Final Memo

Summary

The Brenner article emphasizes the use of language in mathematics classrooms and how the proper use and different uses of language can benefit culturally and linguistically diverse students. The role of language in math classrooms is first discussed, followed by its importance seen in recent studies and educational theories from past and present.  Finally, an overview of the Communication Framework for Mathematics is given that highlights various ways in which all students and teachers can participate and use language in the classroom.

Questions

1.       What issues/ questions about teaching English learners do the readings raise?

One of the most striking facts that Brenner brings out includes: “most research on educational innovation in mathematics has been done with relatively advantaged groups of students while classroom practice and research with less advantaged groups continues to stress mastery of basic skills,” (243). The disparity in the type of education that less advantaged groups, many being English Learners, is great compared to other populations. Some questions that this prompted me to think about included: Why is basic skill mastery over-emphasized in less advantaged groups? and Is it assumed that less advantaged students cannot perform at a higher level, or are these students so behind that basic skills need to be practiced repeatedly?

The article also touches upon student participation within the English Learner population in any given classroom. The idea that culture directly effects how students act in participate was demonstrated in multiple examples. For instance, some students are uncomfortable speaking in a class discussion, but do well in small groups. Often, if students are paired with another student with the same first language, students are more willing to explain their reasoning and insight. However, in the instance where “pairs constituted of an English monolingual student and a student with very limited proficiency in English resulted in very passive behavior on the part of the latter students,” (Brenner 257). These findings made me wonder how I could encourage participation in a classroom full of culturally diverse students. Knowing every students’ personal histories seems somewhat impossible, so developing a strategy that would ultimately help me with student participation is something I would want to look into.

2  How does the reading suggest you can promote these in your classroom:

     a.  Intellectual growth/academic excellence in mathematics and b. equity.

The reading supports me, in my beliefs, that all students should have the opportunity to learn at a higher level; certain populations should not only have a procedural education.  So many students are taught that the final answer of any math problem is all that matters, this type of thinking is the driving force of many basic skill mastery classrooms. Brenner emphasizes “class discussion should focus on why ideas are correct, not whether they are correct or incorrect,” (244).

Intellectual growth will come when students are able to demonstrate and justify their thinking. This article’s main purpose was to support a mathematical framework that utilizes communication and highlights the importance in creating opportunities for all students to communicate and participate in the classroom. Justifying their thoughts and actions, through academic language in English, can be terrifying to any student, even more so to an English Learner. Growth and academic excellence is achieved through justification; as it has been said many times before, one truly knows the material when one can teach it to another. Thus, teachers must create opportunities for all students to demonstrate and justify their thinking. I believe the best advice the article gave was “as the content of classroom discourse changes, some students will need the support of changed participant structures to develop the new communicative competence,” (Brenner 24).

Memo for Week 6

Fostering Algebraic Thinking Chapter 7

1. Summary/ What is Algebraic Thinking?
Algebraic thinking, as mentioned in last week's memo, involves connecting the function and structure of a system and being able to manipulate it using a variety of tools. In Chapter 7 the focus is linking multiple representations and finding connections within them. For teachers, the toughest part in helping students link representations is deciding when to give the necessary help. Understanding how each representation is linked together will help the student grow in their algebraic thinking, so this is something that they should really do on their own.

2. What are the central concepts, connections and habits of mind for teaching algebraic thinking?
The whole chapter emphasizes the importance of students understanding how and why different representations connect together. One connection, out of the many that Chapter 7 said students lacked, involves interpreting graphs at a global perspective and not point-wise. I think this connection is important, because it brings students from viewing points as a single entity (5th grade) to viewing graphs, its data and the function as a symbiotic team (higher level thinking).
By knowing how and why representations connect, students will be able to develop their habit of mind involving doing-undoing. This habit of mind will become easier to establish and do when students know how each representation connects to another; being able to manipulate one representation and see how this manipulation will affect the other representations.

3. What are recommendations for teaching algebra for understanding?
Many recommendations were given to teachers when teaching with multiple representations. One of the most emphasized recommendation was to not give students the answers; building connections must be done on their part because it will have the most meaning when they struggle to find answers on their own (not when they are given them). Because this chapter was on teaching algebra with multiple representations, there are plenty of examples showing the use of different representations in teaching the velocity equation (for example). In the excerpt describing a conversation between the teacher and students, the teacher can be seen scaffolding the material. He may have given them suggestions to approach the problem a specific way, but he never told the students what they should find or the answers. The teacher was also addressing the students' difficulties by asking specific questions that would determine where their misunderstandings lay.

Interview Report

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Thinking about Thinking

After looking at the video of Bart Simpson in the classroom during an assessment I realized how realistic the setting was. I've seen assessments in my own classroom play out similar to that of Bart's experience. Students are asked to sit in "test taking" positions, and students are asked to think to themselves, focus on their own papers and complete the tests. In my own classroom, I see the district check ins (such as benchmarks and CSTs) take control of what is taught in the classroom. I think assessments play an important role in determining what the students do and do not understand; however I think assessments shouldn't control the classroom. Having assessments guide what students are taught on a daily basis makes it harder for students to focus on the process and only worry about getting the right answer. The questions asked on an assessment also can shape how the students think within a classroom. Asking open-ended questions that relate to real-life are more meaningful and allow students to think more abstractly than a multiple-choice, computational question.
Does understanding how students think help you prepare for your as a teacher?
Looking at Bart Simpson's thought process made me realize that some of my students might actually play out a word problem scenario in their head and as a teacher, I shouldn't interrupt their thought process. This reminds me that I need to allow for more wait time. I also need to make sure students know that there is always more than one way to solve a problem and rushing through a problem is not the best idea. When Bart Simpson thought through the problem and visualized what was going on, he was trying to get a better understanding of what was being asked, rather than doing a procedure and finishing the problem automatically. Asking these questions that require higher is also something I would incorporate in my classroom.

Memo #4

Summary
In the FAT readings we learned about developing algebraic habits of the mind. Algebraic thinking involves thinking about the function and structure of a system and having the abilities to do and undo computational/ mathematical processes, creating specific rules for specific situations and abstracting information from a given computation. Overall, the FAT reading dove deeper into these topics, furthering the discussion in common misconceptions/ hardships students deal with in developing their algebraic habits by asking engaging and thought provoking thoughts. Kieran's article further discusses algebraic thinking; represented in two different types. Student misconceptions/ hardships with thinking algebraically is also considered.

1. What is algebraic thinking?
Kieran addresses two types of ways students look at algebraic representations. Students either approach algebraic representations in a procedural or structural mindset. Kieran uses the term "procedural" to illustrate students thinking about algebraic representations in an arithmetic way. For instance, in 2x+5=11, students would substitute a number in for x until the left side of the equal sign was equivalent to 11. Thus a procedural thought process is strictly computational. Another mindset that students may have when approaching an algebraic representation is the structural mindset. Kieran uses the term "structural" to denote students thinking about algebra in terms of the algebraic expression; students operate on the expressions themselves, not the numerical aspects. In referring back to the example of 2x+5=11, students with a structural mindset would subtract 5 from both sides of the equal sign than divide both sides by 2. Algebraic thinking refers to the structural mindset, students having an algebraic thinking are able to see "mathematical entity as an object means being capable of referring to it as if it was a real thing-- a static structure, existing somewhere in space and times. It also means being able to recognize the idea "at a glance" and to manipulate it as a whole, without going into details." (Kieran 253) Students with an algebraic thinking are able to manipulate an expression and treat equations and such as a real thing, without stumbling over the fact that most of what they're working with is unknown.

2. What are the central concepts, connections, and habits of mind for teaching algebraic thinking?
One concept that I really want my students to understand is the relationship between division and multiplication. For instance, many students don't realize that when they are dividing 2x by 2, they are actually multiplying 2x by one half. This connection between division and multiplying by a reciprocal is very important and through this, students would be able to realize why when dividing fractions, you invert and multiply by the second fraction. This concept is a "magical rule" that students think come out of no where so I really want to emphasize that there is a reason we do everything a certain way in mathematics. Many math classes today focus on memorizing a formula, plug and chug computations and getting the correct answer. One habit of mind that I want to instill in my students is the importance of understanding why they are doing something. I don't want my classroom and students to be focused on only getting the answer; a mindset like this only hinders learning and reinforces that only the end product is important. Understanding why they do something will help students develop a structural mindset of algebraic thinking and through this they will be able to fully understand their computations; being able to do and undo their thought processes.

3. What are recommendations for teaching algebra for understanding?
When teaching algebra I should emphasize the importance of knowing why something is being done. I want students to make the connections between non-mathematical ways of solving a problem, and the mathematical ways of solving them. In word problems, I want my students to understand that there are multiple ways to solving them--sometimes using common sense or daily problem-solving skills will get you through a problem. I also want students to understand that there is no correct way to solve a word problem, there is most likely an algebraic way, but anyway that will get them through the question can suffice. Although students look at word problems and proceed to do them with an arithmetic view, I want them to know that this is also correct, however a much easier way would be to set up an algebraic expression. Showing students multiple ways of solving a problem will allow them to make connections through the work they are doing. Another difficulty students have is thinking of the equal sign as something other than "do something." In my placement I've seen a few exercises to help students realize that the equal sign is a relational symbol that says both sides are balanced. Perhaps working with activities that addresses this problem before beginning one-step equations will help students view the equal symbol differently.

Memo #2

Part I: Summary
All three of Lamon's articles focus on proportional reasoning, recognizing it in students work, helping students obtain it and becoming better at teaching it. Chapter one of Lamon's book discusses the topic of the books and breaks down proportional reasoning and what it means to understand it. Chapter two focuses changing student thinking, especially in terms of multiplicative thinking. Even when discussing fractions at the elementary level, Lamon emphasizes that teachers must connect that a number means nothing unless it is compared to something else (5 people in car vs. 5 people in a stadium). Lamon furthers this discussion by bringing up the topics of unitizing and norming. In later chapter Lamon studies the actual work of students on a variety of problems. Dissecting their proportional reasoning and seeing how their responses connect to what they may know and the gaps in what they need to know was also very insightful.
Lowery's article was discussing an activity that students would do help them develop a conceptual understanding in proportional reasoning. The actual activity is mentioned below in  question 3.

Part II: Reading Questions
1. What is proportional reasoning?
According to Lowery, "Proportional reasoning is the comparative relation of one thing to another in part or whole and is expressed in terms of magnitude, quantity, or degree," (Lowery 1). Throughout the Lowery and Lamon readings, examples assessing proportional reasoning have been considered. Proportional reasoning is said to be expressed if students can use a comparative relation (observable in ratios, fractions, etc) to determine the outcome of a similar scenario. As stated by Lamon, developing a proportional reasoning requires skills/reasoning in multiple areas of cognitive thought processes. Students should have skills and knowledge of partitioning, relative thinking, quantities and change, unitizing, an understanding of rational numbers, and a sense of ratios. Because obtaining a keen proportional reasoning involves development in many areas, Lamon admits "the answer is, that we cannot say, in a very concise way what proportional reasoning is, nor can we say how a person learns to reason proportionally," (Lamon 6).

2.What are the central concepts and connections for teaching proportional reasoning?
As mentioned above, Lamon had a harder time defining proportional reasoning because there are so many aspects to consider when teaching for proficiency in proportional reasoning. Tying into last week's reading, the entire network of knowledge gets stronger and more cohesive as more connections are made between bits of knowledge. The same idea applies for proportional reasoning; as students acquire skills in partitioning (and realize a unit can be changed because of this), quantities and change (realizing a question can be posed based on relative or absolute conditions) and so forth in areas of ratios and rational numbers, their proportional reasoning will strengthen.

3. What are recommendations for teaching this topic for understanding?
One of the most emphasized recommendations was for teachers to help their students get comfortable with a unit being a value other than one. Helping students realize that a unit doesn't necessarily have to be one cupcake or one pizza or one soda can will allow them to make inquiries on their own and discover other ways of solving a problem. Also, teachers should state that there are multiple ways to solving a problem and encourage students to explain what they did in multiple representations (a picture, symbols, tables, etc.) this will help the teacher to identify areas of need for each student. Of course teaching with multiple representations is necessary, even more so with proportional problems because so much is involved when solving them.
One useful tool that stood out to me was the Lowery Proportional Activity that involved finding the heights of Mr. Small and Mr.Big using jumbo sized paper clips and smaller sized paper clips. This activity gave the student a drawing of Mr.Small and asked the student to find out how tall he was using the jumbo paper clips as the unit. Mr. Big's height in jumbo paper clips was already given. Next the student was given the smaller paper clips and was asked to determine the height of Mr. Small by using the small paper clip as the unit of measure. Finally, the student was asked to predict the height of Mr.Big in terms of the smaller paper clip. This activity seems like a good activity for developing a proportional reasoning because the student is able to use manipulatives, physically change a unit and measure a drawing without having to deal with the messiness of numbers/rulers/complex units. This activity is very basic but gets to the conceptual understanding of proportionality so as a teacher I would want to use or develop a similar activity that would get students thinking outside the box and focusing on something other than getting the right answer.
Common student difficulties often lay in the ways in which a question was asked. For instance, a question that was asked in the Lamon article involved drinking 75% of a case of Coca-Cola. Student responses were all over the place because students had different ideas of what a case of coke consisted of (24 cans, 12 cans or a 6-pack). I think, especially in the introductory stage of teaching proportionality, I would state the unit being used for the problem and slowly have students progress to working on problems where the unit is not as obvious.
Another common misconception is student thinking of a question too literally. For instance, "two trees  of different heights both grow 3 feet over some time, which one grew more?" can be thought of differently. I would want to make sure my students knew the difference between absolute or relative conditions.

Memo #1

Learning and Teaching With Understanding focuses on the issues of teaching mathematics in such a way that would encourage student learning with a conceptual understanding of the material. Different theories of learning are discussed that connect and represent a framework for understanding. Students' internal framework of knowledge is questionable- whether it develops as an interconnecting web or as a hierarchy of knowledge. The main point of the article expands on how this framework (no matter the form it takes) ties into students' innate ability to make relationships and connections to new knowledge and the knowledge they already have. If students can make connections between models, real-life scenarios, personal experiences, pictures, and other representations to mathematical symbols and then explain their reasoning then a deeper understanding is said to have been developed.
Within the article, Hiebert and Carpenter expand on the idea of developing a conceptual understanding. Developing this deeper understanding involves students internalizing the information and integrating into their internal framework so that it cohesively connects to existing knowledge in different ways. Hiebert and Carpenter relate the strength of each connection and relationship to the level of understanding a student retains. So as students make more connections and as these connections become stronger and cohesive, students are able to expand on their knowledge and their knowledge grows.
Recognizing student understanding/ misunderstanding can be done through viewing different works a student produces. Students who have a conceptual understanding of mathematical material will be able to represent this material in different. An example was given in the reading that described a student applying the subtraction operation to numbers with three digits. Using base 10 blocks a student with a deep understanding of the material can represent 403 in terms of the blocks (4-"100"blocks and 3-"single" blocks) and subtract 156 by replacing a "100" block with 10 "tens" blocks, and representing the act of borrowing a number. Not only can the student demonstrate the act of borrowing on manipulatives, but the student would also be able to explain himself verbally and symbolically (using mathematical notation).
Having a conceptual understanding allows the student to prosper in other areas of academia. Students who generate their own ideas and generate their own connections within their internal frameworks can recall information a lot easier. As noted by Hiebert and Carpenter, when students make connections within their own networks of knowledge, they are less likely to lose key bits of information because "an entire network of knowledge is less likely to deteriorate than an isolated piece of information" (p.75). Also, because this higher level of understanding is dependent on a network of knowledge, less information will need to be remembered. Hiebert and Carpenter made the connection between operations of fractions (addition, multiplication, subtraction, division) all incorporated (somewhat) the procedure of finding similar fractions (and simplifying them). If students tie all their fraction operational "rules" to similar fractions, then remembering each "rule" would be a lot easier to recall.
One take-away lesson that I personally found important was the idea of connecting street math to math found in the classroom. All too often I notice teachers and professionals emphasizing the importance of learning math and placing the education system on a pedestal. I think other types of knowledge, knowledge found outside of the classroom, knowledge used on a daily basis and informally is just as important. I hope that students will be able to realize the importance of all knowledge and as a teacher, I hope I will be able to aid them in connecting both types of math (street and classroom math) in a way that would be most useful and efficient for my future students.