Memo #5

Brenner, M. (1994). A communication framework for mathematics classrooms: Exemplary instruction for culturally and linguistically diverse students.

Summary:
          Brenner's chapter addresses some of the concerns in teaching mathematics to linguistically diverse students.  One of the key problems is that many classrooms spend too much time on independent work and teacher lecture.  Unfortunately, academic discourse skill require more time to develop and more needs to be done in order to foster its learning.  Many females and minorities' attitudes towards math decline as they reach high school.  Aspects of school, context, curriculum, instruction, and institutional support can help support diverse students and improve mathematics communication.  Many students are not adequately prepared in math to pursue technical professions.  As a result, the forms of communication in the classroom need to change in order to develop useful mathematical understanding in students.  Students need to communicate about, in, and with mathematics.  Communicating about mathematics means talking about math from one's own perspective, reognizing when specific methods can be used and why.  Communicating in mathematics means using the academic language and key terms.  Communicating with mathematics means using math as a tool or utility.  Improving mathematics communication and the academic achievement of diverse learners in the curricular context involves giving students access to: problem solving, reasoning, estimation, and of course, communication.  Issues in the instructional context consist of matching instructional group arrangements to student levels, the language of instruction, and the role of technology.  Inssues involved in the institutional contexts include the environment of the school or district, and parent involvement.  This reading will be further discussed in the readings below.
 

Questions

1. What issues/questions about teaching English learners do the readings raise?
   
   Some of the issues brought up from Brenner is how although many student are benefiting from traditional teaching, the increasing number of diverse students in U.S. school demands a change in instruction in order to accomodate all students.  Many minorities are being tracked into lower level classes in mathematics diving many students to failure.  Futhermore, many students are not being adequately prepared in mathematics to pursue technical professions.  This needs to change to adapt to society's change towards technology.  It was also found that many students were uncomfortable in situtations in which they had to individually speak in front of their peers.  
   
   
2. How does the reading suggest you can promote these in your classroom:

(a) intellectual growth/academic excellence in mathematics

(b) equity

Both equity and intellectual grown/academic excellence in mathematics can be promoted through a variety of methods in mathematics classrooms.  Using a wide range of instructional strategies can help accomodate students with differing needs.  One instance of improving communication about mathematics involved using daily journals to write about what was learned in class.  Incorporating student experiences into instruction can further improve students' intellectual growth.  Because communicating mathematics is a key idea, interactive activities are highly important in terms of learning the material and academic language.  This might include the use of pair problem solving, cooperative groups, large group discussions, individual  student journals, and written assignments.  The use of technology can also help students fill in gaps.  Instruction needs to be changes in order to involve all student participation to promote equity because communication is a part of the learning process.  Also, the use of manipulatives and having student write their own problems can foster mastery.

Generalizing Structure and Function Relationships in Algebra

Powerpoint!

https://docs.google.com/open?id=0B0MTyL2FEVkqbDlzaE9tVVJrNWs

Interview Report I: Proportional Reasoning

         I interviewed an 11-year old 6th grader at Branciforte Middle School.  She is a student in my placement classroom studying basic math with Mr. Lammerding.  In this class, she has learned about integers, decimals, and has just been introduced to solving one step algebraic equations.  This class has not gone over ratios and proportional reasoning yet.  During this interview, I introduced the problem to her and gave her the paperclips to work with for Mr. Short.  She did not write down much work to find out Mr. Tall's height in small paper clips.  She got an answer of 10 paper clips.

Me: We're going to do a problem involving Mr. Short and Mr. Tall.  So here, we have Mr. Short.  I'm going to measure how tall Mr. Short is in paperclips.  How many paper clips tall is Mr. Short?
Student: 4 paper clips.
Me: Okay.  Write that down.
Student: (writes it down)
Me: Okay, Mr. Tall, whose picture I don't have with me, he is 6 of these same paper clips tall.  So I'm going to take these away.
Me: I want you to do 3 things.  I want you to measure how tall Mr. Short is with these small paper clips. (hands student paper clips)
Student: 6
Me: Okay, write that down.  Next, I want you to predict the height of Mr. Tall with small paper clips.  I told you earlier that we know Mr. Tall is 6 big paper clips tall.  And now we want to know tall Mr. Tall would be if we used small paper clips.
Student: 10
Me: Now, tell me, how did you solve the problem?
Student:  Because if this is one (Mr. Tall) is 6 (big paper clips), and these ones (small paper clips) are like half of it.  If you measure 6, and then 7, 8, 9, 10. um that tall. (Stacks small paper clips on top of each other)
Me:  Can you explain how you know your solution is complete?
Student: We were trying to find out Mr. Tall’s height using the small ones.  He is 10 small paper clips tall, so um, we are done.

1.      Proportional reasoning:

a) Define proportional reasoning.

Proportional thinking is defined to be the ability to construct and algebraically solve proportions.  In other words, it is "the comparative relation of one thing to another in part or whole and is expressed in terms of magnitude, quantity, or degree" (Lowery 1974).  It involves the strategies that are used to tackle a number of problems in mathematics and plays an important role in students' mathematical development.  Proportional reasoning is used in fractions, word problems, rates, functions, etc. In addition, proportional reasoning is not merely when someone can set up "cross-multiply and divide" algorithm for solving proportions.  It means understanding why the algorithm works, and the mathematical and multiplicative relationships that occur between 2 or more measure spaces in proportional situations.  It also means being able to recognize which situations are proportional and which are not.  It means being able to solve a variety of problem types, including missing value problems, numerical comparison problems, and two types of qualitative situations.   Students with proportional reasoning should also be able to overcome unfamiliar settings and non-integer numbers such as factions and decimals.

b) Describe what you would consider evidence that a student is using proportional reasoning.

Evidence that a student is using proportional reasoning can come in a variety of forms, depending on the student’s stage.  Being able to see that there is a pattern between two measure spaces and relating the two together through a multiplicative relationship can be a form of proportional reasoning.  For example, if the student is introduced to a problem in which 2 miles are run in 18 minutes, they may be able to recognize that each mile is ran in 9 minutes.  From there, if the rate does not change, the student can then also find out how long it takes to run any number of miles.  In other words, the student is able to connect the two measure spaces together and see the multiplicative relationship between the two measure spaces.  In particular, for this problem, evidence of proportional reasoning would be shown after the student can identify that the ratio between the small paper clips and the big paper clips is 4 to 6.  With the given information that Mr. Tall is 6 big paper clips tall, the student should be able to recognize that multiplying 6 by 6/4 would give us height of Mr. Tall in small paper clips.   

I might also describe that the beginnings of proportional reasoning might involve being the ability to compare two different units.  Being able to see the comparison is an important part of building the relationship.  Although visually comparing may not be the most accurate, it demonstrates some understanding of proportions.  If a student is able to show the relationship between two measure spaces as a fraction or ratio, and then use that for find a missing value, then that can also demonstrate proportional reasoning.

c) How did this student use proportional reasoning? Look for evidence of unitizing, relative thinking, absolute thinking, multiplicative strategies, etc.

       My student mostly used visual strategies.  When she first wrote down the height of Mr. Short, she wrote 4 feet.  This shows she was looking at the big paper clips as a unit.  In terms of proportional reasoning, she did notice that the small paper clips were about half the size of the big paper clips, so she visually estimated that Mr. Tall would be 10 small paper clips tall.  In other words, she noticed that it would take about 10 small paper clips to be of equal size to 6 big paper clips.  Although she was able to visually compare the two paper clips are two separate units, she did not show proportional reasoning.  According to Lowery (1974), this student is in pre-stage.  Lowery points out that a student who is in pre-stage “uses proportionality, but is an estimation of length ratio instead of a computation of it from the data” (Lowery 1974).  As noted, this student estimated the relationship between the smallies and biggies.  She did not use computation to reach her answer; instead, she visually used the small paper clips to see how many of them would be equivalent to 6 biggies.

2.      Next Steps: If you were teaching this student’s class, how could you use what you learned while interviewing this child in a lesson plan?  What might be a good lesson to follow up on this interview?

      After this activity, I learned that many students likely use similar visual reasoning strategies because they have not been introduces to proportional reasoning.  As a result, my follow up lesson would include an activity in which students will work with manipulatives in which they will need to compare two different units.  This activity will be similar to the Mr. Short and Mr. Tall activity.  Students will be introduced to a problem in which they will have 2 different units: quarters and dimes.

      Each pair of students will be supplied with 6 quarters, and 8 dimes.  They will be asked to line up the least amount of quarters and dimes so they are of equal height.  The aim of this is for them to notice that the length of 3 quarters is equal to the length of 4 dimes, in other words, they have a 3 to 4 relationship.  Next, they will have to line up 6 quarters to 8 dimes.  During this activity, the pairs will discuss with one another any patterns they are noticing.  After this, I will present a new problem and ask them how many dimes lengths are equivalent to 9 quarter lengths, 12 quarter lengths, 15 quarter lengths?  This data will be recorded into a chart.  By this point, students might discover that there is a multiplicative relationship in which if the quarters are multiplied, the dimes are as well.  Cramer, Post, & Currier (1993) say that all proportional situations have multiplicative relationships.  Therefore, it is incredibly important that students are able to investigate and come to this conclusion on their own.

       I will emphasize the use of relative thinking, which Lamon (1993) explains as the ability to see the ratio as a separate quantity from our data.  Next, I will have time tell me how many dimes lengths are equivalent to 48 quarter lengths.  This will require student to do some computation using the ratio in order to get to their solution.  Furthermore, I will also have them find how many dime lengths might be in 70 quarter lengths so they can see they we do not have to have whole number answers, and that this will work for numbers that are neither divisible by 3 nor 4.  Within this activity, student will be able to work with manipulatives and interact with a pair to explain the reason to their solutions.  After this activity, I will wrap up the activity and ask for what relationships student may have noticed.  I would further emphasize the multiplication relationships proportional problems have between measure spaces.

3.      Reactions: What did you learn? What surprised you? What are you wondering about?

      From this interview, I was able to get some insight on common misconceptions when faced with proportional reasoning situations.  I found that at the 6^th grade level, many students have not been formally taught ratios and proportions.  As a result, my student did not use proportional reasoning to find her solution.  A common strategy a 6^th grader might use is estimation based on what they see.  I found that she was pretty good at visually comparing the two objects to get to a good estimate.  This is a strength that could be used later to help her develop her proportional reasoning skills later on.  I am curious about what steps it takes for students to fully understand proportional reasoning from this stage.

Works Cited

Cramer, K., Post, T., Currier, S. (1993). Learning and Teaching Ratio and Proportion: Research Implications. In D. Owens (Ed.) Research ideas for the classroom: Middle grades mathematics. NY, NY: Macmillan Publishing Company.

Lamon (1993) Ratio and Proportion: Connecting content and children’s thinking. Journal for Research in Mathematics Education, Vol. 24(1), 41-61)

Lowery, L. (1974). Proportional Reasoning. In Learning About Learning Series. Berkeley, CA Univ. Of California, pages 17-20 and 37-38

Week 5 Memo #4

Kieran, C. (1992). The learning and teaching of school algebra

         In summary, Kieran's article addresses some of the issues involved the content, teaching, and learning of algebra.  One of the key points Keiran points out is that many student see algebra as memorizing sets of rules and procedures.  This is a result of the combination of both content difficulty and the way in which algebra is taught in schools.  Before students are taught algebra, they are learning procedures which involve arithmetic operations and numerical results.  Once students begin learning algebra, students need to transition from procedures to structures, which involve different operations on algebraic expressions.  Unfortunately, algebra is taught in such a way that students are not fully understanding how algebra works structurally, and the symbolism behind symbols and letters.  When students are introduced to algebraic word problems, they work backwards using arithmetic backwards strategies.  After a certain point, student may no longer depend on the backwards strategies and must work forward using algebraic structures.  Many algebra teachers felt that their focus was on classroom management and covering all the material required by the school, and guided by the text book.  Many teachers were also unaware that much of what they were teaching were not actually being learned by students.  The procedural process of learning algebra seems to be more accessible to students than the structural process.  Student difficulties in algebra included attempts to convert expressions to equations.  Many students did not have a real sense of the structural aspects in algebra.  As a result, more explicit attention needs to be given to the transition between procedural thinking and structural thinking from arithmetic to algebra.

1.      What is algebraic thinking?

          Algebra itself is defined as the branch of mathematics dealing with symbolizing mathematical structures and numerical relationships.  Algebraic thinking varies depending on which feature of algebra we are addressing.  This includes terms, expressions, equations, word problems, functions, and graphs.  It can mean recognizing the symbol representations and the structural aspects.  Symbol representations include the idea of a variable representing an unknown number, or the equal sign meaning symmetry.  Structural aspects might mean understanding operations can only be carried out onto like-terms.

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

         One key concept, connection, and habit of mind I want students to understand is the equal sign.  Before algebra, the equal sign has been represented as something the generates an answer, but in algebra, it represents symmetry.  In other words, it means the the left and right side must be equal to each other.  To successfully understand the structure of algebra, understanding the equal sign is a highly important concept.  Students need to get in the habit of mind that if something is done to the left side of the equal sign, the same must be done to the right in order to keep the equation symmetric.  I also want students to be able to represent word problems algebraically.  It can be incredibly helpful if student are able to translate from mathematical phrases to algebraic expressions.  Some problems are to complex to work through by using guess and check methods, so understanding how to represent ideas algebraically can help simplify work.  This can be done by showing what phrases can represent a variable and what phrases can represent operations.

3. What are recommendations for teaching algebra for understanding?

          Some recommendations for teaching algebra for understanding is going to to depth as to what a variable means.  I remember when I first saw the use of letters in math and found it incredibly confusing.  It might help for student to see that patterns can be represented algebraically by expressions and equations.  For example, if we have a list of number 1, 3, 5, 7, ... we can use the equation x+2 to represent the next number we can get.  This can also represent many real problems that can relate to algebra.  Something like these can also be represented graphically.  The use of all these methods can help student connect algebraic concepts to patterns, pictures, and real problems.
         Algebra can also be taught with the use of objects, such as algebra tiles.  My cooperating teacher uses algebra tiles to help them work with equations visually.  This helps them see how an equation needs to be kept balanced.  If they add squares to one side, they need to add it to the other.   Having a visual model can help many students see the connection.
         To address student difficulties, I think it is helpful for students to see common errors and try to reason through why it is incorrect.  If students can see common misconceptions, I they are less likely make common errors, and also recognize when they are doing something incorrectly.