Reflection: Standards and Classroom Changes to Deepen Math Learning

As the lead teachers in our classroom it is important that we include the CCSS-M Standards of Mathematical Practice (SMP), NCTM Process Standards, have student engagement, active learning, and math discourse. Through ETE 339 we have learned about each of these and their importance.

CCSS-M Standards of Mathematical Practice (SMP)
I had to work with the CCSS-M standards once before and it was pretty brief compared to the work we have done in this course. I feel much more confident in understanding how each of the eight standards work and their importance. Some of the standards have even caused me to look further and research their meaning to make sure I understand them to the best of my ability. I think when we focus on them or have them known it will help our classrooms become a better learning environment. After working with these standards it became apparent to me that they can be worked with together and really help the students master the material at hand. For example, if we start our students on focusing on using appropriate vocabulary (attending to precision) at younger grades such as kindergarten the students will become more comfortable and will be less likely to confuse terms later.

NCTM Process Standards 
I believe that the process standards are similar to the standards above because these five also work together to help the students master the skills. We have communication, connection, and representation. If a classroom is set up correctly the teacher can incorporate all of these standards in lessons. I also feel more comfortable with these standards and have found myself thinking about them more when coming up with lessons. I want to make sure we are hitting all of them to make sure the students are getting the most out of lessons. When all the standards are hit, mastery of the material is bound to happen. Also, again starting at a young age students using these skills will be able to use them even more as they age and develop.

Student Engagement
Student engagement is the motivational strategies we will use as teachers when presenting information to the students. This involves lessons using technology such as Smartboards and apps. It also includes having lessons that require the students to use inquiry or have self motivation. Class discussions can also help motivate the students and using their ideas for the lessons will stimulate engagement. Any teacher can go up to the class, present the information, write on the board, assign the worksheet. A teacher who makes the lesson rich, and engaging, will have better results. Students will want to learn and pay attention because there is self motivation and excitement. Thankfully we covered strategies in this course on how to make sure we do this. This includes our technology lessons, manipulative work, and group work.

Active Learning
I believe this also goes with the above student engagement. This is where the students are doing hands-on work when using their new information. This can be seen in classroom discussion, experiments, use of manipulatives, projects, etc. Active learning is when the students are doing versus just listening to the lecture. There will be times when students need to just listen but more often than not we can make learning active, especially in math class! Students will appreciate the work more and want to pay attention and learn. Technology helps, we have apps and applets that can be used which are stimulating but also beneficial to the lesson.

Math Discourse and Syntax -
Math discourse is how the students talk and write while syntax would be the symbols and vocabulary the students use. I will be 100% honest and say these are two concepts I am still really learning about. I am reading the edTPA handbook to make sure I understand the two. From what I understand discourse is how we allow our students write and talk in terms of our mathematics lessons. This again can relate to the process standards and CCSS-M Standards of communication and attending to precision. We are asking our students to use precise accurate vocabulary to present their ideas and justifications. Syntax is the words and symbols we use in our classrooms. This includes the knowledge of graphs and tables for example or symbols such as less than or greater than (<, >). Even though I am still working with understanding these concepts I understand that by using them accordingly we provide our students with the chance of deepening their knowledge and appreciating math more.

Coming into this course I wasn't expecting to have a huge change of mind for math class but to be honest I really did. All of these concepts were ones I could have thought about before but never really did. We learn the importance of communication, the importance of the standards and process standards deepening how lessons, I learned about discourse and syntax. If we use these ideas/principles/standards we will provide lessons with such richness our students will be engaged and appreciate mathematics. I am happy to work with this information and it inspires me and motivates me to get back into my own classroom. 

Reflection: Technology

In all of our education classes we hear time and time again how important it is to incorporate technology. I could not agree more. As times are changing technology is making such an impact on our society. Using these tools enriches our assignments and motivates the students to be working with tools students find intriguing and even fun. Over the past six weeks we have used the SmartBoard, Blogs, Apps and Applets. I really appreciated the SmartBoard work because we don't cover it in any of our classes and there are just so many options on it. I love that we each had opportunity to find or show something on the board to the class because there are things that not all of us knew or though about. I loved the idea of using the dice but putting words on them and asking for the definitions. I love the SmartBoard and sadly think a lot of schools that could use them can't afford them and schools that do have them do not use them to their full potential. I think change is hard, so for teachers to adapt and make lessons using the board might be time consuming but I think it is worth it. Next we have our blogs, which provide a lot of opportunity for reflection. This tool has helped me self-assess and really evaluate work and articles. I think this would be a nice tool to have for older students (especially because I think blogging is trendy? maybe I don't really know but I think it's a thing).  The apps and applets we researched were awesome because we got to show each other lots of different ideas we have. I didn't think about games that other people in class did or some of the applets people chose that they made geared towards their domain. These were out of the box ideas I really liked.

I think it is important to have technology in our lessons and classrooms because this generation is very technology stimulating. At the rate technology is going, when used correctly, lessons can really be enriched. 

Reflection: Assessments in Math

Over the last six weeks we have covered many types of assessments with class discussion, reading articles and activities. We learned about traditional, student centered, open-ended, portfolios, conversation, self, peer, formative, and summative.

Traditional assessments is the type of assessment I only knew about before teaching. This type of assessment consists of tests and quizzes. It is an easy cut and paste assessment, the student is either right or wrong and showing all their work on the document shows their understanding. Now, tests and quizzes are not bad but there are many other types of assessment that provide teachers with a deeper understanding to their students and a richer experience for the students.

Student-centered assessment was a new type of assessment for me to really think about. I understood it to be inquiry based, where the students are working to answer their own questions on the material. The teacher is able to see where the students are at with their understanding. Student centered classrooms provides the students with opportunity to be engaged and motivated to work on their work. I still want to look into student-centered assessments more because there are parts that don't make 100% sense but I think this also comes with time and working with the material more.

Open-ended assessments are awesome in my opinion because they are authentic and provide the students with opportunities to communicate their understanding. Communication proves mastery of the concept, if a student can clearly justify and explain their work (that is done correctly) they are on the way to mastery...but I will mention that again in a few paragraphs..when talking about assessment using conversations! When we give students problems that can be answered a variety of ways we are letting them use problem solving as well. The students can see the multiple ways, or later learn about the other ways, and pick which they think is best or which they can do best. Starting the open-ended assessments at a young age helps students build that confidence needed. This really makes the lesson rich and interesting for the students.

Portfolios are also wonderful tools. I think they take a lot of work but are reasonable. Again, as we work with the material more we can build these great templates for portfolios to use year after year. I think it provides the teacher a wonderful group of work to see where the students started and ended, whether we are using it as a unit, a lesson, a quarter, etc. I also love the idea of having students reflect on their own work. Even at younger ages this will provide students with another wonderful opportunity to build their understanding and communication skills.

Conversation is obviously one that I really like but it also can be found in any part of the day, activity, other types of assessment. Teachers can have one-on-one conversations and hear what a student thinks is a good justification or just to identify what the student is thinking. Teachers can allow group conversation and through observation the teacher can hear where the students are at, how they are problem solving, and thinking through the project. The teacher can also have whole class discussions which will help her assess where the students are and what work needs to be taught better or more in depth. Again, conversation helps students work on being able to communicate what they are thinking and practice justifying their work.

Self-assessment is nice because it gives students the opportunity to reflect on their own work. Having done it myself in this class, it's nice to go back, take ideas learned in class, and fix what we want to. It is also a good skill to have in students because they always need to check their work before turning it in. Self-assessments provide opportunities for the students to motivate themselves and strive to give their best work.

Peer-assessment is also nice when we did it in class. Again, we are provided with ideas we might not have thought about. We talked about how this is hard for students at younger levels because the teacher is going to have to go back and re-assess the work if it is a formal assessment such as a quiz. This type of assessment can be done in pairs where the students work together to make their work better.

Formative and Summative assessments will be used. Formative is on-going and regular used, such as through class discussion. Summative assessment would be the tests, quizzes, etc. that provide a grade. It is important to have the formal and informal assessments. Teachers need to make their observations to help justify a student's grade.

As teachers we will be using assessments constantly. It's important for us to remember the variety of options we have to assess our students and the strengths and weaknesses of each. I hope to always provide rich experiences for my students and opportunities to provide and show their growth and development. 

Manipulative Reflection

I really liked the activity in class seeing all the main manipulatives classrooms have. Some I have never worked with and others I have worked a lot with. It's cool to think about the ways you can use one manipulative to cover all or most of the domains. When people see unit blocks they might just think for counting or measuring but it was cool to hear what other people thought of. We even tried to come up with a way to use it and money. A concept that might be confusing but do-able. It was really eye opening and made me appreciate all the manipulatives, the simple ones and complex because we might end up in a school that can't afford every type but seeing what other groups did and my own showed me how we can find ways to make it work.

Students understanding deepens with manipulatives because they make the concept real. Students who have a hard time visualizing can manipulate objects will really grasp the concept. That being said even for students who can visualize, having manipulatives right away helps students see the concept. I could not imagine teaching addition, or even counting without manipulatives. Students can take concepts learned from manipulatives and use them later once the concept is grasped. Students should not always need the manipulatives and one day will be able to do the work without them. Teachers can assess students' understanding and growth by observation of the students and scaffolding. In the beginning the teacher will have to model and guide the students on using the manipulatives appropriately as the students begin to understand the concept they won't need to be guided or follow the teacher. When a student can use the manipulatives independently that shows he or she is understanding the concept. Students will work in groups a lot and the idea or lesson can very easily not be understood by each student and in groups its easy to have one or two students doing the talking, maybe one or two who kind of get it and one who is just sitting there following but not really grasping the concept. As teachers we will need to make sure we are watching all students, and asking questions to specific students. We will know who are leader students are, the ones always trying to ask questions, but we will want to make sure we check in with each student. This can be done through quick conversations and observation.  Teachers can also use this to assess the student's depth of understanding. I think group work is awesome, students can talk, work ideas off of each other but it is important to have individual work on the concepts as well. Students can improve their problem solving skills with manipulatives using inquiry. Teachers can present a new lesson, give manipulatives the students are comfortable with or a variety of manipulatives and ask the students to solve the problem using those manipulatives. 

Reflection: Error Analysis

Using the More Errors document, we had to see students' sample work and figure out where the student went wrong. As I started going through the problems I thought it would be really easy to get the answer I needed, see where the student went wrong. That was not always the case. Even for some, when I could obviously see where the student went wrong I struggled with why he/she did what they did. It was nice talking it over with some classmates because we do see things differently. I think if this were to be a problem in my own classroom I would be much more comfortable because I know the students and the material. It also brought to my intention all the ways students can misunderstand directions. Not to always talk about my experiences, but I definitely did not think about how if a direction is given in a way where any misunderstanding can happen....you will have a handful of students going in the wrong direction. As teachers we have to think ahead of times of instances this can happen and how to avoid them or at least be prepared for them. 

Reflection: Assessments in Math

I think it is really easy to say assessing math is done through critiquing a student's work. An outsider of the teaching major/career would probably assume this is the only way to assess where a student is. To be honest, a few years ago as a Health Science graduate I would have assumed that's mainly what teachers use to assess students. I just never thought about the variety needed to assess students because it was not what I was expecting for my teacher. Even though I taught preschool students it became apparent how multiple options/opportunities to assess the students. First I assessed the students one-on-one multiple times during the school year. This was to see initially where the students were with previous knowledge, how far they have come, and how they finished the school year. I quickly critiqued this process by adding in class activities and observation. The power of observation and conversation. I truly love that through conversation we can see where one student is, or the whole class! I also was working on a portfolio for my class if I were to have stayed an additional year. I just loved the idea and wanted to material all there to analyze and assess. Formative and summative assessments are two names we hear a lot in the major. Formative assessments are on-going, constantly happening. This is through observation, conversations, asking questions, activities. Next summative assessments are at the end of lessons/units and give an opportunity for overall learning. We have worked with rubrics a lot. I have made several for classes and have had them used for my work in a lot, if not all, of my classes. I have a love/hate relationship with them. I do like having all the details in front of me for what I need to give to the teacher. I think in the beginning I struggled making my own so I have rubric anxiety. :) They are growing on me though.

It's definitely crazy how one year of teaching and then a year of being an education major have made me see so much more. I definitely appreciate it all a lot more and always am fascinated to hear the ways we come up with to assess and work with students. 

Assessment-Article Discussions

Article One: Getting Started with Open-Ended Assessment 

Open-ended problems are those with multiple strategies to solve and there can even be multiple solutions. This allows students to show their understanding on the topics. Instead of the student providing a simple answer the student has opportunity to explain their thinking and justify their answers. Open ended questions can be time consuming but they are worth it. As the teacher gets more use to the material coming up with questions will be easier and is beneficial to the students. Starting at a younger age, students can learn to justify their answers and reasoning. The article mentions teachers should not grade these right away, this could make the students uncomfortable for future open ended work. The teacher also used examples on the overhead and talked about the pros and cons of each sample. This helps the students get ideas about strategies they might not have used and work on communicating what they know with their classmates.

I really liked this article because I think open-ended work is important. I do think as we teach longer it will become easier to develop these problems because we know what to expect from the students. For example, a student might come up with a solution we did not think of. I like that the teacher shows examples, the verbal communication of the samples provides the students time to hear and work on using the right language for future problems. I also like that the teacher does not grade them right away. I know that open ended questions are intimidating, especially wanting to teach younger students, so we want them to feel comfortable and confident.


Article Two: A Smorgasbord of Assessment Options 

This article discusses the importance and the benefits of student centered assessments. Student centered assessments help students, acting as tools for learning. Then it helps teachers provide scaffolding for students.  Student-centered assessments therefore provide opportunity for students to set goals for where they should be at the end of the lesson/unit and help the teacher plan the next phase of the lesson/unit. The example provided in the article is about a class working with geometry. The students have to show their knowledge and reasoning to answer the questions. The students have reached mastery when the students can communicate their thinking and reasoning to their peers. The article does note that student-centered assessments should be both formative and summative to be effective.

Student-centered assessment provides a lot to the classroom. I like how the article mentions how it benefits the students and the teachers. I do believe the article was a little confusing on exactly what they are asking for, for a student-centered assessments. It makes sense the article can't give a cookie cutter explanation of what to do for student-centered assessments because it will vary class to class. Again, like the open ended questions from article one, student-centered assessments will be easier as the teacher because more familiar with the work and curriculum.

Article Three: Understanding Student to Open-Ended Tasks 

This article is similar to article one on open ended questions. When teachers use open-ended tasks the students are provided with an opportunity to answer a question and communicate why they did what they did. The students get to provide justification. The article shows four students' work, explaining the various ways students can answer these questions. Students can draw, write, label, etc to answer. The teacher can then gauge the students' level of understanding through their work. Again, the article discussed how students might be intimidated by this process. The teacher was strict/hard on the students about feedback. She also expected students to perform well who did not and vice versa.

I really liked the emphasis on the open-ended tasks through these articles. I think it is important we give the students time to answer the questions and provide their reasoning. Giving the students the ability to draw/write/explain how they want to provides experience for them to be able to work on their communication. To build off the prior article of communication of their thinking showing mastery of the skill. We have to teach students how to communicate and open ended tasks and questions provides an awesome opportunity for the students to work on that.

Article Four: Assessing Students' Understanding through Conversations 

Conversation is often overlooked when thinking about assessment tools but in reality it is the one most used. Teachers can have a conversation with a student, formally or informally, and see where a student is at and how they are thinking. A teacher can quickly see if the student understands the material being worked with. The article provides three examples that demonstrate the power of communication. One teacher used communication to see where the student was in the understanding. Next a teacher talks to students about mistakes made in work over a quiz. Finally, a teacher uses conversation on measurement to see where the students are misunderstanding the material. This helped her see where she needs to back up and help the students understand more.

I love the idea of having these conversations to see where students are lost, what they have mastered , and just creating a dialogue using the information the class is working with. When students can explain the work, they are taking the skill to the top level and will be able to apply their knowledge of the material anywhere. I love the examples provided because at first I thought we would just read about conversation as observation assessment basically. The teachers using it to create a dialogue on mistakes or just to gauge where he class was was awesome! I really liked it.

Article Five: An Experiment in Using Portfolios in the Middle School

The last article discusses implementing portfolios instead of using the traditional quizzes and tests. The teacher collects the students' work over the course of several weeks and has the students discuss their attitudes, growth, writing, and connections. The students have work that goes with each of these.  The students are given time to reflect and justify their reasoning. The teacher mentions writing her lesson plans to provide opportunity to add to the portfolio.

I love the idea of portfolios! It was something I was planning for when starting my second year as a teacher...before I decided to come back to school. I think it is an awesome opportunity to document students' growth and for the older students to be able to reflect on the work is awesome! I think it really ties in the process standards and helps the students grow as mathematicians. Portfolios are also just great for conferences because all the information is right there, the teachers can use it to come up with grade and to have proof of where the student is at for that unit/topic/material.

Journal Summary: STEM Gives Meaning to Mathematics

By: Lukas J. Hefty

Hefty discusses how creating the STEM program (Science, Technology, Engineering, and Mathematics) schools have been able to create a deeper understanding to the five process standards: Problem Solving, Reasoning and Proof, Communication, Connections, and Representation. STEM is able to make real-world, hands-on lessons for students from Kindergarten through middle school. These are skills, lessons, and traits the students can take on to High School and college. Hefty discusses making cars out of K'Nex and rubber bands and how students can problem solve, use mathematics, and make changes to get different results such as how many times to wrap the rubber band. STEM strongly shows students real-world connections which is important because it make the information more fun to learn and retainable. The students will appreciate the subject matter even more. Also by making the lessons hands-on the students are less likely to get frustrated. They are excited to make their experiments work versus getting bored with the experiment or problem. Teachers act as facilitators to help when needed. Engineering helps the students to think critically when using math. It also provides the students with opportunities to communicate and collaborate. The study found that with this approach, the tie between engineering and math, when students struggle or even fail at making something they are more likely to keep trying until they do it right. The article mentions how STEM can't be implemented in all schools but we can take ideas from STEM and incorporate it in our classrooms.

I really liked this article because I love the STEM program. There is even works of a STEAM program, where art is involved as well. I currently tutor at Peoria Academy, where they use STEM. They have an awesome STEM lab where students can work with projects, build cars, and more. So I have seen STEM in action. I like that the article mentions that schools who aren't STEM can still use the ideas sometimes. Teachers can come up with engineering projects and see how the students understand and take it and then adapt it to the students needs. I love that STEM is so real-world, students can relate to it and it's hands-on! This makes math fun. It's awesome to see that students don't want to give up when they are struggling, they want their projects to succeed so they will put the math into to make it work. 

Journal Summary: Linking LEGO and Algebra

By: S. Asli Ozgun-Koca, Thomas G. Edwards, and Kenneth R. Chelst

Linking LEGO and Algebra is about modeling using the Common Core's Standards of Mathematics. The teachers present an authentic problem to the students using the LEGO bricks and from there take time to analyze and use different approaches to understand the problem. Students were first asked to manipulate the given amount of bricks to make a duck and a dog. The students and teachers generated questions on what the lesson would be: how many animals can be built with that amount of bricks, how to make the most amount of animals, how to get more pieces, how to build more, how to make a profit, etc. The class took these questions for the lesson. The students were to see how many ducks and dogs they can build with the given amount and then later plugged this into an algebraic equation to generate a profit. The students kept track of their combinations on a table making up to three ducks (and zero dogs) to a combination of two ducks and one dog. The students then tried to figure out what they would need to make more of a profit. The teacher had an amount for the duck of $18 and $21 for the dog so the students knew not to buy an extra brick for over $3. Next the students were given more bricks and were using an excel spreadsheet to do the math. The students wanted to get the most amount of money possible with this new amount of bricks. They came up with an equation of 18x + 21y = P (x= number of ducks, y= number of dogs, P= total profit). After plugging in numbers the students saw what amount gave them the best profit that they could make with their number of bricks. Technology is what made the algebraic approach possible.

I really liked the approach used for this lesson. The teachers took something fun, the LEGOs, and made it a lesson. This made the students excited to learn and manipulate the toys. Using the generated questions from the class also made it more relatable to the students. I think the project would have been awesome to do when I was in middle school. It incorporated partner work, class discussion, inquiry, and technology. The students were taking a lesson that could be applied to the real-world. The article even mentioned making the students business owners which I thought was cool. I definitely liked the use of excel because this makes that program more relatable to students. I know when I was younger in a technology class our excel lessons consisted of "plug and chug" methods, following the teachers step by step. I love the lesson and even though I want to teach younger students it is motivational to think of fun ways like this to help our students learn and love it! 

Video: Number Operations: Multiplication & Division

For the teacher's opening I like how she extended it to introduce the idea of equal groups, even though I do not think the students really understood or fully grasped this idea. I was getting stressed out through parts with students not using the right words. This has to do with the Precision standard and it really shows in the video how the students can use the wrong words and this will cause confusion. I wish the teacher didn't start with the connection between addition and multiplication even though there is a connection. It's hard..because by comparing the two really helps make multiplication seem easier but causes confusion. There were lots of challenges to get the students to understand what operation to use, especially for the last problem. The teacher does an excellent job asking guiding questions which helped some students. It's hard because we don't know how much they have been taught about multiplication and division or what their class has worked with. I would like to say maybe in an introductory division class have the students work with the ideas of grouping right away. I do believe the teacher was a math coach and not with them all the time but this video helped show me the importance of how ideas are introduced.  

The student debriefing was hard, I feel like the lesson still ended with students being confused and I didn't like that. The student's really struggled with the fact that Maria had more than Wayne and that her money needed to be higher than his. I would have liked to have had more time to have the teacher explain this and the grouping. I did like all the discussion the students had with the coach but it was so hard because one student might get it, hear the wrong one, and then it seemed like back to square one.  I liked hearing what the visitors saw and worked with, with the students. We didn't get to see all of this so it was nice to hear what they saw and see what problems were prevalent. Again we can talk about the use of vocabulary, or the lack really of vocabulary. That idea really overwhelms me because we have to start from square one and break down any misconceptions the students might have with the vocabulary. The teachers have to fix work that has been engraved in the students. 

I really liked the video because it shows us what to expect, it brought a lot of ideas I wouldn't have thought about to my attention. I do feel like the information could be more organized but when asking the students for so much input it's hard to keep it organized because they had many different ideas. I love seeing the discussions at all ages because it is something we as teachers need to use. It is a good gauge to see where students are, what they understand and don't understand. To me I took away the importance of proper vocabulary and the importance of introducing division as equal groups because this concept can help the students grasp the concept. 

NAEP Reflection

When first presented with the problems for this project I was a little worried. It is always nerve wrecking being given a problem for 8th graders and below....or high schoolers, and being asked to solve it. As college students we should hopefully be able to but it is also work we haven't worked with in a long time. My group had the Data Analysis problem where the students were told to pick a graph to represent the information presented to them. When given all of the student work we decided to go through them one at a time and read the rubric and put them in stacks according to where we believe each one belonged. We had a lot of partial and minimum options, the problem with this is the rubric description for both option was pretty similar. We then took each pile and very strictly compared it to the rubric to select our seven best. Again, we struggled and even moved some of our prior choices to different categories. We did select seven. The best strategy we could use was to follow the rubric as strictly as possible to keep consistency. The way we went over each work problem twice and talked through them really helped us grasp what was looked for. I hate the rubrics NAEP uses because they are very vague and can be taken different ways. There are too many variables in my opinion.

Listening and watching the other groups first of all made me appreciate the problem we had. You could see when any leeway was given to the rubric how their is confusion on where the problem should go. We thought our rubric was rough but some of the other group's had really awful ones. With multiple sections to look at. I was getting stressed out just observing them. I also know that since we worked so closely with our material it is easy for us to guess where NAEP would have put them but quickly gazing at the other problems we went back and forth a lot with where they could go. This project helps show us as future teachers the importance of having clear criteria for rubrics. Students should not be as confused as we were when seeing a rubric for an assignment or project. With clear expectations no one will feel lost or have too many questions. 

App and Applets Critique

App: Operations and Algebraic Thinking Plus Counting and Cardinality (K-2)

Source: https://itunes.apple.com/us/app/operations-algebraic-thinking/id827132721?mt=8 

The app is set up to help the student meet the standards of Operations and Algebraic Thinking plus those of Counting and Cardinality. At each level the standard is written in the top corner, so using the game in the classroom will help easily show the teacher what the students are working on exactly. The app is presented with a student in front of a whiteboard with manipulatives to drop and drag. Reviews of the game consist of: "perfecting aligning with Common Core Standards", "easy to use" and "surprised with all you can do and learn from the app!" The design is simple and not overstimulating. I love the idea behind the game but I think the students would get bored from it easily. It is meant for K-2nd. The students will be excited to be on an iPad but it is still designed almost too simplistic. 

I believe the idea behind that app is awesome! If students have technology centers and are allowed time on an iPad the teacher can quickly see what standard is being worked on and what the students are thinking. I do believe the app's design is lacking and will cause the students to not think of the app as fun but as another lesson they just did on paper. The content and set up is great and would be something I would want in my class, hoping to only see the app develop more over time and take a more creative approach. 

Applet 1: Grouping and Grazing (K-2)

Source: https://illuminations.nctm.org/Activity.aspx?id=3526

The applet is a game where the students will use addition and subtraction up to five to see if they have selected five cows before being able to have them get taken away. In the applet the objective can be changed from groups of five, to 10, to addition to subtraction. After the answer is done the students may check their work and it will tell the students if they need to keep working or applaud them on their work. The cows will go upside as the students click them, this will help eliminate double counting, tallies can be written which also help. The game is visually appealing, the cows flipping will entertain the students and aliens taking them away will also be interesting to the students. The game is a little hard to understand at first and took some hands on work with the applet to understand how to answer each question.

When I first opened the applet I was excited to figure it out and play with it. Once I started working with the game I realized it took some work ahead of time to understand how it works. I think if the teacher has the settings for the student done, so levels of addition, subtraction or grouping picked ahead of time. The teacher would also have to take time to explain what the students need to do to answer the addition/subtraction items versus the grouping. This applet could be used during a technology time of the day, during the math lesson, or be available during centers. Students get immediate feedback and get to learn in a fun way. 

Applet 2: Base Blocks Addition 

Source: http://nlvm.usu.edu/en/nav/category_g_1_t_1.html

The applet provides addition problems using representations of the commonly used based 10 blocks. The students can drag the units, flats, longs, and blocks around the columns to help solve the problem. Visually the applet is very basic but is also nicely set up with the amount of columns the student/teacher decides to use. For first grade and kindergarten the option of using 2 columns is available to help keep answers lower. The teacher can even design questions to ask the students. This would be a good applet to open up and use on a SmartBoard. The students can drag the cubes around to make the longs and flats. I do believe the applet was confusing at first but then again they all are until they have been used enough to gain understanding. 

I would use this in my class on a SmartBoard if available. It was harder to move the cubes around with the mouse but on a SmartBoard students can use their hands, or even on an iPad. Being able to design questions is nice because some of them were out of the standards for K to second grade. If it did not have this option I would not want to use it in my class unless we had gifted students in math. This could then be used during center time in the math center. Once the student has the right groupings the answer will show up on the side of the screen. The feedback is immediate but might not make sense to the students. Also getting the blocks to group was difficult at times which could hinder the students experience. I do think the students can grow from this by using visual representations of the base 10 blocks in a fun way such as with technology! a

Article for Discussion-Thinking through a Lesson: Successfully Implementing High-Level Tasks

By: Margaret Schwan Smith, Victoria Bill, and Elizabeth K. Hughes

The article is about how to focus on TTLP (Thinking Through a Lesson Protocol), to provide cognitively challenging tasks. Meaning providing the students with opportunities to select the right path to the answer using past lessons. This provides challenges for the students and the teachers. The students must select the right path and for teachers there is less control. TTLP is divided into three parts: 1 Selecting and Setting Up Mathematical Task 2. Supporting Students' Exploration of the Task 3. Sharing and Discussing the Task. Each step builds off the previous one to provide a path for the students. This will help eliminate the loss of control for the teachers and possible sense of overwhelment the students might feel with all the options. TTLP is not meant to be used everyday for every lesson but periodically. The authors do mention though teachers can form their lesson plans to meet these steps more often than not. When developing a high level lesson teachers need to have clear goal, this helps provide framework as to what to expect from the students and lead them in giving you what you want to see they can accomplish. Teachers need to think of all the possibilities of solving the problem beforehand, this adds to having control of the lesson. Benefits of using TTLP would be getting a deeper meaning of the lesson. Teachers can ask "What if..." and draw on more ideas or changing the paths chosen. The format also helps teachers develop their lesson plans even if they aren't using the full format of TTLP.

I really liked the set up of TTLP. The three parts really set up to have a fulfilling lesson in my opinion. Providing the students with opportunities to use prior knowledge and various ways to answer the problem. I think it would make the students feel very accomplished coming up with a different way than another classmate. I do think it is important for the teachers to think of as many ways it can be solved as possible but also to think of the wrong ways students might think to solve it. This will help the teachers be prepared ahead of time for questions, bumps in the road, and concerns. At first I was overwhelmed thinking of always having to use this but the article said periodically. I do think the set up and questions you ask yourself as a teacher can always be used when making lesson plans. I love the discussion portion to provide students the opportunity to share how they solved the problem and hear the other ways. Students need to see there are multiple ways to do something.

Discussion Questions:
1. Is there a specific structure for performing Part 3 of TTLP? Should the teacher encourage student lead discussion or break the students into groups according to how they decided to solve the problem.
*I actually think if enough students did it a different way it would be interesting to put one of each in a group to teach the others and show the many options.

2. With the many paths to the right answer what should the teacher do when presented with an option they didn't think of? This stresses me out when thinking of the TTLP format. 

Article for Discussion-A Model for Understanding, Understanding in Mathematics

Article by: Edward J. Davis

Davis' article first held my interest because in the last year I have been in the Education Department we hear time and time again to never put "the students will understand..." in our lesson plan objectives which makes sense. The article refers to "kinds of logical things teachers do during mathematics" as moves. After reading this article it is proof what the students really need to have as objectives when we want to say understand..and it's a little longer than just understand and requires work. Having students be able to state the problem and solution in their own words, give examples, find it hidden in other examples, see connections from other facts and it, use it in multiple ways, foresee the consequences, and state its opposite. Teachers should start with physical characters and then move to symbols as representation. The article mentions having students answer certain questions to demonstrate understanding such as using the formula in another problem. The amount of moves will vary from lesson to lesson and teacher to teacher. Understanding on the two levels will also vary, some lessons the students will hit both levels in one try while others it might take longer.

The figures throughout the article really helped me appreciate and understand the material. I like that each section was broken down into two levels providing examples of how the concept works. It was eye opening and just really put the whole idea into perspective on how to use it. I took away the fact that as teachers it isn't just important for the student to be able to solve the problem but also to understand why they are doing and using what they are using. If the students can explain to us why then they have reached understanding. It is easy to present the information to students and have them plug in what they know and present it without really knowing what they did and why. I like that the article said to ask the questions such as recalling facts or explaining patterns.

Discussion Questions:
1. How can teachers start adding moves into their lesson plans to reach the level two from the figures or full understanding?

2. Has there been any research done on using moves in other subject areas? How does it compare to the work done on moves in mathematics? 

Teaching Rich Task Reflection

I really enjoyed brainstorming ways to make our task more rich. It was a hard concept to wrap my head around while planning a lesson and then listening to my group members suggestions. I had an idea of what we wanted to talk about but making sure the task is rich became more complicated than I had thought it would be. It's easy to say make a task better, make sure the students get the most out of it but when it comes down to analyzing pre-made lessons an making sure they are rich enough. It was an awesome experience and gives us the awareness of what to pay attention to when planning.

I loved presenting ours and thinking of what could go wrong or differently in front of a class. It is always eye opening when you go from talking about a lesson to actually doing it. Just preparing for the lesson we learn what techniques would work and what wouldn't. I wish we would have added some addition into the work so the students would see who had the highest total coins at the end. With the Kaboom the students would have lower numbers and so on. Then we could use the idea of money and what we could buy.

I liked the other lessons and thinking of ways to help, ways to use the lessons in different grades and ideas that could come from one lesson. The other two lessons had fun ideas behind them. One thing I learned as a teacher is things will go wrong or not as smoothly and learning to deal with it and prepare better for next time. An example of this would be cutting the tin foil ahead of time to cut time spent standing around or addressing the issue of students like Hallie and I making inappropriate drawings on the wall. :)