Teaching Children Mathematics - Polygon Properties

Summary:

This article focuses on teaching elementary students, grades three through five, the properties of various polygons. The article starts by introducing the activity and explains why it is appropriate for the grade level at hand. It also goes into explain the standards and how the activity is associated to them. The activity gives students a group of many different polygons, both regular and irregular, with various numbers of sides. Students are to sort the shapes by number of sides or other contributing factor. The article explains how the activity can be adapted for other age levels and goes into detail explaining why it is important to teach properties of polygons at this age. Students were able to apply this activity to real life situations, for example, eating crackers that are quadrilaterals.

Application:

Overall, this is a good activity for students and it can be very beneficial when learning shapes and classifying objects. I would use this activity on students of a similar age because it seems appropriate, especially with the explanation of the standards involved. I think this activity is great but can be pretty short in a classroom. This can be used for measurement skills as well as, students can measure length of sides and angles of regular and irregular polygons. The article left out ways to apply this activity to other parts of math and overall seemed slightly repetitive. I feel that this activity should be a small activity of many put together, I would have liked to hear about more examples.

Rodrigue, P, & Robichaux, R. (2010). Polygon properties: what is possible?. Teaching Children Mathematics, 16(9), Retrieved from http://nctm.org/eresources/view_media.asp?article_id=9290

Manipulatives Blog

1. How do you hold all children accountable when using manipulatives for an activity?

When working with manipulatives during class time, teaches need to be aware of students actions and behaviors. Teachers should be doing constant visual checks on students to make sure they are staying on task and completing necessary requirements. A good way of keeping students on task and putting them with a partner, especially one who isn't a buddy they could get in trouble with. This will keep students focused on the problem at hand rather than playing around with something else. Teachers can also give students a chart to document their progress/data with the manipulatives. They can also ask students what they have learned whether it be in a journal or vocally to the class. Lastly, teachers can give students checklists to make sure they stay on tasks with the requirements of the manipulatives.

2. Why do people say not "hands on" but "hands on minds on?"

Manipulatives should involve much more than just being hands on. They need to connect with the student so they can take the knowledge from the manipulative and apply it to something else. This is where minds on comes into play. Students should be cognitively engaged with the manipulative, learning and discovering new things. If an activity is just hands on, students can simply be "playing" and not focused on learning how to do the tasks in front of them. No matter how hands on a manipulative is, it needs to apply to the students learning.

3. How do the process standards fit in with the manipulative activities?

After discovering more about specific manipulatives today, I was able to apply them to the process standards with more understanding. The manipulatives can expand students communication through working in groups and talking with other students about the manipulatives at hand. Problem solving is expressed through students discovery of manipulatives and their ways of working through them. Students use their skills of applying prior knowledge to learn how to solve problems when using manipulatives. When students are left to guess and check using manipulatives, they are using reasoning and proof. This can be done using blocks and discovering their attributes. Some manipulatives cover more than one content area when used. This connects many disciplines and helps students connect all mathematics as a whole rather than specific sections of mathematics. Lastly, students will use representation when working with manipulatives as they organize and record information to solve problems. Representing students work is shown through filling out charts or understanding the relationship between mathematical problems.

Technology Reflection

After participating in math methods for almost a semester now, I have much more knowledge on how to incorporate technology in the classroom. This class emphasizes the use of technology more than any other class I have taken by far. The Smart board is a creative and impressive piece of technology to use in the mathematics classroom. It can be used to do many different mathematical equations and procedures. I had no clue how to use one walking into this semester but, now I feel much for confident using it and feel that I could definitely incorporate it into my own classroom.

After using the math applets in class I found math a lot more exciting and actually fun. Math became more of a game rather than a chore. Also, we created wikis as well. I liked the wikis because it kept us organized and took my writing to a different level. Mathematics as a whole has completely changed and improved with the use of technology.

The Geometers Sketchpad is another handy tool for technology use in the classroom. I liked being able to learn geometry and discover new things about it by using the software. It never got boring and I enjoyed using it and wanted to learn. This is incredibly important in a classroom because students can pay better attention and learn geometry in a much more hands on and creative way. With these and many other technologies like Inspiration and the video camera software, I was able to learn just how important technology in the classroom really is. I also learned that it is becoming more and more dominate and should be adapted into lessons as much as possible.

Error Reflection

Looking at and understanding the errors in Math Methods definitely gave me a better insight on the many mistakes students can make. I realized that students will encounter all types of difficulties in mathematics at all different age spans. Also, students can make errors for many reasons other than the assumed problems of rushing or simply not paying attention. Many students combine rules from other problems or change rules to fit their needs. In a student's mind, these changes make sense because he/she feels like they are getting the right answer.

Teachers need to be aware of the varying needs of each and every student. No student is alike and there will be all kinds of mistakes made. Teachers need to never assume that a student is not trying but always help a student to see if they are making a simple error in work that can be easily corrected. In general, I feel a lot more comfortable with understanding the many problems that can go into a mathematics equation and ways to make students learn easier. I feel that all the hands on activities we did gave me tons of ideas to use in my own class. I especially liked the method Dr. Grant taught us on division. It simplifies the problem and makes everything easier to understand. I will feel comfortable implementing these learning tools in my own classroom.

Mathematics Teaching in the Middle School - Tech-knowledgy and Diverse Learners

Tech-knowledgy and Diverse Learners

This article explains the need for more technology in the classroom and the correct ways to use technology to the fullest. It went into detail about the effects on technology on students with special needs and English Language Learners. When using technology on students, it creates a great way of covering all areas that need to be addressed. Students get a visual and can create things without having to have it physically in front of them. This will really benefit students who tend to get system overload for multiple pieces of work to do. Also, technology in mathematics can be used as a self check when completing problems so students do not have to have as much confusion in the long run. There are numerous programs out there that are advanced enough to greatly benefit an entire classroom of students will all sorts of personalities and backgrounds. The article gave many examples of what to use and when to use it along with visuals.

This article shows math in a new light and really focuses on the future of mathematics. One thing I did not like about the article was that it discussed the problems with ELL students but never went into detail or gave examples of the technology that can help the students like they did for students with disabilities. I feel that this article opened my eyes to a lot of the things schools should be using now rather than worksheets and great ways to implement them into the classroom. I feel though that this article forgets that with all of the budget cuts, some of these opportunities and programs might not be available at every school. In general, I learned the importance of technology on all students no matter what their level in school is.

Suh, J. (2010). Tech-knowledgy and diverse learners. Mathematics Teaching in the Middle School, 15(8), Retrieved from http://my.nctm.org/eresources/view_media.asp?article_id=9197

Teaching Children Mathematics - Teaching with Cognates

Teaching with Cognates

This article focuses on the challenges students and teachers face when trying to facilitate for English Language Learners. It goes into detail about the strict guidelines that are implemented into the school which leaves teachers struggling to find ways to help students be more familiar with the English Language. Many teachers are now relying on cognates, words in two languages that share similar spelling, pronunciation and meaning. These cognates benefit students and tend to help seal the gap between the two languages and help students communicate more effectively. There are many cognates associated with mathematics teaching that can be used to help students progress in mathematics. Teachers can use word walls to help explain the relationship between the two words in the different languages so students have a better grasp on the similarities.

I really liked this article because it made me think about a topic that I am very unfamiliar with. I never really process the likelihood of having a student in my classroom with a different dominate language than everyone else in the classroom. I can only imagine the confusion of all parties involved, especially the child himself. Using cognates seems like it could be a huge break through fro students trying to master English. They could even be put into a card game like memory where students have to match the Spanish word to the English word or used on flash cards. The opportunities are endless but need to be instructed in order for students to achieve at their highest ability.

Gomez, C. (2010). Teaching with cognates. Teaching Children Mathematics, 16(8), Retrieved from http://www.nctm.org/eresources/view_media.asp?article_id=9233

Assessment Article - Journals and Portfolios: Alternative Assessment for Preservice Teachers

Summary

This article expresses the importance of alternative assessment and benefits. The author emphasizes how she uses journals in her own classroom. Journals can be used as open-ended responses or as prompts. Open-ended responses are rewarding because they can have a varity of answers. Students can explain what they have learned, pose more questions, or even look to the future. This is great because students can feel comfortable about their grade without having to fear a test or quiz. Journal prompts can occur anytime or day. They are when the teacher gives students a topic or question about students write on that specifically. This type of journal can show where students are in their learning compared to others and what they understand.

Application

Journals can be beneficial in any classroom and are easy to do. Some pros are that they can be fun and help students not only reflect on their mathematics class but also work on writing skills. These journals will also be great to have to look back on in the future. I feel that some cons may be that students are not going to be assessed over everything they know; just a broad overview. This could leave some students to fall behind in mathematics. In general I think they have a great purpose and help students be more confident with assessment instead of dreading a test.

Liebars, C. S. (1999). Journals and Portfolios: Alternative Assessment for Preservice Teachers. Teaching Children Mathematics , 164-170.

Mathematics Teaching in the Middle School - Shrinking Your Class

Summary

This article explained a great way to introduce seventh grade students to the concept of ratios. Students walked into their classroom to see it taken over by little toys scattered all over. These toys became a math activity! Students learned ration by scaling toys down to a smaller size to model them like an architect would a business or home. Once students got the hang of scaling down these items they got to scale themselves down. Students used a 1:20 ratio to scale down arms, legs, and torsos and document the measurements in a data chart. Teachers discuss how the activity was beneficial because it gave students a creative way to understand ratios and size relationship. One of the big problems was the time management skills of the students; not getting their models done in time.

Application

This activity is a really great tool to use for the introduction of ratios. I like it because it gets students interested in the beginning and they get to look past worksheets and learn in a creative way. I feel that this activity can be used to “shrink” any object in the classroom and shrinking the students is great. I would also use this activity to make objects larger in the classroom so students see both ways ratios work. Overall, this activity is good for discovery and introduction of a new concept. I cannot wait to incorporate it into my own classroom someday.

Herron-Thorpe, F, Olson, J, & Davis, D. (2010). Shrinking your class. Mathematics Teaching in the Middle School, 15(7), Retrieved from http://www.nctm.org/eresources/view_media.asp?article_id=9175

Teaching Children Mathematics - Paint Bucket Polygons - March

Summary

This article focuses on using the paint software on computers to teach polygons to students. Teachers formed a teach/study group to determine how students see and interpret polygons. Together, they created a group of lessons that help students understand geometric shapes and their differences. Fifth graders are introduced to the lessons by being asked, “What is a polygon.” Surprisingly, this had many fifth graders stumped and they needed more background to answer the question accurately. Teachers discovered that students were confused because of the definitions for polygons given to them. They tend to be too simple or too complex. To help with confusion, teachers introduced the polygons to a different class using the paint photo editing software on most computers. They created polygons and non polygons and asked students what would happen if they filled in each shape. Students were more able to visualize this method and determined what a true polygon was. With this software students can gain more knowledge on geometric concepts.

Application

This is a great idea to adapt to any classroom. It is a well thought out hands on approach that would be beneficially teaching many mathematic concepts, not just polygons. This method could be used to teach angles and parallel lines as well. In my classroom, I would definitely use this as a method of teaching polygons. It is visually appealing to students and helps them get involved. By putting the paint “into” the polygon students can realistically see if the paint makes it a polygon or not. This could also be used as an assessment tool to see where student’s level of understanding is about polygons. Although the paint tool is a very old program, some schools may not be able to afford much more and it is convenient since it does not require the internet. Paint is a simple tool which will also be beneficially for students when learning how to navigate through it.

Edwards, M, & Harper, S. (2010). Paint bucket polygons. Teaching Children Mathematics, 16(7), Retrieved from http://www.nctm.org/eresources/view_media.asp?article_id=9163

Video #2 Grade 7 - Lesson on Graphs

Purpose of the Activity:

This activity is a way of getting students to take graphs and create equations to the nth number. With this pattern, students will have a better understanding of how a graph works and what it can represent. Students will create graphs on anything they want as long as it shows a relationship. This is helpful for students so they can relate the math to something physical rather than it being abstract; like a graph can be. By the end of the activity, students learn the importance of graphs and ways to turn their graphs into tables. Tables help the students see their data more clearly and show patterns their data may have. This can teach students how the nth term works and why it is important.

3 Questions:

1. How do you determine whether group work is appropriate and effective?

I personally believe group work is most effective when students stay on task and do not get caught up in other conversations. Group work can be very beneficial when getting others opinions and relying on others for help in solving problem situations. I feel that the example in the video is a great way of using group work and why it is so important. Group work is best when students are actively engaged in the learning rather than letting others do the hard work.

2. Describe the student-teacher interactions during the task debriefing discussions and assess the effectiveness of these interactions.

The student-teacher interactions were very casual. Students asked questions with a purpose and the teacher was quick to answer. Students were challenged further by the teacher asking more questions to their questions. I feel this is an extremely effective method of discussion with your students. The students felt comfortable coming to their teacher for help and the teacher was good at challenging students but helping them at the same time.

3. Describe how appropriate you think the primary task in this lesson is for developing an understanding of the mathematics being taught.

I feel that the primary task of this lesson is the best way this teacher could have taught her students this concept of graphs. With this lesson, students are able to visualize something they are familiar with in their graphs. This makes the lesson more valuable and beneficial to the students because they can relate to their learning. Students then have a solid understanding of the mathematics behind the graph and are given a good base for learning more skills in graphing.

Overall Views on the Video:

I really enjoyed this lesson how helpful the step by step video is at showing it. I think it is a great way to teach the nth term and also show the relationship between graphs and tables. I like that students get to relate their mathematics to something they enjoy rather than given a worksheet to fill out. I think this lesson is appealing and helps students discover more than an average lesson would. I would definitely use this lesson in my own classroom because the students seemed to enjoy it and have fun while learning graphing concepts at the same time.

Math Applet - Investigating the Concept of Triangle and the Properties of Polygons: Making Triangles

Investigating the Concept of Triangle and the Properties of Polygons: Making Triangles - Geometry (K – 2) http://standards.nctm.org/document/eexamples/chap4/4.2/index.htm

Summary:

In this math applet students take what we all did as children on big wooden boards, and do it on the computer. This applet gives a virtual board with dots representing pegs on it. Students are asked to take the rubber bands on the side of the applet and to make triangles. They are encouraged to make many different triangles all of different shapes and sizes. Students can change the colors of the triangles and shape them in to make colorful geoboards. After they have made their triangles, they are to compare their triangles to a friend’s triangles and explain how their triangles are the same and are different. This lesson could introduce many other shapes as well as students learn more about polygons.

Critique:

I really like this applet. It is easy to use and fun to play with, even for me. If I were to use this applet in my own classroom I would not make it the main lesson. I feel as though students still need that object that can grasp and that physical geoboards are still a great way of teaching shapes. I think that combining the physical geoboards and virtual geoboards would make a much better lesson and give students opportunities to compare the two. This lesson is very easy to advance into learning about shapes with more sides and that makes this applet even more helpful. Students can advance to trickier polygons as they master them.

Math Applet - Spin the Big Wheel!

Spin the Big Wheel! Probability (3 - 5) http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=1015

Summary:

This applet gives third through fifth graders a very brief introduction to the way probability works. Students are told to “spin the wheel” to see what they win. They are also told the probability of getting a big prize. This resembles a carnival game that many of us see at a fair or picnic. Students spin the wheel to see what they get while a box on the side of the applet keeps track of what they land on and how many times. After several turns, students can compare results and see which part of the wheel was favored, or if they came out mostly even. Students can also put the data into a circle graph to see it more visually. This is a simply way to introduce probability and is a great way to start students at an early age.

Critique:

This applet has the potential to offer a lot and it has a lot of pros and cons in my opinion. Some pros would be that it is visually appealing for students. Students are going to be more inclined to do this sort of lesson because it is hands on and can become a game for them. They are learning and they do not even know it; which is the best kind of learning. Some cons is that this is a very basic explanation of probability; So basic that I feel it is good for third graders but might come off on the easy side for fifth graders. This applet could bring in more options to land on to make it more challenging. It could also be more interactive and ask students problems and equations rather than leaving it as is. Overall, the applet opened my eyes to the benefits this could add to my classroom to get students thinking about mathematics in a positive way.

MTMS Article - Rubrics as Formative Assessment Tools

Summary:

In this article I learned all about rubrics. Rubrics are one of the best formative assessment tools out there for classrooms because they benefit not only the teacher, but the student as well. There are four types of rubrics and all of these incorporate different approaches to assessing students. The different kinds are; holistic, analytic, specific, or general. Each rubric is explained of how it is created and how it is used, there are also examples listed of each rubric. Teachers gave great information on how they came to creating their own rubrics and the thought process they had to go through to get a well working rubric. There are two ways to make a rubric, by using students work or by using the assessment task. Both of these ways are helpful and can give the end result a teacher needs for assessment.

Application:

This article is very helpful for any new or upcoming teaching, including myself. At Bradley, I feel we learn a lot about different assessment tools but it is always nice to hear it from someone else's perspective. I think one of the most beneficial parts of the article was explaining how to create a rubric. I love the idea of making a rubric based on students work and their understanding of a task. It just makes sense to grade based on what students know rather than a correct set answer. I think I will use this technique in my classroom because it seems fair for all students. I feel more confident in my understanding of rubrics, especially the different kinds and what is put into each of them. In general, this will help me use rubrics to their full potential rather than plopping a letter grade on a students paper without an explanation.

McGatha, M, & Darcy, P. (2010). Rubrics as formative assessment tools. Mathematics Teaching in the Middle School, 15(6), Retrieved from http://www.nctm.org/eresources/view_media.asp?article_id=9096

Teaching Children Mathematics

Summary:

This article deals with preschoolers and there computation of numbers. Children work with an assessor on counting skill and comparing techniques. One on one students and assessors with count with cards and teddy bears determining amounts and corresponding teddy bears to dots. Teachers will have the student pull out however many bears they need to match up with how many dots they have on their card. They continue this process by adding another card and also comparing their amount of bears to the assessors bears. This type of skill is one that is important to assess at this age. Assessors ask questions and from students answers they are able to determine where they are in their counting abilities as well as their number sense. This type of assessment works for students at all ability levels and can be easily adapted to all students.

Application:

This article really helped me get a better sense of how to assess younger students. I feel that a lot of times when students are first brought into school they are at all different ability levels. This technique is a great way of figuring out what that ability level is and ways to help the student later. I feel that being one on one with an assessor is the best way to get correct results for a students number ability and it allows them to be the focus for awhile. I think this is a good technique for any classroom and can be incorporated into different age levels and content subjects. Applying this is easy and beneficial for both student and teacher.

Moomaw, S, Carr, V, Boat, M, & Barnett, D. (2010). Preschoolers' number sense. Teaching Children Mathematics, 16(6), Retrieved from http://nctm.org/eresources/view_media.asp?article_id=9130

PBL Samples Critique

PBL #1 Lounging Around - In the first PBL, students are presented with the problem of creating a new student lounge for the middle school. They are given a set budget and deadline to get it completed by. The activity becomes in depth throughout the process while students have to decide of fabrics and furnitures as well as creating advertisements. Students will create lists throughout the project and holding a debate for the cost and funding for the new student lounge. This PBL takes 16 days and covers many content areas.

PBL #2 Operation "Redo the Zoo" - This PBL is for 5th and 6th graders. Within it students face the problem of redoing the local zoo. Students get to decide what the zoo will look like, what kind of plants and animals are in it, and what building will be in the zoo. This PBL will take 15 days; during those days students will journal about their zoos and keep math logs for the money they spend. By the end of the project students will have a completed portfolio explaining all the work that goes into the zoo.

The Lounging Around PBL has a lot of strengths. I like this PBL because it states a problem that students can relate to extremely well. It goes into specific detail about what needs to be accomplished without getting to wordy and making it confusing. Everything is laid out in a good form that is easy to understand. One weakness I saw in my opinion is that some of the lessons could have held more detail. I felt that a few of them left me hanging and a little confused about what I was suppose to do with my classroom. Operation "Redo the Zoo" has a good general idea; students are interested in the zoo. I also feel that adding journaling is a strength because that is the best way to see students thoughts. One of the biggest weaknesses I see with this PBL is that it seems that there is a lot of unnecessary wordage. I feel like I am being told a lot but not really getting the entire point of the lesson or what the students are actually all going to be doing.

Overall, both PBLs will interest students at the grade level set. I feel like they both can be incorporated into many content areas and take about the same length in time. Also, both require collaboration and work everyday. Some differences are that Lounging Around is more realistic then Operation "Redo the Zoo. " Students could have the opportunity to help create their student lounge but it is a lot less likely that they can completely redesign a local zoo. Also, I feel that Operation "Redo the Zoo" went into a lot more detail than Lounging Around did.

An example of Lounging Around being clear and concise would be when the authors wrote the "description of the problem" it is very short and to the point making it easy for everyone to understand. An example of a lesson being to short is having a "work day" on day 11. I feel that's very broad and should explain better exactly what the students should be doing. In Operation "Redo the Zoo" journaling is mentioned when the authors wrote "Students will have a 15-minute time period to write in their journals about how they think they will apply concepts learned in the previous mini lessons, to their project." This is a great way to get students brainstorming. I see unnecessary wordage in the "brief description," it seems to be to long to be considered brief and many statements can be removed.

In both PLBs I feel that math is a main focus of the project. Students are given budgets and need to keep records of how much money they spend. They also need to measure various items and do algebraic math problems. The level of math expected from both problem based learning projects is right on task with the grade level it is presented to.

I feel that there is a wide range assessment throughout both of the PBLs I have looked over. Not only do the teachers assess the students but the students get to assess each other as well using checklists and rubrics. Teachers use rubrics to grade students which makes it easier for students to see what tasks they need to accomplish. Also, students are assessed both formally and informally. Like I have previously stated, I feel that parts of both lessons have clarity at different parts. Lounging Around has great clarity in describing the overall project and lacks it when going over what is going on each day. Operation "Redo the Zoo" is the exact opposite. It gives far to much detail in the overall description but makes a lot of sense when the daily processes are described. Both PBLs are great about making math more enjoyable. When realating the content to real life examples students are less likely to realize they are doing math and think of it more as an exciting project. This is a great way for students to effectively learn mathematics skills and it is incorporated into both PBLs.

PBL Article Review

I found the webpage "In the Classroom - Problem Based Learning" from the Center for Educational Technologies website. I was very impressed with the page they have set up for problem based learning. The information is very general, but it is great information for someone just starting to experience problem based learning. The webpage began by giving an introduction to problem based learning along with a simple definition. It goes into how a PBL works and what concepts all go into one. It then talks about what the teachers role is in a PBL and ways teachers can overstep their boundaries. Lastly, the webpage gave some great sites to look into for more information on developing a Problem based learning project.

When reading the information the webpage gives, I noticed that it is all very general. What they are explaining is great and very useful but, I think more information would have been better. There is a link to a PBL model which is very helpful, it creates a to-do-list for a PBL. I thought the learning listed made a lot of sense and was accurate. It says that the idea of problem based learning is to teacher students how to learn. Students learn meaningful information that is relevant to their life rather than just memorizing facts.

The strengths of the article include the way it is written and the topics it covers. The article is very easy to understand and provides valuable knowledge about problem based learning and how it strengthens a classroom. The article also covers both students and teachers which is beneficial to everyone involved. A weakness of the article would have to be that it does not provide as much depth as needed to fully understand problem based learning. It simply goes over the basics and leaves it at that.

Problem-based learning. (2002). Retrieved from http://www.cotf.edu/earthinfo/classroom/teachers/FTtopic1.html

PLB Reading/Review

Problem based learning focuses on hands on, student centered learning that involves real world everyday problems. It is a widely used method of teaching and when used correctly, it can be very beneficial for all involved. An essential for creating a PBL is to involve an ill-structured problem. These problems are messy and confusing at first, which is what students are trying to figure out. The problem needs to be create enough to keep students interested and get them hooked into learning about the problem.

During a PBL, students are given roles within their groups. These roles help to maintain students interest and also make sure every student is doing some work on the problem. In the end, the point of a PBL is to get students thinking at a higher level. To apply skills from all areas and apply it to a problem that students need to work through together. Students will work through a thought process and use the scientific method to determine outcomes and guesses. Students will also go through each step one at a time rather than looking for an overall answer.

Problem based learning is used in the classroom over any topic that is challenging to teach or for students to grasp. These problems can be done in groups to establish more peer collaboration and communication. In general, PBLs are important in every classroom

Prove It! Engaging Teachers as Learners to Enhance Conceptual Understanding -- Journal Article

Gathering teachers for a staff development workshop is a great way to get teachers thinking about the way they teach their students. In this article, teachers are asked to prove their answers rather than just telling the answer to a fraction problem. Doing so makes teachers think about the way they ask students to present their answers in their own classroom. Teachers need to learn to teach for understanding and discovery so students can be more comfortable with mathematics.

The activity presented to the teachers asked them to find the answer to a simple fraction problem. When asked how they got the answer they said "I found common denominators and multiplied." The facilitator then asked them to prove how that works; this is where teachers found the problem to be difficult. This opened the activity to other activities all throughout the mathematic spectrum. With these activities teachers realize the most important part of a math problem is discovering how to do it, not the answer to the problem. Learning for understanding is important to teachers also, not just the students. Teachers also need to reflect on their teaching experiences that way they have a better understanding of what to teach and how to teach it. In the end, learning is not a one step process its an entire circuit of questions, answers, and discovery.

Sweetland, J. and Fogarty, M. (2008). Prove it! Engaging teachers as learners to enhance

conceptual understanding. Teaching Children Mathematics 15(2), 68-73.

Reasoning and Proof - The Process Standard

The reasoning and proof standard asks students to be able to do simple things we all really do every day. Asking children questions throughout their mathematical learning is important step in helping them fully understand the problem that is set in front of them. They have to explore deeper and analyze all the evidence. Systematic reasoning is a building block of mathematics and is used at every age. Students learn a lot about mathematics through discovery. Everyone uses informed guessing without even thinking about it and students put it to good use when they apply it to examples and previous knowledge. Teachers can also feed of this guessing by asking students to recall back to their earlier math learning and expand their knowledge to a whole new level.

Once students have figured out the answer to a problem they need to be taught to look even deeper into that problem. Instead of asking for the answer ask how the problem works the way it does and if it always works that way. Given students a reason for math will make it easier on them to understand instead of knowing something happens "just because." As children get older, their arguments and ideas should get stronger. Students should work cooperatively as a class to develop strong mathematical reasoning and a good classroom environment. Lastly, as students get older, they need to be introduced to many types of mathematical forms and situations. With all kinds of different ideas they will become more fluent in learning math and why specific things happen the way they do.

Lessons on Variables - Video Blog

The purpose of the activity in the video was to get students comfortable with using variables. Students had to take variables that they use everyday, their names, and give them a number value. This helps students have a better understanding of what variables do in math and why we use them. The activity also served as a way to keep students asking questions and understanding why we do things rather than just knowing an answer and not being able to explain it. The teacher asked many questions that kept the students guessing and interested in getting the lowest/highest value for their names.

Questions
#1 - Describe how the teacher’s questioning, and the manner in which student responses are handled, contribute or do not contribute to a positive classroom learning environment.
#1 The teachers questioning and handling of responses were great factors in keeping a positive learning environment. When the teacher would ask a question related to the lesson, students would be constantly looking for an answer. She asked questions that were easy for students to understand and kept the lesson tied together. When listening to students answers she constantly gave constructive criticism or asked more questions to keep students moving and never getting bored with the lesson. She also got her students to ask questions and make predictions.

#2 Describe what the teacher does to support learning while students are working in groups.
#2 The teacher is always walking around and focusing on specific students to make sure they are focused and on task. She takes the role on on-looker and feeds on students questions. When one student gets confused she announces the misunderstanding to the entire class in case any other student was making the same mistake. The great thing about the groups was that the teacher almost turned it into a game, so the students were much more willing to participate in the activity.

#3 Describe how effectively the assessment of student understanding was aligned with the mathematical goals of the lesson.
#3 I feel that the assessment was right on track with the goals of the lesson. The best way to assess students in this situation is through communication and listening in on their groups conversations. Students are still learning because they are building off of each others ideas for the assignment. Also, students are comprehension can be easily seen by walking around the classroom.

Overall, I really enjoyed this lesson. I have never heard of learning variables like this and I think it is a great way to help students understand them. Having the variables tie to their name helps the students relate the lesson to their own life rather than thinking "we're never going to use this." I also enjoyed that they did this project mostly in groups but had to work individually as well. This gave students a reason to want to learn because it became a competition for them. Students seem to really enjoy this lesson and I would love to give it a shot in my own classroom someday.

Technology Rich Mathematics Instruction - Thach, K. J. and Norman, K. A

Thach and Norman reiterate the Principle of technology in Technology Rich Mathematics Instruction (2008). Society has greatly changed with the advances technology has given us. Instead of learning how to use technology, students are learning how to further develop their skills through technology. Through the use of computers, students are able to represent concepts and manipulate variables and objects that are difficult to do on paper. This has helped students become more interested in mathematics and ask their own questions rather than answering them out of a workbook.

In this article, Thach and Norman explain a technology based activity they use in their own classroom. To help upper elementary students better understand credit, they use algebra and number operations on electronic spreadsheets. Students are given a financial situation that they can relate to. They try to understand the affects of variables and learn the appropriate tools in solving these problems. Students use calculators to compute information and get to present their data using SMART boards. With an activity like this, students are more interested in learning and can have fun experiencing new technologies.

Thach, K. J. and Norman, K. A. (2008). Technology-rich mathematics instruction. Teaching

Children Mathematics 15 (3), 152-158.

Principle - Technology

Technology surrounds us everyday. Teachers need to take advantage of these technologies and utilize them in their classrooms. Mathematics is a vast subject that can be explored deeper and more fluently with the use of computers and state of the art calculators. Technology will only benefit our future in mathematics with its efficiency and varies ways of teaching. There are many examples of newer technology that can be used at any grade level. From graphing to working with simulations and manipulatives, math has become something that students can become more invloved in at an earlier age. With advances in technology, students are stretching their minds and creativity more than ever.

Teachers also have a big role in the use of technology. Technology never takes over, teachers still have to plan lessons in order for the technology to get used. Teachers are the deciding factor on when and how technology will be used in their classroom. They have also greatly benefited from technology. Now, teachers are able to look deeper into assessing students; rather than grading an answer they can see how students came to that conclusion. Thanks to technology, all parts of mathematics can be covered and in greater detail, not just the basics.