Memo 4

            The Kieran’s article focused a lot about the content of algebra. The author also mentions that if students have troubles with the subject algebra taught by the teacher, then it is more likely that the teacher taught out of the textbook. The textbook may be one of the reasons why algebra can be really difficult for students to comprehend. Kieran (1992) claimed, “Presently, the content of most algebra textbooks does not incorporate a procedural-structural perspective on student learning of mathematics; nor does it appear to reflect how algebra evolved historically” (p, 274). The textbook usually teach the students how to procedurally compute problems, but it does not help students’ gain conceptual understanding towards the subject. Furthermore, the textbooks do not engage the students enough because the textbook merely do not help the students develop more understandings.

            Kieran also mentioned that algebra problems can be written as word problems. There are also different types of word problems, such as traditional word problems, open-ended, and problems dealing with functions. In the algebra textbooks, traditional word problems can often be seen (Kieran, 1992, p. 264). These traditional word problems can often be really challenging for the students because of the word-based text. Furthermore, one of the reasons that they can be confusing is that because these traditional word problems focused on the textbook perspective. They are not easy for students to understand it. Students are just procedurally computing the answers when they see traditional word problems.

            According to Fostering Algebraic Thinking (FAT) by Driscoll (1999), “[people] characterize algebraic thinking as the ability to operate on an unknown quantity as if the quantity was known, in contrast to arithmetic reasoning which involves operations on known quantities (p. 1).  In other words, algebra thinking focused a lot about solving variables, but sometimes we cannot clearly define these variables. For example, solve for x, when the given function is x + 5 = 6. But, what does this really mean? This is when Algebra thinking steps in to help us think about the functions and how they work in our mind.

            To ensure that students understand the content, the teacher has to spend more time teaching conceptually instead of teaching procedurally. The teachers should not spend a lot of time doing traditional word problems because they are procedural in a way. Instead, the teacher should ask more open-ended questions to help engage the students in the topic and content. Furthermore, instead of using textbooks to teach, the teacher can use hands-on activities to engage the students. For example, the teacher can use algebra tiles to teach students algebra. The long stick would represent the variable x, and the small cubicle tile would represent the integer 1. In this case, the students can use manipulative to help them learn algebra, so that the students can have a picture in mind and visualize the main concept of algebra.

            By providing clear and accurate directions and instructions, the students are more likely to grasp the content. Furthermore, if the teacher tries to focus more on hands-on activities, then the students will not have difficulty with traditional word problems that can be found in textbooks. Hence, a good teacher can address the students’ common difficulties by providing hands-on activities and proper scaffolds.

Memo 3

           Cramer et al. (1993) claimed, “Proportional reasoning abilities are more involved than textbooks would suggest” (p. 169). Textbooks usually help students master their procedural fluency on the content instead of helping the students gain conceptual understanding in the content. For example, the authors mentioned that the textbook give out an algorithm that is used to set up and solve missing values using cross multiply, but how does that relate to the context of the problems. The cross-product algorithm simply has no meaning to the students because it is just a procedural way in computing the problem and finding the solutions. It shows no conceptual understandings. Hence, teachers must use hands-on activities that are applicable to real life to help the students get engaged in the contents.

            Furthermore, students would be able to develop deeper understanding of mathematics. Cramer et al. (1993) advised, “Research on various middle school content areas supports the value of a conceptually oriented curriculum over a procedurally based curriculum. Since textbooks are generally procedurally oriented, textbook-dominated programs should become less frequent” (p. 173). The authors believed that textbooks may not be the best resources in teaching students for understanding because textbooks are too procedural. Furthermore, the students will not be critically thinking about the content if they are using the textbooks. Hence, lessons should be taught in a way that the teacher can provide scaffolds to help the students to attain more conceptual understanding.

            To be more concise, proportional reasoning is just a way to help students figure out an algorithm to solve the problems. However, in the process of figuring out the algorithm, the students should be able to gain conceptual understanding upon finding an algorithm. In other words, understanding the proportional situations means that the students understand beyond the procedural aspects of the problems.

            When teaching students proportional reasoning, the teachers can ask questions like why to help emphasize for understanding in proportional reasoning. Instead of just writing out the algorithm, ask the students why they write the algorithm that way. Furthermore, the teachers can use hands-on activities and manipulative to help students learn proportional reasoning. For example, using rods can allow students to experience hands-on activities. Thus, it will be more likely to help students have a better foundation of the content.

Memo 2

            Lamon (1993) mentioned that there are four types of semantic problem in analyzing the problem typically by a portion. The four semantic types are well-chunked measures, part-part-whole, associated sets, and stretchers and shrinkers. These four semantic problems can help students develop meaningful logic in ratio and proportion. It also helps students rethink the key idea of proportion. Furthermore, the semantic problems can be really applicable to real-life. For example, Lamon used the eggs example, which is really applicable to real-life and relevant to the students since most of the students eat eggs for breakfast assuming that they are part of the American cultural.

            According to Lamon (1993), proportion reasoning involved comparison of two or more extensive measures. For example, when Lamon talked about the associated sets, he talked about using pizza to teach the students proportion reasoning. The pizza question asked, “The student is shown a picture of 7 girls with 3 pizzas and 3 boys with 1 pizza. Who gets more pizza, the girls or the boys?” (p. 5). These kinds of questions can help students think critically while figuring out an answer. Lamon (1993) claimed, “Proportional reasoning was said to occur when a student could demonstrate understanding of the equivalence of appropriate scalar ratios and the invariance of the function ratio between to measures spaces, whether or not the student could represent these relationship symbolically.” When the pizza question was being asked to the girl, the girl provided a logistic explanation. However, this would not make sense if the student uses math symbols to symbolize her words. But the key point in proportional reasoning is that the students have to use clear explanation to show their understanding of the question instead of answering the question using mathematical symbols.

           

            Students often have troubles using algebraic expression to translate phrases into mathematical symbols. However, when the students are allowed to use proportional reasoning, the students are able to justify and explain their answer through reasoning. This can address some difficulties of the students that have in mathematics because students only have to explain their reasoning for the problem. It does require conceptual understanding, but the students would not have to write long proofs or detailed math solutions for the abstract problems.

            Pictures and word problems are the best way to teach proportional problems. For example, pizza is one of the best ways to teach proportional because of the feature of a pizza. A pizza is usually in a circle shaped that is cut into equal slices. In this case, students can use fractions in explaining the slices of pizzas. Fractions are really similar to proportion, and this can help the students get engaged using proportional reasoning for the word problems. Also, students can identify patterns through looking at word problems. For example, if three oranges cost as much as four apples, then how much oranges would 8 apples equal to? The students know that if three oranges is equal to 4 four apples, then there is going to be more than 3 oranges if there are 8 apples. The good part is that students can make sense in their mind using proportional reasonings.

Memo 1

Hiebert & Carpenter (1992) Learning and Teaching Mathematics with Understanding (MEMO #1)

           Heibert and Carpenter (1992) start off talking about external representation and internal representation as an introduction for students’ understanding. The authors believe that external representation can be constructed or represented by objects, language, written symbols, and etc. On the other hand, internal representation can be represented what the students already know or what the students already have in their mind of the content. Furthermore, the authors also talk about that understanding can bring a lot of pros in learning, such as promoting remembering, enhancing transfer knowledge, and etc. Most importantly, the authors discuss the importance of conceptual knowledge and procedural knowledge and how they can be linked together in teaching a classroom.

            The authors define skill efficiency as somewhat similar to procedural fluency or knowledge. There need not to be understanding in the content because it is more of a procedure format. An example would be adding and subtracting numbers because only computation will be involved throughout these problems. There will not be much of a conceptual understanding going on. However, procedural knowledge is required for mathematics because procedures allow mathematical tasks to be completed efficiently.

            The authors define conceptual knowledge as knowledge that is rich in relationship. In other word, conceptual knowledge is not stored as information, but it is part of a network. Conceptual understanding in mathematics can also mean that students can make connections with the contents. Furthermore, conceptual knowledge is necessary for mathematical learning because it helps student’s gain conceptual understanding from the content.

            Classroom supports conceptual understanding when the teacher helps the students connect their prior knowledge of mathematics and acquire real-world context. For example, when the students are subtracting two fraction numbers like ½ - ¼, the students can use a real-life problem that is similar to the problem ½ - ¼ to help them visualize the problem. For instance, if I have ½ of a pizza, and I gave ¼ of the pizza away, how much pizza do I still have? The students acquire conceptual understand if the students can have these kind of questions in their mind while computing the difference of the fraction problem. Thus, the students have to make connection to the problems while computing them in order to gain conceptual understanding and procedural fluency at the same time.

            Teaching in US classrooms sometimes involves conceptual understanding and procedural fluency. However, students tend to experience more of procedural activities in mathematics. Heibert and Carpenter’s claimed, “Because students in mathematics classes often are asked to memorize procedures and rules for manipulating symbols as individual pieces information, it is not surprising that many students believe that mathematics is mainly a matter of follow rules, that is consists mostly of symbols on paper, and that the symbols and rules are disconnected from other things they know about mathematics” (p. 77). Students are often asked to use computation and memorize formulas in a math classroom, but they do not have the opportunity to study more about the content in depth, so that they can gain conceptual understanding.

            In some cases, even if the teaching format is very procedural, conceptual understanding can still be gained through the lesson. For example, adding a three digit number to another three digit number can be really procedural, but in order to compute the problem, students have to understand about how to carry and line up place values in order to solve the problem efficiently and accurately. This may sound like procedural fluency, but students can gain conceptual understanding at the same time working on computation problems.