Entry Seven

Howard, C. A. (2009). Mathematics problems from ancient egyptian papyri . Mathematics Teacher, 103(5), 332-339.


People have been studying math for as long as anyone can trace back. Howard's article describes how sometimes we forget that the history of math is important. Teaching this history to our students can help them get excited about the math that they are doing. One area of math history that is applicable to high school students is the Egyptians and their study of pi, arithmetic sequences and volumes of truncated square pyramids found on different Papyrus'. The article goes on to explain all of these problems in depth, and how they can be explained to students. In the pi problem, he shows the circle enclosed by the square, and simply explains how the problem written on the Papyrus comes up with the equations that are used to produce the close estimate of 3.15 for pi. In a further history lesson, and explains how Archimedes uses a bigger polygon than a square to come up with a closer prediction. The other two lessons are explained in a similar fashion. In his conclusion, Howard "effective teachers realize that connections inspire students." By helping students understand connections to history, their curiosity and interested in the subject can be amplified.


I very much agree with the idea that Howard presents in his article. History of math is something that is easily forgotten, yet exciting enough to get get students actively involved in the things they are studying. First of all, it doesn't have to be a confusing advanced concept to be history. Every piece of math was discovered, and there are many high school subjects that can easily be connected to very interesting and exciting historical events. Howard's pi problem is a prime example of this. Second, The evolution of history can help students to see how more complex things are formed. Again, Howard shows this by showing how people were able to form more accurate estimations of pi over time. Lastly, I completely agree with Howard's conclusion. On the surface math can be frustrating and uninviting. However, connections really can help students get involved and excited about math. This article proved a great point and leaves a very good idea for teachers to think about.

Blog Entry Six - First Journal Reading

Daire, S. A. (2010). Celebrating mathematics all year 'round. Mathematics Teacher, 103(7), 509-513.


Having passed the date 3.14.10 many of us realize that this is Pi Day, and weather we celebrated or not, some emotion was felt about this day. Daire writes about how this day is a day for celebrating and getting excited about Mathematics. Her article was more than this though, she goes on to explain how she made math a reason to celebrate all year round. She clearly emphasized and explained how she successfully got her entire school involved and excited about math year round. Each month that has a holiday - there is a math activity centered around that holiday. Examples, February had the title "Love of Mathematics" and they solved equations using hearts and did activities centered around hearts. In the months where there were no major holidays, the school still celebrated math, learning how to do things like Rubik's Cubes and there was even a Monopoly tournament. Daire had the entire school involved, not just those in her math class. Her descriptions of each month portrayed her hard work she put into each celebration, and the success that followed each activity.


Daire's year round celebrations seemed to me like a great idea in theory, they would just take a lot of work. Clearly she was able to accomplish these in her own school, but with a lot of continued and persistent effort. There are a few reasons why I think this task is doable, but would be difficult. From my own junior high and high school experience, and simply being part of student organizations, I realize how extremely difficult it is to get people excited about activities, especially school related ones. Daire's activities varied, some of them allowing many students to get involved, such as posting a problem outside the door, and some were more time consuming, like the Monopoly competition. Doing this she was able to get a wide variety of students. From this I feel like teachers would have to adapt to their own individual schools, to know how to get the most people involved. The next thing that I think would be difficult is getting the administration and faculty on board. Daire had the support of her colleagues. This is key in accomplishing a year round task like this. I know I would have a hard time using my class time for another subject. She was able to convince those around her, so it is possible, again, just difficult. Lastly, I think it would be very time consuming, but rewarding. Daire was able to get the community involved by getting them to give donations as prizes. This extra time allowed the community to see the importance of the celebrations, and the prizes helped the kids be more excited in participating. Overall, I think her ideas were marvelous. It is such a creative way to help math seem less intimidating and more exciting. I commend her on her effort as I notice the extreme amounts it took and I think this is the only way a task like this could be accomplished.

Blog Entry Five

In Warrington's "How Children Think about Division with Fractions" she describes many different positive things that come from teaching through constructivism, without teaching the mindless algorithms often involved in mathematics. One of the advantages that she described was that the children were learning to think for themselves. She recalls one time when one of her students went against both her classmates and even her teacher to state what she believed to be the correct answer. She was confident in her beliefs and figured that the way she got her answer was correct and made sure the class knew. Another advantage she points out is that the children were able to work with each other and think of their own ways to see patterns. When she shared the story problem of seeing how many 1/2 lb bags you could make out of 5 3/4 pounds of peanuts, one girl realized that if you double everything, easier numbers are available to do the math with. Simply comparing and seeing that her answer was right helped her to become confident in her thinking style. I agreed with both of these advantages and one advantage I saw was that, the children were really learning how to do these. Down the road they would not forget the "rules" because they put in the effort to figure them out by themselves.

Although the advantages are strong, there are a few disadvantages to the system. One clear disadvantage, is that the students were not told the right answers. Although this may seem like a great idea, because the students would have to be confident in what they thought, they could never really know if what they were doing was completely perfect. An example of this is 4 2/3 divided by 1/3. This problem is a little bit more challenging, and the students wondered when it was all said and done which answer was correct. Without a concrete answer, they just had to accept what they were thinking. ALthough a majority of the students probably did understand, there is a great chance that some of the students did not completely understand, and that would be what they always remembered. So, there are definite disadvantages to the system as well.

#4 ... Constructivism

In von Glaserfeld's paper "Learning as a Constructive Activity" he defines and explains in detail his thoughts and meanings to the term "constructive knowledge". The key word of this is constructive. He does not believe that people just take in what they hear. Rather, when learning, humans filter things through their mind and use their own experiences to make sense of things. They construct the idea of what they are hearing in their head. One example he used that made it clear to me was with the word 'mermaid'. No one has really met a mermaid, but with a description of it being the head of a human and the fin of a fish, one is able to construct or put together the pictures and create a mermaid in their head. One last important detail of constructing knowledge in the paper is, that the way things are constructed is from prior knowledge. Each time something is learned it can then be used to help construct another idea.

If I whole heartedly believed in constructivism, I would implicate it in my classroom in two ways. First, I would prepare each lesson with the perspective of the age of the students I was teaching in mind. There is a great chance that the things I would know would be drastically different than the things that the students would know. I would need to make sure that the concepts I were teaching would be possible to piece together with the students knowledge. Second, I would be sure to describe things a variety of ways. There is no way I could find a way to teach that each student would be able to construct an idea. So, if I taught things a few different ways, there would be a better chance of everyone understanding.

Entrata tre del blog

In the article "Benny's Conception of Rules and Answers in IPI Mathematics" by S. H. Erlwanger a view of IPI Mathematics is portrayed. The view is a little bit negative as Erlwanger uses the article to show the disadvantages of IPI mathematics as a way of learning and understanding math. To emphasize this point, Erlwanger uses a young student, Benny, and his understanding of mathematics as an example. He show's first how wrong Benny's ideas of the rules of mathematics were. In attempts to understand the assignments and problems he was given, Benny made up his own rules for solving problems with fractions. To drive his main idea home, Erlwanger helped the reader understand why Benny did what he did. All he was learning from was an example and answers to problems. He knew he needed to get from one point to the other, so he created rules to do so, and in this process, he was allowed to teach himself a completely incorrect understanding of mathematics, which Erlwanger proved would be difficult to undo in this young man's mind, along with probably many others.

This study is not just relevant in showing that IPI mathematics is not the best way to teach children this subject. Rather, it shows the important role a teacher plays in a child's education.

From Benny's unfortunate experience, teachers now can see how much of an influence they really have. Not only do they have the responsibility of getting the information to the students, and even more, they must get the information to the students in a way that each student can understand it. Teacher's also have the responsibility and opportunity to make sure children learn concepts and rules correctly. This is because once something is learned, it is difficult to change what ever it may be in the mind of the one who learned it. This is why what Erlwanger emphasized is important to us today.

Second Blog Entry

In Richard R. Skemp's article 'Relational Understand and Instrumental Understanding', he explains both of these ways of learning and points out their advantages and disadvantages. Both of these will be reviewed in this summary. Both of these styles are unique in their own way, yet they are both related. Relational understanding is the big picture, it incorporates everything, including Instrumental learning. In other words, in order to have a completely relational understanding of something, one must need to understand it instrumentally as well. Relational understanding is not only understanding how to do a problem, or applying a formula, but it is understanding why it works the way it does. It is simply knowing why one would do what they were doing. Instrumental understanding is the more basic idea of simply knowing how to do a problem. Skemp reviewed advantages and disadvantages for both. Relational understanding is often times is a great method for helping to understanding something when it is a brand new concept. Also, when trying to remember all of the rules in things (especially math) when the whole picture is understood, it is easy to remember it because a knowledge of why is included. Relational understanding itself can be used as a goal to understand basically the blue prints for different mathematical concepts. On the down side, it is difficult to test if a person actually fully understands the why factor. It can be overwhelming when trying to not that a class will be assessed based on relational learning, it is harder to set and achieve a specific goal, especially when being tested over a basic complete understanding and as a teacher being able to regurgitate what one already knows is quite difficult as well. On the flip side, instrumental understanding has its own positive attributes. One a basic level, it is generally easier to understand something instrumentally. When testing or just trying out a problem, knowing if it is understood instrumentally can be assessed on the spot, and lastly, one can get the correct answer after a short amount of work, because they do not have to think through it. However, this has its downfalls as well. Skemp shows that by simply knowing how to do something, the right answer will not always be obtained. He showed the example through music. It is easy to do parts of music through the basic instrumental way, but without knowing why something is done, it can not be applied in a broader sense. All in all, both styles have their pros and cons, and Skemp does a great job at displaying both.

First Blog Entry

What is mathematics?

Mathematics is the complete study of numbers. Also, it is the application of numbers to any and every situation possible. Math is a way of solving problems through quantifications, and classifying things symbolically and numerically. Math helps to understand everything around us, time, space, and everything in between.

How I learn mathematics best.

I learn math through two ways. First through understanding definitions. By understanding a definition to an equation or even a word, I have a greater idea of how something is to be understood or worked out. The second way is through example. By seeing similar problems, and how they are solved, I am able to understand the underlying concepts of similar types of problems.

How will my students learn math best?

My students will learn math best through example as well. This is true, because in each example, I will be sure that the definitions are understood. Also, by doing multiple examples, I can apply different situations to each problem, so that a great majority of the students that did not understand the first time, will see it in a different light and will be able to pick up on it.

What are some of the current practices in school mathematics classrooms that promote students' learning of mathematics?

I can think of a three main practices in math classrooms that are used to promote students' learning.

First, through independent study. In math classes I have been in, and in others that family members have been in, I have noticed a trend. Often times students are encouraged to read the material before hand or on their own in the classroom to see if sense can be made of concepts through that method.

The next practice I can think of is again, through example. In every class I have seen, teachers will spend a majority of the class doing example problems, to help clarify each of the different concepts.

The last way I can recall is through homework. Basically every math class involves homework. This gives students the opportunity to not only see practice problems, but to apply what they have learned and work on them by themselves.

What are some of the current practices in school mathematics classrooms that are detrimental to students' learning of mathematics?

One way that math is taught in classrooms that is detrimental to the students learning, is through pure lecture. If students are not given the opportunity to work out the problems with the teacher, a lot of them will not have the ability to work out a problem on their own.

Another way is by unclarity. An unprepared teacher can stand in front of a classroom and teach something completely off track, or much to advanced for the caliber of students he or she has in their classroom. By doing this, the wrong way, or the overly difficult way is implanted in a students mind, and they will refer back to this often times before thinking of the correct way to solve a problem.