Virginia Council of Teachers of Mathematics |www. vctm.orgVIRGINIA MATHEMATICSVol. 43, No. 2 TEACHERSpring 2017 Communicating in the Universal Language of Math! 1Virginia Mathematics Teacher vol. 43, no. 2

Editorial StaffDr. Agida Manizade Dr. Jean Mistele MBr.rBiarniaPnrPatrtatt Mr. Liam Downey Editor-in-Chief Associate Editor Assistant Editor Assistant Editor Radford [email protected] [email protected] Radford University Radford University [email protected] Printed by Wordsprint Blacksburg, 2200 Kraft Drive, Suite 2050 Blacksburg, Virginia 24060Virginia Council of Teachers in Mathematics Many thanks to our Reviewers for Spring 2017 President: Jamey Lovin, Virginia Beach Public Schools Dr. Robert BerryPast President: Cathy Shelton, Fairfax County Public Schools Dr. Carrie Case, Radford UniversitySecretary: Lisa Hall, Henrico County Public Schools Dr. Anthony Dove, Radford UniversityMembership Chair: Ruth Harbin-Miles Dr. Kateri Thunder, James Madison UniversityTreasurer: Virginia Lewis, Longwood University Dr. Laura Jacobsen, Radford UniversityWebmaster: Ian Shenk, Hanover County Public Schools Kelly RobinsonNCTM Representative: Betsy Steadman, Hanover County Public Schools Dr. Wendy Hageman-Smith, Longwood University Elementary Representatives: Meghann Cope, Bedford County Public Schools; Eric Dr. Ann Howard Wallace, James Madison University Vicki Bohidar, Hanover County Public Schools Mrs. Anita Lockett, Fairfax County Public SchoolsMiddle School Representatives: Melanie Pruett, Chesterfield County Public Schools; Joyce Xu, Virginia Tech Skip Tyler, Henrico County Public SchoolsSecondary Representatives: Pat Gabriel; Samantha Martin, Powhatan Public SchoolsMath Specialist Representative: Spencer Jamieson, Fairfax County Public Schools Dr. Matthew Reames, University of Virginia2 Year College: Joe Joyner, Tidewater Community College Dr. Kenny Wantz, Regent University4 Year College: Ann Wallace, James Madison University; Karen Zwanch, Virginia Tech Dr. Jay Wilkins, Virginia Tech Robert Berry, University of Virginia VMT Editor: Dr. Agida Manizade This issue had a 20% acceptance rate 2Virginia Mathematics Teacher vol. 43, no. 2

Table of Contents:Note from the Editor..................................................5 HEXA Challenge.....................................................38Message from the President .....................................6 Information for Virginia’s K-5 Teachers................40Note from the VDOE.................................................7 Grant Opportunities................................................40The Role of Skip Counting and PISA Updates...........................................................41Figurative Reasoning................................................8 Polling Data and the 2016 Presidential Election ...42 MathGirls...................................................................46Is the Midpoint Quadrilateral Really a Busting Block Busters!............................................48Parallelogram?........................................................15 Exploring the Solving of Algebraic Equations Through Mental Algebra ........................................49Algebra Achievement of Recursion in Secondary Mathematics Classrooms.53Urban High School Students...................................18 Summer 2016 STEM Camps....................................56 Math Jokes...............................................................56Solutions to Fall HEXA Challenge Problems ........23Upcoming Math Competitions.................................27Concrete, Representational, and Abstract: BuildingFluency from Conceptual Understanding...............28Technology Review..................................................32 The Puzzlemaker .....................................................59Good Reads.............................................................34 Conferences of Interest ...........................................60Key to the Fall 2016 Puzzlemaker Problem............35Call for Manuscripts................................................36Educational Opportunities for VirginiaMathematics Teachers.............................................37Virginia Mathematics Teacher vol. 43, no. 2 3

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Communicating in the Universal Language of Mathematics: Note from the Editor Dr. Agida Manizade As mathematics teachers, one of our main post industrial world that is data intense, and theygoals is to engage students in doing mathematics. will need to be able to comprehend, analyze, andBut how often do we stop to think about, “what is communicate the patterns they encounter.mathematics?” and “what does it mean to domathematics?”. In the regular k-12 curriculum, do The theme of this issue is Communicatingwe engage in doing mathematics? Does adding and in the Universal Language of Math, and we askedsubtracting numbers or memorizing a set of authors to share their ideas about doingformulas qualify as doing mathematics? Does a mathematics, and communicating in this universalstudent do mathematics when using ‘technology as language with their students. We enjoyed puttinga master’ where he or she takes for granted any this issue together, and hope you find inspirationaloutput generated by technology, without engaging and useful ideas for your K-12 classroom. Wein mathematical thinking or without evaluating the invite you to share your thoughts and practices byquality of the the outcome? If we were to ask our submitting practitioner-oriented articles that help tostudents what mathematics is, and what it meant to inform other mathematics teachers. We alsodo mathematics, what would they say? encourage you to participated in the HEXA Challenge and Busting Blockbusters, and to One way to think about mathematics is as a challenge your students to do the same.language to describe and communicate patterns weencounter in our lives. Some of the patterns we see Agida Manizadein nature and human behavior are simple, while Editor in Chief,others seem chaotic. From simple Putnam squares Virginia Mathematics Teacherand golden ratios, to predicting climate patterns or [email protected] of disease, the more mathematics we learn,the more we are able to observe, describe, andpredict the patterns around us. My goal as a mathematics teacher is tocreate classroom environments in which studentsthink critically about these patterns, ask and answercomplex questions, and communicate their ideaswith their peers. Our students will be operating in a Congratulations to the 2017 Winners of theWilliam C. Lowry Mathematics Educator Award: Middle School Awardee: College Awardee: Matthew Reames, Burgundy Farm Andrew Wynn, Virginia State University High School Awardee: Math Specialists Awardee:Jillian Marballie, Mongomery County Tasha Fitzgerald, Culpeper SchoolsVirginia Mathematics Teacher vol. 43, no. 2 5

Message from the President Jamey LovinNow is the time for make connections between grade levels and standards more apparent. The second goal was towelcoming spring – a time provide more support for teachers by including in the Understanding the Standard section of theof transitioning from the Curriculum Framework, more explanations, examples, and definitions of vocabulary words.cold winter to the warm While there, look for professionalsummer. Virginia development materials based on the 2016 SOL Mathematics Institutes! These institutes, framed bymathematics educators are the five process goals, focused on instruction to support the new standards. I know you will findalso in a state of transition them helpful as you seek to implement the standards in your classrooms.as we join district leaders to VCTM is excited to collaborate with you inincorporate the newly welcoming a high quality, challenging mathematics program for our students! As always, please feelrevised Standards of free to contact myself, or any board member, with ideas on how we can better serve you.Learning into local curriculum documents and Jamey Lovin, VCTM Presidentdevelop instructional materials. [email protected] you have not had a chance to visit theDepartment of Education website, try to do sosoon! There you will find timely information fromTina Mazzacane, Mathematics and ScienceSpecialist from the Office of Science, Technology,Engineering and Mathematics, about the focus ofthe new changes. In her webcast, Tina articulatesthe two-pronged rationale behind the 2016revisions. First, there was a concerted effort toimprove the vertical progression of content and Organization Membership Information 6 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:$47/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 Student Membership (For Full-Time College Students) $20 Individual, One-Year Membership $20 Institutional One-Year Membership $39 Individual Two-Year Membership $57 Individual Three-Year Membership Current Virginia Council of Teachers of Mathematics Membership: 750 MembersVirginia Mathematics Teacher vol. 43, no. 2

Note from the Virginia Department of Education:Implementing the 2016 Mathematics Standards of Learning to Support the Process Goals Tina Mazzacane The 2016 Virginia Mathematics Standards understand the changes. Virginia mathematicsof Learning and Curriculum Frameworks were educators are encouraged to take time to unpackapproved by the Board of Education in September, these new standards and determine the curricular2016. The content of the mathematics standards is and instructional modifications needed to ensure aintended to support the following five process goals smooth transition from the current 2009 standardsfor students: becoming mathematical problem to the 2016 standards.solvers, communicating mathematically, reasoningmathematically, making mathematical connections, The Mathematics Team at the Virginiaand using mathematical representations to model Department of Education is focused on providingand interpret practical situations. The theme of the support to educators as they make this importantcurrent edition of the Virginia Mathematics transition. Additional resources to supportTeacher Journal is “Communicating in the implementation of the 2016 Mathematics StandardsUniversal Language of Mathematics.” The of Learning can be found on the Virginiaexpectation for mathematical communication Department of Education Mathematics 2016included in the 2016 Virginia Mathematics webpage.Standards of Learning and Curriculum Frameworkis shown: 2016 Mathematics Standards of Learning Im- plementation Timeline Mathematical Communication Students will communicate thinking and reasoning using the 2016-2017 School Year – Curriculum Development language of mathematics, including specialized vocabulary VDOE staff provides a summary of the revisions to assist and symbolic notation, to express mathematical ideas with school divisions in incorporating the new standards into lo- precision. Representing, discussing, justifying, conjecturing, cal written curricula for inclusion in the taught curricula reading, writing, presenting, and listening to mathematics during the 2017-2018 school year. will help students to clarify their thinking and deepen their 2017-2018 School Year – Crossover Year understanding of the mathematics being studied. Mathemati- 2009 Mathematics Standards of Learning and 2016 Mathe- cal communication becomes visible where learning involves matics Standards of Learning are included in the written and participation in mathematical discussions. taught curricula. Spring 2018 Standards of Learning assess- ments measure the 2009 Mathematics Standards of Learning VDOE Mathematics Standards of Learning, 2016 and include field test items measuring the 2016 Mathematics Standards of Learning. Communication in the mathematics 2018-2019 School Year – Full-Implementation Yearclassroom is essential for students to develop a Written and taught curricula reflect the 2016 Mathematicsdeep understanding of mathematical content and to Standards of Learning. Standards of Learning assessmentsbe able to justify and reason mathematically. In the measure the 2016 Mathematics Standards of Learning.2016 curriculum framework documents, revisionshave been made to improve precision and Tina Mazzacaneconsistency in mathematical language and format. Mathematics Coordinator Virginia Department of Education The implementation of the newly adopted2016 Mathematics Standards of Learning willrequire teachers, specialists, and administrators inpublic school divisions to analyze, discuss, andVirginia Mathematics Teacher vol. 43, no. 2 7

The Role of Skip Counting in Children's Reasoning Jesse L. M. Wilkins and Catherine UlrichSkip Counting and Figurative Material in graders. These solutions will be referred toChildren’s Construction of Composite Units and throughout the article to highlight children’s waysMultiplicative Reasoning of thinking and to make connections to An important part of children’s You have baked 39 cupcakes and you will putmathematical development in the elementary and the cupcakes in boxes of three. How manymiddle school years is the transition from additive boxes will you fill?to multiplicative thinking. This developmentalmilestone affords children necessary tools for (a) (b)understanding more advanced mathematicalconcepts that is limited by additive thinking, such (c) (d)as fractions and proportional reasoning. Animportant part of this mathematical development is (e) (f)the construction and coordination of units, which Figure 1. Examples of sixth graders’ solutions to the Cupcakedoes not occur all at once, but through several Taskhierarchical stages (Ulrich, 2015, 2016).Understanding these stages and the nature of 8children’s ways of thinking about units during eachstage is important for teachers as they plan andprepare instructional activities for their students. Inthis article we discuss the characteristic ways ofthinking associated with these stages andinstructional opportunities for moving childrenthrough these stages. In particular, we discuss therole of skip counting (e.g., counting by 3’s: 3, 6, 9,12, 15, 18, 21…) as both an indicator of, and as away to foster, children’s construction andcoordination of units throughout these stages ofdevelopment. We also discuss the placement ofskip counting within Virginia’s mathematicsStandards of Learning and Curriculum Framework(Virginia Department of Education [VDOE], 2009)and potential extensions and understandings thatcould be helpful for curriculum development forelementary-aged children. Moreover, we discussthe distinction between a child understanding, e.g.,5 × 3 as the result of their skip counting by 3s,from a child who has truly developed amultiplicative understanding of 5 × 3 as 5 times asmuch as 3. To motivate and facilitate our discussion ofthe ideas outlined above we first discuss severalsolutions to the task in Figure 1 produced by sixthVirginia Mathematics Teacher vol. 43, no. 2

instructional strategies for advancing children’s a set, but instead are one part of their countingearly number concepts. All of these children activity. At this point, children may be able toproduced the same correct answer, however, based count a set of objects, but they would interpret theon the solutions, these students all thought very question, “How many?” as a request to say adifferently about the problem. In Figure 1a, the sequence of numbers (e.g., 1, 2, 3, …, 7) whilechild seems to have first drawn 39 “cupcakes” pointing to each object, not as a question aboutwithout attending to groups of 3, then circled how “big” the set is. Furthermore, after counting agroups of 3 to make boxes until they were all used set of objects, if given additional objects and askedup, and then counted the boxes. In Figure 1b, the how many in all, this child would need to count allchild seems to create groups of 3 “cupcakes” in of the objects, first recounting the original set.boxes until all 39 cupcakes are used up, and thencounts the number of groups (notice the single dot A child who has constructed an INS (thein the boxes likely representing this counting act). first number sequence) recognizes that a number,The child’s work in Figure 1d does not represent such as 7, describes the cardinality of a set ofindividual cupcakes, but instead uses skip counting objects and can stand in for counting them, that is,by 3’s to 39, and then counts the number of “counts counting “1, 2, 3, …, 7” (Olive, 2001), an initialby 3” (notice the dots under each number). Finally, number sequence. In this case, the 7 is a numericalthe child in Figure 1f seemed to recognize the composite of units representing the result ofsituation by reversing the context of the problem to counting the seven objects, and can serve as aone asking: “What multiplied by 3 would give me starting point for additional counting. However, the39?” In each of these cases the child’s work 7 is not recognized as a unit that could be used torepresents a different way of thinking about and count with: “with an INS… [the number words]coordinating units. We will revisit these student can only be used to symbolize the results ofsolutions after presenting a hierarchy of how counting acts; they cannot yet be used as input forstudents work with units. counting acts” (Olive, 2001, p. 6). The development of an INS affords a child with the These different ways of thinking may be ability to count-on, that is, if after counting a set ofcharacterized in terms of a hierarchy of four stages 7 objects, a child is given three more objects andcalled number sequences (Steffe & Olive, 2010; asked, “How many altogether?” they would likelySteff & Cobb, 1988; Olive, 2001; Ulrich, 2015, count: “7; 8, 9, 10,” while touching the three2016). These different stages describe how children additional objects, or using their fingers to keepwork with units and coordinate them when working track of the additional three objects; and answerwith the counting numbers. These four stages are “10,” instead of having to count-all. A child withreferred to as the initial number sequence (INS), an INS will often rely on figurative materials totacitly nested number sequence (TNS), explicitly keep track of counting. Consider the child’snested number sequence (ENS) and the generalized solution in Figure 1a. In order to solve this task, thenumber sequence (GNS). In this paper we focus on child needs to represent all 39 “cupcakes,” makingthe first three sequences as they relate to skip groups of three until they are all used up, and thencounting and describe the extension of thinking count the number of groups.required for the GNS which ultimately lays thegroundwork for more advanced mathematical For a child with only an INS their countingunderstanding. Here we briefly highlight the acts are limited to using strings of the numberimportant characteristics of these stages; for a more sequence beginning at 1 (Ulrich, 2015). Later on, adetailed discussion, the interested reader should child begins to recognize that there areconsult Ulrich (2015, 2016) and Olive (2001). subsequences embedded within larger sequencesThe Number Sequences that can be used to aid them in their counting acts. For example, given 7 objects, a child is asked how Prior to a child developing an INS, they are many there would be if they added on 12 objects.considered pre-numerical (Steffe & Olive, 2010; In order to keep track this child might begin at 7Olive, 2001). That is, for these students, numbers and count as follows: “8 is one more, 9 is twothemselves do not represent cardinality, a quality of 9Virginia Mathematics Teacher vol. 43, no. 2

more, 10 is three more, 11 is four more, …, 19 is Steffe and Olive, 2010; Olive, 2001). Here a childtwelve more, the answer is twelve.” This child has is able to recognize the activity of “adding onea tacit awareness of the subsequence 1 to 12 nested more” to the point that these additions arewithin the larger sequence from 1 to 19. This child interchangeable units. That is, number words nohas constructed a tacitly nested number sequence longer only represent the result of counting, but(Olive, 2001, Ulrich, 2015; Steffe & Olive, 2010). represent a multiplicative relationship associatedDouble counting, as in the example, is a with the number of iterable units: for example, “7”characteristic action of a child with a TNS. This no longer only represents “1, 2, 3, 4, 5, 6, 7,” butawareness of the cardinality of the subsequence (1 instead represents 7 ones, or 7 times as much asto 12) irrespective of its location in the larger one unit. At this point the composite unit does notsequence suggests that the child is now able to stand for a subsequence, but stands in for awork with a composite unit (Ulrich, 2015). multiplicative relationship. Olive (2001, p. 7)Different from a numerical composite, in which the distinguishes a child with an ENS from a child withcount stands in for the counting sequence 1 to the a TNS by comparing their activity for solvingnumber, a composite unit is taken to represent the 1+1+1+1+1. For a child with only a TNS theycardinality of the counting acts irrespective of would potentially need to solve this problem inwhere the subsequence occurs in the larger steps by calculating the nested sums: 1+1 is 2, 2+1sequence. Also characteristic of a child with a TNS is 3, …, 4+1 is 5. Whereas, for a child with an ENSis the use of skip counting to solve tasks. For the sums are taken as given, and they recognizeexample, counting by 3’s, each count represents a 1+1+1+1+1 as simply 5 ones, and also recognizecomposite unit of 3. With this ability, children can the reversibility of the relationship, that five onesbegin to answer questions such as, how many are the same as one five.threes are in thirty-nine, by keeping track of theirskip count: “3; 6; 9; 12, …”. Each of the numbers Children with an ENS can reflect onin the skip count represent a subsequence of multiplicative situations (Ulrich, 2016; Olive,cardinality 3, or a composite unit of 3. The child’s 2001) involving multiple levels of units. Foruse of skip counting in Figure 1d is characteristic example, combining 4 groups of 7 objects, can beof this sort of thinking. The child uses skip viewed as making a composite unit of compositecounting by 3’s to 39, recognizing that each 3 units. In other words, 4 groups of 7 objects is seenrepresents a box of cupcakes (composite unit) and as a numerical composite of 4 composite units,then counts the number of “counts by 3” and each of which is a composite unit of 7 iterable unitsindicates “13 boxes” as their answer (notice the of one. Furthermore, a child can view thisdots under each numeral likely representing the combination as a composite unit of 28 iterablechild’s count). Compare this to the solution in units. However, a child with only an ENS has toFigure 1e. Here, too, it is apparent that the child is create composite units of composite units—28 as 4working with composite units, but they no longer groups of 7 objects—in the moment. The childhave to refer to their skip count but recognize that would have trouble operating on 28 without losingthey are interested in the number of units and can track of the 4 groups of 7. Although we will notuse the sequence 1 to 13 to keep track of the units. elaborate greatly, for completeness, a child who has constructed a generalized nested number With a TNS, the composite units remain sequence (GNS) can work fluently with atacit for a child. That is, these units are available to composite of composite units because theirwork with during counting activity, but children are composite units are now iterable in the same waynot explicitly aware of the units prior to counting; units of 1 were iterable for students who have anthey are reproduced through counting. Once ENS.children are able to reflect on units as a given, we Skip Countingsay that a child has developed an explicitly nestednumber sequence (ENS). The defining Skip counting (e.g., counting by 3’s: 3, 6, 9,characteristic of this stage is a child’s ability to 12, 15, …) is often introduced to young children asconstruct an iterable unit of one (Ulrich, 2016; a way to further develop their counting skills and build their knowledge of multiples. Much like theyVirginia Mathematics Teacher vol. 43, no. 2 10

begin their early counting as a sing-song, children encouraged for its own sake. Once children canalso learn to skip count through repetition and relate their skip counting to their multiplicationsong. Based on the Virginia Standards of Learning facts it should not then be assumed that children(SoLs; VDOE, 2009) children in Kindergarten are have developed multiplicative thinking. Skipencouraged to count by fives and tens to 100 (see counting alone does not imply the development ofSoL K.4 in Table 1). Continuing in first grade, the multiplicative understanding inherent in anchildren are also encouraged to count by 2’s to 100 ENS or GNS. By encouraging children to use their(see SoL 1.2 in Table 1). In second grade, children skip counting for solving tasks involvingcontinue to skip count by 2’s, 5’s, and 10’s (see multiplicative situations instead of relying onSoL 2.4 in Table 1). The Curriculum Framework number facts, they may develop a more intentional(VDOE, 2009) highlights the role of skip counting and explicit awareness of the relationship betweenfor the general development of numerical patterns, their skip count and the types of units with whichas well as for use in very specific mathematical they are working. This awareness affords childrenapplications. For example, skip counting by 2’s with increased opportunities to develop morelays the groundwork for understanding even and powerful multiplicative understandings.odd numbers (e.g., SoL 1.2); counting by 10’s lays Skip Counting and Number Sequencesthe groundwork for place-value and money (e.g., Here we discuss the solutions in Figure 1 asSoLs 1.2 and 2.4); skip counting by fives lays the a way to exemplify different number sequences andgroundwork for telling time and counting money the possible role of skip counting. In Figure 1a, the(SoL K.4). In all cases, skip counting is promoted child represents all 39 “cupcakes” before makingfor its relationship to learning multiplication facts. groups of three. This is a possible strategy that aBeyond 2’s, 5’s, and 10’s, skip counting by other child with an INS could use to solve this task.numbers is not explicitly highlighted in the Interestingly, children with only an INS canStandards of Learning or Curriculum Framework. successfully use skip counting to determine theAfter second grade, skip counting is abandoned as cardinality of a set, but would be unable to keepan essential part of the SOLs and replaced with a track of their skip counting (Olive, 2001). Forfocus on the learning of multiplication facts (SoL example, a child with an INS could have used skip3.5, Table 1). counting to recount the cupcakes to make sure that Beyond the specific role that counting by they had created 39, but if asked “how many2’s, 5’s, and 10’s has for developing particular threes?” the question would not make sense sincemathematical concepts as prescribed in the VDOE each “three” stands in for a numerical composite inCurriculum Framework, skip counting, in and of which the 3’s represent three objects instead of oneitself, can represent important developmental shifts countable thing. As seen in the solution in Figurein working with composite units and should be 1a, they would need to create groups of 3 and thenTable 1. Standards of Learning associated with skip counting.K.4 The student will a) count forward to 100 and backward from 10; b) identify one more than a number and one less than a number; and c) count by fives and tens to 100.1.2 The student will count forward by ones, twos, fives, and tens to 100 and backward by ones from 30.2.4 The student will a) count forward by twos, fives, and tens to 100, starting at various multiples of 2, 5, or 10; b) count backward by tens from 100; and c) recognize even and odd numbers.3.5 The student will recall multiplication facts through the twelves table, and the corresponding division facts.Virginia Mathematics Teacher vol. 43, no. 2 11

count these groups (or draw boxes while a focus on multiplication facts alone. Continuedcontinually recounting the number of cupcakes use of skip counting for INS and TNS studentsthey have used up). focuses them on their use of composite structures and allows them to reflect on these composites. The solution in Figure 1b shows a child’s Focusing students on multiplication facts aloneuse of numerical composites to find a solution. hides the iterations of composites involved inNotice that each unit contains 3 visible objects that multiplication and may limit the ability of studentshave been counted one by one, representative of a to explicitly reflect on their number sequences,strategy by a child with an INS. The need to necessary for an ENS and GNS, furtherrecount all the cupcakes after each “box” of constraining their development of multiplicativecupcakes is drawn suggests a lack of a keeping structures (Ulrich, 2016).track strategy, and the lack of a composite unit. The solution presented in Figure 1f In Figure 1c, the child has explicitly linked suggests that the child has constructed an ENS ortheir figurative composite of 3 with their skip GNS as they are able to repose the question ascounting. This awareness of the link between skip “Three of what makes thirty-nine?” Although they,counting and composite units is characteristic of a too, may have counted by 3’s to reach to 39, theychild with a TNS. In Figure 1d, we see a child’s are able to reconceptualize 39 as 13 groups of 3,solution using only skip counting and no which represents a relationship between tworepresentation of individual cupcakes. Here it is numbers, or a multiplicative structure.clear that the elements of the skip count are used torepresent one composite unit of 3, in this case, a A solution that many students gave for thebox of cupcakes. After reaching “39” notice that Cupcake Task was to simply write “13.” Such anthe child seems to have then counted the number of answer, without the apparent use of figurative“boxes” indicated by a single dot associated with materials, is only possible for children who haveeach numeral in the count. By writing down the constructed an ENS; they are able to keep track ofskip count, children with a TNS are able to keep the numerical units of composite units that aretrack of how many times they have applied their necessary to solve this task. Finally, it is importantcomposite unit. In Figure 1e, we see a child who to point out that many students were able to solveeven more clearly interprets each composite of 3 as this task by recognizing it as a division problem,a single countable unit, indicated by the number and carrying out the necessary division to reach ansequence 1 to 13 (instead of the skip count) to find answer of 13. This particular solution strategy isthe number of boxes. As sophisticated as it may not necessarily indicative of a higher stage ofseem for a child to use skip counting to solve a task thinking (e.g., ENS, GNS), as many TNS childrenthat seems multiplicative, we can not necessarily are able to strategically perform the necessaryinfer that the child is thinking multiplicatively. The procedures associated with a given type of taskchildren in 1d and 1e may still be using additive, without using multiplicative thinking.not multiplicative, structures, as they are not Implications for Teachingrepresenting a comparison between two numbers(Ulrich, 2016). Instead, they are describing their Recent research suggests that a significantcounting activity, e.g., “I had to count by 3 thirteen number of children make it to middle schooltimes to get to 39” (Ulrich, 2016). This is not without having constructed an ENS (Ulrich &representative of multiplicative thinking. Wilkins, 2016a, 2016b). This suggests that manyFurthermore, children who can link their children in the sixth grade are still predominatelymultiplication facts to their skip counting have not additive thinkers and thus not poised for handlingnecessarily developed multiplicative thinking, but multiplicative thinking and relative thinking that isinstead may be reinterpreting their multiplication necessary for fractional and proportional thinking.facts in terms of additive skip counting. It is It is thus important for teachers to provideimportant not to de-emphasize children’s use of continued opportunities for children to developskip counting as implicitly suggested by the multiplicative thinking as early as possible. WeCurriculum Framework (VDOE, 2009) in favor of have discussed the importance of skip counting to help children develop composite units, however,Virginia Mathematics Teacher vol. 43, no. 2 12

this level of development does not guarantee that a make groups of 3, because they are not able tochild has developed multiplicative thinking (it does recognize that these composite units stand for 3not imply that they have developed iterable units of objects. However, for children with a TNS these1). Many children like the one in Figure 1d are able rods could afford them the opportunity to developto use their skip counting to solve multiplication the notion of an iterable unit of 1, thus helpingtasks, but cannot work with units as countable them construct an ENS. Again, subtle changes inobjects and thus must depend on their additive the use of manipulatives can provide opportunitiesstrategies to solve such multiplicative tasks. for children to combine their skip counting andWithout further development, children will have notions of composite units to make importanttrouble developing more sophisticated ways of growth in the construction of their numberthinking that require higher levels of units sequences.coordination (Ulrich, 2016). This understanding is Conclusionsnecessary for multiplicative thinking, and thusmany children reach middle school still working Children’s transition from additive towith their additive understandings. multiplicative thinking represents an important shift in thinking that makes it possible to be In addition to encouraging the continued successful with more advanced mathematicaluse of skip counting, helping children develop learning. Children who reach middle schoolnotions of composite units can be aided through the without having made this transition are at a starkintentional use of well-chosen mathematical disadvantage because most of the middle schoolmanipulatives. For example, when students are mathematics curriculum presumes multiplicativeworking with tasks that ultimately involve thinking. In this article we highlighted themultiples (e.g., Figure 1), making available single characteristics of the different stages of numbercounters as well as counters clustered in differentgroup sizes may help promote the use of composite Figure 2. Different models representing different typesunits (see Figure 2). Consider a task similar to the of unit structures for a Cupcake Task with 15 cup-Cupcake Task in which there are 15 cupcakes to be cakes.put into boxes of 3 cupcakes. Figure 2 shows fourmodels of a solution to this task using different 13types of manipulatives. Children with an INS mayuse single objects to represent 15 cupcakes andthen make groups of three cupcakes (see Figure2a). For INS children, providing interlocking cubes(Figure 2b) could make it possible for them tobuild their own figurative composites, a first step indeveloping composite units. Making availableobjects pregrouped by threes (Figure 2c) with thesingle units still visible could stimulate children’sskip counting and use of a composite unit of 3. Inaddition, making available unpartitioned rods thatrepresent 3 (Figure 2d) may help promote thenotion of an iterable unit of one and amultiplicative unit relationship. Subtle changes inthe manipulatives provided to children for solvingtasks can afford or constrain their growth inunderstanding. For example, only having rods likethose in Figure 2d for a child with only an INS mayactually constrain their development. Thesechildren would likely use these “three” rods assingle units and count out 15 of these rods andVirginia Mathematics Teacher vol. 43, no. 2

sequences that children move through as they Ulrich, C. (2016a). Stages in constructing andconstruct their notions of composite and iterableunits on their way to developing multiplicative coordinating units additively andthinking. We highlighted the important role of skipcounting as both an indicator of, and a way to multiplicatively (Part 2). For the Learningfoster, children’s development of composite units.We recommend that teachers continue to of Mathematics, 36(1), 34–39.emphasize the use of skip counting as a way todevelop students’ multiplicative thinking beyond Ulrich, C. & Wilkins, J. L. M. (2016a). Usingjust the connection with multiplication facts.Prematurely deemphasizing skip counting may written work to assess stages in sixth-gradecause children to focus only on the idea that 5 × 3is the result of their additive counting by 3s, students’ construction and coordination ofwhereas continued use may afford them theopportunity to develop the more sophisticated arithmetic units. Manuscript submitted fornotion that these 3s represent composite units, andthat 5 × 3 represents 5 times as much as 3, a publication.composite of composite units. At the same time, itis important for teachers to recognize that the Ulrich, C. & Wilkins, J. L. M. (2016b). Usingproficient use of skip counting does not necessarilyimply that children have developed multiplicative student written work to investigate stages inthinking. By being aware of the characteristic waysof thinking associated with each of the different sixth-grade students’ ways of operatingnumber sequences teachers are better able toprovide appropriate scaffolding to move children with numbers. Paper presented at thethrough the stages. As an example, we feel that theintentional and selective use of manipulatives, both American Educational Researchwith elementary school students and middle schoolstudents who have yet to develop composite or Association, Washington, DC.iterable units, could provide invaluable support forthe students’ mathematical development. Virginia Department of Education (2009). References Mathematics standards of learning forOlive, J. (2001). Children’s number sequences: An Virginia public schools. Richmond, VA: explanation of Steffe’s constructs and an extrapolation to rational numbers of Commonwealth of Virginia Board of arithmetic. The Mathematics Educator, 11 (1), 4-9. Education. Available online at: http://Steffe, L. P., & Olive, J. (2010). Children’s fractional knowledge. New York, NY: www.doe.virginia.gov/testing/sol/ Springer. doi 10.1007/978-1-4419-0591-8Steffe, L. P., & Cobb, P. (1988). Construction of standards_docs/mathematics/index.shtml arithmetical meanings and strategies. New York, NY: Springer. Jesse L.M. WilkinsUlrich, C. (2015). Stages in constructing and Professor of Math coordinating units additively and Education multiplicatively (Part 1). For the Learning Virginia Tech of Mathematics, 35(3), 2–7. [email protected] Mathematics Teacher vol. 43, no. 2 Katy Ulrich Assistant Professor Virginia Tech [email protected] 14

Is the Midpoint Quadrilateral Really a Parallelogram? Zoltan Kovacs Dynamic Geometry software systems geometric facts are obvious and visible when using(DGS) can be useful and challenging tools in the DGS?teaching and learning of reasoning. DGS allow thestudent to formulate certain geometric facts (e.g. as Walter Hickey published a list on the 12intermediate steps towards establishing the proof of most controversial facts in mathematics in Businessa given statement) by drawing auxiliary diagrams, Insider in March 2013 (Hickey, 2013). Such a listand, then, getting convinced of the truth or falsity is always subjective, but surprisingly an easyof the conjectured assertion by checking its validity geometric problem is listed there also, namelyin many instances, after randomly dragging some Varignon's theorem (see Figure 1).elements of the figure. This has already raisedsome concerns: Hickey calls this fact “Midpoint parallelograms” and recalls that by drawing a 4 …increased availability in school sided polygon and connecting its midpoints you mathematics instruction of … dynamic will get a perfect parallelogram every time (see geometry systems… raised the concern that Figure 2). such programmes would make the boundaries between conjecturing and Figure 2. A collection of midpoint parallelograms proving even less clear for students… in GeoGebra Materials (Kovács, 2015). [They] allow students to check easily and quickly a very large number of cases, thus helping students “see” mathematical properties more easily and potentially “killing” any need for students to engage in actual proving. (Lin & al., 2012) Indeed, “dragging” is a characteristicfeature of DGS and, therefore, the above expressedworries apply to all DGS. A natural question arisesas to is really proving necessary in schools if allFigure 1. Varignon's theorem as shown in Wikipedia. Having DGS in use, this fact can be visuallyThe midpoints of the sides of an arbitrary quadrilateral checked quite easily by sketching an arbitrary planar quadrilateral, and constructing its midpoints form a parallelogram. and the polygon they describe. (Here we will use GeoGebra, the free DGS which is an interactiveVirginia Mathematics Teacher vol. 43, no. 2 maths application including geometry and algebra, intended for learning and teaching mathematics and science from primary school to university level. It is available at www.geogebra.org.) Parallelograms, by definition, are quadrilaterals with two pairs of parallel sides. Parallelism is actually easy to check roughly, but more careful investigation is needed when a precise answer is required: how do you check if two lines are parallel exactly? 15

From the numerical point of view it makes Fortunately, a few DGS currently include amore sense to use an equivalent definition of a feature closely related to mathematics reasoning:parallelogram, namely the congruence of opposite that of having implemented an Automatedsides. This property is much easier to measure than Theorem Proving (ATP) algorithm, yielding theparallelism of two objects (see Figure 3). ability to confirm or deny the mathematical (i.e. not numerical or probabilistic) truth of a geometric Figure 3. Measuring the length of two opposite statement. In GeoGebra, some ATP features can sides of a midpoint quadrilateral. also be introduced in the classrooms by using the Relation Tool. This tool in GeoGebra was Even if it is technically easier, the accuracy originally designed to collect numerical equalitiesof such attempts is questionable. Even if it is in a geometrical construction, but the recentperformed on a graphing paper having a regular versions can also be asked to re-investigate thegrid, the easy case with initial points (0,0), (2,0), problem symbolically (see Figure 5).(2,2) and (0,2) yield the midpoints (1,0), (2,1),(1,2) and (0,1) which result in an irrational length Figure 5. Using GeoGebra's Relation Tool tofor each appearing side, namely the square root of numerically check properties (Kovács, 2016).2. On accurate sampling when rounding the lengthsof the opposite sides to 2 decimal places we will When using the Relation Tool, the userclearly get 1.41 for all four sides. By increasing the needs to select two objects to compare. Despite theaccuracy obviously the same results are expected. fact that the numerical computation is inaccurate, GeoGebra assumes that the opposite sides have the This is no longer true for some more same length, and offers a further check tocomplicated cases. By changing the rounding in symbolically prove the congruence of thoseDGS GeoGebra to 15 decimal places (in menu segments (see Figure 6).Options > Rounding > 15 Decimal Places), the lastdigits in the lengths of two opposite sides willdiffer (see Figure 4). Figure 4. A numerical error suggests falsifying a Figure 6. The Relation Tool provides a symbolical true statement. check to prove a property.Virginia Mathematics Teacher vol. 43, no. 2 16

When clicking “More...” GeoGebra starts to the Worldwide Mathematica Conference,translate the geometric figure into a set of algebraic Chicago, June 18-21, 1998.equations with integer coefficients in the Botana, F., Hohenwarter, M., Janičić, P., Kovácsbackground, but assuming the input points to be Z., Petrović, I., Recio, T. & Weitzhofer, S.completely general and arbitrary. GeoGebra then (2015). Automated theorem proving inmanipulates on the equations by usually GeoGebra: current achievements. Journalperforming billions of atomic computations which of Automated Reasoning (Vol. 5, Number 1,are hardly of classroom interest, hence those details pp. 39-59). Springer.will not be shown to the user. Finally GeoGebra Hohenwarter, M., Kovács, Z. & Recio, T. (2016).concludes that the equations can be interpreted as Deciding geometric properties symbolicallyan evidence about the geometric statement on the in GeoGebra. https://www.researchgate.net/midpoint parallelogram, that is, it is true in general, publication/305916853_Deciding_geometriindependently of the initial points. c_properties_symbolically_in_GeoGebra Hickey, W. (2013): The 12 Most Controversial Modern DGSdynamic geometry systems Facts In Mathematics. http://including this ATP feature, can be considered as a www.businessinsider.de/the-most-kind of “geometry calculators”. Their ATP controversial-math-problems-2013-3?algorithms may be hidden for the user with, but just op=0#-63the final result is visible. In this sense the usual Kovács, Z. (2015): Midpoint parallelograms.way of proving is usually substituted by a yes or no GeoGebra Materials. https://answer computed by the machine. www.geogebra.org/m/pBuQA4Eo Kovács, Z. (2016): Midpoint parallelograms. However, the traditional way of proving GeoGebra Materials. https://should not be discouraged at all. When the truth www.geogebra.org/m/BrnY77nEabout a conjecture is already known by mechanical Lin, F.-L., Yang, K.-L., Lee, K.-H., Tabach, M., &computation, the real intellectual challenge will Stylianides, G. (2012). Principles of taskstart: to find an elegant chain of reasons to show design for conjecturing and proving. Inwhy that conjecture is true. Hanna, G. and de Villiers, M., editors, Proof and Proving in Mathematics A well known quote from Paul Halmos Education. The 19th ICMI Study, pp. 305-highlights that “the only way to learn mathematics 326. Springer.is to do mathematics”. What does it mean to domathematics? According to Bruno Buchberger, it is Zoltan Kovacs“knowledge derivation and problem solving by Johannes Kepler Universityreasoning.” (Buchberger et al., 1998) To support [email protected] idea, Buchberger introduced the creativityspiral “algorithms → computational results → 17conjectures → theorems → algorithms → …”, andso forth, to describe mathematics as an infinitechain of recurring activities. In his model provingis the activity which connects conjectures andtheorems. In conclusion we can say that withoutknowing why a conjecture is true learningmathematics is not really possible, either. Acknowledgments. A preliminary versionof this paper was presented at ICME-13 inHamburg, Germany (Hohenwarter et al., 2016). ReferencesBuchberger, B. and the Theorema Working Group (1998). Theorema: Theorem proving for the masses using Mathematica. Invited Talk atVirginia Mathematics Teacher vol. 43, no. 2

Algebra Achievement of Urban High School StudentsAndrew Wynn, Cheryl Adeyemi, Gerald Burton, and Crystal WynnIntroduction civil-rights activist Bob Moses in 1982. Bob grew Student achievement is a major concern for up in a poor family but had a strong academic prowess and was able to attend a competitive highschools, law makers, and other stakeholders. A by school, and eventually was able to earn his-product of this concern is the fact that algebra has Master’s degree from Harvard University. Havingserved as a gatekeeper and barrier for many been instrumental in the civil rights movement, hestudents who were interested in majoring in sought to conquer injustice wherever he saw fit.Science, Technology, Engineering, and After observing the poor quality mathematicsMathematics (STEM) at the collegiate level education that his daughter and her classmates were(Ladson-Billings, 1997). While mathematics has receiving in the public schools in Mississippi, Bobproven to be a barrier for students from all races, sought to immediately impact the curriculum byresearch has shown that upper level math courses offering to teach Algebra to his daughter and herbecome even greater obstacles for students who are classmates(Moses and Cobb, 2002). Armed with aof African-American or Hispanic descent or who strong desire to see equality in both areas of socialhave disabilities (Martin, 2012; Noble, 2011). justice and mathematics, Bob developed the ideaThese students often fail to complete high school. that mathematics education is a civil-right.According to Cortes, Nomi, and Goodman (2013) Therefore, Bob developed his ideas into asuggest, “One theory for these low high-school curriculum to foster change in the urbancompletion rates is that failures in early courses, mathematics classroom. Doing much of his earlysuch as algebra, interfere with subsequent course work in classrooms in Mississippi, Bob was able towork, placing students on a path that makes affect students first hand as a math teacher. Hisgraduation quite difficult”. In fact, “beginning with motivating factor was the thought that “thethe first international mathematics achievement test information age of computers and networks has putadministered to students in the 1960s, the problem advanced mathematics…on the table as anof poor mathematics performance has emerged as education necessity” (Checkley, 2001).an issue of national concern” (Mayfield and Glenn, Algebra Project Pedagogy2008). With respect to algebra courses, reformefforts have been implemented to try to increase The Algebra Project developed astudent performance further improving students’ mathematics curriculum that began with the ideaaccess to upper level mathematics classes. Finding that math is a language that needs to be learnedinterventions that not only improve urban school because it is not a natural language. Today, thestudents’ achievement by making the mathematics Algebra Project method impresses upon studentsmore understandable, but that also help increase that the language of mathematics needs to beurban school students’ desires to want to pursue learned, and that they, as students, are capable ofmath opportunities by giving them a better learning the language replete with syntax, structure,experience is critical. Interventions based upon and conceptual meaning Once students buy in toexperiential and structured experiences may cause the concept of learning the language, the Algebrastudents to grow in algebraic understanding and Project curriculum is centered around theabilities, while developing their own experience of Experiential Learning Cycle (ELC) (Kress, 2005).mathematics. The Algebra Project presents itself Using the ELC, students being taught using the APas one such intervention. pedagogy begin with a concrete experience in theirHistory of the Algebra Project, Inc. community, make observations about their experiences, reflect on their experience and The Algebra Project (AP) was founded by 18Virginia Mathematics Teacher vol. 43, no. 2

observations, then finally formulate an abstract Algebra 1 students in urban school districts. Aconceptualization that is at the heart of Algebra. sample of 152 participants was obtained fromThe ELC in Bob’s words, “…helps kids create a students registered in Algebra 1 courses at the onlyconceptual language by first grounding high school in an urban school district in centralmathematics in the daily life and culture they Virginia. The majority of the students wereunderstand” (Checkley, 2001). Specifically, the African-American with a small mixture of otherAlgebra Project, Inc. has created materials for use ethnic groups. The school district has traditionallyin high school and collegiate level algebra classes not met achievement benchmarks on the Virginiathat focus specifically on exploration of integers, Algebra 1 Standards of Learning test, and manyequations, functions, function representations, and students have had trouble passing stateproperties of functions. These materials are mathematics tests in subsequent courses. Everycontained in the Algebra Project’s Trip Line and participant in the target frame was enrolled inRoad Coloring modules. Algebra 1 for either the first or second time. AMethods Overview convenience sample was used for this study, as the participants in this study attended school in the The purpose of this quantitative study was same urban school district in which the researcherto determine if the use of the Algebra Project worked. Several of the participants were studentspedagogy in Algebra 1 classrooms had a significant in the researcher’s classes and were included in theeffect on the achievement results of students study in order to reach the target sample size andenrolled in Algebra 1 courses at an urban high because of the ease of access to the students’school in central Virginia. Significance was achievement data. The participants were dividedmeasured at the p < .05 level. into three groups for the purpose of this study:Assumptions Group A(experimental group) and Groups B and C (control groups). Group A was enrolled in an According to the Virginia Department of Algebra 1 course that was taught using the AlgebraEducation, all students who graduate from a Project pedagogy as the main curriculum. Group BVirginia Public School must have successfully was enrolled in an Algebra 1 course that was taughtcompleted three mathematics courses at or above using traditional methods of teaching which did notthe level of algebra, and must have one verified involve the Algebra Project curriculum. Group Ccredit in mathematics which is earned by was enrolled in an Algebra 1 course that was taughtsuccessfully passing a mathematics Standard of using traditional methods, however, these studentsLearning (SOL) test in the areas of Algebra 1, were tested in the Spring of 2007, while groups AGeometry, or Algebra 2, which requires a scaled and B tested in the Fall of 2008.score between 400 and 600 (VDOE, 2016). The Variablesstudents who participated in this study werehomogeneous as each participant was enrolled in The dependent variable in this study wasan Algebra 1 course at the only high school in the the students’ achievement. For the purposes of thisschool district. Every student was enrolled in 9th study, achievement was measured by the students’grade and did not have a verified credit in scaled score on the Virginia Algebra 1 SOL test.mathematics at the time of testing. Each student The Algebra 1 SOL test was separated into fourwas taught by a teacher with at least three years of strands, which were Expressions and Operations,experience, and every teacher was trained in both Relations and Functions, Equations andtraditional methods and AP pedagogy. The Inequalities, and Statistics. These strands assessedteachers each had classes taught by traditional students’ understanding of the basic algebraicmethods and classes taught using the AP methods. functions, such as linear and quadratic functions,It is assumed that there were no other significant and their properties, along with solving multi-stepdifferences in the participants or the instructors equations and inequalities, and using matrices towho taught the courses. solve a set of simultaneous linear equations. ThereParticipants was no pretest given at the beginning of the course; The target population for this study was 19Virginia Mathematics Teacher vol. 43, no. 2

therefore, it is assumed that students were district being studied and was given once in thecomparable in their understanding of the material Fall semester and once in the Spring semester. Theassessed by the four strands of the Algebra 1 SOL. overall scaled score for the test is based on a 600 point scale, from 0 to 600, with a score of 400 There are several independent variables representing the minimum proficiency necessary toincluded in this study. The first was the pedagogy be considered passing. Scores of 400 and aboveused within each individual class. Every teacher are rated as Passing/Proficient, scores of 500 andwho taught an Algebra 1 class was trained in the above are rated as Passing/Advanced, and scores ofAlgebra Project Pedagogy but had also taught 600 represent a perfect score.traditional Algebra 1 courses before. The only Data Analysiscontrol was that three teachers signed a contractagreeing to pilot the Algebra Project materials in The researcher was one of the teachers whoone of their classes. These teachers each taught at piloted the Algebra Project material and one of theleast one class that used a traditional curriculum as test examiners for the Virginia Standards ofwell. A second independent variable that existed Learning Test. The results were analyzed ex postwas the Algebra 1 Standards of Learning test itself. facto. The researcher was granted permission toThis test was created and administered by the analyze the results of the students’ tests by theVirginia State Department of Education. school district’s Superintendent and DivisionData Collection and Instruments Director of Testing. The researcher interpreted the data from the assessments and sought any The instrument utilized in this study was statistical significance between the achievementthe Virginia Algebra 1 Standards of Learning test. levels of the group who received the interventionThis test was selected because it was the accepted and the group that did not receive the intervention.standard in the state of Virginia for determining Comparisons of the mean scores on the testmathematics achievement in Algebra 1 courses. measuring achievement were then conducted. TheThe Virginia Department of Education formed a researcher utilized two sample T-tests to comparecommittee that it used to create valid questions for the mean scores on the SOL test between the groupthe test to ensure the test validity. This committee receiving the intervention and students notensures that the tests for the upcoming year are of receiving the intervention during the same testinggood quality and will accurately measure the cycle. This test was appropriate because the meanachievement in an Algebra 1 course, based upon scores of two independent groups, the studentsthe Virginia Standards of Learning, of those taught using traditional methods and the studentsstudents who complete the test. The state had taught using AP pedagogy, were being comparedvarious measures in place to ensure the test was on a common dependent variable (Warner, 2013).valid and the state administered the test to each Independent sample T-tests were also used toschool district in the state of Virginia. compare the mean scores on the SOL test between the group receiving the intervention, and the group The Virginia SOL test for Algebra 1 was not receiving the intervention from the testingdesigned to measure how well the students cycle immediately preceding the introduction of theunderstood the algebra concepts from the intervention. The students’ scores used for theExpressions and Operations, Relations and comparison were taken for the years 2007 to 2008Functions, Equations and Inequalities, andStatistics strands. The test was made up of 60 Table 1. Two Sample t-test for SOL Test Scores for Group A vsmultiple-choice questions, and students were given Group Bas much time as needed to complete the test. Eachquestion had four possible answer choices given, N M S.D. S.E.and the student was responsible for selecting thebest answer for each question. Ten of the questions Group A 25 444.72 38.495 7.699administered on the test were field test items anddid not count towards a student’s scaled score. The Group B 61 435.31 35.339 4.525test was administered by computer in the schoolVirginia Mathematics Teacher vol. 43, no. 2 20

to ensure that the standards used for creating the Table 4. Two Sample t-test for Relations and Functions Strand fortest were the same. This ensured that the tests were Group A vs Group Bcomparable although there were different forms ofthe test, which helped with the reliability of the N M S.D. S.E.instrument used. T-tests were also used to comparethe mean score of the Relations and Functions Group A 25 39.00 5.260 1.052strand of the standardized test between the Group C 61 34.20 7.246 .892Table 2. Two Sample t-test for Relations and Functions Strand for The following statistical tables display theGroup A vs Group B results of the study. They are presented in tabular form to assist the reader in comparing the data. N M S.D. S.E. An independent samples t-test, alpha = .05,Group A 25 39.00 5.260 1.052 two-tailed, was conducted to determine if there was a statistically significant mean difference on overallGroup B 61 34.52 6.324 .810 SOL test score, measured by the Virginia Algebra 1 SOL test. The Levene’s test for equality of‘experimental’ and the ‘control’ groups. variance indicated this assumption was met. TheResults results were not significant, t = -1.092, p = .278. The results indicated the Algebra Project Group The purpose of this study was to use the (M=444.72, SD=38.495) reported higher SOL testresults from the Fall 2008 Algebra 1 Standards Of scores than the Non-Algebra Project GroupLearning test to determine whether or not the usage (M=435.31, SD=35.339). This difference was notof the Algebra Project curriculum in Algebra significant.classes in an urban school district made asignificant difference on Algebraic understanding An independent samples t-test, alpha = .05,and achievement of students who were enrolled in two-tailed, was conducted to determine if there wasthe courses during the 2007- 2008 school year. a statistically significant mean difference on theSignificance was determined at the p < .05 level. Relations and Functions Strand on the SOL test, measured by the Virginia Algebra 1 SOL test. TheTable 3. Two Sample t-test for SOL Test Scores for Group A vs Levene’s test for equality of variance indicated thisGroup B assumption was met. The results were significant, t = -3.121, p = .002. The results indicated the N M S.D. S.E. Algebra Project Group (M=39.00, SD=5.260) reported higher Relations and Functions strandGroup A 25 444.72 38.495 7.699 scores than the Non-Algebra Project Group (M=34.52, SD=6.324). This difference wasGroup C 61 427.53 46.013 5.664 significant.There was a significant mortality rate involved in An independent samples t-test, alpha = .05,this study. Whether it was from suspensions, two-tailed, was conducted to determine if there wasplacement at an alternative school, or students a statistically significant mean difference on overallmoving out of the district, both groups finished SOL test score, measured by the Virginia Algebrawith fewer participants than were originally 1 SOL test. The Levene’s test for equality ofenrolled. To combat this situation, the researcher variance indicated this assumption was met. Theused the data of all Algebra 1 students enrolled at results were not significant, t = 1.659, p = .101.this particular high school for the sessions in The results indicated the Algebra Project Groupquestion. The students completed their respective (M=444.72, SD=38.495) reported higher SOL testcourses then were administered the same end of scores than the Non-Algebra Project Groupcourse test. (M=427.53, SD=46.013). This difference was not significant.Virginia Mathematics Teacher vol. 43, no. 2 21

An independent samples t-test, alpha = .05, mathematics of their localities in the way that thetwo-tailed, was conducted to determine if there is a Algebra Project does, students may gain a richerstatistically significant mean difference on the appreciation for the necessity of math and actuallyRelations and Functions Strand on the SOL test, realize their own capabilities to perform well inmeasured by the Virginia Algebra 1 SOL test. The math classes. Additionally, because students whoLevene’s test for equality of variance indicated this were taught using the Algebra Project pedagogyassumption was met. The results were significant, t scored significantly better on the Relations and= 3.022, p = .003. The results indicated the Algebra Functions strand than their counterparts who wereProject Group (M=39.00, SD=5.260) reported taught using traditional methods, students may behigher Relations and Functions strand scores than able to gain a deeper understanding of functionsthe Non-Algebra Project Group (M=34.20, and their properties which is one of the majorSD=7.246). This difference was significant. foundations of Algebra in general. This in turnConclusion may enable students to proceed into upper level math courses, such as calculus, which relies There have been several studies done on the heavily on students’ understanding of functionsignificance of the myriad of interventions that properties, thus yielding accessibility to otherexist to effect Algebraic achievement of U.S. STEM areas that were at times out of reach forstudents. However, no study had previously been African-American urban students. Based on thecompleted that examined the effects of the Algebra results of this study, the followingProject curriculum on the algebraic achievement of recommendations are made for future study:African-American students enrolled in Algebra 1classes in urban school districts, where African- 1. Future studies should compare the results ofAmerican students are often marginalized, taught all students who participated in the testusing lowered expectations, and prevented from administration, whether the student was areaching their full mathematics potential. In order first time test taker or not.to make the students in the United States morecompetitive with students around the world, more 2. Future studies should compare the attitudesemphasis must be placed on using creative, held towards mathematics of the studentsresearch-based strategies, which will improve the who completed the Algebra Project coursemathematical understanding and achievement to those who did not.levels of students in urban districts who havetraditionally been unsuccessful on standardized 3. Future studies should compare the anxietytests. Therefore, determining which interventions levels of students who are taught usingcan effectively assist U.S. schools in reaching their Algebra Project methods with the anxietygoals of high mathematical achievement and levels of students taught using traditionalunderstanding for all students, especially those of methods.African-American descent who are being taught inurban districts has gained importance in the last Referencestwo decades. The results of this study demonstrate Checkley, K. (2001). Algebra and activism:that although African-American students who weretaught using AP pedagogy did not score Removing the shackles of low expectations;significantly better than students taught using a conversation with Robert P. Moses.traditional methods, they did have comparable Educational Leadership, 59, 6 – 11.scores. Therefore, the Algebra Project pedagogy Cortes, K., Nomi, T., & Goodman, J. (2013). Amay provide an alternative to teachers in urban double dose of algebra. Education Next, 13school districts who are struggling to engage their (1) Retrieved from http://students in Algebra classes, which is a course search.proquest.com/docview/1238139541?where many students develop a disdain for accountid=12085mathematics. By immersing students in the Kress, H. (2005). Math as a civil right: Social and cultural perspectives on teaching andVirginia Mathematics Teacher vol. 43, no. 2 teacher Education. American Secondary Education, 34, 48 – 56. 22

Ladson-Billings, G. (1997). It doesn’t add up: Noble, R., III. (2011). Mathematics self-efficacy African American students’ mathematics and African-American male students: An achievement. Journal for Research in examination of models of success. Journal Mathematics Education, 28(6), 697 – 708 of African American Males in Education, 2 (2), 188 – 213.Martin, D. B. (2012). Learning mathematics while black. Educational Foundations, 26(1), 47 Warner, R.M. (2013). Applied statistics: From – 66. bivariate through multivariate techniques. Thousand Oaks, CA: Sage Publications,Mayfield, K. H., & Glenn, I. M. (2008). An Inc. evaluation of interventions to facilitate algebra problem solving. Journal of Andrew Wynn Behavioral Education, 17(3), 278-302. Professor of Mathematics doi:http://dx.doi.org/10.1007/s10864-008- Virginia State University 9068-z [email protected], R. P., & Cobb, C. E. (2002). Radical *Photos of the other three authors were not equations: Math literacy and civil rights. provided Boston: Beacon Press.Solutions to Fall 2016 HEXA Challenge ProblemsOctober: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 cannot be a per-fect square of any integer number.SOLUTION :20162016=10001*2016The number K can be written as the sequence of digits 2016….2016. If A is a positive integer, then Kcan be written asThere is no A that makes K a perfect square.November:There are N people that live in a city, where there are two main competing companies. Out of thesepeople, n know each other, because they work for the same company. m people also know each otherbecause they work in the same city, but in a rival company. Personal relationships between workers atrival companies are not allowed. What portion of the population of the city, does not work for either ofthese companies, but can knows exactly 1 person from each company.SOLUTIONLet’s estimate the probability of this scenario, as described in the problem.The relevant portion of the population isDecember:Given an infinitely large set of different types of triangles, if one randomly selects a triangle, what arethe chances of this triangle being obtuse?Virginia Mathematics Teacher vol. 43, no. 2 23

Solutions to Fall 2016 HEXA Challenge ProblemsSOLUTION:Let’s label the measures of the angles of the triangle as α, β, and γ.We know thatThe value of any angle may vary between 0 and 180 degrees (In the extreme cases, creating degenerate tri-angles).If one of the angles in a triangle is greater than 90 degrees, then the triangle is obtuse. For obtuse triangles, . The shaded area in the figure below represents the set of all obtuse triangles. The axes are αand β. As you can tell, each of the shaded regions are congruent with each other and represent 1/4th of thetotal area. Therefore, the chance of any triangle being obtuse is , or 75%. January 24 Solve the following equation: SOLUTION: To solve algebraically:Virginia Mathematics Teacher vol. 43, no. 2

Solutions to Fall 2016 HEXA Challenge ProblemsLet’s substitute:Using the quadratic equation:ConsequentlyIn the first case, the roots are 2 and -1. In the second case, there are no real solutions.Graphically, the solution looks as follows: February: 25 Prove the following for any integer number, nVirginia Mathematics Teacher vol. 43, no. 2

Solutions to Fall 2016 HEXA Challenge ProblemsSOLUTION:Let us evaluate each addend in the following expression individually:MarchIn an Isosceles triangle ABC, AB=BC, Angle B is 20 degrees, AC is five units Point D is on BC so thatAngle BAD is 30degrees, and angle DAC is 50 degrees. Point E is on the side AB, so that Angle ECB is60 degrees and angle ECA is 20 degrees. Find the length of DE. See Figure 1.SOLUTION:First, in the triangle ADC, the measure of angle D is 50 degrees (sum of the angles in a triangle equals180 degrees). This means that triangle ADC is isosceles. Therefore side DC measures 5 units. In the trian-gle EAC, the measure of angle E is 80 degrees. This means that triangle EAC is also isosceles, and themeasure of side EC is also 5 units. In the isosceles DCE, the base angles E and D are congruent. Becausethe third angle, DCE, of the triangle is 60 degrees, angles D and E must also measure 60 degrees. There-fore, triangle DCE is equilateral, which means that side DE is 5 units. (see Figure 2)Virginia Mathematics Teacher vol. 43, no. 2 26

Solutions to Fall 2016 HEXA Challenge ProblemsFigure 1. Triangle ABC, in the original problem Figure 2. Triangle ABC, in the solution Upcoming Math CompetitionsName of Organization Website Dates of CompetitionsHuntington University Middle School https://www.huntington.edu/math/mathematics-competition/middle-school- competition April 12, 2017Mathematics Competition April 7-9, 2017Rochester Institute of Technology’s http://www.rit.edu/ntid/mathcompetition/Math competitions for Students who Online competition Mailed to your school are Deaf or Hard-of-Hearing April 18-27SUM Dog online http://www.sumdog.com/enter_contest/ TBA American Math League http://mathleague.org/arml.php Online, month-long contestPurple Comet Math Meet http://purplecomet.org/home/home At your local school Mandelbrot http://www.mandelbrot.org/USA Math Talent Search http://www.usamts.org/TipsFAQ/U_Tips.php http://www.mathcounts.org/programs/competition-series/competition-faq Math CountsVirginia Mathematics Teacher vol. 43, no. 2 27

Concrete, Representational, and Abstract:Building Fluency from Conceptual Understanding Robert Berry III and Kateri ThunderIntroduction explicitly connecting to the previous instruction. Concrete is the first phase, often referred to as “theThe National Council of Teachers of doing stage”, when instruction focuses on using manipulatives or concrete objects. RepresentationMathematics (NCTM) stated, “Effective (semi-concrete or pictoral) is the second phase, often referred to as “the seeing stage”, whenmathematics teaching focuses on the development instruction connects the concrete manipulatives to drawing, pictures, and other visual representationsof both conceptual understanding and procedural of concrete objects. Abstract is the third phase, often referred to as “the symbolic stage”, whenfluency” (NCTM, 2014; p. 42). Conceptual instruction connects the concrete and semi-concrete representations to using only numbers andunderstanding is comprehending the meaning of mathematical symbols or to mentally solving problems. The three phases are flexible andmathematical concepts and reasoning about the reflective of students’ readiness to explain concepts and to fluently apply strategies with different levelsrelationships between concepts. A student with of representation. At every level, there should be parallel modeling of each representation withconceptual understanding is able to explain why mathematical vocabulary and numbers. Rekenreks and Part-Whole Bar Modelsvalues can be described using part-whole Rekenreks and part-whole bar models arerelationships and why values can be equivalent. tools that support developing procedural fluency while deepening and exploring conceptualProcedural fluency consists of accuracy, efficiency understanding. They also make connections among concrete (rekenrek), representational (part-and flexibility (Russell, 2000). A student with whole bar model), and abstract (number sentences) (CRA) modes of representations.procedural fluency is able to select an efficient The rekenrek consists of 20 beads in twostrategy that fits the given numbers to accurately rows of ten, each broken into two sets of five by color (i.e., in each row the first five beads are redsolve the problem. In this way, procedural fluency and the next five are white) (see Figure 1). Adrian Treffers, a mathematics curriculum researcher atis closely related to number sense and requires the Freudenthal Institute in Holland, is credited with developing the manipulative (Fosnot & Dolk,students to do more than just memorizing facts 2001). Directly translated, rekenrek means counting rack. The rekenrek is sometimes mistaken(Baroody 2006). Procedural fluency reduces as an abacus at first glance, but it is not based on place value columns, and it is not used in the samecognitive load while problem solving, which manner as the abacus. Its main characteristic is that it has a five-structure with analogousallows the student to focus on the mathematical representations of the five fingers on each of our hands and the five toes on each of our feet. It isrelationships of the problem at hand. There are 28three phases of procedural fluency development:(1) exploration and discussion of number concepts,(2) use of informal reasoning strategies based onproperties of the operations, and (3) the eventualdevelopment of automaticity (NCTM, 2014).Effective mathematics teaching developsprocedural fluency with number sense whileengaging students in thinking deeply about therelationships among values.Research across grade levels and with allpopulations of students (including students withspecial needs and English Language Learners) hasdemonstrated that CRA is an effective teachingstrategy for developing procedural fluency built onconceptual understanding (Crawford & Ketterlin-Geller, 2008; Hudson, Miller, & Butler, 2006;Paulsen, 2005; Witzel, 2005). CRA (sometimescalled CSA or CPA) is a three-phase instructionalapproach with each phase building on andVirginia Mathematics Teacher vol. 43, no. 2

Figure 1:Rekenrek fluency based on their understanding of part-whole relationships.often used to develop strategies for basic facts suchas working with the five-structure, making ten, one 15more or less than, compensation and stretchingchildren toward using these strategies with the 78support of or in place of counting (Fosnot & Dolk,2001). 15 The five-structure of a rekenrek offers 5 5 23visual support to quickly see the quantity of fivewithout counting. This is a form of subitizing, the 15ability to recognize the number of objects in a setwithout actually counting them (Clements, 1999; 10 5MacDonald & Shuway, 2016). Subitizing with arekenrek is a concrete representation to flexibly Figure 3: 7+8, 5+5+2+3, and 10+5decomposing and composing numbers based on using part-whole bar modelspart-whole relationships. For example, on arekenrek, students can see that seven is composed Part-whole bar models are a visualof five (five red beads) and two (two white beads) representation that show the magnitude of aand eight is composed of five (five red beads) and number as well as the relationships among thethree (three white beads). Figure 2 shows a whole number and its parts. In Singapore, the bar model is used as a tool for problem solving because Figure 2: 7 + 8 on the Rekenrek it communicates graphically complex relationships, such as comparisons, part-whole calculations,rekenrek representing 7 + 8. Students may ratios, proportions, and rates of change (Hoven &subitize and see 7 + 8 is the same as 5 + 5 + 2 + 3 Garelick, 2007). For example, in a bar model, 15or 10 + 2 + 3 or 10 + 5. Students can also see a can be decomposed visually into its parts andconcrete representations of the commutative represented proportionally. Figure 3 shows part-property of addition (7 + 8 = 8 + 7). Using whole bar models representing 7+8, 5+5+2+3, andrekenreks, students can purposefully practice 10+5.building combinations of numbers and develop As students visually represent theirVirginia Mathematics Teacher vol. 43, no. 2 rekenrek work with part-whole bar models and then symbolically represent the models with number sentences, they can connect concrete, representational, and abstract models of basic addition facts. These three levels of representation also provide students with an opportunity to begin an exploration of the concept of equality. CRA in Action In the classroom vignette that follows, Mr. Dominguez, a first-grade teacher, is working with students using rekenreks along with part-whole bar models to build fluency of basic addition facts based on number sense (Virginia Mathematics Standards of Learning (SOL) 1.7) and to explore the concept of equality (SOL 1.15). Mr. Dominguez’s class regularly uses manipulatives to represent and describe their thinking. In a previous 29

Dominguez asks Jaxson, “Can you write that as a number sentence on the board?” Jaxson writes 4 + 3 = 7. 7 1 6Figure 4: Jaxson’s representation of seven on rekenrek Figure 7: Aisha’s representation using part-whole bar modellesson, the class used ten-frames to develop basic Mr. Dominguez engages the class in a discussion of the connections among Jaxson’saddition facts to ten. concrete representation on the rekenrek, his semi- concrete representation of the part-whole barMr. Dominguez says to his students, “I am model, and his abstract representation of the number sentence.thinking of a way to show seven on my rekenrek. Mr. Dominguez sees Aisha’s representationCan you show me a way?” All of the students use on her rekenrek and asks her to describe her representation (see Figure 6).their rekenreks to show seven. Mr. After Aisha’s explanation, Mr. DominguezDominguez takes note of the different ways engages the class to represent Aisha’s representation using the part-whole bar model (seestudents represented seven on the rekenreks. For Figure 7) and then writing a number sentence (6 + 1 = 7)example, he notes that some students show five Again, the class discusses the relationshipbeads on the top and two on the bottom, some among Aisha’s concrete representation on the rekenrek, her semi-concrete representation of theshow four on the top and three on the bottom, and part-whole bar model, and her abstract representation of the number sentence.some show six on the top and one on the bottom. Mr. Dominguez uses Jaxon and Aisha’sAs the students work, he asks them to find more thinking to write the equation 4 + 3 = 6 + 1 on the board. He then asks the class what the numberthan one way to represent seven on the rekenrek. sentence means and how can they use Jaxson’s and Aisha’s thinking to make sense of the numberAfter some time, Mr. Dominguez invites sentence.Jaxson to describe his representation of seven. Marcel says, “If you do 4 + 3 first you will get 7 and then you do 6 + 1 you will get 7…so itJaxson describes that he has four beads on top and means 7 is equal to 7.”three on the bottom (see Figure 4). Many students nod in agreement. Mr. Dominquez challenges the students to find otherMr. Dominguez says, “Let’s show Jaxson’s number sentences that can be written similar to 4 + 3 = 6 + 1. They can use the rekenreks, part-wholethinking another way by writing it on our part- bar models, and number sentences to problem solve. The students work independently to developwhole bar model on the board (see Figure 5). So, their own number sentences while Mr. Dominguez circulates and confers with students.boys and girls, if seven is the whole, what are the Jamal writes the number sentence 5 + 2 = 7parts Jaxson described?” He calls on Donovan + 2. Mr. Dominguez confers with Jamal to learn about his thinking. Then Mr. Dominguez askswho responds by saying the parts are 4 and 3. Mr. Jamal to create two part-whole bar models in which 7 3043Figure 5: Jaxson’s representation using part-whole bar model Figure 6: Aisha’s representation of seven on rekenrekVirginia Mathematics Teacher vol. 43, no. 2

the whole on both models is 7. Jamal makes the conceptual understanding. Mr. Dominguez’sfirst bar model for 5+2 quickly. After thinking a bit lesson is an example of making the CRAand checking with his rekenrek, Jamal creates a connection across all levels in one lesson. It is notsecond bar model for 7+0. Jamal then writes a new necessary that these connections are made withinnumber sentence 5 + 2 = 7 + 0. one lesson; in fact, this not often the case. Rather, connections are made across a unit or series of To bring closure to the lesson the students lessons. As teachers plan instruction, they shouldshare their equations with partners. Then Mr. engage students in using meaningful concrete,Dominguez engages them in whole class discussion representational, and abstract representations andto share and compare equations. Jamal is able to explicitly making connections among them. CRAshare his mistake, his strategy to fix his mistake, is also an effective teaching practice forand what he learned from his mistake. differentiation. Students may be ready for problemDiscussion and Implications solving and explaining their reasoning with different levels of representation (concrete, Time is a critical factor in teachers’ representational, abstract). Students can choose toinstructional decisions about both what and how to use the level of representation for which they areteach mathematics. Effective teaching practices ready or teachers can assign students to use one orimprove student achievement as well as student more levels of representation based on theirretention and application of concepts and skills readiness.over time. Choosing to implement evidence-based,effective teaching practices maximizes Teachers need to develop proceduralinstructional time. fluency that is built upon and reinforced by conceptual understanding using concrete, “Effective mathematics teaching focuses on representational, and abstract representations for allthe development of both conceptual understanding students. Two powerful tools are rekenreks andand procedural fluency” (NCTM, 2014; p. 42). part-whole bar models. These effective teachingProcedural fluency and conceptual understanding practices will maximize instructional time andare both important components of developing grow mathematically proficient learners who aremathematical proficiency. One should not be deeply engaged in the complex, rigorous problemssacrificed for the other; in fact, Mr. Dominguez’s of mathematics.lesson is an example of developing proceduralfluency built on conceptual understanding. As the Referencesstudents problem-solved to become more efficient Baroody, A. J. (2006). Mastering the Basicand flexible with basic addition facts, they reliedon their understanding of part-whole relationships. Number Combinations. Teaching ChildrenTheir fluency worked in tandem with their number Mathematics 13(1), 23–31.sense for composing and decomposing numbers. Clements, D. H. (1999). Subitizing: What is it?The tasks, materials, and discussion within Mr. Why teach it? Teaching ChildrenDominguez’s lesson built upon their conceptual Mathematics, 5(7), 400-405.understanding and continued to deepen this Crawford, L. & Ketterlin-Geller, L.R. (2008).understanding by allowing students to extend their Improving Math Programming for Studentsunderstanding to a new context, new at Risk: Introduction to the Special Topicrepresentations, and further mathematical Issue. Remedial and Special Education, 29discourse. Additionally, students began to explore (1), 5-8.the new concept of equality with an emphasis on Fosnot, C. T., & Dolk, M. (2001). Addition andmeaning-making. subtraction facts on the horizon. In V.Merecki& L. Peake (Eds.), Young CRA (also known as CSA or CPA) is an mathematicians at work: Vol. 1.effective teaching strategy for all populations of Constructing number sense, addition andstudents (including students with learning subtraction (pp. 97-113). Portsmouth, NH:disabilities and English Language Learners). Heinemann.Rekenreks and part-whole bar models are effectivetools for developing procedural fluency from 31Virginia Mathematics Teacher vol. 43, no. 2

Hauser, J. (2004). Concrete-representational- Preparation Curriculum. Teacher Education abstract instructional approach. Retrieved and Special Education, 28 (1). from http://165.139.150.129/intervention/ Russell, S.J. (2000). Developing computational ConcreteRepresentationalAbstractInstructio fluency with whole numbers. Teaching nalApproach.pdf on February 25, 2017. Children Mathematics, 154-158. Witzel, B.S. (2005). Using CRA to Teach AlgebraHoven, J. & Garelick, B. (2007). Singapore Math: to Students with Math Difficulties in Simple or Complex? Educational Inclusive Settings. Learning Disabilities: A Leadership, 65(3), 28-31. Contemporary Journal, 3(2), 49-60.Hudson, P., Miller, S.P., & Butler, F. (2006). Robert Berry Kateri Thunder Adapting and Merging Explicit Instruction NCTM President elect Faculty Within Reform Based Mathematics University of Virginia James Madison University Classrooms. American Secondary [email protected] [email protected] Education, 35(1), 19-32.MacDonald, B. L., & Shumway, J. F. (2016). Subitizing Games: Assessing Preschoolers' Number Understanding: Reflect and Discuss. Teaching Children Mathematics, 22(6), 340-348.Paulsen, K.J. (2005). Infusing Evidence-Based Practices into the Special EducationTechnology ReviewSection Editor: Christophe HirelIn this section, we feature websites, online manipulatives, and web-based applications thatare appropriate for K-12 mathematics instruction. We are looking for critical reviews oftechnologies which focus on both the benefits and limitations of using these tools in a K-12 mathematics classroom. If you use a technological tool and wish to share with us,please respond to the call for manuscripts on page 36. Christophe Hirel Section Editor, Technology Review [email protected] Photomath is an app available for android expressions, logarithmic functions, derivation, andand iOS devices that solves math problems integration. The app was designed as analgebraically by taking a picture. The user points educational tool, but how useful is this app in thehis/her camera towards any mathematical mathematics classroom? And would you considerexpression printed or handwritten and Photomath using it? Why or why not? Here are some of ourshows the solution. In most cases, the app also observations when using the app.provides a step by step outline for how it arrived atthe answer. 1. The app provides a solution, but it might not be the most effective procedure or algorithm. The Photomath app is designed to support One way the app could be used in the classroom isnot only simple addition, subtraction, division, and to examine the solution it provides for the elegancemultiplication, but complex equations as well. It of the algorithm. As we know, there is more thancan solve algebraic expressions, linear equations, one way to solve an algebraic problem, but the appquadratic equations, absolute value equations, only provides a single set of steps to the solution. Itinequalities, systems of equations, trigonometric may be a worthwhile activity to see if students canVirginia Mathematics Teacher vol. 43, no. 2 32

Figure 1. Typed equation 1 solved correctly Figure 2. Typed equation 1 solved incorrectlyfind a shorter and/or different algorithm to reach 3. When solving simple algebraicthe answer. expressions using typed text, we found the app to be unstable. It would read the same expression 2. Unlike other technological tools, the app differently on different occasions (see Figure 1 andonly provides algebraic representation and does not Figure 2). We found the same problem to occurprovide graphical or numerical representations for with handwritten text (See Figure 3 and Figure 4).solving problems. This can limit the possibilitiesfor exploration and rich mathematical discussions If you use this app, or any other app in yourin the K-12 classroom. K-12 classroom, we would like to hear from you. Figure 3. Handwritten equation 2 solved correctly Figure 4. Handwritten equation 2 solved incorrectlyVirginia Mathematics Teacher vol. 43, no. 2 33

Good Reads Section Editor: Dr. Betti KreyeIn this section, we feature mathematics literature that is appropriate for K-12 Mathematicsinstruction. If you use specific literature for your mathematics classroom and wish to share itwith the Virginia Mathematics Teacher community, please respond to the Call forManuscripts on page 36. Algebra in Context Dr. Betti Kreye Piece by: Lynn Foshee Reed Section Editor, Good Reads Algebra in Context: Introductory Algebra [email protected] Origins to Applications is divided into fourparts: Numeration Systems, Arithmetic Snapshots, or making Napier’s Bones. In addition, I think theFoundations, and Solving Equations. The last (and book could provide a different mathematicallargest) section covers linear, quadratic, cubic, and experience for introductory algebra students whopolynomial equations as well as the Rule of 3 and are interested in or at least open to makinglogarithms. Throughout the book, the authors connections across cultures and disciplines. Forpresent concepts, notation, and algorithms central example, Part I on Numeration Systems includesto a typical introductory algebra course. What is discussion of Babylonian, Egyptian, Roman,different however is the non-trivial inclusion of Chinese, Mayan, and Indo-Arabic number systems.historical information that places these concepts, Finally, I would highly recommend this book as anotation, and algorithms in context. For example, resource for teachers of mathematics, both in-in the section on Polynomial Equations in One service and pre-service, experienced or novice.Variable there is a thorough discussion beginningwith Rene Descartes’s notation for exponents and Algebra in Context: Introductory Algebra fromcontinuing through contributions of Fermat, Origins to ApplicationsBhaskara, Leonardo of Pisa, among others. Amy Shell-Gellasch and J. B. Thoo Johns Hopkins University Press, 2015 There are many excellent illustrations (forboth the algebra as well as the history), as well as a 34“Now You Try” and “Think About It” problemsthroughout the text. At the end of each chapter is alengthy exercise list containing both algebrapractice and history analysis. Whereas the algebraexamples and problems in the body of each chapterare at a basic level, the history-infused passagescan be quite dense, with occasional lengthyfootnotes. This leads to my only criticism: somechapters feel a bit uneven in the presentation to andcognitive demands of the reader. Algebra in Context could be used easily andsuccessfully as the textbook for a history ofmathematics course for pre-service teachers.There are opportunities for hands-on activities orprojects throughout the book, such as using abaciVirginia Mathematics Teacher vol. 43, no. 2

Key to the Fall 2016Puzzlemaker ProblemVirginia Mathematics Teacher vol. 43, no. 2 35

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 fascinating mathematical problems that have not yet been reflections on mathematical topics you find solved. The problem itself should be described challenging to teach. If you have a difficult simply so that a middle school student could problem or topic you use for your students, please understand it. It should also include the progress describe the problem, discuss common student difficulties with it, and the way you approach that has been made by the mathematics teaching this topic. Please send to Dr. Agida community on solving this problem. Manizade at [email protected] with the subject Please send to Dr. Agida Manizade at line: Teaching Dilemmas [email protected] with the subject line: Unsolved by: May 30th, 2017. Mathematical Mysteries by: May 30th, 2017. Research or Practitioner Articles We are accepting papers on elementary, middle, Math Girls What does your school district or institution do to and secondary mathematics teaching and encourage girls in mathematics? If you have any education. The Fall 2017 theme will be “Community of Heroes: Educate, Encourage, information or publications related to girls in Inspire.” Articles will be peer-reviewed and mathematics education, please send to comments will be shared with the author. The deadline for consideration for the Fall 2017 [email protected] with the subject line: Math issue is May 30th, 2017. Please provide articles Girls by: May 30th, 2017. in a word document no more than 5 pages (not My Remarkable Student including corresponding images, pictures, graphs etc.). All submissions or questions should be sent Have you had a student that changed the way you to Dr. Agida Manizade at [email protected] with think and teach? If so, please send an article the subject line: Research or Practitioner describing your experiences with this student and Article. the ways they have affected your teaching to Dr. Good Reads Agida Manizade at [email protected] with the Do you know of a teaching resource or literature laden with mathematics content? Please send any subject line: My Remarkable Student literature reviews to Dr. Betti Kreye at by : May 30th, 2017. [email protected] with the subject line: Good Reads Busting Blockbusters by: May 30th, 2017. Technology Review In this section we are accepting suggestions for Do you have a review of an app, website, or scenes from movies for readers to analyze and online resource? Please send your critical review explain the mathematical plausibility of them. to Christophe Hirel at [email protected] with Please describe the scene and provide a timestamp the subject line: Technology Review by: May for it, and send to vmt.radford.edu with the subject 30th, 2017. line Busting Blockbusters Suggestion by: May Reviewers 30th, 2017. If you are willing to serve as a reviewer, please Calling Virginia Authors Virginia residents whose articles appear in the contact us at [email protected] VMT will be granted free membership in theVirginia Mathematics Teacher vol. 43, no. 2 VCTM for one year. 36

Educational Opportunities forVirginia Mathematics TeachersEach Spring issue we will feature an outstanding students. They collaboratively plan, implement,program designed for mathematics teachers and reflect on their lessons with these at-riskwithin the Commonwealth. We will discuss the students.nature of the program, specific details and itshistory. The effectiveness of the program is demonstrated by its students. In the past 10 years, Master’s in Education with 47 PSTs have received Noyce scholarships, seven Concentration in Mathematics PSTs have received VCTM scholarships, dozens of graduates have presented at VCTM Education conferences, and several have been honored as teacher of the year by their school districts. In Facts and Statistics: general, graduates have earned reputations as teachers with strong knowledge of content, - Graduation rate : 100% pedagogy, and technology. They represent Virginia Tech well, as indicated by the following - 46 hours fully synchronous graduate courses quote from an assistant principal whose school district has hired four graduates: “I am convinced - Average Enrollment of 12 students per year that you are doing among the best work in the nation in preparing future math educators. I was a - Number of field experiences: 3 Maryland Terrapin for undergrad, and a Virginia Cavalier for my administration degree, so I do not Virginia Polytechnic and State University say this lightly.” Virginia Tech’s secondary mathematics More information about Virginia Tech’seducation program relies on a partnership between secondary mathematics education program can bethe School of Education and the Department of found here: http://www.mathed.soe.vt.edu/Mathematics. Pre-service teachers (PSTs) beginthe five-year program as mathematics majors and Anderson Nortongraduate with MAEd degrees. Faculty work Professoracross departments to ensure PSTs develop the Virginia Techknowledge they need to become effective [email protected] teachers. The mathematics degree builds fromPSTs’ general mathematics knowledge to supporttheir development of specialized contentknowledge. This effort culminates in a six-hour,4000-level sequence entitled “Mathematics forSecondary School Mathematics Teachers.” The MAEd builds from there to supportthe development of pedagogical contentknowledge. As one special feature of the program,secondary mathematics pre-service teacherspartner with the secondary language pre-serviceteachers for a service learning experience workingwith English Language learners and strugglingVirginia Mathematics Teacher vol. 43, no. 2 37

HEXAChallenge Problems created by: Dr. Oscar TagiyevApril Challenge:Solve algebraically.May Challenge:Construct a right triangle ABC where AC is the hypotenuse. Create a point D, such that BD is the height,perpendicular to AC. In triangles Abd and BCD there are inscribed circles, with radii r1 and r2, respectively.Find the radius of the circle inscribed in triangle ABCJune Challenge:If you were to write all of the whole numbers from 1 to 1,000,000 and add them together, to what wouldthey sum?Virginia Mathematics Teacher vol. 43, no. 2 38

Please be sure to state your assumptions as you solve each problem. Answers to the Spring 2017 Hexa Challenge Problems will be featured in the Fall 2017 Issue of Virginia Mathematics Teacher.July Challenge:A man owes $10,000. At the moment, he has nothing, and has just started a new job. His schedule has himwork only every other month. During the month he is off, it takes him two weeks to cash the check for hisprevious month’s work, and from that amount, at the end of the month, he has $1,000 left. When should thecollection agency next meet with him to settle his debt?August Challenge: The shadows of a three dimensional object cast on three planes perpendicular to each other create a circle, a square, and an equilateral triangle. What is the shape? September Challenge: Contest Alert! It is currently midnight. How long is it until the Virginia Mathematics Teacher is minute hand and the hour hand are perpendicular conducting a contest for educators to each other? and students who can solve theVirginia Mathematics Teacher vol. 43, no. 2 greatest number of problems cor- rectly by 08/01/2.17. 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 39

Information for Virginia’s K-5 TeachersComputer Science Fundamentals (Grades K-5)Code.org offers an elementary school curriculum that allows even the youngest students to explore the limit-less world of computing. Courses blend online, self-guided and self-paced tutorials with “unplugged” activitiesthat require no computer at all. One-day professional development workshops prepare teachers to start teach-ing each of our courses. Each course consists of 20 lessons that can be incorporated into any classroom at anypace. Even kindergarten-aged pre-readers can participate. Visit https://code.org/professional-development-workshops to learn moreAdditional Resources for SOL Test PreparationStandards of Learning Testing resources are available to ensure that students are familiar with the testing envi-ronment and online tools. TestNav 8 has some features with which teachers should become familiar and withwhich students may need practice. The table below provides a list of some of the testing resources available,how to access them, and the intended purpose.Resource Location/Access PurposeElementary School Access to a training center test is via a Available Training Center Tests allowMathematics CAT TrainingTest Student Authorization Ticket created for student practice with:Middle in the Training Center. • signing in using a student test ticket;School MathematicsCATTraining Test The Training Center is located at: • selecting answer choices; www.pearsonaccess.com/ • using the online tools; Select: Virginia in the dropdown • practicing the procedures for exiting menu a test; and • practicing with the CAT format. Select: the Training Center Tab The Training Center test should not be used to review the SOL test content. Grant OpportunitiesThe VCTM Continuing Education Grant program is a grant opportunity of up to $1,000 for a current VCTMmember who is a current K-16 mathematics educator in Virginia. The purpose of the program is to providefunding support to a VCTM member who wishes to continue his/her education in mathematics or mathematicseducation. The funding for this grant can be used for tuition at an institution of higher learning, i.e. a two-yearor four-year institution. Deadline December 1st.For grants through National Council of Teachers of Mathematics, please visit: http://www.nctm.org/Grants/Virginia Mathematics Teacher vol. 43, no. 2 40

PISA UpdatesFigure 1. Student performance on the Mathematics PISA test, retrieved from OECD (2017), Mathematics per-formance (PISA) (indicator). doi: 10.1787/04711c74-en (Accessed on 28 March 2017) PISA stands for Program for International Student Assessment. It is an assessment designed for 15-yearold students around the world, that aims to measure how well their education has prepared them for the post-secondary world. The assessment, which is managed by the OECD, in partnership with national centers andleading experts from around the world, is conducted in over 70 countries and economies every three years. Itcovers mathematics, science, and reading. The PISA mathematics assessment does not measure a set country’scurriculum, instead, it is designed to evaluate students’ ability to apply their knowledge and skills to a realworld context. Students also complete a background questionnaire, describing their homes, schools, and class-room experiences. This information allows PISA evaluators to understand what factors may influence stu-dents’ achievement in mathematics. For more information on this international study, please visit http://www.oecd.org/pisa/. Additional Resources and ReferencesOECD (2016), Ten Questions for Mathematics Teachers ... and how PISA can help answer them, PISA, OECD Publishing, Paris, http://dx.doi.or /10.1787/9789264265387-enVirginia Mathematics Teacher vol. 43, no. 2 41

Polling Data and the 2016 Presidential Election Carrie Case and Jean Mistele To understand the polling data that was critically assess social, economic, or politicalreported prior to the 2016 Presidential elections, itis useful to be statistically literate and to be issues in our world. Next, we discuss the statisticalpolitically literate. In particular, a statically literateperson who understands confidence intervals can knowledge needed to better understand and makemake sense of the polling data. Likewise, apolitically literate person who understands the sense of the reported polling data.election process used in the United States,Electoral College, is better able to interpret the Statistically, we need to understand how tostatistical data situated within the political context.In this article we address these two forms of interpret the polling results so we can make senseliteracy and when they are used in concert, theseliteracy skills allow people to understand and make of them. Specifically, understanding confidencesense of the polling data and the results from thePresidential elections last fall. In the following intervals associated with the polling numbers shedsparagraphs we first address statistical literacyfollowed by political literacy. We close with the light on the projected presidential race outcomes.way these two forms of literacy serve an engagedcitizen. The chart below shows the results as reported by Statistical literacy is a required skill to Jennifer Agiesta in the CNN/ORC obtained frommake sense of our data rich world. KatherineWallman, a former Chief of Statistical Policy in the the Opinion Research Corporation that polledUnited States Office of Management and Budgetduring the Clinton Administration offers her voters on October 25, 2016.perspective on statistical literacy: Clinton 49% Statistical literacy is the ability to understand and critically evaluate statistical Trump 44% results that permeate our daily lives-coupled with the ability to appreciate the contributions With a margin of error, E, of 3.5%. that statistical thinking can make in public and private, professional and personal decisions In this discussion, we contend that valid (Wallman, 1993, p.1). polling organizations understand statistics and Gal (2002) describes statistical literacy ashaving two interconnected parts: knowledge and would implement sound sampling techniques tothe ability to communicate that knowledge.Shaughnessy (2007) also describes statistical ensure the people polled were selected at randomliteracy as two interrelated components, the first asa learner of statistics and the second as a consumer and that each person was asked the same question;of statistics. All three perspectives share the notionthat a person needs to know statistics on a level questions such as, “Are you a likely voter?” and ifthat allows that person to use that knowledge to the person answered “yes,” then the pollster mayVirginia Mathematics Teacher vol. 43, no. 2 ask, “Which candidate will most likely receive your vote?” In addition, we maintain that polling organizations would use large samples. A large sample would include 1000 to 3500 people from the entire population of likely voters, which is approximately 120 million to 130 million people. The polling data shows the results in proportions— the percentage of people favoring each candidate. Specifically, the polling organizations attempt to calculate the range of the true proportion of voters that will vote for a particular candidate. This range is called a confidence interval. This confidence interval estimates the proportion, p, of voters who indicate they will vote for that candidate, which includes a margin of error, E. The margin of error is the amount of variation calculated from all of the responses in the sample. This is the percentage above the proportion p and the percentage below 42

the proportion p. This means that the actual interval that the true proportion of the popular vote for eachfor the proportion, p, of people expected to vote for candidate would lie within their confidence intervaleach candidate most likely lies within this range. almost all of the time. What is unknown is the trueThe confidence interval also has a confidence level proportion of voters within these intervals. Everyassociated with it. The confidence level is the poll the first author examined in the month leadingprobability stated as a percentage that the up to the election showed Clinton leading the race.confidence interval contains the population Yet, understanding the statistics behind the pollingparameter when the sampling is repeated a very numbers suggests that neither candidate is clearlylarge number of times. Typically, pollsters use a ahead.95% confidence level. This means, 95 out of 100times the confidence intervals calculated from 100 Other polling organizations perform a meta-samples would contain the true proportion, p. analysis to identify the proportion of popular vote for the candidates. A meta-analysis combines data Next, these polling results are examined from multiple polling sources. The reason tomore closely to show how the confidence interval combine data is to reduce the margin of error,sheds light on the meaning of the numbers. The while maintaining the same level of confidence,polling results show that 49% of the people polled 95%. The reduced margin of error creates aclaimed they would vote for Clinton with a 3.5% narrower confidence interval, which generatesmargin of error. This means the range of the results with greater precision. For example, theproportion of voters who claim they will vote for organization, Real Clear Politics, performed a metaClinton is from 45.5% to 52.5 %. This suggests that -analysis to determine the proportion of voters forClinton would receive between 45.5% and 52.5% each candidate. The margin of error, E, is smaller,of the popular vote. Likewise, the polling results 1.9% compared to the earlier polling organizationshow that 40.5% of the people polled said they that reported a margin of error, E, of 3.5%. Thewould vote for Trump with a 3.5% margin of error. Real Clear Politics organization polling results areThis means, that it is expected that Trump would shown below.receive from 40.5% to 47.5% of the popular vote.Comparing these two intervals on a number line, Clinton:we observe an overlap (see Figure 1). The overlapis approximately 1/3 of each candidates’ pEconfidence interval. 48.2% 1.9% The confidence intervals state that if thesampling were repeated 100 times we would expect 46.3% to 50.1% Figure 1. Comparing two intervals with an overlap 43 Figure 2. Comparing two intervals with a larger overlapVirginia Mathematics Teacher vol. 43, no. 2

Trump: cause confusion about the recent polling results among the voting age population for the 2016 pE Presidential election cycle. 46.3% 1.9% 1. Clinton had an overall lead over Trump claiming that her gains over 44.4% to 48.2% the past day or two helped her, she was unable to move far enough Again, confidence intervals were calculated ahead to remove the overlapfor each of the candidates. The polling results show between their two intervals.that 48.2% of the people polled claimed they would 2. The number of undecided andvote for Clinton with a margin of error of 1.9%. third-party voters is much higherThis means that it is expected that Clinton would than in more recent elections.receive from 46.3% to 50.1 % of the popular vote. 3. Clinton’s coalition reliesLikewise, the polling results show that 44.4% of increasingly on college-educatedthe people polled claimed they would vote for whites and Hispanics, which isTrump with a margin of error of 1.9%. This means, somewhat inefficiently configuredthat it is expected that Trump would receive for the Electoral College, becausebetween 44.4% and 48.2% of the popular vote. these voters are less likely to live inComparing these two intervals, we again observe swing states. If the popular votean overlap (see Figure 2). This time, the overlap is turns out to be a few percentagea larger even though the intervals are smaller. points closer than the polls projectedAlmost half of each candidate’s confidence interval then, Clinton will be an Electoraloverlaps with each other. College underdog. The polling data does not reflect the actual Evaluating the polling data statistically, election process. The United States voting systemusing confidence intervals, suggests there is no uses the Electoral College process to select theclear leader emerging from the polling data. winner. The pollsters use raw numbers and percentages to project the winners, which is Many times when news organizations, consistent with the plurality voting system, whichpolitical parties, polling organizations, or other is, the person with the most votes wins. Theoutlets report their polling results, they exclude an Electoral College system uses electors. Theimportant aspect of the election process, the electors are proportionate to the population forElectoral College. Statistical literacy is crucial to each state. In addition, all of the states, except forunderstand the polling results, but being political Maine and Nebraska, have a winner takes allliterate is also necessary. policy, which means, the person with the most popular votes captures all of the electors from that Political literacy, in the traditional sense, state. Maine and Nebraska have a proportion policyrefers to an educational goal that is, preparing in which each candidate captures a proportion ofstudents to become knowledgeable and engaged the electors that aligns with the proportion of votescitizens (Cassel & Lo, 1997). Denver and Hands they received. The number of electors needed to(1990) define political literacy similarly, in which win the election is 270. If the polling organizationspeople have the knowledge and understanding wished to report a more accurate prediction, theyabout the political process and political issues so would need to replicate the Electoral Collegethey can effectively fulfil their roles as engaged voting structure using the polling data. This wouldcitizens. We draw on the definition supplied by require polling each state separately, assign theDenver and Hands (1990), in which people need to electoral votes to the candidates and sum up theknow and understand the political process, in this electoral votes for each candidate. Prior to thecase, the political election process. election, there were some swing states that conducted their own state level polls that is It appears that some organizations play onpeople’s weak political literacy skills in an attempt 44to provoke confusion by expounding on extraneousissues surrounding the polling data. For example,an article published by Silver (2016, November 8)in Fivethirtyeight.com cites several reasons thatVirginia Mathematics Teacher vol. 43, no. 2

consistent with the Electoral College process. The pennsylvania-polls/results of these polls also showed overlapping Cassel, C.A. & Lo, C.C. (1997). Theories ofconfidence intervals for Clinton and Trump(Agiesta, 2016, September 26). Unfortunately, Political Literacy, Political Behavior, 19(4).these few states do not offer adequate information Denver, D. & Hands, G. (1990). Doesto accurately estimate the will of the people. To studying politics make a difference? Thedate, polling results do not use the Electoral political knowledge, attitudes, andCollege voting process used in the United States. perceptions of school students, BritishCurrently, they mimic a plurality voting system. Journal of Political Science, 20, 263-288 Gal, I. (2002). Adult’s statistical literacy: In conclusion, the polling data can be Meanings, components, responsibilities.misleading and frustrating. If a person is International Statistical Review, 70(1), 1-statistically literate and politically literate, then he 51.or she have the skills to make sense of the polling Real Clear Politics, (2016, November). Generalresults. In the recent 2016 presidential election, the Election: Trump vs. Clinton vs. Johnson vs.statistically literate person would recognize that the Stein. Retrieved from http://overlapping confidence intervals, based on the www.realclearpolitics.com/epolls/2016/polling data, indicate there was no clear leader. president/us/This same politically literate person would realize general_election_trump_vs_clinton_vs_johthat the polling data at best, is an estimate of a nson_vs_stein-5952.htmlpotential winner because the Electoral College Shaughnessy, J. M. (2007). Research on statisticsprocess is not embedded into the results. In the learning and reasoning, In Frank K. Lester,end, statistical literacy joined with political Jr. (Ed.). Second Handbook on Research onliteracy, allows the citizenry to make sense of the Mathematics Teaching and Learning, (pp.data reported from the polling organizations and 957-1009). Reston, VA: National Councilunderstand that these results they are only of Teachers of Mathematics.estimates at best. Silver, N. (2016, November, 8). Five Thirty Eight. Final Election Update: There’s A Wide References Range of Outcomes, and Most of ThemAgiesta, J. (2016, October 25). Clinton leads by 5 Come up Clinton. Retrieved from http:// fivethirtyeight.com/features/final-election heading into final two weeks. Retrieved -update-theres-a-wide-range-of-outcomes from http://www.cnn.com/2016/10/24/ -and-most-of-them-come-up-clinton/ politics/hillary-clinton-donald-trump- Wallman, K. K. (1993). Enhancing statistical presidential-polls/ literacy: Enriching our society. Journal ofAgiesta, J. (2016, September 26). CNN/ORC polls: the American Statistical Association, 88 Trump, Clinton deadlocked in Colorado, (421), 1-8. Pennsylvania. Retrieved from http:// www.cnn.com/2016/09/26/politics/donald- trump-hillary-clinton-colorado-Carrie Case Jean MisteleFaculty Associate ProfessorRadford University Radford [email protected] [email protected] Mathematics Teacher vol. 43, no. 2 45

MathGirlsNote from the Editor: This article is an opinion piece shared by a fellow mathematics teacher. It has not gonethrough the peer review process. If your are interested in contributing an article related to femaleempowerment in STEM, please respond to the call for manuscripts featured on page 36. The Threat of Stereotypes on Female Participation in STEM Vanessa Vakharia The concern as to why women are not en- a deeper examination suggests that this does nottering STEM fields at a rate proportional to their simply happen by chance, but is instead tied to themale counterparts has been steadily growing over gendered self-concept of certain girls and boys –the past few years, and that growth has been namely those, affected most by an environmentmarked by a seemingly never-ending series of the- which reinforces gendered stereotypes relating toories as to why this is the case. From a lack of fe- intelligence and mathematical ability.male role models to the masculinization of STEM What is stereotype threat and how does it playrelated job prospects to the near-viral theory of out?women’s inevitable emotionally-driven tears (bigthanks to Tim Hunt and Milo Yiannopoulos for this Stereotype Threat is essentially the situa-one), there is no shortage of reasons for why wom- tional predicament in which people feel themselvesen choose to enter any field other than STEM. All to be at risk of conforming to stereotypes aboutof these theories are important and all add a kalei- their social group (Steele, 1997). Stereotype Threatdoscopic fragment to the ever changing lens with is subtle, in that all that needs to be present is thewhich we do and must continue to view the prob- threat of a stereotype regarding a specific group, inlem. In this piece I would like to talk about one order for that group to become anxious about theirwhich I see enacted by my teenage students performance, thus hindering their ability to per-through my work every single day: it’s pervasive, form at their optimal level. In this context, theit’s real, and it’s called “Stereotype Threat.” group in question would be ‘female’ and the asso- ciated stereotype would be that girls are generally Rewind to one of my first teaching assign- bad at math. In one study, two groups of childrenments ever: I was teaching in a Grade 10 math aged 7-8 were given an identical math test. Prior toclassroom and noticed a very strange phenomenon the test, the groups were asked to color a picture,taking place. There was a group of girls in my class one group given pictures containing stereotypicalwho scored almost perfectly on every evaluation, gendered imagery (a girl holding a doll, a boyyet claimed definitively to be ‘bad at math.’ Con- holding a ball), while the second group was given aversely, there was a group of boys in that very neutral image to color (a landscape). Both groupssame classroom, who despite nearly failing ever were then given identical math tests and as ex-evaluation, seemed to suffer no corresponding loss pected, girls whose gender identity had been madein confidence when it came to their innate mathe- salient (by focusing on images depicting stereo-matical ability. While upon first glance this type of typically gendered roles) underperformed com-behavior might be reminiscent of the plethora of pared to the girls in the group whose gender identi-evidence that generically suggests that girls simply ty had not been activated. Conversely, boys whosehave lower self-confidence than boys when it gender identity had been made salient performedcomes to math ability (Brush, 1980; Erlick & better than the boys in the group whose identityLeBold, 1975; Fennema & Sherman, 1976; Kamin- had not been activated (Neuville et al, 2007).iski, Erickson, Ross & Bradfield, 1976; Robitaille,1977; Sherman, 1980, as cited in Hollinger, 1985 ), Other studies have confirmed the finding that gender identity activation leads to reducedVirginia Mathematics Teacher vol. 43, no. 2 46

mathematical performance for women, and consid- grade because she is smart and has a firm grasp ofering that we live in a world where gender stereo- the material.types are constantly reinforced at every turn, it’s nowonder that women not only experience more anx- What’s interesting is that over years ofiety when it comes to mathematical evaluations, speaking to students from different schools andbut that even when they do perform well, that they demographics about their believes in mathematicaldo not attribute their success to innate ability but ability, there have been a few interesting loopholesrather to effort. that have stood out to me as places where we canAn Imposter Among us take action in order to flip the switch. Many of my students from single-sex schools have balked at the The idea of girls performing well yet not idea that stereotypes would ever get in the way oftaking accountability for results is called Imposter their pursuit of a STEM field, in fact claiming thatSyndrome (Spencer et al, 1999). Characterized by a societal stereotypes motivate them to prove thesedeep sense of not-belonging despite proven ability, stereotypes wrong! The culture of their schools isI think that this is what’s really at the heart of the such that these students are constantly being toldgender gap when it comes to STEM. In a sense, that girls can do absolutely anything, that stereo-Stereotype Threat may prevent some women from types that tell them otherwise are false. Going backever performing well on task, but Imposter Syn- to the idea that Stereotype Threat occurs in spacesdrome is what prevents those women who have where the presence of negative stereotypes is mostmanaged to beat Stereotype Threat from ever reap- prevalent, this explains why many girls that go toing their hard earned reward – it’s what prevents these types of schools are not affected by Stereo-women from actually entering the field despite type Threat or Imposter Syndrome. Similarly,how hard they may have worked in order to be able many of my students who are simply not interestedto do so. Achievement has frequently been corre- in mainstream popular media also exhibit the samelated to mathematical self-concept and those who attitudes and performance as those girls attendingbelieve in this correlation argue that there is a re- single-sex schools. I think that this is an interestingciprocal relationship between success/identification point as it showcases the depth and breadth of theand failure/dis-identification. Thus, if a student problem: this is a societal thing, something hugeconsistently succeeds at mathematics, he/she that extends across mass media, that reaches mil-should begin to incorporate math into their self- lions and millions of people. This isn’t somethingconcept. One might presume that high achievement relegated to our classroom walls, and in fact, thein a field of study such as math would lead one to work we do as teachers in order to fight it is met byidentify as a “learner of math,” making math more billion dollar corporations who are trying to tell ourcongruent with one’s self. However, it turns out kids otherwise in order to keep them in their neatthat even when girls achieve and are successful, and tidy gendered boxes. So what can we do asthey still do not incorporate “mathematician” into teachers in order to change the landscape once andtheir identities (Ma and Kishor’s (1997), as cited in for all?Wang, 2006, p.702). Essentially, Imposter Syn- Flipping The Switchdrome leads girls to believe that even if theyachieve high grades in their math classes, that they As teachers, we have many roles to play outsimply just ‘frauds’ waiting to be exposed. I teach in the classroom and I think that it’s especially im-many girls who feel this way, who despite achiev- portant to note that we are the ones with direct ac-ing some of the highest math marks in their classes cess to students, and in turn the ones with a greatnonetheless feel that they had to work harder than degree of ability to change the equation. While of-everyone else, rendering their success illegitimate. ten we get discouraged by evidence that shows usSimilarly, girls will attribute their high grades to that on such a large scale it appears that the perva-their evaluations being too easy, or their teachers siveness and power of these stereotypes trumps ourbeing ‘nice’ to them. Rarely do I see a female stu- work in the classroom, I strongly believe that thedent who proudly boasts that yes, she scored a high opposite is true! Knowing how gender identity af- fects our students empowers us to work on theVirginia Mathematics Teacher vol. 43, no. 2 ground level to educate and inform them, which 47

disempowers these stereotypes. By teaching our Neuville, E. & Croizet, JC. Eur J (2007). Psycholstudents how Stereotype Threat and Imposter Syn- Educ 22, 307.drome work, while simultaneously reinforcing thefact that essentializing stereotypes are not in any Spencer, S. J., Steel, C. M., Quinn, D. (1999).way absolute truths, we have the power to change “Stereotype Threat and Women's Maththe way in which girls approach math and perceive Performance.” Journal of Experimentalthemselves as potential mathematicians. We have Social Psychology. 35, 4 - See more at:the power to re-educate our classrooms as to the http:// www.scientistafoundation.com/harmful effects of stereotyping and the means by imposture-efwhich stereotypes continue to live on with validity. fect.html#sthash.brmHTRKq.dpufUltimately, we have the power to strip stereotypesof their power, by altering our societal belief sys- Steele, Claude M. (1997).\"A threat in the air:tem, one believer at a time. How stereotypes shape intellectual iden tity and perfor mance\". American Psy References chologist. 52 (6), 613–629.Hollinger, C. L. (1985). Self-perceptions of abil Wang, J. (2006). An empirical study of gender dif ity of mathematically talented female ference in the relationship between self- adolescents. Psychology of Women concept and mathematics achievement in a Quarterly, 9 (3), 323. cross-cultural context. Educational Psy chology, 26 (5), 689.Busting Block Busters! Many Hollywood movies have scenes that When accused of having a stray rocket cause seem mathematically inaccurate if not a forest fire, October Sky’s Homer decides to use some math to prove his innocence. Theimpossible. Are these scenes truly impossible, or are they more plausible than they seem? only question is, did he succeed? Are his The goal of the contest is to provide the best calculations accurate, or did he leave some mathematical explanations for the following scene. The solutions that best explain a things up in the air? scene’s possibility or impossibility and the different elements that help form this will https://www.youtube.com/watch?v=udHB3tftPz4 receive an award and the winner will be featured in the Fall issue. Answers may be This issue of Busting Block Busters is submitted by July 31, 2017 to contributed by Dr. Neil Sigmon, [email protected] with the subject line: Busting Blockbusters entry. Department of Math and Statistics, Radford UniversityVirginia Mathematics Teacher vol. 43, no. 2 48

Exploring the Solving of Algebraic Equations Through Mental Algebra Jérôme Proulx For numerous years, mathematics teacher conditional equality, where it is not only the idea ofeducators have attempted to find ways to enrich finding the answers/values that make the equationtheir future mathematics teachers’ understandings true, but also the fact that the equality can be trueof mathematics. This has often been done with the or untrue.intention of deepening and making future teachersmore mathematically flexible, particularly in view When future teachers were givenof their future interactions with students’ own 5x+6+4x+3=–1+9x to solve, some rapidly assertedunderstandings of mathematics. Because doing that there was no solution, because one can rapidlymental mathematics is well recognized as an see 9x on both sides of the equation as well as theoccasion for promoting meaning making and for fact that the remaining numbers on each sides doenlarging one’s repertoire of ways of solving (see not equate. It thus led to the conclusion that theree.g., Reys & Nohda, 1994; Schoen & Zweng, was no number that could satisfy this given1986), it appears as an interesting approach to equation, since no x, whatever it could be, couldexplore for attaining this precise goal with future succeed in making different numbers equal. Thismathematics teachers. I therefore report in this strategy is related to what is often termed “globalshort article on a study undertook with future reading” of the equation (Bednarz & Janvier,secondary mathematics teachers, where we asked 1992), that requires consideration of the equationthem to solve usual algebraic equations of the form as a whole prior to entering in algebraicAx+B=C, Ax+B=Cx+D, Ax/B=C/D, Ax2+Bx+C=0 manipulations, or what Arcavi (1994) calls a prioriwithout paper or pencil or any other material aids inspection of symbols, which is a sensitivity toin a restricted period of time (about 15 seconds). analyze algebraic expressions before making aThe activity was organized in the following way: decision about their solution1.(1) an equation was offered in writing to the group;(2) future teachers solved the equation mentally; Another strategy future teachers engaged in(3) at the teacher educator’s signal they wrote their was one of “solving followed by validation”. Whenanswer on a piece of paper; (4) answers and having to solve x2–4=5, one future teacher rapidlystrategies were orally shared with the group; (5) the transformed it into x2=9, obtaining 3 as an answer.cycle restarted for another equation. However, because he knows being in a mental mathematics context and is aware that his answers As the activity unfolded, diverse strategies in this context are often rapidly enunciated and canwere shared for solving algebraic equations. lack precision, he decided to verify his answer. ByThrough those strategies emerged an interesting mentally verifying if (3)2=9, he realized that (–3)2variety of meanings (implicitly or not) about what also gives 9 and then readjusted his solution. Thissolving an algebraic equation can represent. In manner of solving the equation gets close to thewhat follows, I outline this variety of strategies and idea not only of finding one value that makes themeanings, in order to illustrate how doing mental equation true, but also of finding all values thatalgebra can represent an occasion for the make it true.enrichment of future teacher’ mathematical Meaning 2: Solving an algebraic equation is …experiences. deconstructing the operations applied to anMeaning 1: Solving an algebraic equation is … unknown numberfinding the value(s) that satisfy, make true, theequality This meaning requires reading the equation Underneath this meaning is the notion of a 1Arcavi gives the example of (2x+3)/(4x+6)=2, which has no solution because whatever the value of x, the numerator is worthVirginia Mathematics Teacher vol. 43, no. 2 half the denominator, making futile undergoing additional steps. 49

as a series of operations applied to a number (here graph (Figure 1) offers an illustration of what wasx) and attempting to undo these operations to find done, mentally, by the future teacher.that number. Solving an algebraic equation in this case is When having to solve equations like x2– not about finding the values that make the equation4=5, future teachers would say: “My number was true, but about finding the x that satisfies bothsquared and then 4 was taken away, thus I need to equations for the same y, about finding the xadd 4 and take the square root”. Or, for 4x+2=10, coordinate that, for the same y, is part of each“What is my number which after having multiplied function.by 4 and added 2 to it gives me 10?” These are Meaning 5: Solving an algebraic equation is …similar to inverse methods of solving found in finding the values that nullifies the equationFilloy and Rojano (1989) or of Nathan andKoedinger’s (2000) “unwinding” approach, where This meaning focuses on the equal sign asoperations are arithmetically “undone” to arrive at giving an answer (see e.g. Davis, 1975), but wherea value for x. As Filloy and Rojano explain, when operations are conducted so that all theusing this method “it is not necessary to operate on “information” ends up being on one side of theor with the unknown” (p. 20), as it becomes a equation in order to obtain 0 on the other side. Theseries of arithmetical operations performed on intention then becomes to find the value of x thatnumbers. In this particular case, solving the nullifies that equation, that is, that makes it equal toalgebraic equation is focused on finding a way to 0.arrive at isolating x, in an arithmetic context.Meaning 3: Solving an algebraic equation is … One example of a strategy engaged in wasoperating identically on both sides to find x again related to seeing the equation in a functional view, but finding the values of x that give a null y- This meaning focuses on the idea that is value, or what is commonly called finding theoften called “the balance” principle, where one zeros of the function where the function intersectsoperates identically on both sides of the equation to the x-axis when y=0. For x2–4=5, transformed in x2maintain the equality and obtain “x=something”. –9=0, the future teacher aimed mentally at solvingFor example, when solving 2x+3=5, students (x+3)(x–3)=0, leading to ±3. The quest was mainlywould subtract 3 on each side and then divide by 2. finding the values that nullify the function y=x2–9,Meaning 4: Solving an algebraic equation is … which gave point(s) for which the image of thefinding points of intersection of a system of function was zero. Another way of doing it, less inequations a function-orientation, is to use “binomial expansion” for seeing that for the product to be null This one is about seeing each side of the it requires that one of the two factors be null. Thisequality as representing two functions, and thus said, one needs to use neither a function norattempting to solve them as a system of equations binomial expansion to find what nullifies theto find intersecting points, if any. For example, equation. For example, if x+4=3 is transformed inwhen solving x2–4=5, some future teachers x+1=0, one finds that –1 is what makes the left sideattempted to depict the equation as the comparison of the equation equal to 0.of two equations in a system of equations (y=x2–4 Meaning 6: Solving an algebraic equation is …and y=5) and finding the intersecting point of those finding the missing value in a proportiontwo equations in the graph. To do so, one futureteacher represented the line y=5 in the graph and This meaning was engaged with forthen also positioned y=x2–4. The latter was referred equation written in fractional form (e.g. Ax/B=C orto the quadratic function y=x2, which crosses y=5 at Ax/B=C/D). In these cases, the equation was conceived as a proportion, where the ratio between x = 5 . In the case of y=x2–4, the function is numerators and denominators was seen as the sametranslated of 4 downwards in the graph, and then or consistent. The equality here is not seen asthe 5 of the line y=5 becomes a 9 in terms of conditional but is taken for granted as true, leadingdistances. Hence, how does one obtain an image of at conserving the ratio between numerator and9 with the function y=x2? With an x=±3, where the denominator in the proportion.function y=x2–4 cuts the line y=5. The following 50Virginia Mathematics Teacher vol. 43, no. 2

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