Week Six - Napkins and Place Mats Color Tiles Lesson Plan

CLICK HERE for the lesson plan and the supporting documents.

The objectives of this lesson are to identify and extend patterns, explore the sequence of perfect squares, make predictions based on patterns, and express relationships algebraically.

I had four eighth grade students do the activities in this lesson at lunch on Thursday. They enjoyed the lesson, but they found the part where they had to come up with algebraic expressions to be very challenging. This is likely because the students that volunteered to do the activities are in the lowest eighth grade math level. If they were in algebra then these activities would have been a lot easier for them because they would have been used to expressing problems algebraically. However, with a lot of assistance, the students were able to come up with algebraic expressions to represent the patterns they saw.

On another note, when the students were working on the activities from this lesson I could see that they were very good at identifying patterns and relationships. Thus, they found all aspects other than the part that specifically relates to algebra to be relatively easy.

Also, after the students completed the activities, I felt a sense of accomplishment because I was able to reinforce concepts related to patterns and relationships and help students with the difficult task of learning algebra. In essence, I was able to help students discover math patterns and express them algebraically.

Week Six - Private Universe Project in Mathematics - Workshop Six - Possibilities of Real-Life Problems - My Reaction

The idea of using real-life problems in mathematics is particularly important. Students often cannot relate to discussions involving mathematics. However, by using real-life situations and problems, students can see real-life reasons why they should learn mathematics.

It is not uncommon for mathematics students who are not planning on majoring in mathematics or business to ask why they need to learn high-level mathematics. However, by using a real-life example like calculating the speed of a cat, students can “relate” to the problem at hand. In addition, the speed of the cat problem is a very good introduction to calculus. It provides students with the opportunity to analyze changes in speed.

I really liked how the students were asked to model the actions of a cat. By having the students actually move along a line in the same way that the cat moved, students were able to develop a deeper understanding of what the dots on the graphs that they created actually meant. Honestly, I feel that this deep understanding of what a graph actually means is often lacking for many mathematics students. It is evident in the fact that there is math phobia, and it is even more evident in the fact that many students have trouble interpreting graphs and charts. Basically, what this video shows is that the solution to graphical and tabular interpretation problems that students have is to have them model what is going on in the graphs and tables. In essence, we need to let students bring graphs and tables “to life”. This will allow students to make sense of mathematical information.

In my opinion, the “Rutgers Way of Teaching” is certainly an excellent teaching strategy. Basically, it allows for teachers to be facilitators of learning because it is an inquiry-based approach. By using this way of teaching, students discover new concepts, and they become the teachers for other students. As many people know, if one can teach something that means that they truly understand it. The “Rutgers Way of Teaching” also allows students to develop “ownership” of a problem. It allows students to develop a sense of confidence that they are able to solve a problem on their own. In addition, it allows for students to explain to their peers how to solve a problem. This is particularly important because sometimes the way students go about teaching their peers is more effective then if a teacher shows a student how to do something. Lastly, this strategy of teaching allows for students to see that there are multiple ways to solve a problem. Thus, I feel that the “Rutgers Way of Teaching” is much more effective then a teacher simply telling students information. By telling students information, the chances of them retaining it are not nearly as high. In addition, simply lecturing to students can be boring for everyone involved.

I also agree with the narrator that having students justify their answers is very important. If students can justify their answers then they really understand a topic. This is a great test for educators to use when determining if their students have developed a deep understanding of a particular topic.

The Private Universe Project video series makes it clear that students learn best by working with manipulatives, being challenged, being asked to justify their answers, being required to find their own way of answering a question, modeling mathematical information, and working with real-life problems. If teachers take the lessons learned from this video series and apply them to their own classrooms then student interest levels in mathematics should rise. Ultimately, student test scores should follow suit.

Week Six - Virtual Manipulative - Savings Calculator

Virtual Manipulative - Savings Calculator
Grade Level: 6 - 8
Category: Data Analysis & Probability
Link: Savings Calculator

When I was searching for a virtual manipulative I wanted to select something that could help students in the "real world". Obviously, when I came across this savings calculator I knew that students would learn something that they would need to know for banking purposes. Thus, this virtual manipulative has practical applications, and it would likely spark interest in many students. As a result, this is a "must use" application when the topic of compound interest is discussed.

Developing an understanding of compound interest is important. However, since it is a challenging topic, it is important to clearly demonstrate the results of compound interest. Without a clear demonstration of the results of a compound interest account, students may develop the misconception that whether or not interest is compounded is insignificant.

This application is perfect for demonstrating the significance of compound interest. Not only can students see the result of investing money in an account with compounding interest, but they also can see the difference between interest that is compounded daily, weekly, monthly, quarterly, and annually. In fact, I know many adults who do not understand the different types of compound interest, and this program demonstrates to students the difference in terms of the outcome.

My main recommendation to teachers that they enter large numbers for the deposits. This will lead to a more dramatic difference in interest earned for the different types of compound interest. I also suggest that teachers utilize this application in an introductory lesson as park of a "hook" to grab students' attention. In addition, this application can be used later on in a unit on compound interest to demonstrate the difference between different types of compound interest.

By using an application like this one, it is possible to spark student interest in mathematics. Today's students really need to know that what they are learning could impact their lives. This application can be used to prove to students that what is being taught in math class can affect their lives.

Week Five - Private Universe Workshop Five - Building on Useful Ideas

I was particularly impressed by Melissa Sharp, a first year second grade teacher in the Englewood Public School District. She was able to get second grade students to think at a higher level. In the video, she stated that she does not accept students simply giving an answer. She wants them to be able to convince her that they have come up with the correct solution or all of the solutions. In essence, she wants every student to develop a deep understanding of the problems they are working on so that they can build on what they learn in future grades. If more teachers develop this philosophy of teaching then I suspect that the retention of ideas learned in a particular grade will increase.

After listening to some of Arthur Powell’s ideas, I can see the importance of getting students to “own a problem”. Getting students to think deeply about a problem is so important, and this can be done by getting them to be engaged in working with a problem. In addition, it is important to have students share ideas. Obviously, this requires students to get used to the idea of working together and discussing problems, but doing this process has value in and of itself. The reason that I feel so strongly about this is because so many of my students are able to solve a problem, but they do not completely understand what the answer means. Sometimes they come up with answers that do not make any sense, and they do not realize why their answers are not realistic. For example, they may come up with answers regarding time and speed that are negative. How is this possible? Thus, it is important to get young students to work with problems that are hands-on so that they develop a deep understanding of what numbers actually mean before moving on to more advanced abstract problems in later grades. For the same reason, it is also important to press students to provide a convincing argument for why their answers are correct. In this way, students are required to think like mathematicians, and if they are able to prove that their solution or solutions are correct then they have obviously developed “ownership” of the problem at hand.

I think Gina Kiczek, a teacher in the Jersey City Public School District, made an important realization, which is that students often do not see things the same way that the teacher does. Often there is a “disconnect” between students and their teacher because the students do not think the same way that the teacher does about a problem. This is why it is very important to give students the opportunity to share their ideas about problems. In the type of learning environment where students are given the opportunity to work together and discuss problems, they can not only develop their own line of thinking, but they can also learn from their peers. This is much more valuable then the outdated teaching model where a teacher is a “sage on a stage” and students simply absorb knowledge from him or her. By giving students the opportunity to work together and discuss what they are thinking, there is a much greater chance that students will develop a deep understanding of a problem and what the answer to a problem means. In essence, this style of teaching allows for more students to develop “ownership” of a problem.

I was particularly impressed by the ice cream lesson. This lesson is a testament to the value of a lesson that allows students to work together and solve problems. The reason that I say this is because the lesson was planned for an eighty minute block period. Based on the video, it appeared that students were engaged in the problems provided to them for the full eighty minute block. This is quite an achievement!

I thought the World Series problem was a good problem to use with older students. It is important to choose topics that fit students’ interests. Many students are obviously interested in the World Series, and thus, when one gives students this type of problem they are more likely to become engaged in what they are assigned to do.

Week 5 - Virtual Manipulative - How High?

Name of Virtual Manipulative: How High?
Category: Geometry
Grade Level: 6 - 8 (probably best for grade 8)
Link: How High? Virtual Manipulative

This virtual manipulative is great because it forces students to try to understand spatial relationships. The truly analytical student will figure out very quickly that it is necessary to calculate the volumes of both figures and then compare them. By no means is this an easy activity. The beauty of this program is that the learning curve for an eighth grader would be very steep. At first, they will likely start out making wild guesses. Then, they will make more educated guesses. Finally, they should come to the realization that if they calculate the volume of the liquid and compare it to the volume of the container on the right they should be able to accurately predict how high the liquid will go when poured into the container on the right. Thus, this activity forces students to also study the volume of figures.

The really nice thing about this virtual manipulative is that it is applicable to the real world. It is very common for people to transfer a liquid from one container to another in order to save space in a refrigerator or cupboard. Thus, students will likely see "the point" of doing this activity.

I had my girlfriend do this virtual manipulative. She started out making wild guesses. Then, she graduated to educated guesses, and finally, she began calculating volumes and comparing them. Thus, it is no surprise that, at first, she thought the activity was very difficult. However, by the end, she had changed her opinion because she had developed a mathematical strategy for determining the answers.

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.