Most school systems have installed blockers that are supposed to filter out offensive results and pages, but there is the possibility that some may still slip through the cracks.  On the other hand, the negative effect of these blockers is that informative and educational content sometimes becomes unavailable.  For instance, I was in a classroom one day  where the children were learning about penguins.  I wanted to pull up this wonderful video clip about penguins from a Planet Earth episode that I knew was posted on YouTube.  However, the school system had YouTube blocked so I was unable to do this.

Cyberbullying is another aspect of internet safety that can greatly affect children.  I was not very familiar with cyberbullying until I visited the website dedicated to stopping the form of harassment.  As far as the role of schools with cyberbullying, it was very interesting to learn that the creators of the site encourage schools to word the school’s acceptable use policy so that students can be disciplined for off-campus cyberbullying actions.  I’m not sure whether I agree with this or not.  If all parents monitored their children’s internet usage at home, then parents could be held accountable for disciplining their children when cyberbullying is taking place, however, not all parents do monitor how their child is communicating with others via the internet.  I could definitely understand school involvement if the situation is spilling over to the classroom and causing disruptions at school.  Since electronic communication is done behind the comfort and barrier of a computer, I can understand that children would be braver and more apt to threaten one another than they would be face-to-face.

As teachers, I think it is extremely important to teach and inform children about internet safety as we use it in the classroom.  It is a good idea to go over acceptable uses at the beginning of the year and to remind children of these when they are performing internet searches.  Students need to know that if they see content that they don’t feel is appropriate, it’s not their fault, but they should always notify the teacher.  At this point, the teacher can see about getting the site blocked if necessary.  The teacher must also closely monitor students as they are using the internet.  It may often be beneficial to allow students to work with partners while researching.  This way, the students can hold each other accountable.  The teacher will obviously know which students to partner together so that they aren’t encouraging each other to abuse the internet.

Another classroom safety and legal issue is copyright.  I would venture to guess that many teachers do not know their rights as educators or what is and is not permissible.  The Educational CyberPlayground Teachers Channel provides a very helpful list of acceptable and non-acceptable uses of materials and explanations of these uses.  Most importantly, a teacher needs to be able to meet the four criteria for Fair Use.  If the material really is beneficial to the students, it shouldn’t be very difficult to meet these four criteria.

Most teachers, if they violate copyright, probably do so in photocopying materials.  I would think that it’s common knowledge that a teacher cannot photocopy and distribute consumables, but I’m sure that fewer are aware that they can’t copy more than one work from a single author or no more than three authors can be copied from a single collective work.

It is also imperative that teachers cite their sources when publishing material and that they make sure their students also cite their sources.  I think it’s definitely a good idea for the teacher to research ahead of time what’s necessary when she or her students are creating PowerPoints, blog posts, websites, etc.  This way, she covers herself, her students, the school and the school system.  The Educational CyberPlayground is a great resource when an educator isn’t sure what is allowed or how to comply with Fair Use.

I did find the Fraction Model very helpful.  This would be a great tool for children to use after they’ve mastered concrete manipulatives and are beginning to work with the standard algorithm.  This provides a great way for children to actually see a representation of the fractions.  This is also a great tool to compare fractions and the equivalent decimal.  By using this tool, students can see how fractions and decimals are written completely different but can actually represent the same amount.

I still have a hard time figuring out exactly how to remediate when students have trouble with fractions.  I think it’s a difficult step between the concrete and representational to the abstract.  It’s interesting, though, that even the kindergartners I’m working with are doing fractions in their own way.  They’re learning, as they make art projects, to fold paper into halves and fourths.  It’s a start!

In the video, “Products and Sums,” when the teacher introduces the assignment of finding out if there is a relationship or pattern between the addition problem and multiplication problem for the same number and itself, the students know they should make a model to figure out the relationship. It is evident that they understand the concept of making models because they are able to draw the problems on grid paper and examine them. The teacher really encourages the students to develop their number sense by asking a lot of questions of the students, and she doesn’t shoot down anyone’s ideas. She also doesn’t give them the solutions, but instead she encourages them to explain their reasoning. At one point, she even has a student come up to the board and point out how he found his pattern. The students obviously enjoy their “discovery,” and they feel like they’re really helping each other by coming up with relationships and patterns. One thing that could be done differently… if a student was having problems with drawing on the grid paper to represent the problems, he/she could use concrete manipulatives. That would only be necessary if the representational grid paper was too confusing.

The Ashlock chapter was a needed reminder to me that we as teachers can often figure out not only what a child’s struggling with, but also how he/she is struggling.  This chapter really points out the importance and benefit of looking over a student’s work in order to assess whey they are answering problems incorrectly.  This is common sense, but it’s difficult for a child to express why he/she is wrong if they don’t know they’re wrong or they can’t really tell the teacher how they’re solving problems.  The book also talking about the teacher “interviewing” the student about his/her answers and strategies.  I can see where this would be really beneficial because then the child can actually explain the “method to their madness.”  Then the teacher can make suggestions of ways to adapt the strategies and, together, they can come up with some new ideas.  

I also saw remnants of ideas from our Math CAAMP class.  The book says to show children the importance of estimation, and that was something Dr. Piel talked about a lot.  For every operation we used to work problems, we had a strategy for estimation.  Dr. Piel also talked about the importance of establishing number sense before children move to more difficult paper and pencil exercises.  If this number sense is nurtured, then students wouldn’t conclude with many of the answers they get because the answers simply wouldn’t “make sense.”

The only difference I saw between the two applets was that one tests knowledge of numbers 1-5 and the other expands to 0-10.  After children have mastered the five frame, they can move on to the ten frame.

In response to the children’s classifications of squares and rectangles (mainly incorrect classifications), the teacher asked how the children decided which were squares and which were rectangles.  Without the first activity, she may not have realized that this was a question that needed to be resolved.  Because of situations like this, I often think that it’s difficult to write a lesson plan and stick to it.  I think an effective teacher is able to adapt to what the children express and question, and if one has to stick to the lesson than an opportunity is sometimes lost.  I think it’s impossible to predict the thoughts of all of the children, and therefore, the lesson must be flexible.

The students mostly said that squares were shorter and fatter, and rectangles were longer and skinner.  This is such an interesting explanation, but it’s definitely one that children would likely develop.  The teacher in this situation wondered if the children had seen the shapes mostly from a horizontal orientation, and that was dictating their definition.  This definitely proves that children need to see shapes from different angles and orientations in order to truly understand their properties.

In this case, the teacher presented other four-sided polygons and wanted the children to notice the differences.  Interestingly, some of the children recognized that rectangles and squares were different from this other shapes because of “the corners are different” or because of “crooked” and “straight” lines.  This was a first glimpse into noticing angles.  However, I’m wondering how a teacher introduces the concept of angles to prove the properties of squares and rectangles.  Obviously this is a defining factor, but up until 3rd grade I would guess that the children really haven’t talked about angles.  Thus far, comparing sides is probably one of a few ways they’ve learned to classify.  This teacher realized that all her students may not grasp angles, but she should have presented the concept anyway for those who may catch on.

I like how the teacher started this lesson by reviewing the children’s definitions of a rectangle.  Right away, the children are engaged because they’ve actually helped formulate a working definition, not just had to accept the correct one.  The teacher lets the children discuss and discover, without directly instructing.  She is, however, able to guide and move the discussion by the questions she asks.  When she questions, “So, what’s a square?” she can channel the discussion a certain way and have children think about the particular aspect of the lesson without telling them.  She does this again when she asks the children, “How can we make a definition for rectangles that is more specific?”  Even without being in this classroom, I can really see the wheels turning in the children’s heads.  Their answers may not be exactly correct, but it’s evident that they’re thinking about how to answer the teacher’s questions.  These children have an active role in the lesson, which keeps them interested.

I also thought it was very beneficial for the teacher to have some manipulatives readily available for the children to use during the discussion.  One child examines an orange pattern block, trying to figure out what makes it a square and different from a rectangle.  Even though the children can always verbalize exactly what they’re noticing, the teacher can tell they’re on the right track by “decoding” their language a little.  It’s important to note that the discussion went longer than the teacher planned, but she didn’t cut the children off and move on to the next subject just because her “alloted” time was over.  She recognized that the children were involved in a lively and deep discussion about rectangles and squares.

I also found it interesting that at the end of the lesson everything wasn’t completely wrapped up.  The children were on the right track to a definition of a square and rectangle, but nothing was absolutely decided upon.  This was an important point for me, because I often want to have very noticeable closer in a lesson.  The teacher recognized that the children made some headway, but they would continue working on this topic.  Perhaps time in between gives the children an opportunity to make some more discoveries on their own.

In both the article and the wood staining video, the teachers really let the children make their own discoveries.  The tasks even seemed a little daunting at first (how much stain, creating a polygon), but the teachers really provided the students with the tools they needed to succeed, and that builds great confidence.  In both instances, children were really working in teams to find a solution and then together as a whole class.  I think this builds enthusiasm- you just have to make sure all students are participating.

Supposedly we learn even more and retain knowledge when we teach it to someone else, and these children had just that opportunity.  By explaining their their solutions to the class, they feel more confident in their reasoning.  This also gives the teacher a chance to assess that child’s understanding.  I liked how the teacher had even more children explain their reasoning to emphasize the process and to also check for their understanding.  The teacher could have taught the students a formula for area of a non-rectangular polygon, but I’m sure after the geoboard activity, the process will stick much better because they were actually involved in creating the formula.  Not to mention, they’ve demonstrated that it works!

Secondly, I think it’s just as important that we help students learn which tools/operations to use and when. We often see children miss problems on end-of-grade testing because they don’t know which strategy or operation to use to solve the problem.  This situation shows that it’s no good for a child to be able to complete addition and subtraction facts without knowing when to add and when to subtract.  I think teachers should give students problems that require different operations mixed in together so children learn that decision-making is part of the process.

I was glad to read that researchers don’t think paper and pencil is obsolete, and that students still need to be able to function without a calculator.  Part of this involves estimation.  I was really intrigued to read one child’s reasoning of figuring out an average by rounding up the numbers and then subtracting the differences.  I don’t think I would’ve thought of this approach!  This was a great example of children being about to use estimation skills to solve problems without having to use a standard algorithm or a calculator.  This also shows the value of letting children come up with their own solutions.  I’m sure other students in the class were able to learn a new strategy because of this student’s “outside the box” thinking.

Lastly, I want to talk a little bit about open-ended questions.  While it obviously takes more time for teachers to grade open-ended responses, I think the benefits are great- for both teacher and student.  I do think it is important to make open-ended questions clear so that students understand exactly what is being asked.  I am all for partial grading in these situations.  I think it would be really disheartening for a student to sit and work out a problem, be on the right track but not have the exact answer, and receive no credit.  This undermines the importance of “showing your work” and developing different strategies.

I guess one of my biggest concerns is how I’ll deal with student errors in computation.  I understand that students learn from their errors and that we, as teachers, can also learn a lot about the students.  At the same time, I probably am too “black or white” or “right or wrong” sometimes.  I hope that, as a teacher, I will allow my students to make their own mistakes.  I know I already want to correct them as soon as I see them making an error with their work.  The key is to capitalize on this errors, and use them as teaching tools.  I don’t want my students to be afraid to try new/different approaches because something may be wrong!

I also definitely need more practice with creating and using rubrics as I practically have none right now!  I think rubrics are a wonderful way to assess, but I need to be able to create rubrics that truly measure student progress and knowledge.  I know as a student I appreciate when a professor uses rubrics to grade. This type of grading system lets students know what is expected of them, and there are no surprises when grades are returned.  Children need to know expectations upfront, and I think a rubric is a great way to do that.  I do wonder… should the teacher always show a rubric to students ahead of time or should she just include it with the final grade?  Also, is it more difficult to use rubrics in the lower grades?  I guess I’ll find that out during my diagnostic project!

After reading these articles, I was reminded so much of how resistant we are to change.  I guess that’s why other countries surpass us in math scores and continue to make improvements greater than ours.  Are we too prideful and stubborn to admit that there might be a better way than the way it’s always been done?

I admire the teachers, administrators and school systems that have the guts to try something new.  Granted, I believe we must do our homework about new methods as the “Buyer Beware” article mentions.  However, if there is research to backup a program and there is proven success, why not take it into serious consideration?

Parents most often want better for their children than they had.  With this in mind, they should want improved methods in math.  I was pretty amazed at the parents who were considering home school because they didn’t agree with the Investigations method.  I’ve never seen this method in use, and I don’t know much about it (except what I’ve read on other classmates’ blogs), but I don’t think the argument that it doesn’t involve enough memorization is a very good one!

I looked at the website that the parents have set up, and they do have quotes from a lot of “math experts.”  I know it’s sometimes easy to find a quote to go along with any opinion, but I wonder why these mathematicians feel this way.  Are they right about the research and test scores?  Everything I want to believe from what I’ve learned so far at UNCC says no.  

Also, for those of you who are in schools using Investigations, have you heard of any complaints from parents?