I explored the applet for adding decimals. I really liked this tool because it created a good visual for students to see the decimals that they are working with. Students type in the decimals and a hundreds board pops up with that decimal amount filled in. Then students can type what decimal is to be joined with the first and another hundreds board pops up with the second decimal filled in. Students can then join the two decimals together to find their answer. If they are having difficulty looking at the two separate boards, they can click a button and the two decimals are filled in on one board instead of two separate boards. This is a great way to show students how to get from two separate decimals to one answer.

If I used this in my classroom, I would have this as an option for one way students could solve decimal problems. When we are first learning how to add decimals, I would have students visit this site and practice with it.

I think this is better suited for practice because there is only one way for the students to solve the problems. There is not really any way the students can figure out a different way to solve using this website.

To assess the students using this website, I would look at their answers and make sure they found the correct ones. I would also give them a paper with blank decimal models on it and have them fill in each of the three numbers of their problems (addends and sum).

These videos were very helpful! I really liked the video on ‘Fraction Strips’ where the teacher said she tried to tie literature into everything. I love when integration happens!! The children in the video seemed to be very engaged in the story and were able to answer the questions easily and correctly. I also liked that she started the lesson with writing down what the students already knew about fractions and then she said they would add to the list as the fraction lessons progressed. The fraction strips were really effective as well! The students were able to make their own strips themselves and having each fraction be a different color let the students clearly see each fraction. As they were making the strips, the students were able to correctly guesstimate how many pieces they would have after making a certain number of folds. This was a very hands-on way for the students to learn how to make different fractions!

The video about Arrays & Fractions was really good too. I like how this teacher had a whole class discussion about how they use fractions in their everyday lives. This got her students to put fractions into real world situations. I also like how the teacher used the students’ prior knowledge of arrays for the lesson. Students were able to come up and show the class what an array would look like for a certain number. The one student came up and showed the class what one array for the number 10 would look like (1 x 10). As he was making this, the students were making their individual arrays at their desks. Students were then instructed to divide their arrays into certain fractions (fifths, for example). This way the students were able to see hands-on whether their array (for 14) was able to be divided evenly into fifths. This was so great for the students to do it themselves and could see firsthand the even or uneven divisions.

The final video highlighted fraction instruction in a 4th/5th classroom using geoboards. I was really interested to see a grade integrated classroom in action because I’ve heard of these before but have never seen it.  The geoboards are a great way to get the students using their hands when making fractions. I love how hands-on this instruction was! The students were physically making fractions and could see exactly how they made those fractions. Using the projector, the students could see how to divide the geoboard up with their rubberbands. By having the students give her the steps for making fractions with the geoboards, the teacher was able to see which students understood the concepts associated with fractions. The way the students were able to find different types of halves was amazing! They knew that as long as each half had 8 squares within it, they had equal halves. Some of the patterns they found were mindblowing! I never would have thought to teach fractions this way!!

I really found this article intriguing…once I got into. Honestly, in my pre-college mathematics career, I never ventured farther than a painful semester of Honors Pre-Calculus. Do I remember anything from that year? No. Could I understand the connections that the author was making to Calculus? No. But I really did enjoy the second part of the article.

One big thing I took away from this article was the idea that while math instruction has not really changed over the past century, studies have shown that students still aren’t getting the type of instruction that they need to fully understand the mathematical concepts of fraction manipulation. The author is trying to make us see that a new type of instruction is needed. One that gets students to think deeper about fractions and the part/whole relationship. I really like that the author pointed out that “current instruction in fractions grossly underestimates what children can do without our help.” I think this statement can be applied to the other disciplines besides mathematics!! We need to learn how to foster our student’s learning in a way that they will be able to start thinking for themselves and solving problems on their own and in their own ways.

The part where the author discusses how a student understands new information they are given was really insightful!! Our students are definitely going to “fit new ideas into their existing knowledge.” So we should let them “represent the material [they've learned] in some unique way that shows his or her comfort with the concepts and processes.” I really believe that students should be given opportunities to think for themselves and come up with new ways to solve the problems they are given. The results of the longitudinal study are also interesting since they suggest that “the kinds of instruction children receive may have more impact on their ability to compare [the] fractions than the kind of numbers involved.”

The author does give some advice on how fraction instruction can be improved. The one piece of advice he offers is to give children the “time and opportunity to build understanding without directly presented rules and algorithms.” Such a simple solution, you would think more teachers would adopt it!!!

I found this article extremely interesting!! It was awesome to see how this teacher was using mathematics discussions to foster students’ discovery of the properties of numbers. Beto explains that for students, having “a strong understanding of the base-ten number system helps [them] choose strategies that demonstrate computational fluency.” I think this is so important because Beto stresses the importance of having the understand of the number system before you’re able to choose which strategies will help you solve problems with ease. I think in the past, the main focus has been placed on memorization and the standard algorithm has been pounded into student’s head. No one really took the time to teach students what was going on in that algorithm and why they were doing what they were doing. I was a product of this type of teaching and I can honestly say that I never really “understood” math until I took math CAAMP this summer.

Also, Beto makes an important point about making mistakes. She says that “mistakes are an integral aspect of inquiry-based discussions. Students gain number sense when they make, discover, and analyze mistakes.” I love this quote because it emphasizes the fact that it is okay to make a mistake as long as you can recognize your mistake, analyze it, and then learn how to change what went wrong.

Beto also highlights the students’ interactions with each other. She says that “students consider one another’s ideas and discover even more about the concept.” I think this is brilliant! The students are learning from each other, not just from the teacher!! This can be a really effective tool as long as the teacher is there to foster discussions and discoveries.

Great article with great ideas!

For someone who had never seen a ten frame before reading this article, I was very impressed with this method of teaching number composition. My 1st grade class started using ten-frames right after I read this article which made me so excited! To see my students discover the importance of the number 10 has been very interesting. Barker states in this articles that she knew that she “should be devoting more time in my second-grade classes to developing number sense and efficient computation strategies.” She makes an important point here because if our students don’t have a strong background in number sense, we cannot expect them to develop towards the higher levels of Mathematics. Barker said she focused her class discussions on the relationships of numbers which allowed her students to easily develop efficient computational methods that made sense to them. From there, she was able to move onto discussions about place value. I really like how she incorporated comparisons of the American number system to the Chinese number system. That really helped broaden her students scope of what mathematics means to people around the world. As teachers, we must realize that math is “a set of conceptual relationships between numbers and number symbols.” If we have a correct way of looking at mathematics then we will pass that on to our students. This article really opened my eyes to a new way of teaching Math!!

The readings that we have had so far this semester have been very thought provoking. In the article entitled “Magical Hopes,” Deborah Loewenberg Ball challenged me to reconsider my feigned dependence on math manipulatives. This article taught me that I should make sure that my classroom manipulatives are being used the right way and are being used along side of effective computational instruction so that my students might leave my classroom with a better understanding of the reasoning behind the standard algorithms of mathematics. I felt really inspired when reading the article entitled “Teacher as Architect of Mathematical Tasks.” Reys and Long showed me that teachers across the board could use everyday, real-life classrooms situations to teach math. The article stressed the importance of taking math beyond the pencil and paper, calculator dependent instruction that so many of us are used to. Finally, the article, “Selecting and Creating Mathematical Tasks” clarified a LOT of confusion I had about task writing in general. I took away a lot of valuable information about how to identify “good” mathematical tasks and how to construct engaging tasks that require my students to think on a higher level. I really liked the point that Smith and Stein make about going beyond what the students are doing in a problem and focusing on what they are thinking while they are doing it. I never really thought of math instruction in this way. I now have a lot to chew on!!

There are plenty of ways that students can benefit from technology. Computers have become such an essential part of everyday. Computers in the classroom are no exception. Both teachers and students now have instant access to credible and helpful information which makes lesson planning, researching, and learning smoother processes. Students who are visual learners can greatly benefit from power points, video clips, and graphs/charts. Students who are hands-on learners can have access to activities and resources that were not available before. Interactive websites help kids learn the same information in new and exciting ways. Classroom powerpoint games are a great way for students to test their knowledge of the concepts studied throughout the year.  The introduction of ‘SmartBoards’ in the classroom has proven to be a superior way of letting students have a hands-on experience with learning. Also, teachers can now connect over the internet with other teachers to compare lesson plans and network with each other in order to obtain new ideas about classroom exercises as well as helpful tips. While technology does have some drawbacks (i.e. the availability of false information, consistency issues, personal information released over the internet), no one can aruge that the pros clearly exceed the cons.

As with any technology, there are many issues related to internet use in the classroom.

One issue is security. While many school systems have a network where only certain acceptable websites can be accessed, there is no fool-proof security system out there. Many students are exposed to needless and potentially harmful information without even knowing it. Teachers have to be strict and rigid when it comes to computer usage. Students must be kept on task and shouldn’t be allowed to search on their own without permission from the teacher. That leaves a smaller chance for them to accidentally pull up a website that was never meant for them to see.

Another issue is distraction. I’ve been in many school computer labs where the students immediately start searching the internet for a site they want to look at rather than follow the directions that the teacher has given them. While the internet is an invaluable resource for instant information, teachers must be rigid with the computer usage of their students. Constant supervision must be provided. Also, I think teachers should make sure that they personally visit and use each website or application that will be accessed by their students.

What does it mean to be ethical in the new digital media? That’s a good question. Gardner himself even said that it has never been asked before. I think it’s a really good thing that his group is further studying ethics in the digital media today. Since media is becoming more and more prevalent in everyday life, it’s important to keep those ethical ideals and practices thriving while adapting to the ever-changing technological world.

Another good point Gardner makes is that those things that have been valued in the past are not necessarily going to be valued and picked up easily today. I think it’s important to understand this when thinking about the ethics of the digital youth. They might not easily understand all the values surrounding ethics because that is a completely new world for some of them.

Gardner also points out that with each new promise that technology and the media offer, a slew of challenges arises with them. It does seem to be the case that for every good thing technology does for our life, it does five negative things as well.

All of this must be taken into consideration before technology is used in the classroom.

The story I am responding to describes a teacher, Betty Armstrong, who has two special needs students in her classroom. Jake and Dave are from the non-categorical, special education class and they spend a few hours of their day in Betty’s classroom for math and reading. Sharon Moss is the special education resource teacher and she really feels that Jake and David should be in a general education classroom full time. Betty feels that David would be successful but Jake is not ready.

I feel that part of Betty’s reasoning comes from her type of personality, not from her extensive evaluation of Jake’s classroom behavior. Betty is definitely a ‘Type-A’ personality and has to have her classroom, students included, in order. Jake, through no fault of his own, does not seem to fit in well with her neat and orderly classroom atmosphere. Betty’s concerns with Jake’s classroom behavior, such as him never getting started on time, shuffling his papers, sharpening his pencil all the time, seem to stem from her frustration that Jake doesn’t fit into her idea of how an ideal student should behave.

I would suggest two things in this case. One suggestion would be that Sharon Moss actually come into Betty’s classroom and observe Jake for herself. It is really hard to make the right decision if you’re only getting a second-hand account of how the student is behaving in the classroom. Sharon should go in and observe for herself how Jake behaves in the classroom and make her decision based on that observation. The other suggestion would be to move Jake into a different classroom for awhile and see if that teacher’s has the same concerns about his behavior. If not, then Sharon could work with that teacher to see if Jake was ready to move into a general education classroom.