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.