Week 9; Memo #6

Memo #6: A Communication Framework for Mathematics Classrooms 

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

            The readings raise several issues such as the amount of minorities, females, and English Learners (ELs) taking Pre-Algebra by junior high and taking higher level math classes in high school. Another is tracking and how it affects minorities and their futures. A third issue the readings bring up is math confidence. Minorities and females are entering high school with decreased positive attitudes towards mathematics.

            The overall arching point Brenner is making is how do teachers combat the educational gap in culturally and linguistically diverse student populations. In his article, “A Communication Framework for Mathematics Classrooms,” Brenner offers several solutions for increasing communication within, about, and in mathematics. These solutions will be talked about and analyzed in the next few paragraphs.

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

a               a)               intellectual growth/academic excellence in mathematics

The goal of the reading is to promote Communication in Mathematics which means using the language and symbols in mathematics context. To explore this idea and practice it, teachers have to allow students to communicate their own approach to do tasks and “when mathematics enhances their capacity to communicate about topics that interest them” (Brenner 241). Offering more student autonomy results in a student centered classroom environment in which students benefit from other students in such a way that they are working in groups to solve the same problem. This entails students to work with other students by learning to talk and write about mathematics in their point of view. In order to learn from each other, students must be able to justify and explain the how and why.

Word problems are often the most challenging part about math. Students struggle a lot, but also can learn a lot from word problems. To increase academic excellence, one method is to have them underline the important information and cross out the unnecessary information. Another method is having students rewrite the problem in their own words. A third method involves drawing pictures to represent the word problem in another form for students to see.

                b)          Equity

In order to let all students benefit from my teaching, one strategy to narrow the equity gap is to have students work in pairs where each student in the pairs are given specific roles. One student acts as the problem solver, thinks aloud, and justifies while the other student is the listener and monitors the problem solver to make sure the procedures are justified and explicitly explained. This activity “highlights the metacognitive component of problem solving by giving it an explicit role in peer communication” (Brenner 246).

Minorities tend to participate infrequently. Participation in class is crucial to the learning of minorities because participating allows the exercise of students’ minds and is the chance to question the material. A strategy devise by Brenner that offers to close the equity gap for minorities is to have them write in dialogue journals to increase teacher to student communication. Along with student to student communication, teacher to student communication is just as important because teachers are throwing lots of information forward and it is up to the students to reciprocate that information otherwise it will be hard for the teacher to judge how much the student has learn. Writing dialogue in journals allows teachers to have one on one time to response to the students’ needs. Eventually, this idea can be extended for minorities to grow out of having to rely on dialogue journals so that they can participate fully in class discussions.

Brenner addresses three ways to improving mathematics communication. There are three tiers, curricular, instructional, and institutional, in which teachers have to address for student success. Molding the curriculum in order to aid the needs of students is important in shaping their interests in mathematics. Deciding what to teach gets students invested in math. How to teach it is also important. How we get their interest is done by changing our instructional practices that set up students to excel in the classroom. The bigger picture here is that all teachers including the entire school needs to be involve in social change because curricular and instructional changes address the short term goals for minorities meanwhile the institutional changes address the long term goals.

Week 6; Memo #5

Week 6; Memo #5

1.      What is algebraic thinking?

Algebraic thinking is reaching “a deeper understanding of the underlying mathematics, it is important that students see functional relations in both [globalizing and extending] ways and connect the two ways of seeing patterns to each other” (page 96).

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?

There are two concepts presented in the article that I want students to understand. The two concepts are generalization and extension. Generalization is “making the leap to what is “always true” for a functional relation.” Extension refers to “wondering about and pursuing the mathematical directions that an algebraic result may suggest, beyond the result itself. These two concepts are fundamental in understanding functional relations.

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

As a teacher teaching functional relations, it is important to make connections to what students know and where they need to be. In Chapter 5, recognizing patterns is not a big problem because they know how to do so from elementary years. The problem lies with describing the process of how they recognize the pattern and what it leads to. The teacher must help students understand functional relationships by asking questions that involve solving the problem from a different standpoint. It is important for students to show how they arrive at their answer.

A second thing teachers must do is make the connection between small scale pattern recognition and general case recognition. It is important for students to switch from a table and charts way of thinking to an algebraic way of thinking about functional relations.

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

I want students to be able to ask open ended questions that generate more understanding from one problem. Questions such as “Why?,” “What do we get if we go into higher dimensions?,” and “Can a large pyramid have x number of toothpicks?” Open ended questions promote higher levels of thinking because students have to critically think about how to proceed with solving the problem.

In addition to critically engaging students, I want students to understand that they often can get stuck with a problem. For this reason, I encourage students to work on their social skills. In a classroom, I would do so by allowing group work. The way I see it, group work benefits students in such a way that allows them to acquire higher level thinking by talking about different approaches peers have on tackling the same problem. Exposing students to a variety of methods adds on to their internal representations that can be used to solve future functional relations.

3. What are recommendations for teaching algebra for understanding?
1. What should I emphasize when teaching algebra?

Along with pattern recognition, I want students to learn how to explain various forms of solving algebraic patterns and apply it to real-life questions. Students should be able to formulate a general expression and be able to justify it and apply it to various real-life questions. The goal is to have students think outside the box, in addition to exercising their generalization and extension skills.

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

As a teacher, I provide word problems that require students to make follow up questions for the word problem. The activity involves them creating follow up questions for word problems and having their classmate answer it. Creating and finding solutions for follow up questions works on students’ extending abilities and recursive word problems work on students’ generalizing abilities.

I can use pictures to show students the patterns. It gives them concrete objects to focus on that way the student can better relate to what is being asked. Pictures give a visual of the algebraic problems presented in Chapter 5. Also, pictures are used to describe patterns which are easier for students to do than represent patterns symbolically.

3. How can instruction address common difficulties?

Giving tools for students to use when solving complex patterns lowers the intimidation factor of the complexity of the pattern they need to find. Tools such as pictures, tables, charts, predicting, and explaining with words and written words help break down the challenging problem. Breaking it down allows students to focus on solving one aspect of the problem rather than losing focus and getting overwhelmed with the problem as a whole. I can use instruction to let students know the strategies they have “chunking” pattern problems and how it can be helpful to understanding how to find a general formula.

Interview Report: Proportional Reasoning

http://youtu.be/HBGTrxtSTQs

Interview Report: Proportional Reasoning

Name: Emma Shrank

Age: 11 years old

Grade: 6^th

Subject: Pre-Algebra

Teacher: Molly O’Neil

Pertinent Information: Child’s mother is a UCSC professor

            In the interview, Emma works on the Mr. Tall and Mr. Short proportional reasoning problem. Emma refers a lot to the paper clips because the paper clips gives her a better representation of the problem. Emma arranges the paper clips in a way that measures Mr. Short’s height by positioning one end of the paperclip after another end until enough paper clips cover the height of Mr. Short. The paper clips are the hands-on tool Emma uses to communicate her thoughts along with verbal communication. It helps her quantify what the information given in the proportional reasoning problem.

            During the interview, I notice that Emma understands the problem and knows how to do it. To be certain of her knowledge and to test her knowledge about the question, I take the paper clips away so that she can give me a verbal explanation of her reasoning. Without the use of paper clips or any objects to help her think, I am getting the purest form of what Emma knows. The explanation she gives is based on the paper clips because Emma needs something solid to base her theories and opinions and in this case, paper clips are the objects that solidify her thinking.

            The explanation consists of Emma adding two feet (biggies) to Mr. Tall’s height and her adding three smallies to match the addition of two biggies. As you can see, she uses an additive relation to think proportionally. Although this method works, it only works well with cases that deal with small numbers and not big numbers. In an attempt to squeeze more thought out of her, I try to get her to provide some form of formula for figuring out Mr. Tall’s height given Mr. Short’s height in a general case. Although it is not in the video, she explains to me that she keeps adding to determine Mr. Tall’s height. Procedurally she shows proportional reasoning, but when asked for a general formula, she conceptually does not know two biggies equal three smallies. Emma sees the relationship of for every two biggies added to Mr. Tall, then there are three smallies added as well. The only thing she is missing is that she does not have a strong connection to see both numbers as a fraction or percent. Building the connection between ratios and fractions or percent will be one of the next steps to do with students like Emma learning proportional reasoning.

            My reaction is that she is very dependent on objects and is a student that learns from doing rather than seeing and listening. Emma is a very capable student who knows how to do the proportional reasoning problem. What surprise me is that Emma notice that for every two biggies there are three smallies, but when asked for the general case, she could not produce the same result. I learn that Emma has some background about proportions because she is able to do the problem flawlessly without error. As a result from seeing Emma’s dependence on paper clips, it definitely shows me that most students may rely on concrete objects to get an understanding of proportional reasoning. This can also be applied to other topics in math at the high school and middle school level. A follow up for this lesson is for students to create their own word problems after doing several proportional reasoning problems. I will have students work in groups and come up with one word problem where they show various methods on how to solve it such as paper clips or other objects that help unitize.