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.

Ila White 4 Win-Win Math Games

Ila White Literature Review

Title:  4 Win-Win Math Games

Author:  Marilyn Burns

Source:  http://www.mathsolutions.com/documents/WinWin_MathGames.pdf

Summary:
Marilyn Burns recommends using games to support students’ math learning.  They are ideal for students when they have extra time.  The key to making math games successful is how you introduce them and the classroom management you use to make the time truly valuable.

Math game tips:

a.      Choose games that are accessible to all students; this way students can focus on learning how to play.

b.      Play cooperatively and competitively. Cooperative games foster communication and unity while competitive games help kids learn to be graceful winners and losers.

c.       Choose games that require reasoning and chance. This helps level the playing field.

d.      Teach the game to the entire class at the same time. 

e.      Start a math games chart.  This creates a repertoire of independent math activities that students can play when they have extra time.

Four great games:

Four Strikes:  __ __ + __ __ = __ __
                        0 1 2 3 4 5 6 7 8 9

You have to figure out the number for each blank.  If a correct number is guessed, write it in the correct spaces and draw a line through it.  If an incorrect number is guessed, write an X out to the side.  Four strikes – you’re out!  The answer to this problem is 35 + 10 = 45.

10s	1s
1
2
3
4
5
6

101 and Out   Draw the following game board: 

Students roll a 1-6 cube six times.  With each roll students write the number rolled in either the

10s or 1s column beginning with line 1, then 2,

etc.  After six rolls, put 0 in all the empty 1s

columns.  Next, add to find the sum.  The winner

is the player closest to 100 without going over.

To teach:  Have students discuss as a table where it would be best to put the number.  The second time playing, have each student use their own table and decide on their own.

Seven Up:  This game requires a deck of 40 cards – four each of cards numbered 1 to 10.  A regular deck with the face cards removed works well.  To play, students deal seven cards face up in a row.  They remove all 10, either individual cards with 10 or pairs of cards that add to 10.  Each time a player removes cards, they replace them with cards from the deck.  When it’s not possible to remove any more cards, they deal a new row of seven cards on top of the ones that are there.  The game ends when it’s no longer possible to make 10s or all of the cards are used up.  To manage this game, the kids are in pairs.  One partner holds the deck for the other partner.  Then they switch jobs for the next game.  It could also be played as a game of solitaire.

Target 300:  This game gives students the opportunity to practice multiplying by 10 and multiples of 10.  The object of the game is to be the player whose total is closest to 300 after six rolls of a 1-6 number cube.  The total can be greater or less than 300, or exactly 300, but players must use all six turns.  Example game:           Cindy_________Julie

                                                                        6x10 = 60        5x10=50
                                                                        1x10 = 10        3x10=30
                                                                                    70                   80
                                                                         5x10 = 50       6x10=60
                                                                                    120                140
                                                                        6x10 =  60       5x10= 50
                                                                                    180                 190
                                                                        1x50 =  50       2x50=100                                                                                    230                 290
                                                                        4x20 =  80       2x10 =  20
                                                                                    310                 310

Analysis: I use math games a lot in my classroom.  However, they can be hard to manage.  This article gave me some great management tips.  I really like the idea of a math games chart and plan to incorporate that into my classroom.

Cindy Fonnesbeck Nine Ways to Catch Kids Up

Cindy Fonnesbeck Literature Review

Title: Nine Ways to Catch Kids Up

Author: Marilyn Burns

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

Summary: In every class there are struggling learners.  Ms. Burns list the nine important strategies to help students catch up.

            1.  Determine and scaffold the essential mathematics content. 

                        What is the most important skill and organize it in manageable chunks.

            2.  Pace lessons carefully.

                    Many struggling students need more time and practice to grasp the concept.

            3. Build in a Routine of Support

                        Make sure students understand what has been taught before letting them

                        work independently.

            4.  Foster student interaction.

                        Let students mingle and talk about what they have learned to help them

                        internalize what has been taught.

            5. Make connections explicit.

                        Use explicit instruction to help students make connections with math

                        concepts.

            6. Encourage mental calculations.

                        When students calculate mentally they improve their number sense.

            7. Help students use written calculations to track thinking.

                        Let paper and pencil be a tool to track their thinking.

            8. Provide practice.

                        Provide practice that supports the elements of what you have scaffolded.

                        Practice can include problems and games.

            9. Build vocabulary instruction.

                        Teach vocabulary and use it often when working with your students.

Analysis:  Struggling students need interventions before, during, and after the concept has been taught.  One suggestion from this article I think to be effective would be to teach the concept to the strugglers before the rest of the class.  By doing this the students will be double dipped and they can experience success while learning with the whole class.

Kim Detwiler Nine Ways to Catch Kids Up

Kim Detwiler               LITERAUTRE REVIEW

Title: Nine Ways to Catch Kids Up

Author:  Marilyn Burns

Source: http://www.ascd.org/publications/educational-leadership/nov07/vol65/num03/Nine-Ways-to-Catch-Kids-Up.aspx

Summary:

Three issues must be covered in effectively teaching mathematics to struggling students. 1.) Students must make connections among mathematical ideas so they do not see ideas as disconnected facts. 2.) Build students’ new understandings on the foundation of their prior learning. 3.) To judge mathematical understanding students must accompany their answer with an explanation of how they reason. Students who struggle with math must be provided these concepts/connections as well as the three aspects of numerical proficiency – computation, number sense, and problem solving. She suggests the following nine essential strategies to help struggling students:

1.      Determine and scaffold the essential mathematics content. Teach it explicitly.

2.      Pace lessons carefully. Make sure struggling students understand before going on.

3.      Build in a routine of support. Teacher does work, thinking aloud. Teacher does work, with students input. Students do work in pairs. Teacher and students show work and leave up on board. Students work individually, referring to board when necessary.

4.      Foster student learning. Struggling students must be given time to talk through the problems. Make student interaction an integral part of instruction.

5.      Make connections explicit. Help struggling students see how their previous knowledge will help in new problems. They need time to practice applying these connections.

6.      Encourage mental calculations. When students calculate mentally, they can estimate before they solve problems so that they can judge whether their answer makes sense.

7.      Help students use written calculations to track thinking. Help students see paper and pencil as a tool for keeping track of how they think.

8.      Provide practice. Practice should be directly connected to students’ immediate learning experiences and can even follow the four stage routine allowing for a gradual release to independent work.

9.      Build in vocabulary instruction. Struggling students have a weak understanding of words in math. They need to develop a firm understanding.

Analysis:

I must teach differently to provide my struggling students better scaffolding. I teach with the “I do, We do, You do” method. But I love Marilyn’s suggestion of “I do” with teacher talking aloud. “I do” again with student input. “We do” and then we talk about it and leave it up to see and refer to. “You do” now on your own BUT you can refer to what we have talked about. I know that I must not let the book decide everything I say and do during math instruction. I must make things happen for my struggling students.

Juli Crawford Snapshots of Student Misunderstandings

Juli Crawford Literature Review

Title:  Snapshots of Student Misunderstandings

Author:  Marilyn Burns

Source:  http://www.mathsolutions.com/documents/0210_Ed_Leadership.pdf

Summary:

Marilyn Burns discusses the need for one-on-one interviews with students to find out what they really understand in math.  She points out that written work, even when correct responses are given, does not show possible holes in students’ understanding.  She also mentioned that we conduct one-on-one assessments in reading, but not in math.

            She shares four snapshots of interviews with students and what she learned from it.  She then talks about how the information gained in the interviews could help guide her instruction and help her to be a more effective teacher.

Analysis: 

While reading the article, I thought about the way her snapshots made me think about more effective ways to teach my own students.  I am sure many of my students have similar misconceptions as the students she interviewed.  While interviewing individual students to assess their math understanding sounds like a daunting task on top of what we already do, I cannot help but think that I could apply what I learn from each interview to help the whole group.  It is something I would like to do more of and I can see how it would benefit my teaching and consequently my students’ learning.

Deborah Rolls The Need for Number Sense

Deborah Rolls Literature Review

Title:  The Need for Number Sense: The roots of many students’ math difficulties are evident as early as kindergarten

Author:  Nancy C. Jordan

Source:  Jordan, N. C. (2007, October). The need for number sense: The roots of many students’                    math difficulties are evident as early as kindergarten. Educational Leadership, V. 65, (2),                            63-65.

Summary:

For years educators have believed the relationship between ‘number sense’ and ‘math ability’ is important and intriguing.  This is due in part because we believe that ‘number sense’ is universal, whereas ‘math ability’ has been thought to be highly dependent on culture and language and take many years to learn.  According to Nancy Jordan, the link between the two raises many important questions and issues, including one of the most important ones, which is whether a child can be identified as early as kindergarten and trained in number sense with an eye to improving his future math ability.

            Jordan states that one characteristic of students who displays number sense difficulties is having trouble with simple computationresulting in the use of their fingers and will continue well into the third grade and beyond.  It is assumed that these children have trouble memorizing facts, which results in drill and skill remediation.

            Kindergarten is a time for learning math concepts, number sense, and skills.  Number sense is simply having the basic understanding of numbers, what they are, and how we use them.  The author suggests that kindergarteners be screened for number sense several times during the year.  Her findings found that early screening helps identify students at risk. 

            The article states that “although calculation fluency is not sufficient for succeeding in advanced math, such as algebra, it is a necessary foundation.”

Analysis:

            For me, number sense has long been difficult to define but easy to recognize.  I have known students that have good number sense can move seamlessly between the real world of quantities and the mathematical world of numbers and numerical expressions.  Reading this article has given me hope for those students who enter the upper grades with weak number sense.  I also researched several number-sense screening test and feel that our Trust Lands money that is being spent for a math tutor could be used more efficiently if our focus was on screening and remediating in the early grades. 

           

Suzy Sanders Place Value vs Face Value

Title: Place Value vs Face Value

Author: Rachel McAnallen

Source: http://www.zoidandcompany.com/withoutworksheets_files/%232PlaceValue.pdf

Summary:
      Ms. Math, Rachel McAnallen explains how to teach place value using $100, $20, $10, $5, $2, and $1 denomination bills. Target audience is 4-6th grades.

1. Give pre-test using copied bills and all possible three-digit combinations using these bills.
2. Demonstrate how numbers are adjectives, not nouns.
3. Demonstrate how numbers are written in different forms, related to rubber band.
4. Teach that numbers are spoken in descending order. Keep teens for last.
5. Relate place value to position power using school hierarchy.
6. Have a place value party.
7. Teach numbers vs digits. Increasing or decreasing value.
8. Post-test using same copied bills and all possible combinations as above.

Analysis:

This article would make it very easy for me to teach place value to a class. I taught 3rd grade and using this method would have been fantastic. Students seem to have a better grasp of money than any other sense of number in other forms. By 3rd grade they know that $100 is better to have than $5 etc. Working through this lesson and relating all numbers to money would make it much easier for them to understand that a number is more than just digits written side-by-side.

Suzy Sanders

Shari Smith Nine Ways to Help Floundering Students

Shari Smith Literature Review

Title: Nine ways: How do we help floundering students who lack basic math concepts?

Author: Marilyn Burns

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

Summary:

Marilyn Burns discusses ways to help students who are struggling with mathematical concepts.  She suggests there are three reasons students struggle:

                1. Students have not made connections between mathematical ideas.

                2. New information needs to be built on a foundation of what they already know.

                3. Students don’t have a deeper understanding of how numbers work.

She suggests nine strategies to make interventions successful.

1.       Determine the skills that are essential to the student and then break them into “manageable chunks.”

2.       Watch the students for clues that tell you they may be lost.   Some things may need to be “unlearned before they relearn.”

3.       Provide a routine of support by using four stages.  Stage one - Teacher modeling.  Stage two - Teacher modeling with input from students. Stage three – students work in pairs solving the problem. Stage four – students work independently.

4.       We learn best from teaching; encourage students to interact with each other.

5.       Make sure to explain connection to previous knowledge.

6.       Help students find patterns so that they can do math mentally.

7.       Writing things down helps to track the thinking process.

8.       Practice using written work or games.  

9.       Vocabulary is key to the students understanding.

Analysis:

Reading this article has helped me to understand some of the pitfalls that cause students to struggle mathematically.  It isn’t enough to check a worksheet; it is important to have a conversation to help determine the depth of understanding.

Robin Bailey 10 Big Ideas

Robin Bailey Literature Review

Title:  10 Big Math Ideas

Author:  Marilyn Burns

Source: http://www.mathsolutions.com/documents/2004_10_Big_Ideas.pdf

Summary:  Marilyn Burns gives ten ideas that will encourage students to find a love of math.  Here are her 10 big math ideas, and some of their key points.

1.      Success comes from understanding.

·         Tell students to do only what makes sense to them.

·         Encourage students to explain the purpose for what they are doing, the logic of their procedures, and the reasonableness of their answers.

2.      Have students explain their reasoning.

·         Don’t just look for a right answer.  Have students explain their answer with both right and wrong answers.

3.      Math class is a time for talk.

·         Interaction helps kids clarify their ideas, get feedback for their thinking, and hear other points of view.

4.      Make writing a part of math learning.

·         This is best used after the students have been able to math talk. 

o        What I learned about multiplication so far..

o        Today I learned…

o        I am not sure about…

5.      Present math activities in contexts.

·         It stimulates student’s interests, provides a purpose for learning, and makes math come alive.

6.      Support learning with manipulatives.

·         This helps make mathematical ideas concrete. 

·         Can be used to introduce concepts, pose problems, and to use as tools to figure out solutions.

·         Important not to be used in just the early grades, but also available to older students.

7.      Let your students push the curriculum.

·         Choose depth over breadth. 

·         You don’t want to cover a subject, you want to uncover it.

8.      The best activities meet the needs of all students.

9.      Confusion is part of the process.

·         Don’t expect all students to learn everything at the same time, and don’t expect all students to get the same message from every lesson.

·         Errors are opportunities for learning, and should support children taking risks without fear of failure or embarrassment.

·         Learning should be a long-range goal, not a lesson objective.

10.    Encourage different ways of thinking.

·         There is no one way to think about any mathematical problem.

Analysis:

This article had some great points, that in my opinion, if all math teachers were doing, all kids would love math.  Some of the ideas made me feel great, because they are things I am already doing (math talk, encouraging different ways of thinking), while other ideas reminded me about things I should be doing (writing in math, let students push the curriculum).  This is a good top 10 list I will be striving for.