multiple types of patterns that are shown to her. This would help her to recognize when something is a pattern and when it is not a pattern.

Student Work- Measuring Unit- Part One- Kayleigh Robb

(Pictures coming soon!)

As I taught my students to use rulers for my lesson Wednesday, I noticed several students working in different ways to accomplish their tasks of measuring different parts of a shape.

Students were paired in groups of two or three. Each group contained students of differing levels of mathematical ability but because measurement with rulers is a new concept for all students, I tried to pair students who are "good listeners" wig students who have "trouble focusing."
This allowed me to be able to circulate around the room and offer my assistance if necessary and it also gave me the opportunity to collect some notes/data on how "first time measurerers".

- On their own, many students were able to understand that they needed to start measuring at zero. Students who did not follow this procedure were often corrected by their partners. One explanation from a student of why he started measuring at zero was "you can't just have some before you start counting. You gotta start with none or you might get like, more than it is."

Another student when I asked "why start at zero?", said, "because when we measure with the candies, we start with no candies and then we make more candies to measure the things."

This explanation made me smile because I wanted the students to make connections between measuring with standard and nonstandard units. This student obviously made that connection, and I tried to mention the previous lesson frequently while teaching the ruler lesson in order to show students they had prior knowledge of measurement.

Many students, however could not tell me why they needed to start at zero. In a future lesson I would ask students to theorize about why we needed to begin at the "zero end", instead of just alluding to why in my lesson.

Over all, students did do really incredible with this lesson and with using the rulers and some even liked using the rulers better than using the candy! The following pictures are evidence of how students took on the task. I did attempt to leave it somewhat open-ended, to give them some sort of choice.

Markert-Week 10

The student work that I decided to look at this week is one of my last exit slips I gave the class. I gave this exit slip after teaching the students pennies, nickels and exchanging one for the other. This exit slip was a single question that asked the students to “show me using nickels and pennies 13 cents.” I left this question open-ended so the students could use all pennies or a combination. I wanted the students to feel free to express thirteen cents in a way that made sense to them. The vast majority of the student’s demonstrated understanding of 13 cents by using two nickels and three pennies and a few students used thirteen pennies, both ways correct. The one student that I am looking at used both nickels and pennies. This student decided to draw two nickels and eleven pennies. This shows me that the student may not understand the meaning of thirteen cents. She may think that thirteen cents means thirteen coins. She followed the instructions using the coins but instead of adding their values she did thirteen coins. This would have been correct if it were just pennies but she included nickels in her answer. This could be a mistake on the wording of the exit slip or just a confusion of the question. It could also be that the student does not understand that a nickel is worth five cents.

Looking back on this exit slip there would be a few changes I would make. I would first ask the students to make thirteen cents using nickels, pennies, or both. I think that would make it more clear for the students knowing that they did not have to use both coins if they didn’t think that was appropriate.  I would have also liked to ask another question on the bottom. I would have like to have the students explain their thinking. If I had included this question at the bottom I would be able to see exactly why this student chose to draw thirteen coins using both nickels and pennies. I would also gain information on the students who used only pennies and students who used both nickels and pennies. Including a question at the bottom would have made this task higher level. It may also have been more beneficial if I had asked the students to show me two ways to create thirteen cents. This would allow me to see if the students know that you can create this amount two different ways and also it would show me that the students understand the value of both coins. For some of the students I feel that they use only pennies because they know they equal one so it is easier for them to just draw thirteen pennies and not even bother with the nickel. By forcing the students to think of two ways it will push them to think about using both coins.  

Jones- Week 9

Kallie Jones
Week 9- November 1

This week, I started my Guided Lead Teaching and I am teaching Measurement. The first things that students were asked to do were to find non-standard units of measurements and discuss how they could measure with those to measure different things. For the whole group lesson, students had to use the average shoe shoe size of the class and find things that were the same length. In the essence of time, I tweaked this lesson a bit. I took the heel to toe measurements of 4 students and we did the average of those four students measurements and that became our non standard unit of measure. The average was 46.5 centimeters. The students had to find things in the classroom that were 46.5 centimeter long. This activity proved to be successful because students were able to find things that were 46.5 centimeters. Students asked me if they could find things that were a little bit longer than 46.5 centimeters because most of the things of in the classroom were longer than t46.5 cm. I told them yes. I told them yes because sometimes it is better to over measure so that you can make sure that you have enough space if you went to buy something like carpet so that you can make sure that you have enough.
When students turned in their work, I found one student who seemed to have an issue with the activity. I took the student aside and had them explain to me why they measured what they measured. I wanted this particular student to explain to me why they measured what they measure because everything that they measured was over 70 cm long which was not close to what they were supposed to measuring. When I asked the student why their measurements were so far away from 46.5 cm, she said "I was just measuring". This reveals that when given a certain length to find this student is not able to do it in this instance which means there is a gap in her understanding of using measurements and finding things to fit a certain length of measurement using non standard units of measure. Two ways I might advance this students understanding are to have the student identify their own non standard units of measure and measure with it and have the student work with other students when trying to find things of a certain length. A future task that this student may find fruitful may be finding things of  certain length in their home and measuring them using their own non standard unit of measure. This may make this more concrete for them.

Jones- Week 8

Kallie Jones
Week 8- October 25

As I continue to look at Student Work, it is evident that there are students who want to be challenged more and there are students who struggle to understand grade level concepts. The Student Work that I chose to look at for this week was an a page that Students had to do in their Everyday Math Journal that had to deal with making Ballpark Estimates and then solving the actual problems and comparing their estimate to the actual answer. Students had 6 questions that dealt with this topic. After working with about 4 students to review the work that they completed, their was one student in particular who stood out to me. What the work revealed about this students current mathematical understanding was that this student was having some trouble with estimation. This student showed they he could round to the nearest hundred but not to the nearest 10. The question did not ask the student to round to the nearest ten, however had he rounded to the nearest 10, his ballpark answer would have been closer to the actual answer. The student did ask me why his answer was so far away from the actual answer and I explained to him that the he needed to round to the nearest 10 instead of the nearest hundred. by doing this, it would put his estimate more in the "ballpark" of the actual answer.  Based off of the discussion that I had with this student, two ways that I might advance this students understanding would be to review the concept of estimation with him and discuss the when it is appropriate to round to the nearest hundred or ten when he is not given specific directions. I would also have him solve a series of estimation problem and have him explain his reasoning on why he solved a particular problem the way that he did. Future tasks that this student may find fruitful may be using estimation across subjects. Instead of just seeing estimation in the context of math, using estimation in Science may help this student see how it can be used and how different estimations can yield different results. This could be connected to the ballpark estimation activity.