It has been an eventful and demanding year for

mathematics teachers. With the adoption of Common

Core State Standards, mathematics education discus-

sions have been taken to a new level in many schools. The changes im-

plied by these standards have been cause for reexamination of our

teaching strategies and content knowledge. In the process, we have un-

packed new standards, realigned curricula, and revised lesson plans, all

with the goal of taking our students to a deeper level of mathematical

understanding.

Times of transition may cause anxiety, but transition can also

bring great excitement. KCATM has sought to provide many opportu-

nities for teachers to have rich discussion and voice their concerns,

successes, and curiosities during this time of transition. The Signatures

Series and the Annual Conference, attended by preservice, in-service,

and other mathematics educators, were extremely successful events.

Congratulations to our conference color Nook winner: Sharon Walker!

The Common Core State Standards have provided mutual goals for

Kansas and Missouri educators, removing boundaries as we discuss

our work during such professional development events.

KCATM aims to provide Kansas City area teachers with a place

to feel comfortable to learn and grow. If you have suggestions for this

summer or next fall, please email any member of the KCATM board.

We value your opinions and we want this to be YOUR organization.

April 20, 2013—KCATM Math Contests in Olathe May 2, 2013 - Annual KCATM Banquet

November 9, 2013—Annual KCATM Conference

I N S I D E T H I S

I S S U E :

Brainteaser 2

For the Classroom Elementary Middle Secondary All Levels

3

3

4

5

6

Rita’s Angle 8

PD Opportunities 9

Announcements 10

Upcoming KCATM Events

K A N S A S C I T Y A R E A

T E A C H E R S O F

M A T H E M A T I C S

The Summation W I N T E R 2 0 1 3 V O L U M E 1 2 , I S S U E 3

Letter from the President —Jeanine Haistings, Ph.D.

www.kcatm.net

The Summation is a publication of the Kan-sas City Area Teachers of Mathematics (KCATM).

SOLUTIONS….. Last issue’s brain teaser asked you an age problem. It went like this: Grandpa Jim and Grandson James have

the same birthday. Grandpa Jim’s age is an integer multiple of James’s age for six consecutive years. How old

is Grandpa Jim on the last year of the six-year period? I had one person send in the solution: Sonny Painter

wrote the answer is 66. He provided this very nice table:

James Jim

1 61

2 62

3 63

4 64

5 65

6 66

Mr. Painter also sent a solution to the previous brain teaser that asked for a way to change “white” to “house” if you could change only one letter at a time and if the you had to make a legitimate word at each step. Here is his solution to that problem:

white - write - wrote - wroth - troth - tooth - sooth - south - sough - rough - rouge - rouse - house

NEW PROBLEM…. For this issue I have decided to give you a sequence of numbers. Tell me where in the sequence the number “11” belongs. Here’s the sequence:

8, 5, 4, 9, 1, 7, 6, 10, 3, 2

Brainteaser

My Mind on Math and Math on my Mind

Sequence Solving —Rita H. Barger, Ph.D.

page 2 The Summation ˗ Winter 2013

Rita Barger, bargerr@umkc.edu, is an associate professor (mathematics education) and chair of the division of Curriculum and Instructional Leadership at the University of Mis-souri—Kansas City (UMKC). Her interests include recreational mathematics, motivation, professional development, learning styles, and attitudes and beliefs about mathematics.

Please send your answers to me at bargerr@umkc.edu. I would like to list names of those who solve the teaser and post solu-tions in the next newsletter.

In the world around us we experience three different representations of fractions. Our responsibility as teachers is to help our students understand these rep-resentations deeply enough to see connections and transfer their understanding to other situations. The three representations include:

Regional Models Discrete or Set Models Linear or Length Models

A regional model is an area that is easily divided into congruent parts. Pattern blocks (see the pictures be-low) are one example of manipulatives that can be used as a regional model of fractions in early grades. The red trapezoid is half of the yellow hexagon; the blue rhombus is one-third of the hexagon, and so forth. Later, in upper elementary and middle school grades, hundreds grids may be used to help students make connections between fractions and percentages. Circles, rectangles, and squares are simple shapes that may be used to encourage understanding of fractions. Different shapes should be presented as regional mod-els in an effort to help students see connections among representations.

Discrete or set models show how fractions can repre-sent subsets of multiple items, which is a common use of fractions in our everyday lives. Students can use concrete materials, such as counters or centimeter cubes, or they can draw pictures. For instance, to rep-resent a class with eight students where half of the students are female, a student might draw eight open

For the Classroom

Elementary

circles to represent the class and shade four of the cir-cles to show half the class, or four out of eight, as being female.

Linear or length models provide another visual repre-sentation of fractions. Number lines are most commonly used as linear models of fractions. Additionally, fraction strips and Cuisenaire Rods might help students gain a deeper understanding of fractions while alleviating com-mon misunderstandings associated with number lines. For example, strips of paper can be folded into equal length sections to help students see that it is the length between marks on the number line that makes up the fractional value. Because students must have prior expe-rience with number lines to effectively use them with fractions, I recommend that linear models be used in the intermediate grades.

At the beginning of this newsletter, I discussed the Com-mon Core State Standards for Mathematics. In addition to focusing on content knowledge, such as fraction con-cepts, it is our responsibility to bring as many of the Common Core Mathematical Practices into our lessons as possible. The suggested activities above, which inher-ently involve modeling with mathematics, provide many opportunities to bring all of the Common Core Mathe-matical Practices to life.

The Summation ˗ Winter 2013 page 3

Fraction Models —Jeanine L. Haistings, Ph.D.

Jeanine Haistings, haisitngsj@william.jewell.edu, is an associate professor at Wil-liam Jewell College in Liberty, Missouri. She received her Ph.D. from the University of Kansas. In addition to mathematics methods, her educational interest are inte-grating curriculum, fostering creativity, and effectively utilizing technology.

Data was selected as the topic of the third Signature Series workshop for Middle School teachers because of the increased emphasis the Common Core State Standards for Mathematics places on data concepts, especially in the 6th and 8th grades. A ra-tionale for the importance of data as a topic for middle grades is that students need to become proficient in statistical reasoning as part of informed citizenship. Sixth graders are now being asked to determine and interpret measures of variability as well as measures of center. They will also be asked to summarize and describe distributions of data. In seventh grade, students are being asked to draw inferences about populations through sampling and make comparisons of two populations. Additionally, probabil-ity is a major area of study in seventh grade. Eighth-grade students are being asked to represent and investigate patterns of asso-ciation in bivariate data. Measures of variability provide a picture of the spread of a data set. Range, a common measure of variability, is simple to cal-culate but provides limited information about how a data set is varied. Students in 6th grade will also be asked to calculate the mean absolute deviation and to create box plots in order to analyze variability. At the Signature Series workshop, middle school teachers discussed variability as it appears in graphs. For example, after examining the two graphs below showing test scores for two classes, we discussed which graph shows greater variability in test scores.

If students were to have difficulty determining which graph shows the greater variability, the teacher might ask them to describe what the graphs would look like if there were no variability. (There would be one single bar.) The graph of Class 1’s scores is clos-est to being a single bar, so Class 1 has less variability in scores than Class 2. This is just one example graph that could be used with a middle school class. Other types of graphs may be used to help students think about variability. Another activity, Central Tendency Card Game, was presented as a way for students to think about measures of center. For this game, the teacher will need to prepare a set of cards that includes four 2-cards, seven 6-cards, six 7-cards, four 8-cards, sev-en 9-cards, four 11-cards, four 10-cards, four 12-cards, four 16-cards, and four joker WILD cards. WILD cards may be used to rep-resent any number. Create additional cards for the median and mode values, and a range of values for the mean. For example:

How to play the game: Give each group of students a bag that contains one of the data set description cards and eight Digit Cards. Ask the students to create a data set from their cards that corresponds to the given description card. Initially groups are given 2 minutes to see if the digit cards they have been given fit the description card. At the end of 2 minutes, one person from each original group will meet as a representative group to trade cards. Each representative is allowed to trade a maximum of 3 cards during a trading session. The trading session lasts 2 minutes. At the end of the trading session, the original groups look at their newly acquired cards to determine if they can now make a data set that matches the description card. Additional trading sessions may be convened. Depending on the number of trading sessions held, it is possible that all, some, or none of the groups may be able to create the data set described on the description cards. Following the card game, teachers may choose to have students write about their experiences in terms of the mathematical thinking involved: Why they were able to, or not able to, find a solution using their set of digit cards?

For the Classroom

Middle

Discussing Data —Ann C. McCoy, Ph.D.

page 4 The Summation ˗ Winter 2013

Create a data set of 8 numbers with the following characteristics: Median = 8; Mode = 8; Mean is between 6 and 10

Ann McCoy, mccoy@ucmo.edu, is an associate professor of mathematics education at the University of Central Missouri. Her research interests include the efficacy beliefs of elementary teachers and the preparation of elementary mathematics specialists.

To support teachers and school districts as we tran-sition to the Common Core Standards for Mathematics (CCSSM; NGAC-CCSSO, 2010), the Missouri State De-partment of Elementary and Secondary Education (DESE) organized task forces to develop curriculum units. The task forces offer model curricula designed to engage students and teachers in the Mathematical Practices as districts work to implement CCSSM (NGAC-CCSSO, 2010). You can view and use the curriculum units by follow-ing the icon link (see image below) at http://dese.mo.gov/.

After clicking on the “Model Curriculum Units” icon, select a grade level (e.g., 1, 2 or 9-12) and course (e.g., Math, Algebra 1). Then, hit the blue “Find Unit” button on the far right of the screen. A table of unit titles should appear. Next, click on the red arrows to view a page with links to curriculum unit plans. The unit plans include instructional activities, form-ative assessments, and summative assessments in both Word and pdf formats, which are downloadable! Addi-tionally, unit plans include guiding Essential Questions and alignment to measurable learning objectives.

For the Classroom

Secondary

The Summation ˗ Winter 2013 page 5

Curriculum Unit Models —Sarah J. Hicks, Ph.D.

Want to share your story of assessments or instructional activ-

ities for an active mathematics class? Send it, along with a

short bio, to the editors of The Summation: Clare Bell presi-

dentelect@kcatm.net or Sarah Hicks newsletter@kcatm.net

Sarah Hicks, sarah.hicks@rockhurst.edu, is an assistant pro-

fessor in the School of Graduate and Professional Studies and

Department of Education at Rockhurst University in Kansas

City. Her research interests include teacher knowledge,

teacher learning, mathematics curriculum, and effective

classroom use of technology.

Please consider reviewing and using these model units. After you do, provide feedback by clicking on the red “Provide Feedback” button in the top right corner of the Model Curriculum Units webpages. The survey is provided to offer educators an oppor-tunity to offer constructive feedback on the struc-ture and content of the units and to make sugges-tions for revisions and additions. DESE values this input and encourages teachers from across the state to take time to contribute to this process by sharing their expertise. If you have questions, please contact DESE at modelcurrriculum@dese.mo.gov or 573-522-4003. Thank you to those of you who attended the KCATM Fall Conference informational and discussion session on this topic! We learned a lot from one an-other and I appreciate your questions and com-ments.

REFERENCE

National Governors Association Center for Best Prac-

tices (NGAC), Council of Chief State School Offic-ers (CCSSO). (2010). Common Core State Stand-ards for Mathematics. Washington, DC: Author. http://www.corestandards.org

Current literature in mathematics education, which in-cludes the Common Core State Standards for Mathematics (NGAC-CCSSO, 2010), calls for increasing students’ engage-ment in mathematical practices in order to develop knowledge. Communication is at the heart of mathematical practices. From a sociocultural perspective, knowledge is not constructed individually, but is co-constructed through inter-action within a learning community (Rogoff, 1990, 1998; Vygotsky, 1978). Furthermore, research in mathematics edu-cation indicates that students develop their own concep-tions of competence in mathematics through classroom so-cial interactions (Baxter & Williams, 2010; Gresalfi, 2009; Gresalfi, Martin, Hand, & Greeno, 2009). Math talk, or mathematical communication, is one type of social interaction. In the book Classroom Discussions: Us-ing Math Talk to Help Students Learn, Chapin, O’Connor, and Anderson(2003) focus on ways to increase math talk to im-prove learning. The authors suggest ways for teachers to engage students in mathematical discussions and extend conversations through five “productive talk moves”: revoic-ing, restating, analyzing reasoning, press, and wait time. Revoicing refers to a teacher’s translation of a student’s statement into clearer terms. The point of revoicing is not to correct the student, but to sort out the lack of clarity (Chapin et al., 2003), as in the following example.

Revoicing provides a second opportunity to hear an idea and additional time for students to think. In the example above, Carlos was also given an opportunity to clarify his idea for the teacher if her revoicing did not convey what he had intended.

For the Classroom

All Levels — A Book Review and Recommendation

Restating is much like revoicing, but refers to a student’s (rather than a teacher’s) restatement of what another student said. Restating allows opportunities for other students to add to the discussion and might provide further clarification. Re-stating also provides evidence of listening during discussion (Chapin et al., 2003). By asking students to restate mathemati-cal ideas, a teacher conveys that responsibilities for listening are just as important as responsibilities for speaking. Analyzing reasoning refers to thinking about one’s own or someone else’s reasoning, as illustrated in the next example.

Let’s Talk Math —Clare V. Bell, Ph. D.

page 6 The Summation ˗ Winter 2013

Carlos: I just added eight and eight, and then, um, I needed nine, not eight. So I added one more, and so it’s like taking eight and eight but I got nine, I mean sixteen, seventeen.

Teacher: So are you saying that you knew nine plus eight would be one more than eight plus eight, so you added one more to the sum of eight plus eight?

Teacher: Nikisha, can you tell me what you heard Carlos say, but put it in your own words?

Nikisha: I think so. He said that he already knows eight and eight, so that helped him figure out nine plus eight.

Susan: I think that zero is an even number because it’s two minus two. All the even numbers go by twos.

Teacher: Alec, do you agree with what Susan said?

Alec: No, not all of it.

Teacher: Why don’t you agree?

Alec: Well, she said even numbers go by twos. That’s right. But …with even numbers you can make two groups. You can’t put zero, um, zero is like nothing, into two groups.

Clare Bell, bellcv@umkc.edu, is an assistant professor in the School of Education at the

University of Missouri – Kansas City. Her research interests include the development of

self-regulated learning and creating equitable contexts for teaching and learning mathe-

matics through classroom discourse and connected classroom technologies.

The teacher did not take a position about the correctness of either of the students’ statements. Refraining from judgment allows for the possibility of extending a discus-sion. After a class has fully explored multiple ideas, the teacher might try to have the students come to

agreement (for instance, agreeing on what it means for a number to be even). Alternatively, the teacher might ask the students to think more about the topic for later discussion. Press refers to teacher questions or statements that en-courage students to contribute more to mathematical dis-cussion. Student can be encouraged to make comments or conjectures, provide explanations or reasoning, or engage in argumentation. For example, the teacher could extend the discussion of “evenness” above by asking, “Who else has something to add about even numbers?” In addition, stu-dents could be asked to agree or disagree with previous statements and provide reasoning or alternate explanations (Chapin et al., 2003). Finally, wait time is just what the term implies. Students may need time to put their thoughts into words. Allowing silence gives students opportunities to think. It also facili-tates bringing students who are rarely the first to raise a hand into the discussion (Chapin et al., 2003). For many teachers the urge to fill a silence is so strong that they must consciously think about wait time after asking a question. The examples provided illustrate productive talk moves that can be used during whole-class discussion. The teacher provided opportunities for students to develop as mathe-matical thinkers by refraining from giving answers, accepting a variety of responses for discussion, and encouraging stu-dents to voice their reasoning. Chapin, O’Connor, and An-derson (2003) further suggest that small-group talk formats might encourage students’ engagement in problem solving. While the teacher’s role is somewhat diminished, she/he would spend some time with each group to make observa-tions and occasionally ask questions or offer additional infor-mation. Small-group problem solving could then be followed by whole-group discussion. Additionally, pair-share or part-ner talk could be utilized to provide a safe space for students to try out voicing their ideas before engaging in whole-class discussion. I recommend Classroom Discussions (Chapin, O’Connor, & Anderson, 2003) to anyone trying to increase mathemati-cal discussions in mathematics classrooms. Norms for en-gagement in mathematical discussions can be cultivated at all grade levels. Children benefit from approaching mathe-matics in sense-making ways from the earliest grade levels.

For the Classroom

All Levels — A Book Review and Recommendation

Providing opportunities for students to construct their own knowledge through mathematical discussions may help them develop positive feelings of competence and open up doors of opportunity as they progress through school and beyond.

Let’s Talk Math, continued —Clare V. Bell, Ph. D.

page 7 The Summation ˗ Winter 2013

REFERENCES

Baxter, J. A., & Williams, S. (2010). Social and analytic scaf-

folding in middle school mathematics: Managing the dilemma of telling. Journal of Mathematics Teacher Edu-cation, 13, 7-26.

Gresalfi, M. S. (2009). Taking up opportunities to learn: Con-structing dispositions in mathematics classrooms. The Journal of the Learning Sciences, 18, 327-369. doi:10.1080/10508400903013470

Gresalfi, M. S., Martin, T., Hand, V., & Greeno, J. (2009). Con-structing competence: An analysis of student participa-tion in the activity systems of mathematics classrooms. Educational Studies in Mathematics, 70, 49-70. doi:10.1007/s10649-008-9141-5

National Governors Association Center for Best Practices (NGAC), Council of Chief State School Officers (CCSSO). (2010). Common core state standards for mathematics. Washington, DC: Author. http://www.corestandards.org

Rogoff, B. (1990). Apprenticeship in thinking. New York: Ox-ford University Press.

Rogoff, B. (1998). Cognition as a collaborative process. In D. Kuhn & R.S. Siegler (Eds.), Cognition, perception and language [Vol. 2, Handbook of Child Psychology (5th ed.), W. Damon (Ed.)] pp. 679-744. New York: Wiley.

Vygotsky, L. (1978). Mind and society. Cambridge, MA: Har-vard University Press.

FEATURED BOOK….

Chapin, S. H., O’Connor, C. & Anderson, N.C. (2003). Classroom discussions: Using math talk to help students learn, grades 1–6. Sausalito, CA: Math Solutions Publications.

Rita’s Angle

NCTM joined with the Mathematics Association of Amer-ica (MAA) to write a new Calculus Position Statement. You can find it on NCTM’s website with more details, but the main statement is:

Although calculus can play an important role in secondary school, the ultimate goal of the K-12 mathematics curriculum should not be to get students into and through a course in calculus by twelfth grade but to have established the mathematical foundation that will enable stu-dents to pursue whatever course of study inter-ests them when they get to college. The college curriculum should offer students an experience that is new and engaging, broadening their un-derstanding of the world of mathematics while strengthening their mastery of tools that they will need if they choose to pursue a mathemati-cally intensive discipline.

I encourage all of KCATM’s high school teachers and coaches to go to the website and read more about the position statement and consider what a calculus course in high school should look like and include.

NCTM and KCATM Updates —Rita H. Barger, Ph.D.

KCATM’s New Fall Conference a Rousing Success

KCATM tried something new this year. We held our an-nual conference in November rather than February, and the feedback was all positive. It appears that our mem-bers prefer to have this conference earlier in the school year so that they will have more time to try things they’ve taken from the conference in their classrooms. This year 126 attendees chose from 32 different sessions arranged in four grade bands plus general sessions. The mean overall rating of the conference from the 72 peo-ple who turned in evaluations was 4.6 out of 5. Based on this feedback we have decided to continue to hold the conference in the fall. Please save Saturday, November 9, 2013 for our next conference. Put in your request for PD money early; we guarantee that you’ll find the conference valuable no matter what grade level you teach. If you have a favorite lesson that you’d be willing to share with other teachers, please share that with our conference chair, Rita Barger (bargerr@umkc.edu). We’ll see that you receive a pro-posal form. Nothing improves the teaching and learning of mathematics more than sharing experiences with one another.

page 8 The Summation ˗ Winter 2013

Rita Barger, bargerr@umkc.edu, is an associate professor (mathematics education) and chair of Curriculum and Instructional Leadership at UMKC. Her interests include recreational mathemat-ics, motivation, professional development, learning styles, and attitudes and beliefs about mathematics.

For more information about membership

with KCATM or NCTM, go to

www.kcatm.net and www.nctm.org or

contact Rita Barger.

PD Opportunities

The Summation ˗ Winter 2013 page 9

Summer Opportunity - Rita H. Barger, Ph.D.

Are you looking for a summer course to take to help you move up on the salary schedule? Are you wanting to earn credits toward a Master’s Degree so you can qualify as a mentor teacher for student teachers starting next year? This may be the course for you. This July Dr. Barger will be teaching a two-week course on Creative Problem Solving for Teachers. The class will meet from 1:00 to 5:00 during the weeks of July 15 and 22nd. We will be addressing problems faced by classroom teachers and developing new and exciting ways to solve them. After taking this class you will find yourself applying its processes to all aspects of your life. Earn three hours of graduate credit in two weeks this summer. For more information, see the flyer below, or contact Rita directly at bargerr@umkc.edu.

Announcements

MCTM Middle School Math Contests –

March 23 & April 27, 2013

Looking for a contest for your middle school students? Middle school teachers in the Kansas City area are invited and encouraged to register their students to participate in the Missouri Council of Teachers of Mathematics (MCTM) regional 7th and 8th grade mathematics contest. Each school may enter 5 students in each of three individual levels – 7th grade without algebra, 8th grade without algebra, and 7th/8th grade with algebra. In addition, up to 4 students participating in the individu-al levels may also compete in a team competition. Awards are given to the top ten students at each level and to the top 3 teams. The participation fee is$7 per student. The contest will be held in the Education building on the UMKC campus on Saturday, March 23. The top 3 students at each level will be eligible to compete in the state finals contest to be held on the campus of the University of Central Missouri on Saturday, April 27.

More information about the contest and a registration form are available on the MCTM website (moctm.org). If you have questions, contact Ann McCoy, mccoy@ucmo.edu.For more information and to register for the conference, visit the MCTM website at https://www.moctm.org.

2012-2013 Final Board Meeting

Apr. 6, 2013 9:30 a.m. – 11:30 a.m.

www.kcatm.net

Jeanine Haistings, President president@kcatm.net

Clare Bell, President Elect, Newsletter Co-Editor presidentelect@kcatm.net

Rena Shull, Past President pastpresident@kcatm.net

Alan Gilmore, Executive Secretary executivesecretary@kcatm.net

Thomas Sullivan, Treasurer treasurer@kcatm.net

Sarah Hicks, Newsletter Co-Editor newsletter@kcatm.net

JoAnn Hiatt, Contest contest@kcatm.net

Rita Barger, NCTM Representative, Membership Chair, Conference Chair nctmrepresentative@kcatm.net

Randy Peterson, Publicity publicity@kcatm.net

Ann McCoy, Signature Series signatureseries@kcatm.net

Mike Round, Web web@kcatm.net

KCATM Board Officers

The Summation ˗ Winter 2013 page 10

For more information about member-

ship with KCATM, go to

www.kcatm.net or contact Rita

Barger at bargerr@umkc.edu.