Virginia Mathematics Teacher Spring 2017

Virginia Council of Teachers of Mathematics |www. vctm.org V IRGINIA M ATHEMATICS T EACHER Vol. 43, No. 2 Spring 2017

Commun i ca t i ng in t he Un i ver sa l Language of Ma t h!

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Editorial Staff

Dr. Agida Manizade Editor-in-Chief vmt@radford.edu Radford University vmt@radford.edu

Brian Pratt Assistant Editor Mr. Brian Pratt

Dr. Jean Mistele Associate Editor Radford University jmistele@radford.edu

Mr. Liam Downey Assistant Editor Radford University

Radford University

Printed by Wordsprint Blacksburg, 2200 Kraft Drive, Suite 2050 Blacksburg, Virginia 24060

Virginia Council of Teachers in Mathematics

Many thanks to our Reviewers for Spring 2017

President: Jamey Lovin, Virginia Beach Public Schools

Dr. Robert Berry

Past President: Cathy Shelton, Fairfax County Public Schools

Dr. Carrie Case, Radford University

Secretary: Lisa Hall, Henrico County Public Schools

Dr. Anthony Dove, Radford University

Membership Chair: Ruth Harbin-Miles

Dr. Kateri Thunder, James Madison University

Treasurer: Virginia Lewis, Longwood University

Dr. Laura Jacobsen, Radford University

Webmaster: Ian Shenk, Hanover County Public Schools

Kelly Robinson

NCTM Representative: Betsy Steadman, Hanover County Public Schools

Dr. Wendy Hageman-Smith, Longwood University

Elementary Representatives: Meghann Cope, Bedford County Public Schools; Eric Vicki Bohidar, Hanover County Public Schools Middle School Representatives: Melanie Pruett, Chesterfield County Public Schools; Skip Tyler, Henrico County Public Schools

Dr. Ann Howard Wallace, James Madison University

Mrs. Anita Lockett, Fairfax County Public Schools

Secondary Representatives: Pat Gabriel; Samantha Martin, Powhatan Public Schools

Joyce Xu, Virginia Tech

Math Specialist Representative: Spencer Jamieson, Fairfax County Public Schools

Dr. Matthew Reames, University of Virginia

2 Year College: Joe Joyner, Tidewater Community College

Dr. Kenny Wantz, Regent University

4 Year College: Ann Wallace, James Madison University; Robert Berry, University of Virginia

Karen Zwanch, Virginia Tech

VMT Editor: Dr. Agida Manizade

Dr. Jay Wilkins, Virginia Tech

This issue had a 20% acceptance rate

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Table of Contents:

Note from the Editor..................................................5

HEXA Challenge.....................................................38

Message from the President .....................................6

Information for Virginia’s K-5 Teachers................40

Note from the VDOE.................................................7

Grant Opportunities................................................40

The Role of Skip Counting and Figurative Reasoning................................................8 Is the Midpoint Quadrilateral Really a Parallelogram?........................................................15 Algebra Achievement of Urban High School Students...................................18

PISA Updates...........................................................41

Polling Data and the 2016 Presidential Election ...42

Math Girls ...................................................................46

Busting Block Busters!............................................48

Exploring the Solving of Algebraic Equations Through Mental Algebra ........................................49

Solutions to Fall HEXA Challenge Problems ........23

Recursion in Secondary Mathematics Classrooms.53

Upcoming Math Competitions.................................27

Summer 2016 STEM Camps....................................56

Concrete, Representational, and Abstract: Building Fluency from Conceptual Understanding...............28

Math Jokes...............................................................56

The Puzzlemaker .....................................................59

Technology Review..................................................32

Conferences of Interest ...........................................60

Good Reads.............................................................34

Key to the Fall 2016 Puzzlemaker Problem............35

Call for Manuscripts................................................36

Educational Opportunities for Virginia Mathematics Teachers.............................................37

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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 they will need to be able to comprehend, analyze, and communicate the patterns they encounter. The theme of this issue is Communicating in the Universal Language of Math, and we asked authors to share their ideas about doing mathematics, and communicating in this universal language with their students. We enjoyed putting this issue together, and hope you find inspirational and useful ideas for your K-12 classroom. We invite you to share your thoughts and practices by submitting practitioner-oriented articles that help to inform other mathematics teachers. We also encourage you to participated in the HEXA Challenge and Busting Blockbusters, and to challenge your students to do the same.

goals is to engage students in doing mathematics. But how often do we stop to think about, “what is mathematics?” and “what does it mean to do mathematics?”. In the regular k-12 curriculum, do we engage in doing mathematics? Does adding and subtracting numbers or memorizing a set of formulas qualify as doing mathematics? Does a student do mathematics when using ‘technology as a master’ where he or she takes for granted any output generated by technology, without engaging in mathematical thinking or without evaluating the quality of the the outcome? If we were to ask our students what mathematics is, and what it meant to do mathematics, what would they say? One way to think about mathematics is as a language to describe and communicate patterns we encounter in our lives. Some of the patterns we see in nature and human behavior are simple, while others seem chaotic. From simple Putnam squares and golden ratios, to predicting climate patterns or spread of disease, the more mathematics we learn, the more we are able to observe, describe, and predict the patterns around us. My goal as a mathematics teacher is to create classroom environments in which students think critically about these patterns, ask and answer complex questions, and communicate their ideas with their peers. Our students will be operating in a

Agida Manizade Editor in Chief, Virginia Mathematics Teacher vmt@radford.edu

Congratulations to the 2017 Winners of the William C. Lowry Mathematics Educator Award:

Middle School Awardee: Matthew Reames, Burgundy Farm High School Awardee: Jillian Marballie, Mongomery County

College Awardee: Andrew Wynn, Virginia State University Math Specialists Awardee: Tasha Fitzgerald, Culpeper Schools

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Message from the President Jamey Lovin

Now is the time for welcoming spring – a time of transitioning from the cold winter to the warm summer. Virginia

make connections between grade levels and standards more apparent. The second goal was to provide more support for teachers by including in the Understanding the Standard section of the Curriculum Framework, more explanations, examples, and definitions of vocabulary words. While there, look for professional development materials based on the 2016 SOL Mathematics Institutes! These institutes, framed by the five process goals, focused on instruction to support the new standards. I know you will find them helpful as you seek to implement the standards in your classrooms. VCTM is excited to collaborate with you in welcoming a high quality, challenging mathematics program for our students! As always, please feel free to contact myself, or any board member, with ideas on how we can better serve you.

mathematics educators are also in a state of transition as we join district leaders to incorporate the newly revised Standards of Learning into local curriculum documents and develop instructional materials. If you have not had a chance to visit the Department of Education website, try to do so soon! There you will find timely information from Tina Mazzacane, Mathematics and Science Specialist from the Office of Science, Technology, Engineering and Mathematics, about the focus of the new changes. In her webcast, Tina articulates the two-pronged rationale behind the 2016 revisions. First, there was a concerted effort to improve the vertical progression of content and

Jamey Lovin, VCTM President Jamey.Lovin@vbschools.com

Organization Membership Information 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 Members

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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 of Learning and Curriculum Frameworks were approved by the Board of Education in September, 2016. The content of the mathematics standards is intended to support the following five process goals for students: becoming mathematical problem solvers, communicating mathematically, reasoning mathematically, making mathematical connections, and using mathematical representations to model and interpret practical situations. The theme of the current edition of the Virginia Mathematics Teacher Journal is “Communicating in the Universal Language of Mathematics.” The expectation for mathematical communication included in the 2016 Virginia Mathematics Standards of Learning and Curriculum Framework is shown: Mathematical Communication Students will communicate thinking and reasoning using the language of mathematics, including specialized vocabulary and symbolic notation, to express mathematical ideas with precision. Representing, discussing, justifying, conjecturing, reading, writing, presenting, and listening to mathematics will help students to clarify their thinking and deepen their understanding of the mathematics being studied. Mathemati- cal communication becomes visible where learning involves participation in mathematical discussions. VDOE Mathematics Standards of Learning , 2016 Communication in the mathematics classroom is essential for students to develop a deep understanding of mathematical content and to be able to justify and reason mathematically. In the 2016 curriculum framework documents, revisions have been made to improve precision and consistency in mathematical language and format. The implementation of the newly adopted 2016 Mathematics Standards of Learning will require teachers, specialists, and administrators in public school divisions to analyze, discuss, and

understand the changes. Virginia mathematics educators are encouraged to take time to unpack these new standards and determine the curricular and instructional modifications needed to ensure a smooth transition from the current 2009 standards to the 2016 standards. The Mathematics Team at the Virginia Department of Education is focused on providing support to educators as they make this important transition. Additional resources to support implementation of the 2016 Mathematics Standards of Learning can be found on the Virginia Department of Education Mathematics 2016 webpage. 2016 Mathematics Standards of Learning Im- plementation Timeline 2016-2017 School Year – Curriculum Development VDOE staff provides a summary of the revisions to assist school divisions in incorporating the new standards into lo- cal written curricula for inclusion in the taught curricula during the 2017-2018 school year. 2017-2018 School Year – Crossover Year 2009 Mathematics Standards of Learning and 2016 Mathe- matics Standards of Learning are included in the written and taught curricula. Spring 2018 Standards of Learning assess- ments measure the 2009 Mathematics Standards of Learning and include field test items measuring the 2016 Mathematics Standards of Learning. 2018-2019 School Year – Full-Implementation Year Written and taught curricula reflect the 2016 Mathematics Standards of Learning. Standards of Learning assessments measure the 2016 Mathematics Standards of Learning.

Tina Mazzacane Mathematics Coordinator Virginia Department of Education

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The Role of Skip Counting in Children's Reasoning Jesse L. M. Wilkins and Catherine Ulrich Skip Counting and Figurative Material in Children’s Construction of Composite Units and Multiplicative Reasoning An important part of children’s graders. These solutions will be referred to throughout the article to highlight children’s ways of thinking and to make connections to

mathematical development in the elementary and middle school years is the transition from additive to multiplicative thinking. This developmental milestone affords children necessary tools for understanding more advanced mathematical concepts that is limited by additive thinking, such as fractions and proportional reasoning. An important part of this mathematical development is the construction and coordination of units, which does not occur all at once, but through several hierarchical stages (Ulrich, 2015, 2016). Understanding these stages and the nature of children’s ways of thinking about units during each stage is important for teachers as they plan and prepare instructional activities for their students. In this article we discuss the characteristic ways of thinking associated with these stages and instructional opportunities for moving children through these stages. In particular, we discuss the role of skip counting (e.g., counting by 3’s: 3, 6, 9, 12, 15, 18, 21…) as both an indicator of, and as a way to foster, children’s construction and coordination of units throughout these stages of development. We also discuss the placement of skip counting within Virginia’s mathematics Standards of Learning and Curriculum Framework (Virginia Department of Education [VDOE], 2009) and potential extensions and understandings that could be helpful for curriculum development for elementary-aged children. Moreover, we discuss the 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 a multiplicative understanding of 5 × 3 as 5 times as much as 3. To motivate and facilitate our discussion of the ideas outlined above we first discuss several solutions to the task in Figure 1 produced by sixth

You have baked 39 cupcakes and you will put the cupcakes in boxes of three. How many boxes will you fill?

(a)

(b)

(d)

(c)

(e) (f) Figure 1 . Examples of sixth graders’ solutions to the Cupcake Task

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instructional strategies for advancing children’s early number concepts. All of these children produced the same correct answer, however, based on the solutions, these students all thought very differently about the problem. In Figure 1a, the child seems to have first drawn 39 “cupcakes” without attending to groups of 3, then circled groups of 3 to make boxes until they were all used up, and then counted the boxes. In Figure 1b, the child seems to create groups of 3 “cupcakes” in boxes until all 39 cupcakes are used up, and then counts the number of groups (notice the single dot in the boxes likely representing this counting act). The child’s work in Figure 1d does not represent individual cupcakes, but instead uses skip counting by 3’s to 39, and then counts the number of “counts by 3” (notice the dots under each number). Finally, the child in Figure 1f seemed to recognize the situation by reversing the context of the problem to one asking: “What multiplied by 3 would give me 39?” In each of these cases the child’s work represents a different way of thinking about and coordinating units. We will revisit these student solutions after presenting a hierarchy of how students work with units. These different ways of thinking may be characterized in terms of a hierarchy of four stages called number sequences (Steffe & Olive, 2010; Steff & Cobb, 1988; Olive, 2001; Ulrich, 2015, 2016). These different stages describe how children work with units and coordinate them when working with the counting numbers. These four stages are referred to as the initial number sequence (INS), tacitly nested number sequence (TNS), explicitly nested number sequence (ENS) and the generalized number sequence (GNS). In this paper we focus on the first three sequences as they relate to skip counting and describe the extension of thinking required for the GNS which ultimately lays the groundwork for more advanced mathematical understanding. Here we briefly highlight the important characteristics of these stages; for a more detailed discussion, the interested reader should consult Ulrich (2015, 2016) and Olive (2001). The Number Sequences Prior to a child developing an INS, they are considered pre-numerical (Steffe & Olive, 2010; Olive, 2001). That is, for these students, numbers themselves do not represent cardinality, a quality of

a set, but instead are one part of their counting activity. At this point, children may be able to count a set of objects, but they would interpret the question, “How many?” as a request to say a sequence of numbers (e.g., 1, 2, 3, …, 7) while pointing to each object, not as a question about how “big” the set is. Furthermore, after counting a set of objects, if given additional objects and asked how many in all, this child would need to count all of the objects, first recounting the original set. A child who has constructed an INS (the first number sequence) recognizes that a number, such as 7, describes the cardinality of a set of objects and can stand in for counting them, that is, counting “1, 2, 3, …, 7” (Olive, 2001), an initial number sequence. In this case, the 7 is a numerical composite of units representing the result of counting the seven objects, and can serve as a starting point for additional counting. However, the 7 is not recognized as a unit that could be used to count with: “with an INS… [the number words] can only be used to symbolize the results of counting acts; they cannot yet be used as input for counting acts” (Olive, 2001, p. 6). The development of an INS affords a child with the ability to count-on , that is, if after counting a set of 7 objects, a child is given three more objects and asked, “How many altogether?” they would likely count: “7; 8, 9, 10,” while touching the three additional objects, or using their fingers to keep track of the additional three objects; and answer “10,” instead of having to count-all. A child with an INS will often rely on figurative materials to keep track of counting. Consider the child’s solution in Figure 1a. In order to solve this task, the child needs to represent all 39 “cupcakes,” making groups of three until they are all used up, and then count the number of groups. For a child with only an INS their counting acts are limited to using strings of the number sequence beginning at 1 (Ulrich, 2015). Later on, a child begins to recognize that there are subsequences embedded within larger sequences that can be used to aid them in their counting acts. For example, given 7 objects, a child is asked how many there would be if they added on 12 objects. In order to keep track this child might begin at 7 and count as follows: “8 is one more, 9 is two

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more, 10 is three more, 11 is four more, …, 19 is twelve more, the answer is twelve.” This child has a tacit awareness of the subsequence 1 to 12 nested within the larger sequence from 1 to 19. This child has constructed a tacitly nested number sequence (Olive, 2001, Ulrich, 2015; Steffe & Olive, 2010). Double counting, as in the example, is a characteristic action of a child with a TNS. This awareness of the cardinality of the subsequence (1 to 12) irrespective of its location in the larger sequence suggests that the child is now able to work with a composite unit (Ulrich, 2015). Different from a numerical composite, in which the count stands in for the counting sequence 1 to the number, a composite unit is taken to represent the cardinality of the counting acts irrespective of where the subsequence occurs in the larger sequence. Also characteristic of a child with a TNS is the use of skip counting to solve tasks. For example, counting by 3’s, each count represents a composite unit of 3. With this ability, children can begin to answer questions such as, how many threes are in thirty-nine, by keeping track of their skip count: “3; 6; 9; 12, …”. Each of the numbers in the skip count represent a subsequence of cardinality 3, or a composite unit of 3. The child’s use of skip counting in Figure 1d is characteristic of this sort of thinking. The child uses skip counting by 3’s to 39, recognizing that each 3 represents a box of cupcakes (composite unit) and then counts the number of “counts by 3” and indicates “13 boxes” as their answer (notice the dots under each numeral likely representing the child’s count). Compare this to the solution in Figure 1e. Here, too, it is apparent that the child is working with composite units, but they no longer have to refer to their skip count but recognize that they are interested in the number of units and can use the sequence 1 to 13 to keep track of the units. With a TNS, the composite units remain tacit for a child. That is, these units are available to work with during counting activity, but children are not explicitly aware of the units prior to counting; they are reproduced through counting. Once children are able to reflect on units as a given, we say that a child has developed an explicitly nested number sequence (ENS). The defining characteristic of this stage is a child’s ability to construct an iterable unit of one (Ulrich, 2016;

Steffe and Olive, 2010; Olive, 2001). Here a child is able to recognize the activity of “adding one more” to the point that these additions are interchangeable units. That is, number words no longer only represent the result of counting, but represent a multiplicative relationship associated with the number of iterable units: for example, “7” no longer only represents “1, 2, 3, 4, 5, 6, 7,” but instead represents 7 ones, or 7 times as much as one unit. At this point the composite unit does not stand for a subsequence, but stands in for a multiplicative relationship. Olive (2001, p. 7) distinguishes a child with an ENS from a child with a TNS by comparing their activity for solving 1+1+1+1+1. For a child with only a TNS they would potentially need to solve this problem in steps by calculating the nested sums: 1+1 is 2, 2+1 is 3, …, 4+1 is 5. Whereas, for a child with an ENS the sums are taken as given, and they recognize 1+1+1+1+1 as simply 5 ones, and also recognize the reversibility of the relationship, that five ones are the same as one five. Children with an ENS can reflect on multiplicative situations (Ulrich, 2016; Olive, 2001) involving multiple levels of units. For example, combining 4 groups of 7 objects, can be viewed as making a composite unit of composite units. In other words, 4 groups of 7 objects is seen as a numerical composite of 4 composite units, each of which is a composite unit of 7 iterable units of one. Furthermore, a child can view this combination as a composite unit of 28 iterable units. However, a child with only an ENS has to create composite units of composite units—28 as 4 groups of 7 objects—in the moment. The child would have trouble operating on 28 without losing track of the 4 groups of 7. Although we will not elaborate greatly, for completeness, a child who has constructed a generalized nested number sequence (GNS) can work fluently with a composite of composite units because their composite units are now iterable in the same way units of 1 were iterable for students who have an ENS. Skip Counting Skip counting (e.g., counting by 3’s: 3, 6, 9, 12, 15, …) is often introduced to young children as a way to further develop their counting skills and build their knowledge of multiples. Much like they

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begin their early counting as a sing-song, children also learn to skip count through repetition and song. Based on the Virginia Standards of Learning (SoLs; VDOE, 2009) children in Kindergarten are encouraged to count by fives and tens to 100 (see SoL K.4 in Table 1). Continuing in first grade, children are also encouraged to count by 2’s to 100 (see SoL 1.2 in Table 1). In second grade, children continue to skip count by 2’s, 5’s, and 10’s (see SoL 2.4 in Table 1). The Curriculum Framework (VDOE, 2009) highlights the role of skip counting for the general development of numerical patterns, as well as for use in very specific mathematical applications. For example, skip counting by 2’s lays the groundwork for understanding even and odd numbers (e.g., SoL 1.2); counting by 10’s lays the groundwork for place-value and money (e.g., SoLs 1.2 and 2.4); skip counting by fives lays the groundwork for telling time and counting money (SoL K.4). In all cases, skip counting is promoted for its relationship to learning multiplication facts. Beyond 2’s, 5’s, and 10’s, skip counting by other numbers is not explicitly highlighted in the Standards of Learning or Curriculum Framework. After second grade, skip counting is abandoned as an essential part of the SOLs and replaced with a focus on the learning of multiplication facts (SoL 3.5, Table 1). Beyond the specific role that counting by 2’s, 5’s, and 10’s has for developing particular mathematical concepts as prescribed in the VDOE Curriculum Framework, skip counting, in and of itself, can represent important developmental shifts in working with composite units and should be

encouraged for its own sake. Once children can relate their skip counting to their multiplication facts it should not then be assumed that children have developed multiplicative thinking. Skip counting alone does not imply the development of the multiplicative understanding inherent in an ENS or GNS. By encouraging children to use their skip counting for solving tasks involving multiplicative situations instead of relying on number facts, they may develop a more intentional and explicit awareness of the relationship between their skip count and the types of units with which they are working. This awareness affords children with increased opportunities to develop more powerful multiplicative understandings. Skip Counting and Number Sequences Here we discuss the solutions in Figure 1 as a way to exemplify different number sequences and the possible role of skip counting. In Figure 1a, the child represents all 39 “cupcakes” before making groups of three. This is a possible strategy that a child with an INS could use to solve this task. Interestingly, children with only an INS can successfully use skip counting to determine the cardinality of a set, but would be unable to keep track of their skip counting (Olive, 2001). For example, a child with an INS could have used skip counting to recount the cupcakes to make sure that they had created 39, but if asked “how many threes?” the question would not make sense since each “three” stands in for a numerical composite in which the 3’s represent three objects instead of one countable thing. As seen in the solution in Figure 1a, they would need to create groups of 3 and then

Table 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.

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count these groups (or draw boxes while continually recounting the number of cupcakes they have used up). The solution in Figure 1b shows a child’s use of numerical composites to find a solution. Notice that each unit contains 3 visible objects that have been counted one by one, representative of a strategy by a child with an INS. The need to recount all the cupcakes after each “box” of cupcakes is drawn suggests a lack of a keeping track strategy, and the lack of a composite unit. In Figure 1c, the child has explicitly linked their figurative composite of 3 with their skip counting. This awareness of the link between skip counting and composite units is characteristic of a child with a TNS. In Figure 1d, we see a child’s solution using only skip counting and no representation of individual cupcakes. Here it is clear that the elements of the skip count are used to represent one composite unit of 3, in this case, a box of cupcakes. After reaching “39” notice that the child seems to have then counted the number of “boxes” indicated by a single dot associated with each numeral in the count. By writing down the skip count, children with a TNS are able to keep track of how many times they have applied their composite unit. In Figure 1e, we see a child who even more clearly interprets each composite of 3 as a single countable unit, indicated by the number sequence 1 to 13 (instead of the skip count) to find the number of boxes. As sophisticated as it may seem for a child to use skip counting to solve a task that seems multiplicative, we can not necessarily infer that the child is thinking multiplicatively. The children in 1d and 1e may still be using additive, not multiplicative, structures, as they are not representing a comparison between two numbers (Ulrich, 2016). Instead, they are describing their counting activity, e.g., “I had to count by 3 thirteen times to get to 39” (Ulrich, 2016). This is not representative of multiplicative thinking. Furthermore, children who can link their multiplication facts to their skip counting have not necessarily developed multiplicative thinking, but instead may be reinterpreting their multiplication facts in terms of additive skip counting. It is important not to de-emphasize children’s use of skip counting as implicitly suggested by the Curriculum Framework (VDOE, 2009) in favor of

a focus on multiplication facts alone. Continued use of skip counting for INS and TNS students focuses them on their use of composite structures and allows them to reflect on these composites. Focusing students on multiplication facts alone hides the iterations of composites involved in multiplication and may limit the ability of students to explicitly reflect on their number sequences, necessary for an ENS and GNS, further constraining their development of multiplicative structures (Ulrich, 2016). The solution presented in Figure 1f suggests that the child has constructed an ENS or GNS as they are able to repose the question as “Three of what makes thirty-nine?” Although they, too, may have counted by 3’s to reach to 39, they are able to reconceptualize 39 as 13 groups of 3, which represents a relationship between two numbers, or a multiplicative structure. A solution that many students gave for the Cupcake Task was to simply write “13.” Such an answer, without the apparent use of figurative materials, is only possible for children who have constructed an ENS; they are able to keep track of the numerical units of composite units that are necessary to solve this task. Finally, it is important to point out that many students were able to solve this task by recognizing it as a division problem, and carrying out the necessary division to reach an answer of 13. This particular solution strategy is not necessarily indicative of a higher stage of thinking (e.g., ENS, GNS), as many TNS children are able to strategically perform the necessary procedures associated with a given type of task without using multiplicative thinking. Implications for Teaching Recent research suggests that a significant number of children make it to middle school without having constructed an ENS (Ulrich & Wilkins, 2016a, 2016b). This suggests that many children in the sixth grade are still predominately additive thinkers and thus not poised for handling multiplicative thinking and relative thinking that is necessary for fractional and proportional thinking. It is thus important for teachers to provide continued opportunities for children to develop multiplicative thinking as early as possible. We have discussed the importance of skip counting to help children develop composite units, however,

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this level of development does not guarantee that a child has developed multiplicative thinking (it does not imply that they have developed iterable units of 1). Many children like the one in Figure 1d are able to use their skip counting to solve multiplication tasks, but cannot work with units as countable objects and thus must depend on their additive strategies to solve such multiplicative tasks. Without further development, children will have trouble developing more sophisticated ways of thinking that require higher levels of units coordination (Ulrich, 2016). This understanding is necessary for multiplicative thinking, and thus many children reach middle school still working with their additive understandings. In addition to encouraging the continued use of skip counting, helping children develop notions of composite units can be aided through the intentional use of well-chosen mathematical manipulatives. For example, when students are working with tasks that ultimately involve multiples (e.g., Figure 1), making available single counters as well as counters clustered in different group sizes may help promote the use of composite units (see Figure 2). Consider a task similar to the Cupcake Task in which there are 15 cupcakes to be put into boxes of 3 cupcakes. Figure 2 shows four models of a solution to this task using different types of manipulatives. Children with an INS may use single objects to represent 15 cupcakes and then make groups of three cupcakes (see Figure 2a). For INS children, providing interlocking cubes (Figure 2b) could make it possible for them to build their own figurative composites, a first step in developing composite units. Making available objects pregrouped by threes (Figure 2c) with the single units still visible could stimulate children’s skip counting and use of a composite unit of 3. In addition, making available unpartitioned rods that represent 3 (Figure 2d) may help promote the notion of an iterable unit of one and a multiplicative unit relationship. Subtle changes in the manipulatives provided to children for solving tasks can afford or constrain their growth in understanding. For example, only having rods like those in Figure 2d for a child with only an INS may actually constrain their development. These children would likely use these “three” rods as single units and count out 15 of these rods and

make groups of 3, because they are not able to recognize that these composite units stand for 3 objects. However, for children with a TNS these rods could afford them the opportunity to develop the notion of an iterable unit of 1, thus helping them construct an ENS. Again, subtle changes in the use of manipulatives can provide opportunities for children to combine their skip counting and notions of composite units to make important growth in the construction of their number sequences. Conclusions Children’s transition from additive to multiplicative thinking represents an important shift in thinking that makes it possible to be successful with more advanced mathematical learning. Children who reach middle school without having made this transition are at a stark disadvantage because most of the middle school mathematics curriculum presumes multiplicative thinking. In this article we highlighted the characteristics of the different stages of number

Figure 2. Different models representing different types of unit structures for a Cupcake Task with 15 cup- cakes.

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sequences that children move through as they construct their notions of composite and iterable units on their way to developing multiplicative thinking. We highlighted the important role of skip counting as both an indicator of, and a way to foster, children’s development of composite units. We recommend that teachers continue to emphasize the use of skip counting as a way to develop students’ multiplicative thinking beyond just the connection with multiplication facts. Prematurely deemphasizing skip counting may cause children to focus only on the idea that 5 × 3 is the result of their additive counting by 3s, whereas continued use may afford them the opportunity to develop the more sophisticated notion that these 3s represent composite units, and that 5 × 3 represents 5 times as much as 3, a composite of composite units. At the same time, it is important for teachers to recognize that the proficient use of skip counting does not necessarily imply that children have developed multiplicative thinking. By being aware of the characteristic ways of thinking associated with each of the different number sequences teachers are better able to provide appropriate scaffolding to move children through the stages. As an example, we feel that the intentional and selective use of manipulatives, both with elementary school students and middle school students who have yet to develop composite or iterable units, could provide invaluable support for the students’ mathematical development. References Olive, J. (2001). Children’s number sequences: An explanation of Steffe’s constructs and an extrapolation to rational numbers of arithmetic. The Mathematics Educator , 11 (1), 4-9. Steffe, L. P., & Olive, J. (2010). Children’s fractional knowledge . New York, NY: Springer. doi 10.1007/978-1-4419-0591-8 Steffe, L. P., & Cobb, P. (1988). Construction of arithmetical meanings and strategies . New York, NY: Springer. Ulrich, C. (2015). Stages in constructing and coordinating units additively and multiplicatively (Part 1). For the Learning of Mathematics, 35 (3), 2–7 .

Ulrich, C. (2016a). Stages in constructing and coordinating units additively and multiplicatively (Part 2). For the Learning of Mathematics, 36 (1), 34–39 . Ulrich, C. & Wilkins, J. L. M. (2016a). Using written work to assess stages in sixth-grade students’ construction and coordination of arithmetic units . Manuscript submitted for publication. Ulrich, C. & Wilkins, J. L. M. (2016b). Using student written work to investigate stages in sixth-grade students’ ways of operating with numbers . Paper presented at the American Educational Research Association, Washington, DC. Virginia Department of Education (2009). Mathematics standards of learning for Virginia public schools . Richmond, VA: Commonwealth of Virginia Board of Education. Available online at: http:// www.doe.virginia.gov/testing/sol/ standards_docs/mathematics/index.shtml

Jesse L.M. Wilkins Professor of Math Education Virginia Tech Wilkins@vt.edu

Katy Ulrich Assistant Professor Virginia Tech culrich@vt.edu

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Is the Midpoint Quadrilateral Really a Parallelogram? Zoltan Kovacs

Dynamic Geometry software systems (DGS) can be useful and challenging tools in the teaching and learning of reasoning. DGS allow the student to formulate certain geometric facts (e.g. as intermediate steps towards establishing the proof of a given statement) by drawing auxiliary diagrams, and, then, getting convinced of the truth or falsity of the conjectured assertion by checking its validity in many instances, after randomly dragging some elements of the figure. This has already raised some concerns: …increased availability in school mathematics instruction of … dynamic geometry systems… raised the concern that such programmes would make the boundaries between conjecturing and proving even less clear for students… [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 characteristic feature of DGS and, therefore, the above expressed worries apply to all DGS. A natural question arises as to is really proving necessary in schools if all

geometric facts are obvious and visible when using DGS? Walter Hickey published a list on the 12 most controversial facts in mathematics in Business Insider in March 2013 (Hickey, 2013). Such a list is always subjective, but surprisingly an easy geometric problem is listed there also, namely Varignon's theorem (see Figure 1). Hickey calls this fact “Midpoint parallelograms” and recalls that by drawing a 4 sided polygon and connecting its midpoints you will get a perfect parallelogram every time (see Figure 2).

Figure 2. A collection of midpoint parallelograms in GeoGebra Materials (Kovács, 2015). Having DGS in use, this fact can be visually checked quite easily by sketching an arbitrary planar quadrilateral, and constructing its midpoints and the polygon they describe. (Here we will use GeoGebra, the free DGS which is an interactive 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?

Figure 1 . Varignon's theorem as shown in Wikipedia. The midpoints of the sides of an arbitrary quadrilateral form a parallelogram.

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From the numerical point of view it makes more sense to use an equivalent definition of a parallelogram, namely the congruence of opposite sides. This property is much easier to measure than parallelism of two objects (see Figure 3).

Fortunately, a few DGS currently include a feature closely related to mathematics reasoning: that of having implemented an Automated Theorem Proving (ATP) algorithm, yielding the ability to confirm or deny the mathematical (i.e. not numerical or probabilistic) truth of a geometric statement. In GeoGebra, some ATP features can also be introduced in the classrooms by using the Relation Tool. This tool in GeoGebra was originally designed to collect numerical equalities in a geometrical construction, but the recent versions can also be asked to re-investigate the problem symbolically (see Figure 5).

Figure 3 . Measuring the length of two opposite sides of a midpoint quadrilateral. Even if it is technically easier, the accuracy of such attempts is questionable. Even if it is performed on a graphing paper having a regular grid, the easy case with initial points (0,0), (2,0), (2,2) and (0,2) yield the midpoints (1,0), (2,1), (1,2) and (0,1) which result in an irrational length for each appearing side, namely the square root of 2. On accurate sampling when rounding the lengths of the opposite sides to 2 decimal places we will clearly get 1.41 for all four sides. By increasing the accuracy obviously the same results are expected. This is no longer true for some more complicated cases. By changing the rounding in DGS GeoGebra to 15 decimal places (in menu Options > Rounding > 15 Decimal Places), the last digits in the lengths of two opposite sides will differ (see Figure 4).

Figure 5 . Using GeoGebra's Relation Tool to numerically check properties (Kovács, 2016). When using the Relation Tool, the user needs to select two objects to compare. Despite the fact that the numerical computation is inaccurate, GeoGebra assumes that the opposite sides have the same length, and offers a further check to symbolically prove the congruence of those segments (see Figure 6).

Figure 4. A numerical error suggests falsifying a true statement.

Figure 6 . The Relation Tool provides a symbolical check to prove a property.

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When clicking “More...” GeoGebra starts to translate the geometric figure into a set of algebraic equations with integer coefficients in the background, but assuming the input points to be completely general and arbitrary. GeoGebra then manipulates on the equations by usually performing billions of atomic computations which are hardly of classroom interest, hence those details will not be shown to the user. Finally GeoGebra concludes that the equations can be interpreted as an evidence about the geometric statement on the midpoint parallelogram, that is, it is true in general, independently of the initial points. Modern DGSdynamic geometry systems including this ATP feature, can be considered as a kind of “geometry calculators”. Their ATP algorithms may be hidden for the user with, but just the final result is visible. In this sense the usual way of proving is usually substituted by a yes or no answer computed by the machine. However, the traditional way of proving should not be discouraged at all. When the truth about a conjecture is already known by mechanical computation, the real intellectual challenge will start: to find an elegant chain of reasons to show why that conjecture is true. A well known quote from Paul Halmos highlights that “the only way to learn mathematics is to do mathematics”. What does it mean to do mathematics? According to Bruno Buchberger, it is “knowledge derivation and problem solving by reasoning.” (Buchberger et al., 1998) To support this idea, Buchberger introduced the creativity spiral “algorithms → computational results → conjectures → theorems → algorithms → …”, and so forth, to describe mathematics as an infinite chain of recurring activities. In his model proving is the activity which connects conjectures and theorems. In conclusion we can say that without knowing why a conjecture is true learning mathematics is not really possible, either. Acknowledgments. A preliminary version of this paper was presented at ICME-13 in Hamburg, Germany (Hohenwarter et al., 2016). References Buchberger, B. and the Theorema Working Group (1998). Theorema: Theorem proving for the masses using Mathematica. Invited Talk at

the Worldwide Mathematica Conference, Chicago, June 18-21, 1998. Botana, F., Hohenwarter, M., Janičić, P., Kovács Z., Petrović, I., Recio, T. & Weitzhofer, S. (2015). Automated theorem proving in GeoGebra: current achievements. Journal of Automated Reasoning (Vol. 5, Number 1, pp. 39-59). Springer. Hohenwarter, M., Kovács, Z. & Recio, T. (2016). Deciding geometric properties symbolically in GeoGebra. https://www.researchgate.net/ publication/305916853_Deciding_geometri c_properties_symbolically_in_GeoGebra Hickey, W. (2013): The 12 Most Controversial Facts In Mathematics. http:// www.businessinsider.de/the-most- controversial-math-problems-2013-3? op=0#-63 Kovács, Z. (2015): Midpoint parallelograms. GeoGebra Materials. https:// www.geogebra.org/m/pBuQA4Eo Kovács, Z. (2016): Midpoint parallelograms. GeoGebra Materials. https:// www.geogebra.org/m/BrnY77nE Lin, F.-L., Yang, K.-L., Lee, K.-H., Tabach, M., & Stylianides, G. (2012). Principles of task design for conjecturing and proving. In Hanna, G. and de Villiers, M., editors, Proof and Proving in Mathematics Education . The 19th ICMI Study, pp. 305- 326. Springer.

Zoltan Kovacs Johannes Kepler University Zoltan@geogebra.org

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Algebra Achievement of Urban High School Students Andrew Wynn, Cheryl Adeyemi, Gerald Burton, and Crystal Wynn

Introduction

civil-rights activist Bob Moses in 1982. Bob grew up in a poor family but had a strong academic prowess and was able to attend a competitive high school, and eventually was able to earn his Master’s degree from Harvard University. Having been instrumental in the civil rights movement, he sought to conquer injustice wherever he saw fit. After observing the poor quality mathematics education that his daughter and her classmates were receiving in the public schools in Mississippi, Bob sought to immediately impact the curriculum by offering to teach Algebra to his daughter and her classmates(Moses and Cobb, 2002). Armed with a strong desire to see equality in both areas of social justice and mathematics, Bob developed the idea that mathematics education is a civil-right. Therefore, Bob developed his ideas into a curriculum to foster change in the urban mathematics classroom. Doing much of his early work in classrooms in Mississippi, Bob was able to affect students first hand as a math teacher. His motivating factor was the thought that “the information age of computers and networks has put advanced mathematics…on the table as an education necessity” (Checkley, 2001). Algebra Project Pedagogy The Algebra Project developed a mathematics curriculum that began with the idea that math is a language that needs to be learned because it is not a natural language. Today, the Algebra Project method impresses upon students that the language of mathematics needs to be learned, and that they, as students, are capable of learning the language replete with syntax, structure, and conceptual meaning Once students buy in to the concept of learning the language, the Algebra Project curriculum is centered around the Experiential Learning Cycle (ELC) (Kress, 2005). Using the ELC, students being taught using the AP pedagogy begin with a concrete experience in their community, make observations about their experiences, reflect on their experience and

Student achievement is a major concern for schools, law makers, and other stakeholders. A by -product of this concern is the fact that algebra has served as a gatekeeper and barrier for many students who were interested in majoring in Science, Technology, Engineering, and Mathematics (STEM) at the collegiate level (Ladson-Billings, 1997). While mathematics has proven to be a barrier for students from all races, research has shown that upper level math courses become even greater obstacles for students who are of African-American or Hispanic descent or who have disabilities (Martin, 2012; Noble, 2011). These students often fail to complete high school. According to Cortes, Nomi, and Goodman (2013) suggest, “One theory for these low high-school completion rates is that failures in early courses, such as algebra, interfere with subsequent course work, placing students on a path that makes graduation quite difficult”. In fact, “beginning with the first international mathematics achievement test administered to students in the 1960s, the problem of poor mathematics performance has emerged as an issue of national concern” (Mayfield and Glenn, 2008). With respect to algebra courses, reform efforts have been implemented to try to increase student performance further improving students’ access to upper level mathematics classes. Finding interventions that not only improve urban school students’ achievement by making the mathematics more understandable, but that also help increase urban school students’ desires to want to pursue math opportunities by giving them a better experience is critical. Interventions based upon experiential and structured experiences may cause students to grow in algebraic understanding and abilities, while developing their own experience of mathematics. The Algebra Project presents itself as one such intervention. History of the Algebra Project, Inc. The Algebra Project (AP) was founded by

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