Fall 2016 Final

Virginia Council of Teachers of Mathematics |www. vctm.orgVIRGINIA MATHEMATICSVol. 43, No. 1 TEACHERFall 2016Special Issue: Virginia Mathematics and Science Partnership GrantsVirginia Mathematics Teacher vol. 43, no. 1 1

Editorial StaffDr. Agida Manizade Dr. Margie Mason Brian Pratt Liam Downey Editor-in-Chief Associate Editor of the Assistant Editor Assistant Editor [email protected] Radford University Radford University Special IssueRadford University [email protected] The College of William and Mary Printed by Wordsprint Blacksburg, 2200 Kraft Drive, Suite 2050, Blacksburg, Virginia 24060Virginia Council of Teachers in Mathematics Many thanks to our Blind Peer Reviewers for Fall 2016 President: Jamey Lovin, Virginia Beach Public Schools Alfreda Jernigan, VCTM Board MemberPast President: Cathy Shelton, Fairfax County Public Schools Dr. Steve Corwin, Radford UniversitySecretary: Lisa Hall, Henrico County Public Schools Dr. Robert BerryMembership Chair: Ruth Harbin-Miles Dr. Betti Kreye, Virginia TechTreasurer: Virginia Lewis, Longwood University Mrs. Anita Lockett, Fairfax County Public SchoolsWebmaster: Ian Shenk, Hanover County Public Schools Jean Mistele, Radford UniversityNCTM Representative: Betsy Steadman, Hanover County Public Schools Dr. Matthew Reames, University of Virginia Elementary Representatives: Meghann Cope, Bedford County Public Schools; Eric Dr. Wendy Hageman-Smith, Longwood University Vicki Bohidar, Hanover County Public Schools Dr. Maria Timmerman, Longwood University Dr. Kateri Thunder, James Madison UniversityMiddle School Representatives: Melanie Pruett, Chesterfield County Public Schools; Skip Tyler, Henrico County Public SchoolsSecondary Representatives: Pat Gabriel; Samantha Martin, Powhatan Public SchoolsMath Specialist Representative: Spencer Jamieson, Fairfax County Public Schools Dr. Ann Howard Wallace, James Madison University2 Year College: Joe Joyner, Tidewater Community College Pam Bailey, VCTM Board Member4 Year College: Ann Wallace, James Madison University; Joyce Xu, Virginia Tech Robert Berry, University of Virginia Karen Zwanch, Virginia Tech Dr. Anthony Dove, Radford University The acceptance rate for VCTM journals is 20% 2Virginia Mathematics Teacher vol. 43, no. 1

Table of Contents: Awards Page 5 Message from the Editor 6 Note from the President 7 Virginia Department of Education 8 Technology-Enhanced Inquiry of Light and Optics Concepts 9 HEXA Challenge Fall 2016 14 Solutions to Spring 2016 HEXA Challenge Problems 16 Secondary Mathematics Professional Development Center 20 Key to the Spring 2016 Puzzlemaker 25 Technology Review 26 Enhancing Pedagogical Practices Through PD 27 Grant Opportunities 32 Good Reads 33 Upcoming Math Competitions 34 Interactive Mathematics Institute for Middle School Teachers 35 Math and Science Partnership Grants 40 Call for Manuscripts 44 VISTA ELIS Professional Development 45 Unsolved Mathematical Mysteries 51 Integrating Mandates for the Benefit of Professional Learning 53 Math Jokes 58 Evaluation of Lesson Study Based Teacher PD Model 59 Busting Blockbusters 64 The Puzzlemaker 67 Conferences of Interest 68Virginia Mathematics Teacher vol. 43, no. 1 3

Virginia Mathematics Teacher vol. 43, no. 1 4

Featured Awards 5 Congratulations to the 2016 Winners of the William C. Lowry Mathematics Educator Award: Middle School Awardee: Becky Pierce, Smyth County High School Awardee: Amy Lamb, Northumberlamb County College Awardee: Betti Kreye, Virginia Tech Math Specialists Awardee: Corrine Magee, Arlington County Congratulations to our Reigning Champion Hexa Challenge Winner Veronica Moldoveanu! Teaches AP Calculus AB and Geometry at Falls Church High School. With two consecutive victories, in the Hexa Challenge, she is now our Reigning Champion!Be sure to attempt our Hexa Challenge problems featured in this issue!Virginia Mathematics Teacher vol. 43, no. 1

Sharing the Knowledge:A Message from the EditorDr. Agida ManizadeWe are pleased to introduce this special issue of the 64, and the Hexachallenge, page 14. We also inviteVirginia Mathematics Teacher journal, focused onMathematics and Science Partnership (MSP) grants you to read the Technology Review and Goodsponsored by theDepartment of Reads columns. Please consider applying for ourEducation forthe past six featured grant onyears. Theseawards have page 32, and joinbeen granted to25 institutions, us at thefrom universitiesto public conferencesschools. Morethan 70 teams of listed here. Ifeducators havereceived this you have neverfunding andhave worked participated intirelessly toimprove the the journal,state ofmathematics consider actingeducation in theCommonwealth. as a reviewer forAs funding willcease in 2018, a future issue, orour editorialstaff have writing an articledecided thatmathematics teachers should hear about these based on yourprojects, and have access to the products created bythe teams. This can allow teachers across the experiences. WeCommonwealth to access and effectively utilize theeducational products in their own classrooms. We believe sharinginvited every Principal Investigator (PI) to submitan article briefly describing their project and any experiencesinformation which would be helpful to practicingteachers. You can find a list of the projects, PIs, based onand their contact information, provided by theDepartment of Education, on page 40. In addition reflectiveto the articles related to the special issue, thenormal recurring sections and challenges are practices is oneincluded in this issue. Please participate and inviteyour friends to answer Busting Blockbusters, page of the mostVirginia Mathematics Teacher vol. 43, no. 1 effective ways of distributing professional knowledge. If you are interested, please see the call for manuscripts on page 44. Thank for your continued interest in our publication, and please enjoy this special issue. Agida Manizade, Ph. D. Editor in Chief, Virginia Mathematics Teacher [email protected] 6

Note from the President Jamey Lovin Welcome Back! I outstanding support for our teachers in its hope everyone was able to implementation. It is an exciting time and I am enjoy time with families glad to be part of that as an educator in Virginia. and friends this summer. So, I will end where I started. We your Your VCTM Board board, and me, your President, want to be a source did spend some of our of encouragement to you, the Virginia Mathematics summer together, meeting Educator community. When speaking of our to talk about our plans for membership, one board member shared a most the organization in the fitting t-shirt she saw worn by a 2015 conferenceupcoming year. We examined our vision statement, attendee. It said, “I am a Math Teacher, what isgoals, and objectives and renewed our commitment your Super Power?” We all agreed Virginiato them. We want our focus to be on YOU, our teachers are super heroes! It shaped what we didmembers, and creating opportunities for you to be that day and became the theme for our 2016actively engaged with the organization, not just at Conference: We Are a Community of Mathconference time, but also all year long. Heroes! Teachers of mathematics in Virginia face a Please feel free to contact myself, or anychallenging year thinking about how we will board member, with ideas on how we can bettertransition to the new Standards of Learning. I am serve you.optimistic about the changes and believe our stateleaders have developed an exceptional plan that Jamey Lovin, VCTM Presidentprovides a first rate education for students and [email protected] Organization Membership Information 7 National Council of Teachers of Mathematics Membership Options: Individual One-Year Membership : $93/year, full membership Individual One-Year Membership, plus research journal: $120/year Base Student E-Membership:$46/year Student E-Membership plus online research journal: $61/year Pre-K-8 Membership: $160/year with one journal Pre-K-8 E-Membership: $81/year with one digital journal -$10 for per additional teacher Current National Council of Teachers of Mathematics Membership: 70,000 Members Virginia Council of Teachers of Mathematics Membership Options: $20 Individual, One-Year Membership $20 Institutional One-Year Membership Current Virginia Council of Teachers of Mathematics Membership: 750 MembersVirginia Mathematics Teacher vol. 43, no. 1

Note from The Virginia Department of Education:The Mathematics and Science Partnership Michael Bolling The Mathematics and Science Partnership funded since 2003, providing growth opportunities(MSP) program is designed to improve the content to an estimated 7,500 mathematics and scienceknowledge of teachers and the performance of teachers.students in the areas of mathematics and science byencouraging states, institutions of higher education Michael Bolling(IHE), local education agencies (LEA), and Virginia Department of Educationelementary and secondary schools to participate in [email protected] that: Improve and upgrade the status and stature of 8 mathematics and science teaching by encouraging IHE to improve mathematics and science teacher education; Focus on the education of mathematics and science teachers as a career-long process; Bring mathematics and science teachers together with scientists, mathematicians, and engineers to improve their teaching skills; and Provide summer institutes and ongoing professional development for teachers to improve their knowledge and teaching skills. Partnerships between high-need schooldistricts and the science, technology, engineeringand mathematics (STEM) faculty in institutions ofhigher education are the core of these improvementefforts. Other partners may include state educationagencies, public charter schools or other public orprivate schools, businesses, and nonprofit or for-profit organizations concerned with mathematicsand science education. The MSP program is a formula grantprogram to the states, with the size of individualstate awards based on student population andpoverty rates. Each state is responsible foradministering a competitive grant competition, inwhich grants are made to partnerships to improveteacher content knowledge in mathematics andscience. As a part of MSP grant requirements, IHEare required to provide public access to productsand professional development resources developedthrough grant funding. The Virginia Department ofEducation (VDOE) provides access to theseresources through the VDOE MSP website.http://www.doe.virginia.gov/federal_programs/esea/title2/part_b/index.shtml Over 80 MSP grant projects have beenVirginia Mathematics Teacher vol. 43, no. 1

Technology-Enhanced Inquiry of Light and Optics Concepts: Teachers’ Professional DevelopmentJennifer Maeng, Richard A. Lindgren, Jesse T SenechalAbstract PD Model The Developing Science Teachers’ Previous research has identified fiveUnderstanding of Light and Optics Professional components of PD likely to influence teacherDevelopment (PD) provided an integrated quality and student achievement (e.g. Loucks-approach to teaching science through inquiry and Horsley et al., 2010). These components include (1)educational technology for upper elementary, immersing teachers in inquiry, questioning, andmiddle, and high school teachers with the goal of experimentation to model inquiry teaching, (2)increasing their content and pedagogical engaging teachers in concrete teaching tasksknowledge for teaching physical science. Below, based on their experiences with students, and (3)we describe the PD model employed as well as focusing on subject-matter knowledge andteacher and student outcomes. Results indicated deepening teachers' content knowledge. Further,teacher’s understandings of light and optics content effective PD should be (4) intensive, long­ term,and their pedagogical knowledge for teaching and coherent and (5) be grounded in a common setthrough inquiry and technology improved of PD standards in order to show teachers how tofollowing participation in the PD. These results connect their work to specific learning standardshave implications for the implementation of PD for student performance (Loucks-Horsley et al.,that supports middle and high school teachers’ 2010). These components informed the Developingunderstanding of light and optics content. Products Science Teachers’ Understanding of Light andof the PD include teacher-generated lesson plans. Optics PD.Introduction POE Inquiry Model. Inquiry can be defined The Developing Science Teachers’ simply as “Students answering a research questionUnderstanding of Light and Optics Professional through the analysis of data” (Bell, Smetana, &Development (PD) provided an integrated Binns, 2005). Several models of inquiry instructionapproach to teaching science through inquiry and exist. This PD taught light and optics contenteducational technology. This MSP project, led by through a predict-observe-explain (POE) inquiryDr. Richard Lindgren, was a collaboration between model (Haysom & Bowen, 2010). The POE modelthe University of Virginia (UVa), Jefferson involves eliciting student ideas, discussing studentNational Laboratory (JLab), the Virginia School predictions, students making observations andUniversity Partnership, Albemarle County Public explaining their observations, and the teacherSchools, Charlottesville City Schools, Newport supporting students’ explanations with theNews City Schools, and Hampton City Schools. scientific explanation. Goals of the project were to (1) support Technology-enhanced inquiry. Researchupper elementary, middle, and high school indicates integrating computer-based models intoteachers’ content knowledge and conceptual inquiry instruction promotes students’ conceptualmodeling instructional skills to effectively teach understanding of scientific phenomena (NRC,science content outlined in Virginia’s Science 2011). The role of computer simulations is not toSOLs and (2) to support teachers in integrating replace inquiry investigations but to providetechnology-enhanced inquiry to improve student students with supplemental contact with theachievement in science. To accomplish these goals, variables tested in a real experiment or to visualizethe program held two summer institutes, one at the process that occurs at sub-atomic scale (Luft,UVa and one at JLab, during the summer of 2014. Gess-Newsome, & Bell, 2008). Research alsoThe summer institute focused on increasing indicates that computer simulations are useful forteachers’ pedagogical knowledge for teaching simulating labs that are impractical, expensive, orscience through technology-enhanced inquiry and too dangerous to conduct in a school-based setting,light and optics content knowledge. contribute to conceptual change, and provide toolsVirginia Mathematics Teacher vol. 43, no. 1 9

for scientific inquiry and problem solving (e.g. refraction and wavelength. They summarizedMaeng, Mulvey, Smetana, & Bell, 2013; NRC, patterns in their data to explain their observations,2011; Windschitl, 2000). which resulted in a formal statement of Snell’s LawPD Context (i.e. the Law of Refraction). PHET Simulations, developed by UC-Boulder, are a repository of free, Morning sessions during the summer downloadable simulations that address a variety ofinstitute involved teachers engaging in modeled math and science concepts. In a parallel morninghands-on activities to build their content activity, teachers used a plastic block, a laser beamknowledge and that they could easily modify to and a protractor to measure angles and verifyinclude in their own classroom instruction. Across Snell’s Law (Figure 1). This activity required somethe 10 days of the summer institute, teachers knowledge of sines and cosines, but could haveengaged in 43 different hands-on investigations of also been completed just by making use of thelight and optics. Many of these investigations Pythagorean Theorem. Knowledge of sines andrelated to the reflection and refraction of light (e.g. cosines has the potential to help teachers in otherVA SCIENCE SOL 5.3, 8.9, PH.8) and relied mathematical applications of science (e.g.heavily on an understanding of angles (e.g. VA determining the work done when pulling a cartMATH SOL 3.15, 5.11, G.3, G.4). For example, along a road with a handle inclined at a 45° angleteachers explored the ray model of light, in which from the road).light is represented as straight lines emanating froman object. Using a laser beam as a ray of light and a Teachers discussed the affordances andsingle plane mirror teachers investigated the Law limitations of simulations and generated resourceof Reflection (the angle of incidence is equal to theangle of reflection). They also used multiple plane Figure 1. A laser beam is used to illustrate how a ray ofmirrors to track a ray of light over several light incident on a glass block is reflected from the block asreflections to locate the final image. Later, they a faint red ray and the main ray is refracted through theused these same mirrors to understand how images block. A protractor is used to measure the angles involved.are actually formed, which required additionalmath skills. Using protractors, they measured the banks of simulations and animations to supportangles of incidence and reflection and looked for their teaching of light and optics content. They alsopatterns in how the angle of reflection varied with developed and received feedback on lesson plansthe angle of incidence (e.g. VA MATH SOL 3.19, that incorporated these strategies that they could5.17, 7.13). Each activity teachers engaged in directly implement into their classroom scienceinvolved some combination of making instruction. Finally, they discussed how they couldmeasurements, performing calculations, creating integrate the POE inquiry model and educationaland interpreting graphs, describing patterns, and technology in cross curricular ways. For example,made use of their knowledge of geometry and because the PHET simulation included a protractortrigonometry. 10 Afternoon sessions addressed practicalaspects of classroom implementation of light andoptics content using a technology-enhanced POEmodel. During this time, teachers learned strategiesto effectively integrate simulations and animationsto support students’ scientific investigationsthrough modeled lessons designed to reinforce thecontent they explored during morninginvestigations. For example, in one afternoonactivity, teachers used a PHET simulation (https://phet.colorado.edu/en/simulation/bending-light) toexplore the Law of Refraction. First, they predictedan answer to the question, “What happens to thespeed of light as a light ray passes through differentmediums (for example air into water)? Why? Howmight this affect what we see?” Teachers used thesimulation to make observations by manipulatingthe media (i.e. water, air, glass) through which thelight waves traveled and measured the angle ofVirginia Mathematics Teacher vol. 43, no. 1

to measure angles as the light rays passed through Content Knowledgevarious media, teachers discussed ways in which Results indicated teachers’ contentthey could reinforce students’ understanding ofangles and protractor use to both science and knowledge significantly improved from pre- tomathematics content. post-instruction on both Light and Optics content assessments; assessment 1 pre (M = 10.3), post (M For participating in the PD, teachers = 15.9) (t = 5.883, p < .001), assessment 2 pre (Mreceived physics graduate course credit, all = 28.8), post (M = 32.1) (t = 3.776, p = .001).materials needed to implement the modeled light These results suggest the PD positively influencedand optics activities in their classrooms as well as teachers’ understanding of physics content.generate new activities, feedback from peers andinstructors through follow-up sessions, and the Themes in the qualitative data supportedopportunity to attend and present lessons they these findings. For example, teachers perceiveddeveloped through the project at the annual their content knowledge to be limited prior to theVirginia Association of Science Teachers PD and that the PD helped them develop a deeperProfessional Development Institute. understanding of the content. For example, oneResearch Design teacher noted, “I haven’t honestly had physics since college, so it was really good for me to Participants included 24 teachers from 22 refresh my memory of physics and my knowledgeschools in 15 divisions. Twenty (80%) of the of physics and even go beyond what my studentsteachers taught in middle school, 28% of teachers need.” They also perceived that the content of thehad 5 or fewer years of experience and 64% held PD went beyond what they needed to know toMaster’s degrees in education. address grade-level physical science standards. For example, a teacher indicated, “There were some The PD was evaluated through a quasi- things that I won’t teach in eighth grade, like allexperimental pre-/post-test design in which teacher the understanding of the distance of the lens fromand student pre-assessments served as their own the focal point and all that. That’s high school andcontrol. The design assessed changes in teachers’ while I feel I need to know that, if it werecontent and pedagogical knowledge, and their professional development it goes beyond what Iperceptions of the PD as well as their students’ need.” Finally, developing their content knowledgescience achievement. Teachers’ content knowledge through the PD appeared to provide teachers withwas assessed on two light and optics content- confidence in their ability to explain physics ideas,knowledge assessments pre- and post-summer answer questions, and design lesson plans. Ainstitute as well as year-end. Teachers also teacher described how the PD supported hercompleted surveys related to pedagogical understanding the mathematics behind the physicsknowledge. Changes in their students’ science involved in the content: “It taught me more of thecontent knowledge were assessed at the beginning physics and the mathematical part behind things soof the year and following light and optics as I’m doing labs I’m not just following directionsinstruction via a researcher-developed instrument. and getting through but I actually understand the reasoning behind some of the things that we’re Quantitative data were analyzed doing.”descriptively and inferentially and qualitative datawere thematically analyzed. Two major limitations Pedagogical Knowledge pedagogicalneed to be considered when interpreting the results Teachers’ self-rateddescribed below. First, all data related to knowledge was statistically significantly higherpedagogical knowledge were self-reported by following participation in the PD for all assessedparticipants. Second, the research design did not pedagogical skills (Tables 1 and 2). Qualitativeemploy a control group, therefore, causal data support these findings. As exemplified by theinferences regarding the impact of the PD must be interview response below, many teachers noted theinterpreted with caution. use of simulations was a novel instructional toolResults that they would use to support their physical science teaching. “I honestly have not used Results suggested the PD positively simulations in the past and so being able to find ainfluenced teachers’ knowledge related to light and wide variety of simulations as a result of that classoptics content, pedagogical knowledge for teaching was great.”light and optics, and their students’ light and optics Others noted the use of the POE modelcontent knowledge. In addition, teachers had supported their thinking about how to integratepositive perceptions of the PD. inquiry into their physical science instruction:Virginia Mathematics Teacher vol. 43, no. 1 11

When we talk about the predict, and then teachers suggested more modeling prior to labs. the observe and then the explain; when I do For example, one teacher wrote, “Some of the lab other lessons now I keep that in the back of activities were difficult to follow through just by my mind. Like, Ok, when they should reading the directions. Having the teacher model predict something then they need to be the the procedure before more complicated activities ones that actually observe it, and then we would have been helpful. This would also model need to kind of regroup and explain and for the teachers how to model procedure for their make sure we kind of revisit the predictions students.” and make sure that they have the Conclusions and Implications knowledge that they need before they more on and so that model has really been Overall, results of this investigation suggest helpful with other lessons as well. that PD that supports middle and high schoolStudent Achievement teachers integrating technology through a POE A subset of teachers submitted pre/post model to teach light and optics content has thestudent achievement data using a researcher- potential to positively influence middle and highdeveloped 14-item Light and Optics content school students’ understandings of these concepts.instrument. The mean student score increased 17 Teacher-generated lesson plans to teach light andpoints from pretest to posttest, from 53% (SD = optics, including alignment with Virginia SOLs,20%) correct to 70% (SD = 17%) correct. Paired assessment plans, and associated student handouts,samples t-test indicated this was a statistically were the primary product generated through thesignificant gain, t (278) = -17.196, p < .001. UVa-JLab project. These lesson plans as well as In interviews, teachers indicated they PD materials from the summer institute (labperceived that the POE instructional model, activities, PowerPointTM slides, instructionalcombined with their enhanced content knowledge videos and photos) are available at: http://facilitated their students’ learning. For example, galileo.phys.virginia.edu/outreach/one teacher indicated: ProfessionalDevelopment/UVa-JLab/ I think they seemed like they understood it, teacher_institute/2014-labs.html. These materials based on hearing their explanations of have the potential to support teachers in defending their answer with the initial implementing light and optics content through a prediction and trying to support it, and also technology-enhanced POE model as well as just when they were explaining their facilitating teachers’ considering the potential to reasoning for why they thought a certain integrate science and mathematics content (e.g. way, they were able to support the answer. patterns, data collection, angles).Other teachers responded similarly, that theinvestigative nature through which they were able Referencesto teach the content had a positive impact on Bell, R., Smetana, L. K., & Binns, I (2005).student learning.Perceptions of the PD Simplifying inquiry instruction. The Overall, the majority of teachers (95%) Science Teacher, 72, 30–34.indicated the PD program met or exceeded their Haysom, J. and Bowen, M. (2010). Predict,expectations. Further, 87% of participants said observe, explain: Activities enhancingthey would recommend or highly recommend the scientific understanding. NSTA Press:program to other teachers. Arlington, VA. Most participants responded that the hands- Loucks-Horsley, S., Stiles, K.E., Mundry, S.,on labs including the integration of technology, Love, N., & Hewson, P. (2010) Designingand the collaborative nature of the projects were professional development for teachers ofthe most effective components of the PD. For mathematics and science. (3rd Ed.)example, one teacher wrote, “Getting all the hands Thousand Oaks, CA: Corwin Press.-on experience in doing the labs. It made all the Luft, J., Gess-Newsome, J. & Bell, R.L. (eds).concepts ‘real’ for me, and I'm glad that we got to (2008). Technology in the secondarydiscover them on our own.” The “talented and science classroom. NSTA Press: Arlington,knowledgeable professors” were also identified as VA.a program strength. When asked to make Maeng, J.L., Mulvey, B.K., Smetana, L.K., & Bell,suggestions for improvements, a number of R.L. (2013). Preservice teachers' TPACK:Virginia Mathematics Teacher vol. 43, no. 1 12

hgtechnology to support inquiryinstruction. Journal of Science Educationand Technology, 22, 838-857. DOI:10.1007/s10956-013-9434-z NationalResearch Council (NRC). (2011).Learning science through computer gamesand simulations. Washington, DC:National Academy Press.Windschitl, M. (2000). Supporting the development of science inquiry skills with special classes of software. Educational Technology Research and Development, 48, 81-95.Jennifer MaengCurry School of EducationUniversity of [email protected] A. LindgrenDepartment of PhysicsUniversity of [email protected] T SenechalMERC School of EducationVirginia Commonwealth [email protected] Mathematics Teacher vol. 43, no. 1 13

HEXAChallenge Problems created by: Dr. Oscar TagiyevOctober Challenge:It is 2016. If we continue writing the digits, 2, 0, 1, 6, in this order N times, we’ll GET a different numberK=201620162016...2016 where the digits 2, 0,1, 6 are repeated N times. Prove that K can not be a perfectsquare of any integer number.November Challenge:There are N people that live in a city, where there are two main competing companies. Out of these people,n know each other, because they work for the same company. m people also know each other becausethey work in the same city, but in a rival company. Personal relationships between workers at rivalcompanies are not allowed. What portion of the population of the city, does not work for either of thesecompanies, but can knows exactly 1 person from each company. December Challenge: Contest Alert! Given an infinitely large set of different types of Virginia Mathematics Teacher is triangles, if one randomly selects a triangle, what conducting a contest for educators are the chances of this triangle being obtuse?Virginia Mathematics Teacher vol. 43, no. 1 and students who can solve the greatest number of problems cor­ rectly by 2/29/2017 The winner will receive a prize and will be featured in the next issue of the VMT. Send your solutions to [email protected] with the email subject line: Hexa Challenge 14

Please be sure to state your assumptions as you solve each problem. Answers to the Fall 2016 Hexa Challenge Problems will be featured in the Spring 2017 Issue of Virginia Mathematics Teacher.January Challenge:Solve the following equation: February Challenge:Prove the following for any integer number, n: March Challenge: Figure 1 In an Isosceles triangle ABC, AB=BC, Angle B is 20 degrees, AC is 15 five units Point D is on BC so that Angle BAD is 30degrees, and angle DAC is 50 degrees. Point E is on the side AB, so that Angle ECB is 60 degrees and angle ECA is 20 degrees. Find the length of DE. See Figure 1.Virginia Mathematics Teacher vol. 43, no. 1

Solutions to Spring 2016 HEXA Challenge ProblemsApril Challenge:In a trapezoid ABCD. E is the midpoint of AB, F lies on CD, and EF is perpendicular to CD.Estimate the area of the trapezoid if EF= h and CD = b.SOLUTION :Draw line GE parallel to CD. G lies on line AD. H is the point of intersection between lines GE and BC. Thearea of the parallelogram GHCD is equal to the area of trapezoid ABCD, because the triangles EHB and AGEare equal to each other by angle-side-angle congruence condition. Thus the area of the trapezoid equals b timesh. May Challenge: In a triangle ABC. AD and CF are medians, D is on BC and F is on AB. The medians intersect at O. What is the ratio of the triangular areas DOF to ABC?Virginia Mathematics Teacher vol. 43, no. 1 16

Solutions to Spring 2016 HEXA Challenge ProblemsConstruct the third median BG. The three medians AD, CF and BG split ABC into six triangles (AOF, AOG,GOC, DOC, BOD, and BOF) whose areas are equal to one sixth of the ABC area. The area of the triangle BDFis one fourth of ABC area, as those triangles are similar. Thus the area DOF equals area of DOFB - area ofBDF = 2 x one sixth - one fourth = one twelfth of ABC. The area of DOF equals the area of DOFB minus thearea of BDF. The area of DOFB equals 2 times 1/6th the area of ABC, as it is the sum of the areas of the trian­gles BOD and BOF. As the area of BDF equals 1/4th the area of ABC, the area of DOF equals ofthe area of triangle ABC.June Challenge:AB is the diameter of a semicircle. AC is the diameter of smaller semicircle which completely inside the firstsemicircle.The line FG is tangent to the smaller semicircle and parallel to AB. The length of FG is 10 units. Find the ar­ea of the bigger semicircle that does not include the area of the smaller semicircle.SOLUTION :Let’s translate the smaller semicircle to the right, so the center points of both semicircles will overlap.The shaded area will remain the same, as the area of the smaller semicircle is not changing: Therefore: Shaded area equals:July ChallengeMr Saver has $ A, spends nothing and saves all money he has been paid by the clients. If a client pays any­thing, Mr Saver estimates the \"significance\" of the payment by calculating the ratio of this payment to the totalamount he has saved, including the last payment. During this month, Mr Saver has had N clients, who paid him$ B in total. Assume R - the maximum \"significance\" ratio. Find the minimum value of R.SOLUTION :Assume so far Mr. Saver has had N clients who have paid him the amounts x1, x2, x3, respectively. If R is the\"significance\" of their payments, then:... 17Virginia Mathematics Teacher vol. 43, no. 1

Solutions to Spring 2016 HEXA Challenge ProblemsThe inequalities could be rewritten in the following way:...Let’s multiply all of the inequalities, then:August Challenge:Four spheres are all tangent to a plane. Three spheres with the radii 2 units, 1 unit, and 1 unit respectively, arestanding on the plane around the fourth sphere which is smaller. If all four of the spheres are tangent to eachother, what is the radius of the smaller sphere?Note that If two spheres with radii R and r are on the same plane and touch each other, the distance betweenthe points at which the spheres are touching the plane can be expressed through the Pythagorean theoremThen, all the arrangement of the four sphere centers O1, O2, O3 and O4 could be projected on a plane aspoints A, B, C, and D respectively. The projection of the centers on the plane is the figure below. Where dis­tance AB is the distance between the projected centers.Below on the left is the side viewConsider the top view, of the four spheres with centers O1, O2, O3, and O4 (above right). We can see that The solution to this equation can then be calculated as follows:We have to choose the smaller root because x must be less than 1. Therefore 18Virginia Mathematics Teacher vol. 43, no. 1

Solutions to Spring 2016 HEXA Challenge ProblemsSeptember Challenge:Inside a sphere (the original print had a typo that said circle) there is a polyhedron where all n vertices are onthe sphere. Prove that the number of points where the diagonals intersect each other, cannot exceed:SOLUTION :The number of quadrilaterals that can be formed out of n points on the circle is:Each of the quadrilaterals has two diagonals, therefore they can intersect at only one point. As a result, thetotal number of points of intersection for the polyhedron can never exceed this number.Virginia Mathematics Teacher vol. 43, no. 1 19

Secondary Mathematics Professional Development CenterDr. Agida Manizade, Dr. Laura Jacobsen, Christine Belcher, Robert Thien, Jamey Lovin, Stephanie Brady, Dee BakerAbstract based on work related to our projects in state, The Secondary Mathematics Professional national, and international professional outlets. TheDevelopment Center (SMPDC) has been funded by purpose of this paper is to present materialsmultiple Math and Science Partnership grants developed by secondary mathematics teachers, forthrough the Virginia Department of Education secondary mathematics teachers.(VDOE) since 2010 through the present. TheSMPDC serves high school mathematics teachers Introductionas well as middle school teachers teaching high- The SMPDC was originally funded in 2010school-level mathematics classes. The main goal of by a Math and Science Partnership grant from thethe Center is to provide a professional development VDOE and has received six consecutive years ofprogram to teachers interested in improving their funding through MSP grants. The individualmathematical knowledge and pedagogical content projects’ foci varied each year, based on the goalsknowledge (PCK). Our research has shown that as identified by VDOE, however, they shared theteachers who participate in these programs have common themes of: 1. Improving teachers’significantly improved their subject matter mathematical knowledge and pedagogical contentknowledge as well as their PCK in algebra, knowledge; 2. Developing and improving an onlinestatistics, probability, geometry, and Algebra PD model; and 3. Developing resources forFunctions and Data Analysis. The SMPDC has mathematics classrooms in critical areas identifiedpartnered with public school systems across the by the VDOE. In addition, the project PrincipalCommonwealth, several institutes of higher Investigators (PIs) published dozens of researcheducation in Virginia, the Virginia Math and papers (e.g., Corey, D.et. al., 2016; Manizade &Science Coalition and NASA. The SMPDC has Jacobsen, 2013; Manizade, & Martinovic, 2016) inserved 352 mathematics teachers since its original national, international, and state outlets. Ourfunding in 2010, providing over 700 contact hours purpose for this paper is not to focus on theof graduate-level professional development research findings, but instead the outcomes mostcoursework to relevant andthe teachers. The useful forparticipants have practicingcreated several mathematicshundred teachers. Weproducts intend to useavailable to this paper toteachers across disseminate thethe knowledgecommonwealth, gained byincluding lesson teachers duringplans, unit plans, these projectsperformance- and provide anbased opportunity toassessments, and use theclassroom materialsvideos. SMPDC produced byfaculty have their peers.produced 39 These include,publications and but are notpresentations limited to Figure 1. Sample performance-based assessment taskVirginia Mathematics Teacher vol. 43, no. 1 20

hundreds of unit and lesson plans, problem based Figure 3. Sample of student workassessments, classroom videos, and teacherinterviews. All of these products are published on Mathematical Education of Teachers II (AMS/the project website and are freely available for use CBMS, 2012) report recommended continualby teachers across the Commonwealth. Several improvement of mathematical knowledge andexamples will be discussed later in this paper. teaching skills, with such improvement promoted by regular interactions between teachers, The online PD model for secondary mathematicians, and mathematics educators in theteachers was developed through a collaborative creation and analysis of mathematics lessons, texts,effort with public schools, state universities, andother institutions across the Commonwealth suchas Virginia Commonwealth University, the Collegeof William and Mary, Longwood University; theVirginia Math and Science Coalition; and NASA’sLangley research facility. Courses are transferablebetween partnering institutions. Online classeswere designed to be hands-on and interactive, seean example at http://lecture-play.radford.edu/Mediasite/Play/84c88e2983c44e4eb35189c4f6e4ee911d . Thefocus of the model was to improve secondarymathematics teachers’ subject matter knowledge inAlgebra 1, Algebra 2, AFDA, Geometry,Modeling, and Probability and Statistics. Since2010, six grant projects have been conducted,serving 352 teachers, and providing more than 700hours of instruction. The number of partners eachgrant year varied between 25 and 52. The completelist of partners and details on each project areavailable on our website at http://www.radford.edu/rumath-smpdc/PartnersMap.html.Literature Review and Research Overview Teachers’ subject matter knowledge andpedagogical content knowledge have long beenwidely acknowledged as essential components ofteaching expertise (e.g., Shulman, 1986). TheFigure 2. Rubric for student evaluations 21Virginia Mathematics Teacher vol. 43, no. 1

the American Mathematical Society (AMS) and the Conference Board of the Mathematical Sciences (CBMS) indicated, “Although high school mathematics teachers frequently major in mathematics, too often the mathematics courses they take emphasize preparation for graduate study or careers in business rather than advanced perspectives on the mathematics that is taught in high school” (p. 5). The high school curriculum prepares students for sophisticated and often abstract understandings, but may not prepare teachers to see the mathematics through students’ eyes (McCrory, Floden, Ferrini-Mundy, Reckase, & Senk, 2012). Teachers need both advanced mathematical knowledge and an understanding of how that advanced mathematical knowledge relates to theFigure 4. Example worksheet high school curriculum (AMS & CBMS, 2012; McCrory et al., 2012). Ourand curriculum materials as well as examinations primary underlying assumption is that by providingof the underlying mathematics. The SMPDC teachers with quality educational experiences, theprovides mathematics teachers with extensive corresponding increase in their subject andopportunities such as online graduate level courses pedagogical content knowledge will translate intoin mathematics designed for secondary teachers, changes in classroom practices that ultimately havesummer institutes at NASA, and classroom a positive impact on students’ learningobservations and feedback. These programs help The SMPDC’s research examines thedevelop teachers’ knowledge bases and ultimately Center’s impact on measuring: (1) gains in teacherincrease students’ mathematics achievement. content knowledge, (2) impact on student achievement, and (3) progress towards meeting the Teachers’ mathematical knowledge for assessed needs of partnering school divisions.teaching contributes significantly to gains in Toward these ends, assessment measures includestudents’ mathematics achievement (e.g., Hill, pre- and post-assessments of teachers’Rowan, & Ball, 2005; Sample McMeeking, Orsi, & mathematical content knowledge and mathematicalCobb, 2012; Telese, 2012). The Mathematical knowledge for teaching Algebra I, Algebra II, andEducation of Teachers II (2012) co-publication byVirginia Mathematics Teacher vol. 43, no. 1 22

Figure 5. Provided temperature data These products then went through two to three rounds of blinded peer-review with externalAFDA; multiple classroom observations of reviewers, and are currently available for use byselected participating teachers using Instructional teachers across the Commonwealth. The productsQuality Assessment (IQA) rubrics (Junker et al., implement best practices for teaching mathematics,2006); teacher surveys addressing content or and feature hands-on, interactive activities withpedagogical content knowledge; assessments of various STEM applications. In this paper we willgains in high school students’ understanding and share two example products including a unit planknowledge of Algebra I, Algebra II, and AFDA; and a performance-based assessment.surveys of participating school administrators;participating teacher surveys to evaluate the Our first example is a performance-basedstrengths and to offer suggestions for the SMPDC; learning and assessment task that was designed byand course fidelity assessments to evaluate whether teachers Dee Baker, Jamey Lovin and Robert Thiencourses are designed and taught in ways that (http://www.radford.edu/rumath-smpdc/correspond with identified needs of partnering Performance/src/Dee%20Baker%20-%school divisions. Based on the SMPDC project 20Building%20a%20Recreation%20Center.pdf ).evaluation reports from the past six years, teachers In this task, learners were prompted to explorehave shown a significant gain in their knowledge of constructions of triangle centers to determine thealgebra, geometry, and probability and statistics as best location for a recreation center with respect towell their ability to implement new educational three communities in a geographic area (Figure 1).technologies in their classrooms. Readers who wishto engage with the data and analysis portions of the To solve the task, students usedprojects can find full evaluation reports for each of applications such as GeoGebra, Google Maps, asthe six projects at the SMPDC website. (http:// well as standard construction tools. The taskwww.radford.edu/~amanizade/projects.html) included two assessment components and rubricsProducts Generated which assess students’ understanding of the content (Figure 2). Students were expected to select a site One of the main outcomes of the MSP and justify their selection using credible research,projects are the hundreds of products generated and geometric principles, and constructing trianglesshared by the participating teachers. These include (Figure 3).lesson plans, unit plans, and performance-basedassessments and have been made available in the Our next example is a unit plan by ChristineTeacher Resources section of our project’s website, Belcher and Stephanie Brady (http://http://www.radford.edu/rumath-smpdc/. www.radford.edu/rumath-smpdc/Units/src/A% 20Change%20in%20the%20Weather.pdf). ThisVirginia Mathematics Teacher vol. 43, no. 1 unit, “A change in the weather”, guides students through data analysis, finding equations of the curve of best fit, mathematical modeling that includes polynomial, exponential, and logarithmic functions, and making predictions, as well as teaching them about climate as one of the real world applications of mathematics. Students create charts and models using provided data of 17 different locations across the world (Figure 4, Figure 5), and make scatterplots and trend lines of the data (Figure 5), as well as other indicators of climate change such as global mean sea level and temperature. We also have samples of classroom videos of our participants implementing different approaches to teaching mathematics. These approaches include the Structuralist, Integrated- Environmentalist, and Formative approaches. They also feature the teachers’ educational knowledge and insights gained from the program. They can be watched at the SMPDC website http:// www.radford.edu/rumath-smpdc/ VideoCourse.html (see Figure 6). Our project 23

website includes hundreds of more products like address the major obstacle of online professionalthe examples included in our papers. We encourage development courses, the lack of interaction andreaders to examine and utilize these free resources. hands-on experiences. Finally, we suggest that allImplications and Recommendations of the teacher-generated products created during the projects be available for use by teachers Providing teachers with an online, ongoing throughout the Commonwealth, as this is one of theprofessional development program that involves most effective ways of distributing the knowledgepartnerships between multiple institutions of higher generated in each program.education, and which utilizes the strengths of these Conclusionsinstitutions, allows the creation of an effective, During the past six years of the MSP projects atstatewide program for the continuing improvement Radford University, we were consistently able toof mathematics education. An important element of show growth in the following areas: 1) Teachers’this model is simple and easy credit transfers content knowledge and pedagogical contentbetween partner institutions, allowing teachers knowledge in Algebra, Geometry, and probabilityflexibility in taking graduate-level classes. This and statistics. 2) Students’ mathematicalalso allows institutions and their faculty to focus on knowledge in the aforementioned areas. We alsotheir strong subject areas, increasing the overall met the needs of the participating partners, andquality of the professional development. An addressed the goals identified by theimportant part of our professional development Commonwealth. The major challenges of the MSPprojects over the past six years has been our projects relate to the following: 1) The availabilitypresence in participants’ classrooms following the of reliable and valid assessment measures ofcourses. We observed and interviewed teachers and teachers’ professionally situated knowledge at theprovided them with critical feedback related to the high-school level. 2) The rate of turnover of teachers in the partner divisions. 3) The length ofFigure 6. Screenshot of an online video the award periods that prevent longitudinal studies. We would also like to invite readers to explore thequality of their mathematics instruction. This not products and resources created as a result of theseonly allows us to see if the theoretical knowledge projects, which can be found on our website,learned during the course has been implemented in (http://www.radford.edu/rumath-smpdc/the participants’ classrooms, but also provides an Resources.html).opportunity for additional professionaldevelopment in the field. References American Mathematical Society & Conference We also encourage the continueddevelopment of online courses that feature hands- Board of the Mathematical Sciences (2012).on, interactive instructional practices that allow The Mathematical Education of Teachersteachers to engage with the ideas of the content II. Providence, RI: AMS and CBMS.area through exploration and discovery. We Corey, D., Jacobsen, L., Manizade, A., Dove, A.,recommend the use of all available technological Galeshi, R. & Younes, R. (2016).tools such as Adobe Connect, Geogebra, Best Practices: Lessons Learned From anCinderella, Mathematica, and others to aid in the Online Statewide Collaborative Master's increation of an interactive online environment. We Mathematics Education Program. Inalso recommend providing participants with Proceedings of Society for Informationmanipulatives and corresponding explorations to Technology & Teacher Educationcomplete as part of their coursework. This helps International Conference 2016 (pp. 2497-Virginia Mathematics Teacher vol. 43, no. 1 2498). Chesapeake, VA: Association for the Advancement of Computing in Education (AACE). Hill, H., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. A merican Educational Research Journal, 42(2), 371- 406. Junker, B., Weisberg, Y., Matsumura, L. C., 24

Crosson, A., Wolf, K., Levinson, A., et al. 1_14 (2006). Overview of the instructional McCrory, R., Floden, R., Ferrini-Mundy, J., quality assessment. CSE Report 671, National Center for Research on Reckase, M. D., & Senk, S. L. (2012). Evaluation, Standards and Student Testing Knowledge of algebra for teaching: A (CRESST), University of Los Angeles, Los framework of knowledge and practices. Angeles, CA. Journal for Research in MathematicsManizade, A. G., Jacobsen, L. (2013). Education, 43(5), 584-615. Access to professional development Sample McMeeking, L. B., Orsi, R., & Cobb, B. R. opportunities for mathematics teachers in (2012). Effects of a teacher professional rural USA. In Lindmeier, A.M. & Heinze, development program on the mathematics A. (Eds.) Proceedings of the 37thConference achievement of middle school students. of the International Group for the Journal for Research in Mathematics Psychology of Mathematics Education, 5, Education, 43(2), 159-181. 243. Keil, Germany: PME. Shulman, L. S. (1986). Those who understand:Manizade, A.G., Martinovic, D. (2016). Knowledge growth in Developing interactive instrument for teaching. Educational Researcher, 15, 4-14. measuring teachers’ professionally situated Telese, J. A. (2012). Middle school mathematics knowledge in geometry and measurement. teachers’ professional development and In P. MoyerPackenham (Eds.), International student achievement. Journal of Perspectives on Teaching and Learning Educational Research, 105(2), 102-111. Mathematics with Virtual Manipulatives (pp.323-342).Switzerland: Springer Publishers DOI 10.1007/978-3-319- 32718-Agida Manizade Key to the Spring 2016 Puzzlemaker ProblemDepartment of Mathematics and StatisticsRadford [email protected] JacobsenGraduate CollegeRadford [email protected] BelcherActing Coordinator, Math & Science 6-12Math Science Innovation [email protected] ThienMathematics TeacherPlaza Middle [email protected] LovinMath SpecialistPlaza Middle [email protected] BradyMathematics [email protected] BakerMathematics [email protected] Mathematics Teacher vol. 43, no. 1 25

Technology ReviewSection Editor: Christophe HirelIn this section, we feature websites, online manipulatives, and web-based applications Christophe Hirelthat are appropriate for K-12 mathematics instruction. We are looking for critical Section Editor,reviews of technologies which focus on both the benefits and limitations of using these Technology Reviewtools in a K-12 mathematics classroom. If you use a technological tool and wish to [email protected] with us, please respond to the call for manuscripts on page 46. Youtube’s Reinforcement of Mathematical MisconceptionsDr. Amanda Sawyer Youtube boasts that over 1 billion unique Many videos on Youtube teachusers visit their website every month, with over mathematics skills through songs. For example, in100 hours of video being uploaded every minute Figure 1 the video states, “Ten times any number.(https://www.youtube.com/yt/press/statistics.html, You put a zero after the number. You get2014). With this kind of popularity, it is no surprise it” (https://www.youtube.com/watch?that many elementary teachers use this website v=zCJug1WlYJs, 2014). The 10 TIME TABLESregularly to find videos to help teach specific “ROCK” teaches students the mathematicalmathematical topics. However, this venue does not misconception that you always put zero after aensure the videos posted are mathematically valid. number when multiplying by ten, but this rule does not hold true for multiplying 10 by a decimal. Figure 1: 10 TIME TABLES “ROCK” Figure 2 shows another example of a video teaching a mathematical misconception. The video reinforces the misconception that subtraction is always “take away,” teaching students that subtraction has only one meaning because, “When you subtract with a pirate, you always take away” (https://www.youtube.com/watch? v=qWEm5xozyK8 , 2014). A mathematical misconception is defined as a mistaken idea or view resulting from a misunderstanding of a mathematical concept (Allen, 2007). One way to avoid developing these misconceptions is to critically review a video intended for classroom use, as not everything posted in social media has gone through content review. Social websites where individuals share their ideas also share individuals’ misunderstandings. Today’s technology and social media is a great teaching resource, so long as we all act as well informed critical consumers.Figure 2: When You Subtract With a Pirate Referencesmath song for kids https://www.youtube.com/yt/press/statistics.html 2014 Allen, G. (2007) Student thinking. http:// www.math.tamu.edu/~snite/MisMath.pdfVirginia Mathematics Teacher vol. 43, no. 1 26

Enhancing Pedagogical Practices ThroughProfessional Development in Proportional ReasoningPadmanabhan Seshaiyer and Jennifer SuhAbstract higher level mathematics. Teachers have also been There is a continuous need to develop more urged to focus students’ attention on the meaning of problems and to help students value differentcontent-focused professional development (PD) mathematically correct solutions to a singleprograms for teachers that can help lead to problem (NCTM, 2000). There is a great need forimprovements in teacher content knowledge in research in evaluating the effect of solving oneproportional reasoning. This work presents how the proportional situation via multiple solutionpedagogical practices of a group of 85 elementary strategies for example using unit rate strategy;and middle grades teachers that participated in a repeated subtraction strategy; equivalent fractionssummer institute through a Mathematics and strategy; size-change strategy; cross multiplicationScience Partnership (MSP) program, were using equal rates or ratios strategy, relative andimpacted in proportional reasoning through absolute thinking strategy and; reasoning up andproblem solving activities. The observations on down strategy (Lamon, 2007). Teachingtheir understanding of proportional reasoning proportional reasoning through problem solvingthrough poster artifacts, the reflections on their therefore requires depth of mathematicalwork through journals as well as misconceptions in knowledge for teaching that not only includestheir problem solving found in this work are understanding of general content but also havingpresented. Impact on student learning through domain specific knowledge of students. Researchfollow-up lesson study are also discussed. has shown that a content-focused PD leads toProject Overview improvements in teacher content knowledge with a focus on student learning goals, highlighting the Over the last seven years, the Center for concepts being addressed, how they are developedOutreach in Mathematics Professional Learning over time, difficulties students may encounter, andand Educational Technology (COMPLETE) at how to monitor student understanding (Suh andGeorge Mason University (GMU) has been Seshaiyer 2013, 2014a, 2014b). To evaluate thesupported by the Virginia Department of Education collaborative nature of designing PD, Suh,MSP Program to coordinate projects that has Seshaiyer, Freeman and Jamieson (2011) used thehelped to provide PD to teachers from various collective self-study method to examine howschool districts in the Northern Virginia area. purposively designed experiences such as theTopics in these PD have included Building content-focused institute in the summer withnumbers and number sense for elementary grades; school-based follow-up Lesson Study cycles in theRational numbers and Proportional reasoning in fall encouraged vertical articulation of algebraicmiddle grades and; Expeditions in Science connections. In this work, we present one such PDTechnology Engineering Education through program that systematically introduced the conceptMathematics at high school level. The projects of proportional reasoning through problem basedwere coordinated by a mathematician and a learning activities. Specifically, the paper presentsmathematics educator from GMU in collaboration how the pedagogical practices of a group of 85with partnering districts that included Alexandria, elementary and middle grades teachers thatArlington, Fairfax, Fall Church, Frederick, participated in a summer PD institute wereLoudoun, Manassas City, Manassas Park, Prince impacted in proportional reasoning. The studyWilliam and Virginia Council for Private explored the following research question: How canEducation. The project website provides more professional development for mathematics teachersdetails: http://completecenter.gmu.edu. be designed and implemented so that the teachersIntroduction develop deep understanding of proportional reasoning? Proportional Reasoning is fundamental tomany important mathematical concepts and is often The goals for our project were forregarded as the pathway to performing well in elementary and middle grades teachers to relearnVirginia Mathematics Teacher vol. 43, no. 1 27

proportional reasoning through multiple Data Analysis through Poster Artifacts andrepresentations in problem solving and linking Reflectionsrelated problems and concepts. This project wasdesigned based on the current research and needs in The teachers worked in groups attemptingmathematics education in the state. The program to solve the problem multiple ways and were askedincluded a content-focused summer PD institute to create a poster representing their solution. Eachand a follow-up Lesson Study (Suh and Seshaiyer, teacher was also asked to reflect on how they2014c) throughout the academic year focused on participated in the problem solving process andengaging teachers in active learning through how they would take this problem back to theirrational numbers and proportional reasoning tasks, classrooms to present to their students.exploring pedagogical strategies, utilizing Photographs of all the posters created by themathematics tools and technology, and promoting teacher groups of the cathedral problem were takenconnections aligned and coherent to the elementary and the data from the posters were analyzed forand middle school curricula. Daily activities in the content, connections between concepts, and anysummer institute included modeled lessons using a possible differences related to the time alreadyvariety of mathematics tools and technology and in spent in the seminar. Data was also analyzed from-depth conversation about the proportional the teacher reflections for common themes as wellreasoning, pedagogical strategies such as using as individual perspectives. Some of the solutionsproblem solving and multiple representations. The from the various poster artifacts created by teacherpurpose of the PD was the development of groups are indicated in Figures 1-4. The analysismathematical teaching knowledge through a and discussion of these poster artifacts indicatedcollaborative network of pre-service and in-service strategies employed by the teachers including guessteachers who collaboratively plan lessons and and check, tabular and pictorial representation,exchange best instructional practices and effective linear addition, working backwards,uses of technology tools to design instructional partitioning, and comparison strategies.tasks which promote algebraic conceptual thinking. (Figure 6). As these poster illustrations clearlyTeacher collaboration enhances their professional indicate the teachers exhibited a wide variation inpractice which then affects students' learning. The their thinking. They generated a lot of interestingteachers were engaged in content-focused activities conversation that helped the instructors to bring ato help them become aware of the specific math nice closure to this problem using proportionalcontent topics in rational numbers and proportional reasoning. Teachers reported that the reasoning upreasoning (Lamon, 1999): Relative and Absolute and down helped them to break problems intoThinking; Measurement; Quantities and Co- chunks and build on those chunks. They saw howVariation; Reasoning up and down; Unitizing; building on known concepts or known quantitiesSharing and Comparing; Proportional Reasoning; gave them a sense of control as opposed to the lostEquivalence; Reasoning with fractions; Part-whole feeling we sometimes experience during thecomparisons with unitizing; Partitioning and introduction of a completely new idea. TheQuotients; Rational numbers as operators; Rational teachers realize that the latter is a source ofnumbers as measures; Ratios and Rates; Distance-rate-time relationships; Similarity and percents; Figure 1. Solution by Guess and CheckChanging fractions. Each of these content topicswere motivated through sample benchmark 28problems that aligned with the changes in the 2009VA Standards of Learning. While we consideredseveral during the institute, we include here onespecific problem that was provided to the teachersas an opening problem for the day, the cathedralproblem that is adapted from Burns, S. (2003):While building a medieval cathedral, it cost 37guilders to hire 4 artists and 3 stonemasons, or 33guilders for 3 artists and 4 stonemasons. Whatwould be the expense of just 1 of each worker?Note that guilders here refers to currency used inthe Netherlands from the 17th century to 2002.Virginia Mathematics Teacher vol. 43, no. 1

Figure 2. Solution by tabular and pictorial Figure 5. Solution by partitioningrepresentation Figure 3. Solution by linear addition Figure 6. Solution by comparison strategies Figure 4. Solution by working backwards concern, frustration, and fear in their students. One teacher commented that she never realized howVirginia Mathematics Teacher vol. 43, no. 1 emotional the process could be and that she was gaining a new perspective on her students and how she interacts with them. Another teacher wrote that she would use reasoning up and down to help her students focus on what they already know and then guide them in building on that knowledge. Several teachers remarked on the importance of labeling processes so that students have a clear picture of how the concepts tie together; this leads to the development of conceptual understanding and the internalization of concepts and processes for the students. Another teacher reflected, “I am also starting to think differently about analyzing student work. When problems have the opportunity of yielding a variety of correct answers, it is important to consider what the student is doing and what math they can do and understand.\" 29

Along with in-class activities, the teachers Figure 8. Sample of student work 2had the opportunity to reflect on all the problemsthat they worked on each day. The teacher included baseline data on participating schools,reflections enabled us to focus on the teachers, and students including the numbersunderstanding, reactions, and feelings of the served, qualification levels of teachers, proficiencyindividual teachers. While the posters showed how levels of students; program results for teacherspeople in a group approached problem solutions in related to changes in content knowledge anda variety of ways, the reflections gave us insight highly qualified status and for students related tointo how the individual teachers were feeling about changes in academic achievement. For thethe sessions, about their own competence, and teachers’ content knowledge, a pre- and post-about their classroom practices. Several themes assessment of teacher content knowledge wasemerged including the value of the struggle, the administered before and after the summer institutejoy of using conceptual thinking, the importance of and following the final follow-up meeting. Specialclarity, the advantage of building, the benefit of efforts were made to create these assessments tocollaboration, and recognizing that there are reflect VA SOLs at various grade levels. A pre-multiple valid ways in which to approach problem survey solicited information regarding theirsolving, which leads to viewing student work with opinions, their preparation, their teaching practice,new eyes. In the institute, there was also a focus on and the quality and impacts of their professionalformative and performance based assessment and development experiences. To assess the project’serror analysis to improve student learning. impact on classroom instruction and studentFollowing this, in the fall semester the teachers achievement, data such as student assessment andworked on Lesson Study as a team with K-12 math artifacts of student work was collected during theteachers, special educators, LEP teachers led by a academic year at the follow-up Lesson Studymathematician, a mathematics educator and a math sessions. Examples of student sample work from acoach (Suh and Seshaiyer, 2014a, 2014b). The Lesson study is illustrated in Figures 7 and 8follow-up lesson study was one mechanism used to where students demonstrate their thinking ofsustain the learning and keep the focus on the solving problems using techniques in proportionalcontent, helping teachers understand how the tasks reasoning such as unitizing. These included havingthey are using in their classrooms are intended to teachers develop and implement assessments ofcontribute to student understanding. their students learning. These assessments wereProgram Evaluation and Assessment aggregated as part of the report of impact on student achievement. To assess impact on Data collected from the MSP programs classroom instruction, teacher reports and activities related to their dissemination plans was aggregatedFigure 7. Sample of student work 1 during the follow-up sessions throughout the school year. The primary indicator of studentVirginia Mathematics Teacher vol. 43, no. 1 30

achievement was the change in students’ strategic Referencescompetence in math. Students learnt to think about National Council of Teachers of Mathematics.situations in relative rather than absolute terms. Forexample, the students in a particular lesson study (2000). Principles and standards for schoolwere asked to compare two situations of increase in mathematics. Reston, VA: NCTM.numbers from 500 to 800 versus 300 to 600. Lamon, S. J. (1999). Teaching fractions and ratiosStudents thinking in absolute terms may answer for understanding: Essential contentthat these situations have the same amount of knowledge and instructional strategies forincrease, however, a relative multiplicative teachers. London: Lawrence Erlbaumthinking helped the students to think proportionally Assoc.to help them understand and justify the differences Lamon, S. J. (2007). Rational numbers andin the two situations. Reports were prepared each proportional reasoning: Toward ayear that identified the major findings of the theoretical framework for research. In F.evaluation documenting evidence of increased K. Lester, Jr. (Ed.), Second handbook ofknowledge and skills and improved classroom research on mathematics teaching andpractice as a result of teachers’ participation in the learning. (pp. 629-668). Charlotte, NC:project. A written evaluation was conducted at the Information Age Publishing.end of the program year to determine how well the Suh, J. & Seshaiyer, P. (2014a). Developingoverall goals and objectives have been achieved. Strategic Competence by Teaching UsingThe summative evaluation measured the extent to the Common Core Mathematical Practices,which the project goals were met, the degree to Annual Perspectives in Math Education,which the participants gained the desired skills and Chapter 8, pp 77-87.knowledge, and the project’s effectiveness in Suh, J. & Seshaiyer, P. (2014b). Examiningimproving classroom practice and student teachers’ understanding of the mathematicaloutcomes. We were fortunate to have support for learning progression through verticaldoctoral students and lead specialists from the MSP articulation during Lesson Study, Journal ofaward that allowed them to serve as knowledgeable Mathematics Teacher Education, pp: 1-23.others to sustain a collaborative coaching effort to Suh, J. & Seshaiyer, P. (2014c). Mapping teachers’continue to engage the teachers after the understanding of the mathematical learningcompletion of the summer institute. progression through vertical articulationImpact and Outcomes: during Lesson Study. A merican Educational Research Association Online Our results have revealed that the design of Repository, Philadelphia, PA.the PD and Lesson Study offered opportunities for Suh, J.M., Seshaiyer. P., Freeman, P. & Jamieson,teachers to become inspired by what their students T.S. (2011). Developing teachers’were capable of doing. Despite encountering representational fluency and algebraicchallenges as they transformed their teaching connections. In Wiest, L. R., & Lamberg, T.approach, the MSP program has greatly impacted (Eds.). Proceedings of the 33rd A nnualthe teachers to find ways to rethink their Meeting of the North American Chapter ofmathematics instructional and pedagogical the International Group for the Psychologypractices in order to promote higher ordered of Mathematics Education. 738-746.thinking skills in their students. The successful Suh, J. & Seshaiyer, P. (2014b). Examiningcompletion of the summer PD followed by teachers’ understanding of the mathematicalcollaborative coaching & Lesson study also gave learning progression through verticalthe opportunity for the 85 participating teachers to articulation during Lesson Study, Journal ofearn 3 credits of graduate coursework at George Mathematics Teacher Education, pp: 1-23Mason University. Struggling through problem (2014).solutions with colleagues, analyzing their Suh, J. & Seshaiyer, P. (2013). Mathematicalapproaches, questioning their reasoning, Practices That Promote 21st Century Skills,understanding how to develop open tasks, Mathematics Teaching in the Middlecomparing their work with others and, contributing School, NCTM, 19(3), pp 132 - 137.to group efforts were noted by the teachers as beingvery beneficial to their discoveries in the Summer 31PD institute and the follow-up Lesson study.Virginia Mathematics Teacher vol. 43, no. 1

Seshaiyer, P., Suh, J., & Freeman, P. (2011). Unlocking the locker problem with technology”, Teaching Children Mathematics, NCTM, Vol 18(5), pp 322- 325.Jennifer Suh Padmanabhan SeshaiyerAssociate Professor of Mathematics Professor of Mathematical SciencesGeorge Mason University George Mason [email protected] [email protected] Grant Opportunities Continuing Education Grant ProgramThe VCTM Continuing Education Grant program is a grant opportunity of up to $1,000 for a current VCTM member who is a current K-16 mathematics educator in Virginia.The purpose of the program is to provide funding support to a VCTM member who wishes to continue his/her education in mathematics or mathematics education. The funding for this grant can be used for tuition at an institution of higher learning, i.e. a two-year or four-year institution.The application requires applicants to attach a short explanation of why the applicant in teaching mathematics or mathematics education, how it will benefit the applicant in teaching mathematics or assisting teachers of mathematics, and how the applicant can share the knowledge learned from the course with VCTM. The application deadline is December 1, 2016 http://www.vctm.org/Grant-Continuing_Education 7 Karen Dee Michalowicz First Timers Grant.The First Timers Grant Program is named for Karen Dee Michalowicz who actively promoted and supported conference development and attendance. The purpose of this grant is to provide funding support for: VCTM members who have NOT previously attended but wish to attend a regional or annual NCTM meeting or Any Virginia teacher (including non- members) who wish to attend a VCTM Annual Conference or VCTM Academy for the first time. Threegrants are available. One award is given for the regional or annual NCTM conference of $500 and two awards of $250 each for the state conference or academy. The funding for this grant is to be used toward registration, hotel room, banquet dinner, and transportation expenses. The application requires that applicants attach a short explanation of why the applicant wishes to attend the conference and how it will benefit the applicant’s classroom instruction. http://www.vctm.org/Grant-First_Timers Other Grant Opportunities available in the Commonwealth at www.vctm.org/Awards The NCTM has numerous grant opportunities available at https://www.nctm.org/Grants/Virginia Mathematics Teacher vol. 43, no. 1 32

Good ReadsSection Editor: Dr. Jean MisteleIn this section, we feature mathematics literature that is appropriate for K-12 Mathematics Dr. Jean Misteleinstruction. If you use specific literature for your mathematics classroom and wish to share it Section Editor,with the Virginia Mathematics Teacher community, please respond to the Call for Good ReadsManuscripts on page 26. jmistele@Radford .edu Eyes on Math: A Visual Approach to Teaching Math Concepts I was drawn to this for the reader. In another example, I was a bit book, because many of my disappointed with the explanation for a students claim, “I am a visual mathematics concept that was shown in the chapter learner.” This book responds titled, 2-digit by 2-digit multiplication, on page 86. to these students preferred The author demonstrated the distributed property learning style. The author for multiplying binomials using a grid. I wished the shows teachers how to bring author extended this concept to include the FOIL mathematics concepts, taught method because this example shows why the FOIL in the K-8 mathematics method works. Likewise, on page 150 when classroom, out of the darkness addressing algebraic thinking with growing and into the light. The visual patterns, the author could have identified this as andemonstrations are great for your visual learners to example of the arithmetic sequence and include thehelp them understand the mathematics concepts geometric sequence. Including these two pointsand answer their why questions. The author links would broaden the notion of growing patterns. Thethe mathematics shown in each chapter to the next example I found troubling occurred in theCommon Core Standards, which makes this book a chapter titled, Equivalent Rates on page 166. In thegreat resource for teachers. extension section the author has students research The book is 230 pages with 215 pages filled some typical rates used in the real world. However,with mathematical content divided into 3 sections before the students can do such research, I believefor each of the three grade bands: K-2, 3-5, and 6- there needs to be a discussion on the definitions of8. Each section focuses attention on the most ratio and rate such as: how are they the same, howcrucial mathematical concepts taught in that grade do they differ, is there agreement amongband. Each chapter has the same format; a picture mathematicians, educators, or use of these terms inor diagram accompanied with a question. The the real world?author explains the mathematical conceptassociated with the picture, then provides a second The author promotes the notion thatpictorial example followed by several discussion discourse in the classroom facilitates the learningquestions. The author includes expected student for all students, which made this omission moreresponses and/or explicitly identifies the obvious to me. Overall, the author does a good jobmathematics the students should notice for each promoting classroom discourse. The last exampleexample. Each chapter ends with an extension that I found a bit troubling was the chapter,section, which is typically stated as an activity for Addition: Changing addends, but not the sum. Inthe student that reinforces the mathematics concept this chapter the author demonstrates thataddressed in the chapter. decomposing a number in different ways will not Although I enjoyed the book, there were a change the sum. She claimed this demonstrated thefew places that were troubling. I was confused by associative property for addition. The author givesthe wording used in the chapter, Multiplication: the following example: 9 + 2 = 7 + 4 demonstratesThe distributive principle. In the explanation, the the associative property because the first addendauthor referenced a second picture and then was decreased by 2 (to get the 7 on the right handimmediately referred back to the original picture in side of the equation) and the second addend wasthe explanation without indicating this redirection increased by 2 (to get the 4 on the right hand sideVirginia Mathematics Teacher vol. 43, no. 1 of the equation). Actually, the associative property 33

refers to the two different ways of grouping 3 measures of a circle are related (i.e. diameter,addends, which does not change the total sum. For radius, circumference, area-if you know one of theexample, (9+2) + 3 = 9 + (2+3), which becomes 11 measures you can find the others), while the+ 3 = 9 + 5 and reduces to 14=14, a true statement. measures of a rectangle are not (i.e. side measures,Perhaps, this is what the author intended to show. area, perimeter-if you know one of the measures you cannot find the unique values of the others). I particularly liked how many of thesevisual representations are presented before Overall, the book has many great ideas forformulas are presented to the students. Research the mathematics teacher to engage all of thesuggests this is the best way to build mathematics students in meaningful mathematics conversationsconceptual understanding. I also like the way the as students keep their eyes on math. I enjoyed theauthor explicitly states mathematics ideas that are book and will use it when teaching my K-8many times implied. For example, in the chapter preservice teachers.titled, How measures are and are not related onpage 188, the author explains that all of the Upcoming Math CompetitionsName of Organization Website Dates of CompetitionsHuntington University Middle School https://www.huntington.edu/math/mathematics-competition/middle-school -competition TBAMathematics Competition November 15, 2016American Math Competition 8 http://www.maa.org/math-competitions/amc-contests/amc-8SUM Dog online http://www.sumdog.com/enter_contest/ Online competitionAmerican Math League http://mathleague.org/arml.php Mailed to your schoolAmerican Math Competition 10/12 http://www.maa.org/math-competitions/amc-contests/amc-1012 February 07, 2017Math Club at Radford University http://www.radford.edu/mathclub/events.html See website for future eventsUSA Math Talent Search http://www.usamts.org/TipsFAQ/U_Tips.php Online, month-long contestMath Counts http://www.mathcounts.org/programs/competition-series/competition-faq At your local school Continental Math League http://www.cmleague.com/ December 1, 2016 Conrad Challenge http://www.conradchallenge.org/ And many moreHarvard/MIT Math Tournament https://www.hmmt.co/ TBA November 12, 2016 The Math League http://www.mathleague.com/ 2017 Math Con http://www.mathcon.org/ Month depends on grade level http://www.moems.org/ Math Olympiads January 16, 2017 Math Kangaroo in the USA http://www.mathkangaroo.orgNoetic Learning Math Contest http://www.noetic-learning.com/mathcontest/ Mailed to your school March 16, 2017 November 10, 2016 Purple Comet http://purplecomet.org/ April 18, 2017Rocket City Math http://www.rocketcitymath.org/ November 7, 2016USA Mathematical Talent Search http://www.usamts.org/ TBAWho Wants to Be a Mathematician http://www.ams.org/programs/students/wwtbam/wwtbam Oct 27, 2016Virginia Mathematics Teacher vol. 43, no. 1 34

Interactive Mathematics Institute for Middle School Teachers Kristina Anthony and Heather NunnallyAbstract complete the program over 3 years. The Interactive Mathematics Institute for Content for these courses aligns with theMiddle School Teachers (IMI-RMST) is designed Virginia 2009 Mathematics Standards of Learning.to improve mathematics content, pedagogical Each curriculum was developed, under thestrategies and assessment practices of middle direction of lead investigator, Dr. Aimee Ellington,school teachers and principals in an urban school through teams consisting of mathematics educationdistrict. This project consists of course work and faculty, mathematicians, mathematics instructionalfollow-up experiences focusing on rational staff, and lead teachers with strong mathematicsnumbers, proportional reasoning and algebraic backgrounds. An underlying assumption of ourthinking through the use of inquiry-based activities project and, more specifically, the foundation ofinvolving multiple representations of mathematical the work of the course development team was thatconcepts, engaging teachers in rich mathematical students learn mathematics primarily through theconcepts and developing. Furthermore, workshops experiences that teachers provide. As a result,for building level administrators were developed to teachers must have a deep understanding of thefacilitate discussion among administrators on content as well as effective pedagogical skills tosupporting good mathematics instruction. This help students develop their mathematicalpaper reports on the results from the first year of a understandings (Hill, Rowen, & Ball, 2005). Tothree year project and is applicable to mathematics that end, course instructors model research-basedteacher educators and district level mathematics instructional practices to help participants makesupervisors. connections between content and pedagogy. The courses provide many opportunities for participants The Interactive Mathematics Institute for to engage in mathematical dialogue and hands-onMiddle School Teachers (IMI-RMST), developed activities.by Virginia Commonwealth University inpartnership with a large urban school district, is During the summer of 2015, the firstdesigned to improve mathematics content and cohort consisting of 17 teachers from 5 differentpedagogical practice of middle school teachers. schools began the professional developmentThe urban school division is a high needs school program. The first course, Rational Numbers, wasdistrict with a significant amount of teacher taught by a 2-person team consisting of a universityturnover from year to year. This program is faculty member and master teacher with middledesigned to provide mathematical and pedagogical school experience. Small and whole groupsupport for these teachers. In order to achieve discussions focused on sharing ideas learned notpositive change to teacher practice, IMI-RMST only in the summer institute, but participants alsorecognizes that PD must be content specific shared their own successes using different forms of(Campbell & Malkus, 2011; Hill et. al, 2005; classroom instruction. During the institute,Garet, Porter, Desimon, Birman & Yoon, 2001) teachers worked in professional learningand provide opportunities for teachers to apply communities (PLC) led by mathematics specialiststheir new knowledge with follow-up support as from surrounding districts to design richthey do so (Roberts & Pruitt, 2003). mathematical tasks for classroom use. PLC’s provide an ongoing up forum for discussion and To that end, teachers complete two courses: reflection to deepen their understanding ofRational Numbers and Proportional Reasoning for mathematics and make positive change to theirMiddle School Teachers and Algebraic Thinking pedagogy (Jackson & Cobb, 2013; Campbell &for Middle School Teachers. Each course consists Malkus, 2011; Olson & Barrett, 2004; Zelpada,of a 2-week intensive summer institute and school 2012). Figures 1 and 2 contain sampleyear follow-up sessions that feature activities mathematical tasks developed by participants.designed to help teachers increase their conceptualunderstanding of mathematics and develop the 35pedagogical strategies to help their students betterunderstand the mathematics. Two cohorts willVirginia Mathematics Teacher vol. 43, no. 1

STAGE 1: Desired Results ~ What will students be learning? Task 1 (see Figure 1) asks students to applySOL/Learning 6.1 The student will describe and compare data, using ratios, and will ratio concepts theyObjective use appropriate notations, such as, a to b, and a: b. learned to a real world situation. 7.4 The student will solve single-step and multistep practical prob­ lems, using proportional reasoning. Task 2 (see Figure 2) requires students toEssential Ques- What is a ratio? How can ratios be used for comparison? compare and ordertions & Under- fractions.standings/BigIdeas STAGE 2: Assessment Evidence ~ What is evidence of mastery?Assessment Students will find relationships among parts of ratios and make com­ After the summerStrategies parisons based on the ratios by drawing models of those compari­ institute, teachers used the sons. tasks with their students. Students will explain their mathematical understanding of equivalent During four follow-up ratios. sessions, mathematics Students rearranging the order of the ratios. specialists facilitated For example: Flipping 7/5 to 5/7 to change it from an improper discussion around fraction to a regular fraction not understanding the relationship participants’ experiences between two sets. using the tasks with theirPossible mis- Comparing incorrect parts of ratios. students as well as otherconceptions or For example: Assuming because 7 is a larger number that it topics related to currentlearning gaps would be the greatest ratio. 7/5 and 5/4 mathematics instruction. In addition to Understanding equivalent ratios. providing professional For example: Students may not understand that a ratio of 5:4 is development for teachers, equivalent to a ratio of 10:8. another aspect of this Relationship of part-part-whole within ratios. program focused on the needs of building For example: A win-loss ratio of 2 to 3 represents 5 total games administrators. During theSTAGE 3: Learning Plan ~ What are the strategies and activities you plan to use? 2015-2016 school year, workshops for principals The teacher will set up the problem by discussing how sports teams were held to address the use ratios to determine conference standings. The teacher will then administrative supports break the class into groups of 2-3 and pose the following scenario: At the middle of the season, the Wildcats and the Vikings have each that are necessary for played 36 games. The Wildcats have a win to loss ratio of 7:5 and the meaningful mathematics Vikings have a win to loss ratio of 5:4. instruction. TheseTeaching and Which team will make the playoffs? Explain how you determined sessions were offered toLearning Activ- your solution. Draw a picture/diagram that compares the ratios. all principals, assistantities (Task) principals, and lead mathematics teachers within the division whose teachers participated in Students will discuss in groups of 2-3 how they came up with their the Rational Numbers answer. Groups must come to a consensus before they explain it to course in year 1. Kanold, the rest of the class. Briars, & Fennell (2012)Checking for believe that in order toUnderstanding build high quality Teacher will monitor groups by walking around and asking students instruction “principals to explain their models. must consider what STAGE 4: Closure ~ What did the students master & what are they missing? content issues to address in the mathematicsTask Closure & curriculum” (p. 9).Student Sum-marizing of Students will share with the class the different ways in which they Participants engaged intheir Learning solved the problem. mathematical tasks to understand how these skills were related to theFigure 1. Mathematical Task Example 36Virginia Mathematics Teacher vol. 43, no. 1

STAGE 1: Desired Results ~ What will students be learning? NCTM process andSOL/Learning Ob- 8.1b - Compare and order fractions content standardsjective for mathematics.Essential Questions Videos of effective& Understandings/ How can we compare and order fractions in a real life situation?Big Ideas mathematics instruction were STAGE 2: Assessment Evidence ~ What is evidence of mastery? shown to illustrateAssessment Strate- Students will compare and order fractions on a number line. the use of richgies mathematical tasksPossible miscon- Students may struggle with creating fractions from a given situation. in the classroom. Discussionceptions or learn- Students may struggle with mentally comparing fractions when looking at revolved arounding gaps the relationship between the numerator and the denominator. mathematicsSTAGE 3: Learning Plan ~ What are the strategies and activities you plan to use? proficiency and Prior to activity: Teacher will create number lines with 0, ½, and 1 as what good benchmarks. Teacher will need to make “paper basketballs” or purchase mathematics looks balls to be used during the activity. Each student will need 2 or 3 sticky like in a middle notes. school classroom. Guided Practice: Teacher will prompt students to explain how numbers Results are organized on a number line. Teacher will provide an example such At this point as: If you get an 8/10 on a quiz, where would that fall on the number line? in the project, quantitative data has been collected from teachers and Task: W ho Has the Best Shot? principals who Students should be placed in small groups of 2-3. participated in the Students will each be given 2 numbered cubes (both numbered 1 – 6). first year ofTeaching and Students will roll the number cubes to determine the number of “free activities. ToLearning Activities throw attempts”. determine the(Task) Students will shoot paper basketballs according to the number of at­ impact of the. tempts rolled, and record their baskets made. program, we Students will write their fraction on the sticky note. In their groups, students must determine how to organize their free gathered data on throws on a number line from least to greatest. participant self- Groups will explain their order. efficacy (i.e. one’s belief in his/her Questions: ability to produce How did you order your numbers on the number line? desired student What if we took all of the groups and compared the fractions, which outcomes for group has the highest free-throw average? Explain. learning), and Who is the closest to a free throw average of 50%? Explain. content knowledge. Who had the same number of attempts? Teachers completed a pre-post content assessment, a self- reported knowledge growth survey andChecking for Un- Students will explain their methods for comparing and ordering “free a self-efficacyderstanding throws”. survey. The STAGE 4: Closure ~ What did the students master & what are they missing? surveys used in this study wereTask Closure & Once the entire class has placed their fractions on the number line, there designed by theStudent Summariz- will be a whole group discussion about where and why they placed them VCU Metropolitaning of their Learn- in that location. There will also be dialogue about the various shot at­ Education Researching tempts and why some students had more or less attempts and what that Consortium who did to their fractions. The teacher can do one for him/her and have the class discuss where to place it on the number line. created the survey items and testedFigure 2. Mathematical Task ExampleVirginia Mathematics Teacher vol. 43, no. 1 37

them for validity and Table 1: Teacher Self- Knowledge Growth Reportreliability. The pre-postcontent assessment was Please rate your under- Time point None Some A A lotdesigned by the curriculum standing of the following littledevelopment team. The topics both before thisinstrument was tested with a Math professional develop- ment and aftergroup of middle school Modeling fractions effective­ Before … 5.6% 27.8% 66.7% 0.0%mathematics teachers. Based ly.on their responses to items After … 0.0% 5.6% 0.0% 94.4%and feedback on the Modeling fractional opera­ Before … 16.7% 27.8% 55.6% 0.0%instrument adjustments were tions.made before the instrument After … 0.0% 5.6% 0.0% 55.6%was used with project Teaching students to relate Before … 0.0% 50.0% 38.9% 11.1%participants. Interested fractions to decimals.readers may contact theauthors if they would like Modeling decimal opera­ After … 0.0% 11.1% 11.1% 77.8%more information about the tions. Before … 5.6% 44.4% 50.0% 0.0%evaluation instruments used Connecting fractions, deci­ After … 0.0% 11.1% 55.6% 33.3%in this project. mals and percents. Before … 0.0% 33.3% 55.6% 11.1% After … 0.0% 5.6% 22.2% 72.2% The self–reportedknowledge growth surveyasked teachers to rate theirunderstanding of rational Misconceptions of percentsnumber and proportional students have. Before … 0.0% 61.1% 33.3% 5.6%reasoning concepts with the After … 0.0% 5.6% 27.8% 66.7%descriptors None, Some, A Teaching ratios and propor­ Before … 5.6% 27.8% 61.1% 5.6%Little, or A Lot. Self-efficacy tions.survey questions focused onperceived changes in practice. After … 0.0% 5.6% 22.2% 72.2%Both surveys used a 4 choice Teaching proportional rea­ Before … 5.6% 38.9% 50.0% 5.6%Likert scale. Principal soning. After … 0.0% 5.6% 16.7% 77.8%workshops were evaluated Helping students solve pro­ Before … 27.8% 38.9% 27.8% 5.6%using feedback gathered from portional problems without After … 0.0% 5.6% 16.7% 77.8%open-ended questions at the computation.end of each session. With respect to the Developing tasks that en­mathematics content courage student discourse Before … 5.6% 38.9% 50.0% 5.6%knowledge scores on the and engagement. After … 0.0% 11.1% 22.2% 66.7%assessment instrumentincreased from a pre-test mean of 46.47 (SD = and connecting fractions, decimals, and percents.22.5) to a post-test mean of 84.61 (SD = 20.78). There was a 66.6% increase in the number ofBased on a paired sample t test, there was a participants that reported knowing “a lot” aboutsignificant difference between the pre- and post-test teaching ratios and proportions. Findings suggestmeans (t =-8.107, n=17, p<.01). Teachers increased that overall teachers believed they increased theirtheir understanding of rational number and knowledge growth by the conclusion of the summerproportional reasoning concepts by completing the institute.course. Table 2 contains data from the self-efficacy The self-reported content knowledge survey survey. In a similar manner, by the completion ofwas given at the beginning and end of the summer the summer institute participants felt moreinstitute (see Table 1). At the beginning of the confident about their abilities to help studentssummer institute, no one reported knowing “a lot” understand rational number concepts. Specifically,about how to effectively model fractions. At the all participants selected either “agree” or “agreecompletion of the workshop, 94.4% of the strongly” to many of the statements. For example,participants felt they could effectively model participants responded more positively tofractions. Similar trends were seen in modeling statements like “when a student has difficultyfraction operations, modeling decimal operations, understanding a math concept, I am confident IVirginia Mathematics Teacher vol. 43, no. 1 38

Table 2. Pre- and Post-Workshop Responses Relating to Teacher Self-Efficacy “Teacher mindsets and habitsPlease respond to these items with Time Disa- Disa- Agree must change to facilitatean indication of how much you Point gree gree Agree Strong teaching that will promoteagree or disagree with the state- Strongl deeper [student] understanding.”ments about your own teaching ly Participants responded to what yI know the steps necessary to teach Be­ 0.0% 22.2% 61.1% 16.7% three things stood out in theirmath effectively. fore mind as a result of the workshop After 0.0% 5.6% 77.8% 16.7% experience. Responses included: (1) studentI understand math concepts well Be­ 0.0% 5.6% 38.9% 55.6% communication is important inenough to be effective in teaching foremath. the classroom; (2) classroomGiven a choice, I would invite a col­ After 0.0% 0.0% 38.9% 61.1% discussions reveal studentleague to evaluate my math teach­ Be­ 0.0%ing. fore 0.0% 0.0% 44.4% 55.6% understands of concepts; (3) After students need to be supported in 0.0% 33.3% 66.7% their study of mathematics; and (4) teachers can plan engagingI am confident that I can answer Be­ 0.0% 5.6% 27.8% 66.7% lessons that do not require themstudents’ questions about math. fore After 0.0% 0.0% 44.4% 55.6% to do all of the talking. These findings suggest a change inWhen a student has difficulty under­ Be­ 0.0% 16.7% 38.9% 44.4% principal views on instructionalstanding a math concept, I am confi­ foredent that I know how to help the practices in a mathematicsstudent understand it better. After 0.0% 0.0% 61.1% 38.9% classroom.I am confident that I can teach math Be­ 0.0% 11.1% 33.3% 55.6% Productseffectively. fore IMI-RMST has After 0.0% 5.9% 41.2% 52.9% generated two new middle Be­ 0.0%When teaching math, I am confident fore 0.0% 5.6% 33.3% 61.1% school curricula and professionalenough to welcome student ques­ After development models for usetions. 0.0% 47.1% 52.9% with middle school teachers.I know what to do to increase stu­ Be­ 11.1% 16.7% 55.6% 16.7% Rational Numbers was useddent interest in math. fore during the summer of 2015 with After 0.0% 22.2% 38.9% 38.9% RPS. During the summer of 2016 a second cohort of teachers will take Rational Numbers. The secondknow how to help them understand it better” and “I curriculum, A lgebraic Thinking, was developed inam confident that I can answer students’ questions the spring of 2016. This course will be offered forabout math” at the end of the institute. Overall, the the first time in summer of 2016. These coursesfindings suggest that gains in teacher efficacy were written to include professional learningoccurred after completing the IMI-RMST institute. communities facilitated by mathematics specialists with the hope that teachers and administrators will At the completion of each principal continue to initiate mathematical discussionsworkshop, administrators were asked to identify among staff. Both curriculums are also being usedwhat resonated with them as well as what ideas with another Virginia public school system throughfrom the session they would take back to their a grant awarded by the Virginia Department ofschools. The principals provided various responses Education.that illustrate their views on mathematicsinstruction in their buildings. In response to the The tasks that were developed byquestion “What resonated with you?” a middle participants during the summer institutes andschool principal stated, “Thinking of math as more refined during follow-up discussions followed athan just finding the right answer. The importance modified division lesson plan to allow for easyof students really understanding instead of just implementation within the school division. Oneputting numbers in formulas.” Another participant purpose of this activity was to provide tasks thatreplied, “The ‘look-fors’ in the classroom for other teachers in the school district could use fordiscussion, highly effective teaching and classroom discussions and student evaluations.encouraging teachers not to say what a kid can These tasks have been given to districtsay.” To the question, “What do you want to take mathematics instructional staff and will beback to your school?” one principal replied, uploaded to their online database for future use inVirginia Mathematics Teacher vol. 43, no. 1 39

division curriculum and instruction. American Educational Research Journal,Conclusion 38(4), 915-945. Hill, H., Rowan, B., & Ball, D.L. (2005). Effects of Though we are only one year into this three teacher’s mathematical knowledge foryear project, findings suggest positive changes to teaching on student achievement. A mericanteacher content and practice, as well as, Educational Research Journal, 42(2), 371-administrative supports that will enhance 406.mathematics instruction in the schools. By Kanold, T.D, Briars, D.J., & Fennel, F. (2012).simultaneously working with teachers and What principals need to know aboutadministrators, this project hopes to build a school- teaching and learning mathematics?.wide, collaborative effort for deepening teacher Bloomington, IN: Solution Tree Press.understanding of mathematics content and Roberts. S. & Pruitt, E. (2003). Schools aspedagogical strategies. We believe this project professional learning communities:adds to the existing knowledge base on effective Collaborative activities and strategies forprofessional development models for working with professional development. Thousand Oaks,middle school teachers and administrators within CA: Corwin.schools and across the school division. The resultsof the first year of our project reveal that Heather Nunnallyprofessional development of this type does have a Instructor of Mathematicspositive impact on teachers and their students. We Virginia Commonwealth Universityencourage mathematics teacher leaders, school [email protected], mathematics coaches to engageteachers in similar activities in the school buildings Kristina Anthonyin which they work. For more information Instructor of Mathematicsregarding this program and products, please contact Virginia Commonwealth UniversityDr. Aimee Ellington ([email protected]) or theauthors of this article. Anthhonykc@vcu ReferencesCampbell, P. & Malkus, N. (2011). The impact of elementary mathematics coaches on student achievement. The Elementary School Journal, 111, 430-454.Garet, M., Porter, A., Desimone, L., Birman, B., and Yoon, K.S. (2001). What makes professional development effective? Results from a national sample of teachers.Math and Science Partnership Grants Years 2010-2016 Alexandria City Public Schools Chesterfield County Public SchoolsScience Academy for Improving Learning in Science (SAILS) Building Teacher Capacity in Mathematics Instruction to Science 2015-2016 Support English Language Learners Mathematics 2015-2016 Mr. Daniel Alcazar-Roman and Gregory Tardieu Sharon Hoffert [email protected] [email protected] [email protected] [email protected] [email protected] Caroline County Public Schools Emory & Henry College Rational Numbers and Proportional Reasoning for Middle School Teachers https://sites.google.com/a/ehc.edu/emt/home E&H Regional Mathematics Professional Development Ctr. Mathematics 2015-2016 Beth Burnap Grades 4-6 Mathematics 2010-2012 [email protected] [email protected] Douglas Arnold [email protected] Mathematics Teacher vol. 43, no. 1 40

Contemporary Teaching of Science and the Nature of Sci- Proportional Reasoning: Teaching and Assessing Virginia's ence Grades K-5 2009 Grades 5-9 Mathematics SOL Science 2012-2013 Mathematics 2015-2016 Padmanabhan Seshaiyer and Jennifer Suh Douglas Arnold [email protected] [email protected] [email protected] EMT (Elementary Math Teaching) Developing Rational Numbers and Proportional Reasoning Mathematics 2015-2016 through Math Modeling and Performance Based Assess- Douglas Arnold [email protected] ments: Teaching and Assessing Virginia's 2009 6-8 Mathe- matics Standards of LearningGeorge Mason University Mathematics 2015-2016http://completecenter.gmu.edu/index.htmlCenter for Excellence in Mathematics, Grades 7-8 Padmanabhan Seshaiyer [email protected] 2010-2012 Henrico County Public SchoolsPadmanabhan Seshaiyer and Jennifer [email protected] [email protected] Supporting Math Process Goals through Research-Based Teaching PracticesCtr for Excellence in Mathematics Professional Learning &Coaching in Northern VA Grades K-3 Mathematics 2015-2016Mathematics 2010-2012 Cheryl Gray Ball and Skip TylerPadmanabhan Seshaiyer and Jennifer Suh [email protected] [email protected]@gmu.edu [email protected] Middle School Science Teachers Soar (MSSTS) Science 2015-2016ESTEEM: Expeditions in Science, Technology and Engineer- ing Education through Mathematics: Grades 9-12 Cheryl Gray Ball and Laura Casdorph STEM: Focus on Mathematics 2011-2012 [email protected] Padmanabhan Seshaiyer [email protected] [email protected] Fundamental Mathematics for Number Sense: En- James Madison Universityhancing Virginia K-2 Standards of Learning and AssessmentCenter for Outreach in Mathematics Professional Learning http://www.jmu.edu/stem/outreach/mathk-6.html Developing Specialized Content Knowledge for Teachers of and Educational Technology Mathematics 2012-2013 Mathematics Grades 4-6 Mathematics 2010-2011 Padmanabhan Seshaiyer [email protected] LouAnn Lovin [email protected] Modeling Mathematical Ideas for Rational Numbers and Developing Specialized Content Knowledge for Teachers ofProportional Reasoning: Teaching and Assessing VA Grades Mathematics Grades K-3 Mathematics 2010-2011 6-8 Mathematics SOL Mathematics 2012-2013 LouAnn Lovin [email protected] Padmanabhan Seshaiyer [email protected] Rational Numbers and Proportional Reasoning Enhancing Student Achievement Across Virginia Throughthrough Math Models and Performance Based Assessments: Modeling Instruction Science 2014-2015 Teaching and Assessing Virginia's 2009 6-8 Mathematics Brian Utter [email protected] Standards of Learning Mathematics 2013-2014 Longwood University Collaborative for Math Professional Development (CoMPD): Padmanabhan Seshaiyer [email protected] Number and Number Sense through Math Models Teaching the 2009 Grades 4-6 Mathematics SOL in Southern and Performance Based Assessments: Teaching and As- VA Mathematics 2010-2012sessing V.A.'s 2009 K-2 Mathematics Standards of Learning Manorama Talaiver [email protected] Mathematics 2013-2014 Collaborative for Math Professional Development (CoMPD): Padmanabhan Seshaiyer [email protected] Teaching the 2009 K-3 Mathematics SOL in Southern VA Mathematics 2010-2012Building Number, Number Sense and Computational Fluen- Sharon Emerson-Stonnell [email protected] through Math Modeling and Performance Based Assess-ments: Teaching and Assessing Virginia's 2009 3-5 Mathe- STEM Kairos: Engineering Everywhere (K-3) STEM 2011-2012 matics Standards of Learning Manorama Talaiver [email protected] Mathematics 2014-2015 Padmanabhan Seshaiyer [email protected] of Argumentation Skills in Science (SASS) Integrating Nature of Science and Physical Science in In-Science 2014-2015Erin Peters-Burton [email protected] struction in Rural Schools (INSPIRS) Science 2012-2013https://sites.google.com/site/acpssails1516/ Manorama Talaiver [email protected] Transitions: Transforming Mathematics InstructionThrough Mathematical Modeling, Algebraic Thinking andVirginia Mathematics Teacher vol. 43, no. 1 41

Teaching the 2009 Mathematics SOL in Grades 3-5 Across Melani Loney [email protected] Southern Virginia Portsmouth Public Schools Mathematics 2012-2014Sharon Emerson-Stonnell [email protected] SAIL: Scientific Argumentation and Inquiry for Learning Science 2015-2016VA STEM Collaborative Nurturing Network to Enhance Content-focused Teaching (VA STEM CoNNECT) Laura Nelson, Brena Daniels, and Sherry Umphlett STEM 2012-2014 [email protected] Manorama Talaiver and Paula Klonowski [email protected] [email protected] [email protected] [email protected] Collaborative for Innovative and Enhanced Content Radford UniversityExcellence: Middle School (SCIEnCE)Science 2013-2014 http://www.radford.edu/rumath-smpdc/Suzanne Donnelly [email protected] Southwest & Southside VA Secondary Mathematics Profes-Science Collaborative for Innovative and Enhanced Content sional Development CenterExcellence: K-5 (SCIEnCE) Mathematics 2010-2011 Agida Manizade and Laura Jacobsen Spielman [email protected] [email protected] 2013-2014 Southwest & Southside VA Secondary Mathematics Profes-Suzanne Donnelly [email protected] sional Development Center Mathematics 2011-2012PREP: Proportional Reasoning Enrichment Project Mathematics 2014-2015 Agida Manizade and Laura Jacobsen [email protected] [email protected] Virginia Lewis [email protected] Mathematics Partnership for Rural Elementary Secondary Mathematics Professional Development Center Schools (IMPRES) Mathematics 2012-2013 Mathematics 2014-2015 Agida Manizade and Laura Jacobsen Paula Leach [email protected] [email protected] [email protected] Lunenburg County Public Schools Secondary Mathematics Professional Development Center Grades 9-12Improving Mathematics of Teachers in Lunenburg County (IMathTLC) Mathematics 2013-2014 Laura Jacobsen and Agida Manizade Mathematics 2015-2016 [email protected] [email protected] Sidney Long [email protected] City Public Schools Secondary Mathematics Professional Development CenterNumbers & Number Sense and Computation and Estimation Mathematics 2014-2015Mathematics 2015-2016 Jane Carter Laura Jacobsen, Agida Manizade, Kristan [email protected] tspra­ [email protected] [email protected] [email protected]@nortoncityschools.org [email protected] Secondary Mathematics Professional Development Center Old Dominion University Mathematics 2015-2016 Darryl Corey, Agida Manizade, Laura Jacobsen, Rayya Younes,Partnership for Teaching and Leading a Quality Science Pro- and Roofia Galeshi gram, K-5 [email protected] [email protected] [email protected] [email protected] Science 2011-2012 [email protected] Karen Sanzo [email protected] Enhanced through the Nature of Science (LENS): Regent UniversityAn Inter-Disciplinary Sustainable Professional Development http://www.regent.edu/acad/schedu/mathandsciencecenter/ grants.cfm Model for High School Science Inquiring Young Minds Want to Know, K-5 Science 2012-2014 Science 2011-2012 Jenny Sue Flannigan [email protected] Joanna Garner [email protected] Old Dominion University Research Foundation Flipped Out for ScienceProject D 'n' A: Building Blocks for Middle School Science Science 2013-2014 Jenny Sue Flannigan [email protected] Literacy Science 2014-2015 Standards Within Real-World Learning Joanna Garner [email protected] Science 2015-2016 Building Bridges Science 2015-2016 Jenny Sue Flannagan [email protected] Joanna Garner [email protected] Knowledge through Experience for Youth in Science KEYS 42 Science 2015-2016Virginia Mathematics Teacher vol. 43, no. 1

Spotsylvania County Public Schools PCRRC: A Process Standards Approach to Build Bridges from Arithmetic to Algebraic Thinking (High School)Integrated Professional Learning Communities (K-8) Cen- Mathematics 2010-2012tered around Applied Math & Applied Science to Support Vickie Inge (PI) and Loren Pitt (Co-PI) [email protected] [email protected] Vertical Best Practices Mathematics and Science 2015-2016 Virginia Commonwealth University Amber Belako [email protected] Interactive Mathematics Institute for Middle School Teach-Sweet Briar College ersCentral Virginia Collaborative for Developing STEM Les-sons to Improve Learning in Grades 4 and 5 Mathematics 2015-2016STEM 2011-2012 Aimee Ellington [email protected] Nelson Granger [email protected] VISTA ELIS at VCU Science 2015-2016Introduction to Inquiry: A Professional Development Modelto Reform Teacher Practices Elizabeth Edmondson [email protected] 2012-2013 VISTA MELIS at VCUJill Nelson Granger [email protected] Science 2015-2016A Successful Teacher Professional Development Model for Elizabeth Edmondson [email protected] Teaching in STEM Virginia TechScience 2013-2014Jill Nelson Granger [email protected] Inquiry by Engineering Design in Middle School Science and Mathematics Grades 6-8 The Institute for Advanced Learning & Research Mathematics 2011-2012Collaborative for Math Professional Development (CoMPD): Mary Kasarda [email protected] Teaching the 2009 Grades 7 - 8 Mathematics SOL in Southside VA Virginia Tech/Buchanan County Mathematics Partnership Mathematics 2015-2016 Mathematics 2010-2011 Penny McCallum Julie Brown/As of Jan 19, 2011, Dr. Brown was no longer em­ [email protected] [email protected] ployed at IALR. There was no replacement. [email protected] [email protected] Rector and Visitors of the University of Vir- The College of William & Mary ginia The Tidewater Team Middle School Mathematics Profession- al Development Ctr (High School) VISTA ELIS at UVa Mathematics 2010-2011 Science 2015-2016 Marguerite Mason, Ph.D and David Lutzer, Ph.D. [email protected] [email protected] Jennifer Maeng [email protected] of Virginia Tidewater Team to Improve Middle School Mathematics In-Teaching Scientific Inquiry and the Nature of Science,Grades K-5 struction Mathematics 2013-2014Science 2011-2012 Marguerite Mason [email protected] Bell and Jennie Chiu [email protected] Optics, Electricity, and Magnetism Home Lab Activi- Tidewater Team to Improve K-2nd Grade Mathematics Achievement ties Mathematics 2014-2015Science 2012-2013Richard Lindgren [email protected] Marguerite Mason [email protected] Grades 6-12 Science Teacher Leaders' Under- Southern Virginia Elementary Mathematics Coalition to En- hance Student Achievement through Teacher Professionalstanding of Electricity, Magnetism, and Light via Modeling Developmentand InquiryScience 2014-2015 Mathematics 2015-2016 Marguerite Mason and George RubleinRichard Lindgren [email protected] [email protected] [email protected] Teacher-Leaders in Integrated Math and Science Tidewater Team to Enhance STEM Education in Grades 4-5 for Inclusive Classrooms STEM 2015-2016 Science 2014-2015 Marguerite Mason [email protected] Candace Lutzow-Felling [email protected] Century Teaching Leads to 21st Century Learning:Functions Algebra Project (High School)Mathematics 2010-2012Vickie Inge (PI) andLoren Pitt (Co-PI)[email protected] [email protected] Mathematics Teacher vol. 43, no. 1 43

Call for ManuscriptsAll submitted articles should be single spaced, with 12 pt. Times New Roman font, and should be in APA style. Teaching Dilemmas Unsolved Mathematical Mysteries We are accepting articles that include teachers’ We are accepting articles about fascinatingreflections on mathematical topics you find challenging mathematical problems that have not yet been solved.to teach. If you have a difficult problem or topic you use The problem itself should be described simply so that a for your students, please describe the problem, discuss middle school student could understand it. It should also common student difficulties with it, and the way you include the progress that has been made by the approach teaching this topic. Please send to Dr. Agida mathematics community on solving this problem. Manizade at [email protected] with the subject line: Please send to Dr. Agida Manizade at [email protected] with the subject line: Unsolved Mathematical Mysteries Teaching Dilemmas by: Oct 30th, 2016. by: Oct 30th, 2016. Research or Practitioner Articles Math Girls We are accepting papers on elementary, middle, and secondary mathematics teaching and education. The What does your school district or institution do to encourage girls in mathematics? If you have any theme will be “Communicating in the universal language of math.” Articles will be peer-reviewed and information or publications related to girls incomments will be shared with the author. The deadline mathematics education, please send tofor consideration for the Spring 2017 issue is Oct 30th, 2016. Please provide articles in a word document no [email protected] with the subject line: Math Girls by: Oct 30th, 2016. more than 5 pages (not including corresponding images, pictures, graphs etc.). All submissions or My Remarkable Student questions should be sent to Dr. Agida Manizade at Have you had a student that changed the way you think [email protected] with the subject line: Research or and teach? If so, please send an article describing your experiences with this student and the ways they have Practitioner Article. affected your teaching to Dr. Agida Manizade at Good Reads [email protected] with the subject line: My Remarkable Do you know of a teaching resource or literature laden with mathematics content? Please send any literature Student by : Oct 30th, 2016. reviews to Dr. Betti Kreye at [email protected] with the Busting Blockbusters subject line: Good Reads by: Oct 30th, 2016. In this section we are accepting suggestions for scenes Technology Review from movies for readers to analyze and explain the Do you have a review of an app, website, or online mathematical plausibility of them. Please describe theresource? Please send your critical review to Christophe Hirel at [email protected] with the subject line: scene and provide a timestamp for it, and send to vmt.radford.edu with the subject line Busting Technology Review by: Oct 30th, 2016. Blockbusters Suggestion by Oct 30th, 2016. Reviewers Calling Virginia AuthorsIf you are willing to serve as a reviewer, please contact Virginia residents whose articles appear in the VMT will be granted free membership in the VCTM for one us at [email protected] year.Virginia Mathematics Teacher vol. 43, no. 1 44

VISTA ELIS Professional Development Jennifer L. Maeng, Elizabeth W. EdmondsonAbstract model employed in the VISTA Elementary Science The Virginia Initiative for Science Teach­ Institute (ESI) that was funded by the U.S. Depart­ ment of Education Investing in Innovation granting and Achievement (VISTA) Elementary Litera­ awarded to George Mason University. This re­cy Integrated with Science (ELIS) Professional De­ search-based PD model (e.g. Bell, Maeng, Konold,velopment (PD) provides an integrated approach to & Whitworth, 2015) employs a Learn-Try-teaching science and literacy through problem- Implement approach in which problem-basedbased learning (PBL). The VISTA ELIS PD, de­ learning (PBL) provides a meaningful context forsigned for elementary and middle school teachers, teachers to integrate science, mathematics, literacy,began spring of 2015 and will end fall of 2018. In and engineering to engage students in solving real-this article, we describe the PD and report changes world problems.in teacher confidence and understandings as well asparticipant perceptions of the PD after the first year The Learn-Try-Implement VISTA ELIS PDof the project. Results indicated that while their model is well-aligned with components of PD like­content knowledge around matter and energy did ly to change teachers’ practices and improvenot change, their confidence in using a PBL ap­ student achievement (e.g. Desimone, 2009;proach to teach science, integrating literacy strate­ Loucks-Horsley et al., 2010). For example, the PDgies, nature of science, inquiry, and technology into is sustained and ongoing, well-aligned or coherentinstruction improved following their first year in with school and division goals and state Standards,the PD. Teacher-generated PBL units were the pri­ content-focused, engages teachers in active learn-mary product of the PD. ing by modeling instructional approaches andIntroduction providing opportunities for reflection and discus­ sion, and provides just-in-time support through ex- The Virginia Initiative for Science Teach­ pert coaching. Finally, it included collective partic-ing and Achievement (VISTA) Elementary Litera­ ipation, promoting opportunities for teachers col­cy Integrated with Science (ELIS) Professional De­ laborate in their professional practice (Birman etvelopment (PD) provides an integrated approach to al., 2000). Research suggests these components ofteaching science and literacy across three years be­ effective PD can support the professional learningginning in the spring of 2015.VISTA ELIS repre­ of teachers in all content areas (e.g. Desimone,sents two parallel Mathematics and Science Part­ 2009; Loucks-Horsley et al., 2010).nership grants (MSP) awarded by the Virginia De­partment of Education to the authors. The VISTA The VISTA ELIS PD includes a summerELIS PD supports K-8 teachers in Richmond City, institute that comprises content and pedagogy com­Henrico County, Chesterfield County, and Caroline ponents, academic year follow-up, principal in­County Schools and K-5 teachers in Albemarle volvement, and just-in-time coaching to supportCounty and Waynesboro City Schools. In this arti­ elementary and middle school teachers with neededcle, we share our findings after year one of the pro­ science content knowledge, literacy approaches,jects. and relevant pedagogical strategies. The PD engag­ es teachers in a process of learning science content Goals of VISTA ELIS were to increase (e.g. life processes and living systems, force, mo­teachers’ science content and pedagogical tion, energy and matter) and pedagogical strategiesknowledge, increase teachers’ reflective instruc­ (e.g. inquiry, nature of science, PBL, literacy, andtional practice, develop teacher-leaders in problem- mathematics) during summer institutes, tryingbased learning instruction and science and literacy these strategies with in-class coach support duringintegration, increase students’ achievement in sci­ the academic year, and implementing teacher-ence, increase principals’ awareness of and support developed PBL units in their classrooms. Each yearfor integrated science and literacy instruction, de­ focuses on a different NGSS Crosscutting Concept.velop a robust K-5 STEM education, and increase In year 1, this was matter and energy. In year 2, thetime spent on science. focus will be cause and effect and the focus of yearLearn-Try-Implement PD Model 3 will be patterns. The VISTA ELIS PD model is based on the Specifically, the model includes 1) a 10-day 45Virginia Mathematics Teacher vol. 43, no. 1

institute each summer that consists of integrated across the previous summers in their culminatingscience and science pedagogy coursework, 2) four schoolwide PBL unit.follow-up PD days across the academic year, 3) in- Research Designclassroom coach support during the school year, 4)completed PBL unit plans and assessments for use The VISTA ELIS PD is being evaluatedin their own classroom, 5) use of NSTA Learning overall through a quasi-experimental approach de­Center , and 6) opportunities for reflection and signed to assess changes in teachers’ content andfeedback from peers, coaches, and instructors. pedagogical knowledge and students’ scienceThese components are integrated throughout the achievement compared to a control group as wellPD with the goal of improving elementary teach­ as teachers’ overall perception of the quality of theers’ science content knowledge and capacity to in­ PD. At the conclusion of the final year, treatmenttegrate science and literacy instruction. The model teachers and students outcomes (e.g. contentalso includes principals and facilitates the building knowledge, pedagogical knowledge) will be com­of school-wide instructional supports for the inte­ pared to control teachers and students outcomes.grated approach. The present study, which includes only data During the first summer institute, content from the first cohort of teachers, reports changes inexperts led sessions specifically designed to build teacher confidence and understandings as well asteachers’ content knowledge around matter and en­ participant perceptions of the PD. The followingergy concepts. Teachers were also introduced to research questions guided the present investigation:pedagogical strategies related to inquiry, nature of  How does teachers’ content and pedagogicalscience, hands-on, PBL, and literacy by engagingthem in model life, physical, and earth/space sci­ knowledge change following participation inence lessons related to matter and energy that used the PD?these strategies. In these lessons, teachers acted asthe students so they experience the lessons as their  What content do teachers choose to teachstudents would. While not a primary project goal, through a PBL approach following the PD?mathematics was integrated into the science experi­ences and highlighted throughout the PD. For ex­  What are teachers’ perceptions of the PD?ample, during model lessons teachers collected da­ To answer these questions, teachers’ wereta and created data tables and graphs, made quanti­tative measurements, and performed calculations assessed pre- and post-summer institute on theirwhile analyzing data. Teachers then selected from a understandings of and confidence in incorporatingnumber of model PBL unit overviews aligned with PBL, inquiry, NOS, and literacy instruction, theirtheir grade level standards and matter and energy. content knowledge, and their perceptions of theThey modified these PBL units and developed dai­ summer component of the PD. Thus, in this quasi-ly lesson plans that included each of the strategies. experiment pre-/post-test design, teachers served asDuring the academic year, they received feedback their own control. Data were analyzed using de­from instructional coaches as they implemented the scriptive and inferential statistics and systematicunit. They also participated in follow up PD to sup­ data analysis (Miles & Huberman, 1994).port integration of other literacy and science peda­gogies (e.g. discourse circles). Below, we discuss changes in Cohort 1 treatment teachers’ science content and pedagogi­ During the second summer, returning teach­ cal knowledge following the summer institute, theers (cohort 1) will refine their initial PBL unit and content topics and pedagogical strategies embed­develop a new PBL unit for their grade level based ded in PBL units developed, and teachers’ percep­on the crosscutting concept of cause and effect. tions of the effectiveness of VISTA ELIS PD. TheThe teachers will also begin exploring authentic first cohort included 21 teachers from 10 schools inassessment and create an authentic assessment for one district, 9 teachers from two schools in anothertheir PBLs. Teachers new to the PD (cohort 2) will district, and 9 teachers from one school in a thirdcomplete summer one activities, described above. district (Table 1).During the final summer, teachers from both co­horts will work as school teams to develop a Resultsschoolwide PBL unit focused around a single Results indicated teacher confidence of thecrosscutting concept (i.e. matter and energy, causeand effect, patterns). They will also experience and targeted pedagogical strategies increased as a resultlearn about other types of authentic assessment and of the summer institute. However, their contentembed multiple pedagogical strategies learned knowledge related to matter and energy did notVirginia Mathematics Teacher vol. 43, no. 1 change following the summer institute. Teachers developed PBL units that addressed a variety of elementary and middle school science content and 46

Table 1. Cohort 1 Participant DemographicsDistrict # Teacher com- % Female % Minority Years Experience(# schools) pleting (M SD) Summer InstituteA (10 schools) 21 100% 9.5% 15.3 (8.9)B (2 schools) 9 100% 11% 7.4 (6.3)C (1 school) 9 78% 100% 8.3 (9.4)integrated literacy instruction, nature of science, of the survey and the post-assessment was longereducational technology and inquiry into these units. than the pre-assessment; participant fatigue mayMany of these units also embedded mathematics have contributed to the lack of difference in scorescontent. Finally, teachers reported positive percep­ pre- to post.tions of the summer institute component of the PD. Teacher Perceptions of PD EffectivenessWe elaborate on these results below.Teacher Confidence and Understandings Pre/post summer institute results indicated teachers were very satisfied with opportunities to Results indicated a statistically significant practice integrating PBL, inquiry, NOS, and ofincrease in teachers’ confidence regarding the tar­ practice literacy strategies (all means>4.0). Further,geted pedagogical approaches following participa­ teachers perceived they were part of a communitytion in the summer institute. However, no differ­ (M = 4.38, SD = .60) and reported being very like­ence existed in teachers’ content knowledge fol­ ly to implement what they learned in the near fu­lowing the first summer institute (Table 2). Several ture (M = 4.76, SD = .50). For example, one teach­reasons may exist for the lack of observed change er noted about her experience:in teachers’ content knowledge. First, the assess­ment was out of 16 questions. The majority of par­ This workshop was very effective becauseticipants missed between 3 and 5 questions on both of the range of concepts covered, the varie­the pre and post-assessment with one participant ty of experiences, and the opportunity toscoring perfectly on the preassessment and two an­ interact and collaborate with other teachers.swering all questions correct on the post- One of the best parts was the focus, whichassessment. It is possible, given the teachers’ rela­ was to create a science unit that we will ac­tively high beginning mean content knowledge tually do; it was purposeful, and not simply(M=12.6), that there was a ceiling effect on the as­ theory. I also like the fact that we will besessment. Second, the questions selected were de­ able to build on this summer's work nextsigned to get at misconceptions about matter and year, and that we are involved in somethingenergy concepts and were highly conceptual in na­ that is ongoing, and that we will haveture. Given the number of misconceptions related coaching and more collaboration throughto matter and energy documented in the literature the year. Most PD is finite, but here we(e.g. Driver, Squires, Rushworth, & Wood- have become part of a new 'community' thatRobinson, 1994), it is possible that these were not will continue to work together. (Kim, Post-fully addressed during the summer institute. Final­ PD Survey)ly, the content questions were presented at the end PBL Units GeneratedTable 2. Changes in Teachers’ Pedagogical Confidence and Content Knowledge. Pre-SI Post-SI Paired Samples Sign. M (SD) M (SD)Confidence (n=37) 4.05 (.62) 3.86 (.53)PBL 2.51 (.90) 3.67 (.58) <.001 4.30 (.66) <.001Inquiry 2.76 (1.1) 4.40 (.64) <.001 12.4 (1.7) <.001NOS 2.35 (.95) <.001 .627Literacy 3.30 (1.2)Literacy into science 2.89 (.71)Matter/Energy Content Score (n=34) 12.6 (1.6)Virginia Mathematics Teacher vol. 43, no. 1 47

Table 3. Science and Math Topics Addressed in Participant-generated PBL Units.Grade Level Science Topics Addressed Science SOL Alignment Math SOL AlignmentKindergarten Gardens K.1, K.6, K.7 K.8, K.15First Sun’s energy 1.1, 1.2Second Changes in matter 2.1, 2.2 1.1, 1.9, 1.10 Ecosystems 2.1, 2.4, 2.5, 2.7 2.11, 2.14, 2.17, 2.18, 2.19 2.17, 2.18, 2.19, 2.20Third Simple machines 3.1, 3.2 3.17FourthFifth Properties of matter 3.1, 3.3 3.19, 5.18 Weather 4.1, 4.6 4.7, 4.13, 4.14 Ecosystems 4.1, 4.5 4.14 Force/Motion/Energy 4.1, 4.2 4.7, 4.14 Sound 5.1, 5.2 5.18 During the first summer institute, teachers tic student role, culminating activity, and resourcesworked in teams to generate entire PBL units in­ (Figure 1).cluding an overview document, question map, anddaily lesson plans. Selection of the topics of their For example, in a 4th grade PBL unit onPBLs was informed by their grade level Standards, Living Systems, the problem question was, “Oh,alignment with the theme of Matter and Energy, Honey: It’s a bad time to be a bee, does it have toand teacher topical interest. Table 3 describes the be?” Teachers posed the following scenario to stu­grade-level and dents:content of thePBL units created Bees affect yourby Cohort 1 daily life a lotteachers as well more than youas science and might think. Asidemath SOL align­ from giving usment. These honey and wax,teacher-generated they pollinatePBL units, which plants that pro-include unit over­ vide a quarter ofviews as well as the food eaten byeasily modified Americans, ac-daily lesson plans counting for morewill be available than $15 billionbeginning Fall in increased crop2017 via a link value per year,from the VDOE according to theScience Re­ U.S. Departmentsources website. of Agriculture.Unit overviews But bees aroundand question the world havemaps are current­ been dying inly available droves for theat http:// past severalbit.ly/29DwkSt. years, and scien- Overview tists are stilldocuments in­ struggling to un-cluded an over­ derstand why.arching question, The problemscenario, authen­ Figure 1. Example PBL overview. seemed to be get- ting better lastVirginia Mathematics Teacher vol. 43, no. 1 year. Now, how- 48

ever, things seem to be getting worse again. tion of the VISTA ELIS PD suggest that a Learn- U.S. beekeepers saw annual losses of 42.1 Try-Implement PD model can build K-5 teachers’ percent between April 2014 and April 2015, confidence and understanding of pedagogical ap­ according to a new federal survey. The proaches that increase emphasis on interdiscipli­ Central Virginia Beekeepers Association nary (e.g. literacy, math, science) instruction. Inte­ needs our help. They want us to develop grating content across different disciplines using a community outreach program to inform the PBL approach helps students understand how con­ public about the importance of bees and tent is interconnected in the real world and not iso­ what we can do to help bee thrive. lated into silos. This is particularly true for scienceIn this unit, students explored ecosystems, food and mathematics as evidenced by the ways inwebs, niches, and human impacts on ecosystems by which mathematics content was naturally integrat­taking on the role of a Junior Bee A dvocate. Litera­ ed into the teachers’ science PBL units. Further,cy skills were integrated throughout the unit. For teachers perceived PBL as a way to hook studentsexample, students read expert texts and answer into learning in authentic ways, which research in­guiding questions and completed a vocabulary ac­ dicates is important in supporting student learningtivity. Mathematics skills could include making (e.g. MacMath, Wallace, & Chi, 2009). Finally,charts of collected data and analyzing population teachers reported positive perceptions about thedata collected from a simulation (Figure 2). opportunity to collaborate and plan a unit together In another example for a 3rd grade unit on within or across schools; many of the PBL unitschanges in matter, teachers presented students with were developed in teams from different schools.the problem question, “What is matter? What are Thus, opportunities for teachers to learn and planthe properties of various types of matter?” through together remains critical. Future research will ex­the scenario: plore student achievement following participation The country of Fluvania has stolen the con- in the teacher-generated PBL units developed dur­ fidential Hasbro play-doh recipe. We have ing the VISTA ELIS PD. purchased the last samples of play-doh available in the United States. It is our mis- References sion to duplicate Hasbro’s original recipe Bell, R.L., Maeng, J.L., Konold, T., & Whitworth, to the best of our ability.In this unit, students took on the role of Hasbro In­ B.A. (2015). Professional development tovention Consultants and explored what is matter, support elementary teachers’ understand-distinguishing characteristics of solids, liquids, and ing and implementation of reforms-basedgases, and changes in phases of matter. Literacy science: Randomized controlled trial. A pa­skills were integrated through read alouds, turn and per for the annual meeting of American Ed­talks, use of OWLs, and use of Anchor Charts ucation Research Association, Chicago, IL.throughout the unit. Mathematics skills were inte­ Desimone, L. M. (2009). Improving impact studiesgrated throughout the unit as students made meas­ of teachers' professional development: To­urements of the mass and volume of solids and liq­ ward better conceptualizations anduids using balances and measures. Educational Researcher, 38, 181-beakers and recordedthese in data tables.More advanced studentscould calculate the densi­ty of the solid objectsfrom these data based onthe equation D = mass/volume to determine iftheir playdoh recipe hasa similar density to theactual playdoh.Conclusions and Impli-cations Results from the Figure 2. Example population simulation data and student graph. (Simulation screenshot re­first year of implementa­ trieved from www.explorelearning.com.)Virginia Mathematics Teacher vol. 43, no. 1 49

199. 50Driver, R. Squires, A. Rushworth, P. & Wood- Robinson, V. (1994). Making Sense of Sec- ondary Science: Research into Children’s Ideas. New York: Routledge.Loucks-Horsley, S., Stiles, K.E., Mundry, S., Love, N., & Hewson, P. (2010) Designing profes- sional development for teachers of mathe- matics and science. (3rd Ed.) Thousand Oaks, CA: Corwin Press.McMath, S, Wallace, J., & Chi, Xiaohong. 2009. Problem-based Learning in Mathematics. What Works? Research into Practice. Re­ search Monograph #22 (November). Avail­ able at: literacynumeracy/inspire/research/ WW_problem_based_math.pdfMiles, M. B. & Huberman, A. M. (1994). Qualita- tive data analysis: An expanded sourcebook (2nd Ed). Thousand Oaks: Sage Publica­ tions. Jennifer Maeng Curry School of Education University of Virginia [email protected] Elizabeth W. Edmondson Principal Investigator Virginia Commonwealth University [email protected] Mathematics Teacher vol. 43, no. 1