Manipulative Reflection

Manipulatives offer students a way to physically see and grasp mathematical concepts in front of them. They are a beneficial tool for teaching new skills and also creating a deeper understanding of previously learned knowledge. The use of manipulaitves can be helpful for all students, specifically those who prefer the use of hands-on learning. In math methods we have used manipulatives on several occasions. Some examples of manipulatives activities we have done in class are the use of geoboards, snap cubes to explore volume and surface area, and the use of yarn to measure. Deciding what manipulatives to use in order to create the best possible understanding can be a difficult challenge. Students develop their understanding by seeing the information they are learning about first hand and demonstrating it on their own. Students have the opportunity to rationalize math concepts, discuss, and describe what they are seeing in front of them. Through description students begin to understand the mathematics and what the concepts mean versus simply memorizing facts. 

Students should be given different assessments and chances to demonstrate the knowledge they have gained through the use of manipulatives. When a student can demonstrate the skill in several situations, his shows that students have truly assimilated the knowledge. In order to assess the growth students can be given pre-assessments as well as an assessement after the use of manipulatices. 

When determine how to determine the work of all group members, students should use reflection, as well as, self, and peer evaluations. To check and assess individual understanding group members can be required to fill out an exit sheet. 

The use of manipulatives helps improve problem solving skills because students are given something tangible that they can see. They can relate the mathematical terms and problems to something real and hands-on. This provides students with the opportunity to truly understand the problem step by step. 

More Errors Reflection

The error analysis activity was really an interesting activity to look at. It piggy packed on some of the same ideas we discussed previously in the NAEP project. Often times it was difficult to understand what the child was doing incorrectly without having them there in front of us to ask, "where did you get your answer?" Many times it was very difficult to tell what pattern the students were following. As a teacher I could see myself overlooking many of these patterns and simply marking them as incorrectly and it is easy to me to see why these types of process errors often go unnoticed. Many teachers do not take the time to stop and truly analyze where the students is coming up with these solutions. It shows how important it is for us as future educators to be vigilant of our student's work. Students expressing these kinds of patterns and errors in their work would have to be re-taught those math skills, it would be just like them learning a whole new math concept and have to forget everything they thought they had previously learned. One other thing I found interesting about this activity is our difficulty expressing in words what the student was doing wrong. We could identify the problem but then have a whole new challenge of putting the error into words. Sometimes, that was an extremely difficult process. To us, what these students had done incorrectly made no sense, we had to change our way of thinking in order to do so. However, in the students mind the correct way of solving the problem was the incorrect way. I imagine that a student being told to change their way of thinking in order to find the correct solution would feel similar to them. I think that the difficulties and challenges we felt when learning to do math their way is something we should all make note of and remember when teaching a student that, 'just doesn't get it.' 

Journal Review 3

Teaching Children Mathematics

Capturing Thinking on the Talk Frame by: Tutita M. Casa 

This article discussed the importance of talking in the classroom in order to help students engage in discussion, enhance reasoning, draw conclusions, and improve the mathematics classroom community. For those who struggle with analytical concrete thought, it is easy to see why discussion would help improve their understanding. By giving students a way to talk through their thoughts and share the viewpoints of others mathematics is become more relatable and easier to assimilate. 

One negative component of moving towards a more discussion based classroom is the possibility of teacher losing control of the discussions. Talk can begin to drift off topic, become too loud, or even be on topic but students begin to accept misconceptions and incorrect answers from their peers. Due to these possibilities this article was designed to create a 'talk frame,' a layout for teachers to follow when pushing their class in the direction of more discussion in the classroom. 

According to the article, a talk frame captures three portions of evolving discussion: Think, Talk Ideas, and We Understand. The Think portion consists of: students contemplating what is being asked before they solve the problem. Students rework what the teacher has told them to change the objective of the lesson in to an answerable question. Example- What do they want to know? What should you explain? Then, how would you ask this same question using your own words? In the Talk portion: The teacher first addressed common misconceptions with the students as a whole. In this particular lesson the teacher was addressing triangles compared to chevrons. She created with the help of the class a working definition for both triangle and chevron that highlighted their differences. If their is a discrepancy in a child's thought the students then regrouped and discussed it in partner pairs. After the students discussed the group came back together as a class to talk about what they had decided. When the class together reach a clear and correct consensus they have reached the We understand portion of the talk frame. 

I think the idea of moving mathematics towards more discussion based learning is a great idea that would truly benefit many students. Some analytical information associated with mathematics is hard for students to grasp, by incorporating discussion and talk, students will be able to take a more in depth look at a concept and explore different ways of thinking. I think the article is correct in the idea that teachers must have a framework they follow when incorporating this kind of 'talk' it would be very easy for the number of student misconceptions to multiply in this type of setting. I liked the way this framework incorporated group work but made the whole class discussion home base. In this way students will begin to become more confident sharing their opinions. One thing I think is important to note is this is not something that can be done on every concept every day. This activity has the potential to take quite a bit of time and if it were to be used on every lesson students would get through maybe one specific portion of a larger concept in a single day. The quantity of knowledge learned would go down but the quality of the child's understanding would increase. 

Mathematics Teaching in the Middle School 

Mathematical Explorations: Interpreting Box Plots with Multiple Linked Representations by: S. Asli Ozgun-Koca & Thomas G. Edwards 

This article talked specifically about an activity that used student's ability to use box plots and linking this with their ability to use higher-order thinking. This activity was compared in five sixth-grade classrooms. Students were asked to work in pairs, each pair had one calculator. Students were observed helping each other with the activity. What they found was that many students had difficulties reading box plots in general because it was not a skill that was extensively covered in many of the classes. They also found that when asked a question that could be found in the data, "which whisker is longer?" Students referred to the actual size of the data set (the whisker is longer because there are more values in the range)  instead of specific data points that determined the length. This types of misconceptions were common throughout even when the box plots were changed to histograms. 

I thought this was an interesting concept, but I did not feel that any real conclusions were drawn or that any real information was given. I found it interesting that students had several misconceptions when it came to reading the charts and data. I believe that that is probably because students do not ever really analyze what the answer they gather from the chart actually means. Students generally know what answer they are looking for on the graph find it and move on without looking at the information first, as a whole. Most have probably not covered analyzing charts and data extensively in their classrooms. I however, felt like the activity did not make a very clear link of student's box plot interpretations and higher-order thinking. From my understanding of the reading, students were too confused by the interpretation of the information to truly be able to participate in higher order thinking. I also found it funny how there was an advantages and disadvantages portion that really only gave disadvantages then the only advantage they gave was 'there are advantages and disadvantages to any representation of data.  Overall, I was not a huge fan of this article.  

Journal Review 2

Teaching Children Mathematics 

Can We Really Count on Frank? By: Jennifer Marston, Tracey Muir, and Sharyn Livy 

This first article was about incorporating literature and picture books into mathematics. The article identifies three different types of mathematical picture books, those with perceived content, those with explicit content, and those with embedded content. Books that fall into these three types are then assessed pre-assessed by framework that has been set in place to help guide teachers. This framework consists of seven different categories. The article then goes on to talk about how to use the framework through the book, One is a Snail, Ten is a Crab, Then talked similarly about components in the story, Counting on Frank. Importance of accurate math portrayed in stories is also a important factor in book selection. As a teacher it is crucial that students are not developing misconceptions about mathematics based on the materials we are selecting. 

I liked the overall idea of using picture books in mathematics. Using literature to teach concrete ideas can be a very valuable tool, especially, when working with younger students, the article left me wanting something more however. It began to feel a little repetitive after a while and I would of liked to hear more about how the stories were actually incorporated and a bit less about the framework. I also often felt like this article did not follow one cohesive thought. One idea would start to be developed and explained then something else would come up so that would be explained. I enjoyed the use of student work in the article as well in regards to the first book. I think this demonstrated the potential to not only use this particular story for teaching mathematics, but also the potential for many other picture books. 

Mathematics Teaching in the Middle School 

Using Aviation to Change Math Attitudes, By: Jerra Woods 

This article, discussed the implications of using high-intensity, interactive, flight simulations, and aviation to help teach mathematics. It begins by talking about careers in aerospace and the lack of qualified professionals in this field currently, leading to increased need for individuals in this field. Aerospace inside the classroom offers teachers the chance to add real life applications to their curriculum in a fun, new, and engaging way. The article explained how this program introduced several lesson plans to students at the middle school age level. These lesson plans included the use of reading linear graphs and their equations, how to copilot an airplane through simulation, reading flight instruments, graphing, data gathering, and processing. At the end the article gives resources to teachers about how they can incorporate aerospace and aviation into their lesson plans, where to get different ideas, and how to gain access to simulation technology. 

I really enjoyed this article and I thought it gave a cool new way to teach linear graphing and formulas. As a teacher, I think I may find a way to incorporate a lesson or two of this nature but I do not think I could maintain a unit or program of such extensive detail. I think although the activity is of high-interest it is not high-interest to every student and many students may have a hard time connecting it to their own life. Overall though, I thought it was a really cool idea and I think the idea of using simulations, not just for flight, in the classroom has potential to be highly valuable. I also thought this article was great for giving teachers so many resources and options to adapt this idea into their own classroom. 

Group Content Standard Textbook Work: Geometry

For this assignment my group was given the task of analyzing several textbooks used currently or previously to teach the grade level concepts of geometry. We were each given a range of grades k-8 to find textbooks for and analyze different aspects of the text. After looking at textbooks across grade levels from my content standard textbook group mates, I have established that their are several good aspects of many textbooks, as well as, many faults in the texts.

Starting with grades k-2 that I analyzed. The text I look at for the kindergarten level was very good at teaching the concepts and several skills related to identifying shapes, however, it did not give much real world application or application problems for students to use to help assimilate their newly learned skills and concepts. The text I looked at for the first grade level covered little to no geometry skills. The text for the second grade level was the only text that really touched on application in the content of the text, as well as, concepts, and skills.

At the third to fifth grade level, there was similar difficulty finding application within the texts. There was only two texts out of nine texts for this band of grades that were viewed as textbooks that were completely irrelevant to our standards. Most texts covered some but not all of the areas of geometry. Some were stronger in areas of symmetry while others focused on graphing.

The final grade band sixth through eighth grade showed much more evidence of application within the textbooks at these higher grade levels. However, at the seventh grade level many concepts were not addressed in the texts and did not differentiate between 2D and 3D shapes which is mentioned in the standards for this grade.

Overall this assignment showed the importance of reviewing the textbooks you choose to incorporate into your classroom. If teachers are not given the option of choosing their own texts it at least shows the importance of predetermining what will need to be covered within supplementary texts and activities within the classroom.

Video Blog Two

Just like the first video the amount of planning put in to the actual lesson plan about word problems was extensive. I always appreciate the way he videos really portray that the teacher has an invested interest in their lesson plans. This video was a nice one to watch after we had completed the NAEP project. The idea, that the teacher could not always understand what the student meant without having them directly in front of her, was something we struggled with as well in class. In this instance however, the teacher also found that when asked about their work her students also had difficulty verbalizing what exactly they were trying to show with their work, which is something I found interesting. The students struggled with decoding clear meaning, both in their answers, and also in how they read the problems. This brought the teacher to her goal hoping to have someone be able to pick up one of her student's work and not have to ask questions. I also had an easier time following this teachers initial train of thought than I did in the first video.

I felt that this lesson was extremely well done, and once again I think a lot of the success of the lesson comes down to planning. There is no way to plan for every situation and the lesson obviously did not go exactly as the teacher had originally planned but she was prepared enough to adapt and improve as she went. She made appropriate use of whole, partner, and individual work time. I liked her use of other student work, it felt more appropriate than the first, however, I still am not sure how I feel about using a student who is currently in the class even if they do remain anonymous. I always wonder about the child who realizes the work is their own and how it alters their classroom experience. When other kids look at student work they do not always choose sensitive language. Not because they are trying to be malicious or hurtful, but they're kids. They often say the first thing that comes to their mind. The teacher did a wonderful job of leading discussion and she really helped support students as they explored their reasoning. The students did a great job of identifying areas that should be improved in the work they were analyzing. However, I did notice that students continued to struggle with the word, 'more' often meaning the use of addition, they had a hard time seeing how, 'how much/many more' often means the use of subtraction. Overall, I felt this lesson supported the use of critical and higher order thinking skills.

In the teacher debriefing the teacher noted how she felt the lesson plan went. She addressed how much she felt students had improved in their execution of reading word problems as well as, students simply being able to admit they struggle. Students admitting they have a problem will make it easier to work on specific skills, and finding out where students struggle in the future. She also talked specifically about why she choose the student work she choose. I felt like this just goes back to the idea of planning. She realized the similar problems students commonly had in her classroom and choose to select problems that would allow her to address these issues head on with her class. She also talked about the student misconceptions that remained and recognized that her students still struggled with word decoding in their initial reading of word problems.

Overall I enjoyed this video. To me the video demonstrated a good example of teacher planning and performance. This video was also very appropriate for the information we have been talking about recently in class. It helped me transfer the information about analysis of student work from a standardized testing format to work in the classroom setting and how I might approach word problems with my students. It was easy to say in class if I had the student in front of me... But this video just demonstrated that sometimes even when the student is right in front of you they can not clearly convey their way of thinking.

Journal Review 1

Teaching Children Mathematics

Using Technology to Teach Equivalence by: Rochelle Goldberg Kaplan and Sandra Alon 

This article was all about technology and specifically the importance of knowing how and when to use different technology appropriately. This article talked specifically of different examples where technology was used. The main use of technology that was introduced was using a pan balancing activity online to teach the concept of the "equal sign" as a meaning of showing that the equal sign in fact means equivalency and does not just signify the end of a solution. the program was first introduced to teachers as a partner pair in which they were given a short time to explore the activity. After that as a whole group the teachers addressed the activity and what they found, along with what problems arose initially. After that the teachers regrouped in their partner pairs and try to balance the pans. This article gave examples of how teachers may implement a similar strategy into teaching their students using this activity in a similar way. Teachers put into practice the skills they learned and tried them on elementary students. Then the teachers reflected on what they thought about the experience and whether they believed the students were impacted by the lesson. 

Overall I liked the initial idea of talking about how teachers should be familiar with the technology they're asking students to use. I felt like there are probably better ways to teach the idea of equivalency and I also wished the article went a little farther to talk about the uses of technology and less specific about this particular lesson. I thought students might be helped to grasp the concept because of the interaction and visual representation but any one will learn any type of content better given those supports. I didn't find anything extremely profound about the article most of the information was pretty standard. 

Mathematics Teaching in the Middle School

The Engineering Process in Construction and Design by: Melissa A. Stoner, Kristen T. Stuby, and Suzanne Szczepanski 

The second article I read, was pretty interesting and took the concept of using high-impact activities like designing skate or amusement parks to teach engineering principals which is are usually thought of as more challenging and dry skills to teach. This sort of thinking uses things that are fun and relevant to students to peak their interests in a topic that students may typically shy away from. Students through the use of demonstrations were asked to develop thoughts on how math concepts they had learned might be turned into practical application in the building of high-impact structures. Students were asked to translate their ideas through a scale model project. Where school could make a scale model of the school and a nearby skate park. Students were not given specific guidelines about how big their school should be but were given a specific piece of cardboard that was 3ft by 3ft in which it had to be built. From there students had to decide how large that piece of cardboard was representative of and create the size of their school based on that scale. Students similarly used finger skateboards and full size skateboards to determine the ratio of the finger skateboards to full size to complete their scale of the skate park in comparison. The students also explored ferris wheels and carousels and talked about how relative size changed relative motion and the speed at which the rides would be moving at different scales. 

I really enjoyed this article I thought it gave great ideas for classroom implementation. This would definitely be an activity that would engage students in the process of engineering. This kind of lesson would be wonderful for children at the middle school age because they are often so difficult to engage in the subjects of math and science. Choosing high impact activities give the children a relation to these concepts and help them make connections to their life outside of the classroom. The teachers seemed to have good results by doing this activity and probably managed to intrigue some students who would have never thought they were interested in the engineering process. Hopefully theses students would continue to be interested in engineering throughout the years and possibly even consider studying or making a career out of engineering in the future. 

Math Applet Review

Geometric Solids

http://illuminations.nctm.org/ActivityDetail.aspx?ID=70

The first applet I reviewed was Geometric solids. This applet is an interactive tool that students can use to explore the different properties of 3D shapes. Students have the capability to rotate the shape and select/count number of faces edges and vertices. Students may enlarge the shapes and also change the color of the parts of the shape. This tool would be useful for students both when they are becoming familiar with the terms faces, edges, and vertices, but also when students are becoming familiar with specific types of shapes like cubes, tetrahedrons, and dodecahedron. As well as the shapes in 3D students can choose to view the shapes as a net, or basically 'the unfolded version of the 3D shapes.' There is also a tool on the applet that allows the student to create their own shape net.

This tool would be useful for varying ages of students and is also great for students to explore at different levels of understanding 3D shapes. This applet would be fairly easy to incorporate into the use of a lesson but I do not believe the applet its self could be the center of a entire lesson simply because it is not very complex and does not take much time to explore the applets full capabilities.

Tessellation Creator

http://illuminations.nctm.org/ActivityDetail.aspx?ID=202

The next applet I chose to review was called, Tessellation creator. Tessellations are shapes that combine over and over again in order to form a pattern. The use of tessellations are useful because they help students better visualize how different types of shapes have different size angles hence why certain shape combinations do not fit snug together. The use of tessellation is a creative concept that allows students to explore shapes in a different way. This type of activity would be beneficial for students who are creative or in to art and would help get them engaged.

This applet let students choose several different shapes (triangles-dodecahedrons) in an attempt to fit them together to form a tessellation. The shapes can be rotated copied and morphed in shape in order to make them fit together the way the student wants. Eventually the end product results in a colorful tessellation. I remember doing a tessellation of my own in 8th grade but we used our own original shape or object to create a tessellation. This type of applet would be a great way to introduce that kind of project or be used in place of physically building a tessellation on paper. This applet has the potential to be really engaging for students but I wish it had a feature where you could create your own shape or import a clipart or photo to experiment with tessellation in a different way.

Overall I like the idea of using math applets in lessons and trying to incorporate the use of these interactive activities into math learning. Technology is something that really intrigues students and anytime you can convince students that the learning process is fun or like a game in anyway they tend to be more willing to get on board.

Looking at PBL

https://www3.imsa.edu/

I have used PBL time and time again from the perspective of the student, but admittedly I have never taken the time to truly consider the amount of work a teacher puts in to developing an enriching, well put together PBL. I never quite understood why teachers choose to use PBLs when at a glance they appear not to cover that much content. Now I'm beginning to see that PBLs have the capability of pushing our students to cover a much wider range of content in their own way! This type of learning does not follow the cookie cutter mold. It is meant to be challenging, thought provoking, hands-on learning that can not be memorized and repeated back. There is hardly ever a cut and dry solution, and that can leave students feeling frustrated at times. 

PBLs can be unorganized at times but thats okay! They're suppose to be! Students are going to be met with a lot of challenges but that's one of the great things about PBL. These challenges lead to a million different solutions. Students have to use their skills of reasoning and decision making to decide which solutions will benefit their intended outcome for the better, and which for the worse. It also promotes team work and shows students the benefits, and the difficulties, of working as a cohesive group. Developing these types of skills are really what every teacher is hoping there students develop by the time they complete their education. These kinds of skills are relevant real-world skills that will help students in the future work force to become contributing members of society. 

The Sakai resources did a nice job of summing PBL up at a glance and IMSA laid it out in a little more detail. the IMSA was well organized, clear, and easy to navigate. This is a great site for teachers who are considering doing a PBL, especially those doing one for the first time with little exposure to PBL in the past. I enjoyed the presentation style format, and the continued message of Real-world problems creating Real-life connections. I realize that there are those out there who know very little about PBL and I feel very fortunate that by the end of my college education I will have had the opportunity to look at PBL from several different angles. 

After viewing the examples and looking more at the websites it is easy to see why some teachers may choose to avoid PBL. The amount of time and effort it takes on behalf of the teachers can be overwhelming. Having the ability to create an enriching PBL that is meaningful, creative, effective, engaging to students is a lot of work. To me however creating a PBLs benefits far outweigh the amount of time and effort that is needed to be put in. If a teacher is really concerned about the amount of time PBL would be a great thing to collaborate on in developing with your teaching team. Teachers in the same grade and content area could help in the development of a PBL. Even those in different content areas could help create a PBL that was cross-curricular. As teachers it isn't really fair of us to ask our students to work and develop a solutions in teams if we can't do so ourselves! Teachers also have the capabilities of using outside resources when developing their own PBL. Ask other teachers about PBLs they may have done in the past, look at books, and even look online to find resources. 

A look at 2 PBLs

Both the PBLs I looked at consisted of many of the same parts but were very different in their effectiveness and final product....

The first PBL I looked at was from the 6th example and was called, Safety in Community Parks. I felt like the general idea of this PBL was well thought out. This problem would be a very relevant real world problem for students in the Peoria area. However, I felt that the mini lessons lacked relevance to the problem at hand. For example, My understanding of the budgeting lesson left me feeling like the students were not actually being taught how to budget but more of the importance of it. This lesson also discussed whether or not a student had ever owed anyone money, in general I felt like many of the guiding questions were given little thought and were simply there as space fillers. I felt similarly to many of the standards they selected as well. For instance, the same budgeting lesson had an NCTM standard from Geometry which said students would be applying transformations and using symmetry. Nowhere in the lesson did I find where that standard was actually being put to use. For their adaptations for special needs they did not state which special needs category was directly being helped and their adaptation simply stated they would provide clear instructions about how to create the budget. As a teacher would you not already be trying to provide clear instruction for all of your students? In my opinion this was in fact not an adaptation and gifted learners were just given extra work. It was apparent to me that not much thought was put into this component throughout the PBL. For the second mini lesson I did not see how creating a scale model of the classroom was aiding the students in their planning to solve the issue of park safety? Overall I felt like the PBL guiding questions were a bit of a stretch and did not necessarily help to guide students in the development of their solutions. I also had a hard time finding the relevance of why they choose the mini lessons they did. There did not appear to be much thought out into the intended outcome and providing the students with the tools and background knowledge to develop an appropriate solution. Although the final product did appear to contain all the main components, there was a lack of cohesiveness throughout the PBL and left me feeling like the authors of this PBL rushed to put it together. The presentation of the PBL was very professional looking however and easy to navigate. Math seemed to be a focus but the way the PBL was presented made it difficult to understand where students would be applying math into their final solutions. 

The second PBL I reviewed was Combating Literacy through Improvements to the Community Library. Immediately the first thing I noticed after reading the Rationale is that there was a lot more effort placed in to writing the rationale for this PBL. When I first read the problem I found it less relevant than the first topic but after reading the groups rationale I found it to be even more meaningful than the first. One comment I would like to make however is that the focus of the initial problem became a little confusing at times as to whether making improvements to the library or improving literacy was the real problem students were combating. I know that the hopes were that the improvements to the library were suppose to then increase the literacy but some areas where the problem itself was discussed were a little vague. As a teacher I would just want to make sure that the students realized that the students had a full understanding of how there changes could benefit illiterate community members from the very beginning and that there decisions on improvements should take this into account. The guiding questions in this PBL seemed more relevant to the topic and would actually serve their purpose of guiding students to think of different components they may overlook when finding their solutions. This group also went as far as to include the guiding questions for the mini lesson and journal prompts. I'm not entirely sure that step is necessary since the guiding questions are meant for the students to look at during the course of their project it is good that they at least took the time to consider how their mini lesson would be guiding learners, something I felt the first PBL lacked. The adaptations felt well thought out and did not include just placing learners with disabilities in groups or just giving gifted students more work but more enriching and challenging work. The only issue I had was with their adaptation for those with ADHD/ADD too me, making a list of transition to keep the student, "focused," does not meet the students need. I would instead suggest a visual aide for transitions that the student could view independently. Depending on the students individual needs the student may also be provided with a fidget toy or have the ability to stand/walk in a specified area when was needed. Rarely is the lack of focus associated with those with ADHD/ADD due to a disinterest in the topic, but rather simply a lack of stimulation. This group also provided a clear daily schedule which would be of benefit to the teacher, as well as the students. Overall this unit felt like it had a better flow to me, and it also appeared to be better thought out. The guiding questions, standards, mini lessons, and adaptations all showed that genuine thought was put into their development and the total PBL supported the students on their journey to create their independent solutions. 

Video Blog Number 1

At first I was somewhat confused what the teacher was trying to accomplish with the button project until I actually saw it in progress. You can tell the large amount of work into planning this lesson for his students. I liked the concept of a re-engagement lesson and I feel like this teacher did it in a really good way. I feel like as a teacher you need to be careful when deciding to use re-engagement. You do not want to run a subject into the ground by over teaching it and at the same time you have to make sure the timing of when you reintroduce a lesson is appropriate. The students still have to remember the general idea of what the lesson was but not every detail. When used correctly re-engagement provides teachers with the opportunity to uncover deeper meaning to previously covered topics. In this lesson the teacher wanted students to use their patterns and adapt them to graphic organizers. I liked how hands on this activity was and I also enjoyed how students had to use their own logic and reasoning to form an outcome. I did also find the input/output activity to be a interesting activity that required a lot of thought and logic from the students, but as far as connecting to the button project I didn't see how that worked. Both required the use of student logic, so on that level it relates. To me however, the button project was much more hands on.

It was cool to see the teachers planning and hopes for what the students would accomplish with this activity come to life. There was a lot of discussion happening amongst the students. The discussions appeared to be really meaningful to the students as well. During the input/output activity it was awesome to see the little boy figuring out the pattern and wanting to share his thoughts with all of his classmates around him. When comparing the two learners strategies to solving the same problem I also appreciated the amount on analysis and understanding that was occurring. Students were really attempting to determine why the learners chose the route to solving the problem that they did. There was a ton of higher order thinking going on and the discussions were all very student led which I loved.

While watching the teachers debriefing there were certain aspects I agreed and disagreed with. The teacher felt his introduction was too long, and that the students were not understanding. Like I stated earlier the introduction was an interesting activity but I did not see it's relevance to the button project, that would be something I could do without. I noticed in watching too that it felt a little long and drawn out. I however, did not feel like the students did understand but more that the instruction was a little unclear. There were obviously some students who did understand as you could see in the clip.  It was a good idea in theory but I think it didn't work as well in practice as he had hoped. There was also a portion where he made it seem that the students had difficulty describing when it came to learn B's solution and I would have to say I disagree. There understanding and descriptions came from their own perspective and their answers were not necessarily incorrect but creative. The students to me were very clear about what they were trying to convey.

Reflecting on, "A Case of Units"

For my reflection I selected the article, "A Case of Units," by Christopher Kribs-Zeleta and D'lynn Bradshaw. I spent a lot of time searching for an article I thought really expressed the mathematical practice of, attending to precision. After searching for quite some time I stumbled upon this article which I felt expressed the importance of precision in some of the earliest and simplest forms. 

As explained in the mathematical practices, math proficient students understand the importance of using precise language to communicate their thoughts, ideas, and quantities. Without the use of labeling, clear direction, and units much information would be lost in mathematics. Suppose a student were to say 1 is larger than 5. Automatic response tells us that this student would be incorrect but what if we now added the units of pounds and ounces, 1lb is indeed larger than 5 oz, so in this context 1 is in fact larger than 5. 

"A Case of Units," Explained the challenges and importance of introducing the principals of precise language at an early age so that students begin to incorporate the habits of using exact communication into their everyday lives. In this article three students had been drawing snakes at recess their teacher wanted them all to devise a way of measuring their snakes on their own. Some of the students used their entire body lengths, others used estimated marks, while others used feet to measure their snakes. Students were surprised to find that although they could clearly see that one snake was much longer than the others it was reported to have a measure of 7 when the next smallest snake had a measure of 20. What the students failed to realize when they reported their measurements was that their measures lacked units and therefore could not be realistically compared on the same scale. 

I enjoyed this article because it took the idea of precise communication and the importance of using it down to a very basic level. I thought it was a great activity to use when talking about the importance of units and appropriate language in math and could potentially be used as a future activity for introducing measurement into my own classroom one day. 

Citation:
Kribs-Zeleta, C., & Bradshaw, D. (2003). A case of units .Teaching Children Mathematics, 9(7), 397. Retrieved from NCTM.org

Understanding CCSSM Standards

The Common Core Standards for mathematical practice provides teachers with a blueprint of teaching students to develop expertise and confidence in their skills, while providing students with quality and meaningful learning experiences in mathematics. This is the first time I have read through the mathematical practices in great detail and chose to summarize what each practice meant to me, and how I believe teachers are able to develop these practices in their students...

1) The first of these standards, Make sense of problems and persevere in solving them, calls for students to create a total understanding of mathematical problems. Students assess a problem using various mathematic skills before attempting a solution. Automatically, students have the ability to rule out several routes and outcomes to a problem based on their knowledge of several mathematic components. Students who demonstrate such proficiencies will posses the ability to prove and disprove their solutions using several different approaches.

2) Reason abstractly and quantitively, means students have the capability of picking numbers from a problem (story problems) and see how they are, or are not relative to helping them find a solution. Students can take these quantities and use them to form mathematical equations, and knowing how interchanging numbers within the equation changes the solution, as well as what these changes mean. 

3) Next students ability to, Construct viable arguments and critique the reasoning of others, means proficient students can use assumptions and solutions to creating logic and form meaningful opinions based on information. Students also have the ability to analyze and refute other findings and opinions based on their analysis of information by comparing different sets of findings and results. With this skill students develop the capability of establishing truth from fictitious, or incorrect findings. 

4) To, Model with Mathematics, Students must use their understanding of mathematics to address day to day problems that arise. This may be directly related to school and homework, but may also apply to planning and organizing events. Students constructing a model using mathematics have the capabilities of seeing which mathematical principals will be most beneficial in aiding their success. They also have the ability to use hindsight to see how these skills may have been perfected or improved upon in order to make adjustments to future problem solving. People often times use this skill without thinking because problem solving is deeply intertwined with day to day living. 

5) Use appropriate tools strategically, students must assess the tools provided to them in any given situation and decide which tools to use, and in what ways can they be best assembled to achieve success. Proficient students are responsible with their tool selection and know what tools will best fit their needs in achieving their desired outcome. Students posses the understanding of how tools may not only enhance but also limit their final product. 

6) Students are able to, Attend to precision, when they understand the importance of communication. These students realize that information is lost in the details and that careful communication can prevent confusion. Students using precise language are careful to include units, and direct clear instructions when communicating information to others. 

7) Look for and make use of structure, means that students are trying to categorize and find patterns within mathematics. Categorizing and making connections is a natural part of human learning. From a very young age children learn to assimilate knowledge by creating schemas that find likeness in the world around them. Similarly, when children begin working towards proficiency in mathematics they find similarities between current and previously found problems when searching for solutions. As children progress in age, they grow from simple shape and number patterns, to understanding when to apply certain theorems.

8) Students will, Look to and express regularity in repeated reasoning, similar to making use of structure, students are trying to understand mathematics by making connections. At the very basis of mathematics children will learn to find understanding in the quantities being expressed in their work. An important part of gaining this understanding will come from students abilities to recognize repeated calculations. Division and fractions are good early examples of this process. Students begin by realizing that division is the repeated process of separating one larger quantity into small equal quantities. For example 56 divided by 7, students realize that the same calculation is being completed over and over again in order to fin a solution. This kind of oversight allows students to check their calculations as they go and know precisely where an error has occurred and the problem may then be reevaluated.

Citation:

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