Assessment Articles

Getting Started with Open-Ended Assessment

This article is addressing some different ways of assessing math in the classroom. The articles focused on the difference between multiple choice and performance based assessment and how performance based assessment is better. It talks about using open-ended problems to assess math students' work. I like this idea, because with the many possible solutions to the problem, students may think outside the box to find a strategy to solve the problem. Open-ended problems allow teachers to see the students thinking process and allows them to help the students where they struggle.

Sbord of Assessment Options

The article is addressing problem solving. Problem solving is a huge part of the math curriculum. It also talks about the different ways math can be assessed. This is often referred to as the menu choice. This article mentions how even though multiple choice may be an okay assessment, performance assessments with accurate rubrics are a much better tool to assess whether or not a student is able to apply the information in a real life situation.

Understanding Student to Open-Ended Tasks

This article is addressing how open-ended questions can work in a math classroom. I really like this because I believe that it allows students to really vocalize their thought process for the math concept. This also allows the teacher to feel like they are better able to understand and student’s understanding.  The teacher in the article had students explain what area was in their own words. This allowed the teacher to see what each student was thinking and how they were able to think or reason throughout the problem based on what the student wrote.

Assessing Students' Understanding through Conversation

This article is addressing the importance of conversation in the math classroom. The article is addressing how based on conversations in the classroom the teacher might need to change the plans for the day. For example, this teacher was able to realize that her students were struggling with some of the properties of a rectangle, so instead of moving on she created a small mini-lesson on the spot to help improve her students’ understandings of the topic. I really like the idea of assessing the conversations that take place in a classroom I believe that it allows for the teacher to reflect on her teaching methods based on how well the students are speaking about the topic they just learned. I believe that this is allows for the teacher to create stronger lessons and for the students to speak up about their understandings. 

An Experiment in Using Portfolios in the Middle School Classroom

This article was talking about a teacher who decided to implement portfolios as a way of assessment in her math classroom. Portfolios are an important tool that allow teachers as well as students and parents to see the mathematical growth of the child. I love this idea of having a folder where the student can reflect back on their work. I also liked how this teacher used a certain criteria to assess the portfolio so that this was a project that encouraged the students in mathematics, instead of discouraging them. She allowed the students to pick the materials they felt should be in the portfolio and then they had to so some sort of reflection with them.

Assessing for Learning

This article is all about research-based strategies that uses the different levels of mathematical cognition to design assessment items. To develop meaningful assessments, it is helpful to organize expectations of students into cognitive types, or different learning levels. These learning levels include: constructing a concept, discovering a relationship, simple knowledge, comprehension and communication, algorithmic skill, application, and creative thinking. These learning levels describe the different kinds of thinking that are typically required in learning mathematics. they are placed in order according to a learning progression that is meant to help teachers plan instruction.  There are different assessment strategies and designs for each learning level. When students are constructing a concept, give students prompts that require sorting or categorizing. By sorting examples into categories, students display how they have built a concept in their minds. Other approaches for testing the construct a concept learning level include asking students to describe concept attributes and create examples or nonexamples of their own. When students are at the discovery learning level, teachers can require students to report narratives of the experience of discovery. Many times at this level students have used inductive reasoning to make mathematical discoveries and they may have had a personal and unique experience. Students are more likely to understand what they have done after writing it all out. For simple knowledge level’s assessment, think stimulus and response and ask students to respond based on what they remember. Assessment at the comprehension and communication level requires a prompt that induces students to formulate explanations involving literal and/or interpretive understanding of a technical mathematical expression or message. This helps to assess students’ understanding of mathematics and fluency with their use of the relevant mathematical language. To demonstrate algorithmic skill learning, students should be able to recall a sequence of steps and execute the specified mathematical procedure. Teachers need to be sure to emphasize the PROCESS and not the outcome. Students show achievement of application level learning if they demonstrate proficiency deciding how to solve problems. They show that they can decide if a particular situation is an example of a broader mathematical principle or not, exhibiting deductive reasoning. Assessment items at this level should be created to avoid clue words that tell students what they need to do. Assessment design at the application level can also be done using shorter-format problems. To design items that test your students at the creative thinking level, we recommend using synectics, the juxtaposition of seemingly unrelated ideas. Open-ended application prompts is an example for an assessment that can be used to assess creativity. I believe that by adapting and using these strategies in your assessment and test design, students will actually learn from taking the test.  

Kohler, B., & Alibegovi'c, E. (2015). Assessing for Learning. Mathematics Teaching in the Middle School, 20(7), 424-433. Retrieved June 28, 2015, from http://www.nctm.org/Publications/mathematics-teaching-in-middle school/2015/Vol20/Issue7/Assessing-for-Learning/ 

Moving Beyond Brownies and Pizza

Fractions are a hard concepts for students to understand. Students struggle with the idea that fractions are numbers in and of themselves, not a composition of two, distinct, whole numbers. It is also likely that students fail to recognize fractions as discrete numbers because many math classes focus on understanding fractions as parts of wholes or parts of sets. The Common Core State Standards for Mathematics expect students to understand arithmetic operations on fractions by the time they leave fourth grade. Students may not be well prepared to meet this expectation if they think about fractions exclusively in terms of brownies, pizzas, or other area models. Students need to have opportunities to develop and explore their understanding of fractions as measures, that is, understanding both the relative size of fractions and understanding how fractions measure specific intervals. The number line model is a logical context within which these explorations can take place. There are three goals when it comes to student comprehension of fractions. these include: building students’ understanding of fractions as numbers with a definite magnitude, increasing students’ understanding of measuring with fractions, and developing fraction number sense by avoiding early introduction of traditional fraction algorithms. Students are already familiar with the number line from earlier grades, so they understands it as a concept, a representation, and a tool. This is why teachers should use the number line as a way to help teach students fractions. Looking at a fourth grade classroom, for a period of five weeks, students completed many different groups of fraction problems. Students worked in pairs to compare the fractional quantities in each problem set. At first, many students used part-whole concepts and diagrams to aid them in their work. However, a number of students noticed the linear context of each problem and began shifting to linear representations. The teacher encouraged these efforts and asked students to share their work during our whole-group discussion. The teacher continued to use the number line exclusively during whole-group discussions. She displayed student-drawn number lines on the walls and used them as reference points during lessons. Within a few weeks, nearly all the students began to demonstrate an understanding of fractions as actual quantities on the number line. All students in this class made progress in relation to the three goals the teacher had set for them. Students began to build an understanding of fractions as numbers with a definite magnitude. They were beginning to understand how to use the number line as a tool for representing fractions. In particular, students were using ideas connected to the measure subconstruct as a way to reason about the size of fractions in comparison to other fractions and to one whole. Finally, students were developing fraction number sense as they determined the relative size of fractions without resorting to traditional methods of comparison. I believe that using the number line is a great approach and allows students to better visualize the number. I will definitely be using a number line in my classroom!

Freeman, D., & Jorgensen, T. (2015). Moving Beyond Brownies and Pizza. Teaching Children Mathematics, 21(7), 412-420. Retrieved June 28, 2015, from http://www.nctm.org/Publications/teaching-children-mathematics/2015/Vol21/Issue7/Moving-beyond-Brownies-and-Pizza/ 

Video 2- 4th grade division

This lesson is taught in a fourth grade classroom and is directed for students who were struggling with story problems requiring division. Students tended to be challenged by interpreting the math language in the word problems. The step of drawing a “math picture” or model of the problem, poses a challenge for many students who have limited exposure to models. I think presenting a model to students to help guide them in solving the word problems was a good idea. It is an engaging challenge for students of all levels to attempt to make sense of someone else’s model or strategy. There were several teachers and administers that offered help and opinions to Mrs. Sherman. It is sometimes hard to know what to do or what to change when students are having trouble understanding a concept, so Mrs. Sherman did the right thing having a small intervention! The main purpose of the lesson was to introduce the “Singapore Bar Model” to students who had never seen it before to try and help improve students problem solving. They used the previous math knowledge of solving “division” story problems to introduce the model. They taught a similar lesson in a neighboring classroom with the same content, but a “direct teaching” instructional style. The intent was to compare student responses from the two lessons. They instructed the multiplication/division similarly in both lessons. I thought this was a really interesting concept and think it is a great idea. All children learn in different ways, so looking at which teaching strategy was more effective was a great idea. The goal in this process was formative assessment of students’ understanding of these concepts and opportunity to engage in the main idea of equal parts. For this lesson only, we designed the exploration of one or more story problems, but they were only able to get to one of the problems due to time. I believe that this tends to happen often, so teachers should really be aware of time constraints. 

NAEP Reflection

I learned a lot from completing the NAEP project. It is a lot harder than it looks to grade students work. A lot of things need to be considered while evaluating. We looked at the radio station problem. A lot of the work samples we had to work with were incorrect, so it was hard to choose which samples we wanted to use in our table. The easiest category to select was off-task or incorrect. It was easy to spot when a student had an incorrect answer. It was very difficult to differentiate between minimal and partial. We went back and forth on whether a sample was minimal or partial. On the rubric, there were only slight differences that made a work sample either partial or minimal. I would say that was the most difficult part of the task. It was fairly easy to spot the satisfactory and extended responses because not much was incorrect. It was difficult coming up with what we should have them work on. Even if a student got an extended response, there is always something they can improve on. I enjoyed analyzing different examples of student work and it really opened my eyes on how difficult it can be. The second part of the NAEP project was also beneficial. I liked completing this because it will get me prepared to do edTPA. We had to write out an objective of what students should be able to accomplish and then identify which students were not able to do it and what they need to work on. I believe this is extremely important because teachers need to be able to identify what they need to work on with students. They can not just keep chugging along if students are struggling. This project was very beneficial and I learned a lot!!

Rich Activity Reflection

I learned a lot from doing the rich activity lesson. You have to consider many different things while picking a rich activity. A lot of planning is involved in order for the lesson to be successful. A rich activity needs to include higher level cognitive demand, meaning you need to really get students minds moving. We actually had a little trouble picking a rich activity we wanted to present to the class. We finally decided on a great problem we could do, but we weren't too sure if it could be solved with more than one strategy. After really reviewing the problem, we found out we could solve the problem on paper using guess and check and also using manipulatives. Our group then came up with questions we could ask students throughout the lesson. It was hard to come up with questions that would help students but would not give them the right answer. We want to guide students in the right direction, but it is important to not just hand them the right answer. I believe that our presentation went well. We didn't have many people in the class, so we could not use the scribe. The students did work very hard and one group eventually came up with the right answer and could explain the steps they took to solve the problem. We do need to work on asking questions throughout the presentation and not just at the end. Other than that, I believe we did a great job and this project was very beneficial.

Math Applets and App Review!

IXL Learning-Applet

IXL Learning applet is an awesome app for students of all ages. It is separated into grade level categories, so teachers would just need to select their particular grade level. Once a grade level is selected, it shows all of the math standards for that grade. This makes life a lot easier for teachers, because they will know exactly what material needs to be covered. After you pick a standard and practice area, it will direct you to practice problems. They give a variety of problems for students to work on. I believe this is a great site for students to practice their math skills. 

NCTM- Illuminations Applet Page

This is a great compilation of applets for many different math skills. All of the applets are fun and interactive for students. It would be a great place for students to go to practice their math skills. I would recommend this applet page to other teachers! An example of an applet on this page is Algebra tiles. Students have to solve the equation, substitute in variable expressions, and expand and factor. 

Middle School Algebra- App

This is a great app for middle schoolers to practice their algebra skills. This is fun and different way to look at algebra. This app includes instructional teaching, quizzes, and games on many different math topics. Some of the topics include:

• Words to equations
• Evaluating expressions
• Solving 1 to 2 step equations
• Linear Equations
• Exponential functions
• Rational functions
• Systems of equations

After reading the reviews for the Middle School Algebra, I did not find one review that was negative. The app is free, so many students would be willing to purchase the app on their iPhones or iPads. Many reviews say the the app is very helpful and gave them a better understanding of the math concepts they were learning in school. I would encourage my students to purchase this app for extra practice!

June 18th Articles

A Model for Understanding Understanding in Mathematics

Understanding is a hard concept for many teachers and students to grasp. Both Holt and van Engen view understanding as a continuum. Students can have partial understandings. They may be able to answer a question correctly, but they aren't quite sure why they took the steps they did or how to explain their solution. In order for teachers to promote a better understanding for students, they can make "moves" in teaching different kinds of mathematics. Moves will help teachers to identify objectives that students will need to accomplish in order to have a complete understanding of the material. In addition to planning to make moves, teachers can use moves to evaluate and assess the level of understanding of students for a given mathematical knowledge. There are for different types of mathematical knowledges: concepts, generalizations, procedures, and numerical facts. For each of these categories, there are different moves that teachers can make. There are also 2 different levels of understanding. In order to understand a concept, students must be able to give examples and non examples of the concept and also understand different characteristics of the concept. Statements of relationships between concepts are called generalizations. In order to fully understand generalizations, students must know what the generalization is saying and give proof of why it is true.  The next type of mathematical knowledge is procedures. To prove understanding, student must make moves such as: accurately carrying out the procedure, showing another student how to do the procedure, or paraphrasing the procedure step by step. They also need to be able to explain why the procedure works. Many numerical facts are generalizations, so many of the same moves for generalizations apply to number facts. In order to fully understand numerical facts, students must be able to recall the fact and understand what it is saying and also understand why the fact is true and realize its significance. It is important for teacher to implement and rely on these moves to promote students understanding of different mathematical knowledge. Moves in teaching mathematics can serve as a basis for defining understanding of mathematics. I know it is difficult to see if students truly understand a problem, or if they just wrote down the answer. Students need to be able to understand the steps they took to solve the problem and why those steps helped them to come up with a correct answer. 

Davis, E. (2006) A Model of Understanding Understanding in Mathematics. Mathematics Teaching in Middle School.  12 (4). 190-197. Retrieved June 15, 2015.

Thinking Through a Lesson: Successfully Implementing High-Level Tasks

Tasks that give students the opportunity to think critically and use reasoning skills are the most difficult for teachers to implement into the classroom. This is because procedures for solving high-level problems are not usually specified in advance. There are different ways to solve high-level tasks so it is hard for a teacher to present this to the class without confusion. It is important for teachers plan in detail prior to the lesson. Teachers can use TTLP, or Thinking Through a Lesson Protocol to help further the use of cognitively challenging tasks. The TTLP helps to develop lessons that make students use their mathematical thinking as the major ingredient for understanding a concept. There are three different sections in the TTLP, which include: (1) selecting and setting up the mathematical task, (2) supporting students explanation of the task, and (3) sharing and discussing the task. Part 1 is where the teacher should lay the groundwork and identify goals for the lesson and also set expectations for how students will work. Part 2 focuses on monitoring the students during the task. Teachers need to me sure to ask students throughout the lesson to make sure they understand the concept and are on their way to complete the goal for the lesson. Part 3 focuses on a class discussion of the task. Students will need to share their results with the class and explain how they got their answer and justify why they believe it is correct. I believe that the TTLP is a great tool to use in planning for the classroom. Although it can look very overwhelming, I think that it will help in the long run. I always struggle with coming up with questions to ask students throughout a lesson, so brainstorming and having questions ready is something I plan on doing. It also helps not only the students think deeper about a concept, but also the teacher. This is vital because the teacher needs to have a solid understanding of the concept before conveying it to students. I believe the TTLP will be very beneficial and will help promote students problem solving and critical thinking skills while working on high-level tasks. 

Smith, M., Bill, V., Hughes, E. Thinking Through a Lesson: Successfully Implementing High-Level Tasks. Rich and Engaging Mathematical Tasks: Grades 5-9. 11-18. Retrieved June 15, 2015. 

   

Three Strategies for Opening Curriculum Spaces

Teachers need to open curriculum spaces in order to develop students multiple mathematical knowledge base, or MMKB. Children’s MMKB includes mathematical thinking and children’s home and community-based mathematical funds of knowledge (Carpenter et al. 1999; Gonzalez, Moll, and Amanti 2005; Turner et al. 2012). Students make sense of problems and develop multiple solution strategies by connecting problems to their own experiences both in and out of school and by using and building upon their MMKB. It doesn't matter where you teach or what grade level, all teachers need to make changes to the curriculum to open spaces for MMKB. There are 3 strategies to guide teachers on how open curriculum spaces. The first strategy is rearranging lesson components. Most curriculum lessons have several components, including: opening routines or messages, many student tasks, differentiation suggestions, and then finally homework. Teachers can open spaces for children’s MMKB by moving these components around or omitting some of them and focusing on others. The goal of this strategy is to find those components that focus on (1) having students make connections between the task and their prior knowledge and experiences, (2) providing support for students to develop their own strategies, and (3) encouraging students to share and explain their strategies. There are two different ways that teachers can incorporate this strategy into the classroom. The first is foretold problem solving. This is when teachers engage students in complex problem solving in the beginning of the lesson before introducing the preferred solution strategy. The second way is cutting components. Teachers should omit sections that either tell, direct, or show students how to make sense and solve problems. The second strategy to open curriculum spaces is adapting tasks. Teachers can adjust numbers or offer choices. This can be done by providing multiple number choices for a single problem and allowing students to work on the numbers that are “just right” for them. This will open access for struggling learners and fast finishers while maintaining the cognitive demand and mathematical goals of the lesson. Teachers should also encourage students to use multiple representations and strategies. This will help increase their capacity to solve problems, make them practice justifying their solution, and then comparing and contrasting their solutions. The third and final strategy is making authentic connections. Students will be more engaged and willing to work when a problem is realistic. It is extremely important that teachers remember to incorporate all three of these strategies into their classroom because students need to be exposed to their MMKB!

Drake, C., Land, T., Gau Bartell, T., Aguirre, J., Foote, M., Roth McDuffie, A., & Turner, E. (2015). Three Strategies for Opening Curriculum Spaces. Teaching Children Mathematics, 21(6), 346-353. Retrieved June 15, 2015.

 

 

 

 

A Blizzard of a Value

Have you ever wondered which blizzard size is the best value? Ms. Bosetti and her class were determined to answer this question. This question is all about modeling with mathematics. The “model with mathematics” text within the Common Core’s Standards for Mathematical Practice states: “Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace” (CCSSI 2010, p. 7). This focuses on engaging students with problem solving that is focused on realistic situations. Students LOVE ice cream, so who wouldn't be excited to complete this problem? Modeling with mathematics brings up two key instructional questions: What should students engaged in modeling with mathematics be doing? and What should teachers do to promote modeling with mathematics? Ms. Bosetti decided to do a practice run first with her colleagues. They completed an airplane problem and decided that modeling with mathematics had three parts: presenting students with a realistic problem, talking with and about math representations, and re-examining math models after receiving feedback. The class was then ready to start the Dairy Queen Dilemma. Ms. Bosetti first presented the problem to the students. Students worked in groups of 3 or 4 and were just given the problem, the 4 different size cups, and a measuring tape. They worked with there group and had to come up with a ratio explaining which size blizzard was the best value. After given plenty of time to work, each group presented their work to the class. Multiple strategies were used to solve the problem such as: tables, pictures, and graphs. Students had to explain their reasoning for the solution they came up with and the class was allowed to ask them questions. Ms. Bosetti then let them go back to their groups and re-examine their solution. I believe that modeling with mathematics is a great strategy to incorporate into the classroom. It will keep students engaged because the will be working on realistic problems. It also teaches students that there was multiple strategies to solve problems and there may actually be more than one possible answer to a problem. It teaches students how to use appropriate tools to solve problems and makes them think abstractly. I enjoyed this article and will use modeling in my future classroom!


Bostic, J. (15). A Blizzard of a Value. Mathematics Teaching in the Middle School, 20(6), 350-357. Retrieved June 15, 2015.

Video: 2nd Grade Math- Word Problem Cues

2nd Grade Math- Word Problem Cues

Ms. Lewis was working with her students on word problems. Many of her students were struggling with how to read and process these problems and how to choose which operations to get the correct answer. Students' tended to see 2 numbers and as a first instinct just dd them together. She met with some of her colleagues to explain her class's situation and what she wanted to gain from their observations. She mainly wanted to know what her students were thinking during the problem solving process and what they mutter under their breaths. It is very hard for her to help all of her students at once, so having extra people in the classroom observing her students will help her gain insight into what she needs to work on. For this lesson, she went over the first 2 word problems as a class. She chose samples of students work and went over how they solved each problem and went into detail about each step they took. This allowed them to really think about the problem and decide if the operation they chose was correct. She discussed what students should be looking for while reading the problem, such as key words and vocabulary. For example, if students read "how many more" they should automatically know that they will be subtracting the numbers. I believe that the class discussion on the carpet was very useful. Ms. Lewis made sure to have all students participate and took the problem step by step to make sure ALL students were on the same page. She did frequent checks on where students stood with understanding the problem. For example, they would either have to put their thumb up or down or shake their hand if they were still a little bit confused. Once Ms. Lewis was done reviewing the problem with her students, she passed back their work. I loved how she photocopied their papers so they could not erase their original work. Many students will want to erase their work right when they get it back. I think it is important for students to see what they did wrong and then look at it and see what changes need to be made. Giving them a colored pen was a great idea to show that the correct answers or changes were in the different color. After that, students were told to complete numbers 3 and 4, which were the 2 word problems they had not gone over yet. Some students could look at their work and automatically see if they had made mistakes and then know what to do to change them. Other students were still somewhat confused and Ms. Lewis took time to further explain the concepts. I was intersted to see how Ms. Lewis thought her lesson went. Like all lessons, she believed there were some positives and negatives. She didn't get to show her students all of the sample student work that she wanted to. She strategically picked out each sample to show her students multiple ways to solve the word problem and she was upset she didn't have time to show each of them. She did express that she was so happy that her students participated so much and were engaged throughout the entire lesson. She thought that it went well overall and her students ended up understanding the word problems better. This video was very beneficial. It was interesting to see the planning that needs to go into each lessons. I also liked how she collaborated with other colleagues to get other opinions and ideas. It is sometimes hard to know what to do or what to change when students are having trouble understanding a concept, so Ms. Lewis did the right thing having a small intervention!

CCSM Reflection

I learned a lot while researching my 2 mathematical standards. My group was given standards 1 and 2. After reading the CCSM article and the 3 PLC articles for each grade levels, I learned everything I need to know about standards 1 and 2. I mastered the main points of each standard and how they can be incorporated into each grade level. I also found a great article that correlated with standard 1. It gave 4 different strategies that teachers can use in their classroom to help enrich the problem solving process. The article I found for standard 2 was a sample lesson plan and I went through and picked out where I saw standard 2 and how it was incorporated into the lesson. It was very interesting to see how you could incorporate each standard in so many different ways.
Coming together with my group and viewing each Prezi and listening to the Jing's was also beneficial. It is very important to learn about each standard and how it fits into a math classroom. Each Prezi provided the main points for each standard and also gave several examples. After listening to each Jing, I can say that I fully understand the math content standards. I am confident that I will be able to keep these standards in mind and incorporate them into my math instruction. I also enjoyed learning about the process standards. It was interesting to see how all of the process standards correlated with the content standards. It was fun to think of different ways to connect them and come up with a poster design; it definitely got our creative minds going!!

June 10th Articles

Rich Activities

Teachers should be sure to plan rich activities into their lesson plans. We must consider our students grade, age, prior knowledge, and experiences when considering rich activities. The expectations of each class will very, so planning is very important. Another thing that must be considered when planning rich activities is the cognitive demand. This MUST be higher level. The four cognitive demands include: memorization, procedures without connections to concepts or meaning, procedures with concept or meaning, and doing math. Teachers need to be sure to not have students just memorize the material. When we ask students to just memorize, we are not giving them a chance to actually understand why the procedure has one step before another or how we get from one fact to another. Instead, teachers should encourage students to think conceptually. By doing so, we are assisting students in developing different thinking processes. 

It is crucial for teachers to plan ahead when considering specific lessons in order to select a rich activity. There are three main parts, which include: selecting and setting up the task, supporting students exploration of the task, and sharing and discussing the activity. There are five questions teachers should ask when selecting and setting up the task:

1. In what ways does it build on prior knowledge, previous life experiences, and culture?
2.  What are all the ways the problem can be solved?
3.  What particular challenges might the activity present to struggling students, ELL students?
4.  What are your expectations for students as they work on and complete the problem?
5. How will you introduce students to the activity so as to provide access to all students while maintaining the cognitive demands of the task?

It is important for teachers to support students exploration of the task. We need to have questions ready to help them get starts and to stay focused and engaged throughout the activity.  The final part is sharing and discussing the task. This final step is so important because we need to make sure that all students are on the same page. The goal is to make sure all students comprehend the material, so sharing and discussing our ideas and work will benefit the entire class. 

From Smith, M. S. and Stein, M. K. (2012). Selecting and creating mathematical tasks: From research to practice. In Lappan, G., Smith, M. S. & Jones, E. (Eds.). (2012). Rich and engaging mathematical tasks Grades 5-9. Reston VA: The National Council of Mathematics Teachers, Inc.

Smith, M. S., Bill, V. & Hughes, E. K. (2012) Thinking through a lesson: Successfully implementing high-level tasks. In Lappan, G., Smith, M. S. & Jones, E. (Eds.). (2012). Rich and engaging mathematical tasks Grades 5-9. Reston VA: The National Council of Mathematics Teachers, Inc.

Groupworthy and Idea About Math

There are eight different math habits of mind. The habits include: exploring ideas, orient/organizing, thinking in reverse or being able to work backwards, generalizing, representing, justifying, math language, and checking for reasonableness. Our goal as teachers is to have our students acquire each of these math habits of mind and also making our students mathematically competent. We need to push our students to learn math more deeply. Teachers need to make math fun and engaging for students so they are excited to learn. It is important to have students interact with one another and work in groups during math instruction. In order to have an equitable classroom, we must establish norms for group interaction; for example, being respectful, taking turns, and listening to others' examples. 

Teachers also need to be sure that all students know there are multiple ways to contribute to math problem solving. This includes: posing interesting question, making astute connection, representing ideas clearly, developing logical explanations,working systematically, extending ideas

Like I stated earlier, it is important for students to engage in groupworthy tasks. When preparing for a lesson, teachers need to:

1. Focus on central mathematical concepts or ideas 
   
2. Involve some interpretation 
   
3. Provide multiple ways of being competent in problem solving
   
4. Perform in group bolsters students’ interdependence 
   
5. Require individual and group accountability 
   
6. Establish clear evaluation criteria
   
7. Present tasks laid out clear, simple directions, and has directions, diagrams, probing questions, and evaluation criteria

Although students are working with their groups, the teacher still plays a vital role. Before the lesson, teachers must think it through, consider challenges that may arise, and compile the various resources students may need. During the lesson, the teacher will need to walk around and check in on groups and ask spontaneous questions throughout. They must also make sure all groups remain on task. After the lesson, the class will need to come back together as a whole and discuss the task. It is also important for teachers to hold students accountable for their work. This can be done through group interaction or through formal assessments. 

From Horn, I. S. (2012). Strength in numbers: Collaborative learning in secondary mathematics. Reston VA: The National Council of Mathematics Teachers, Inc. 

Content Standards 1 & 2

1. Make sense of problems and persevere in solving them.

Proficient math students will look at a math problem and start by explaining the meaning to themselves and figuring to ways they can solve the problem. They will make conjectures and make a plan to solve the problem. They will consider different problems they may have seen and try to gain insight into finding a solution. Students will monitor and evaluate throughout their solving process and make changes if needed. Students will also check their answer to their problem and use various other mathimatical methods to make sure their solution is correct. Students will continually ask themselves "Does this make sense" to make sure they fully understand the problem and solution.

a. K-2 (PLC)

The teacher plays a major role in developing younger students problem solving skills. The task provided and the guidance a teacher provides will help students build confidence and persevere in solving problems. Teachers need to be sure to provide good problems to students. There are six questions teachers need to ask themselves while planning lessons and picking problems. These include: Is the problem interesting to students, does the problem include meaningful mathematics, does the problem provide an opportunity for students to apply and extend mathematics, is the problem challenging for students, does the problem support the use of multiple strategies, will students' interactions with the problem reveal information about the students' mathematical reasoning.  It is also important to facilitate student engagement in the problem solving process. Students' may not always answer a question correctly, but if they genuinely make an effort to solve a problem and can explain why they took the steps that they did, they are still learning. This will help them to persevere to then try and solve the problem correctly. Teachers will need to help younger learners to unpack problems, check reasonableness of solutions, and to make connections. To unpack a problem, students will need to learn how to dissect the problem to help them to better understand what they are trying to solve. Teachers will need to incorporate strategies related to reading comprehension to help students' relate to known material. These strategies include: identifying relevant details, noting relationships, predicting, making inferences, and distinguishing between mathematical terms and general vocabulary. A problem solving organizer will also benefit young students to help them organize their thoughts on how they want to tackle the problem. Many students will struggle with the problem solving process, but that is normal and will help them build perseverance. 

Understanding Questions for Standard 1: 

1. The goal is for students to become successful problem solvers of word problems and operations.

2. To facilitate this standard, teachers will need to select appropriate problems and guide students in the problem-solving process. 

3. To know that students are successfully demonstrating the standard, they should be actively pursuing solutions to a variety of problems and be able to explain the outcomes of their problem-solving process. 

b. 3-5 (PLC)

Many of the same ideas and strategies are the same for the 3-5 grade levels. The only difference I saw in this article was that they gave more sample problems. This helps teachers to visualize how they should be approaching each math problem in their classrooms. The understanding questions are the same for grades 3-5 as grades K-2.

c. 6-8 (PLC)

When students are solving a problem, they are using a four step process. This process includes: (1) understanding the problem, (2) making a plan, (3) carrying out the plan,  and (4) looking back. Teachers at the middle school level play a key role in supporting students' ability to problem solve. The tasks, guidance, and classroom environment all contribute to the students progression in this process. To learn how to persevere, students must be given many opportunities to meet challenges, but be sure not to overwhelm them. They must present appropriate problems and rich mathematical tasks. Teachers must also examine students' interactions while solving the problem so that they will be able to help answer any questions that arise. Teachers need to remember that successful problem solving does not always mean that students' will conclude with the correct answer; but, they must make a genuine effort to engage in the problem solving process. This article provides a great example of how teachers should guide students during the problem solving process. 

d. Article: Classroom Strategies to Make Sense and Persevere

This article was written by three middle school teachers and a university professor. It talks about four different strategies that are effective when helping students during the problem solving process. These four strategies include: asking Does it make sense?, the process of elimination (POE), perseverance logs, and analyzing incorrect response. In order to use the first strategy, Does it make sense?, teachers must select student work that will require the class to really think hard about why the solution is unreasonable. The teacher will present the problem and the solution to the class and have students reflect on it individually. They should then find a partner and discuss the reasonableness of the solution. Come back after a few minutes and conduct a class discussion about why or why not a solution makes sense. The next strategy is process of elimination. Teachers should use this strategy to help students identify possible versus impossible solutions. Make sure to provide 4 or 5 choices as possible solution to the problem. Have students think about the problem and determine which of the choices could be correct. Teachers should encourage them to plan out a solution pathway to solve the problem. The next strategy is very important and is called perseverance logs. Teachers should discuss what perseverance means and share various examples of how students will become successful after persevering. Ask students to keep a Solving/Perseverance journal or log which will include problems and how they persevered to solve them. This will them recognize that struggling and perseverance are part of the process. The final strategy is analyzing incorrect responses. Students should practice highlighting common errors that lead them to incorrect answers. One example on how to do this is having all students solving the same problem and then collecting them. The teachers should then collect one incorrect solution and ask the class to discuss why it is incorrect and also offer ways to solve it correctly. By implementing the strategies into the classroom, students will become better problem solvers. Students will become more confident with their answers and learn to not give up so easily.

Wilburn, J., Wildmann, T., Morret, M., & Stipanovic, J. (2013). Classroom Strategies to Make Sense and Persevere. Mathematics Teaching in the Middle School, 20(3), 144-151. Retrieved June 9, 2015, from http://www.nctm.org/Publications/mathematics-teaching-in-middle-school/2014/Vol20/Issue3/Classroom-Strategies-to-Make-Sense-and-Persevere/

2. Reason abstractly and quantitatively

Proficient math students will make sense of quantities and their relationships in the problem. The possess two complementary abilities to use while solving quantitative problems. These include: the ability to decontextualize- to abstract a situation and represent it symbolically and manipulate the representing symbols, and the ability to contextualize- to pause as needed during the manipulation process in order to decode the symbols involved. Students that use quantitative reasoning have habits of creating a representation of the problem, considering the units involved, attending to the meaning of quantities and not just how to solve them, and knowing and using different properties of operations and objects. 

a. K-2 (PLC)

Reasoning aids students' mathematical understanding and ability to use concepts and procedures in meaningful ways. It also helps students reconstruct faded knowledge, which is knowledge that is forgotten but can be restored through reasoning with content. Students specifically need to learn how to reason with quantities and their meaning. Once again, teachers play a critical role in developing this standard. They must provide distinct opportunities for students to develop number sense. Students must have a strong number sense in order to develop a strong foundation for reasoning. Number sense can be viewed as an understanding of numbers that empowers students mathematically in at least  6 ways. These include: expressing interpretations about numbers, applying relationships between numbers, recognizing magnitudes of numbers, computation, making decisions involving numbers, and solving problems. It is also important for teachers to draw students' attention to numbers and their applications. It is helpful to consider numbers to be a source of exploration for learners. Numbers are found everywhere and students should be able to explore these numbers! Teachers need to regularly encourage students to demonstrate and deepen their understanding of numbers and operations by solving interesting, contextualized problems and discuss the strategies used throughout the process. Discussion is another key aspect to promote reasoning. Teacher-to-student communication will get students minds moving and make them think harder and out of the box about different problems. 

Understanding questions for standard 2:

1. The goal is for students to learn how to reason with and about mathematics.

2. Teachers should provide students space to think and reflect on mathematical content and support students in communicating and refining their thinking.

3. To know students are successfully demonstrating this standard, they should be able to share and justify their mathematical conceptions and adjust their thinking based on mathematical information gathered through discussions and responses to their questions.

b. 3-5 (PLC)

Many of the same ideas and strategies are the same for the 3-5 grade levels. The only difference I saw in this article was that they gave different sample explanations and problems. This helps teachers to visualize how they should be approaching each math problem in their classrooms at each particular grade level. The understanding questions are the same for grades 3-5 as grades K-2.

c. 6-8 (PLC)

Many of the same ideas and strategies are similar in the 6-8 grade levels. Teacher-to-student communication as well as student-to-student communication is so important in helping these older students develop reasoning. This can include questions that teacher ask that probe students' thinking beyond what they might normally believe. Communication should involve discussions emerging from students' hypotheses about a certain mathematical concept and how students believe it should be done. Talking with their peers will also help students' to provide justification and reasoning for their thinking. Inferences about students' ability to reason can be determined through analyzing students work, discussions with peers,  and performance on mathematical tasks. Students' must also be able to consider the units involved in a problem and discover the meaning of those units in context, not just be able to compute them. Teachers need to be sure to give students regular opportunities for students to practice this advanced form of reasoning.

d. Article: Building Squares and Discovering Patterns

This article is a sample lesson design that integrates several of the mathematical standards. I went through and read the article and was able to recognize which parts covered reasoning abstractly and quantitatively. Students were told to use many different ways to solve the problem. They were told to strategically use both geometric and numeric representations as tools for reasoning. This will help them to better understand why they are taking the steps they are to complete the problem. Students were given exploration time which I thought was very interesting and effective. Students were able to use manipulatives to explore solutions to the pattern. Students' also had to explain why a particular pattern was occurring which forced them to reason about the steps they took and if they were correct or not. This lesson gave students' many different opportunities and I believe it was a great way to teach them how to work through problems and reason abstractly and quantitatively.

Whitin, D., & Whitin, P. (2014). Building Squares and Discovering Patterns. Teaching Children Mathematics, 21(4), 210-219. Retrieved June 9, 2015, from http://www.nctm.org/Publications/teaching-children-mathematics/2014/Vol21/Issue4/Building-squares-and-discovering-patterns/