Week 4 - True Blue Color Tiles Probability Lesson Plan

True Blue Color Tiles Probability Lesson Plan and Supporting Documents

This lesson requires the use of Color Tiles.

I tried this lesson with four of my eighth grade students at lunch. It is very interesting to note that their predicted probabilities did not match up with either their experimental probabilities or the theoretical probabilities. Based on the discussion that the students and I had, it became apparent that they had trouble understanding the concept of not replacing a Color Tile. In essence, if there are six tiles for the first draw then there are only five tiles for the second draw. Obviously, this caused many of the students to come up with predicted probabilities that were not close to what one would expect them to be.

On the other hand, students developed a deep understanding of the idea that as the number of trials increases the experimental probability becomes closer to the theoretical probability. This is a very important concept for students to learn because it is something that can be applied to the real world quite often. In addition, students must use this concept quite often in the scientific laboratory setting.

Week 4 - Reaction to Private Universe Workshop 4 – Thinking Like a Mathematician

One very important suggestion that was made in the video with regard to the Tower of Hanoi Problem was to try a simpler problem. This is a very important strategy to teach students because often problems in mathematics look very difficult. Trying to solve what appears to be a very difficult problem can quickly become frustrating. However, if students can simplify a problem, learn something new from the simpler problem, and then apply what they learned to the original more difficult problem then the problem becomes easier to solve. The nice thing about having students use a strategy like this is that it forces them to look for patterns in simplified problems that can be applied to more challenging problems. Thus, this strategy encourages students to think creatively because it requires students to create new simplified problems that they can solve to determine if there is a pattern that can be applied to a more difficult problem. It also encourages students to think critically because students need to determine what the pattern is.

The narrator of the video points out that the thought process that the students were using is much like the thought process of a mathematician. This is because mathematicians typically analyze simpler problems in order to develop a pattern that can be applied to more complex problems. Since this is what “real world” mathematicians do, I feel that this is the type of mathematical thinking that students really need to learn in school. Thus, it is important that videos like this one exist to show prospective teachers how they should teach mathematics to meet the needs of twenty-first century learners.

I particularly liked the teaching strategies that Janet Walter employs. This is because I believe that it is important to encourage students to become active learners. When students become active learners they rarely become distracted. Also, by becoming a part of the learning experience, students generally develop a deeper understanding of the material. Unfortunately, I have experienced many mathematics courses that are not engaging. In the type of mathematics class that I am describing, the teacher typically acts like a “sage on a stage”. He or she gives students the information necessary to solve problems, and then the students are asked to take notes and solve problems using those notes. In this type of mathematics courses, classroom discussions about mathematics questions and problems are a rarity. In essence, students are not asked to think deeply about problems in this type of class. They are simply asked to do a process that was clearly mapped out by the teacher. Since this style of teaching simplifies the job of a student, it is not common to hear of a student who complains about this method of instruction. However, this style of teaching is not the most effective way to get youngsters to learn mathematics. Thus, I am glad to see that this video encourages new teachers to break this mold and get students to think about mathematical problems.

Lastly, I think the most important thing that this video demonstrates is that if teachers give students the opportunity to become active learners then there is a good chance that the students will have fun. I have heard many students tell me that math is boring, math is hard, and math is inapplicable to their daily lives. This is likely because students are often not asked to participate in mathematics lessons in ways that encourage them to think deeply about a problem. I think the strategies presented in this video have the potential to change this type of reaction to mathematics. By employing the strategies demonstrated in the video, teachers can get students to think like mathematicians, develop a deep understanding of the material, and enjoy what they are learning.

Week 4 - Virtual Manipulative - Number Puzzles

Hi Everyone,

Number Puzzles Virtual Manipulative

I love the Number Puzzles Virtual Manipulative for several reasons.

First, it challenges students to solve relatively difficult math puzzles. For each puzzle, students need to come up with a strategy. For example, they must consider whether it is better to put a high or a low number in the vertex of a triangle? Keep in mind that this changes depending whether the sum for each side of the triangle is large or small.

Second, this virtual manipulative forces students to do computation. I feel that most students will find it annoying to have to keep typing numbers into a calculator. Thus, they will challenge themselves to do the computation in their heads. Obviously, the ability to do computation in one's head is a very important skill, and thus, this virtual manipulative helps students build this skill.

Third, these puzzles are fun! I allowed my middle school students to play these puzzles during lunch, and they loved them. I do not think they even realized that they were doing math. Thus, if you want to trick students into doing computation in their heads then this is certainly one of the best ways to do it.

I suggest that you try this virtual manipulative at home. You will find that it is actually very entertaining.

Sincerely,

Marc

Yola Website

Hi Everyone,

Up to this point, I have been using a Google website. I like Google because I am familiar with it. However, I just created a Yola website, and I realized that Yola websites are much better than Google websites. You will be surprised just how user friendly Yola websites are.

Check out my new Yola website:

Marc Vogel's Math Manipulatives 1 Yola Website

I have already posted the two Pattern Blocks lesson plans that I created on my Yola Site. I also posted the supporting documents. Thus, you can find everything that you need on my Yola site to conduct two lessons with Pattern Blocks in your respective classrooms.

I will post the lesson plans that I create in the future on this site, so I suggest that you check it regularly.

Sincerely,

Marc

Updated Blog

Hi Everyone,

Throughout my blog, I converted the web addresses that I posted to links. I hope this makes it easier for you to find everything.

Sincerely,

Marc

Week Three - Backyard Improvements Lesson Plan

This lesson plan, like the lesson plan from week two, uses Pattern Blocks.

Marc Vogel's Week 3 Lesson Plan

Marc Vogel's Yola Website (Contains Lesson Plans and Supporting Documents)

If the link above does not work then copy and paste the following link into the address bar:

http://sites.google.com/site/mathmanipulativeslessonplans/home/lesson-plan-for-week-three

I based this lesson plan on the Backyard Improvements Lesson Plan, which you can find under the Patterns and Functions link for Grades 7 - 8 on the Super Source CD-ROM.

Reflection on Perimeter Search Lesson for Week Two

As you know, I demonstrated the Pattern Search lesson in class this past Monday. When I tried it with two of my high school students that stopped by after school it was very successful. I feel that I was able to meet all of the objectives of the lesson. The interesting thing is that the students never seemed to realize that they were doing mathematics. To them, the activity was like a game. One would think that high school students would not want to play with blocks, but the reality is that they enjoy working with manipulatives as much as elementary school students.

For anyone who wants to become a mathematics teacher, this lesson is a good way to reinforce the concept of perimeter. Students generally enjoy a challenge, and they enjoy a challenge that involves manipulatives even more. To students, the activities in this lesson are more like "fun time" than they are work. Thus, it is easy to get students engaged by using this lesson.

Week Three - Reaction to the Private Universe Project Video - Workshop 3: Inventing Notations

One thing that I thought was very interesting in the video was the high level of support that the Englewood teachers’ received. It is not uncommon in the field of education for teachers to be required to attend a workshop that is labeled as professional development and then not receive any further professional development on the topic discussed in the workshop. While teachers are obviously expected to implement new ideas that they learn in a workshop, they are often not given any support from facilitators coming to their classrooms or meeting with them after school. Thus, the end result is often that when a teacher gets frustrated he or she often gives up because there is nobody with whom to discuss the situation that he or she has encountered. As a result, continuous professional development is the most effective way to get teachers to learn and implement new things. Fortunately, for the Englewood Public School System, the teachers did, in fact, receive continuous professional development. As a result, the ideas seemed to be implemented in classrooms effectively.

On a different note, I particularly liked that the students in the video were given the opportunities to share their ideas. This is the best way for students to learn from one another. I found it interesting that students came up with vastly different strategies for solving problems. For example, some students made lists, other students created diagrams, and some students made tables. Allowing students to create their own representations of their solutions provides a teacher with the ability to see how students “see” problems. In essence, by allowing students to share their solutions, a teacher can learn how students think about a problem and how they go about solving it. This is very valuable information that is not always easy to acquire.

One must remember that a student’s level of interest generally correlates with his or her level of engagement. Thus, I particularly liked the decision that the teacher from Redshaw Elementary School in New Brunswick, New Jersey made, which was to allow students to work on problems that interested them. The video informs the viewers that the students were not interested in solving for the number of possible combinations of two different colored blocks to form five-block towers and six-block towers. However, the students were interested in the number of combinations of four-block towers that could be made from three and four different colors. Thus, choosing to use the pizza problem was a great decision because it is similar to the type of problem that the students wanted to solve. This demonstrates a teaching strategy that is flexible and adaptable to student interests. Students are more likely to learn when they enjoy what they are doing. Thus, this method of teaching is admirable because it is more likely to be effective. As a teacher of science, I try to choose topics that interest my students to teach them the major themes of biology. For example, we are currently learning about cell division, and several students have asked questions about cancer. While cancer is only a subtopic of the unit on cell division, I will now spend additional time discussing cancer its relationship to cell division because the students are interested in it. In essence, I can use the students’ interest in cancer to get them to learn the information about cell division that is required by the New Jersey Core Curriculum Content Standards. Thus, when I become a mathematics teacher I hope to incorporate this strategy of teaching into my teaching of the new subject matter.

I was really surprised by one Conover Roads Elementary School fourth grade student’s method of solving the pizza problem. The student was able to use a system of binary numbers to come up with the solution. I felt that it was a better strategy than the ones that I used to solve this problem on the first day of class. The student’s method was very efficient, and I think I could have solved the problem faster had I used his strategy. In addition, his chart made it much easier to determine if there were any duplicate solutions. My own realization that I learned from this elementary student provides me with additional motivation to include the sharing of ideas in as many mathematics lessons that I teach as possible. This way the students can learn from one another.

Also, the discussion between Amy Martino and this same student from Conover Roads Elementary School demonstrated that students will not necessarily notice isomorphisms unless one actually encourages them to think critically and creatively about two problems that are basically the same. This type of activity where students are forced to think critically and creatively is so important at a young age. All too often, students struggle with being able to come up with original ideas. They want to be told what to do and how to do it. By encouraging students to think critically and creatively in elementary school, students will be more capable of dealing with high level thinking questions in high school and in college.

On a sad note, I agree with the narrator that the world is full of “Brandons”. What she meant by this is that there are many very intelligent students who “slip through the cracks”. In my opinion, these students are capable of thinking critically and creatively; however, they do not fit the mold of a typical “smart” child. In other words, they may not do well on standardized tests or some performance-based assessments. However, when these students are given free reign to solve a problem in their own way they can be impressive. I have seen this time and time again in my biology classes. For example, some students who rarely do well on tests, quizzes, and labs shine when I assign my Cell Parts Project. Basically, I require the students to teach the class about a cell part in any way that they see fit. Many students use PowerPoint Presentations. Other students create songs, games, or movies. However, my point is that some students who fail assessment after assessment are capable of demonstrating a high level of understanding in an assignment like this. Thus, we must give students the opportunity to think about problems in their own way. This will allow students the opportunity to make the material make sense to them. I plan on incorporating these ideas into my teaching when I become a math teacher.

In conclusion, I felt that the video made several important points. Students should be given the opportunity to think critically and creatively in mathematics class. Also, at least at first, they should be given free reign to determine what notation to use and how to use it. In addition, students should be able to convince someone that their answer is correct. Lastly, students should share their ideas with their classmates, so that they can learn from one another. In my opinion, anyone who incorporates these ideas into their teaching of mathematics is likely to be more effective.

Week Three - National Library of Virtual Manipulatives Pattern Blocks Program

On Tuesday at lunch, I asked six students to find the largest perimeter and the smallest perimeter of a shape created by any six pattern blocks. Basically, I asked the students to do the activity that I demonstrated on Monday night in class. I divided the students up into three groups of two. I allowed one group to work on the problem using the National Library of Virtual Manipulatives program titled Pattern Blocks under the measurement section of the grades 6 - 8 tab. For this, the students were allowed to use the Smart Board that I have in my room. I asked the other two groups to solve the problem using the "real" pattern blocks. Interestingly, the two groups that used the "real" pattern blocks solved the problem faster.

However, the students who used the virtual manipulatives program seemed to enjoy the activity just as much as the students who worked with the "real" pattern blocks. In addition, the students who worked with the virtual manipulatives seemed to have fun changing the colors of the blocks. This is something that you obviously cannot do with the "real" pattern blocks.

In conclusion, I would recommend using the "real" pattern blocks over the virtual pattern blocks program if you are limited by the amount of time that you have to complete a lesson. However, since students naturally gravitate toward technology, if you have the time to devote a few extra minutes to an activity then it might be well worth your while to use the virtual pattern blocks program. In addition, keep in mind, if your district has the technology available then it is cheaper to use the virtual pattern blocks program. I looked up the cost of "real" wooden pattern blocks on the Discount School Supply website, and the cost is $10.99 per case. Thus, if cost is an issue then consider using the pattern blocks program on the National Library of Virtual Manipulatives website.

Perimeter Search Lesson Plan

Dear Classmates and Professor,

I hope you enjoyed the lesson on perimeter that I taught today. Feel free to use it.

The lesson that I taught was based on a lesson plan titled Perimeter Search, which can be found on the Super Source CD-ROM. You can find this Super Source lesson plan by clicking on the measurement link for grades seven and eight.

Sincerely,

Marc

Perimeter Search Pattern Blocks Lesson Plan

Dear Classmates and Professor,

Please go to the website below to see my lesson plan for this week.

Marc Vogel's Week Two Lesson Plan

If the link above does not work then copy and paste the the following web address into the address bar of your search engine:

http://sites.google.com/site/mathmanipulativeslessonplans/home/lesson-plan-for-week-two

Marc Vogel's Yola Site (Contains Lesson Plans and Supporting Documents)

I was unable to format the document the way that I would like to on the website. If you would like a copy of the lesson plan as a Microsoft Word file or a PDF file then e-mail me at the following address: mvogel1023@optonline.net.

I will see you on Monday.

Sincerely,

Marc

The Peg Puzzle and the Towers of Hanoi: Two Great Virtual Manipulatives

The Peg Puzzle Virtual Manipulative

The Towers of Hanoi Virtual Manipulative

I asked a math teacher to allow me to challenge her eighth grade students to try to solve the Peg Puzzle Virtual Manipulative and the Towers of Hanoi Virtual Manipulative over the course of two class periods. She agreed because she saw that these activities had a lot of academic value. In addition, she thought that the skills the students would develop by working on these activities would help them on the NJASK.

First, I started by putting the Peg Puzzle Virtual Manipulative that had eight pegs on the Smart Board. Then, I explained the rules. When I asked for volunteers at least seventy-five percent of the class volunteered. Not surprising, after several people tried to complete the activity, they realized that it was much harder than it looked. I let the students continue to work with the problem. When I saw that students were getting very frustrated I gave them the two peg problem. I called on a volunteer, and he was able to get it on the first try. Then, I put the four peg problem on the Smart Board. I asked the students if they thought this one was going to be easy, and they said that they thought it would be. However, the first three students that tried it got it wrong. Finally, someone got it correct. Then, I put the six peg problem on the Smart Board. It took about ten tries before one of the students got it correct. Then, I put the eight peg problem back up. I called on volunteers once again. Finally, after about fifteen tries a student got it correct; however, it took three more tries for him to repeat it.

Once students solved the problem, we began discussing the patterns. However, the period was almost over. Thus, I gave students the website, and told them to try the eight peg problem at home until they could complete it over and over again. I also told them to write an explanation of the pattern that they saw once they became “experts” with regards to the Peg Puzzle. We discussed the patterns in class the next day, and interestingly, they were very similar patterns to what we came up with in class.

Then, I put the six ring Towers of Hanoi virtual manipulative problem on the Smart Board. Then, I explained the rules. As expected, the students struggled with it. Then, I gave them the same problem with only two rings. The students had no trouble with this. However, I had to call on three different students to solve the three ring problem and seven students to solve the four ring problem. Unfortunately, no students were able to solve the five or six ring problems in the allotted time. However, they made many attempts. At the conclusion of the activity, we discussed the patterns that they saw.

Both the Peg Puzzle virtual manipulative and the Towers of Hanoi virtual manipulative are great activities for students. In addition, since they are online, students can work on these activities individually or as a class using a Smart Board. Considering that students love to play games and love to work on a Smart Board, every teacher should consider putting these two activities in their “bag of tricks”.

In addition, I prefer both of these virtual manipulatives to the “real” manipulatives. Obviously, one reason for this is because students love to work with technology, and the virtual manipulatives accommodate this. Also, I really like that the Peg Puzzle and the Towers of Hanoi virtual manipulatives count the number of moves for you. Believe it or not, I found it difficult to count the number of moves and complete the problem with a “real-life” apparatus. In addition, the virtual Peg Puzzle problem and the virtual Towers of Hanoi problem do not let you make “illegal” moves, which is nice because there is no way of knowing that you followed the rules when the problem is solved using a “real” manipulative. In addition, the virtual Peg Puzzle problem tells the students when there are no more moves left. This is nice because it avoids the confusion of whether there are moves still available. Also, the Towers of Hanoi puzzle informs students if they have solved the problem and if they have done so using the fewest number of moves. Lastly, I felt that the “undo” button was very valuable when completing the virtual Towers of Hanoi problem. It saved several students from having to start over. Thus, as a future middle school math teacher, I would recommend using either of these activities when teaching about patterns.

Is It Possible to Find a Balance?

Dear Classmates and Professor,

As fellow educators and aspiring educators, do you think that the number of standards in all subjects should be reduced? As the second workshop in the Private Universe Project series demonstrates, it is important to foster critical thinking skills and creative thinking skills. In addition, there are many experts in the field that feel that these are the types of skills that students need to learn in a twenty-first century school. How do we balance teaching many standards and teaching students so that they develop a deep understanding of the material? Being able to articulate one's thoughts and justify one's answer is so important in many subjects and mathematics is certainly no exception. However, there are state tests that loom over everyone’s heads, and they cover all of the standards. I see this as a dilemma that many educators currently face. What do you think about this?

Marc

Can You Justify That? - A Response to the Second Private Universe Project Workshop

I found the second workshop in the Private Universe Project series to be on target with the direction that math educators need to take. All too often, educators provide students with a problem, and they require students to solve the problem. While problem solving is a valuable skill, this video prompted me to think about the value of problem solving without the ability to understand an answer or justify an answer.

From my experience as a science teacher, I can tell that students do not understand why the answer to a scientific problem that involves math is what it is. They have trouble justifying an answer, and this leads to them coming up with solutions that make no sense. For example, I have seen students come up with lengths of times that are negative. How is this possible? Well, if students do not understand what they are looking for and do not have the ability to justify what they are doing then it is very easy to get an unreasonable solution. In essence, many students can do math, but they lack strong fundamental math thinking skills.

This video reassured me that the ability to think critically about mathematics can begin to develop at a young age. I remember when I was in school learning mathematics. Rarely was I asked to explain my answer or justify why my answer was correct. I was often expected to show my work, but this is not the same as proving that my answer is the only possible correct answer. This led me to have difficulty when I had to prove things in geometry. In fact, I remember saying on multiple occasions that I hated geometric proofs. However, in the video, students in the fourth grade were able to prove things by using inductive reasoning and breaking a problem down into cases. Honestly, I was amazed. In fact, it appeared that the fourth graders were able to prove their answers better than many high school students could or for that matter better than some teachers could.

The video made me think a lot about what students want and what is in their best interests. Students often want someone to model a problem for them so that they can copy the behavior over and over. However, this type of skill is not valued in industrialized nations like ours. High paying and well-respected jobs are likely to require students to think for themselves and be able to justify their thinking. Most people can do what they are told to do. However, it takes individuals with creative thinking skills and critical thinking skills to be able to form a justification for a solution. Thus, students who are taught mathematics in the way it was being taught in the video are likely to have a competitive advantage over students in schools where mathematics is taught in a traditional manner.

Another thing that I liked about the video was the fact that students were able to go beyond learning just the basic mathematical concepts. In school, we all learn about patterns, and in due time, we learn how to identify them. However, these students learned what patterns are, how to identify them, and how to justify an answer using patterns by the fourth grade. In my opinion, this is remarkable.

In addition, the level of engagement in the classroom was great. Not once did I see a student who was not engaged in the task at hand. Considering that getting everyone interested in a task is an extremely difficult endeavor, this fact alone indicates that the strategy of teaching in the video is an excellent one.

Lastly, I am shocked that students in the fourth grade were able to justify their answers to the four cube tower problem using almost the same logic as the logic used by teachers. Think about the level of education that a teacher has as compared to a fourth grader. Then, compound that with the fact that students’ minds are nowhere near being fully developed in the fourth grade, and one can see that what took place in the video was truly amazing. Thus, I am sold on the fact that math must be taught in such a way that students are given an opportunity to grapple with problems, discover ways to solve problems, and justify their solutions.