Unsolved Mathematical Mysteries: Rectangles Covering Rectangles Matt HarveyA rectangle with dimensions can For a rectangle R, there are threebe covered by three slightly reduced copies of restrictions that limit how small x can be (seeitself, rectangles of sizes where x<1. figure 2 on the left).For a square, one such covering was identified bythe mathematician Henry Dudeney and published 1. In order for the left two rectangles to cover R from top to bottom, , soin his book Puzzles and Curious Problems in 1931. .It is shown in figure 1 on the left. In hisconstruction, the scaling factor x is the square root 2. In order for the right rectangle to cover R from top to bottom, .of , so . 3. In order for all three rectangles to cover R from left to right, , so . The three conditions are graphed in figure 2 on the right. To cover R, all three conditions must be met, so the minimum value of x that the arrangement will permit, in terms of h, isFigure 1. (left) Dudeney's covering of a The graph of x (as well as the pointsquare by three smaller squares; (right) acovering of a non-square rectangle by threesmaller copies of itself.For any non-square rectangle, a covering corresponding to Dudeney's square construction) is shown in figure 3 on the left. Note that the pointsuch as the one shown in figure 1 on the right, with on the graph corresponds to a goldenone rectangle turned , works. It is natural toask: how small can be in this construction? rectangle, for which , so restrictions 2 and 3 are met exactly, as shown inSince similar rectangles will have the same figure 3 on the right.minimum value, and since every non-squarerectangle is similar to a rectangle (with ), we may restrict our attention to rectangles, where Figure 3. (left) The graph of x as a function of h; (right) covering the golden rectangle. Figure 2. (left) To cover a This function describes only one possible rectangle, x must satisfy three conditions; arrangement, however, the one with a rectangle (right) graphs of the equations that govern turned . Other arrangements might give the three conditions. covers with smaller x values. The significant dropVirginia Mathematics Teacher vol. 43, no. 1 from the graph to the Dudeney point is evidence that there is room for improvement for rectangles 51
that are almost squares. Perhaps, for instance, thereis a way to morph from Dudeney's construction tothe -turn construction. The open question is: Given a rectangle R (where ), what is the smallest value x so thatthree rectangles can be arranged tocover R? Coverings of sets play an importanttheoretical role in many fields of mathematics,particularly those that are related to topology.However, in those fields, it is usually onlynecessary to know that the right kinds of coveringsare possible. It is not particularly important thatthey be optimally arranged, so this problem has notbeen studied extensively. Recently, there has beensome interest in related problems of coverings bycircles, squares, and other regular polygons. Agallery of best-known results may be found at thewebpage www2.stetson.edu/~efriedma/packing.html. Matt Harvey Associate Professor of Mathe matics The University of Virginia’s College at Wise [email protected] Mathematics Teacher vol. 43, no. 1 52
Integrating Federal, State and LocalMandates for the Benefit of Teachers’ Professional Learning Joanna K. Garner and Melani A. Loney,Abstract based PD features, and effective strategies for This article is intended to provide school building school-university partnerships. Project 1, Learning Enhanced through the Nature of Sciencedivision mathematics and science administrators (LENS), began in 2011 and served two cohorts ofand instructional specialists as well as university science teachers in high schools. Project 2,faculty members with a summary of three Virginia Discourse ‘n’ Argumentation: Building Blocks forMathematics and Science Partnership projects to Middle School Science Literacy (D’n’A), began inillustrate how projects can align federal, state and 2013 and served two cohorts of science teachers inlocal level mandates and goals. Project 1, LENS, high-needs middle schools. Project 3, Buildingdeveloped Virginia Beach City Public Schools’ Bridges Across the Curriculum Using Argumenthigh school teachers of Earth Science, Biology, Driven Inquiry (Bridges) began in 2015 andChemistry and Physics teachers’ content currently serves two cohorts of teachers from high-knowledge and nature of science understandings, needs elementary schools. The first author servedassessment literacy, capacity for data-informed as the Principal Investigator of all three projects.instruction, and peer coaching skills. Project 2, Old Dominion University was the lead IHE in eachD’n’A, provided professional development (PD) partnership in collaboration with the College offor middle school Life, Physical and Earth Science William and Mary (LENS), Tidewater Communityteachers in Portsmouth Public Schools and Virginia College (LENS, D’n’A) and the University ofBeach City Public Schools. The project highlighted Texas at Austin (D’n’A, Bridges). Divisionthe Argument Driven Inquiry (ADI) instructional partners include Virginia Beach City Publicmodel to increase teachers’ instructional emphasis Schools, Portsmouth Public Schools, Hampton Cityon argumentation. The project also resulted in Schools and Newport News City Public Schools.spontaneous transfer of ADI to mathematics Mapping MSP Priorities on to School-lessons in one school. Project 3, Bridges, is University Collaborationsongoing. It provides PD for teachers in high needselementary schools in Portsmouth Public Schools, Selected from VDOE MSP Request forHampton City Schools and Newport News Public Proposal documents (VDOE, 2011, 2012, 2013,Schools. Activities provide grade 4 and 5 teachers 2014)1, the project priorities shown in Table 1 alignwith the knowledge and skills to bridge or integrate with the federal MSP theory of action, which isinquiry-oriented science with mathematics and that student achievement can be impacted throughEnglish content through Argument Driven Inquiry, teacher professional development. Also shown arewhile PD for grade 3 teachers focuses on the research-based strategies included in the projectsdevelopment of performance based assessments (Borko, 2004; Darling-Hammond, Wei, Richardsonthat measure student progress in scientific & Orphanos, 2009) and implementation strategies.investigation and mathematics-related scienceskills. Descriptions of these projects, are enhanced Project 1: Learning Enhanced through theby the addition of tables and figures that Nature of Science (LENS): An Interdisciplinarysummarize a collaborative approach to partnership Approach to High School Science (2012-2014)formation and maintenance that was implementedthroughout all phases of each project. The general LENS responded to the 2011 call frommodel is provided as a reference for those seeking VDOE to integrate Earth Science and Physicalto design PD in science and mathematics. Science with other sciences and nature of scienceOverview principles (Flick & Lederman, 2004). PD activities saw multiple cohorts (48 teachers in total) working In this article we situate three VirginiaMathematics Science Partnership projects within 1More information can be found on the United States Departmentthe broader contexts of MSP priorities, research- of Education website for the MSP program, at http:// www2.ed.gov/policy/elsec/leg/esea02/pg26.htmlVirginia Mathematics Teacher vol. 43, no. 1 53
Table 1. Alignment of Federal/State Priorities with Research-Based Principles and Project ExamplesPriorities Research-based Principles ExamplesDesign: Partnerships be- Multi-stakeholder leader- Regular planning meetings throughout the project; divisiontween school divisions ship team for project de- input into content and format of professional development;and institutions of higher cision making ‘debrief’ meetings at institutes and school year work dayseducation to provide PD (LENS, D’n’A)in high needs schools Collaborative learning Interdisciplinary groups of teachers from within the same among teachers within schools conducted field investigations; discipline specificDesign: Data-informed and across schools groups among teachers created authentic assessmentsproject design originating (LENS, D’n’A, Bridges)from a comprehensive Analysis of school divi-needs assessment sion characteristics; SOL scores by strand and question determine PD content teacher input (Bridges); teacher survey determines needs and interest in On-going professional PD models (LENS) development Teachers participated in multiple years of summer insti- tutes and school year work days (LENS, D’n’A, Bridges)Implementation: School- Use of instructional Coach co-facilitated institute to establish rapport; observed teaching and debriefed afterwards; gathered feedbackbased approach coaching (Bridges) Teachers worked in grade level teams when learning andImplementation: Atten- Grade or subject level practicing the new instructional model (Bridges)tion to connections across teamsthe curriculum, including Teachers designed and conducted their own college-levelvertical alignment and Opportunities to deepen field investigations, the content of which was aligned withapplication across sub- specific curricular con- specific SOLs (LENS)jects tent knowledgeImplementation: Teaching Argument Driven Inquiry (ADI) trained faculty designed SOL-of content through discipli- Use of a specific instruc- based investigations; teachers were taught content throughnary practices tional model to integrate the strategy; teachers implemented investigations during content with practices; the school year (D’n’A, Bridges)Implementation: Assess- deepen pedagogical con-ments provide feedback tent knowledge Cohorts of teachers identified performance standards, SOLand measure student pro- Teacher assessment liter- content and cognitive levels, created rubrics, tables of speci-gress acy for improved instruc- fications, and items. (LENS, D’n’A, Bridges) tional decision making Teachers reviewed products of pilot work to inform imple- Use of student work to mentation and classroom management recommendations reflect on practices (LENS, Bridges)in heterogeneous groups (Physics, Chemistry, (Novak & Middendorf, 2004). Open-endedBiology, and Earth Science) to design and conduct questions without a single correct answer or afield-based investigations2. An interdisciplinary quantitative or “yes/no” answer were used toteam of college faculty acted as facilitators. The structure the content-based investigations andopen-ended, participant-centered nature of the guide participants’ thinking during key activitiesprojects and variations in participants’ prior such as data analysis (Stenmark, Penmark &knowledge necessitated just-in-time teaching, a Asturias, 1994). Discussions and written promptsmethod involving a cycle of questioning and permitted reflection for practice (Clarke, 2000) tocontent exploration to target gaps in understanding promote transfer to the classroom. The project also addressed local priorities of developing2A project description and video is available at http:// assessments to measure student progress. Teacherswww.odu.edu/education/programs/tcep/project/lens used SOL documents to generate performanceVirginia Mathematics Teacher vol. 43, no. 1 54
standards in the areas of scientific investigation and Science Literacy (2013-2015)science-embedded mathematics skills. Standards Project D’n’A served 35 middle schoolarticulated the expectation that the student wouldbe able to design an experiment, analyze sample science teachers. It targeted 2012 VDOE prioritiesdata, and draw content-based conclusions (see of collaborations with high needs, low-performingFigure 1). Embedded mathematics skills included schools, PD around a specific instructional model,multi-step problem solving, data tabulation and attention to teachers’ capacity to develop students’analysis, graphing, solving equations, estimating, skills in scientific discourse and argumentation,and using scientific notation.3 A teacher-developed mathematics-science integrations, and progress-5E format curriculum unit (Bybee et al., 2006) was based assessments. Two cohorts of teachersalso created to teach performance standards skills participated in summer institutes and school yearin the context of field-based investigations. The 5E meetings and collectively developed lessons thatunit progression included: Engagement, targeting use the Argument Driven Inquiry (ADI)students’ perceptions of unit relevance and instructional model (Sampson, Grooms & Walker,requiring a pre-assessment of students’ content 2011)4. ADI involves an eight-step process thatknowledge; Exploration, requiring students to scaffolds inquiry and provides opportunities forgenerate a research question and collect data; students to develop, evaluate and revise content-Explanation, requiring students to analyze and based arguments. Arguments are comprised of acommunicate their data; Extension, requiring guiding question, (e.g. for science, “How does lightstudents to connect their findings thematically; and interact with different surfaces?” or forEvaluation, requiring students to present their mathematics, “How does altitude affect thefindings through a ‘science fair’ classroom period. percentage of hemispherical area that is visible byThe unit is available upon request. an observer?”) followed by a claim, presentation ofProject 2: Project Discourse ‘n’ Argumentation evidence, and a justification. Students work in(D’n’A): Building Blocks for Middle School groups to develop a proposal for investigating a question, gather experimental or observational data, and prepare a whiteboard that includes eachFigure 1. Performance Standards for Secondary Sciences used to generate Measures of Student Progress3To access the assessments, single sign on (SSWS) permission 4See also www.argumentdriveninquiry.comis required. Go to http://www.doe.virginia.gov/administrators/superintendents_memos/2014/290-14.shtml. Tutorials on thedesign and utilization of data from the assessments can befound at https://sites.google.com/site/vasecassessscience2/home Sample assessments can be obtained from the authors. Ageneric version is shown in Figure 1 (a), (b), and (c).Virginia Mathematics Teacher vol. 43, no. 1 55
element of the argument. After discussing one included. Teachers have worked with universityanother’s boards, students receive content-based faculty to learn how to develop and pilotdirect instruction and collect additional data or performance assessments that assess progress inrevise their boards. The final product is an observation, classification, hypothesis generation,individual report that has been double-blind peer quantification, measurement, graphing, andreviewed prior to submission. drawing conclusion. Lessons and sample assessments are available upon request. PD events used the ADI strategy as the Key Findings and Limitationsinstructional model for teacher-level content. Thisallowed teachers to strengthen their own content Project evaluations measured teachers’knowledge and experience prior to using the model science content knowledge and argumentationinto their classrooms. LENS assessments were used skills (LENS, D’n’A, Bridges), and beliefs aboutas the basis for nine middle school assessments that the nature of science (LENS only). Statisticallymeasure student progress in experimental design, significant pre- to posttest differences were foundscientific reasoning, and argument evaluation. ADI in content and pedagogical content knowledgeassistance for mathematics and science is available (p<0.05), with the exception of science contentupon request. knowledge in the first year of the Bridges projectProject 3: Building Bridges across the (p>0.05). Teachers’ motivation for their subjectElementary Curriculum Using Argument area was positively impacted by the PD focus onDriven Inquiry (Bridges). (2015-present) hands-on, teacher-designed investigations. Broader impacts included additional collaborations between Currently in its second year, Bridges has project partners, faculty adoption of the ADIserved 37 grade 4 and 5 teachers and 28 grade 3 instructional model in college courses, improvedteachers. At institutes hosted by Old Dominion STEM-pipeline related communication betweenUniversity and Hampton City Schools, grade 4 and high school and college faculty, and the5 teachers learn science content through ADI. The development of a generalizable approach toelementary version of ADI prioritizes opportunities workflow. Despite anecdotal reports andfor students to apply mathematics skills such as observations of high student engagement, a generalmeasurement, quantification, data tabulation and limitation of the projects is the low number ofgraphing, in science-oriented contexts. During the teacher participants from SOL-tested grades andschool year, an instructional coach supports the resulting lack of evidence regarding impact onimplementation of pre-developed lessons. Once SOL scores. An additional limitation of the LENSagain, local assessment priorities have beenFigure 2. A General Model of Professional Development through School-University Collaboration 56Virginia Mathematics Teacher vol. 43, no. 1
and D’n’A projects was the low number of Sampson, V., Grooms, J., and Walker, J. (2011).classroom observations. This challenge is currently Argument-Driven Inquiry as a way to helpbeing prioritized in the Bridges project. students learn how to participate inRecommendations – A General Model of scientific argumentation and craft writtenProfessional Development Focused arguments: An exploratory study.Collaboration Science Education 95 (2), 217-257. A general model of PD design is presented Stenmark, J.K., Beck, P. & Asturia, H. (1994). Ain Figure 2 and reflects a workflow for those room with more than one view.seeking to form and maintain school-university Mathematics Teaching in the Middlecollaborations for mathematics or science. The School 1, 44-49.workflow reflects a collective commitment to SOL-based teacher PD as well as SOL-based classroom Virginia Department of Education (2014).instruction, a focus on collaborative PD design Mathematics and Science Partnershipbetween school divisions and university faculty, Program Request for Proposals. Retrievedcareful selection of research-based PD practices from http://www.doe.virginia.gov/based on project goals and division needs, and the federal_programs/esea/title2/part_b/deployment of evaluation and feedback gathering implementing_msp.shtmlmechanisms to inform refinement of the project atall stages of its implementation. Virginia Department of Education (2013). Mathematics and Science Partnership References Program Request for Proposals. Retrieved Borko, H. (2004). Professional development and from http://www.doe.virginia.gov/ federal_programs/esea/title2/part_b/ teacher learning: Mapping the terrain. implementing_msp.shtml Educational Researcher 33 (8), 3-15. Bybee, R.W., Taylor, J.A., Gardner, A., Van Virginia Department of Education (2012). Scotter, P., Powell, J.C., Westbrook, A., & Mathematics and Science Partnership Landes, N. (2006). The BSCS 5E Program Request for Proposals. Retrieved Instructional Model: Origins, Effectiveness, from http://www.doe.virginia.gov/ and Applications. Colorado Springs, CO: federal_programs/esea/title2/part_b/ BSCS. implementing_msp.shtml Clarke, D. (2000). Time to reflect. Journal of Mathematics Teacher Education 3, 201– Virginia Department of Education (2011). 203. Mathematics and Science Partnership Darling-Hammond, L., Wei, R.C., Andree, A., Program Request for Proposals (first Richardson, N., & Orphanos, S. (2009). round). Retrieved from http:// Professional Learning in the Learning www.doe.virginia.gov/federal_programs/ Profession: A Status Report on Teacher esea/title2/part_b/ Development in the United States and implementing_msp.shtml Beyond. Stanford, CA: National Staff Development Council. Yoon, K. S., Duncan, T., Lee, S. W.-Y., Scarloss, Novack,, G. & Middendorf, J. (2004). Just-in- B., & Shapley, K. (2007). Reviewing the time teaching: a 21st century pedagogy. evidence on how teacher professional Retrieved from https://serc.carleton.edu/ development affects student achievement introgeo/justintime/what.html (Issues & Answers Report, REL 2007-No.Flick, L.B. & Lederman, N.G. (2004). Scientific 033). Washington, DC: U.S. Department of inquiry and nature of science: Education, Institute of Education Sciences, Implications for Teaching, Learning, and National Center for Education Evaluation Teacher Education. NY: Springer. and Regional Assistance, RegionalPopham, J. (2006). All about accountability/ Educational Laboratory Southwest. needed: A dose of assessment literacy. Retrieved from http://ies.ed.gov/ncee/ Educational Leadership 63 (6), 84-85. edlabs/regions/southwest/pdf/ rel_2007033.pdfVirginia Mathematics Teacher vol. 43, no. 1 57
Joanna K. Garner Melani A. Loney, Ed.S.Research Associate Professor Prog Manager Military FamOld Dominion University Old Dominion [email protected] [email protected] Math JokesThe math teacher could never figure out why their students kept callingtheir cat mean. All it ever did was mu at them.Parallel lines have so much in common. It’s a shame they’ll never meet.Jimmy was in the lunch room, ready to take a bite of his six inch sandwich,when it slipped from his hands. His math teacher just sighed and shook hishead. Jimmy needed to work on his subtraction.Math problems? Call 1-800-[(10x)(13i)2]-[sin(xy)/2.362x].The United States celebrated it’s bicentennial 1976, but in the year 2090, itwill celebrate it’s first pi-centennial.Virginia Mathematics Teacher vol. 43, no. 1 58
Evaluation of Lesson Study Based Teacher Professional Development Model for Inquiry Teaching in STEMJill Granger, Arlene Vinion Dubiel, Hank Yochum, James Alouf, Tim LoboschefskiAbstract similar professional development projects funded by both Math Science Partnership grants and grants Middle school science and math teachers from the State Council of Higher Education forengaged in the process of inquiry/investigation andused inquiry strategies of teaching in their Virginia, the Science by Inquiry (SxI) at Sweet Briar College project team sought to improve andclassrooms. Our professional development (PD) identify the critical programmatic components thatmodel is based on effective practice that helpsstudents learn to question and analyze, as a means lead to success within a professional development project. The acronym INoVATR (“Innovator”) is aof understanding science and math content and the tag line that brings attention to the model’snature of science. Through two graduate courses,teachers reflected on their inquiry teaching emphasis on inquiry, vision, assessment, teaching, and reflection; and to identify our teacher-experiences and learned to use effective assessment participants with classroom innovation.strategies to gauge student understandings,misconceptions, and skills. The project The project team composed of faculty members in science, engineering, math, andincorporated a lesson study approach in which education, has focused for over a decade onreflection and assessment outcomes guidedrevision. Teachers worked in Peer Learning training science and math teachers to utilize an inquiry approach to teaching and learning. InquiryCommunities (PLCs) with designated inquiry is defined as a lesson that begins with a testablecoaches (peers with prior experience and success ininquiry teaching) and the college STEM faculty to question and through data collection and analysis, answers that question. The level of inquiry isfacilitate the use of best practices. The project had determined by the extent of student direction in abuilt-in dissemination plans involving teachers inlocal, regional, and state-wide workshops on lesson progressing from structured, to guided, to open (Bell, 2005).inquiry teaching and learning led by a team of The INoVATR project utilized bestscience and education faculty at the university leveldedicated to facilitating meaningful reform of practices from both research and from previous teacher professional development projects toinstructional practice in STEM education. A support middle school math and science teachers inrigorous project evaluation monitored the cohort of13 teachers, over time, in terms of typical program their efforts to teach through inquiry strategies in their classrooms. This project, like previous ones,effects (and compared with a group of intermittent utilized a modified lesson study in which lessonprogram attendees) on outcomes such as attitudestoward inquiry teaching, and the understanding and plans are continuously improved based on assessment results (Arani, 2010; Rock, 2005) withpractice of inquiry teaching. The outcomes of this targeted lesson reflection (Moseley, 2008) as theprojectcontributed to a primary professionalmore developmentgeneralizableapproach to activity (Figure 1). This projectteacher PD by also adopted twobuilding upon abasic model that well-recognized professionalthe project team developmenthas beencontinually strategies: Peer Learningimproving. Communities Aftersuccesses with (PLC) of groups of similar Figure 1. Lesson Study cycle used for each of four lesson plans.Virginia Mathematics Teacher vol. 43, no. 1 59
teachers collaborating on lesson development and number of knots in the rope?” In an open inquiryimplementation (Lieberman, 2011) and formal lesson, students articulate the question formentoring of the PLCs by an experienced inquiry investigation, design the procedures, and analyzeteacher (Counsell, 2011). These additional data to answer the question. An example isstrategies were incorporated into the established Transformations. For a more comprehensiveprofessional development program to provide description of inquiry-based instruction, see theadditional support to teachers as they progress article by Bell, Smetana, and Binns, 2005.through the project. The teacher-participants engaged in a In 2014 and 2015, middle school science summer course focused on teaching science andand math teachers from the Central Virginia math through an inquiry strategy with activitiescounties of Amherst, Appomattox, Bedford, across all levels of inquiry (Bell, 2005) andCampbell, Nelson, and Prince Edward and from targeted discussions on questioning and assessmentLynchburg city participated in the INoVATR strategies. During this initial course, teachers wroteprofessional development project. Teacher- a structured inquiry lesson plan to be implementedparticipants were grouped into PLCs of 2-4 during the first weeks of school. Four subsequentteachers based upon location and content areas. course meetings followed by lesson planning,Three additional teachers with previous experience implementation, and reflection, were spacedin inquiry teaching and assessment were identified throughout the academic year (Table 1).as project Coaches to work as formal mentors with Implemented lesson plans progressed fromthe PLCs to support implementation and structured to guided to open inquiry with the fourthdissemination of best practices locally and and final inquiry being at a level of the teachers’regionally. The goals of the project were to choice based upon lesson goals and objectives.increase teachers’ understanding and practice of Other professional development activities includedinquiry approaches to teaching science and math readings and discussions from course textscontent. Thirteen (13) teachers completed the full (Carlson, 2003; Llewellyn, 2007) and attendance atyear-long project and evaluations reported here. the Virginia Association of Science Teachers Professional Development Institute (VAST-PDI). Inquiry approaches to teaching math andscience begin with a testable question followed by In collaboration with the STEM facultyan investigation to generate data that students then members, project coaches worked with the PLCs atanalyze to answer the question. The level of each stage, troubleshooting lesson challenges andinquiry depends upon who provides the question providing workshops for teachers in response toand investigation procedures. In a structured common challenges. For example, coachesinquiry, students are provided with the question presented a workshop on helping students to writeand the procedures as described in the lesson testable questions in response to teachers’Volume and Surface Area of Rectangular Prisms. apprehensions in engaging with open inquiry.Only the question for investigation is provided in aguided inquiry with students designing the The INoVATR model of professionalprocedures and analyzing data to answer the development has been demonstrated to bequestion. An example is Linear Functions where successful by multiple project measures. Results ofstudents investigate the question “What is the two of the measures from this project arerelationship between a rope’s length and the highlighted here.Table 1. The INoVATR Lesson Study Schedule Study curriculum Inquiry Lesson 1 Inquiry Lesson 2 Inquiry Lesson 3 Inquiry Lesson 4and formulate goals Structured Inquiry Guided Inquiry Open Inquiry Teacher’s-Choice … before and during following previous following previous following previous summer course. inquiry. inquiry. inquiry. Plan the lesson … at the end of the after the lesson 1 after the lesson 2 after the lesson 3Implement the les summer course. class meeting. class meeting. class meeting. son … during the 1st nine during the 2nd nine during the 3rd nine during the 4th nine weeks of school. weeks of school. weeks of school. weeks of school.Analyze and reflect follows soon after follows soon after follows soon after follows soon after … implementation. implementation. implementation. implementation. at the end of the 1st at the end of the 2nd at the end of the 3rd prior to the project Class meeting is dissemination con held … quarter of school. quarter of school. quarter of school. ference.Virginia Mathematics Teacher vol. 43, no. 1 60
The effect of the professional development shown by comparing average scored for theon the participating teachers as a function of time, required level of inquiry with the level of inquiryover the course of four inquiry lesson chosen by the teachers for the final lesson planimplementations was measured using a revised (Figure 3). This indicates that over the course ofversion of the Science Lesson Plan Analysis the PD, teachers increased their ability to write andInstrument (SLPAI) (Jacobs, 2008). The instrument implement successful inquiry-based lesson plans.was revised by the project team to align withproject goals and for single lesson plans covering Teachers’ beliefs about inquiry teachingscience and math content and re-dubbed the SxI- were measured as a pre-/post- comparator using aSLPAI-R (Granger, 2011). Teachers’ lesson plans, validated self-report survey, the Teaching Sciencepost-implementation reflections, and teacher- as Inquiry (TSI) (Smolleck, 2006, Smolleck &designed classroom assessment data were used as Yoder, 2009). Teachers demonstrated significantevidence for the SxI-SLPAI-R to assess the quality gains across all five subscales suggesting anof the lesson plans and the degree to which increasing awareness to and appreciation forinstructional practices aligned with inquiry as inquiry based strategies in their teaching (Figure 4).aligned with the National Science Education Gains on subscales, indicative of the teacher’sStandards (NRC, 1996). changing beliefs, were similar to those observed for previous professional development projects Teachers’ lessons were considered to align (Granger, 2011). This increase in TSI scoreswith an inquiry strategy if the scores on the SxI- suggests teachers buy-in to the process of teachingSLPAI-R were above 50 (Granger, 2011). By this using inquiry strategies.measure, teachers across the board implementedinquiry-based lessons from the first lesson and SxI- Teachers, in their last reflection papers,SLPAI-R scores increased as a function of time, confirmed successes of inquiry teaching andalthough there was increased variance which revealed a commitment to continuing to use inquiryexplains the modest gains in overall score (Figure as a teaching strategy. The following quote is2). One might expect that the increase could be due similar to sentiments expressed by almost all of theto the level of inquiry as teachers moved from 13 teachers involved in the project:structured to guided to open. However, this “This whole process has allowed me to grow as a teacher it has pushed me to do things that I did notFigure 2. Distribution of SxI-SLPAI-R scores for each lesson cycle. Figure 3. Average SxI-SLPAI-R scores by type of inquiryA score of 50 or above is consistent with an inquiry strategy. (structured, guided, open) and overall. The first, required inquiry level is compared with the last inquiries of the same type. Theincrease was not due to the level of inquiry as gray are first inquiry implementations of that type and the blackVirginia Mathematics Teacher vol. 43, no. 1 bars are last inquiry implementations of that type. 61
feelasFigure 4. TSI subscale results pre- to post- project. All subscales had significant gains pre- to post- using repeated measures t-tests.comfortable with. I have given up some of the mentors” Tenets of quality professionalcontrol in the lesson and it has actually worked development and how they can reinventvery well for me and the students. I plan on to early science learning experiences. Sciencecontinue using all of these types of inquiry in my and Children, 49(2), 52-56.classroom for now on.” Granger, J. N., Alouf, J. L., Loboschefski, T., Vinion-Dubiel, A., Yochum, H., Although the number of teachers in the Herrington, D., & Yezierski, E. (2011).study is small, only thirteen (13) completed the full Assessing Inquiry in STEM Lessons: use ofyear-long, intensive project, the gains were obvious lesson plan analysis to determine thein both measures and through end-of-project impacts of a teacher professionalinterviews and conference presentations. These development program. The Gordoninterviews and presentations can be viewed on the Research Conference on ChemistrySTEM4Teachers.org website under Inquiry Education Research and Practice.Teaching in STEM Videos (http:// Davidson, NC.www.stem4teachers.org/inquiry-teaching/videos/). Jacobs, C.L., Martin, S.N., & Otieno, T.C. (2008).Of particular interest to the readers would be the A Science Lesson Plan Analysis Instrumentvideo “Meeting STEM Content Goals Through for Formative and Summative ProgramInquiry” where two math teachers present math Evaluation of a Teacher Educationcontent topics that work well with an inquiry Program. Science Education, 92, 1096-teaching strategy, as well as the video “Inquiry as a 1126.Habit of Mind” where partner math and scienceteachers share how they work together to Lieberman, A. & Miller, L. (2011). Learningencourage inquiry thinking in their shared students. Communities: The Starting Point for Professional Learning is in Schools and References: Classrooms. Journal of Staff Development. 32(4), 16-20.Arani, M.R.S, Keisuke, F., & Lassegard, J.P. (2010). \"Lesson Study\" as Professional Llewellyn, D. (2007) Inquire within (2nd ed.). Culture in Japanese Schools: An Historical Thousand Oaks, California: Corwin Press. Perspective on Elementary Classroom Practices, Japan Review, 22, 171-200. Moseley, C. & Ramsey, S. (2008). Elementary Teachers' Progressive Understanding ofBell, R.L., Smetana, L., & Binns, I. (2005). Inquiry through the Process of Reflection. Simplifying Inquiry Instruction. The School Science and Mathematics, 108(2), Science Teacher, 72(7), 30-33. 49-57.Carlson, M.O.B., Humphrey, G.H., & Reinhardt, NRC, National Research Council. (1996). National K.S. (2003). W eaving Science Inquiry and Science Education Standards. Washington, Continuous Assessment: Using Formative DC: National Academy Press. Assessment to Improve Learning. Thousand Oaks, CA: Corwin Press.Counsell, S.L. (2011). Becoming Science “Experi-Virginia Mathematics Teacher vol. 43, no. 1 62
Rock, T.C. & Wilson, C. (2005). Improving Science Teacher Education, 17(2), 137- Teaching through Lesson Study. Teacher 163. Education Quarterly, 32, 77-95.Smolleck, L. D., & Yoder, E. P. (2009). Further In Table 2, a select set of sixteen teacher’s lesson development and validation of the teaching plans are also found on the STEM4Teachers.org science as inquiry (TSI) instrument. School website under Inquiry Teaching in STEM Lessons . Science and Mathematics Journal, 108(7), These lessons incorporate objectives that are 291-297. science-only, math-only, or combined science and math. These lessons all align to Virginia StandardsSmolleck, L.D., Zembal-Saul, C., & Yoder, E.P., of Learning (SOL) and include robust classroom (2006) The Development and Validation of assessment plans. an Instrument to Measure Preservice Teachers' Self-Efficacy in Regard to the Teaching of Science as Inquiry. Journal ofTable 2. Lesson plans on STEM4Teachers.org designed by middle school math and science teachers through the Inquiry Teaching inSTEM project. Title Subject Level of In- Description or Question for Investigation quiryAdding Integers Mathematics Explore adding integers of different signs to de Structured velop rules for adding integers.Slope of Parallel and Per Mathematics Exploring relationships between slopes of perpendicular Lines Structured pendicular lines, parallel lines, and intersecting lines that are not perpendicular.Volume and Surface Ara Mathematics Structured What are the relationships among the dimenof Rectangular Prisms sions, volume, and surface area of a rectangular Guided prism?Rock, Paper, Scissors Mathematics Explore relationship between trial outcomes and Guided fairness of a game with an emphasis on probaTriangle Inequality Mathematics bility. Guided What is the relationship between the longestLinear Functions Mathematics Guided side and the sum of the remaining sides of a triTemperature and Bounce Guided angle?Waves and Density Science and What is the relationship between a rope’s length Mathematics Guided and the number of knots in the rope? Science and How does temperature affect the height at which Mathematics Guided a tennis ball bounces? How does the density of a medium affect waveWhat’s Eating the Moon? Science Open speed? Open Investigate positioning of the sun, Moon, andHeat and Water Science and Open Earth to create a model for the phases of the Technology Open Moon. Open Investigating kinetic energy and states of matterTransformations Mathematics Open using an online simulation. Students apply transformations to polygonElectromagnificent Science shapes on a coordinate plane and define types of transformationsSolubility Science Exploring strength of electromagnets Investigating solubility by changing features inAcids and Bases Science the solvent, the solute, and the environment. Exploring acids and bases and testing of pHWeathering Science Investigating mechanical and chemical weather ing processes.Exploring Projectile Mo Science and Investigating projectile motion with catapulttion Engineering construction and manipulationVirginia Mathematics Teacher vol. 43, no. 1 63
Hank Yochum (far left) Jill Granger (middle right)Director of Engineering Project Principal InvestigatorSweet Briar College Dean of the Honors [email protected] Western Carolina UniversityArlene Dubiel (middle left) [email protected] Manager James Alouf (far right)Formerly of Sweet Briar Col Professor Emeritus of Educalege [email protected] Sweet Briar College [email protected] Tim Loboschefski (not shown) Professor of Psychology Sweet Briar College [email protected] Busting Block Busters! Many Hollywood movies have scenes that Speed (1994) seem mathematically inaccurate if not In the movie a bus is shown to jump a gap inimpossible. Are these scenes truly impossible, a bridge, that is missing due to construction. or are they more plausible than they seem? In the last shot of the odometer before the The goal of the contest is to provide the best jump, the bus is shown at 67 miles per hour, mathematical explanations for the following and still accelerating, so let’s assume that at scene. The solutions that best explain a the time of the jump, it was traveling at 70 scene’s possibility or impossibility and the different elements that help form this will mi/h. The portion of the bridge that was receive an award and the winner will be missing was approximately 50 feet. Is thisfeatured in the Spring issue. Answer s may jump possible? What math proves it, or what be submitted by November 31st to additional factors would be needed to make [email protected] with the subject line: Busting Blockbusters entry. the jump possible? https://www.youtube.com/watch?v=dKJa- KQNjQUVirginia Mathematics Teacher vol. 43, no. 1 64
“Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding.” -William Paul ThurstonVirginia Mathematics Teacher vol. 43, no. 1 65
Virginia Mathematics Teacher vol. 43, no. 1 66
The PuzzlemakerDr. David Shoenthal Rules:Associate Professor, Chair 1. The digits 1 through 9 appear in each row and each column exactly once.Longwood University 2. Digits can appear in a shaded region multiple times as long as Rule 1 isn’[email protected] broken. 3. Digits in each region must either add or multiply to the indicated number. Both the operation and the sum/product are indicated in each region. 4. Squares with multiple colors (for example, the square in row 2, column 1) contain a number that’s used in the operation for adjacent regions of each of those colors. 5. The puzzles are rotationally symmetric--the board can be rotated by 90 degrees and the design remains the same. Answers will be featured in the Spring issue of the Virginia Mathematics Teacher.Virginia Mathematics Teacher vol. 43, no. 1 67
Conferences of Interest Virginia Council of Teachers of Mathematics | www.vctm.org 2016 Regional NCTM Conference: 68 Philadelphia, PA: October 31-November 2, 2016 2016 Annual PME-NA Conference Tucson, AZ November 3-6, 2016 2016 ATCM Conference Bangkok Thailand, December 14-18, 2016 2017 AMTE Conference Orlando FL, February 9-11, 2017 2017 SITE Conference Austin TX, March 5-9, 2017 2017 VCTM Conference James Madison University, March 10-11, 2017 2017 National NCTM Conference San Antonio, TX April 5-8, 2017 2017 PME Annual Conference Singapore, July 17-22, 2017 2017 MAA MathFest Chicago, IL July 26-29, 2017Virginia Mathematics Teacher vol. 43, no. 1
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