Week 4- Jones

The student work that I brought in for Week 4 was an assignment that my students had to do on identifying a bar graph and decribing the mean, median, mode and range for a set of data. My Mentor Teacher bought a pet for the class (a Geko) and the students had the opportunity to submit a name that they thought would be a good name for the class pet. Of the names that were submitted, my Mentor Teacher chose 4 that the class would vote on. As part of the whole group activity on bar graphs, my mentor teacher took the votes and turned it into a bar graph that the whole class could see on the board. The names that the students had to choose from we Gabriel, Glen, Geffrey and George. Of the four names, Gabriel won. Also as a part of this the students were introiduced to the x and y axis on a graph. They had to correctly identify what information was on the x-axis and what information was on the y-axis. The students then went back to their desk and did a worksheet in their Everyday Math Journal on the concepts that were presented. They were asked to construct a  bar graph using data from the letters in their classmates names.

After talking with students and going over the activity with the students individually, it seems, for the students I met with, that they understand bar graphs. They understand the vocabulary that was introduced as well. In addition to checking the worksheet they were asked to do, I asked them what the meanings of mean, median, mode and range were without looking at their books. I did this because, I felt that they were able to write the definitions of the vocab words because they were given hints about what the answer should be. They students that I met with were able to describe the words correctly. Also from what I observed there seems to be no gap in student understanding. I also spoke with my mentor teacher about the students he met with. He informed me that they students he met with were also able to complete the tasks successfully as well.

Two potential ways I could advance my students' mathematical understanding are having them explain the significance of a lesson. My mentor teacher gives objectives but it is important that the student knows how to apply what they are learning. The second way would be to ask students what other subjects could bar graphs be applied to. By doing this they are making connections across subjects which is also important. For the students that I met with future tasks in involving finding the mean of various sets of data may be fruitful.

Week 5-Bode

          My class is still on the first unit in Everyday Math for first graders, we practice our numbers a lot and my teacher has then turn these numbers in so she can see where the students abilities are. The sample work I have a picture of displays the different levels of ability among the students. The student work on the top left shows a low to average student work, the bottom left shows the struggling student work, while the piece of work on the right shows an average/above student’s work. The directions for this assignment were “write the numbers 1,2,3,4 on the piece of paper.” The student’s work on the top left is interesting because all of the numbers are backwards and he understood the directions to mean that you have to write the numbers in order 1,2,3,4 repetitively. The directions confused this student and therefore his thinking was that the numbers have to be written how the teacher said. I think the gap in his thinking would be that the numbers are backwards but all exactly how they should look if you put a mirror to the piece of paper, which is normal for a first grader at the beginning of the year. Compare this piece of work to the one below it, which is the struggling student. This student I observed during this time and she was not writing anything at first. I saw her scribble down what looked like a backwards “s” and then immediately erase it. She asked me for help and so her whole line of 2’s on this piece of paper for actually written by her but only because I sketched the number out. It was interesting though that she started off with the number 2 at first, not with 1 like others did. She was able to complete writing all the numbers but sadly only because she was looking at her neighbors work and mimicked the way that he wrote it. This is good that she’s learning but scares me because she is very behind for a beginning of the year first grader. She needs to be told to begin assignments and by guided by the teacher or myself. As far as seeing “gaps” in this piece of work to the right, I really don’t see any except maybe that he needs to neaten up his numbers a little more. The range of ability in Ms. Hall’s class worries me at times because I’m not sure how you scaffold for such a wide range of students.

            As far as how I would approach these students and advancing their cognitive thinking, I think the above average ability student would without a doubt be ready to continue on to other units. However, as far as the two other students go. I believe using examples from newspaper clippings or coupons to show how numbers look maybe they would be able to see how numbers are used in their everyday life and that would help them learn how to write the numbers correctly. Another way I could help both students is by having the student watch my writing of the numbers and then mimic that. This may be good for students to see how their teachers write the numbers and the movement of their hand. Just tracing the number may not be enough help for students to learn what their hands should be doing while they write.

Week 4- Robb Student Work Blog

This past week, my students were introduced to the concepts of comparison and measurement. My MT and I created a large, 4 foot tall pencil and students estimated whether they were taller than it or not.  After the estimation on the first day, students actually compared themselves to the pencil to see if they were taller, the same size as or shorter than the pencil. They then drew themselves in relation to the pencil and wrote "shorter", "taller" or "same size as".

The artifacts that I collected were their drawings. Some students were able to conceptualize the ideas of "taller", "smaller" and the "same size as", very well. Students who understood that they were taller than the pencil drew themselves larger than the pencil. Some students even made sure that their feet and the tip of the pencil lined up exactly, a key concept in measuring length. All too often, mistakes made with measurement calculations happen because the student does not begin measuring at zero. One student even told me "If it isn't even, then how do I know if I am really taller than it (the pencil)?".

On the other hand, some of my students drew themselves larger than the pencil no matter what they actually measured. Even if they knew they were shorter than the pencil, and wrote that they were shorter than the pencil, they still represented their findings as if they were taller. I think this could have been avoided with a little more modeling, or even if we had not used a giant pencil. The students work with pencils everyday, and are obviously larger than the pencils. This became a difficult concept for one of my ELLs to comprehend. He kept confusing his pencil with the big pencil and saying, "I bigger, I bigger!" I might have used an animal instead.

In general, the lesson definitely needed more concrete modeling for the students. I might have asked three students to come up the the board and do a quick draw of how they might show being taller, smaller and the same size as the pencil after modeling. Overall, I liked the lesson a lot and the kids enjoyed it too. I think it is a great way to expose children to measuring and comparing.

Berger- student work week 4

I brought in a few samples of students' two-digit multiplication problems. I brought in samples from two different students. This was their exit slip and there were two different problems. One of the students got both of the answers correct, following a correct method of multiplication. The other student got both of the problems wrong. It was clear to see that this student did not understand the correct way to multiply two-digit numbers. The student would do the steps correctly, multiplying the two numbers furthest right, but instead of carrying the tens-place number, the student would put the full number below the line. It was easy to see where and why the errors were being committed which was good in that there was some mathematical sense behind the student's thinking. This makes it easier to help the student when we can see what he/she was thinking. After class, I took the student aside and did a complete problem while he/she watched in order to illustrate the correct way. I highlighted where I carried a number and left the ones number under the line. Then I did a few problems with guided practice so that the student could get used to the pattern of the method. Then I had the student do a few problems on his/her own. Another way that I could have advanced the student's understanding was have another student help him/her out. The students love working together and not only would this benefit the student in need, but it would also benefit the student that was doing the teaching. I could ask this student why he/she is doing what they are and after further instruction if they understand why this works and how to do it independently. There are many tasks this student could find fruitful because in 5th grade, multiplication is in a lot of the other topics so it is important to grasp the concept early on.

Andrea Delise- Week 4

The students were given a math message on the board to start off the lesson. The questions on the board were 10, 12, 14, _, _, _.& 34, 36, 38,_,_,_. & 124, 126, 128,_,_,_. I walked around and observed the students. One girl, Maya, was crying and so upset that she didn't know what to do. I helped her to notice what the numbers were doing from each one to the next. She looked at the number grid at the front of the classroom and was able to figure out that the numbers were counting by two. I noticed that she was fine with the first two problems, but struggled with the third problem. She wrote down: 124, 126, 128, 129, 130, 131. I assume that she had a harder time with this problem because the number grid only goes up to 100. I believe that Maya still knows how to count up, however counting by twos was a struggle for her.

To advance Maya's thinking I would circle a grouping of even numbers on the number grid (2,4,6,8). I would ask her to tell me all of the things in common with those numbers. She will get to the fact that they are all even numbers. I will tell her that when we count by twos, the numbers can only be even, so to check our problems, if the number is odd, then we messed up along the way.

Another way to further her thinking would be to give her manipulatives and have her group them in two. Then I would have her tell me to count the whole group of items, by twos. This would help her practice counting by twos while having something concrete to use as a guide.

Blog of Student Work-Montague

The image above is from Karen’s kindergarten class in which (to my understanding) the children were working on one-to-one correspondence and the children were asked to draw a picture that had 5 items. Ways that I anticipate children to solve this is counting one, drawing one thing and so on (like the task was intending children to do). Another way that I anticipate children to solve this would be an incorrect way in which I see several children in my own class do. They count 1-5 out loud while they are making a dot on the paper but are not keeping in mind the one-to-one correspondence. It looks to me like this student started drawing hearts and may have started counting but lost track of that one-to-one correspondence at some point within his/her counting. However, when they finished, they realized that they had too many and went back and counted using one-to-one correspondence, therefore determining that they drew too many hearts. To me, that is why he/she went back and crossed off the hearts. There could be other reasons, maybe the student didn’t notice themselves that this happened. I wasn’t there so I am not sure, maybe the teacher brought it to their attention and told them they had 3 too many. I am not sure what else to say to this child to have them use a better strategy than to reinforce the importance of pointing to one item at a time to “assign” it a number (so to speak)—one-to-one correspondence. I know that this is only kindergarten so a lot of children this age struggle with this and it is something that they need lots of practice with, eventually I’m sure they will get it! On the basis of this child’s understanding, I would give them more practice with this with bigger numbers. It looks like they understood the concept, maybe they know how to draw 5 items. Maybe they would need more support with a bigger number? If they have the one to one correspondence completely down, I may suggest doing a simple story problem. Then again, I am not sure because I do not know what is too much for kindergarten yet. A simple story problem could be, there are 5 frogs on the log, 2 frogs hopped away to the pond, How many are left? I would include a picture and have them see if they know to cross off two of the frogs to get their answer. This could also be done with addition. 

ERIKA BEHRMAN student work blog

                  For this week’s student work analysis, I chose a worksheet completed by one of our lower-level math students, Fernando*, that focused on greater and less than. The first problem of the sheet provided students with a visual representation using hundreds flats, tens rods, and ones units, which although Fernando miscounted the units, his number was close and he chose the correct symbol to represent the comparison with another given number. However, the rest of the sheet consisted of various number pairs with a blank space for the student to select the correct symbol of >, <, or =. The only problems that Fernando chose the correct symbol for were those where both numbers were equal. This reveals to me that he understands that the equal sign means that both sides are the same.

Since he switched the symbols for all the other problems, I would like to work with him one-on-one to see if he does not remember what each sign means and therefore understood conceptually which number was smaller/larger, but just chose the wrong sign. If so, that is a misunderstanding that could be clarified with a mini-lesson. Or, it could be that his number sense is not developed to the point where he can compare them. If this is the case, a helpful next step for working with Fernando could be to use a number line that helps him to visually compare where the two numbers fall on it. He would be able to observe the trends of number getting bigger or smaller depending on the direction one is moving on the number line. To reinforce the greater than and less than symbolic representations, we could create these symbols out of craft supplies and have him choose numbers on the number line and then place the appropriate symbol in between. Between these two approaches, I feel that Fernando’s understanding of greater than and less than would improve; if they didn’t succeed, I feel that information gained from observing how he worked through these two options would provide me with a direction in which to proceed with assisting him. 

ERIN MAXWELL Student Work Blog

          I have noticed many of my students are quite uncomfortable reading aloud. Additionally, when reading with my students in a one-on-one setting, many are seemingly unfamiliar with common sight words.  Words such as “because” and “for” are often skipped over.  The first time I read aloud with a student and they struggled over such words, I thought is was perhaps nerves or an individual challenge with reading.  However, I have heard nearly every student read in some capacity, and this issue is quite widespread.  

          The students who attended Jackie Robinson for second grade were in a rather inconsistent learning environment, as several teachers and long-term substitutes came in and out of their class.  I believe that such an environment may have been distracting for some students learning, especially for those who were still struggling with reading after completing first grade.  Additionally, my students have been testing these past few weeks to determine their lexical levels and reading fluency.  Based on the significantly below grade-level scores assigned to the majority of the students, I know this year will be challenging when it comes to creating literacy lessons that challenge without overwhelming those who are struggling.

Marie Lewis Student Work Blog

The other day I was working with a small group on finding the mean of a certain set of numbers.  Before the students completed this task they were given two steps to solve for the mean.  They were instructed to first add the values to find the sum, and the divide the sum by the number of values in the set of data.  When the student went to solve the problem, he added the numbers, and then added them again.  He proceeded to divide by the correct number of values, only to arrive at the incorrect answer.  I was confused by his method of problem solving, seeing as we went through the “how to” of solving the problem. 

            As I thought about his thinking I first arrived at the conclusion that maybe he just had absolutely no understanding of what an average is, causing him to mindlessly add and then add again.  It was also brought to my attention that maybe he was looking for a number that was easily divisible by the sum of the number set.  Lastly, the idea arose that he was looking for two number sets to divide by, just as the average is between a high and low number. 

            One way to advance the student’s thinking would be to ask him to explain his thinking when performing the task.  I think it would be beneficial to hear his reasoning, which could potentially trigger him to realize where his own misconceptions lie.  I could advance his thinking by having him explain what an average is or talk about where the word average is seen in everyday living.  This might help spark his understanding of what an average is and lead him to a better goal of finding the correct answer.  This student would benefit from using manipulatives to see how the sum is divided into equal groups.  He would also benefit from a real life application of averages, such as batting averages or temperatures for the week. 

Student Work in Mathematics

     This week, we have been working on subtracting, using tables and graphing. Although these topics seem unrelated, a homework assignment asked the student to analyze the graph and answer questions. While grading papers, I noticed a common theme among the homework assignments and the questions the students had a misunderstanding about. The question says, "Looking at the graph, how many more students like chocolate milk than white milk?" Due to our current work on subtraction, the students were supposed to subtract the amount of students who like chocolate milk (11) from the students who like white milk (4). However, many of the students answered the question by simply saying the answer was 11. These answers showed me that either the student did not understand the question (because the student simply wrote chocolate milk had 11 more students), or the student does not know what operation to use between the two different types of drinks to either count up from 4 or count back from 11. I think this artifact reveals gaps in the student's current mathematical understanding. Since my third graders have been reviewing subtraction facts, this may reveal that the student does not recognize familiar language when dealing with subtraction problems.

     Due to this misunderstanding, there are two ways to advance this student's mathematical thinking. I would ask the student to explain why he/she approached the problem the way that they did. From there, I would be able to better understand the student's thinking and work. However, from my own assumptions, I would push the student to understand what more means. How many more students like chocolate milk than white milk? Or, framing the question a different way: What is the difference between the amount of students who like chocolate milk and white milk? In order for the students to better understand this concept, future assignments such as analyzing graphs and word problems will help this student grow in this area.

Student Work 1

My sample of student work comes from a second grade female student named “Jane”.  The second graders in my class are reviewing (some learning for the first time) how to tell time on an analog clock.  The artifact I’ll reference is a homework sheet that asked students to draw the hour and minute hand on six clocks with a given time, and tell what time it is from three clocks that are filled out.  The first clock on the page asks for the hands to be drawn to show 9:00.  This clock, as an example/reference, also has the minutes labeled outside the clock (ex: 00 above the 12, 05 above the 1). 

              Jane drew the hands correctly on the first six clock problems, but then recorded the time incorrectly on the three problems where she had to look at the hands to tell what the time was.  Interestingly enough, she recorded all the minutes correctly with just the hour hand off by 1.  The strategy that she used for finding the minutes is clear when looking at all the clocks on the page.  She used jumping arches from number to number on the clock to count by 5’s.  Drawing these lines is the strategy my mentor has been working with them on and proved to be effective for Jane in all nine problems on the page.  These examples show me that she has a strength and understanding of the fact that there is 5 minutes represented between two numbers on an analog clock.  However, just from looking at this sheet that remains an assumption because of the example clock I mentioned above.  Though I can see the made the arches to assume she counted by 5’s on her own, I do not know for certain (considering the sheet was done at home) that she didn’t just look at the example to give her the adequate hand to draw.  She may understand that we count by 5’s to find what goes in the minutes, but not actually comprehend that it represents five minutes of passing time within an hour.   However, one concept I observe that Jane shows full understanding of is the difference in lengths between the minute and hour hands.  She clearly shows one long and one short when depicting the time in the given problems with no confusion.

              Another interesting part of Jane’s responses that I mentioned briefly above, is that she interpreted the hour hand on the last three clocks consistently an hour before what is correct.  For example, the first clock depicts 11:15 and she responded 12:15, the second clock depicts 4:45 and she responded 5:45.  I have two initial thoughts on why Jane possibly did this.  First, the minutes in the examples isn’t at the “top of the hour position” so since the hands are already drawn in properly, the hour hand falls between two numbers rather than pointing exactly to one (how a child would draw 9:45, with the hour hand exactly on the 9).  When first teaching students time, the hands point exactly to the numbers 1-12 and any conception of “in-between” is absent initially.  So, this is a logical source of confusion for a young learner who is just getting used to telling time.  I believe that Jane was confused which number to depict as the hour because it was not pointing to just one.  This then lead her to choose the number it hadn’t reached yet making her responses 1 hour early.  My second thought pertains to why she would choose the number the hand hadn’t reached yet versus the number it had just passed.  It is possible that Jane doesn’t understand that as time passes on an analog clock, the hands move around to the right in a clockwise motion.  My students haven’t been exposed to manipulating a clock themselves yet.  Instead my mentor’s focus for them right now is to just be able to read/draw given times on a clock.  This leads me to believe that Jane has some confusion about how a clock actually operates signaling that she may not be comprehending “what time it is” at all. 

              To be able to advance Jane’s thinking I would first and foremost need to ask her to explain why she completed the problems in the way that she did.  I would need to know what her thought process was in writing the hands the way she did to really know what misconceptions she has.  Since this assignment was completed for homework I am unsure.  The next advancement I could make would be to actually start using clock manipulatives to model and show consistently that in a given hour the hour hand moves between two numbers and to the right.  I would show her that (in 1 hour) while the minute hand moves all the way around once, the hour hand only moves slightly and that all the minutes within that one cycle pertain to the number it has passed, even if the hand is closest to the next number on the clock. 

Katelyn McCormick Student Work Week 4

During this week, I observed math in my first grade classroom, but I also saw student math homework.  I chose to bring in a sample of math homework from Student A so that I could see how students complete work at home.  The homework assignment was from the Everyday Math book. The directions  said "Cut examples of numbers from scrap papers you find at home and glue them onto the back of this page".  Student A's homework assignment that was brought back had pictures of numbers in a grocery advertisement.  Some of these numbers were 2/$3, $6.99, and 1.98 per lb.  The student showed that she knew there were numbers in advertisements.  The student did not have any other number representations besides those from the grocery ad. 

            Student A's artifact may reveal that she only thinks numbers are in terms of money in advertisements or that she did not want to search for numbers elsewhere.  Assuming the former, Student A does not know that numbers can represent other things besides money.  The artifact reveals that this student has a gap in her understanding of where numbers can be found.  One way that I could advance her thinking is to show her samples of numbers in magazines, newspapers, and other papers that I have at my home.  This way she could see that numbers are everywhere!  Another way that I could advance her thinking would be to ask her where else she sees numbers on a day to day basis besides at the grocery store.  Simply by asking her this question, she may think of other places that she sees numbers.  My way of adding to the homework assignment would be for students to cut examples of numbers from at least 3 different papers at home so that they are seeing numbers in a variety of places.

            A question that I might ask Student A is what are the numbers used to tell us?  Another question that I may ask is why do some numbers look different than others?  By this I mean why do some numbers say 6 and some say $6.  This would get the student to think about numbers in terms of money and numbers in terms of other concepts.

            A future math task that this student might find helpful is writing a list of where she sees numbers while she is in public places and at home.  This way she will see many places where numbers are used and will not have to focus only on ones that she can cut out from a magazine.  The student could complete this task on her walk home from school by noticing numbers around the school community.  This student will benefit from constant exploration of numbers inside and outside of the classroom.

Bode--

            For week 2’s student sample work I brought in a transcript of a math conversation my mentor teacher had with the whole class while helping one student. The students were playing monster squeeze as a class. Someone will pick a number and the students have to guess what it is by asking, “is it bigger/smaller than [number]?” In this particular situation a student was having a hard time understanding that if he asked is it bigger than a certain number, why would either Ms. Hall or myself move? Wouldn’t just Ms. Hall move since she’s near the bigger number and he asked if it was bigger. This sample work shows the use of language in a math conversation and how visually showing students can aid their understanding when it comes to using a number line and understanding the concept of bigger/smaller than. In the sample work Ms. Hall allows another student to try and answer this confusion before she explains. The student was able to understand what the other student said and comprehend that the actual word bigger/smaller doesn’t mean the person near the bigger number moves when asked if its bigger and visa versa for the smaller number.

            This artifact reveals that the current mathematical understanding of this student may have been confused by the visual aid of using the number line on the wall. The concept of guessing if a number is bigger or smaller than the number being guessed has to do with being able to test students abilities understand which number value is greater or less than others. It may be safe to say that this student knows which number has a high value compared to others but that the visual use of the number line and yard sticks on the wall confused him because you had one teacher near the bigger numbers and another, myself, near the smaller numbers so he thought that that connected with the BIGGER and SMALLER question. Maybe this student is a auditory learner, some students need visuals while others it might throw them off from something they already understand. This shows to me that there may be a gap in this student’s ability to see a number line and still use it for concepts of bigger or smaller number values when guessing a number. When students learn greater or less than signs they have to be able to know which number value is greater or less than others, sometimes while looking at a number line. There may be a gap in this student’s ability to visualize these number values with a number line.

            One way I would aid this student is by utilizing a number line for some activities. One activity I would want the student to do is have a number line in front of them, ask the student to point to any number. When the student picks a number (lets say it’s 9), I will ask a series of questions like “is 6 greater or less than 9?”, and so on with different numbers. By doing this I want to get the student comfortable with using a number line and recognizing the value of a number associated with the number line. Another activity I would do with this student is then sit one on one with them and play the monster squeeze game with them except this time have the student be in charge of covering both ends of the number line. Having the student be in charge of moving the number line will help the student learn first hand how to move each side based on the questions he’s asking “if its bigger or smaller than ___?” I would ask questions like “how do you know that ___ is bigger than ___? Why did you move that side of the number line down and not the other side?” By asking the student along the way to explain themselves it will help him understand why he’s doing and also myself to understand maybe where the student is getting confused or what he is misunderstanding.

Student Math Sample Transcript

[Playing a game called monster squeeze where the teachers hold yard sticks over two numbers on a number line and the class tries to guess what number the teacher, or a student, is thinking of by asking questions like “is it bigger/smaller than…?”]

Ms. Hall: “The number is between 1 and 20, so Miss Bode and I are going to cover the 0 and 21. Who wants to guess what number were thinking of?”

Student: “Is it bigger than 14?”

Ms Hall: “No, so I move the stick to cover 14 while Miss Bode stays on 0 because the number is between 0 and 14.”

Student: “But why? Doesn’t Miss Bode have to move closer to the number 14 too?”

Ms. Hall: “Who can help ____ understand why I moved down from 20 to 14 and Miss Bode didn’t move from 0?”

Student: “Because ____ asked if the number was BIGGER than 14 and since Ms. Hall said no well then I think that the number is going to be smaller than 14 because its not bigger and it could be anywhere from 1 to 13 so Miss Bode don’t move.”

Ms. Hall: “Good. Miss Bode doesn’t move because the number is smaller than 14 meaning it could be any of the numbers between 1 and 13. Who wants to guess now?”

Student: “Is it bigger than 5?”

Ms. Hall: “Yes.” [Miss Bode moved the yard stick from 0 to cover 5]

Student: “Wait wait wait…now why did Miss Bode move? I thought only Ms. Hall was supposed to move if the question asked if it was BIGGER?”

Ms. Hall: “Ahh I see some confusion here. When a student asks if it’s bigger that doesn’t always mean that I am going to move…and if its smaller that Miss Bode will only move. What matters is if my number that I’m thinking of is bigger or smaller than the number your guessing. Let’s say I am thinking of the number 10…if you asked is it bigger than 14, id say no so then I move because 10 is less than 14. Now what if you ask is it bigger than 5, then I say yes. Miss Bode moves now because 5 is closer to 0 and she is on 0 still.”

Student: “So Miss Bode has to move since the number is gonna be between 5 and 13 now and you Ms. Hall stay there cus you have to cover 13 and Miss Bode is closer to 5 than you are.”

Ms. Hall: “You got it!”

Student Work on Exploring Square Numbers

The sample of student work I obtained this week was a worksheet the students completed for homework as an extension from the previous day’s lesson on square numbers. I took a picture of one of the lower level math students’ completed worksheet to analyze. The worksheet had 3 sections. The second section built off the first section and the final section was “practice” of math procedures multi-digit addition, subtraction, and multiplication. The first section required the student to solve number sentences to find square number equivalent. For example, one question was 4*4= ___ and MJ filled in the blank with 16, which is the correct answer. Another format for the questions in this section was 5²= ___. MJ correctly completed this section of the worksheet. The second section asked students to provide the number model for two rectangular arrays then determine which one of the arrays represents a square number and explaining their choice. MJ struggled with this section of the worksheet. She gave the correct number model for both arrays, but struggled to determine which array represented a square number and why the array represented a square number. The third section of “practice” was 4 questions of multi-digit basic math; 2-digit by 2-digit multiplication, 4-digit plus 4-digit addition, and 3-digit minus 3-digit and 4-digit minus 4-digit subtraction. MJ was very successful with this section and showed all of her work for this section.

            MJ’s answers on this worksheet reveals that she has an understanding of how to solve for square numbers, but does not fully understand what it actually means for a number to be square. This gap in her understanding is evidenced by the fact that she identified the wrong number as a square number and her explanation of why the array represented a square number used addition instead of multiplication to prove the number was square.

            To advance MJ’s mathematical understanding I would ask her to verbally explain to me what it means for a number to be square because she may simply be struggling to write down clearly her thought process. If she is unable to clearly explain this concept, which I suspect she will struggle with, I would then have her work through a series of equations to determine the pattern and connection between squaring a number and a square number (ex. 4² and 16.)