Entry #7

Kaufmann, M.L., Bomer, M.A., & N.N. Powell. (2009). Want to play geometry? Mathematics Teacher, 103(3), 190.

In this article, the authors argue that students can learn to understand proofs through playing games. They explain that by setting, following, and disputing rules in a game, students can learn the main ideas covered by proofs. When students play games, they understand that rules are true, unchangeable, and if the rules contradict one another, the game is not fair. Also, by looking at the statement of the rules, students can prove whether or not a play or move was fair. Games draw the attention to students because they are familiar and they allow students the chance to play and be creative. In the article, the authors suggest game ideas that will help students be engaged while still allowing them to develop the skills to write proofs.

I think that games are a great idea to engage student’s interests while still helping them develop the skills necessary to write proofs. Many students have played lots of games in their lives, and they would be able to understand how rules work, and what is considered to be against the rules of a game. When students can recognize when someone has contradicted the rules of the game, they can also begin to understand that some things can contradict proofs. Also, they can see that all possible cases that could happen in a game need to be confronted in the rules. For example, in a dice game, all the possible rolls and combinations need to be discussed in the rules, so no holes are left in the explaination. This helps students understand that proofs need to cover all possible cases, as to leave no gaps in the proof. By comparing proofs to something familiar, the students can better understand the questions proofs need to cover and the information they should contain.

Entry #6

Lege, J. (2009). Fences, forms, and mathematical modeling. Mathematics Teacher, 103(3), 184-189.


In his article, Jerry Lege focuses on the idea that when students use models to represent mathematics, they can better understand functions and what they look like. He begins by explaining that students can look at different kinds of fences, and see what kinds of functions they represent. The four main types he talked about were constant functions, step functions, absolute value functions, and the sine function. He explained how students could better understand the functions when they can connect the ideas to a familiar image in their heads. A second big point he made, was that students can work from the graph of a function to the fence. When they understand the graph, students can model a fence from that information. He says that these two ideas help students develop the ability to apply mathematics in different contexts, and use models to identify relationships in mathematics.


Mathematical modeling does help students gain a greater understanding of the concepts addressed. When students learn to use models, they can more easily recall the images in their head, so they can better remember the concepts involved. Thus, they can also explain their reasoning to other people better because they have a more solid understanding of the concept. I think that when students have a concrete mental image, they can connect the ideas to others more easily. Also, when students use models, they can see mathematical ideas in real world contexts. This causes the students to gain a better understanding of the importance of mathematics. When students can see mathematical ideas in real world situations, they find mathematics more interesting and feel more that math has a purpose. Modeling mathematical ideas helps students have more interest in understanding mathematical concepts, and therefore the students become more engaged in learning and they have a deeper understanding of the concepts taught in class.

Entry #5

Some advantages of teaching mathematics without giving the students specific algorithms or procedures or even correct answers include the students become more independent and they gain the ability to explain their reasoning to their peers in order to conclude when they believe they have found the correct answer. When the students are not thinking of algorithms to find the solutions, they have more freedom to find their own methods to solve the problem. There are many different strategies to find the solution, and this way the students find the one that makes the most sense to them. Furthermore, when the students find their own method of solving a problem, they also have some sort of justification as to why their method works. In Warrington’s class, the students would find a method to divide the fractions, and then they had to explain how they found their answer and why their answer is correct. The students work together to determine when a certain method works. Overall, the most important advantage of teaching mathematics this way is that the students rely on themselves to find the answer. They are confident in their own skills to find their solution and justify their reasoning.


Some disadvantages of teaching mathematics without specific algorithms or procedures is that this type of teaching requires a large amount of time and without giving the students the answers, they may never know when they have come to a correct solution. Warrington spent lots of time with her students, and they engaged in lengthy discussions in order to solve the problems involving fractions. So these types of classes would be extremely slow paced, and the student would not cover as much material in a class that teaches the algorithms. This could cause a problem if a student moved into a class that taught procedures. Although the student would have a deeper understanding of some of the things covered, the student would not be as far in the curriculum as this new class might be. Also, teaching mathematics without giving the students answers could cause confusion among the children. If one student’s method seems to work and make sense, and so does another student’s, but they have two different answers, this may cause confusion among the other students in the class. It might be better for the teacher to step in and explain which is right and way, so the ideas would be clear to the students, and the class could move past it without any students who would still be confused.

Entry #4

I think what Von Glasersfeld meant by constructing knowledge is that our knowledge is built upon things that happen in each of our lives and through prior experiences that we have encountered. One thing that Von Glasersfeld strongly believes is that all of the information we are given is filtered through our own personal experiences, so it is impossible for everyone to have the same knowledge and information. All knowledge is subjective, and there is no way to know if what we see is automatically correct. We draw conclusions from our previous experiences. We can only know if things are incorrect, and we come to know that things are incorrect when something from our prior experience contradicts what we think is true. All of our knowledge is constructed and filtered through our prior experiences.

Knowing that personal experiences help construct our knowledge tells me that I can never assume that the students have the same picture or concept in their head as I do in my head. As a teacher, I think it will be important to ask the students questions so they can describe to me what they are understanding so I can get a better idea of what they know. It is the student who takes in the information I give them and filters and conceptualizes it in their head. I need to make sure to take responsibility to work with the students and have them tell me what they are learning so I can try to make sure that we are all on the same page.

Entry #3

The main point that Erlwanger is trying to get across is that with the IPI program, Benny just searched to find that rule that would get him to the correct solution that the program was looking for. Benny’s purpose in completing the IPI exercises and assessments was to find the rule, not necessarily to know the reason behind his procedures. Erlwanger argues that this way of reviewing and assessing the students causes the students to come up with their own rules for what they think the procedure is because the students were not taught the concepts behind the procedures. Since this is an individualized program, the students are expected to work mainly on their own. This seems to be causing the problem, especially for Benny. Benny is doing what is necessary to pass the assessments and do well, but he is not discussing with his teachers or peers what he has learned, so his crucial mistakes were not caught early on. Because the students have too much independence, they make up their own rules that fit the examples they see in the IPI programs, and they think this is true for all other problems involving the same mathematical concepts. Erlwanger explains throughout his argument that just because a student can master the skill and content does not mean that they have the correct understanding.


In today’s mathematical teaching, it is still important to remember to make sure that the students understand the concepts that are behind the procedures. When students only understand the procedures, they may be able to master a certain area of mathematics, but they will not be able to make connections using the information they had previously learned. For all mathematical content, it is always important to have the underlying concepts so that one can understand fully the procedures that are taking place. Math does make sense as long as the correct concepts are understood as well as the procedures.

Entry #2

Although instrumental and relational understanding have both advantages and disadvantages, relational understanding is more beneficial to students, especially in the long term. Relational understanding includes knowing not only how to do something, but also knowing the reasoning behind the action. Skemp explains, throughout his argument, that instrumental understanding, the ability to know how to do something, is actually contained within relational, meaning that when a student is learning through relational understanding, that student is still gaining the instrumental understanding as well. Skemp examines why teachers choose to teach instrumental mathematics when clearly relational is the better option because it incorporates both types of understanding and has a longer lasting effect. He found three advantages associated with instrumental understanding. He says that it is easier to understand, the results are more immediate, and the answers could be found more quickly. Along with this, the main disadvantage that Skemp emphasizes in his article is that instrumental understanding only lasts for the short term and is only useful within a limited context. However, he states that relational understanding had four basic advantages. He explains that relational understanding is easier to apply to new tasks, can be an effective goal in itself, and causes the students to reason and understand things for themselves. The last advantage of relational understanding is that it is easier to remember. It takes more time and effort to learn as well as to teach relational understanding, but once it is learned, the result is more lasting. Skemp does not identify any real disadvantages associated with relational understanding, but he explains some of the factors that add to the difficulty of using relational understanding. He says that it is hard to test whether or not a student actually understands relationally just by their test answers, and that sometimes the curriculum is so large that it is hard to be able to make the time to teach everything using relational understanding. Also, he explains that it is difficult to change the ways of both teaching, and the ways for students to learn. The focus of mathematics in schools is often placed on instrumental understanding, so it is difficult to make that switch to relational understanding, even though relational understanding is more beneficial to students. Students will be able to use what they learned from relational understanding because the effects are longer lasting and are easier to connect to other ideas and new tasks.

Entry #1

Mathematics is a way to solve problems. Math is the basic foundation for almost anything you need to do in life. Mathematics is essential knowledge in any sort of education. Many people use numbers, figures, and equations in their everyday life. This is mathematics.

I learn math the best by following examples. If I am taught the basic concept first, and then shown some examples to illustrate the particular concept I am able to apply that concept to many other problems even though those problems might be more intricate. I am a visual learning, so watching someone complete a similar problem helps me learn to solve problems.

I think most students are also visual learners. I think a lot of people learn by examples. When people see how to complete a problem, I think it helps them to see that the problems are just done step by step, and it gives them the confidence to know that they can also complete problems that deal with the same concepts.

One thing that helped me in my high school were the workbooks that we completed while the teacher was lecturing. I don’t know if these workbooks are still out there, but it was really helpful because we had to listen to the teacher while writing down the important concepts that were being taught. Also, there were many examples in these workbooks, so we could practice the concepts with the teacher’s help before attempting our actual homework. These workbooks really helped me to follow along with what the teacher was talking about, and they helped me understand what concepts I had a more difficult time with.

One thing that I had trouble with in high school is when the teacher would only do examples. This sounds contradicting to what I have previously said, but sometimes the teachers just get up do examples without explaining the concept behind it. I had a hard time in high school when my teachers would do this. I think it is important that the teacher explain the underlying concepts before delving into examples and letting the kids loose on the homework problems. It is important that the students understand why they are doing certain operations and steps when working their problems. They need to have that foundation, or else they won’t really learn the material, they will just get enough information to roughly complete the homework.