mathbits february 2008

Mathbits February 2008

President’s message

2

MDE Math Specialist Column

3

CONNECT 5

Matt Mentor 6

Overcoming miscon-ceptions

7

Spring conference poster

10

Focus on elementary 13

Focus on middle grades

14

Conference registration 18

Inside this issue:

State Finalists for the 2007 Presidential Award for Excellence in Mathematics and Science Teaching Announced

The National Science Foundation has announced the three state finalists for the 2007 Presi-dential Award for Excellence in Mathematics and Science Teaching (PAEMST).

In mathematics, the finalists are:

Liza Conzemius from Detroit Lakes High School Donna Forbes from Mahtomedi High School Karen Hyers from Tartan High School

They will be honored at the Minnesota Spring Mathematics Conference in Duluth this April. The Minnesota Council of Teachers of Mathematics honors each of the state finalists with a $500 cash award, a commemorative gold coin, a paid two-year membership in MCTM, and a complimentary registration for the 2008 Spring Mathematics Conference. One of the three state finalists will be selected by a National Selection Committee as the Presidential Award winner for Minnesota. The state Presidential Award winner will be announced by the White House this spring. The state Presidential Award winner will receive a $10,000 cash award from the National Science Foundation and will be invited to Washington, DC for a week of recognition. The 2007 award was open to teachers in grades 7-12. The 2008 award is open to teachers in grades K-6 with at least five years of teaching experience.

Election Results The Nominations Committee thanks the MCTM membership! We had a record number of ballots this year for our election. We thank all the candidates for their leadership and exper-tise in mathematics education and we congratulate the following people who won the elec-tion for our Board of Directors.

President-Elect Terry Wyberg

Vice President At-large Sara VanDerWerf

Vice President High School Lisa Conzemius

District 1 Director Joan Rustad-Huisman

District 4 Director Mary Roden

District 7 Director Jane Reck

We look forward to working with these MCTM members, and thank all the candidates for their hard work in math education.

Remember to check the MCTM website for the most current information about upcoming events! www.mctm.org

2008 MCTM Spring Conference

A+ in MATHEMATICS: Algebra and Much

More

April 25-26

Wow, it is already second semester. Every school year seems to go faster than the last.

In the September issue I asked for areas interested in hosting the MCTM Board meeting next September and did not get any takers. I am not sure if no one is interested or no one read that part of Mathbits.

This month I am making two more requests. First, we are trying to get one contact person in each district and eventually in each school to disseminate information for us. We can do a mass email to our members, but there are other math teachers out there we would like to include. If you would be willing to get a periodic email from MCTM and forward it to the teachers in your building, please contact your district director.

The second request is for you to start thinking about what you need to get your students ready for the new standards, especially the algebra requirements. If you have any ideas on how MCTM can assist you in these areas email those to your district director and encourage others to do the same.

See you in Duluth on April 24 for the symposium on algebra and on April 25-26 for the spring conference.

Nominations and applications for the 2008 Presidential Award for Excellence in Mathemat-ics and Science Teaching (PAEMST) are now being accepted. The Presidential Awards for Excellence in Mathematics and Science Teaching are the Nation's highest honors for teach-ers of mathematics and science. Teachers in grades K-6 with at least five years of teaching experience are eligible. Teachers applying for the 2008 PAEMST award must be nominated and self-nominations are accepted. Nominations can be submitted on-line through the PAEMST website at www.paemst.org. Contact Tom Muchlinski, the state program coordi-nator for mathematics at [email protected] or 612-210-8428 if you have any ques-tions. Applications must be postmarked by May 1, 2008, so an early nomination will pro-vide the applicant with sufficient time to complete and submit a high quality application.

Three state finalists will be chosen by a State Selection Committee. The Minnesota Council of Teachers of Mathematics honors each of the state finalists with a $500 cash award, a commemorative gold coin, a paid two-year membership in MCTM, and a complimentary registration for the Minnesota Spring Mathematics Conference. One of the three state final-ists will be selected by a National Selection Committee as The Presidential Award winner for Minnesota. The state Presidential Award winner is announced by the White House. The state Presidential Award winner will receive a $10,000 cash award from the National Sci-ence Foundation and will be invited to Washington, DC for a week of recognition.

Please consider nominating an outstanding mathematics teacher in grades K-6 for this pres-tigious honor. The 2009 award will be open to teachers in grades 7 – 12.

District Name Email 1 Bill Putnam [email protected] 2 Heidi Boerboom [email protected] 3 Betty Johnston [email protected] 4 Deb Guthrie [email protected] 5 Kristin Johnson [email protected] 6 Kathleen Miller [email protected] 7 Sonja Goerdt [email protected] 8 Paula Bengston [email protected]

District Name Email 1 Bill Putnam [email protected] 2 Heidi Boerboom [email protected] 3 Betty Johnston [email protected] 4 Deb Guthrie [email protected] 5 Kristin Johnson [email protected] 6 Kathleen Miller [email protected] 7 Sonja Goerdt [email protected] 8 Paula Bengston [email protected]


President’s Message

Judy Stucki

Nominate an Outstanding K-6

Teacher for the 2008 Presidential

Award for Excellence in

Mathematics and Science Teaching

Contact your District Director

Locate your district at

www.mctm.org

It’s About the Thinking! Ellie exclaimed, “I know, ten one, nine two, eight three, seven four, six five. It’s 5!”

What was the question that Ellie had answered?

She was asked, “What is 11 take away 6?”

Wow, that’s not thinking you can just walk away from—especially when knowing that Ellie is only five years old!

Needless to say, I was anxious to hear more of her thinking. I learned that she used many different approaches and relationships when she was adding or subtracting. Some of her pro-cedures were more efficient than others, and perhaps most interesting to me was the fact that she understood what she was doing and could describe and represent her thinking so clearly.

I found myself thinking about the five strands of mathematical proficiency (conceptual un-derstanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition) cited by the National Research Council’s 2005 report, Adding It Up: Helping Children Learn Mathematics. This report notes that memorization is not enough, and a key factor in learning with understanding and thus being able to use knowledge for problem solving is how learners represent and connect pieces of knowledge. Learning for under-standing requires that the five strands be interwoven throughout the learning experiences.

I found myself wondering about the kinds of experiences Ellie had been provided that had enabled her to develop such confident thinking around numbers. She has a firm foundation in counting that is very connected to her work with addition and subtraction; she has con-ceptual understanding of addition and subtraction; her procedures are remarkable; she is inquisitive and eager to learn more. Since her brothers had been working on multiplication, she was very interested in learning about that also. Listening to her skip count as she climbed the stairs, I knew she was well on her way!

It appears that Ellie is having experiences with mathematics that are enabling her to become increasingly proficient with her thinking, but what about all the other Ellies and Eddies? Given that improving students’ learning is dependent upon the learning experiences that they are provided, how can we as teachers more effectively help our students to develop their mathematical thinking? Surely this cannot be accomplished by the “perfect workshop.” Rather it is an exciting challenge that will outlast any individual teacher’s career.

Mathematics instruction “continues to emphasize the execution of paper and pencil skills in arithmetic through demonstrations of procedures followed by repeated practice” (Adding It Up, p. 4). Until we are able to change the learning experiences that we provide, we will con-tinue to get the same results we have always gotten, and the MCTM spring conference in Duluth is a great opportunity to get connected with your colleagues on this professional journey.

As an organization of professionals dedicated to promoting the teaching and learning of meaningful mathematics for all students, the MCTM is here to support educators in their efforts to improve mathematics instruction. With increasing challenges in teaching and newly revised standards in mathematics, I encourage you to become more involved in this organization and invite your colleagues to join MCTM. In addition, look for more informa-tion regarding professional development and technical assistance to be available around the state with funding appropriated by the 2007 Minnesota Legislature to help schools success-fully implement the state’s new math standards.

The Minnesota Academic Standards in mathematics have been revised to reflect new expec-tations for the achievement of all students. The new legislative requirements are in line with

(Continued on page 4)


Mathematics Specialist Report

Sue Wygant MN Dept. of Education

Until we are able to

change the learning

experiences that we

provide, we will

continue to get the

same results...


the following quote from Adding It Up (p. 1),

The mathematics students need to learn today is not the same mathematics that their parents and grandparents needed to learn. When today’s students become adults, they will face new demands for mathematical proficiency that school mathematics should attempt to anticipate. Moreover, mathematics is a realm no longer restricted to a select few. All young Americans must learn to think mathematically, and they must think mathematically to learn.

Key changes from the 2003 standards to be aware of are as follows:

• The progression of skills and knowledge has been changed to prepare all students to satisfactorily complete an algebra I credit by the end of 8th grade (a graduation require-ment for the class of 2015). This requirement raises the expectations for mathematics in 8th grade and impacts other grades, as well.

• Math reasoning is integrated into other content strands.

• College and work readiness skills are integrated into all strands.

• There is increased attention to appropriate uses of technology such as calculators, and dynamic geometry software.

• There is greater coherence in the development of skills and concepts within grade levels and among grade levels.

It is very important that teachers are aware that the standards document is not a curriculum document. Minnesota’s standards are written at the mastery level. Curriculum documents developed by districts provide much more information regarding when concepts are intro-duced and developed so that they are mastered at the appropriate grade level. Hence, teach-ers need to be very familiar with the standards at least two grade levels below and two grade levels above their own grade level.

Please refer to the standards document as well as a frequently asked questions document at http://education.state.mn.us/MDE/Academic_Excellence/Academic_Standards/Mathematics/index.html for more details.

Thank you for the daily commitment you make to work with your students and colleagues in teaching and learning mathematics. I look forward to working with all of you in the coming years, pursuing a common agenda of raising student achievement in mathematics.

Look for information coming out from the National Math Panel in the next issue of Mathbits.

(Continued from page 3)

It is very important

that teachers are

aware that the

standards document

is not a curriculum

document.

Cactus (noun): a Latin word, borrowed from Greek kaktos, a prickly, artichoke-like plant. The biologist Linnaeus applied the word to a different kind of prickly plant that we now know as cactus. By a second analogy, mathematicians have used the word cactus to refer to a type of connected graph in which no line segment lies on more than one cycle (closed path).

Examples of mathematical cacti.

Not a cactus.

http://education.state.mn.us/MDE/Academic_Excellence/Academic_Standards/Mathematics/index.html�





Mentoring New Teachers In September 2007 NCTM issued a position paper regarding mentoring early career mathe-matics teachers. The paper points out the high attrition rates of new teachers – nearly half leave the profession in their first five years and around 30 percent leave in their first three years. Causes include new math teachers receiving challenging teaching assignments for which they are unprepared, some do not have strong math backgrounds, they are often iso-lated from professional involvement with colleagues and they often receive little content-specific professional development.

The NCTM position is that states, school districts, and colleges and universities share the responsibility for providing a structured program of induction and mentoring for beginning math teachers.

MCTM CONNECT agrees that this is the ideal and we know that many good mentoring pro-grams are in operation in Minnesota, yet other districts are not providing meaningful help for early career teachers. While some of the attrition is no doubt due to salary, district politics, and the continuing struggle for raising the professional status of teachers, many of our newest colleagues could be retained if good mentoring was available to them. While this is best done at the district level, MCTM has several initiatives available to support new teachers. Partici-pants in our Virtual Mentoring program receive emails every two or three weeks that include lesson suggestions, problem ideas, useful websites and information about upcoming events.

To be included in this mailing, contact Ann Sweeney at the College of St. Catherine, [email protected]. We also offer a real live mentor, a teacher with experience at the grade level or content course that the beginning teacher needs. To arrange for a mentor, con-tact Larry Luck at [email protected]. We also are interested in helping district mentors de-velop a network for mutual information and support. Some of the professional development that new teachers need is available through our Spring Conference, Fall Conference and CONNECT activities.

So, if your district offers a mentoring program make sure that the mentors are making use of what MCTM has to offer. And each of us can take it upon ourselves to befriend, help and professionally support new teachers in our buildings. To paraphrase Cathy Seeley’s keynote talk at the Spring Conference a few years ago, “You can do it, we (MCTM) can help!”

To see the complete text of the NCTM position paper, go to www.nctm.org.

CONNECT Committee to Orient and Network New/Novice Educators into a Community of (math) Teachers

Mathematics Vocabulary—Where Does it Come From?

Steven Schwartzman’s The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English (1994, published by the Mathematical Association of America) explains the meanings and origins of over 1500 mathematical terms in plain English that is suitable for use with elementary, secondary, and college students. Arithmetic (noun, adjective): From the Greek arithmos “number,” from the Indo-European root ar- “to fit together.” A related borrowing from Greek is aristocrat, presumable a person in whom the best qualities are fitted together. Arithmetic must once have been conceived of as fitting things together or arranging or counting them. An arithmetic (stress on the third syllable) series is one in which each term is a fixed number apart from adjacent terms, just as the counting numbers of arithmetic are equally spaced. Interestingly, the same Indo-European root found in arithmetic appears in native English read, since when you read you have to fit the sounds together into words. So of the so-called three R’s—reading, (w)riting, and (a)rithmetic—two of them are etymologically related. Because arithmetic is a foreign word, English speakers have sometimes misconstrued it. In the 14th and 15th centuries it was known in Eng-land by the Latin-like name ars metrik “the metric art,” out of confusion with metric. It has similarly been called arith-metric.

mailto:[email protected]�



Dear Matt Mentor:

My eighth grade students seem to “glaze over” whenever we generalize a process by creating a formula, such as the laws of exponents. One of my colleagues said something about stu-dents having trouble with the concept of a variable, but didn’t say what to do about it. Can you give me more details about how to help students understand that a variable represents a number?

Dear Eighth Grade Teacher:

First, let’s take a look at the idea of variable. This may help us understand why students are often confused about this idea. Until 1650 AD, variables were only used as a placeholder for a specific unknown quantity. Since then, they have been used in many different ways. This table lists a few of these.

Note that in several of these equations, there are several different variables, and they have different meanings. For example, in the last example, the m and b are identified for particu-lar lines, while the y and x vary for every line. There are other examples where the variable represents an angle (sine x) or a statement (if p then q). So it is not surprising that students are confused about variables, since we use them widely and with multiple meanings, often without distinguishing very carefully among these meanings.

So now that we understand that formulas, such as the laws of exponents, are only one of the many ways we use variables, maybe we can find some ways of planning instruction so that students make sense of these generalizations and are able to use and apply them. The way we may have been instructed in these laws went through a sequence that goes something like this: give an example of a number – usually 2 or 3 – multiplied by itself several times, state the symbolic way to represent this, show what happens when you multiply say four 2s by three 2s, giving seven 2s, generalize this to the first law of exponents, then practice some problems and assign homework.

As an alternative, let’s look at the recent work of Robert Moses. Moses, who was well known as a leader during the Civil Rights movement of the 1960s, is now devoting his time and energy to helping students learn algebra. He calls algebra the new Civil Right, since it is the gatekeeper to all high wage employment in today’s economy. Moses believes that stu-dents learn algebra with understanding through classroom activities that lead them through the following steps, which I summarize in a table with some examples.

Use of variable Example

In a formula A = lw

To state a property a + b = b + a; b = an

As a placeholder or an equation to solve 15 = x – 5

To indicate a relationship, in this case, linear y = mx + b

Use of variable Example

In a formula A = lw

To state a property a + b = b + a; b = an

As a placeholder or an equation to solve 15 = x – 5

To indicate a relationship, in this case, linear y = mx + b

Ask Matt Mentor!!

Step in learning algebraic concepts Example of step 1) A physical event, an activity in a context that makes sense to students Measure a box or container

2) Represent this situation with a picture or model Use cubes to model filling the box or container

3) Talk about the situation with informal “people talk” Discuss methods of counting the cubes and finding how many it takes to fill the container

4) Moving to more structured and mathematical language Discussion leads to terms like dimension, and area, and stating formula in words

5) Representing the ideas using symbols Develop a generalization for finding the volume for the given figure, finding formula in symbols

Have a Question for Matt? Send your questions about teaching math topics to [email protected] and watch for Matt’s re-sponse in the next issue of Mathbits.






When you plan a rich activity incorporating these ideas, students can progress through these steps, perhaps over several class periods. Though it seems to be time-consuming, both for planning and instruction, it allows students to make the idea their own and remember it. The students’ statement of the generalization – in your case the first law of exponents – may not be in the same format as a textbook, but it will be remembered if it has meaning to stu-dents. Some students may not be ready to use symbols; however, if they can generalize us-ing language, they will still be able to remember, use and apply the formula when called for.

Finally, remember that generalization is one of the important reasoning processes of mathe-matics, one with which your students should be very familiar, which will come from multi-ple experiences. So, to summarize:

1) Give experience before formulas 2) Let students develop the formula rather than the teacher 3) Generalize away and take that glaze out of their eyes!

Good luck! Matt

Overcoming Misconceptions about the Equals Sign Despite all the reform and improvements suggested by NCTM and others, students still have misconceptions concerning basic concepts in math. One common misconception students carry into middle school and beyond is related to one of the most elementary concepts, the equals sign. According to the authors of Teaching for K-12 mathematical understanding using the Conceptual Change Model (2005), “many students read the symbol = as an action verb, such as ‘makes’ or ‘gives’; therefore, they may use the symbol to connect nonequiva-lent expressions.” This is because most elementary students only see the equals sign used after an operation with nothing behind it, for example 3 + 4 = ____. Students rarely under-stand the concept of an equation as something that has equal quantities on each side of the equals sign, but instead believe that the symbol simply tells them where to place the answer.

The concept of equality is first addressed in the Principles and Standards for School Mathe-matics (National Council of Teachers of Mathematics, 2000) in the Algebra standard at the Pre-K – 2 level:

Equality is an important algebraic concept that students must encounter and begin to understand in the lower grades. A common explanation of the equals sign given by students is that “the answer is coming,” but they need to recognize that the equals sign indicates a relationship – that the quantities on each side are equivalent. (p. 94)

Equivalency is not specifically addressed again until the Algebra standard in grades 9 – 12. At this point students are expected to:

• understand the meaning of equivalent forms of expressions, equations, inequali-ties, and relations;

• write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency – mentally or with paper and pencil in simple cases and using technology in all cases. (NCTM, p. 296)

In the time between, students encounter many uses and misuses of the use of the equals sign, and may develop misunderstandings related to the meaning of the equals sign. Misconcep-tions related to the use of the equals sign tend to follow two main patterns – operational use and relational use. Operational misuse of the equals sign can be seen in the work of middle, high school, and even in college students. The article 2 Algebra lists the example in figure 1


Feature Article Contributed by

Rochelle Lehman & Dr. Rhonda Bonnstetter Southwest Minnesota State University Marshall, MN

Equality is an

important algebraic

concept that students

must encounter an

and begin to

understand in the

lower grades.


which has the correct solution; however, the equals signs on the left of the second and third lines should not be there. The sign is being used incorrectly because it is not joining equal

things. Students often misuse the equals sign by writing it whenever they get a part of the answer like in figure 1, where they used it to describe what hap-pened after an operation on both the right and left side of the equals sign. Instead of using the equals sign to describe the chain of steps, students should record logical equivalent statements that lead them to toward the solu-tion. For example 2x + 6 = 17, then 2x = 11, and so x = 5.5. By requiring

students to use this method they will properly use equals sign and possibly prevent new mis-conceptions from forming. According to Stacey and MacGregor (1997) in Building Founda-tions for Algebra, “Contrary to work in elementary school, when students read algebra, the interpretation of “=” as denoting the answer is next is not appropriate.” Students often mis-use the equals sign by writing it whenever they get a part of the answer (as in figure 1) to de-scribe what happened after an operation.

The second misuse of the equals sign often isn’t noticed until students get to their first course in algebra. There they find problems in which there are variables and constant values on both sides of the equals sign. They may have problems solving for the variable because now they don’t see where to put an answer. The entire sentence looks confusing to students that do not understand the equals sign because they do not understand what they need to do to solve the problem or even what they are solving. According to the authors of Does Understanding the Equal Sign Matter? Evidence from Solving Equations (2006), “The association between equal sign understanding and equation-solving performance was significant.” Students that understood the equal sign knew that what they did to one side of the equals sign, they had to do the same to the other because equality denoted a relationship which needed to be kept in balance (Mann, 2004). Even though students are exposed to the equals sign beginning in 1st grade, they may graduate from high school with plenty of experience with the operational uses of the equals sign but without understanding the relational uses of the equal sign. It is assumed by texts that students will figure out the relational uses through the operational prac-tice, but students who do not discover this on their own or are not taught it will struggle with solving equations. Elementary teachers can help to prevent this misconception by thoroughly explaining to their students the meaning of the equals sign and giving examples of equations where the equals sign does not always lead to the answer, for example 9 = ___ + 7 and 4 + 5 = 6 + ___. Teachers of all grades can lead a class discussion on the meaning of the equals sign during classroom interactions as well as intentionally creating opportunities to discuss this misconception.

Resources

Jennings, M. (2007). 2 Algebra. Retrieved April 3, 2007, from http://www.maths.uq.edu.au/courses/MATH1040/math1040_notes.pdf

Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does Understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathe-matics Education, 37(4), 297-312.

Mann, R. L. (2004). Balancing act: The truth behind the equals sign. Teaching Children Mathematics. 11(2), 65-69.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

Stacey, K. & MacGregor, M. (1997). Building foundations for algebra. Mathematics Teach-ing in the Middle School, 2(4), 252-260.

Stephans, J.I., Schmidt, D.L., Welsh, K.M., Reins, K.J., & Saigo, B.W. (2005). Teaching for K-12 mathematical understanding using the conceptual change model. St. Cloud, MN: Saiwood Publications.

Figure 1

2x + 6 = 17 = 2x = 11 = x = 5.5

Even though

students are exposed

to the equals sign

beginning in 1st

grade, they may

graduate with plenty

of experience with

the operational use

of the equals sign

but without

understanding the

relational use.

Just for Fun—Find your way through the maze


Note: The following center pages (10-11) provide a poster with informa-tion about the MCTM Spring Conference. Pull the poster out and hang on a bulletin board at your school to share information with colleagues. The poster was designed by the MCTM Publicity Com-mittee.

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Geometric Grove Develop students’ visual and spatial sense with this one-of-a-kind task. Provide each student with a set of pattern blocks and a copy of dot grid paper. Give each student a predetermined amount of time to form a design using yellow hexagons, green triangles, red trapezoids, and blue parallelograms. Then challenge each student to copy his design on the grid and color it in.

Shed light on congruent figures with this technique. Divide students into groups of four and supply each student with a set of pattern blocks. Use pattern blocks to form a unique shape on an overhead projector. Then turn on the projector so that students will see the outline of the shape. Challenge each student in the group to use pattern blocks to form the shape shown. Then have group members compare their shapes to see if they are congruent (the same size and shape) or similar (same shape but different size). Have one person from each group report the group’s findings. Then repeat this process by forming a different shape on the overhead projector.

Source: Math Skills Workout by The Mailbox

A Formula for Success As we approach the hope of spring, every Minnesota math teacher’s head turns to … test preparation! How do we intelligently help our middle school students a) review concepts, skills and vocabulary; b) develop into savvy test-takers, and c) become motivated to do their best?

One issue that addresses all three goals above is students’ familiarity with and ability to use the Formula Sheet that accompanies the state mathematics test. All middle school math stu-dents need to • review measurement concepts (length v. area v volume, etc.) • practice measuring, using rulers and protractors, and • differentiate vocabulary terms related to measurement (radius v. diameter, pi, etc.)

Giving students practice using this helpful resource on the test will make them better test-takers and finally, providing some appropriate measurement practice in a fun format will in-crease their motivation to do well and their belief that they can be successful.

One strategy that has been used successfully in the past by many middle school math teachers is “Measurement BINGO”*. Students receive one version of a BINGO card (see next page for example), a copy of the grade appropriate MCA Formula Sheet, a calculator, and chips or counters. Each version of the BINGO card has the same 16 measurement problems on it, but arranged in a different order. At least six versions are necessary for a normal-size classroom.

Students work independently to find the answers to the 16 problems and write their answers in each of the 16 boxes on the card. They use the Formula Sheet for reference and the calcu-lator, if desired, for the computing. This typically takes about 30 minutes. Teachers may also encourage students to form small groups to compare answers and discuss any disagreements about them.

BINGO follows this work time. The teacher or some other leader calls off the answers from the Answer Key in random order and checks them off to keep track of which ones have been called. (You might also wish to cut the answers up and place them in the proverbial ‘hat’.) Students call out BINGO when they have four in a row vertically, horizontally, or diagonally. The leader verifies the validity of the winner’s claim by having the student tell the letter and


Focus on the Middle Grades Anne Bartel Minneapolis Public Schools (retired)

Focus on the Elementary Grades

Judy Hansen First Grade Teacher Brown Elementary Pipestone, MN

*This activity was origi-nally shared with Minnea-polis teachers by Karin Gabrielson.


answer of each square in his/her row. (Since there are multiple copies of each version of the card, more than one person could win at the same time—so be prepared for multiple winners in each round.)

Challenge #1:

Explain to the students how you made the different BINGO cards. How many differ-ent BINGO cards are possible to make with these same 16 problems?

Challenge #2:

Have students write measurement problems for another game of BINGO. Have them record and illustrate their problems on pieces of paper congruent to the spaces on a blank BINGO card.

Challenge #3:

Have a student select one answer from the game and challenge students to write measurement problems with that answer.

Go to www.mctm.org for an online copy of this BINGO card, one addi-tional version of the BINGO card, and the Answer Key.

http://www.mctm.org/_________�


There are many ways that higher education faculty can support K-12 teachers in implement-ing new and revised standards and curriculum. One way is to take a careful look at our own mathematics teaching practice. In this spirit, Christopher Danielson (MSU, Mankato), Anne Bartel (Minneapolis Public Schools, retired and occasional adjunct instructor), Lucy Payne (University of St Thomas), Cathy Wick (Normandale Community College) and Nancy Des-mond (Hamline University) invite others to join a developing conversation about the mathe-matical preparation of elementary licensure students.

Our initial conversations have been focused on the work of Philipp, et al. (2007) at San Diego State University. Their work has taken up the challenge of incorporating field and video experiences into the mathematics content courses for elementary teachers. They have made modest gains in content knowledge and had substantial impact on beliefs about the na-ture of teaching and learning mathematics.

We have recently received funding from Minnesota State Colleges and Universities (MNSCU) to gather our colleagues in mathematics and in education for a one-day meeting. At this meeting, we plan to develop a research agenda and a support network in taking on the challenge of carefully examining our courses with the ultimate goals of better preparing our students to teach mathematics in elementary schools, and of making the teaching of mathe-matics (such as through middle school licensure) a more appealing prospect to this popula-tion. We imagine this work building on the collaboration of the former Transforming Teacher Education (TTE) network.

We anticipate sending formal invitations to this event shortly. We hope anyone interested in joining the conversation-either in person or long-distance-will contact Christopher ([email protected] or 507-389-6401).

Reference

Philipp, R., Ambrose, R., Lamb, L., Sowder, J., Schappelle, B., Sowder, L., Thanheiser, E. & Chauvot, J. (2007). Effects of early field experience on the mathematical content knowl-edge and beliefs of prospective elementary school teachers: An experimental study. Jour-nal for Research in Mathematics Education, 38, (5), 438-476.

Conference on CAS in Secondary Mathematics Computer algebra systems (CAS) have the potential to revolutionize mathematics education at the secondary level. They do for algebra & calculus what calculators do for arithmetic: simplifying expressions, solving equations, factoring, taking derivatives, and much more! Attend the Fifth U.S. Conference on CAS in Secondary Mathematics. Come explore the fu-ture of mathematics education. Discover how secondary and middle school teachers are us-ing CAS in their own classrooms. Get classroom tested ideas developed for CAS-enhanced classrooms. Learn what other countries are doing with CAS. Interact with prominent CAS pioneers from the USA and beyond. Saturday, June 28, 8:15 AM - 4:15 PM & Sunday, June 29, 8:00 AM - 1:00 PM New Trier High School, Northfield, IL (approx. 20 mi. from O'Hare airport) Registration $195 before May 2 ($250 after May 2) On-line registration, updates, and hotel information available at http://meecas.org For more information or questions, contact: Ilene Hamilton [email protected] Dan Hall [email protected] Pat Bowler-Johnson [email protected] Sponsored in part by: Texas Instruments, Wright Group/ McGraw Hill Organized by MEECAS (Mathematics Educators Exploring Computer Algebra Systems)

Professional Opportunity for Teacher Educators

Professional Opportunity for Secondary Teachers


“Algebra is a student’s first experience with higher-level mathematics." "Algebra is the serious study of the last three letters of the alphabet." "All students should be doing algebra by grade 8."

Algebra is talked about a lot these days. Alan Schoenfeld (in Lacampagne, Blair, and Kaput [1995]) describes algebra as "an academic passport for passage into virtually every avenue of the job market and every street of schooling." Hyman Bass (2006) notes that algebra is viewed as foundational for all mathematics and science. Currently, about 40 percent of eighth-grade students in this country are enrolled in first-year algebra or an even higher-level math course (for example, geometry or second-year algebra).

At a time when maintaining our nation’s competitive edge means encouraging more students to con-sider math- or science-related majors and careers, should we address the challenge by moving more students into higher levels of mathematics earlier? Well, I am not so sure.

Yes, we have more students taking higher-level courses in mathematics, and yes, the path to a good job often begins with algebra. But is mandating algebra for all seventh- or eighth-grade students a good idea? Teachers of algebra frequently tell me that far too many of their students are not ready for algebra, regardless of how it is defined (first- or second-year algebra, integrated mathematics curricu-lum, etc.).

I regularly ask teachers, "What do you wish your students knew—and knew well—before taking their first course in algebra?" Although I was initially surprised, I have grown accustomed to hearing some teachers reply, "Basic multiplication facts."

Actually, most teachers indicate that their students don’t know as much about fractions as they would like. By fractions, I mean fractions, decimals, percents, and a variety of experiences with ratio and proportion. Another major topic on the wish list of algebra teachers is problem solving, but that’s on every teacher’s list.

So, if teachers could wave a magic wand, they would ensure that students beginning to study alge-bra—whether in a course with algebra in the title or an integrated curriculum—bring with them a strong background in the mathematics that precedes this first experience with higher-level mathemat-ics.

As Chambers (1994) notes, algebra for all is the right goal—we just need to make sure that we’re all targeting the right algebra in our teaching. This algebra would focus on topics like expressions, linear and quadratic equations, functions, polynomials, and other major topics of algebra. (Note that these ideas will be discussed in the National Math Advisory Panel’s report on algebra topics.)

If students were better prepared for introductory algebra courses, their teachers could think more seri-ously about how and when to have them use technology or solve problems that engage them and help them connect algebra to everyday situations. Who knows? Such experiences might eliminate the age-old question, "When am I ever going to use this stuff?" Furthermore, these opportunities will allow students to see the need for reasoning as they learn how to generalize relationships.

Of course, we must not overlook the importance of integrating the essential building blocks of alge-bra in pre-K–8 curricula, especially during the middle grades. Work with patterns is probably over-emphasized in some quarters as the defining component of algebra with younger learners, but early experiences with equations, inequalities, the number line, and properties of arithmetic (such as the distributive property) are foundations for algebra. Silver (1997) notes that integrating algebraic ideas into the curriculum in a manner that helps students make the transition from arithmetic to algebra also prepares them for what occurs later in algebra.

So is early access to algebra a good idea? Sure—for some—probably for many. More importantly, however, all students who are working to secure this valuable "passport" should begin their study of algebra with all the prerequisites for success, regardless of when the opportunity comes their way. [President’s Message reprinted from NCTM News Bulletin, January/February 2008, Vol. 44, Issue 6]

MCTM is an Affiliate of NCTM. If not a member of the National Council of Teachers of Mathemat-ics, consider the benefits of NCTM membership. Visit http://www.nctm.org/membership/

References

Bass, Hyman. Presentation to the National Mathemat-ics Panel, Chapel Hill, NC, June 29, 2006.

Chambers, Donald L. "The Right Algebra for All." Edu-cational Leadership 51, no. 6, (1994); pp. 85–6.

Lacampagne, Carole, Wil-liam Blair, and Jim Kaput (eds.), The Algebra Initia-tive Colloquium: Papers Presented at a Conference on Reform in Algebra. De-cember 9–12, 1993, vol. 1–2, Washington, D.C.: U.S. Department of Education, Office of Educational Re-search and Improvement. 1995.

Silver, Edward A. "Algebra for All—Increasing Stu-dents’ Access to Algebraic Ideas, Not Just Algebra Courses." Mathematics Teaching in the Middle School 2 (February 1997), Reston, VA: National Coun-cil of Teachers of Mathe-matics, pp. 204–7.

What Algebra? When?

Francis (Skip) Fennell

NCTM President

Minnesota Council of Teachers of Mathematics

TENTH SYMPOSIUM ON MATHEMATICS EDUCATION

"Algebra for All: What does this mean for Minnesota Schools?" Thursday, April 24, 2007

Duluth Entertainment and Convention Center – Duluth, MN

Algebra has long been regarded as the gateway for students in their study of advanced mathematics. Students’ lack of success in their study of algebra as well as a lack of success in their preparation for a formal algebra class is one of the defining characteristics of the achievement gap. Legislation passed by the state legislature and signed by the governor in the spring of 2006 requires the eighth grade students of 2010-2011 to success-fully complete the study of Algebra I as defined by the 2007 edition of the Minnesota Academic Standards for Mathematics. In addition, this same group of students will be the first group required to complete an Algebra II requirement (again, as defined by the 2007 edition of the Minnesota Academic Standards for Mathematics) for graduation. These new requirements, by themselves, will not close the achievement gap. In fact, they have real potential to widen it, unless we thoughtfully employ appropriate instructional strategies that are ef-fective in helping all students develop a deep understanding of mathematics.

Who should attend?

• Administrators • District curriculum leaders • Teacher leaders • Teacher educators • Teams from districts or schools

Why should you attend?

The symposium will provide practical tools and suggestions for addressing the achievement gap by focusing on the teaching of algebraic concepts throughout the grades K - 12. The keynote speaker and grade-level break-out sessions will:

• Examine the K - 12 implications of the new state algebra requirements and how they can help close the achievement gap.

• Actively engage participants in activities that illustrate effective instructional strategies for helping all students develop a deep understanding of important algebraic ideas.

• Discuss the importance of viewing algebra not as a one time course but as a recurring theme through-out the curriculum at each grade level.

• Explore how important mathematics concepts at each grade level, from number and operations in the elementary grades, proportional reasoning in the middle grades, and functions in the high school grades can play a role in preparing students for a successful experience with algebra.

9:00 Registration 9:30 Welcome Judy Stucki, MCTM President 9:45 Keynote Address Dr. Don Balka 10:45 Break 11:00 Algebra Breakdown Dr. Don Balka 12:00 Lunch 1:00 Grade level breakouts Navigating through the Minnesota Standards 2:30 Break 2:45 Closing Session Dr. Don Balka 3:30 Reception sponsored by McDougal Littell Publishing

REGISTRATION FORM REGISTER EARLY – Registration is limited

Minnesota Council of Teachers of Mathematics

TENTH SYMPOSIUM ON MATHEMATICS EDUCATION

“Algebra for All – What Does This Mean for Minnesota Schools?” Thursday, April 24, 2008

Duluth Entertainment and Convention Center - Duluth, MN

NAME__________________________________________________________________________________ TITLE__________________________________________________________________________________ SCHOOL or DISTRICT____________________________________________________________________________ ADDRESS_______________________________________________________________________________ _______________________________________________________________________________ CITY___________________________________________ STATE _____________ ZIP _______________ PHONE___________________________ E-MAIL _____________________________________________

INDIVIDUAL REGISTRATION $140.00

Includes one copy of NCTM’s “Navigating through Algebra” for your specified grade level, lunch and reception.

______ Vegetarian lunch required

Circle One: Grade Level Choice for “Navigating through Algebra” book: PreK - 2 3 - 5 6 - 8 9 - 12

DISTRICT or SCHOOL TEAM REGISTRATION $140.00 (Please complete the information for each additional team member—form available on MCTM website) (first registration)

Includes one copy of NCTM’s “Navigating through Algebra” for your specified grade level, lunch and reception for each team member. $90.00

(each additional registration) ______ Number of vegetarian lunches required

TOTAL REGISTRATION FEE ENCLOSED _____________________ METHOD of PAYMENT

__________ Check Number _____________________ Purchase Order Number ______ Credit Card (Purchase Order must be attached)

Credit card number _________________________________________ Expiration date _________________________________________ Type of card ______ VISA ______ MasterCard ______ Discover Signature _________________________________________

Lodging information is available on the website at www.mctm.org If you have questions concerning the symposium contact:

Karen Coblentz 507-934-3260 x600

[email protected]

Mail the completed registration form and payment to: MCTM P.O. Box 289 Wayzata, MN 55391

April 9-12 NCTM Annual Conference, Salt Lake City April 25-26, 2008 MCTM Spring Conference, Duluth, MN May 1, 2008 2008 PAEMST applications due November, 2009 NCTM Regional Conference, Minneapolis, MN

Non-Profit U.S. Postage PAID Permit No. 1967 Minneapolis, MN

Forwarding and Return Postage Guaranteed Address Service Requested

Mark Your Calendar

MCTM strives to provide membership with current information regarding mathematics educa-tion in the state of Minnesota. To accomplish this goal, we need an accurate, permanent address for each member. Is your correct address printed on the label of this issue of Mathbits? If not, contact Executive Director Tom Muchlinski at [email protected] or visit the MCTM web site (www.mctm.org) membership page to make your change. Student MCTM members and members in transition are encouraged to provide a permanent address. Newsletters mailed to student members will not be forwarded. Thank you for helping us stay in touch! FYI: In an effort to be cost effective, MCTM sends newsletters at USPS bulk rate. As a result, delivery times may vary between postal districts.

Do we have your correct address?

Please submit items for publication in the April issue of Mathbits to [email protected] by March 15, 2008. Email or call 651-631-5228 with any questions. - Teresa Gonske, Editor

The MCTM is an organization of professionals dedicated to promoting the teaching and learning of meaningful mathematics for all students by supporting educators in their efforts to improve mathematics education.

Check the mailing label for your membership renewal date. Renew online at www.mctm.org

Mission Statement:

Published by Minnesota Council of Teachers of Mathematics P.O. Box 289 Wayzata, MN 55391 www.mctm.org

Judy Stucki, President 952 - 544 - 1198 [email protected] Tom Muchlinski, Executive Director 612 - 210 - 8428 [email protected] Teresa Gonske, Mathbits Editor 651- 631- 5228 [email protected]

mathbits february 2008

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