CCSSM SMP Reflection

What I learned while doing the reading and work for our standards:

         I thought that this project was a great way to gain a better understanding of the standards and how those standards could actually be implemented into the classroom for various grade levels. Rather than just reading the standards, I liked that we had to actually break down the standards we were assigned, process them/get comfortable with them, and teach them to our peers; this approach provided me a more valuable learning experience for understanding the math standards. Lauren and I were assigned two standards: 1. Model with Mathematics (SMP #4) and 2. Look for and express regularity in repeated reasoning (SMP #8). I learned that modeling with mathematics is all about providing students with real-life situations in math. Some people think modeling is just manipulatives but it's more than that; it's also providing students with visual and structural representations. I learned that the more students can use different visual and structural materials/representations to help them solve a real-life math problem, the more engaged and meaningful their learning experience will be. Look for and express regularity in repeated reasoning was a little more challenging to understand. I learned that this standard is about having students look at math problems/math situations and students are consistently able to recognize skills within the problem and can learn to apply shortcuts. I also learned that if students use a shortcut, they should understand and be able to explain why they used it.

What I learned while listening to others present their standards:

       After watching the other groups' video presentations, I learned more about the other math standards in addition to the two that I researched. One main point that I noticed when learning about the other groups' standards was that many of the standards relate in some way with each other. For example, I learned that in many of the standards, students had to explain their reasoning or make sense of the problem. Also, I realized that many of the standards fit into multiple NCTM process standards (problem solving, communication, connections, reasoning and proof, and representation). For instance, I think students are problem solving in all of the CCSSM SMP standards. One standard in particular that I found interesting was SMP #5, Use appropriate tools strategically. There are so many math tools (including math technology tools) that can help students, but it's key that students understand when to use which materials. I enjoyed watching and listening to everyone's Prezis and Jings, and I learned a lot about the standards so I can effectively apply them in my future classroom.

Video Analysis 1: "Word Problem clues"

"Word Problem Clues" Video-2nd Grade

The "Word Problem Clues" video in a 2^nd grade classroom with Tracy Lewis was intriguing to watch.  After watching all the videos (planning, lesson, debrief), it caused me to reflect a lot about different strategies and approaches to teach students math in order for them to be the most successful. I appreciated that the students were going back to analyze and change their work if necessary. This shows that making mistakes is a part of the math process, and they can learn to go back and read the problem more clearly as well as use pictures, words, and, labels to help explain their process.

            The Planning:

Ms. Lewis explained that using pictures, numbers, and words to explain our thinking could be a challenge especially when trying to relate all of the information together. She describes that when she picks up a piece of paper to see if a problem is more or less accurate, she wants to see how they thought about the problem. Without any information, it’s hard to clear up misconceptions or see if the students discovered a different strategy/approach to solve the problem.

These students are used to working with numbers and adding them together. A misconception is that the students just look at two numbers and simply add them together and think that’s the answer but then have trouble explaining their answer (If you ask, “Why did you add?,” they don't know). However, in 2^nd grade, these students actually need to start reading through the problem. The core math idea is using addition and subtraction (up to 100) to solve word problems. After reading through the problem, students need to recognize and understand the words (i.e. “more” or “how many”), and figure out what the problem is asking (i.e. “all together?” or “how many more?”). The students need to look at the word problem, break it down, and have some kind of representation (pictures, numbers, and words or a combination).

The goal of this lesson is for Ms. Lewis to understand what they did/how they thought about the problem and for students to process/break down their work and analyze it to see if it needs to be made clearer or needs adjusting. In other words, the goal is to study the impact of the lesson on student thinking, learning, and understanding. Teachers should look at: Are students able to identify the math strategy (i.e. addition, subtraction, etc.) and then are they able to use this strategy effectively? (Is their process correct? What strategies are they using to solve the problem and are they using these strategies correctly?)

The focus of this reengagement lesson is for students to go back and look at their math on the first two words problems that were discussed as a class as well as review problems 3 and 4 (not addressed in the reengagement lesson) to see if they can apply what they learned. The students need to reflect and possibly fix their mistakes/clarify their work.  After this lesson, students will hopefully understand that they may need to re-work a problem with a different strategy or modify an existing strategy.

           The Lesson:

At the carpet, Ms. Lewis explains that the students are now detectives and they need to look for evidence to solve math mysteries. She and the students discussed that mathematicians use evidence, labels, pictures, words, number sentences, and answers. Ms. Lewis reviewed some students’ previous work, which was displayed on chart paper.

Throughout this reengagement lesson, Ms. Lewis used the students’ work to emphasize how the numbers, pictures, words, and answer needs to correlate with one another. In other words, they all need to support each other; if you drew a picture, words should explain that picture and the answer should reflect what was shown in the picture and described in the explanation.

In the first student example (field trip), the student used numbers and words to explain how they arrived at the answer. While their answer is correct, the student’s numbers, words, and explanation did not match with each other. In the second student example (apple farm), the student subtracted rather than added (maybe they thought the “how many more” meant adding or maybe they simply saw numbers and added them together (as mentioned earlier as a misconception)). For the second student example (field trip), the student used pictures, words, and numbers and got the correct answer but once again, it was confusing as to how they arrived at their answer; What are the words describing? (the pictures, words, and numbers didn’t correlate with each other). In the second student example (apple farm), the student used pictures, words, and numbers and it seemed to correlate better with each other. In this problem, the student got the correct answer (realizing that it was asking for “How many MORE?,” which is a code for subtraction).

            At the carpet, a common theme I noticed throughout the problems was that many students had difficulty identifying the strategy to use (i.e. addition or subtraction) and then had trouble explaining why they chose that strategy. I liked how Ms. Lewis took the time to go through the students’ work and ask the students what they observed. Rather than telling and showing the students how to correctly solve the problem, I think that it was valuable to have the students themselves observe, reflect, think, and discuss errors and successes amongst the student problems. This self-analysis strategy allows students to understand that errors are okay, but by working through those mistakes they could get a more accurate answer.

            If I were Ms. Lewis, I probably wouldn’t have had the students sit on the carpet as long as they did. She could have had them look through the samples of the first problem, go to their desks to review it with a partner, come back on the rug to discuss samples of the second problem, and then go to their desks to review it with a partner. This way, the students would probably be more focused and engaged because the instruction is alternating between teacher-centered and student-centered (as opposed to all teacher centered and then student centered). Despite this suggestion, I thought it was great that the students were able to use a pen, work with a partner to discuss their work (dyad), and then make changes or additions to their work right on the paper or on a new sheet of paper. It was good that the students were able to reflect upon, share, and verbally explain their strategies to a partner, and see if their strategy was appropriate. I noticed with one student that when he verbally explained the problem to his partner, he realized a mistake he made and was able to fix it on his paper. Students worked through problems 1 and 2 as well as 3 and 4. Ms. Lewis and some of the other observers walked around and interacted with the students one-on-one to aid them in discussing the strategy they were using and why. I think that ALL students were able to recognize the importance of going back and revising their work to make it clearer. However, I think some students still struggled with the idea of picking the correct strategy and/or explaining why they chose that strategy.

           Faculty Debriefing:

            During the debriefing of the lesson, I liked how Ms. Lewis was able to easily identify her challenges as well as successes. First, she discussed how the lesson did not go as planned, she ran out of time, and she wanted to dissect more student samples in order for her students to see different ways to answer the question. Next, Ms. Lewis discussed some strengths of the lesson. She was pleased with the students’ level of participation and contribution to share their thinking. It was nice to see that they were comfortable talking about their own work, even if it wasn’t perfect. Also, she appreciated how students were able to recognize and express if they were lost or stuck.

            In the future, Ms. Lewis wants to continue enforcing her students to go beyond the pictures, words, and numbers and actually start looking at how those things work together. She wants them to take apart the problem, figure out what the question is asking, identify the strategy, and be able to explain why they did what they did. I think her students understand what she is asking: they need to identify a strategy, show that strategy using words, pictures, and numbers, and be able to explain what they did. However, I think they got stuck actually employing those strategies; they were either stuck on identifying a strategy and/or explaining their strategy (and I think Ms. Lewis realized this, too).

            As previously mentioned, Ms. Lewis also wished she could show more student samples. For instance, she wanted to show one that displayed a number line and one that used base ten blocks. For future instruction, maybe Ms. Lewis could show some more of these student samples for students to evaluate and discuss, and she could further stress that there are various strategies students can use to solve a problem (as long as the student has evidence-pictures, words, and numbers-which are aligned with each other, many strategies could be used).

           Overall Thoughts:

            I think it was a meaningful learning experience to have students go back and analyze their work. Ms. Lewis promoted a respectful learning environment (it was okay for students to make mistakes and she wanted to help guide them through how to correct their mistakes). I also liked that the poster samples were anonymous so no student could judge another student for their work.

            I found it interesting that even after Ms. Lewis’s reengagement lesson, the students still had difficulties breaking apart the word problems.  After reading and discussing the problems as a whole-class, many students still thought you just add the numbers together that you see. I also was captivated at the one-on-one work with Ms. Lewis and one of her female students when working through the pen problem. Even though Ms. Lewis tried explaining through verbal and physical explanations, this student still struggled identifying that she needed to subtract rather than add. I noticed this with some of the other students as well. This showed me that maybe Ms. Lewis needed to modify her method of instruction (instead of asking the same questions (i.e. How much more?), maybe she needed to provide more sample problems the students could work through as a class and then have them dissect their work). I felt Ms. Lewis sometimes repeated the same questions and wanted her students to understand it so badly, but when they didn’t understand it, she often still repeated the question rather than altering her instruction. Also, this showed me that breaking down a problem may be harder for students than it appears, but it’s worthwhile for students to do because they are learning to work through the process. In other words, I feel so many teachers give students a problem but don’t have students analyze and explain the strategies they are using. Even though some of the students were struggling, it was amazing that they were able to self-reflect and determine what they could change/add to improve the accuracy of their work. I believe understanding why is a key part of education, and if students don’t understand why (i.e. Why did you add? Why did you subtract?), then they are missing out on a good chunk of their learning process.

Hopefully Ms. Lewis will continue to guide her students in breaking down word problems so they can grow and develop in their thinking and understand/explain why they did what they did. In addition, I hope to implement self-reflecting strategies in my future classroom and providing opportunities for students to be able to explain their strategies.

Journal Summary #1: The Story of Kyle & Fracking: Drilling into Math and Social Justice

Journal Summary #1

Article: The Story of Kyle

-NCTM, Teaching Children Mathematics, February 2015, Vol. 21, No.6 (p. 354-361)

Summary:

            Kyle is a kindergarten student from a low-income family. The article showed that Kyle was successfully able to perform nonverbal math calculation activities but struggled to successfully perform on story problems and number combinations. It’s important that children start linking their understanding of numbers to number symbols/representations and number operations. However, many children, especially many low-income children like Kyle, struggle making these connections. Students need to be making these connections, though, in order to be more mathematically proficient and more successful in the future. Therefore, a number sense intervention program (NSI) was developed. It’s an RTI for kindergarteners who may be at risk of math failure. The whole NSI approach is centered on whole-number concepts specifically the number, number relations, and number operations. NSI focuses on representations of numbers and explicit instruction to help improve children’s number sense specifically with story problems and number combinations (which was what Kyle struggled with). There are 24 lessons about 30 minutes long consisting of fast-paced activities in a game-like format. First, students are introduced to a few numbers at a time (i.e. lesson 1-0, 1, and 2, lesson 2- 3, 4, and 5) and they focus on just these numbers. A cardinality chart is used where one block is added for each number (plus one principle) and after that, a number list is used to show the plus-one principle in a different way. Next, students are introduced to part-part-whole through partner cards, and they practice the problem with a farm scene (story problem). Furthermore, children learn how to count effectively using their fingers (count on one hand and then add using the other). This intervention allowed Kyle to be successful and fluent in his math performance.

My Thoughts/How Useful it would be for a Classroom Teacher:

            I think the NSI intervention program is an excellent tool for teachers to use to help kindergarteners (or even other ages) be more proficient at story problems and number combinations. I believe these young students need to learn math through explicit instruction using manipulatives in order to become more fluent in math. The problem I foresee with this intervention is that it appears to be time consuming (there are a lot of steps). However, although it is time consuming, it may be worth it if the students are truly gaining an understanding of numbers. I found this intervention program really interesting. I believe this step-by-step model allows students to break down and process the numbers. I liked the fact that this process was like building blocks (slowly adding another, slightly more challenging concept); as I believe teaching should be, it seemed to go from more physical to more mental/fluent as well as simple to more complex (i.e. cardinality chart-bricks, then number list-circles, partner cards, farm story problem/fingers). In addition, I think this step-by-step model is helpful for lower-achieving students and/or students with learning disabilities because it is clear, slow, breaks down instruction, and provides visual and oral models. I think this intervention would be useful for teachers in a classroom because it allows students to connect with and understand numbers and link that information to symbolic representations of the numbers. They are not only seeing numbers and pictures, but they are learning to make sense of those numbers and pictures. If teachers use this intervention, I think many students would understand/be more proficient in math, less repetition of concepts would be required from the teacher, and more students would progress in their math skills.

Journal Summary #2

Article: Fracking: Drilling into Math and Social Justice

-NCTM, Mathematics Teaching in the Middle School, February 2015, Vol. 20, No.6 (p. 366-371)

Summary:

            The main goal of the teacher in this article was to help her students create a math model to better understand a community issue. She wanted them to see the value of math and how math can help them better understand societal issues. Through models, students can explore real-life situations and make sense of it. This teacher presented students with a relatable society issue question (Should the community ban fracking, or should individuals take advantage of the money?). After learning background information about fracking through videos and articles, a class list of questions related to fracking was brainstormed. Then, she had each student explore/research one aspect of the issue and create a mathematical model. Next, she had them make a presentation (including the student’s question and answer and describing the math they used).  Many students came across challenges when researching their question (information was not available or their question was too broad) or they struggled creating the mathematical model. On the presentation day, students questioned each other about their models and then completed a written reflection showing what they learned about fracking in math and how it can be applied to the real world. Rather than a step-by-step math problem with lots of teacher guidance, the students were engaged in a rich math experience where they were the ones to make math decisions. They were the investigators and had to use the information they researched to create a math model. Furthermore, they needed to decide what information to include and show what their numbers meant.

My Thoughts/How Useful it would be for a Classroom Teacher:

            My favorite aspect about this article was that it was student-centered. The teacher could have simply provided a worksheet about fracking and percentage problems, but this teacher recognized the importance of increasing student engagement and understanding. This teacher wanted students to use a math model to personally connect with the information to gain a deeper learning experience. After discussing general information about fracking, the remainder of this lesson was pretty much student-oriented. They chose the question to research, created a model, and showed what their model meant. While their models may not have been perfect, I appreciated that the students were the explorers. Plus, I think students find it more meaningful when they are truly physically and mentally involved in their learning process (as opposed to being lectured at or handed a worksheet). I think exploring a social issue through a math model would be useful for teachers in a classroom because it allows students to explore math concepts, understand what their numbers mean, and relate to the world around them. I believe that the more teachers provide opportunities for their students to explore social issues through math, the more they will engage, connect with, and understand their findings at a deeper level.

Main Ideas for Standards (Model with mathematics & Look for and express regularity in repeated reasoning)

 Standards

Read the material in the CCSSM Standards for Mathematical Practice and the PLC

articles on Sakai about your standards. Be sure to pick out the 2 to 5 important points about

the standards - and put it in your own words. Make a blog entry regarding the standards

that you have read about so that you and your partner(s) can see what each of you believe

is important.
 
4. Model with Mathematics

-Students can use prior math to help them solve problems in real-life situations
-Students can identify the important variables in a situation and draw out their relationship (i.e. diagrams, 2-way tables, graphs, formulas, flow-charts, etc. /or physical or pictorial manipulatives and symbols), and they are able to interpret, reflect upon, and understand those relationships.
-The goal is for students to become more skilled in mathematics
-Teachers can question students after they found a solution in order to have students determine if their solutions make sense within the context.

8. Look for and express regularity in repeated reasoning

-When computing calculations, students are able to identify repetition and generalize methods or shortcuts for procedures
-Students are able to evaluate/check their results to see if they make sense/are reasonable
-Teachers often simply instruction/provide shortcuts too early. It's important that teachers provide examples for students where they are able to identify regularity and answer questions where they can express the process they used (define their reasoning/methods).



PLC Articles 
 
Next, each member of your group is to read a recent article related to the Standard of

Mathematical Practice assigned to your group. Each person must choose a different article.

You may find these articles in the earlier NCTM journals. Use your email to let your

partners know what you are reading – early so they do not read the same material, each

person must read a different article. Use a blog entry to explain the material and examples

from your article. Now you make blog entry about the journal article - as a new entry for

the article on your blog.

1. Model with Mathematics 

Article: Spotlight on Math: Strategies for Addressing the Most Challenging Math Standards – Modeling

http://gettingsmart.com/2015/03/spotlight-on-math-strategies-for-addressing-the-most-challenging-math-standards-modeling/

Summary:

• Model is a representation of a real-world object or situation
• Not one correct model, but rather, there can be many models
• Ex. Numbers/symbols, geometric figures, pictures or physical objects, diagram (i.e. number line, table, graph)

• Teachers should ask/question students to explain their work and to relate it back to the original problem context. This way, students are constantly looking for connections/making sure their results are logical/make sense.
• Students need to use modeling to apply their learning on a regular basis. They need to look at real-world situations, identify a problem, collect data, devise a mathematic expression, check data, and revise.
• Modeling steps (may or may not be sequential):

• Identify a problem situation
• Make a representation of one or more elements of the situation
• Create a mathematical expression
• Compare results or predictions from the mathematical model with the real situation
• Make revisions to the model if needed

Examples:

• Bar diagrams to represent contextual problems requiring multiplication or division
• Using rectangular block diagrams to represent/solve problems with content from addition of whole numbers to ratio and proportion
• Making a presentation about area/perimeter of a school to prove school is too crowded.
• Book model (ex. adding 9 books and 15 books)
• Weather forecast models to predict weather

2. Look for and express regularity in repeated reasoning

Article: Reason Why When You Invert and Multiply

http://www.nctm.org/Publications/Mathematics-Teaching-in-Middle-School/Blog/Reason-Why-When-You-Invert-and-Multiply/

Summary:

• Students need to reason in math. They need repeated practice to reach realizations/discover patterns.
• Teacher's should guide students in their understanding of how problems make sense.
• Students need time to explore what something means. They should know why we do that. (i.e. What does it really mean to divide by a fraction?)
• When teachers and students explore a problem, students will likely identify a pattern, which the teacher can then explain (i.e. reciprocal) 

Examples:

• Fractions
• They should know why we do that. (i.e. What does it really mean to divide by a fraction?)
• Students can recognize what a problem is really asking (i.e. What does 30 divided by 3/4 really mean? or What does it mean to divide two fractions?). Students will note patterns/new terms and actually understand the problem versus just applying shortcuts. In other words, shortcuts/generalizations are a great tool, but students should know what these shortcuts mean.