Curriculum Plan

Although this project was very challenging I believe that I benefited a lot from it all. I learned what it takes to develop a strong curriculum plan that covers all the common core math standards and is also inquiry based learning. I think that the most challenging part was all the thinking that went into it. We started off over thinking a lot of the material. We were trying to create too challenging of a project. Once we realized what we needed to do the rest of it was a bit easier. However the hardest part of it all was the semester long projects for all grade levels. It was really hard to think of something the students could truly be working on for a whole semester and not become bored with the task.

I got super lucky, I had amazing group members. We all got a long so well and we all understood the math background at the same level, which made our plan much stronger.

In terms of all the presentations I believe that what some people believe is a good math curriculum is different than what others believe to be true. I enjoyed watching and learning about all the other grade level curriculum plans. However for more than one I didn’t understand the structure that they were using. I felt that the way they organized their plans was confusing and would also confuse the students as well. I think as teachers it is important to make sure that we are creating strong math lessons at every grade level. I have saved all the other plans on my computer and I plan on adjusting them and using them later in life in the teaching field.     

Assessment Articles

Portfolio: This article was talking about a teacher who decided to use portfolios as a way of assessment in her math classroom. 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. I really like the idea of portfolios I think it is a great way to allow the student to be proud of the accomplishments and advances they have made throughout the year and to showcase their skills. I think it allows will cause the students to put out better work. It is using a hidden message to better the efforts of students without them even knowing that is happening. 

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.

Open-Ended: 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.  

Reasoning: The article is addressing problem solving. The students are asked to define and explain area in their own words to the teacher. This is yet another way for the teacher to hear and better understand their students’ understandings of a certain math topic.

Options: 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. 

Classroom Changes Reflection

The math classroom in the 21^st century in taking on some very different changes compared to some of the years prior. Math is becoming more inquiry based. Students learn best when they are discovering and exploring. This is hard to do for every math topic and for every lesson; however as future teachers it is important to provide our students with sufficient lessons that will challenge them and allow them to explore and investigate the different math problems. The math environment is important as well. We need to make sure we are providing students with a variety of different communicating tasks as well. For example students should be given the chance to work with groups, partners individually and also have the chance to present their knowledge to the class. We are not only building strong math learners we are build strong life skill learners. A math classroom needs to have a variety of different teaching and learning tasks and these should be altered quite frequently. This will keep the students more engaged and on task and hopefully excited about learning.  

Assessment Reflection

I think that one of the most influential impacts that I got out of this class is that math does not always need to include a math test to test your skills and understanding. Out of all the different assessment ways we have learned about in this class I think that my favorite might be the portfolio. I really like that the student gets to decide how they which to showcase their skills. I think this also allows the student to take pride in their learning and truthfully see how they have evolved with their skills and knowledge.

In addition I also really liked how we did some recording to voice our knowledge of different math skills. I think that this is a great way to really hear and understand a student’s process skills and understand on a topic. If you ask the student to prove you with visuals as well this can make the voice recording assessment even stronger.

Lastly I like that the class is very reflective. At sometimes I feel that it is almost too reflective; however I believe that when students are given time to reflect is it also time for them to really digest what they learned or experienced. This will allow them to process their skills and hopefully understand them to a level that they will be able to use the next them they are presented with a different challenge or math problem.   

Technology Reflection

I really liked that we spent some time making sure we knew different ways to use technology in a math classroom. Personally I think that class was the first time I have ever learned so much about the smartboard and all the different things it is capable of; they usually are just used for projections purposes in my other classrooms. I wish more of the professors spent more time teacher us about them. I also really liked the apps and applets presentations; however I would say for next time I think it might be more helpful to have the students present two apps and one applet. We live in such an apple/app run society these days, and apple run schools are becoming more popular. I just think that might be more helpful as the years go on. Also I think that Westlake has an ipad cart. I think that it would be really cool to just spend half a day working with the ipads and some of the apps that people present (the free ones at least). Personally I believe that a technology day would be really helpful and should take the place of one of the other activities that this class has.   

Manipulatives Reflection

I really enjoyed the manipulative activity. I felt that it was really useful to work with and think about them all in a different light. It is very true that we might be hired somewhere where they don’t even have one math manipulative so we need to learn how to use what we have the fullest potential to better the understanding and knowledge  level for our students to the best of our abilities.

How do you know students deepen their understanding while using manipulatives?

They are able to work with the manipulatives to physically see the math applications that are occurring. This allows them to understand what they are seeing to help make better sense of the applications when they are not using the manipulatives.

How do you know if the students can transfer their understanding from manipulatives to other situations?

You have to slowly teach the students how to do the application with less and less manipulatives. If they can solve the correct answer to the math problem posed without the use of manipulatives you have worked with their level of proximal development and scaffolded them to the zone in which they are currently in for the use of that specific math application to be done without the use of manipulatives.

How can you assess that understanding or growth?

I believe that this is done in a variety of ways.

   a.       Simply walking around and observing the students’ working with the manipulatives. Try and              hear what they are saying to themselves while they are using their processing skills.

   b.      Provide them with a task and see how they perform.

   c.       Ask them to teach you the concept and see how well they are able to accurately teach you.

   d.      Their performance level on classroom or state level testing.

When students work in groups, how do you hold each youngster accountable for learning?

I believe that this is a challenging task. However I have learned that if students are given a roll to fulfill and carry out with their group members there is a better chance each student will be learning and participating.

When students work in groups, how do you assess each youngster’s depth of understanding?

You can ask the group to present what they working on to the class or to you individuals and asses the body language and verbal language from each student. I believe you can also have the students complete a reflection; which will allow them to tell you what they did and what they believe they got out of the experience.

How are you improving students’ problem solving skills with the manipulatives?

I believe that the use of manipulatives allows students to improve problem solving skills because it is allowing them to physically see the reasoning and proof behind the math application/task. Then later the student will be able to draw from their experience with the manipulatives to help them make sense of and understand a more difficult math problem. They will be able to visualize what they were working with and possibly use that knowledge to solve the new/ more challenging math problems.  

Errors/reteach reflection

I really enjoyed this activity. I found it fun and interesting. It was like playing a little math game. It was also really eye opening because I never realized some of the strange ways that my future students could possibly misunderstand math applications. I really liked that we were asked to think of a way to reteach the coordinating topic for one of the students. This is an important practice that I think some teachers forget. If a student does not fully understand the math concept at least 90% percent of the time I believe a reteach is a must. I know of plenty of people in my high school who were placed in a lower math class because they needed to be retaught a lot; which is a weird concept. However this is how my school made sure everyone was getting the right math education at the right level. I think teacher avoid re-teaching because they feel that it will take away time from other material that needs to be coved. Which is the truth, but how will the student understand the new material if they can’t even understand the current material? As a future teacher I will look for these possible errors or misunderstandings with my future students and reteach when necessary.      

"I don't really know how I did that!"

By: R. Scott Eberle

This article was about geometry, specifically the use and idea of tessellations. The teacher provided students with different shapes and asked them to create different kinds of tiled patterns. The teacher explained that most students understand how to make the tiled patterns with actual tiles but sometimes when they are asked to draw the pattern they struggle with how to make it look like how it should. The teacher then went on to explain how it is important that teachers allow their students to try tiling with different shapes, rather than just squares. When he allowed for his students to do this some of the discoveries they found we pretty cool. For example one boy discovered that tiling with triangles with make hexagon figures as well. Where another little girl discovered that the way she ordered her L shaped title was different than the way her friend did and they were able to create two completely different patterns. The teacher also explained that with tessellations it is important to talk about the different lines of symmetry that the students are able to draw from their pattern. He also mentioned at the very end that this is a good activity to use for an inquiry lesson with geometry.    

I thought that this was a good article to read about geometry. I think that a lot of times teacher push aside the importance of this topic in the classroom. I really liked that it went of to explain how different patterns can be made within shapes. I also like that the teacher provided his students with the chance to use inquiry skills. The only thing I didn't like about the article was that I felt like it started to become a little repetitive and the end of it. Overall though, I want to do an activity like this with my students in the future I think they would really benefit from this knowledge.  

Taking it to the next level: Students using Inductive Reasoning

By: Jaclyn Murawska and Alan Zollman

This article started off with the difference between inductive and deductive reasoning. Inductive reasoning is when student understand some thing possible because they have seen it through multiple examples. Where deductive reasoning is using reasoning assuming it it true to arrive at a conclusion. The article then goes on to spend most of its time explain math inquiry, the different levels and each level's significance in importance. For example the 4 levels of inquiry from lower to higher are: 1. Confirmation, 2. Structured, 3. Guided and 4. Open. The article went on to give examples of each level of inquiry; it was dealing with a geoboard and line segments on the board to then develop their reasoning in of pythagorean theorm. They stated that in order for a teacher to be good at using inquiry in her classroom she needs to be really good at asking and developing questions. They talked about how this was a science approach to learning that is now being adapted and developed in the math words for teaching as well.

I am a huge advocate for inquiry math usage in the classroom, so I found this article to be really interesting and explanatory. It was cool to learn about the different levels of inquiry. I never really knew their was different levels, I just thought that inquiry was inquiry. I enjoyed that they provided and explaining of each level and also showed some student work throughout the article. The only part that I did find a little confusing was how they expected the students to make the connection to the rule of right triangles. Even after reading the article I tried to think about it more and I feel like I am still unsure as to how that would happen. Overall I found this article to be interesting and informative. I am excited to share what I learned with my group and also to use all the different levels of inquiry throughout my time teaching in the future.    

Video Blog Post II:

number operations: multiplication & division

Planning:

This video did not have the planning part like the other video had. Personally I liked the way the other video did that, I felt like that also helped me as a viewer to better understand what was going to be happening for the class time. However I really liked that she used the term active listening with the students. Really using that academic language with them is important.

Lesson:

I liked that she made sure she wrote what the students were saying to on the board, she was not just assuming. However I did not like that she asked a student to draw a picture and she would get back to it and then she never got back to it. To me that tells the student that she didn’t feel that she had time to hear about his thinking. Also I don’t think she provided enough wait time after her questions. I think she did a good job listening to the students but I also feel like she didn’t really stick with anything that they said. Every time a student made a suggestion she would go back to the boxes and writing the numbers in the boxes; which personally even I was confused and distracted by her doing that. I also did not like when she called a student to the board to write something and she erased what the student wrote and rewrote it. I could read what the student wrote, why did she erase it?

I also think she jumped the gun on their knowledge. She talking about multiplying and dividing single digit number and then she jumped to a two digit number by a single digit number. I don’t this the students were ready for that big of a jump so quickly.

Also I think that she should have made the lesson only about multiplication and do division a different day. There was just too much being spoken on. Also of the students were having a hard time figuring out the equal groups for multiplication what is there to say you should move on to division. They also never did any word problems with division so I just think it made a long lesson longer… Do division on a different day.

One of my favorite parts in of the whole lesson was when the student counted “20, 40, 60, 80” “6, 12, 18, 24” equals 104. While watching this I said “WOW” out loud. I was so blown away by that student’s thought process. To me I would have never thought to think about it that way. It was amazing to me that a fourth grader was able to use that skill and process the right answer. This was an ah-ha moment for me, really making it clear that all students really do process math skills differently and as teachers we need to work with them on that and not tell them their thinking is wrong.

The biggest thing that I did not like about the lesson was that she kept asking them how THEY would draw a picture. She spent so much time on that. And I am a firm believer that when you asked student to draw a picture you are going to get a variety of different kinds of pictures, but as long as it makes sense with the math problem it is not wrong. But it’s as if she wanted every student to draw the same picture when she should them the Charlie model. It was like she said “yeah your picture is nice, but this is what I want your picture to look like”; which I believe that this is really where she lost them. That picture just did not make sense to the students.

At the very end I did like that she explained the camera crew more to the students and told them thank you and also told them they would be rewarded for their good behavior.  

Faculty:

I felt like she did a good job addressing the positives of her lesson, but she never addressed the negatives of her lesson. I also really liked how the one teacher drew what she saw on the students’ papers. And she even kind of pointed out that the students really had no idea how to make sense of the Charlie model. Which I thought was really important because the main teacher thought that the Charlie model was awesome and so perfect, but if it doesn’t make sense to the students… It is not awesome. Also personally I found the Charlie model to be weird as well. That is not how I would have represented my picture for that question and also not how I would have wanted it to look after I saw that model as well.

Thoughts:

My overall thoughts have been address throughout the course of this blog post. 

NEAP Reflection

I actually really enjoyed the NEAP project. With being a math concentration I obviously enjoy math. I have taken my fair share of math courses, but this project was different than anything else I had done before. It was very interesting to look at the student's work and try to figure out their process skills form the notions they provided; just like teachers do on a regular biases. I was also cool to think about how your would assign the student with a grade, what parts are required and what parts acan be assumed knowledge?

Also for part two I really liked the part where we got to write the comments to a few selected students. I think that it really helped us to understand what makes a good comment that the student will actually be able to understand and be able to make improvements for next time. The one thing my group did comment on was that it would take a long time to hand write all the comment for a classroom, so maybe we would type them and staple them to each student's work.  

Overall I really enjoyed this project and I feel that me knowledge of teacher math and all other classroom related math procedures have improved.

Rich Task Reflection

Although I was unable to be there on the day that my group taught the lesson I felt that it was still necessary to write a blog post on my reflection.

We had lesson in mind but when it came the to start typing the lesson one of the other girls showed us the barbie lesson idea and we all loved it. I really enjoyed finding that lesson, we all agreed that it would be much more fun for everyone in the class than the other lesson we had in mind.

I typed most of the lesson plan because I knew I would not be there the day to teach the actual. It was really cool to think about the inquiry with the math lesson we had created. I really liked that the lesson used a common child's toy and applied math to it. I think this is something that every student could relate to and could also enjoy.  

Applets and Apps

Applet:http://illuminations.nctm.org/DeepSeaDuel/

Deep Sea Duel

This applet allows for students to pick what level of difficulty they wish to practice during the game. They are then given a screen with a “goal” number and bubbles with numbers inside each of them. They will then be asked to build the “goal” number in a 3 number combination. This allows students to understand parts of a whole and positive and negative values.

I like this applet because it allows for the students to visually see the math steps they are computing in their heads. I also allows for students to better practice their knowledge on decimal, positive and negative numbers.

Applet: Kahoot

https://play.kahoot.it/#/?quizId=9f8af42b-d67d-40f3-988a-ef61fceb2529

Kahoot is a website where teacher can create their own interactive games for their students or they can browse the library of games and use a premade game. The Kahoot can have as many questions as you please. It also allows you to decide how much time it will allow for the students to pick an answer. As the creator you are able to decide if you want to make it a points game or not. With Kahoot you are also able to insert pictures and videos and create questions off of that as well. Kahoot also tracks the students’ right and wrong results and will make a stats chart for you at the end of you request such.

I really like Kahoot because I believe that it allows the teacher to have full control over the material that her students are benefiting from. I love that you are able to personalize every aspect of the game creation. This game could be used in lue of a test for your students, by it tracking the right and wrong answers you would be able to place an accurate grade to each one of your students.

App: Explain Everything http://explaineverything.com/

This is an app that provides you with a blank page and tools to create a situation on that page. It allows you to place animations, add text and add voice recordings. In a math setting the teacher could as the student to show her what they learned about functions and expressions by creating an Explain Everything.

I like this app because it allows students to demonstrate their learning at a higher-level because by asking them to explain what they know or have learned is at one of the highest level on the Blooms taxonomy Pyramid. This can also be in lue of a test. The teacher will be able to go back and watch the student’s explain everything and she can assess the student's knowledge based off of that.

Thinking through a Lesson: Successfully Implementing High-Level Tasks

This article addressed the idea and concept of Thinking Through a Lesson Protocol (TTLP). This is a concept that is used to help teachers create higher-level thinking situations for their students. Teacher want to create higher level thinking situations for their students however this can cause some teachers to be more nervous about the situation because it means that they will have to set back from the direct instructional method of teaching. This article addressed the different parts that a teacher should think about before, during and after implementing a lesson. TTLP is a way for teachers to think about possible situations that could occur with their students during the lesson. This is a way for teacher to feel more confident when working with higher-level tasks with/for their students. TTLP also allows for teachers to better understand the possibility of different method of actions that different students could possibly take. As a teacher it is important to understand that different does not mean wrong.

I really liked this article because it helped me better understand how to make sure I am helping my student achieve higher-level thinking skills. I liked how one of the personal testimonies stated how she does not write down answer to each of the questions but just simply thinks about it so that she feels better prepared or is able to better understand a student’s possible thinking process on the problem. I also really liked this article because I think that TTLP mirror the mathematical process. Math is completed in steps or parts and developing a proper math higher-level thinking lesson should also be thought-out and completed in parts. This also helps us to make sense of each part to better make sense of the whole. I look forward to using this method in some of my math lesson to come in the future.       

A Model for Understanding Understanding in Mathematics

This article was about the use of "moves" or in other words steps to help students understand their processing skills. This article addressed how teachers can phrase certain questions to better understand their students' current understanding of the topic. The article provided many different detailed examples of how certain "move" questions would look and to what goal they are aiming for. The article addressed different levels of moves and different focus points; for example, examples of the concept and understanding what the generalization and why that generalization is accurate. They also discus moves that will help students understand the process to the problem and why that process is correct for the situation at hand. The last moves the article addresses is the understanding of the outcomes and why the outcome makes sense and is accurate to the findings of the problem. The article explains how moves help teachers from questions of understanding for their students in both a lesson setting and a testing setting. Research shows that the moves do not need to be done in a specific order; however they should follow a logical order and be at the student's level of cognitive development with the situation at hand.

I really liked this article. With being a math concentration I have taken my fair share of math courses here at Bradley; however we have never really discussed how to teach our students or measure their level of understanding. I really found these moves to be very informative. They better showed me how to phrase questions to understand my students' understanding of the topic at hand. I think that these moves should be placed at the end of a teacher's lesson plan. I think it would be a good idea for her to come up with a few possible moves that the students could benefit from. She could come up with a list of moves in the order in which she would as them and if she had a student with a math learning disability she should compile a different set of moves developed around that student's cognitive ability level. I am excited to keep this information with me and to use these moves in my classroom in the future.

Group Reflection on Standards Project

I am really grateful for this project. I had learned about the different standards before but it was nice to actually dive into a couple and really get to know them. I learned that modeling with mathematics included really life situations and the use of manipulatives. It was cool to read articles how the article have related situations so closely to the lives of their students. Next Hallie and I have look for an express regularity in patterns. I really liked this standard although it can be hard for some people to really understand what it is aiming for. I think this is such an importation skill for learning math. I like to look at it in terms of building block. As the student is learning different math shortcuts their level base for math is increasing and increasing. I look forward to using and applying these standards along with the rest of the standards with my future students.  

Video Clip: Word Problem Clues

1) The planning, including tasks, core math and challenges, and questions for teacher reflection

I thought it was really cool that before the teachers were just observing in Tracy's classroom that they all meet and had some time to learn about and hear about what they were going to be observing during that lesson. They were provided with background knowledge on the overall classroom performance with math. I really liked that they actually took some time to look at and try the math problems themselves. I think this is something that some people forget to do, it is important to make sure what we are asking them to do is doable and makes sense. Without doing the math problem themselves maybe the teacher would not have realized how hard it really is for some students to draw pictures for some of the problems. It was interesting to hear her explain how the students just see the numbers and automatically just start adding, they are not taking the time to read the actual problem at hand. This was interesting to me because I never really would have thought that they would just skip the reading part of the world problem. I could understand if they misunderstood the message but just not reading it at all is something new or me to keep on my radar. Tracy really stressed that she wanted her students to make sure they knew what they were writing down and if what they wrote down made sense. She explained how she wanted the students to understand how their numbers, pictures, and words need to all make sense and connect together. This is something that the students were having some difficulty with. I also really liked how she said that she tries to avoid using the word "wrong" because she realizes that this can discourage students to keep thinking and trying. I really agreed with her on this point. Yes there is such thing as a right and wrong answer but we as teachers need to keep our students motivated to learn and show them that they are headed in the right direction. If we just say "No, wrong" well maybe they were wrong but they were close... and now that student has lost their drive to keep going. 

Tracy asked the observers to document any muttering they heard among the students. She stated that this would help her to better understand the students thinking process later when they were shared with her. I thought that was a really cool request from her, because it is hard to catch every students side conversations and outside thoughts. Having more people in the room can make this an easier task.    

2) The lesson including student debriefing, along with success and misconceptions and student work

I really like how much time she spent with the students walking them through the different thinking and processing skills that this math problem was asking them to use. This was the first time that I saw a math lesson start out on the carpet. This really stuck with me because I feel like a lot of teacher always think that math needs to be done at their desks while you stand at the board and write on it. I think doing it the way that Tracy did sets the students up for a better thinking palate. They are sitting at the carpet and have that time to think and verbalize their thoughts. Then when they go to their desks they get to use those thoughts and turn them into process skills and thoughts. 

I also really liked how she pulled examples from students in the classroom and how they all knew to respect each other's work. This is something that Tracy had to have worked with them on before hand and this is something I would like to do with my future students as well. I think it helps them really notice there is more than one way to think and process problems and by seeing it done amoungst their peers with hopefully stick with them better. I also like that while see what doing that she used post it notes show the students what was where and to make the students really be specific with what they were saying they were seeing. 

When Tracy asked them to look back at their work, I thought it was a really cool idea that she used photocopies of the original work of the student and also asked them to use a different color pen. This allowed the students to see their base and how they need to build and improve from what they have already provided. This is also some thing I want to use with my students. More often than not I have seen where a student just erases all the work they have done when they know it is wrong, instead of building off what they have discovered. It actually drives me crazy when I see students do that... 

I liked that she then provided the students with some new and different math problems to see if they could apply what they just learned to accomplish a new task. This is something important, because instead of just ending the lesson with "oh yes now they get it more", Tracy built on it and made the lesson to come to "let me make sure they understand it more by having them show my and apply what we have discussed through different but similar problems". I loved that!!!     

3) The faculty debriefing and implications for instruction

I found it really powerful that Tracy was able to be an accurate reflector of the lesson. She stated exactly what she thought and how the lesson went. I found it very cool that she was able to right off the bat notice what did not go as well as she planned and also how the lesson formed into a little bit of a different lesson. I think as teachers we need to always be able to look at a lesson and understand that it is okay for that lesson to be morfied a little bit as long as the overall main goal is still on track. Which is what I would say happened with Tracy's goal. She wasn't able to get through as much as she had wanted, but that is because she needed to spend more time on certain specifics with some students to make sure they understood and we comprehending the problem. As teachers we need to recognize when we need to slow the pace down to make sure we are keeping all of our students on track. 

4) Explaining your thoughts on the overall use of the video. 

I found this video to be really useful. I have provided my individual comments on specifics throughout my entire reflection. I learned a lot from this video and can not wait to incorporate these math techniques in my future lesson as well. I would also just like to comment on the fact that Tracy used outstanding Math vocabulary with her students. This is so important! Students need to know the proper way of speaking the math language and they are never going to learn that if it is not being used around them. Tracy did an excellent job with that!!! 

I

Count On It: Congruent Manipulative Displays

By: Joe Morin and Vicki Samelson

Summary of Journal 

This article's focus was placed on the use of math manipulatives in the classroom. Most of the time people think that math manipulatives in the classroom could only better a student's understanding of a topic, it would never render their learning. However the authors of this journal go into great detail as to how manipulatives CAN have downsides. This article goes on to explain how manipulatives should and should not be used in the math classroom. They start out by saying that it all depends on the cognative level of the student. If the student does not understand the idea of "representation" through the use of manipulatives, you will need to build the student to that level of congative ability first. They then develop the understanding of how students transition from standardized manipulatives to semiconcrete manipulatives. The article address how the use of manipulatives can lead to major distractions with younger children because depending on what you pick to use as the item they can have a hard time seeing it as a learning tool rather than a toy. Another idea that they spent a lot of time on was the idea of Conceptual Congruence; which deals very closely with students understanding the base 10 units. They provided an example of how to properly use base 10 manipulatives with younger students. Long story short all aspect of the 10 units need to visually present for the student to understand. The present ones and the empty spaces that would build a full unit of 10.  

Reflection of Journal 

This was a very interesting article, because I never really knew so much thought went into the use of manipulatives in the classroom. I kind of always thought what the majority of people think: manipulatives always help. However after reading this article that is not always the case. I learned some very valuable concepts from this article. For example, I did not know that for younger students if you provide them with two different types of manipulatives, one for which is physically larger than the other, but smaller in quantity; some students might have a hard time getting past that if asked to state which group has the larger amount. I never really took the time to take a step back and think about the congative development that needs to be present in students when dealing with manipulatives in the classroom. Lastly I also really liked how they stressed the importance of base 10 manipulatives for students. How it is important that all aspects that form a unit of ten are present visually for the student to see and grasp how many units make a "base" of ten. I look forward to carrying this insight with me as I progress in my teaching career. I will always think about this information when it comes time to pick out manipulatives in my future math classroom.  

An Authentic Task That Models Quadratics

By: Lorraine M. Barron 

Summary of Journal

I had the pleasure of reading this journal article that was written in February of 2015. This article was placing its focus on the Common Core Standard number 4: Modeling with Mathematics. There was a team of teachers that wanted to create an authentic problem situation for their students. They also wanted the problem to fulfill the Common Core Standard for High School math dealing with non-linear functions. The teachers wanted to make sure that the problem was relateable to the students current lives, so they based it off of a school fundraising project that many of the students shared a passion in. The problem had to deal with selling muffins for the fundraiser. The students were placed into groups and then presented with the problem by their teachers. They were then given 90 minutes of work time to figure out the problem and create a visual representation to go along with their findings. The teachers created a very inquiry based learning environment for their students. If the students asked questions the teacher did not give an answer but yet guided them in a different direction by posing more questions for the students to think about. Once the time was up the students were then asked to present their findings to the class and explain what, why and how they did what they did. Once all the groups had presented the teacher followed it up with a closing of the topic. They explained what non-linear graphs are and how they are important to know about. They told the students about maximum and minimums. The last thing the teachers asked the students to do before their time together came to an end, was each student did a quick write. Their version of quick write is each student writes a question that they either still have about the material or a question they have about the material they learned and how it is related to future learning. The very last part of this reading is the teacher's and their self-reflection. They reflect on the on going thought: is it worth spending the extra time with your students to do an inquiry lesson? They then go on to say: yes it is. Throughout their later lessons for this topic they felt that their students really remembered and knew the information better than if they would have presented it in a different manner.  

Reflection of Journal 

I found this journal to be very interesting. I had learned a lot about inquiry skills and how to write an inquiry lesson for science topics, but I had never looked at it from a math standpoint. I really liked that the teachers picked an inquiry based problem that was not only relateable for the students, but also a current issue with their fundraiser. I believe that this allowed the students to be dedicated to finding the answer because they wanted to figure out how to help save their fundraiser in real life. I also really liked that the teachers stated that they picked the problem they did because it can be solved in a variety of different ways. This allowed the student to make sense of the problem in whatever way works best for them and their group members. Another aspect that I really like what that they allowed each group to present their findings to the class. While presenting the students had to explain why they did what they did. The reason I liked this was because when students are able to explain what steps they took and why this demonstrates higher level learning, which is usually closely related to a better retention level as well. I also really liked that the teachers explained how if a group was unable to reach an understanding or conclusion while presenting they would "parking lot" the idea and come back to it later at the end of class. Lastly I never would have thought to do a quick write or an exit slip for math class, but I really liked how they used them. They just asked the students to write a question that was still on their mind about the topic they worked on for the day; all the other quick write I have seen require more that the students write more than one sentence. However for a math lesson that allows the teachers to see how their students feel I think that the way they did the quick writes was perfect. This is something I will use with my future math students someday.       

Article Related to Standard: Look for an Express Regularity in Repeated Reasoning

Why Play Math Games? 

http://www.nctm.org/Publications/Teaching-Children-Mathematics/Blog/Why-Play-Math-Games_/ 

Summary and Examples: 

- Games helps students develop reasoning and and understanding 

- They create engaged learners in a positive learning environment

- Allows for the students to explore other math patterns, values and concepts 

- Guide students towards computational fluency 

- Games support a School-to-Home connection. The parents can play the games at home with their child to help improve the child's skills. 

- Group games can allow for the teacher to walk around and observe the students' methods and reasoning. She can take note of each student's progress and ability level. 

- Fluency requires a balance and connection between conceptual understanding and computational proficiency. 

- Drilling students with tests and worksheet all the time will not allow them to learn the skills as well as using a math game. 

- However games without efficient strategies being used are useless and should not be used in the classroom. 

- Be cautious of games that would cause the students to memorize the answers and not develop skills and concepts. 

- After playing the game have the students complete/do some sort of reflection. 

Important Ideas about Standards

4: modeling with mathematics 

- model real wold applications to mathematics 

- choosing and using appropriate math methods and skills for the problem at hand 

- visually representing mathematic questions, reasonings and solutions 

- engaging students with a variety of different of modeling problems 

8: look for and express regularity in repeated reasoning

- patterns in mathematics help to develop students' reasoning 

- helping students to find efficient methods to problem solving 

- developing regularity in calculations can help students recognize irregularities due to mistakes.