Angles Case Study

Chad: I think that Chad understands that there are different size angles. I would like to draw some pictures with him and see if he means the lines are long and short (lines in regards to sides) or if when he says long and short he is talking about the width of the angle. We need to determine what he is referring to before we can help him proceed with his definition.

Cindy: I think that Cindy’s definition was the most vague out of all the student examples. I think that she is referring to the sides instead of angles. I would want her to show me what part she is talking about. I think having her define side and make sure that she is not thinking about that instead of angle would be helpful.

Nancy: Nancy gave a good definition. I think it would be helpful to have her draw an example of an obtuse and an acute angle to see if she really understands the difference. Also, I would like to ask her what she thinks a right angle is.

Crissy: Chrissy gives the most detail out of the students. I would like to see her prove that an angle is obtuse and/or acute. We would want to make sure that she understands the correlations that a 90 degree angle, which she speaks of, has in regards to these other two angles. 

Measurement Case Study

In Barbara’s case study her students know that when you are measuring the length of something that you must start at one edge and measure straight across to the other edge. I think that they understand that your units need to touch and that you should not leave space in between. However, they seem to think that you can use different types of units as long as you get from one edge to another. For example, they are trying to use books and baskets at one time.

Often time’s students get confused and think that bigger numbers in measurement mean bigger units. In the Rosemarie and Dolores case studies they are working through this misconception to help student’s understand why that is not correct.

In the Dolores’ case, the students were comparing a measurement at home with their parents. Students were identifying the correlation between parents and students that had about the same size feet and this caused their measurements to be close. They also noticed that one of the biggest measurements that differed came from one of the smallest students in the class. They understood why that was possible because they knew he had a small foot and that would be a big difference from his father’s foot.  

At the end of the study Les finally gathers his thoughts on this topic and proudly states “big hands and feet get low numbers and little hands and feet get high number because little things take up less room.” This is a great definition that proves to his teacher that he has established a firm understanding between a unit size and the measurement that it results in.

In the Rosemarie study, Miriam also displays knowledge of the correlation between unit size and the overall measurement. She states “I think Gita has the biggest foot because it took fewer of her foot lengths to measure the doorway. The smaller your foot is the more feet you need to measure something.” While her classmate Courtney display the misconception of the bigger the unit the bigger the measurement number. She states “I just looked at the numbers, and Miriam has the highest number of foot-lengths, so I think that her foot is the biggest.”

In Mabel’s case study the students are focusing on ways to use a ruler to measure something that is longer than a ruler. They are working hard to make sure that they do not add any extra space when moving the ruler from one spot to the next. They determine that it is easier to use two rulers because then you do not waste space in between. In Jose’s case study they have moved on and are working to make sure that they start measuring at the correct spot on the standard measuring tool. They see that you must start at the correct spot and end at the correct spot, while taking in to consideration any extra space on the measuring tool. If you do not do it correctly each time your measurements will not be the same.

Barbara’s students are going to have to understand the correlation between the size of the unit and how that effects the overall measurement. They will then work to make sure that when they are using a tool if they do not have enough to go across the entire length, they will have to figure a way to get the entire measurement without leaving spaces. They will also have to learn how to properly use standard measurements. It was interesting to see this alignment across grade levels. 

Ordering Rectangles

1.       Smallest to Largest Perimeters: D, E, G, B, A, C, F

2.       Smallest to Largest Area: C, D, E, B, F, A, G

3.       Perimeter by Measurement: D, B, G, A, E, C, F

I did a good job getting the smallest and the biggest, but I did not do to good at getting the ones in between. I don’t know that I had a good strategy. I think I just kind of eye balled it if you will and guessed.

4.       Area by Measurement: C, D, B, A,E, F, G

Again I did well with the smallest and the biggest, but I did not do well with the ones in between.

5.       It seems that a lot of them were similar in height and that caused the perimeters to be close, but did not affect the areas as much. I am not sure the best way to gauge them without measuring. I did not have a ruler at my house, so I used two colored squares that I cut off the end of Post-Its that you would use to bookmark a page.  

Mean Case Study

Trudy says : Putting it on one of the 4 foot 8’s. Because then you could even all the heights out. And you might get close to the same answer.

She is understanding that if you spread all the numbers out to make a height that is the same for every person that would be your average. So she is using the numbers she has and adding or subtracting a few to each one until they are all the same.

Celica says: I think so because, like, if all the fourth graders stood on each other’s shoulders, and the third graders did that too, I think that the fourth graders would be higher.

She is understanding that you need to add up all the heights to see which group is taller, but is not thinking to figure out an average for each person. She is going in the right direction, but is not quite there yet.

Laural seems to have an understanding of how to arrive at the mean, because she states “we got the average of 13.2, but we rounded it down to 13 because we can’t have a point something.

She knows how to calculate the mean, but does not fully understand what it means. She is just getting rid of the .2 and this tells us that she needs to continue working on a deeper understanding of what the mean stands for. 

Space and Shape

I thought that the shadow activity was the hardest. I thought that it was interesting to see the shadows that it casted. I was not good at visualizing all the different ways that it could make a shadow. I did not think that it would make a hexagon.  

It is important that students become proficient in spatial visualization because it helps students when they are reading a map, giving directions, and packing efficiently. Last year I went to Charlotte to the distribution warehouse where they pack up Christmas shoeboxes to be sent around the world. A man from my church and I were in charge of putting the shoeboxes into bigger boxes and it became a competition of who could pack the most in one box. His spatial reasoning was clearly sharper than mine! As we continued on I did slowly get better and figured our strategies to use to make sure that I was packing in as many boxes as possible.

I think that students can begin working on their spatial reasoning skills as early as kindergarten. When I was teaching kindergarten my student enjoyed doing pattern block puzzles without the shape lines on them. They would have to figure out which shapes would fit where. I think that they were starting to practice spatial skills through this type of activity.  

I like the pentominos activity that we did the other day and I could see how this would help students with their spatial reasoning skills. I also think that nets that we discussed in the webex meeting would be a great way to practice as well. I was not very good at these activities, but I do think that I am better at them now than I was when I was in school. I did well on the first two activities “Trip on a Train” and Plot Plans and Silhouettes.” Providing student’s time to work through activities like this I think would be very beneficial. 

Geometric Definitions Case Studies

Susannah knows that triangles have 3 sides and 3 vertices. I am thinking though that in her head she visualizes what the standard triangle looks like. If a triangle fits or is close to her vision she considers it a “real” triangle. If is too different than she places it in the triangle family, but it is not a “real” triangle. She makes clear that her definition of “real” triangles are similar to triangle B. Near the end of the discussion she is proposing the idea of turning the triangle to see the sides in different ways, leading another student to understand that no matter which way you turn a triangle it is still a triangle.

In Mrs. Rivera’s class the students are working on developing definitions of a square and a rectangle. They are using square as part of their rectangle definition. They state “two squares make a rectangle.” They have not yet figured out the distinguishing quality between a rectangle and a square. At the end Brett is becoming close because he is turning an orange square pattern block around and around in his hand. He is thinking through what makes that shape different from a rectangle. Roberto’s definition makes it clear that he is at a level one on the Van Hiele Model. He is using properties to describe a rectangle. The students are describing the properties that are visual and describe a rectangle, but they are not considering what makes it different from a square. They are engaging in productive talk though that is leading them closer and closer to discovering the difference between the two shapes. Once they can see that I think it will be easier for them to come to a clear definition of both shapes.

I was especially intrigued by Zachary’s comments as it seems to be very insightful to me. He seems to really understand the properties of a triangle and is articulating this thoughts very well. I think that his point is very valid and reflects the thinking of many of the students in that class and many other classes as well. I like how he is trying to a conscience effort to make sure his thoughts coincide when he says “we just have to make our eyes and our heads meet.” In Dolores’s study the students are making it clear that they understand what a triangle is, but because they are not use to seeing it that way it is much harder to classify irregular triangles, as triangles.

The purpose of a definition is to give us something to go by when trying to classify shapes correctly. It is gives us an understanding of if a shape meets the classifications to be called by a certain name. 

Geometry 

What are the key ideas of geometry that you want your students to work through during the school year?

In geometry for the upcoming year a big feature that we work on in first grade is shape names and spatial reasoning skills. We want students to understand the attributes of shapes, how to compose shapes (2D & 3D), and how to partition shapes. We often use manipulatives, specifically, pattern blocks and tangrams to help students become proficient in the geometry strands.

Designing Data Case Study

When Chad begins a conversation with Sally he is displaying that he understands the value in defining and classifying your data categories. He understands that if one has not developed a clear definition then participates maybe responding in any way.  This means that their response could be incorrect for the data he desires to obtain. He wants his data to be a true picture of the question that was proposed.

Students understand that when you represent given data in a chart or graph that it needs to be accurate. Therefore, their questions needs to be made clear to all participates, so that they can responded in the correct manner.

I enjoyed reading the Nadia’s study and thought that the teacher did a wonderful job questioning the students about their survey questions. I actually got a kick out of the sports question. Students are learning that they must have a well-defined question if they want to receive accurate data. Luke displays his understanding of survey questions when he states the difference between to moving questions. This group gave a clear definition of moving by asking the question “how many times have you moved from house to house with all of your belongings?” One group gave a clear question of “how many languages do you speak fluently?” Then they defined what they meant by fluently.

In Andrea’s case study they should say “how many people are in your family that live in your house?” As far as the questions about “how many houses are on your street, I am not sure what exactly the students are looking for and how you would redefine it. That would be a hard question to answer.

Natasha is wanting to know how many states the person being interviewed have visited and they were intentional about being there.  She may could word it like this “how many states have you visited where you spent your vacation or visited family members? She is looking to know how many states her participates have actually spent a given amount of time in. The survey question that they decide on does not reflect the information that she is looking to gather. I think that the teacher in these studies are doing an excellent job of getting their students to think about what their survey questions are really asking. They ask a lot of open ended questions that get their students thinking and questioning others.

                                                               Number of Pockets Data

What are some important features of what’s happening in this classroom?

The students in the video are active participates of collecting the data. They have each counted how many pockets they have and watch as their teacher records it on a line plot.

What did you notice?

Many of the students are unaware of what the x’s represent and what the numbers represent. I think that maybe if the teacher would have used a picture to show the pockets and record the students names above the number of pockets that they have the may have been able to interpret the data better. This would have given them more practice with graphs and prepared them for more symbolic use of x’s and numbers on graphs later.

What struck you about the students’ thinking?

They were determining the correct responses, but they were having to focus so much on what the x’s and numbers represented that they were getting confused.

What struck you about the teachers’ moves?

I think the way that she was having them repeat each other to see if the students had an understanding of what their classmates were saying. She was giving them time to develop their own conclusions and not just telling them the answer before they had time to think.

As you think about each of these, what’s they idea the students are working on?

They are working on collecting and interpreting data. They are working to determine facts about a given set of data. They are learning to use the data that they have collected to determine facts about a group of people. 

Overview of Module 3

Since the NCDPI MOY and EOY benchmark assessments have changed and now include many different ways to use the equal sign my views on how to teach the equal sign has been evolving. This module has challenged my views even more. One statement that particularly resonated with me was in the chapter on equality they addressed the fact that many students see the equal sign as a sign of calculations and not a sign of relation (Carpenter, T., Franke, M., & Levi, L, 2003, p. 21). In many ways I think that many teachers themselves see the equal sign as a way of calculation and this is where many of our children’s misconceptions come from.

As we begin teaching students to develop a concrete understanding that the equal sign is a relationship between numbers and demonstrates equality on both sides, I think that we need to first engage them in an understanding of balance. We want them to understand that for something to be equal it must be the same on both sides. I can see myself using the example of engaging students in a human balance scale. This example comes from “Algebraic Thinking” and is located on page 67, activity 14.3 (Van de Walle). Students in the primary grades are very concrete learners and engaging them in this type of activity would be meaningful to remembering the activity and referencing it in later lessons.

I also think that the activity in “Algebraic Thinking” called “Capture Ten” would be a great way to get young students thinking about using symbols in different context (Van de Walle). Students enjoy playing games and this game would be helpful in developing their understanding of equal as the same instead of thinking of equal as the answer that comes next.

In the “Equality” chapter the first grade teacher states that she engages her students in problems with missing addends and where the equal sign is in the middle of the equation from the very beginning of school (Carpenter, T., Franke, M., & Levi, L, 2003, p. 18). I think that this was a great idea to get students accustomed from the very beginning to seeing the equal sign in different locations of the equation.

I noted that a good sequence for teaching students about the equal sign would first be to involve them in balance scale activities, then true/false equations, then open ended equations, and then allowing them to develop their own equations. Through this process students will see the equal sign in many different locations and become accustomed to seeing the equal sign in different locations of a given equation. Their misconceptions will be challenged from the beginning and they will have time to manipulate and make sense of problems that do not display the equal sign in the traditional way.

I am not sure that I used rational thinking to come up with the correct solutions, but I was able to come up with the solutions.

1.        In the problem with the baseballs I used a guess and check method. I used the “if this then that” strategy. I found the possible numbers to the first equations and then how knowledge/number to find the amount of the soccer ball. However, then I did not have the correct answer to the last one given the answers I had come up with. Given my answer though I was able to figure out how much I needed to manipulate the numbers to come with the correct answer to the last problem. I came up with the answers .35 for the baseball, .9 for the football, and 1 for the soccer ball.

2.       

For the first balance problem in the power point I think that the cylinder weighs the most and the cubes weigh the least. I think this because the cylinder is equal to two spheres and the three cubes are equal to only one sphere making the cubes weigh the least because there are three of them and only one cylinder.

3.       

For this problem, my first step was to determine the value of one sphere. I determined that one sphere is 12 divided by 3 and each one equals 4. Then you have four for the sphere and divided the remaining four on the other scale divided by two would leave each cube at a value of 2. Then 2+2+4=8.  

Critical thinking is involved in finding the solution to all of the problems presented. I think that I have a better understanding of the importance of the equal sign and how important it is to start developing the conception that an equal sign represents the same quantity on both sides at an early age. If done correctly students will have a better understanding algebraic equations from a younger age and will possibly run in to less misconceptions as they advance in their mathematical career.

Work Cited

Carpenter, T., Franke, M., & Levi, L. (2003). Equality. In Thinking Mathematically. Portsmouth, NH: Heinemann.

Walle, V. (n.d.). Algebraic Thinking: Generalization, Patterns, and Functions. In Elementary and Middle School Mathematics: Teaching Developmentally. 

Handshake Blog

Part A

This problem took some thinking. I first decided that I would multiple 19X20, because there are 20 people and they will each shake 19 hands. However, then I decided to break it down and make sure that this was correct. First, I did it with 3 people which would be 3X2=6, but when I drew a picture of each person shaking a hand it was only 3 handshakes. Next, I tried 5 people. The multiplication problem would be 5x4=20 and when I drew a table to check it there were only 10 handshakes. I recognized a pattern each was half of the multiplication problem because each person is only shaking every person’s hand once and not twice. Person A is shaking person B’s hand, but then person B is not going to shake person A’s hand again. In the first multiplication problem you are accounting for AB and BA. This is not the case.

When I charted it out it looked like this.

1-2          2-3          3-4          4-5

1-3          2-4          3-5

1-4          2-5

1-5

4+3+2+1= 10

5x4=20/2= 10

I then decided to start with one less than is in the group (19) and add all the numbers up.

19+18+17+16+15+14+13+12+11+10+9+8+7+6+5+4+3+2+1= 190

Which is also the same as 20x19=360/2=190

Part B

Manuel is not correct. I do understand what he is trying to do, but like I explained above he is accounting for everyone shaking each other’s hand twice. His strategy alone does not work for this problem. His reasoning is not correct. He does try to explain himself. I think that when Manuel sees a chart or a picture he will understand why his thinking alone does not solve this problem.

Beatriz is very much on the right track. I like the way she drew a picture. Her strategy does work, but she made a mistake and did not get the correct answer. Beatriz drew a great chart and justified her answer well. Yes we can fix her mistake. At number 15, she was supposed to add 15, which would be 15+105= 120 and she recorded it as 130. This mistake caused her to get the wrong answer.  Yes, we could use her reasoning for other problems.

I enjoyed Carmel’s reasoning because I did not come up with anything close to this! It was a great pattern that I did not see at all. I am not sure if you could use this strategy for other problems. He did a great job explaining and showing his work. I can see that he arrived at the correct answer.

Katia’s way is very similar to the way that I chose to solve the problem and the way that I checked Beatriz’s work. We used different charts. I did understand what she did and it does work for this problem. Her reasoning is correct and she did a good job explaining it. I do not see any mistakes that she made and I think we could use her chart to solve other problems. 

Teaching for Mastery of Multiplication

How else is there to teach students multiplication facts besides the traditional memorization? In the article they promote teaching students multiplication through contextual situations, so that students will improve their computational skills. Just as we have learned about addition and subtraction, students must have an understanding of what the “multiplication sign” means and what it represents, so that they can develop strategies to solve problems.

Very few students are able to memorize all of their multiplication facts and then use them in a problem solving situation. This is why it is so important for teachers to teach meaningful ways for students to understand and be able to manipulate multiplication problems. The article suggest four steps to teaching multiplication, “introducing the concepts through problem situations and linking new concepts to prior knowledge, providing concrete experiences and semi-concrete representation prior to purely symbolic notation, teaching rules explicitly, and providing mixed practice” (Wallace and Gurganus, p 29). We must begin by teaching students understand multiplication through real world problems, which is the opposite approach from the traditional method of memorizing and then looking at real world problems. It is important that students have opportunities to use manipulatives to grasp the concept of what is taking place in a multiplication problem. They must be taught the rule of 0 and 1 as well. The most effective way to teach these rules are examples paired with strong teacher language, for example the rule of the 1’s in multiplication. Last, teachers can provide some mix drill practice to promote fluency.

We want to teach student to understand and not just memorize. If we change our multiplication teaching and allow students opportunities to develop understanding and “encourage them to use personal strategies for learning the facts and developing automaticity” (Wallace and Gurganus , p 33) students will develop a positive attitude toward multiplication. A positive attitude will in turn have an effective impact on their mathematics education.

Work Cited

Wallace, A. H., & Gurganus, S. P. (2005).  Teaching for mastery of multiplication. Teaching Children Mathematics, 12(1) 26 – 33.

Storyline Online

Chester’s Way by Kevin Henkes

11. Chester and Wilson like to play baseball. At the game Chester got 5 hits when he was at bat. Wilson got some hits too. Together they hit the ball 8 times. How many hits did Wilson get when he was at bat?

22. Chester, Wilson, and Lily like to eat watermelon on a hot day. There were many seeds in Wilson’s watermelon slice. Wilson swallowed 10 watermelon seeds. He spit out 7 seeds. How many seeds were in Wilson’s watermelon slice?

33.  Lily like to wear a lot of Band-Aids in the story, so that people thought she was brave. On Monday she wore 7 Band-Aids on both legs. On Tuesday she words 12 new Band-Aids on her arms and legs. How many more Band-Aids did Lily wear on Tuesday than Monday?  

44. Chester, Wilson, and Lily are raking leaves in Chester’s back yard. Chester raked some piles by himself. Lily and Wilson rake 6 piles of leaves together. They raked 15 different piles of leaves altogether. How many leaves did Chester rake by himself? 

Place Value

Developing Whole- Number Place-Value Concepts

This chapter really challenged me to reflect on my teaching practices when it comes to teaching place value. This is a big part of first grade and like the chapter stated it is a concept that students are going to continue to build on in their future math classes. It was good to see the alignment of this topic throughout grade levels.

“In first grade, students count and are exposed to patterns in numbers up to 120 and they learn to think about groups of ten objects as a unit. By second grade, there initial ideas of patterns and groups of ten are formally connected to three-digit numbers, and in grade 4 students extend their understanding to numbers up to 1,000,000 in a variety of contexts. In four and fifth grades, the ideas of whole numbers are extended to decimals (CCSSO, 2010) (Van de Wall and Karp and Bay-Williams, p 222).” 

The three ways of counting that are referred to in this text are unitary, base ten, and equivalent counting. Unitary counting is when students are counting each single unit by ones. They can count all of the objects in a set and tell you what number is represented in the set. However, they do not have a understanding of the value that each digit holds in the number.The base ten counting is when students organize a set of objects in a set into the maximum amount of tens and then provide the other counters as left overs. They understand what the digits in a two digit number represent and can connect that to their counters. They can count by tens and ones and know that 5 tens and 3 is the same as 53. Equivalent counting is when students are shown a set with some groupings already made, but there are more than 9 single units left over. Students would know to continue grouping sets until there are nine or less single units left before counting. They understand that just because they change the grouping does not mean that they are changing the number. For example, 3 tens and 23 ones is the same as 5 tens and 3 ones.  

The three different models used to help students grasp place value are groupable models, pregrouped models, and nonproportional grouped models. Groupable models allow students to put the groups together themselves. This model allows students with limited exposure to place value to have the opportunity to manipulate and create groups to develop an understanding that you are placing single units together to make a group and then you can take them back apart. Items that could be groupable are beans, straws, and digiblocks. Pregrouped models are items that are already put together and cannot be physically taken apart they must be traded for single units. I think that this is a common manipulative used in many elementary classrooms, including my own. I know that when teaching place value in my classroom we spend many weeks using and manipulating our place value blocks. I know understand that to help children understand why there are ten little blocks glued together we must first allow them to use the groupable manipulatives. Nonproportional models should not be used in an elementary classroom as this model can be very confusing for students. They need to be able to see the visual difference between a group of ten and a single unit. A group of ten should be ten times bigger than a single unit.

Students can use place value pictures, manipulatives, cards, and mats to help them write two and three digit numbers. I have used many of these methods in my classroom, such as having students draw pictures and then write the number or use mats to lay out their number and then record the written numeral in their journal. However, I have never used the place value cards before and think that they would be helpful to have in your classroom. They would look something like this.

Hundreds Cards would look like this.

 

            9               0                        0

Tens Cards would look like this.

 

                               7              0

Ones Cards would look like this.

                          

                                                                        7

(I am going to ask you to imagine that they numbers are in the box like that are for the first one I am having formatting trouble and I am not sure why it is not working correctly, but I think you can understand what I am trying to demonstrate.)                                                                                 

                        

Students could then stack them on top of each other to see the written numeral they are trying to make.

Hundreds charts can be used to help students become fluent at skip counting, identifying two digit numbers, recognizing patterns when counting by 10’s, filling in missing numbers, and identifying relationships among numbers. They are great visual to help students understand numbers and where they are located. There are many different ways to engage students in learning with hundreds boards.

Benchmark numbers are numbers that students learn to use as they learn informal methods of computation. Common benchmark numbers are multiples of 10 and 100 and occasionally 25. When students are learning to do computation you want them to easily be able to create a benchmark number. For example, when adding 74+126 you would want them to add 6 to 74 to make 80 and then add 80+120= 200. This is why it is important for younger students to become fluent in making 10. If students can make ten then when they being adding and subtracting bigger numbers they will be able to do it more fluently.

One of the activities that was describe to help children with using benchmark numbers was to give them a set of three numbers and then ask questions to guide discussion about the given numbers. See page 241 for more in depth instructions. 

Understanding Our Students’ Thinking

Case Study Reflection

Case Study #11

In the first case study that we read about this week Andrew’s knowledge of rote counting reflects the understanding of many young elementary students. He has mastered the skill of understanding and counting to 10 in order. He sees that all the numbers in the row of 50 begin with a 5. He observes that the second number is in order from one to nine and therefore makes the connection that 10 comes after nine and this is the reason he thinks that “fifty ten” comes after fifty-nine. He understands that the ones are going in a consecutive order; however, he does not understand that when you have more than nine the 10 becomes a single unit of ten. He has not grasp the concept of making a group of 10 and where it belongs in the place value system.

Case Study #15

In Danielle’s case study she is asking her students if “anyone knows how to write the number 195 using numerals?” She records these answers that the students have come up with.

1095

10095

195

1395

1295

In the first response I would say that the students has some knowledge about what they number 100 looks like. However, they do not demonstrate a full understanding because they only give one 0 in between the 1 and the 95. The second student has a full understanding of how to write the number 100, but doesn’t understand how to record numbers bigger than 100. They are demonstrating literally putting together 100 and 95.  They have put these numbers together to math what they think forms 195. The last two response I am not completely sure about. I would say that these students were just guessing. I am not sure where the 2 or the 3 came from.

I enjoyed Nathan’s (which is the student that provided the correct answer) explanation when asked “how he figured out how to write 195 correctly, using numerals?” He said that he knew when you write 101 you replace the last 0 with a 1 and as you keep going up you replace the 0’s so that is how he came up with 195. I think that this is a very clever observation and demonstrates that he has some understanding of bigger numbers and their different values. This explanation could help other students see similar patterns and relationships between numbers.

Case Study #14

I enjoyed the discussion that this 2^nd grade class took part in about the number zero. They are genuinely  thinking and trying to make sense of this number for themselves and for their classmates. They refer to zero as being different kinds of zeros.  Lamont says “There are two ways to make zero. This is the 7 for the tens and it (the zero) makes 70.” He understands that depending on where the zero is placed in the number will determine what the number is. If the 0 is before the 7 then the number is just 7, but if the 0 is after the 7 then that make the 7 seven groups of 10 making the number 70. If this concept was just explained by the teacher without allowing students to discover and come up with understandings of what zero is, then I can see where many students would easily become confused. By allowing them to think through the problem and listen to other students ideas they engaged in a deeper understanding of the number zero.

Case Study #12

The the activity presented in case study 12 is a great way to get students thinking about the patterns that can be observed when grouping sets into 10. This is a pattern that I think we take for granted as adults. I loved this activity and am eager to try it in my classroom. I think that it is perfect for introducing first grade students to place value.

The math skills that students are practicing through this activity are rote counting, skip counting, grouping, problem solving, representation, and reasoning. As they participate in this pre-place value lesson they are thinking about connections and patterns that I believe will lead them to a more comprehensive understanding of place value. This frim understanding will lead to a firm foundation that will carry them through their mathematical career.

The activities presented in these case studies were very meaningful to students and allow students to being thinking critically about math. I enjoyed thinking about the ideas presented and loved how students were engaged in math conversation. I am eager to try some of these activities in my own classroom. 

The Vision for an Engaging Mathematical Classroom

Throughout my undergraduate career at UNCW I am very thankful for the professors that prepared me to look at math through a new lens and push me beyond my comfort zone. Teaching to the common core standards and incorporating the process standards into my daily lessons was not something that I was accustomed to in my educational journey as a student. It was something that was very new to me and took time for me to process. This became even more valuable to me when I entered into the educational field. When I began teaching the North Carolina teachers were learning about these new standards and experiencing the challenge and shift that I had already been exposed too. It was nice to feel prepared. With that being said, I view myself as a lifelong learner and know that there are always areas that I can improve on.

The common core standards are the guidelines that we are to teach and what our students should be able to do when they leave our classroom. The process standards are what we should reflect on when are planning a math lesson for our classroom. We should be asking ourselves “are we guiding our students through an understanding of the content in a way that they will experience problem solving, communication, reasoning and proof, connections, and representation?” These documents are intended to help teachers engage students in critical thinking when participating in a mathematical lesson. They are intend to help teachers engage students in lessons that will prepare students to use mathematics on a daily basis and feel comfortable doing so. The process standards are the aspects that should be included in a meaningful lesson when teaching the content of a specific or multiple common core standards. We are wanting to prepare students to have the capacity to think critically and solve mathematical problems that they are faced with in their everyday lives, as well as in their professional careers.

In order to prepare students to have lifelong critical thinking skills in mathematics educators must look at the way their classroom is being structured. We do not want our students sitting and listening to us talk throughout the whole mathematical lesson. Our students should be challenged with problems that encourage them to activate prior knowledge and participate in collaboration with other classmates. They should be given opportunities to represent the given problem with objects of their choice. Teachers are becoming more of a facilitator in the mathematics classroom. They are guiding students using questioning and support. Teachers are not the only one doing the talking and students are not just practicing mathematical skills in isolation.

This type of classroom requires careful planning and preparation. We are still giving students examples of what good problem solving looks like, but we are allowing them the time and opportunities to use the skills we have taught them in many different math lessons. We are connecting math to the real world, so that they can understand the relevance and importance of math in our daily lives. I think this can be a challenging shift for some, but it is very necessary to the future of our students.