Wendy Kidd-In Defense of Mathematical Foundations

Wendy Kidd Literature Review

TITLE: In Defense of Mathematical Foundations

AUTHOR:  W. Stephen Wilson

Source: EDUCATIONAL LEADERSHIP/MARCH 2011

SUMMARY:

1.  The problem of the disconnection of elementary school math and college math requirements.

2.  Elementary students need to be taught the understanding and fluency with addition, subtraction, multiplication, and division of whole numbers, fractions and decimals.  They also need to be taught complex multistep problems involving operations.

3.  Teachers shouldn’t be teaching alternative algorithms or supporting student invented algorithms. He states,  “A choice of a 10-year-old might not be a good choice for a college student.”

4.  The use of calculators shouldn’t be used in elementary school or even high school. The children only learn to push buttons and are inadequately prepared for college math.

5.  Arithmetic is the foundation of mathematics.  The standard algorithms for whole numbers are the only really big theorems that students can be in elementary school.

6.  Math standards need to recognize the importance of fractions.  “The problem is that without a solid foundation in fractions, students have little hope of succeeding in college-level mathematics.”

7.  The common ore standards do put mathematics a priority.  Students do have to memorize some number facts and learn algorithms.  The Common Core really makes the difference in teaching fractions. “Children are taught  a clear definition of fractions as numbers, and allows the usual four arithmetic operations on fractions to make sense.”

ANALYSIS:

This article has made me realize the importance of teaching the foundational skills in the elementary grades so that kids are prepared for further education in college.

Michelle Gardner-Service for Learning

Michelle Gardner Literature Review

TITLE: Service for Learning

AUTHOR:  Rahima Wade

Source: EDUCATIONAL LEADERSHIP/MAY 2011

SUMMARY:

1. “ Far from being a distraction, service learning can help students build academic skills while they become more involved in the community.”

2.   High stake assessments have  driven schools to adopt a rigorous and focus approach to reading, writing, and math instruction that takes up the majority of the school day.

3.  Many see service learning as a practice that produces mainly civic outcomes, so many educators have decided that there is no time for it.

4.  Two related facts to challenge this conclusion is that first high quality service learning engages students and second engagement is critical to academic achievement.

5.  Service learning is our avenue for keeping students interested.

ANALYSIS:

Service learning not only helps the community, but also helps our school community.  The article told of several projects that students completed that supported math skills.  Sometimes when students can’t understand mathematics from the book, when their math skills are used in real live situations they suddenly grasp the concepts taught. 

Shaylyn Ekins-Mental Math

Shaylyn Ekins Literature Review

Title:  Mental Math

Author:  Marilyn Burns

Source:  http://www.mathsolutions.com/documents/2007_Instr_MentalMath.pdf

Summary:  We use mental math in everyday situations…figuring tax or a tip, estimating the total at a grocery star, how much to pay the babysitter, doubling recipes, etc.

·         We should use mental math regularly in the classroom because it is a life skill.

·         “Hands on the table” math

o        Make this  a class effort, not a test

o        Model my mental math process

o        Debrief as a class and explain solutions

·         Focus on multiple ways to solve problems

·         Money is a good context for mental math

·         Allow students to self-correct

Analysis:  This article struck a chord with me.  As a 5^th grader, I remember my teacher doing a lot of mental math drills and I feel like I learned how to calculate mentally very well.  I plan on adding more opportunities for my students to use and practice their mental math skills, including modeling my thinking when I use mental math.  Mental math is also one of those areas of math that is very useful in the real world.

Blaine Norris-Building Confidence with Calculators

Blaine Norris Article Review
Title: Attitude Adjustments: Building Confidence with Calculators
Author: Lesa M. Covington Clarkson
Source: This is one of the articles that Mary Kay found for us. I am unsure of the source
Summary: The author states that there is a gap between black and white students on the National Assessment of Education Progress. And in order to reduce this achievement gap there must be high-qualified teaching and even higher expectations, culturally relevant mathematics environments, and 21st-century technology and curriculum. 1. Teachers of the 6
th grade in a 99% black elementary school with a 70% free and reduce lunch population used graphing calculators to instruct students in math. 2. Students used the graphing calculators to discuss patterns and data.
3. The calculators were used to interest the students in more sophisticated mathematics and the questions that could be answered using technology.
4. The teacher expected the students to do the higher level math, so the students assumed that they could do it to.
5. The students first became familiar with the calculators by entering numbers, performing operations, and then learning about the function keys.
6. Students first sorted m&m’s according to color before performing real-world tasks.
7. Students later searched newspaper ads for prices on similar makes and model of vehicles. They recorded and graphed the data. They could then predict reasonable prices for the given cars.
8. Students worked in pair and presented their finding to the rest of the class.
9. Once students could see how calculators could help them with data that they were interested in they could then move on to more abstract mathematics.
Analysis: I enjoyed reading this article. It presents the ideas that calculators can both motivate student achievement and point their thinking to higher educational goals. Many of the students would never have considered college or upper-level degrees leading to careers in technology or other advanced fields. Because the students were introduce to sophisticated tools at a basic and useful level they were able to gain confidence in solving real-world problems with them.
One thing I did not like about the article was its lack of detail in what the students were asked to do.

Michael Pedersen From Arithmetic to Algebra

Michael Pedersen’s Literature Review

Title: From Arithmetic to Algebra

Authors: Leanne R. Ketterlin-Geller, Kathleen Jungjohann, David J. Chard and Scott Baker

Source: Leanne R. Ketterlin-Geller, Kathleen Jungjohann, David J. Chard and Scott (2007, Nov.). From Arithmetic to Algebra. Making Math Count, 66-71.

Summary:

This article discussed key components needed to teach algebra. It also talked about how teachers can move from just teaching arithmetic to combining arithmetic and Algebra.

1.      Variables and Constants: The same rules apply for constants and variables.

a.       For example, the distributive property 4 x (3 +2) = 4 x 3+ 4 x 2. Then do the same problem but replace constant with a variable and solve.

2.      Represent and Decompose word problems algebraically.

a.       “Start by identifying the unknown” (p 69). What they are solving for is the variable.

b.      Write out the steps you need to find the unknown as Math expressions.

c.       “Check the problem using different Constants to verify the equation” (p 69).

3.      Symbol Manipulation

a.       Help students understand the equal (=) sign does not mean solve, but to balance each side

4.      Functions

a.       Find patterns

b.      “Sort and classify objects on the basis of unique properties” (p 69). For example all students wearing a blue shirt is a function or rule for sorting students.

Analysis:

Students need to start learning algebra, problem solving skills, and math concepts starting in Kindergarten. Students need to be explicitly taught these concepts and how to verbalize these concepts. In class teachers need to explicitly teach concepts using the algorithm, as well as, bringing in pictures and models to help students understand why the algorithm works. A teacher can use think-a-louds to verbalize how they are thinking about the problem and working through problem. After modeling and talking through the problem, students need to be given a chance not only to solve the problem with pen and paper but also be able to verbally discuss the problem. Students should be able to orally share what the problem is and the steps they will use to solve it. Students need to see and understand that problem solving is more than filling in a formula. Problem solving is creating math expressions and solving them using math properties.

LeVae Heiner What's Right About Looking at What's Wrong?

LeVae Heiner Literature Review

Title:  What’s Right About Looking at What’s Wrong?

Author:  Deborah Schifter

Source: http://www.ascd.org/publications/educationalleadership/nov07/vol65/num03/What%27s-Right-About-Looking-at-What%27s-Wrong%C2%A2.aspx

Summary:

Deborah Schifter shows how both students and teachers gain new mathematical understanding by examining the reasoning behind a student’s incorrect answers.

1. A 5^th grade teacher asked her students to find at least two ways to determine the products of several multi-digit multiplication problems.  They then met as a group at the end of the period to discuss their work.  She asked Thomas, (a student) to write his strategy for solving one of the problems (36 x 17) on the board.  Then, for a homework assignment, she asked the class to use what he was thinking in the 1^st steps, then revise it and come up with the right answer.

36 + 4 = 40     17 + 3 = 20      40 X 20 = 800 -4 = 796 – 3= 793

2. James envisioned the problem in a context, where the calculation could be used.  He thought of 35 x 17 as 36 bowls, each holding 17 cotton balls.  He explained that first he arranged the balls into groups of 10.  Each group of 10 had 170 cotton balls (10 x 17 = 170).  Then he added the three groups (170 + 170 + 170).  In addition to the groups of 10 bowls, there were another 6 bowls with 17 cotton balls in each (6 x 17).  To simplify that calculation, James thought of each bowl as having 10 white and 7 gray cotton balls, which yielded 60 white balls (6 x 10) plus 42 gray balls (6 x 7) for a total of 102 cotton balls in the 6 bowls.  He used the distributive property to find the answer: (10 + 10 + 10 + 6) x 17= 170 + 170 + 170 + 102 = 612.

3.  They took this problem to a teacher seminar and had the teachers work in pairs to find ways to approach the problem.  Again they looked at a wrong answer to make more discoveries.  800-(4 x 20) – (3 x 40) = 600.  As a result, they discovered that the answer was off by 12.

4.  Finally they drew an array, to think though the problem more clearly.  There were able the see their error and came to a better understanding of the procedure FOIL and why you multiply the first terms, outer terms, inner terms and last terms.

Analysis: I realized that as a teacher it is important to view math as a realm of ideas to be investigated rather than a set of facts, procedures and definitions.  This article made me reflect on a teacher I had in Junior High who helped me to love math and directed me in the mathematical course I should follow.   She helped us understand the important math ideas.

Jordan Oyler Writing in Math

Jordan Oyler literature Review

Title: Writing in Math

Author: Marilyn Burns

Source: http://www.mathsolutions.com/documents/2004_Writing_in_Math.pdf

Summary:

Usually when you think of mathematics you don’t often correlate writing with it. That’s how I’ve felt in the past and don’t remember writing much about math when I was in school.  That’s the same thing Marilyn thought when she first chose math education as her major. It took her quite a few years to bridge that gap and integrate writing into her daily math teaching, but she now doesn’t teach math without it.  In her article she says, “Not only did I see how writing helped students think more deeply and clearly about mathematics, but I also discovered that students’ writing was an invaluable tool to help me assess their learning.” She mentions an important thing about writing in math is she’s not worried about how correct it is, but rather the content that is important and the ideas the kids are writing down. She talked about four different writing assignments she uses.  First keeping a journal, she provides the students with some type if paper or notebook to be used as their journal. This provides the students an ongoing record of their learning. She gives them a prompt and they go with it. Second is writing about solving math problems. This allows the students to build up their problem solving strategies as well as trying multiple strategies to solve a problem. Sharing is big here too so they hear other ways to get the right answer. Third is explaining mathematical ideas. During a specific unit she asked students t explain what they know about that topic then once she reads them she can guide her lessons to correct misconceptions. And last writing about learning processes. For this she often has students write about what they’ve been doing by writing a letter to parents or guests in their class about the topic. Or she has them write what they like or dislike.

Analysis:

With the common core there has been a big push on writing, in every subject. I chose this article for that very reason so I could get some ideas on how to better myself with writing in math. She broke it down nicely into four writing assignments she uses and I would like to see myself start integrating those ideas into my teaching. It is very useful and can help me better asses if my students really understand the subject.