Weekly Blog

For this week's blog post, I was unable to obtain any student work.  I usually have my placements on the Monday before the blog is due, and the students are on spring break this week.  In my classroom, the students are being introduced to addition and subtraction.  They have been working on adding numbers and taking away numbers during a variety of different activities during math workshop.  One activity has students work with a partner where they draw cards and have to add chips to a board to find a new number.  Another activity is a worksheet that students fill out and it introduces the idea of "difference", which relates to subtraction.  A lot of the students have been finishing the activities successfully and it seems that most of the students are understanding the basic concepts of addition and subtraction.  This is definitely a harder task and with kindergarten, it is important that each student will understand the process involved with addition and subtraction.  After spring break, my MT and I are going to introduce story problems to students.  With this, we are going to have the students do simple addition problems first; such as, "Two bears were playing outside, one bear joined them outside. How many bears are playing?"  They will be able to use counting chips or counting bears, pencil and paper, etc. to solve this problem.  A lot of the students cannot count up/down in their heads, so they need to use tools to help them get the correct answers.  It will be interesting to see how the students will do with story problems when they return from spring break!

Measurement Task

This week's task coincidentally actually tied in with my lesson. The task called for the children to trace the outside of their hand and measure it using unifix cubes. The teacher left it open ended--she did not tell the children how to measure. The "big idea" of this is to define what it means to measure something, and are there different ways to measure the same thing? This task gets to the root of measurement and can serve as a base assessment and introduction to words like width, length, how tall, how wide, and possibly even area.


I predicted 3 ways the children could approach this problem. The first way would be for the children to use cubes to comepletely fill the white space of their hand--basically to find the "area" of their hand. With this method the child would be measuring how much space their hand takes up (in relation to unifix cubes). The second way the children could measure is to put the cubes one after another along the width of either the wrist, the palm, or the finger. This would show children that not all parts of the hand are the same measurement, and would start to use mathematical vocabulary like width, how wide, same/different, and comparisons. The third way would be to put cubes starting at the wrist/bottom of the picture and continuing up to the tip of a finger. This would show how long the fingers are and introduce vocabulary like how long, length, same/different and comparisons. The student's whose work is pictured above chose to use the first method-he used cubes to measure the entire space his hand took up. He said
my hand is 14 cubes" though he did not specify a unit (length, width, etc). This shows me that he understands the basic concept of measurement as the amount of space an object takes up and assigning a numerical value to that, but not the vocabulary of length, width, height, etc or the way to distinguish between the different values of measurement.

Some questions I have to follow up or extend this task:

1. The students were often frustrated because the cubes were often thicker than their fingers so they crossed the lines they traced. How could this activity be modified (either the task itself or the materials given) to approach the same concept and limit the frustration of the children involved?
2. How could a discussion be used to extend the concept taught in this task? The students pretty much traced their hand, measured it, and then turned in their work. Would a large group discussion and comparison be beneficial with this type of task?
3. What other objects could the children measure?

Problem Solving Task

The learning task for this problem is to individually solve the problem to track the path of a boy as he collects his friends on his side of the block. The students have to solve the problem to identify how many houses there are on one side of the street. The students have been working on problem solving in different contexts such as time elapse problems. I think the big idea of this problem would be to be able to track a path backwards and forwards. Instead of just having this problem be add 5, subtract 2, add 13 subtract 1. The problem puts a math concept into a real life situation.

2 effective ways to approach this problem is to draw a picture to help represent the word problem. By drawing a number line with the starting point being the first house on the block and the ending point being the last house on the block, it allows the students to follow the path of the person picking up his friends. Another possible effective way to approach this problem is for the teacher to ask questions like, what house do we start with? What does down the hill mean versus up the hill? Would it be easier if we were able to follow the path in which the person is picking up all his friends? What kind of picture can we draw/use to help us visualize that? One possible student approach (one that majority of my students thought) was to add all the numbers in the problem together. The last sentence in the problem ask how many houses, so the students assumed they  had to add. Another possible student approach would be to draw pictures of houses on their paper. Some of my students drew pictures of houses to represent the houses on the side of the block, but the problem was that the houses are not all in one line. This makes it hard to follow a path because they are in multiple lines.

This student decided to draw a number line for the image and each "notch" he drew on the number line represented a house. He followed the steps in which the word problem asked and labeled each notch with the first letter of the neighbor's name. As he went through the appropriate path he made sure to label on top the number house so he can keep track of how many houses there are. When he got to the last step he was able to count from the 4th house up 13 houses to get to the last house on the block. He then was able to label on top the number of houses and got to the correct answer of 17. Instead of counting to then number 13 he knew to count up 13 houses to 17 instead of stopping at the 13th house. One thing about his mathematical thinking that comes through is how he is able to follow a path by understanding directions step by step. He took time to analyze the direction of what way the person is going on his block. It also shows that he was able demonstrate that there are multiple directions that you can move on the number line block as well as a number line in general. To advance his thinking I would give him other word problems that included the same concept but with a different real world situation. In this context I worked with students in a small group, but I would be interested to see if this student could accomplish this task without any teacher support.

Student Work Blog 3/26

This week my students took a test based on subtraction. As you can probably tell, the last three or four posts I have uploaded have dealt with double digit subtraction. This is the test my teacher just recently gave her class, and this in particular is a photo taken of the a graded test. The students had to solve these subtraction problems using different methods. This particular student did a fairly good job on the first page. When he was supposed to subtract and fill in the empty boxes, he had no issue. He also did a great job with counting backwards and subtracting using photos of the base ten cubes. However, when it game to specifying if he had to regroup and work on word problems, it was clear that he began to struggle.
The first problem he struggled with was 72-3. He realized that he needed to re-group, but when asked "How many tens and ones are left?" he said 7 tens and 9 ones. I have looked at this particular problem and realized that he knew he had to re-group. What he forgot to do, however, was change the 7 tens to 6 tens, because he got the correct number for the ones place. He had no issue with the first question in this format, but that was probably because he did not have to re-group.
For the word problem, this student solved the problem correctly. He just did not fill in the boxes to show the addition problem he completed. It was clear that he did the problem mentally, but he must have either forgot to fill out the box or just could not figure out how to put it into writing. I don't really like the format of this test, mainly because it is directly out of their math book. The students have been working on this exact same format on every worksheet, so the teacher knew what struggles her students were having. Forgetting or choosing not to fill out the boxes is a common occurrence with this type of worksheet, so I think that if my Mentor Teacher would have written her own test, catering to these struggles, the students may have had an easier time. I also noticed that she would randomly hand these tests out to students during free time. This was only if they had not finished their test, but she didn't try to seclude these students away from others, so they were asking their classmates for help. I think that she should have had them work in a particular corner of the room, or tried to have them do it during their lunch or recess time to help them focus.

Student Work 3/26

This was a worksheet the students were given today for their warm up. This math work sheet is very repetitive, and requires the students to do long division for many problems, and then at the bottom complete 8 multiplication problems. This reminded me of what we discussed in class last week about worksheets with a hundred of the same problems on it. Like we discussed in class, I believe this worksheet could be narrowed down to only a few different type of long division problems. I feel that we would have been provided with enough information on the students understanding of long division if they only complete the first 8 problems of this worksheet. Another problem I have with this worksheet is the amount of space that is given to the students to complete the first 8 problems. I noticed the students were becoming very frustrated with this, as they kept having to erase their work and try to rewrite it.

I am curious as to what my mentor teacher uses this worksheet for. I wonder if she looks over the types of mistakes the students are making and tries to create a lesson that helps them fix where they are struggling. I could simply ask her what she does with the worksheets once they have completed them. Does she grade them and hand them back? Does she look over and compare the struggles of the students? Does she just give them points for completion?

I am also curious to know how the students do on the mixed review portion. Do all of these worksheets have this type of review? Does my MT occasionally review these types of problems? If so, I do not think that this section is necessary, as I feel like it is just busy work. The students would benefit more from a challenging problem, one that makes them think and be creative, as that will take up time, rather than something that just makes them go through the process over and over again. This is how we lose students engagement.

I also wonder how she has instructed the students about long devision. When I worked with students today, they did not understand what it meant to divide something. I tried to explain that it is how many times one thing can go and fit into another. I had to repeat this over and over again. I then felt that giving these students these problems without teaching them what it meant to divide, was unfair to the students. I want to know what steps she takes in teaching them how to divide. I could figure this out once again by having a conversation with my MT or by coming in later in the day to observe a math lesson.

Time

For this task students were asked to read the description of the time and then place the clock hands in the correct spot on the clock. Most students in the class have a good understanding of how to place the hands if the time was easy to understand for example twelve o’clock. The students started to become more confused and made more errors when the time said half past one or a quarter past one. They have no understanding of what the terms quarter and half mean. If they do understand what these terms mean that are unable to relate them to telling time. This particular student had no trouble writing the time on the analog clocks but needed some assistance when writing the hands on the other clock. They often got confused about which hand went where and needed reminding that the long hand was the minute hand. Another common mistake was that some students struggle with counting by fives. If the student can’t count by five they have trouble knowing where to place the minute hand on the clock. Even the students that have mastered counting by fives still sometimes forgot that they needed to do this in order to place the minute hand in the correct location. I think having more practice with reading clocks would be beneficial to these students.  

1.     How can the teacher incorporate telling time into other activities in the classroom rather than only filling out worksheets?

2.     Should the students be learning about fractions while learning about time? Would this help the gain a better understanding of quarter and half?

3.     Do the students in this class understand what each hand represents or are they only memorizing the procedure of how to place the hands on the clock? Would they know that the hour hand moves after 60 minutes have passed?

Liz Slusher- Coins

This is actually one of the artifacts that I collecting from the teaching of my lesson. (It isn't one of the ones that I analyzed in the reflection). I thought that this student's take on the task was very interesting though, and would be a good choice to question and analyze. Maybe even talking through her thinking as a class would be interesting, as I really feel she thought of this task in a unique way.

In this lesson, we discussed money, what we use money for, and we talked about pennies and nickels--their values, ways in which they look different, but that they are both coins. Finally, the students discussed that coins are worth or represent certain amounts. Each coin is worth or represents its own amount. The students were asked to complete this worksheet and then we had a discussion of the various responses students had. The worksheet had a set of 6 pennies. The first question was "How many coins do you see?" This student answered "6." The next question was, "How many cents do these coins represent?" This student answered "1." The second set of coins was 1 nickel and 1 penny. When asked, "How many coins do you see?" The student answered "2." When asked, "How many cents do these coins represent?" The student answered, "1     5."

1. Did the wording of the question "How many cents do these coins represent?" confuse this child?
 I'm wonder if this was confusing for this student because for the group of pennies she indicated that she knew a penny represented 1 cent. In the second question she also indicated that she knew the correct value of a nickel and a penny. She many have thought I was asking about the given coins, not the group of coins that was shown in the picture. Many if I had worded the question, "How many cents is this group of coins worth?" she would have understood to add the amount of each coin. Although, through the way in which she answered the question on the paper and explained the answer it is clear that she understands the value of each of these two coins, and that those values are independent of one another.

2. Going back to how she answered the question discussed above, this also makes me wonder, "Does the student understand that values of coins can be added together to make larger values? Or is she still at the phase of simply understanding that each coin has a value of its own?
In order to explore this I think I would have to do more money activities with this child. If it were my classroom I think I would set up a pretend store. I might make an item cost 12 cents. This way the students would realize through this game and exploration that you can put coins together to pay for things that don't have a coin to represent that specific amount.

3. I have a question about teaching money in general...Once all of the coins are introduced, it seems like it would be very difficult to differentiate, on a piece of paper, between a nickel, dime, and quarter--how do teachers do this?
My thinking is to of course discuss the different pictures on each coin, what they mean, and why they are important enough to be on our money. This way the students would look closely at each coin and truly understand the differences in the ways each one looks. I also think it would be a really good idea to make sure that all of the coins on a paper are "life size." This would be that if you put that coin on top of the picture of it on the paper it would fit exactly. I think this would be an important learning tool for kids because realistically, when I am searching through my wallet for a certain coin, I'm mainly focusing on the size and width, not the picture.

Cookie Counting

This was part of the math lesson that my kindergartners had for today. I saw the students working on their counting skills, working on their writing numerical values skills, and associating a counted number with the actual written number. I think by this point the teacher is looking for the students to have mastered counting up to 20. Majority of the class is getting to the point where they have mastered this. 

I see that the wks has ten cookies in a row before starting another line. I think that the ten cookies in a row help the student see ten and in their mind bundle those ten cookies (or what they have been practicing is bundling straws for counting the days of school). 

I see this as just a basic worksheet. I interpret it and completely believe that it's purpose is simply practice of counting. Something that I saw the substitute teacher do was have the students make more on the back when they were done. I think it was just to keep the faster students busy while the others were finishing up their worksheet. What I did a few, but not all students, was they would make the cookies, I would give them my answer and they then would have to count them to check and make sure that I got it correct. 

My three questions about this task would be:

1. How can this task be furthered or built off of?

2. How does this help all the students in the class (high level math students are not challenged while the low level students are struggling)?

3. What could be a more hands on task added to this task?

I think that the substitute teacher's having the children create their own idea could have been a way to build off of it, if there was more structure added and planned. 

I think that the worksheets needed to be altered to higher numbers for the high level and lower numbers for the lower level students.

Having the kids find different object that represented the same number could be a fun, hands on way to help the students visually associate the quantity with the actual number.

Student Work 3/26

This week I observed my kindergarten class during the centers. This week at my particular center they played a math game. They got into small groups of 2 or 3. There was a dice that had + or - a number the highest number being 3. They were to use the blocks and either add or take away the amount they rolled. They were also supposed to count how many blocks they had left after the took added or took away blocks.  The big idea of this task was for students to be able to understand adding and taking away as well counting.

While I enjoyed this activity and think it worked with those students who could focus on the math part of it. However, many of the students were more focused on whether or not they had the most blocks or they were more focused on building the blocks then using them for the game. Students started cheating and just rolling numbers in which they added blocks. They were accusing one another of cheating and complaining that people weren't following directions. It got really out of hand. I told my mentor teacher this and she actually made certain students stay in from recess and try playing it again correctly.  I'm not sure if there is another way to play this "game" in which the students don't feel the need to compete but I think that would work better. However, for the students who could stay on task and follow directions, they really enjoyed this activity.

One thing I noticed with some of the higher academic level students in the class was they would start counting up or down from the last number they got instead re counting all of the blocks. For example if they had five their last turn and rolled -2 they would say 5,4,3 and come up with the answer three. I thought that was really cool and loved how they didn't need to recount from the very beginning.

To further advanced the students who were able to stay on task during this task's learning I would put all whole numbers on the dice maybe 3-8 and have them make that number of blocks and then say how many they had to add or take away. I think this will make things a bit more challenging. As they advanced, you could change the numbers to larger numbers.  However, for those who could not stay on task I would have them continue to work on adding and taking away small numbers.
 

Casey Droste Student Blog 3.36

We had a substitute today so we did not have normal math class.  We had a “quiz” with key terms and the students had to match the word with the definition.  The terms were like mean, median, mode, range, bar graph, line graph, outlier, double bar graph, double line graph.  The students were to make note cards on Friday and then they were told to study because the quiz would be today.  The class I was with was our most difficult class due to attitude/discipline problems as well as learning disabilities so it was a rough quiz. 

One student in particular with a large track records of detention and suspension due to attitude told me “I don’t know any of this, I didn’t study” then we other classmates started to ask for help she exclaimed “you’re stupid this is easy” So I told her if it was so easy why couldn’t she do it and this got her to do a few problems.  The most interesting part of walking her through the quiz was she said “I know this one but I cant find the right definition” she was talking about range and she explained that “it’s when you take the smallest number away form the biggest one.” So I helped her find a definition that had the word largest and smallest in it, she still was unsure about the wording but figured that must be the right fit.  As I walked around the room to help other one boy who had earlier told me he studied said, “these are the definitions we learned, they aren’t the ones we put on our flashcards” (he pointed to the board where it told them to get their information from the glossary)

This information really confused me, why would my teacher expect students to study specific definitions and then change them on the quiz?

These students rarely have to actually study for things in class so when they actually do they should not be confused and somewhat punished for that.  If they had a lot of experience with these words and were able to focus less on the definition and more on the overall meaning of the word they would have been fine but I just find it ridiculous to have them study a specific definition and then not test them on that.  These students are not ready for something like that!

I also am concerned about how this kind of quiz shows they actually know the material, again it would just be memorization.  Wouldn’t a better way to test this knowledge be to give them real math question and they have to solve for these terms or create a bar graph rather than give a definition.

I understand that the teacher is out of town and this is just a simple quiz but the students put a lot of time into studying and creating note cards for them to be this confused and disjointed by it.  The other problem with this is that the teacher who assigned this is not present so I have no clue what the teacher was expecting of them or how I could assist them in coming to the original definition they wrote to the one on the page.  I was really quite lost when helping them, I helped them find some clue key words but other then that I could not give them the answers for I was not even sure on all of them.  I really think a better way to administer this kind of test would be to create problems and have them identify what is being asked for or solve for the word, etc.  Example: Find the range in the bar graph for number 4.  or Draw a bar graph for the information given in problem 1.

Student Work Blog - Amy T.

This past week I watched my students participate in math stations again. One station they do regularly is one involving puzzles. I don't have a picture, but I can describe what they do for the station. They get together into a group of three or four students and put together puzzles of different difficulties. Some are easy, some are harder. All the puzzles are from the 90's, so the students don't really know the pictures on the front. This is better in my opinion, because then they can't cheat by just putting the picture together. The goal, my mentor teacher told me, is to have them use critical thinking skills and basic geometry skills to figure out how to put the pieces together. This invokes not only their skills in geometry, but also their communication skills. I know that's not specific to math, but it's very important that they learn to communicate their thoughts and ideas about problems. Watching them do these puzzles is so interesting. They first try to put the pieces together just willy-nilly, and then, once they've realized that won't work, they get down to figuring out which puzzle cut-out piece goes with which puzzle jut-out piece. Once they've figured out which certain pieces go with which other pieces, it's smooth sailing from there. I liked watching them do this task, and I think it's a good communication task, but I don't think it's a high-level task. It does not require them to draw on any background knowledge, and it is essentially just guess-and-check for finding an answer. If I were to try to make it into a higher level task, I would put numbers on each puzzle piece and then have the students add or subtract the numbers when they get two pieces to fit together. It would not be extremely high-level even with the modification, but it would be more mathematics than they are currently getting out of the puzzle-building exercise.

Student Work

The task that these students were doing was called "build it, change it".  The students worked in partners and the first partner would draw a card.  In this instance, the student drew a four so she put four chips down onto the mat.  The second student had to change it, so she drew a seven and had to turn the four chips into seven chips on the board.  They continued this until they were finished with the entire deck.  The point of this task is to introduce "taking away or adding" to a number to get something different. I liked this task because it made students think about why they had to add chips to the original number or take away chips from the original number.  When I was working with these two students, I asked them how they knew which to do and they responded with, "seven is bigger than four, so you need to add chips to get up to seven".  Their reasoning showed me that they understood the purpose of this task and understood the basic concepts of addition/subtraction.

Three questions
1. Do all of the students understand that this task introduces adding/subtracting?
2. How can this task be altered to help the students that may be struggling?
3. How could this task be expanded to further their knowledge?

1. It would be helpful to walk around to every group, and ask how they know you need to add/take away chips to get to the second number.  It also may be helpful to use different ojects or different variations of this concept to see if they can still do the same things, but with different tools/resources.
2. to help students that may be struggling, it may be helpful to use smaller numbers or to walk them through the process one-on-one. Sometimes, students learn better when they can see how the task is supposed to be completed.  I could also be a partner with one of the students and state what I am doing as we are doing the task.
3. To further their knowledge and learning with this task, you could have students add the two numbers together, to find the sum.  One student could draw a card and place the chips down, then the second student could draw another card and add that many chips to the board.  Together, the students could count the total number of chips to see what X + Y equals.

Student Work Post #7

This task is called “The Raft Game”. The learning goal of this task is how to exchange 5 of something to get one of something. For example 5 planks to get a whole raft. This task is designed to elicit student thinking by providing them with different ways to think about getting to one plank, then to two planks and so on until they reach 5 planks and then they can exchange that for a raft. The students have to figure out how many beans they can exchange to get one plank. If the student has left over beans then they have to figure out how many more beans they have to get to get another plank.

Two possible ways that this task could be represented is by explaining it to the class in a group setting. This could be done for the first time so that everyone gets the instructions on how to play the game. The second way to represent this task is by having the children actually play the game in centers. In the picture above there are two boys that were playing the game during centers time. 

Two ways I would anticipate a student might solve this task would be if the student rolls a 4 and then a 5. They now have 9 beans. They may exchange all 9 beans for one plank, thinking this was correct.  Another anticipated student approach would be to “rig” the dice when it is their turn. I have seen some students do that, and that is taking away from their practice to add numbers or subtract numbers to get to 5.   

First, the student would roll the dice, then they would pull out the number of beans out of the bag.  For example, if the student rolled a 3, they would then pick up 3 beans out of the plastic baggie. Next, the partner goes; let’s say they roll a 4. The first partner now goes, and they roll a 2. 3+2= 5 so they now have 5 beans. Now they are able to exchange the beans for one plank. As soon has they get 5 planks then they exchange that for one raft. Whoever gets the most rafts is the winner.

If the student were to struggle with the exchange of the beans, I would help them to recognize certain amounts of objects that are in sets.  For example if you have 10 beans, how many planks get you get from 10 beans. The answer is 2!  I would have them count it out. If the student exchanged all of the beans, then I would explain to them they we are only suppose to exchange a certain amount. This type of thinking tells me that he struggle with knowing what 5 looks likes in different representative ways.

Two approaches that could advance student thinking on “The Raft Game” could be to add a second dice into the game. This way if the students get ten beans then they will have to figure out if they have 10 beans then how many planks can they grab. Another approach could be to turn this game in to a money game for example instead of beans we could have pennies and nickels to exchange for each other.

Three questions that I still have about the artifact:

1.     How can the students experiences this exchanging task in other ways?

2.     Is this task helpful to their practicing of exchanging 5 of something to 1 of something?

3.     What are some other ways we can extend this learning experience?

Answer:

1.     The students could experience this task through a penny and nickel exchange game. This game would entail the same ideas, but instead of beans there would be pennies and nickels. My next student work analysis will focus on this type of game.

2.     I think this task is helpful to their exchanging practice. The students are able to practice exchanging 5 beans for 1 plank and 5 planks for one raft.

3.     Some other ways to extend this game could be to add two dice, for Kindergarteners this may be tricky but I think for some it could be a nice extension. 

Alyssa Berger Post #8

This week, I chose to use a picture of the worksheet I gave my students with the Pizza Comparison Task on it for my lesson to analyze. I told the students that once they came up with one answer to the problem, they should try to come up with another possibility that way they didn't sit and become disruptive to the students who were still working out their first solution. A lot of students came up with more than one answer pretty quickly, but upon looking at and/or talking to their neighbors, they began working on more (even though they weren't supposed to be talking!!). This particular student chose to come up with his second answer so that it contradicted his first answer. He clearly was thinking in a very shallow mathematical way when he came up with the answer that Jose had the same amount as Ella because they both had half of a pizza. But then, when he thought about the problem more deeply, and attempted to figure out how it could be possible that Jose did eat more pizza, he discovered that if the pizzas were different sizes, Jose could be correct and he in fact ate more pizza than Ella. The thing I find most revealing in his example is when he draws the pictures of the different sized pizzas, then he drew a greater than sign between them to show that the pizza on the left has more than the pizza on the write, but he also chose to write next to the drawings "1/2 big" and "1/2 small." I found this very telling of his current mathematical understanding. It was a relatively common answer for the students to discover that if the pizzas were different sizes, Jose ate more pizza overall, but this student was the only one to add the description of "1/2 big" and "1/2 small." I think with some scaffolding, this student would be able to quickly understand area and how comparing the pizzas when adding area and/or diameter to the pizza problem, he would be able to much more clearly talk about how the bigger pizza is more. Similarly, I think this student is very close to being able to multiply fractions by whole numbers and understand how the numbers relate to each other in these problems (i.e. multiplying by a fraction has a result that is smaller than the initial number). This is a much more complex mathematical concept and I am curious if this student is already ready to move on to these problems or if he would need more help with these problems.

The three questions I would ask as a result of this piece of evidence are:
1. Does this student already have an understanding of area without explicitly knowing the term area and how to calculate it?
2. Has the student already experienced problems where he needs to multiply a whole number by a fraction to arrive at the understanding that the whole number would be smaller even though he was multiplying?
3. How could I modify this task to help answer my questions about his understanding?

To answer the first two questions, I would have to provide problems for the student in a one-on-one setting so that I could observe his thinking while he worked out the problems and see what he comes up with as an answer. I could find these problems in the 5 Practices book or in the CGI book (or at least a similar problem that I could modify). I think I would have to do a lot of thinking about what a meaningful addition to the problem would be that would also answer my questions about what the student is thinking. I could also do some research problems in curriculum books and on the internet to find similar problems that could help me modify the task in a meaningful way. I could also discuss with other math educators.

Blog # 8

This is a game that my Kindergarteners sometimes play during literacy centers. This is called the “raft game” and requires the students to be paired into groups of two. The rules are that one person rolls the dice and the number determines how many beans they get out of a bag. Once a student gets 5 individuals beans, they must grab a plank which equals 5 beans. Then when the student gets 5 planks, then they are allowed to swap it out for a raft, which is 5 planks. The purpose of the game is for the partners to try to build a raft first. The tricky thing about the game is for them to determine when they are allowed to get a plank. If the students roll a 5 they can quickly decide they need to swap the individual beans out. But if the students roll a 6, I usually have to help them determine that they still need to switch out the individual beans for a plank. Some students are faster at learning the pattern, while other students struggle more and need more support from a teacher. I think that the game is beneficial for students to work on addition and subtraction and working with skip counting in groups of five. This could also relate to money because 5 pennies equal a nickel and so on.

I do have a couple of questions about this game.

1.     Is it beneficial for all students? Or are some students just messing around and not understanding the purpose of the game?
The teacher could do a quick check for understanding after the students finish playing. They could have a quick discussion and talk about what things the students found easy and challenging. Then the teacher could figure out which students need more practice and which ones need more difficult practice.

2.     Are students seeing the patterns that the teacher wants them to see while playing the game?

The teacher could hold a discussion to see if the students are noticing any patterns while they are playing. This could take more scaffolding because they are in Kindergarten. You could also check that they are benefiting from playing this game.

3.     When can the more advanced students move to higher number like planks of 10 or even 20?

This is probably for higher grades, but I could read into the common core standards and check what grades this game could be used in. This could help students with grouping groups of 10 or other higher numbers. 

Number Jump- Jamie Blankenship

One part of the kindergartners' daily schedule is something called the number jump.  First, my mentor teacher will call out a certain number, on Friday the number was 82. Then, there are strips of tape on the floor that form a large square.  The students stand on this tape in a square and the helper of the day begins. He starts by saying the number 1 while he jumps.  The student next to him has to jump while saying number 2. This continues counting up and going from student to student until they reach 82.  If a student miscounts, he or she  must sit down.  Eventually, a student will say the number of the day, in this case, 82, and win a prize. This activity really gets students moving around while they practice counting.  They also have to be able to know what number comes next, by listening to their classmates count the numbers. I think this activity, along with other counting practice has really brought the students a long way from how they counted at the beginning of the year.  Most of the students are able to participate in the number jump for the majority of the time.  Students who get "out" will usually say the wrong tens place or will say a number that is completely unrelated to the number that is supposed to be said.  For example, with the tens place, a student might say 19 when the number should be 49. I wonder why this happens? It could be possible, since these students who make this error often have trouble counting above 20 or 30, that they know what the number should end in, but that they don't know how to say that number.  To find out more about why this is happening, you could ask the student, How did you get from 48 to 19? Are those first numbers in both of those numbers the same? Should they be the same when you are counting up? Although this is a very interactive task that students really enjoy, it is not a high level task.  Some questions I would ask is, how would you make this a high level task? and also, Does this activity need to be a high level task? When answering these questions, I honestly don't know if this task could be made into a higher level task. Counting during this activity seems to be a way to add in a little bit of some extra practice to the rest of the counting activities that the students do.

Weekly Blog of Student Work 3/25

This photo is a picture of one of the students in my classroom's worksheet from this past week. She is one of the higher students in the classroom, and although the worksheets had yet to be corrected, I went over this one myself to see how well she had done. She got a majority of the questions right except for a few division problems at the top. Although the directions ask the students to make or draw a model to solve, I don't think that the students do so. So the questions that I would ask would be:

1) Did the students have the opportunity to make or draw models?

2) If so, with what and where? 

3) Because only the answers are given and no work is shown, I'm curious as to what this student's train of thought was when completing this worksheet. Was it difficult for her? 

To answers these questions, I think that the best source would be to ask the student as well as the mentor teacher. I'm not sure what the exact instructions were that the teacher gave the students, and I could talk to the student to see how well she followed those instructions. 

Student Work Blog

Today I did not get a picture of the student's work in math because they focused mostly on literacy today, except for at the very end of the day. For the last 10 minutes of the day, my mentor teacher had me do "minute math" with them. This is basically a small booklet that has story problems on each page that can be read aloud. The way my teacher works it is that the students sit in a circle and she starts with a student and goes around the circle reading aloud a math problem for each student. I thought this was a pretty neat task because the problems are all word problems that ask children to decode the problem such as ones in our CGI booklet. An example of one that I read was "There were seven cookies in a basket after Lucy baked cookies. She gave two of them to her friend Sam. How many cookies did Lucy have left?" The students were able to answer the questions fairly quickly and the only help from me that they needed was maybe a repeat of the question for clarity. Because this was a fairly quick task and something I am used to seeing from the CGI books, I only really have one question which would be:
How often are the students exposed to problems like these?
I am wondering this question because when asking my student interview questions from the CGI, I noticed that most of the students did not have trouble with word problems read in this way. Another reason I ask this is because I would assume from the ease of students answering these problems is that the teacher must do these at least once or twice a week - but I would have to ask to be sure. It seemed to be more of a "when we have time" sort of thing, but judging by their answers, I feel like it might be more often then she is leading on. The students also had no manipulates or anything to write on either which is another conclusion leading me to believe that these problems are done often enough to allow students adequate mental math practice.

Blog of Student Work

The students recently have been introduced to fractions. This worksheet is an overview of the things they have been working on. The first box asks them to color in fractions of a shape, the second asks them to draw a line of symmetry, the third asks them to complete the table, fourth asks them to show 1 way to make $1.28, fifth box has them circle an event that is likely to happen, and the sixth asks them to choose what unit makes sense. I don't think there is an overall big idea on this entire page but just testing the things they have learned and see how much they remember. For the second box I am curious as to how the teacher explained "lines of symmetry." I had a very difficult time showing it to the students without a piece of paper that I could fold in half. That would be one of the questions I would want to ask the teacher. Another question I would like to ask is how the students think about completing the chart without the knowledge of multiplication. When I completed the chart while I was helping the students I had a hard time trying to teach them without using multiplication. When the teacher went over it, she described it using grouping. Another thing I am curious about is how they teach science in the classroom. Do they do science and math together to teach the concept of weight? I have not seen science in the classroom but I think it would be a great visual if the students were allowed to use scales and different objects to determine weight. One of the things I would like to see to further this thinking is with shapes. I think the students would benefit from more practice with fractions and dividing them with different shapes. I am curious as to why the student did not color in half of the shape so that they could see "half" more visually. Many of the students had a hard time with that problem and I think it would benefit them to have more exposure to fractions. I also think they should be exposed to more lines of symmetry with different shapes as well. I think it is important for them to realize that some shapes have many different lines of symmetry while others, such as certain triangles only have one.

Blog 7

3.25.13


Last week in my placement classroom students were reviewing counting with base ten blocks and bits. This particular question ask students what is the number, students were then supposed to count the number of ten blocks and then add the bits. Since this was a review many students did not have a problem with solving it. This student solved this problem in a really unique way. Unlike most of the students who counted the blocks in their head. Cindy decided to write down what she knew. When I asked her why she wrote the numbers on top of each image she said it was to help her count them all up. Cindy said she remembered that the long ones where 10 and there were 5 of them so its 50 and then  since the bits are worth one she added them on to the 50.

Questions:

1. Are the base ten blocks and bits really help these students learn place holder or has this assignment just become a math problem they need to solve?
2. Could the students still solve the problem if there were a hundredth block?
3. Do the students really know why they add the tens first and then the ones, or do they just perform this action because that is what the teacher told them to do?

I would try to figure these questions out by giving the students some more challenging problems. I feel the students are given tons of problems like the one above. However, would they still know what to do if I decided to challenge them with a hundred block 12 ten blocks and 24 single blocks. This would then require regrouping. To figure out my third question I might do a little interview with a couple of students and ask them why they count the tens first then the ones. From their answer I can then tell if they understand the logic behind counting.

Weekly Blog #7

This task is asking the students to color in the fraction that is given. The students have to understand what the fraction says then color in the corresponding checkers. The big idea of this task is to see if students understand fractions and what certain fractions look like by coloring in the fraction that is given, so they were asked to color in 3/4 so the student colored in 3 checkers of the 4. This is correct. But, when the students moved on down the page to questions 7-10 they became much more confused. This particular student ended up coloring the right checkers after some help and errors at first. At first this student, along with more than half the class, did not understand this particular set. Let's take number 7 for example. It states 1/3 are red. The students understand that, oh one of these 3 should be colored in so they colored in one checker out of all 9 that were there. The student saw the fraction as 1/3 so one should be colored in but did not understand or recognize until after further help that one whole set of three checkers should be colored. I helped explain this to the student by drawing a square split into three parts. I drew three checkers in each part for a total of 9. I then asked, "how many parts do we have?" The students responded with 3. I then asked how many checkers are in each part, again they said 3. I then helped them understand what the problem was asking by explaining, "ok we have 3 parts and it is asking us to color in one of those three parts. So if I want to color in ONE part, how many checkers and what part would I color?" The student(s) then understood and said to color in the one part that had three checkers. I helped them further their understanding by explaining that when there are sets like this pretend a line splits them into parts, when it asks to color 1/3 we know to color one part of the three parts total.
The students current mathematical thinking about this problem reveals to me that she understands fractions as a whole but not as part-whole. She understands that if there are 4 pieces and it says to color in 2/4 she would color in 2 of the 4. But she is struggling with understanding that if there is a set, then you will have to color in ONE SET of the total amount of sets, like problems 7-10.

The Questions I have about this problem are:

1. The student understands the parts and fractions now, but if this were to pop up in another worksheet or on their math test will she understand how to do it? If a similar thing showed up will this student realize the problem is dealing with parts and not just a whole pizza or 4 checkers total?
2. Why did the teacher not explain this to the students before about the parts? Was it not in the content dealing with fractions?
3. How can we teach students about fractions without simply having them color in 2/5 of a group of objects? How can we further their understanding into a deeper meaning of what fractions are?

Ways I might answer them:

1. I suppose the only way I could find out this answer is by observing other worksheets or seeing what the test will look like for them. I have no other way of seeing if the student will remember this information unless I provide them with a similar worksheet to assess their understanding or experience them completing another page similar to this one.
2. I could find the answer to this question by asking my mentor teacher if she taught them about fractions and dealing with parts of a group. 
3. I think that the answers to this question can vary. We can further their understanding by having them complete word problems and creating their own drawings and fractions from the numbers within those problems. Students will then be solving a higher level task because it is not just simply saying, "color 1/4." The students will have to see what numbers are in the problem, what numbers are placed in the numerator, the denominator, and how they know that. This will aid in their mathematical thinking about fractions and help them understand more to them.

Student Work #7

This photo is from a worksheet done by my 1st grade placement class during a math exploration activity.  For this worksheet, the students were to fill the blank shapes (after the equals sign) with pattern blocks to count how many blocks fit inside.  After this was determined, the students were to then use a stencil with the same sized shapes as the pattern blocks to fill the larger image.  This showed them that smaller parts of something fit and go together to form a whole.  After the students drew in their shapes, they were to just determine how many fit into the larger shape, then color it.  For this reason, a few questions I thought of to ask would be:

1) What is this lesson looking to teach students?
2) How could this have been used to teach fractions?
3) Was this assignment just busy work given to students?

If I were to answer these questions, I would say that this lesson wasn't necessarily looking to teach the students any concrete mathematical concept.  The students were only required to place the pattern blocks inside the large shapes to see how they fit, figure out how many fit inside, then draw those parts, and color the whole larger shape.  If this were going to teach them anything specific about mathematics, I think that fractions would be the way to go.  I think that the first 3 steps could remain the same (place pattern blocks in large shape, see how many fit, draw) BUT the worksheet would ask the students to only color a portion of (1/2, 2/6. etc) the shapes.  In my opinion, this worksheet seems to be nothing more than busy work for the kids, but if there was one more step added to it, this worksheet could be very helpful to the students' understanding of fractions and smaller parts fitting into a whole.

Student Work Blog #7

These pictures show two worksheets that were given to the class at the same time. The students were instructed to solve each of the problems using whatever they needed. They had unifix cubes and their individual boards with beads on them. One worksheet had vertical addition problems and the other worksheet at horizonal subtraction problems.

My questions:

1.     Do the students really understand that both ways to write problems (horizontal and vertical) mean the same thing?

2.     Do they have a deeper mathematical understanding of addition and subtraction besides putting blocks/beads together and taking them apart?

3.     What is the point of having a worksheet with a bunch of problems on it without really talking about why they are doing it?

I noticed that many of the students would just go through the procedure of moving beads to one side of the board and then moving beads away from or to that side according to whatever number was being added or subtracted. It seemed as though most students were rushing right along with these problems, moving beads back and forth and moving on to the next question without stopping to think about what was actually happening in the problem.

            I might try to answer these questions by talking to the students and asking them what 8-6 means. I would like to see if the students respond using examples of real life things to show that these numbers can represent actual objects. I would also ask them why some questions were written up and down while others were side by side. I would ask them to explain if it matters or not and if how they solve each is the same or different. I could also ask my mentor teacher what other activities she does (if any) that leads up to or follows these worksheets that might help answer my questions. Since I am not there every day, she might go through a bunch of other activities and actually talk to the students about the ideas in these questions I have.