Reflection for CCSS-M SMP Project

The CCSS-M SMP project was a very valuable learning experience for me. It is one thing to read a set of standards, but it is an entirely different experience actually finding activities and examples of implementing each standard in a classroom. My partner and I focused on SMP #1 and #2, make sense of problems and persevere in solving them, and reason abstractly and quantitatively, respectively. While doing the reading for each of these standards, I tried to focus on the main points and relate them back to my past experiences working in classrooms. This helped me to visualize the example better, and apply it better to my life. I also learned that the standards build on each other throughout the grade levels. I thought it was interesting how a single standard is implemented through grades K-8, but the intensity and difficulty of the standard is increased with each grade level. This provides a sense of flow for students moving from grade level to grade level.

Additionally, I learned a lot from viewing the videos that other groups created. There really is an abundance of information and examples that can be found for each standard. One of the standards that I liked learning about the most was the standard involving modeling with mathematics, SMP #4. Modeling in mathematics is a very effective way to get students engaged in their work, while also helping them to understand their task and problem solve. I liked how all of the groups used a Prezi and narrated it with a Jing. I know for myself, I tend to get nervous speaking in front of my peers. The Jing helped with this, because I was able to get all of the pertinent information about my standard across to my peers without having to feel nervous. Overall, I took a lot away from learning about my assigned standards and the other standards that were presented.

Video 1 Analysis - "Word Problem Clues"

Video 1 Analysis- "Word Problem Clues"

The Planning

This re-engagement lesson was planned accordingly to Common Core State Standards for 2nd grade, and also kept in mind the CCSSM SMP guidelines. The lesson consisted of 4 separate tasks, some of which involved addition and some which involved subtraction. The core math idea was using addition and subtraction up to 100 to solve word problems. In addition, another core math idea was for students to be able to use their understanding of place value in order to assist them in adding and subtracting for the word problems. Ms. Lewis, the teacher of the 2nd graders shown, made note in the planning period beforehand that her students had a tendency to just assume they were adding in all word problems. She believed this was due to the fact that they had focused so much on addition during 1st grade and the beginning parts of 2nd. Many of her students did not even read word problems, and instead automatically added the numbers within the problem. When she would ask these students why, they would have no response to back up their chosen process. This was a clear indication to Ms. Lewis that additional work on adding and subtracting in word problems was needed. She also made note that she promotes her students to use words, numbers, and pictures in all of their word problems. However, she noticed that the first time this lesson was done, many of her students had words, numbers, and pictures but they did not correlate to one another. The plan for the lesson was to discuss problems 1 and 2 as a class, and then break out so students could change or add to these problems and also look at problems 3 and 4.

The Lesson
The lesson began on the carpet with all of the students. Ms. Lewis reminded students of the problems they did recently, and explained that they were going to look at those again. She had several examples of actual student work blown up onto chart paper. In the first sample, the student used numbers and words to show how they solved the first question. They got the right answer; however, their numbers did not match their sentence. Ms. Lewis used this to show how your words, numbers, and pictures all have to connect to each other and say the same thing. In the second problem, this student added the two numbers together instead of subtracting. This was a common error among all of the students. The second student example used pictures, words, and numbers for the first problem. They got the right answer; however, their three items did not align with each other. Their picture did not display how they found their answer. In the second problem, they also used a picture. This time, they did find the right answer and the words, numbers, and pictures seemed to align better. At this point, the kids were becoming very restless, so Ms. Lewis had them move to their desks. She explained that they were going to get photocopies of their work back. They had the option of using a blue pen and making changes/additions on their photocopied sheets OR they could work on a new, blank worksheet. First, though, students were to form a dyad with a table partner to discuss what they did and why. Students did this for a bit, and gave feedback to each other about strategies and answers. Students worked on the two problems that were discussed, and some also went on to look at problems 3 and 4. Ms. Lewis helped many of the students who still seemed unable to grasp the concept. Some students seemed to be getting a better grasp on the idea, but others were still struggling.

Faculty Debriefing
Once the faculty joined together again after the lesson, Ms. Lewis made some points about the lesson. She believed that it definitely did not go as she planned. She felt that she ran out of time, and more time would have been beneficial to her. She was happy, although, with the high amount of participation and willingness of her students to admit if they were lost or confused. She explained that one of her goals moving forward was to have her students realize that their words, numbers, and pictures ALL need to be connected to one another. She wants her students to look back at all of these elements and make sure they all lead to the right answer. Her students have these 3 items instilled in their heads, but often have them representing different things. Ms. Lewis also said that she would have liked to show more student samples, including one that utilized a number line. There are multiple strategies that could be used to solve each of these problems, and Ms. Lewis said that she really wanted to emphasize these to her students.

Overall Thoughts
I thought that this was a very interesting video to watch. It was clear that many of the students automatically assumed that all of the word problems involved addition. Even after reading the problems, many still believed addition was the process to be used. I thought it was interesting how Ms. Lewis' students knew to focus on "code words" for processes, like "how many more" meaning subtraction. I read an article recently saying that teachers should deemphasize the use of these "code words" because they are not always correct in suggesting the process that should be used. I think that Ms. Lewis did an excellent job of explaining throughout, and allowed for students to do a lot of the thinking. Her students definitely need to focus on reading word problems carefully, and breaking them down into the most important pieces.

Journal Summaries

A Blizzard of a Value 

(Found in Mathematics Teaching in the Middle School - Vol. 20, No. 6, February 2015)

One of the most important, yet most difficult tasks of creating mathematical problems is making them so that students are engaged, interested, and motivated to solve the problem. As teachers, how do we create effective problems that are mathematically challenging while also relating to students and keeping them engaged in problem-solving? This article was about Ms. Bosetti, a 7th grade math teacher in Ohio. This was a question that she faced regularly, and even more so after participating in over 100 hours of professional development in mathematics. She came to the realization that modeling in mathematics, one of the CCSSM SMP standards, could be realistically achieved by keeping students engaged in real problems that affect their lives, deeply discussing their models and thought processes, and giving and receiving feedback about models. With these concepts in mind, Ms. Bosetti created a realistic problem for all of her students to solve. Her problem asked students which size blizzard at Dairy Queen was the best value. She listed prices for each size, and additionally gave students a cup of each size to work with. Ms. Bosetti knew that this was a challenging task, but felt confident that students could figure it out. She made it clear at the beginning of the school year that her math class would NOT be about solving a problem, finding a solution, and moving on. Instead, students would continually work to find alternate ways of solving problems, justifying their work, and critiquing other's ways of solving problems. Once the DQ problem was presented to students, they self-divided into small groups. Most of the groups began by measuring each cup's height and diameter. From there, many students began to notice that the bases and tops of the cups were not identical in size. Some also questioned exactly how much ice cream was in each blizzard (filled to the top, overflowing, under the rim, etc.). When groups were ready to share their problem-solving methods, Ms. Bosetti asked different groups to explain their findings using different types of displays. One group presented using a graph, another using a table, another using pictures. Once students began presenting, they came to the realization that not all groups came up with the same measurements and numbers. As a class, they discussed that this was due to precision error during measurement. Ultimately, students came to the conclusion that a medium sized blizzard was the best value at DQ. After the problem-solving, Ms. Bosetti asked students to reflect on the activity. For the most part, the students said that they felt more motivated to find the solution because it actually pertained to their lives. They also said that they liked the use of multiple representations (graphs, tables, pictures) to solve the problem. This showed them that multiple strategies could have been used. Additionally, students learned about precision error and how it affected their measurements.

I thought that this was a very interesting article, and an ingenious way to implement effective problem-solving in a middle school mathematics classroom. I think that it is very difficult to make problems that are not only meaningful, but also relatable to students. Ms. Bosetti did an excellent job of this, and even her students noted that they felt more motivated to solve the problem because they ate at DQ frequently. I think that this article will greatly apply to me as a future teacher. Regardless of grade level, I will need to create meaningful problems for my students that they can relate to and feel engaged to solve. I believe that this is a difficult task at some times, because it may be hard to relate to every single student. However, I will need to find out my students' interests, hobbies, likes, etc. to be able to create problems like the one Ms. Bosetti created.

Three Strategies for Opening Curriculum Spaces 

(Found in Teaching Children Mathematics - Vo. 21, No. 6, February 2015)

In most school districts, teachers are provided a mandated textbook or teacher's guide for each subject that is to be taught. In general, teachers are required to go by these textbooks. However, it is not necessary for teachers to follow these guides exactly. In fact, most curriculum materials are created so that teachers have "open space", or additional room, to engage students and relate more to their lives. This article began with a hypothetical question about a teacher reviewing the day's lesson in the teacher's guide, and noticing that the lesson is about subtraction and begins with 10 practice problems followed by 2 word problems. She knows that her students can solve those problems using at least three different strategies. So, how can she use this within the lesson? The CCSSM requires that students utilize multiple strategies when solving problems, in addition to connecting problems to their own lives. This is an issue that many teachers face, and this article provided three suggestions for modifying curriculum materials to promote more learning. First, it was recommended that teacher's rearrange the components of the lesson. Many textbooks have students problem solve at the end of the lesson, but this article suggested that students problem solve throughout. In addition, the teacher should consider getting rid of any parts that include the teacher telling, directing, or showing students how to problem solve. The second suggestion was for teachers to adapt the tasks within the lesson to promote their students' Multiple Mathematical Knowledge Bases, or MMKB. This could be done by adjusting the numbers in word problems for different students, and also encouraging the use of multiple representations and strategies while solving problems. The third suggestion was for teachers to continually look to make authentic connections. Changing word problems to be more relatable to students should be done regularly. In addition, beginning a lesson with problem-solving may be more effective than ending the lesson that way.

This was a very interesting article to read. I have always questioned how we, as teachers, are supposed to make our lessons engaging and relatable to students while also utilizing the provided textbook or teacher guide. After reading this article, I have realized that the teacher guide is supposed to serve as more of an outline of the lesson. Teachers, in turn, are then to modify those outlines to have students more involved and able to connect the lesson to their lives. I took great interest in the second suggestion of adapting the tasks within a lesson for individual students. I have never thought to adjust a single word problem using different numbers for lower and higher level students. This is a wonderful idea, and definitely something that I will do to differentiate mathematics learning in my classroom. In addition, making mathematics relatable to students is extremely important, and ultimately what I think leads to more motivation in solving problems. This is something that I plan to work hard to achieve in my future classroom.

References:

Bostic, J.D. (2015). A blizzard of value. Mathematics Teaching in the Middle School. 20(6). 350-357.

Drake, C. Land, T.J., Bartell, T.G., Aguirre, J.M., Foote, M.Q., McDuffie, A.R., Turner, E.E. Three strategies for opening curriculum spaces. Teaching Children Mathematics. 21(6). 346-353.

Articles Related to CCSSM Standards #1 & 2

Tasks, Questions, and Practices 
by Chandra Howley Orrill, Associate Professor and department chairperson in STEM Education and Teacher Development at the University of Massachusetts–Dartmouth

The 1st CCSSM Standard is make sense of problems and persevere in solving them. Many teachers will create their own problems for students to solve, and it is important to recognize and monitor three items while doing this. Chandra Howley Orrill writes that tasks, questions, and practices are three very important things to take note of when creating problems.

Tasks are described as the actual problem, and require a certain level of competency in order to solve the problem. There are lower-demand tasks and higher demand tasks. Orrill explains that lower demand tasks are problems that require simple memorization to solve. On the other hand, higher-demand tasks are problems that require students to think outside of the actual question and connect it to other areas of mathematics in order to solve.

Questions are another important component in creating problems. Lower-demand questions are those that can be answered with one-word or short answers. Orrill gives examples of these questions being similar to, "What is your answer?" Higher-demand questions require students to explain their process for finding their answer. Higher-demand questions are more effective in allowing students to make sense of the problems that they are handed.

When using these higher-demand tasks and questions, the processes that students use to solve problems, in turn, become more higher-demand practices. Students must be able to pull sense from the problem, while identifying meaning and determining how to solve the problem. All of these higher-demand practices allow students to make better sense of problems and persevere to solve them.

Additionally, this article provided two tasks, or problem solving examples questions, that could be asked to a group of students. One was considered a lower-demand task, while the other was a higher-demand task. It was interesting to see the difference and how the higher-demand task required much more thinking and cognitive abilities to be used on a student's part.

Reasoning and Sense Making - Expanding our NCTM Initiative 
by J. Michael Shaughnessy, NCTM President

The 2nd CCSSM Standard is reason abstractly and quantitatively. To the surprise of many educators, reasoning is not a new concept in education. In fact, books published in the last two hundred years have made mentions to student reasoning, and the importance of young learners actually thinking about their thinking and processes. Even in these dated publications, reasoning was a student-centered activity. In today's classrooms, reasoning is still just as important and useful to students' understanding in mathematics. However, it has grown and created a more insightful and in depth type of reasoning. 

Current reasoning practices include the use of student metacognition, discourse, and ample opportunities for students to share and explain their reasoning to peers (Shaughnessy, 2011). All of these are examples that can be used in a classroom to promote reasoning among students during mathematics. Reasoning in mathematics has become so well-known and discussed that half of the CCSSM Standards deal with reasoning in some way. This is a clear indication that reasoning in mathematics is a very important concept, and teachers need to encourage students to practice it.

References:
Orrill, C.H. (n.d.) Tasks, questions, and practices. National Council of Teachers of Mathematics. Retrieved from http://www.nctm.org/Publications/Teaching-Children-Mathematics/Blog/Tasks,-Questions,-and-Practices/

Shaughnessy, J.M. (2011). Reasoning and making sense - expanding our NCTM initiative. NCTM Summing Up. Retrieved from http://www.nctm.org/News-and-Calendar/Messages-from-the-President/Archive/J_-Michael-Shaughnessy/Reasoning-and-Sense-Making%E2%80%94Expanding-Our-NCTM-Initiative/

CCSSM Standards for Mathematical Practice - #1 & 2

Standard #1: Make sense of problems and persevere in solving them.
Important Points:

• Problem solving is notoriously a difficult task for students of all grades levels. Teachers have been known to create more basic or simplified problems for their students to solve, in hopes of eliminating confusion. However, this is not the goal of the CCSS Mathematical Practices. In fact, this simplification of problems does not allow students to be challenged or take away any meaning from problem solving.
• It is crucial for teachers to create effective problems for students to solve. The CCSSM recommends that problems are interesting to students, utilize meaningful mathematical situations that are attainable for students to solve, allow students the opportunity to apply the concepts that they are currently learning or have previously learned, are challenging enough, allow for multiple solving strategies to be used, and have the ability to gauge a student's understanding of the problem. 
• Problem solving involves many comprehension strategies that students learn in reading. These strategies can greatly assist students while problem solving. Encouraging students to break a problem up into smaller, more understandable pieces may help them to identify the most important information, decide useful strategies to solve, and reflect on problems they have previously solved that may be similar to the one at hand.
• Some students get so caught up in finding the right answer to a problem, that they do not think about anything else while problem solving. It is important for a teacher and students to remember that getting a correct answer is great, but just going through the process of problem solving can be very useful and students can take a lot of learning away from it to apply to future problems.

Standard #2: Reason abstractly and quantitatively.
Important Points:

• Reasoning in mathematics can help students to better comprehend and find effective ways to use concepts and procedures. In addition, reasoning can help students to look back on past information that they have learned, and apply it to their current learning.
• Helping students to build a strong number sense can lead them to have better reasoning abilities in mathematics. Some things that can assist with this are helping students to apply relationships to numbers, realizing the largeness and complexity of certain numbers, and learning different ways to compute numbers.
• Promoting the discussion of reasoning as a whole class or in small groups can also benefit students in building their reasoning skills during problem solving.
• Students who can reason effectively look at mathematics as more than just solving number problems and getting a correct answer; they see it as a way to understand their own lives and the world that they live in.

References:

Briars, D. J., Austurias, H., Foster, D., and Gale, M. A. (2013). Common core mathematics in a PLC at work: Grades 6-8. Bloomington IN: Solution Tree Press.

Larson, M. R., Fennell, F., Adams, T. L., Dixon, J. K., Kobett, B. M. & Wray, J. A. (2012). Common core mathematics in a PLC at work: Grades 3-5. Bloomington IN: Solution Tree Press.

Larson, M. R., Fennell, F., Adams, T. L., Dixon, J. K., Kobett, B. M. & Wray, J. A. (2012). Common core mathematics in a PLC at work: Grades K-2. Bloomington IN: Solution Tree Press.

National Governors Association (2010). Common Core State Standards Initiative - Mathematics Standards. Retrieved from http://www.corestandards.org/