Week 5: Memo #4

Memo #4: Handbook of Research in Mathematics Teaching and Learning

1.      What is algebraic thinking?

Algebraic thinking refers to “the capacity to represent quantities situations so that relations among variables become apparent” (Driscoll 1). It includes “being able to think about functions and how they work, and to think about the impact that a system’s structure has on calculations” (Driscoll 1). In other words, algebraic thinking is important because it helps students work on recognizing patterns and finding a shortcut to the system. Essentially, students are playing around with numbers (undo and do mathematics or going forward and backwards and decomposition) and making discoveries (trying new methods and building new rules) that play a role in how they think about algebra.

2.      What are the central concepts, connections, and habits of mind for teaching algebraic thinking?

a) Concepts: What concepts do you want students to understand?

I want students to recognize structural arithmetic. Structural arithmetic can include a + b – c = a – c + b, but is not limited to 1/2*(a) = a/2. Having a strong foundation in structural arithmetic will allow teachers to build off a strong foundation to learn complex abstract concepts such as the quadratic formula or graphing.

b) Connections: What connections (between representations, between procedures and concepts, etc.) do you want students to develop?

Even though I am a big fan of teaching for conceptual understanding, I believe procedural understanding is just as important. To emphasize procedures in a coherent way, I want students to develop the ability to learn how to apply an algorithm to a problem. Instead of teaching students all these different algorithms to solve different things, as a teacher, I have to show the differences and similarities among the algorithms so that students do not see mathematics as a bunch of disconnected algorithms.

c) Habits of mind: What habits of mind do you want students to develop?

I want students to learn how to be able to transfer concepts from one body of knowledge to the next. For example, students who know how to simplify expressions may have a hard time simplifying radical expressions or polynomials.

What are recommendations for teaching algebra for understanding?

a.       What should I emphasize when teaching algebra? Can be seen on page 15.

As a teacher, I should emphasize three things. First is the property of reversibility. I want my students to be able to know the ins and outs about the processes they use. Using a process to reach a goal is only half the battle. The other half is to start at the end and work backwards which would expand their understanding of the process. Second is to build rules to represent functions. Being able to recognize patterns and organize data is crucial to a student’s algebraic learning because it works on their information processing skills. Linking ideas to patterns helps students grab a better picture of the concept. The third, is abstracting from computations. I want students to be able to get as much information as they can by thinking about computations independently rather than dependently. It helps fixate the abstractness of algebra into something more comprehendible.

b.      How can I teach algebra using objects, pictures, and word problems?

I can teach algebraic thinking by using word problems to show that there are multiple ways of approaching one problem. In addition to this, as a teacher, I have to show a variety of ways that are range from one end of the spectrum to the other end. The two ends I am talking about are doing direct and indirect methods for solving word problems. Solving the word problem using forward operation and backward operations is an alternate way of representing and solving word problems.

Most of the time algebra is represented in the form of numbers, variables, and symbols. All these are abstract things students have a hard time grasping. To solidify the important concepts algebra brings, pictures and objects are needed to bridge the connection between abstract thinking and practical thinking. This connection must be made so that students do not fall in a deep hole where they lose understanding of algebra and eventually lose interest in learning the subject. For examples, drawing graphs lets students physically see how changes in functions work.

c.       How can instruction address common difficulties?

Having clear instructions let students know what is expected of them to do. They know that they have to get from point A to point B provided that the teacher assists them.

Another way instruction addresses common difficulty is by incorporating English into math lessons. Instead of having students procedurally solve problems and use verbal explanations to justify their method, I will have students write out their reasoning and use sentence structures to formulate defending arguments. Having to write out their explanations instead of verbally communicating them accesses a different part of their mind so that they become use to the written math literacy which will help them in higher level math such as calculus.

Week 4; Memo #3

Summary: The article is about defining the meaning of proportional reasoning and proportional problems and making relationships in proportional situations.

1.      What is proportional reasoning?

“Proportional reasoning involves an understanding of the mathematical relationships embedded in proportional situations” (Page 395). Proportional reasoning also involves being able to differentiating the difference between proportional and nonproportional problems by having to recognize the properties of proportional problems and non proportional problems. In addition to finding the properties of proportional problems, students obtain proportional reasoning by being able to solve a variety of types of proportional reasoning problems. This includes knowing what method to apply, identify what is being asked, being able to use a numerical comparison, and quantifying units.

2.      What are the central concepts and connections (between representations, between procedures and concepts, etc.) for teaching proportional reasoning?

Students must conceptually understand proportional reasoning problems before solving the problem procedurally. Teachers must show students a way to better identify what is being asked either by showing examples or drawing pictures. On page 393 of Learning and Teaching Ratio and Proportion: Research Implications, the article shows that teachers should represent the word problems pictorially and through the use of examples by looking at the problem from a different point of view. The example given relates proportion to strength of the orange juice. The teacher is building a relationship between ratio/fractions to the concentration of orange juice. Students can relate to this problem because they most likely encountered a situation similar to this problem. Adding more orange juice to water will make the solution more concentrated. What this problem also addresses is the fact that teachers are making proportional reasoning connections from accessing students’ prior knowledge. Relating the new knowledge to old knowledge makes learning proportional reasoning more meaningful.

3.      What are recommendations for teaching this topic for understanding?

a)      What should I emphasize when teaching proportional reasoning?

On page 396, teachers need to emphasize postponing teaching algorithms and focus more on internal understanding. By giving students the algorithms to solve various problems, I, as a teacher, am teaching my students to become masters at using the algorithm, but not masters at applying the algorithm. The first step to solve proportional reasoning problems is knowing what algorithm to apply. This is done by teaching for conceptual understanding before teaching for procedural understanding.

b)      How can I teach proportional reasoning using objects, pictures, and word problems?

“Teachers need to step outside the textbook and provide hands-on experiences with ratio and proportional situations” (page 396). These hands-on experiences include activities, drawings, objects, and word problems. To teach proportional reasoning via external representations, I can show how to represent numerical proportional reasoning expressions using objects. Showing is still not enough because I have to use words to explain the connection between proportional reasoning and the external representations by using objects to clarify what is being asked, pictures to explain problems, and word problems to connect to real life situations so that students are able to draw more concepts out of the topic.

c)      How can instruction address common student difficulties?

Providing clear instructions lets students know what they need to do so that they can succeed. Being clear about the instruction makes the curriculum more organized for the teacher in the way that I know what I have to do each day to get students to understand proportional reasoning.

I can also provide instructions on misconceptions and common mistakes that students make when doing proportional reasoning. This is to show students what not to do so that they do not repeat the mistakes others have made. In addition to these instructions, I will have students reflect upon their methods and correct mistakes. This is so that they can learn from their mistakes and learn how to reason through the processes that lead up to the answer.

Week 3: Memo #2

Part I: Summary Questions

a)      MAIN POINTS or CLAIMS: What main points, claims, or arguments is the author making?
There are four semantic types students and teachers encounter which are well-chunked measures, part-part-whole, associated sets, and stretchers and shrinkers. The point Lamon stresses is that teachers “need to move beyond the level of identifying…..that affect problem difficulty” instead they should move “toward the identification of components that offer more explanatory power for children’s performances in the domain.”

b)      EVIDENCE: What does the author use as evidence?
Lamon does case studies on 69 males and 69 females that are just starting sixth grade. The sixth graders complete a series of tests that test each semantic type. Lamon gathers data on the test by recording conversation and showing a report of the work of students did on each semantic problem. The semantic type problems are designed to explore children’s mathematical process and reasoning which is essential to proportional reasoning.

Part II: Discussion Questions

1.      What is proportional reasoning?
Proportional reasoning is the recognition of structural similarities involving ratio and proportion. Lamon states some of the students in his case study achieve correct answers without using proportional reasoning. There is a misconception about proportional reasoning.  Proportional reasoning does not consist of being able to construct and algebraically solve proportions.

2.      What are the central concepts and connections (between representations, between procedures and concepts, etc.) for teaching proportional reasoning?
Students draw pictures, tables, charts, and groups to represent their mathematical processing of the problems. Students are drawing upon their pictures and connecting it to the problem. By verbally explaining their pictures or procedure, students are making connections to the problem and pictures. Having to verbally explain their steps on how they solved each problem means they have to work on their mathematical reasoning. A few students have difficulty explaining their thought process which does not exclude them from not conceptually understanding the problem. Out of those few students who did not offer explanation, some of them could not offer explanations to the problems because they were not able to find the correct words to explain, but they were able to show on paper how they arrived at their answer.
In the stretcher and shrinker problems, Lamon uses the term relative thinking. Relative thinking is being able to relate all the essential features in the stretcher and shrinker problems. Students must understand that changing the length, for example, will proportionally affect the width given a ratio change. If students are not able to connect a number changing a length, then they are not able to do the problems. Most students give up in the article when they come across the stretcher and shrinker problem.
There is no direct transfer between the knowledge of semantic problems. Within types, the variations in context and modes of representations affected students’ responses.

3. What are recommendations for teaching this topic for understanding?

a)      What should I emphasize when teaching proportional reasoning?
Allow for flexibility and choice of methods to use when solving proportional reasoning problems. Allowing more choice for students to use does not limit the students’ mathematical processing. Accepting only one method as answers to one problem will hinder students’ learning because students will have to close off their original ideas and work with the idea I, as a teacher, offered. I believe learning does not happen this way and I strongly believe in using different methods to tackle the same problem. My reasoning behind using multiple representations to represent one problem is to reach out to different types of students. Some students may not understand how to solve a math problem when I show them using one method, so I use a second and a third method in hopes to conceptually and procedurally teach them.

b)      How can I teach proportional reasoning using objects, pictures, and word problems?
As a future teacher, I can use objects to show understanding for kinesthetic learners. To show understanding of proportional reasoning, I can use pictures for visual learners. I can point out the important facts and relations within word problems so that students can isolate what is being asked. The use of these multiple representations allows me to reach a variety of learners. It also gives me the ability to explain one problem using different methods. I am able to teach sophisticated strategies and multi-step methods using multiple representations, therefore, I am making sure students are challenged and at the same time, teaching for mathematical understanding.

c)      How can instruction address common student difficulties?
Instruction helps coordinate all the essential relationships in proportional reasoning. Teaching students to look for key words makes students struggle less with procedures and give more time focused on conceptually understanding what they problem asks. Finding key words such as “miles per hour” relates the amount of miles traveled within an hour or finding key words such as increased relates to addition. Having clear instructions makes the problem easier because they struggle less with processing the question and more with working on the problem.

Prezi Link for Week 2: Hiebert and Carpenter

http://prezi.com/sdcfmeud12w7/educ-228/