Algebraic Reasoning Interview

Algebraic Reasoning

            The student I interview is a geometry student, whose name is Alex and he is sixteen years of age. He currently is a sophomore in Pajaro Valley High School and is maintaining a C+ in Geometry class. The interview lasts twenty five minutes. During the interview, Alex is able to list the sum of two consecutive numbers from twenty to thirty five and make conclusions from his results. With extra time left, I am able to get Alex to move on to the sum of three consecutive numbers, four consecutive numbers, and even touch upon five consecutive numbers. Let us not stray too far away from the point of the interview. I believe any student is able to list out consecutive numbers which does not let the student show much algebraic reasoning. To deviate to the point, the point of this interview is to test the student’s algebraic reasoning by analyzing and apply the results, generalizing, and assuming.

            Alex starts by listing the sum of two consecutive numbers. After creating a chart of numbers from twenty to thirty five, I ask him a series of questions to test his algebraic reasoning. One question I ask is if he can make a theory base off his results. From looking at the list of two consecutive numbers on the list, Alex immediately notices a pattern that the sum of two consecutive numbers gives only odd numbers and not even numbers. I proceed to ask a follow up question and the question I ask is “Are we able to make the number eighty using two consecutive numbers?” The response he gives is no because if we show that we are not able to make even numbers using the sum of two consecutive numbers, then we are not able to make eighty with two consecutive numbers. Alex’s response shows that he is able to apply his conclusion base off his results and use it to predict what will happen with bigger even numbers. He draws conclusions because there are certain things that he looks for such as commonalities among numbers in the two consecutive sums, patterns in adding two consecutive numbers, and numerical results of odd and even numbers.

            The final series of test comes down to Alex being able to provide a formula to find the sum of two consecutive numbers of any numbers. As he thinks aloud, he tells me he starts by dividing the number in half. If the number gives two whole numbers, then subtract one from one number and add it to the second number. If it the number gives a decimal number, then subtract half from one number and add it to the second number. The wordy algorithm he writes down and shows me works brilliantly and I think it is very creative. Alex shows me his algebraic reasoning is strong because he is able to come up with his own algorithm for figuring out the two consecutive numbers of any numbers. The author of “Fostering Algebraic Thinking” by Driscoll describes the Alex’s thinking process as recognizing regularities in the sum of consecutive number systems in order to find shortcuts that help solve future issues. In this case, Alex formalizes a theory base off results and verbally explains his thoughts as he is solving the problem.

            After noticing I can get no more reasoning out of him from using the sum of two consecutive numbers, I move on to questions about the sum of three consecutive numbers. Alex lists the sums of numbers through twenty to thirty five. Upon finishing, he notices that there are a lot more numbers that cannot be written using the sum of three consecutive numbers. He supports his reasoning by saying that adding more numbers limits his results from twenty to thirty five. What he means by this is that he has to add three consecutive numbers and doing so, gives us really large results which mean that there are bigger gaps in between results from twenty to thirty five. I ask him if he can draw a conclusion using his results. He is unable to generalize a conclusion. Other than that, he is able to say that higher degree of consecutive numbers gives fewer results (bigger gaps) between the numbers twenty to thirty five. At this point I ask him to list the sum of four consecutive numbers and five consecutive numbers for numbers between twenty to thirty five. I get ask the same question and again I get the same response from Alex.

            What could have been done differently to get more reasoning out of Alex is to have him organize his results. In the video, the results are all over the place and it is to see patterns in the sum of consecutive numbers. Another change I have is to work with smaller numbers, but not too small. Ideal numbers are one to twenty five. I choose to use these numbers because they are big enough to be made up of three or even four consecutive numbers, but not too big as to confuse Alex on his calculations. Also, using smaller numbers means Alex has an easier time calculating to divert more attention and concentration to thinking rather than spending his efforts on calculating for the correct answer.

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.

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.