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WISCONSINMATHEMATICS COUNCIL, INC.2018 | Issue 2Wisconsin

TEACHERof MATHEMATICS

Wisconsin Teacher of MathematicsThe Wisconsin Teacher of Mathematics (WTM) is the official journal of the Wisconsin Mathematics Council

(WMC). Annual WMC membership includes a 1-year subscription to the journal. WTM is a forum for the exchange of ideas. The opinions expressed in this journal are those of the authors and may not necessarily reflect those of the Council or editorial staff.

Journal GovernanceWTM is overseen by an Editorial Board, which consists of the Editorial Team and the Editorial Panel. The

Editorial Team, led by the WTM Editor, leads the review, decision, and publication process for manuscripts. The Editorial Panel reviews manuscripts as requested by the editor, assists in setting policy for the journal, and gathers feedback from readers. Serving on the Editorial Panel is a 2-year commitment.

The Editorial Board is chaired by the WTM Editor and has two regular meetings during the academic year: a virtual meeting in Fall/Winter and an in-person meeting at the WMC Annual Conference in May.

Editorial Team• Joshua Hertel, Editor, University of Wisconsin–La Crosse• Matthew Chedister, Associate Editor, University of Wisconsin–La Crosse• Jenni McCool, Associate Editor, University of Wisconsin–La Crosse

Editorial Panel• Bhesh R Mainali, University of Wisconsin–Superior• Timothy M Deis, University of Wisconsin–Platteville• Jennifer Kosiak, University of Wisconsin–La Crosse

The National Council of Teachers of Mathematics selected the Wisconsin Teacher of Mathematics to receive the 2013 Outstanding Publication Award. This prestigious award is given annually to recognize the outstanding work of state and local affiliates in producing excellent journals. Judging is based on content, accessibility, and relevance. The WMC editors were recognized at the 2014 NCTM annual meeting.

Manuscript Submission GuidelinesThe WTM Editorial Board encourages articles from a broad range of topics related to the teaching and learning of mathematics including submissions that focus on:• Engaging tasks that can be implemented in the preK–12 classroom,• Connecting research and theory to classroom practice,• Showcasing innovative uses of technology in the classroom,• Work with preservice teachers in the field, and• Current issues or trends in mathematics education.

Other submissions not focusing on these strands are also welcome. Articles should be 2000–4000 words and follow APA 6th Edition guidelines. Figures, images, and tables should be embedded. Submit articles in .doc/.docx format to [email protected].

mailto:[email protected]

Wisconsin TEACHER of MATHEMATICS

Table of Contents

2018 | Issue 2

Editorial 1

President's Message 2

Exploring Triangle Similarity Using AngLegs and GeoGebraBhesh Mainali, University of Wisconsin–Superior

3

Using Number Talks to Develop Student Understanding of Equality and the Equal SignJill Leffler and Mindi Stoneman, School Districts of Greenfield and South Milwaukee

8

The Struggle is Real: Anticipating Student MisconceptionsKatelyn Albright and Derek Pipkorn, Mequon-Thiensville School District

15

Stop Counting By Ones: Seeing Twos and Threes as a Foundation for Kindergarten MathematicsLaura McNelly, Milwaukee Public Schools

20

Engaging Our Students in Upper-Level Mathematics: Finance ApplicationsDave Ebert, Oregon High School, Oregon, WI

24

1 Wisconsin Teacher of Mathematics | 2018, Issue 2

Greetings WMC Members,Welcome to another issue of the Wisconsin Teacher of Mathematics! This issue features a

range of articles outlining ways to engage students in the classroom. No matter your classroom context, we hope that you will find ideas to further mathematical discussion and reasoning. If an article strikes a particular chord or sparks your interest, we would love to hear about it!

Looking to 2019, we are hoping to assemble a future issue along the theme of Equity and Access in the Mathematics Classroom. The Editorial Panel encourages submissions that address this theme and related questions such as:• How do we ensure that all students have access to a challenging mathematics curriculum?• How do we monitor student progress and make appropriate accommodations within the

mathematics classroom?We also welcome submissions focusing on other topics related to the teaching and learning

of mathematics (e.g., engaging preK–12 tasks, current issues in mathematics education). If you have questions or wish to submit an article for review, please email the editor Joshua Hertel ([email protected]) or visit the WMC website for more information (http://www.wismath.org/Wisconsin-Teacher-Mathematics-Journal). Additionally, if you have an article idea and would like some direction or feedback, we encourage you to stop by our session on Friday at the WMC Annual Greenlake Conference.

In Mathematics,From

the E

dito

rs

Joshua Hertel Editor

Matthew Chedister Associate Editor

Jenni McCool Associate Editor

2Wisconsin Teacher of Mathematics | 2018, Issue 2

Mathematics Colleagues,

The task of improving mathematics instruction and achievement in our districts, our state, and our country will take collaboration among those of us teaching mathematics. We know that by planning together in school or district professional learning communities, we can share ideas that are based on our collective understanding of the math we are teaching. We can brainstorm interventions as we come together to examine student work and data from formative and summative assessments. None of us have all of the answers individually, but together we can meet more and more students’ needs.

I want to thank those of you who go beyond your local PLC discussions to share experiences and expertise with a larger mathematics community. Many of you participate in CESA workshops and coaching sessions, WMC Twitter Chats, and WMC conferences. Some share time by participating on the WMC Board of Directors or the NCTM Board of Directors. And others publish their ideas in newsletters, e-blasts, and journals. Specifically, the authors of the articles in this fall’s WMC journal have included ideas to help students from ages 5 to 18 better understand and use the mathematics of the Wisconsin and Common Core State Standards. I believe all of the strategies shared in the lessons on the following pages will actively engage our students in important mathematics.

Laura McNelly connects young students’ conceptual understanding of the smallest whole numbers to how students experience those numbers beginning in Kindergarten. Katelyn Albright and Derek Pipkorn remind us that anticipating student errors in our planning helps us to create a safe environment for students to learn from those errors. When we can make students comfortable and help them engage in a struggle with ideas that is truly productive, we assist in deepening students’ understanding.

Many of us think of using number talks and manipulatives in our primary mathematics classrooms to help students develop number sense and calculation strategies. Jill Leffler and Mindi Stonemen explain how number talks can be used with middle school students to increase their conceptual understanding of equality and the equal sign. And Bhesh Mainali provides examples of using manipulatives with geometry students to deepen students’ understanding of similarity.

Engaging our high school students in productive struggle with mathematics concepts can be accomplished when the students see applications of those concepts in life situations. Dave Ebert combines the financial literacy standards with high school mathematics standards by having students apply the advanced mathematics of exponential functions to the concept of interest and credit card use.

Thank you to our authors for engaging the mathematics teaching community outside their own districts. Thank you to our members for reading and sharing these ideas in your local PLC discussions. Together, we can make a difference for all students learning important mathematics.

Lori WilliamsWMC President

President's Message


Exploring Triangle Similarity Using AngLegs and GeoGebra

Bhesh Mainali, University of Wisconsin–Superior

BackgroundInstructional practices in mathematics often

focus on computational skills rather than conceptual understanding. Only focusing on computation skills, rules, and mathematical procedures using a drill-and-practice approach does not lead to a deeper understanding of mathematical content. As a result, students will have a limited mathematical understanding. However, utilizing manipulatives and technologies would enhance conceptual understanding along with procedural skills in which students explore, discover, and understand mathematical content. Furthermore, manipulatives would enhance mathematical learning by providing connection and visualization between numeric and visual representations (Suh, Johnston, & Douds, 2008).

Utilizing manipulatives in math lessons enhances conceptual understanding by providing hands-on learning experiences for students. Integration of technology such as dynamic geometry software (DGS) in math lessons further reinforces the mathematical concepts by supporting students’ learning and fostering the acquisition of mathematical knowledge and skills (Hohenwarter, Hohenwarter, & Lavicza, 2008). The unique features of DGS, such as its interactive and dynamic nature, help students to check and test mathematical knowledge in a multiple ways that they learn from manipulatives in a static way. For example, students can play with dynamic figures, move and drag objects, and see visual pictures of a mathematical scenario in DGS. Thus, the multiple ways of representing a mathematical task with the help of DGS along with manipulatives can lead learners to obtain both conceptual and procedural understanding.

Refining Bruner’s (1966) three modes of mental representation, Tall (1994) states that integrating hands-on activities and technological tools fosters three modes of mental representations. Bruner (1966) proposed three modes of mental representation: enactive (by actions), iconic (by summarizing images), and symbolic (using language and mathematics symbols). According to Bruner, these mental representations grow in sequence during an individual’s learning. From

this point of view, a student must first get a chance to actively work with concrete objects before making steps towards visual and then symbolic representations while learning mathematical concepts. Teaching mathematical concepts by utilizing manipulatives and then reinforcing the understanding by integrating DGS fosters the mathematical learning process described by Bruner .

Exploration With AngLegs and GeoGebraThe Common Core State Standards for Mathematics

(CCSSM; National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010) emphasize using different tools such as a compass and straightedge, string, reflective devices, paper folding, DGS for geometric construction. Utilizing manipulatives and DGS in teaching geometry concepts also aligns with the CCSSM. In contrast to the drill and practice instructional approach, using manipulatives and technologies in math lessons helps to build conceptual understanding as well as procedural skills. For example, when we teach the triangle similarity concept, we provide the rules and relationship rather than providing a learning environment in which students explore and discover the rules and relationships between two similar triangles. Although students might be able to memorize the relationship, they might not understand why the rule is always true and where it comes from. Thus, to provide conceptual understanding of the triangle similarity concept, AngLegs can be used. In order to further reinforce the triangle similarity concept, we can use the mathematical software GeoGebra in connection with AngLegs.

AngLegs can be used for various purposes. They are handy and easy to use. AngLegs are available in six colors (orange, purple, green, yellow, blue, and red) with six different lengths (5 cm, 7.07 cm, 8.66 cm, 10 cm, 12.24 cm, and 14.14 cm). Various types of geometry concepts can be explored with AngLegs. For example, properties of triangles, quadrilaterals, triangle similarity and congruency, the Pythagorean theorem, and other concepts can be easily explored with


AngLegs. Furthermore, different types of polygons can be made shaping different AngLegs with each other. A protractor can be attached to the AngLegs in order to measure the angles as shown in Figure 1. The length of each AngLegs is designed on purpose in order to explore various mathematical ideas. A set of six AngLegs and a triangle formed by using AngLegs are shown in Figure 1.

How AngLegs can be used to explore the triangle similarity concept is described in the following steps.• Step 1: Provide a set of AngLegs, and ask the

students to make two different triangles. Let form a triangle ABC using three orange AngLegs and a triangle ∆PQR using three yellow AngLegs, as shown in Figure 2.

• Step 2: Now, ask your students to measure the length of each side of the triangle made in step one and complete Table 1.

• Step 3: After students complete Table 1, ask them to complete Table 2.

Table 1Sides Measurement

Sides AB BC CA PQ QR RP

Lengths

Table 2Side-Ratio Measurement

Side-ratio

Ratio

After students complete Table 2, have them discuss this in their group. What did they notice? What did they discover, or what pattern did they find? We can pose several follow-up questions in order to deepen understanding. We can repeat the same steps but now with a different set of AngLegs to see if the same rule

Figure 2. Similar triangle formed with orange and yellow AngLegs.

Figure 1. A set of AngLegs, and a triangle with AngLegs.


would be true even for different sets of similar triangles. As a follow-up activity, we can ask students to make set of similar triangles using different AngLegs, as shown in Figure 3. Repeat Steps 1 to 3.

After students complete the steps, have them discuss this again in their group. As an extension of the above activities, we can make various types of nonequilateral triangles (isosceles and scalene) using different sets of AngLegs. For example, a set of two isosceles similar triangles formed by different set of AngLegs is shown in Figure 4. Similarly, we can also make two similar scalene triangles using different sets of AngLegs.

We can also repeat Steps 1–3 for these non-equilateral triangles. After doing these different triangle similarity activities, we can pose various questions. Does the rule still hold true with a different set of similar triangles from the one we utilized in the activities above? In doing so, students get chance to explore relationships between the corresponding sides in different sets of similar triangles. Several sets of these AngLegs similar triangles activities help students to discover the underlying rules based on the triangle similarity concept themselves.

Exploration With GeoGebraThe AngLegs triangle similarity task above provides

a learning environment in which students can explore and discover the relationship between the corresponding sides in similar triangles, but it is still not an interactive and dynamic way to explore the concept. To reinforce the similarity concept in an interactive and dynamic way, we can use the mathematical software GeoGebra. GeoGebra is a free and multiplatform dynamic mathematics software for all levels of education that joins geometry, algebra, tables, graphing, statistics, and calculus in one easy-to-use package. It has received several educational software awards in Europe and the United States. It is an open-source software that is freely available from www.geogebra.org. There are tons of GeoGebra ready-made applets that can be found in the GeoGebra website. Someone who wants to integrate this software in teaching mathematics lessons does not necessarily need to be an expert in it because the ready-made applets can simply be used in the mathematics lessons.

What we have been done with AngLegs can be done with GeoGebra but in more dynamic and interactive ways. We can construct similar triangles using GeoGebra as shown in Figure 4. In order to see the interactive applet, please visit https://www.geogebra.org/m/MqkHBgbw. We can see the options to change the angles of the given triangles with the help

Figure 4. Set of nonequilateral similar triangle.

Figure 5. Screenshot of the interactive GeoGebra applet.

Figure 3. Set of new similar triangles with blue and red AngLegs.


of two sliders (Beta and Lambda) in the given applet. As we drag the sliders in the GeoGebra applet, we can see how changes in the two angles yield various types of similar triangles. The triangle shown in Figure 5 all have angles that are 60°.

In order to see the relationship between the corresponding sides in the triangles, we need to check the box “show sides ratio,” as shown in Figure 5. Once we check the “show sides ratio” button in the GeoGebra applet, we can see the relationship between the corresponding sides in the two similar triangles, as shown in Figure 6. We can see that the ratio of corresponding sides of the similar triangle is always constant, in this case 0.5 units. This example is similar to the one done earlier with AngLegs. However, as we drag the slider in the GeoGebra applet, we can see several sets of similar triangles and how each of them holds to the same rules of side proportionality.

ConclusionUsing AngLegs would provide a concrete

exploration and discovery of the triangle similarity concept and relationships between their corresponding sides. Rather than being taught the rules and relationships between the corresponding sides of similar triangles, students get a chance to explore and discover the rules and relationships themselves, yielding conceptual understanding of the relationship between two similar triangles. Students also get to see in different ways why the relationship is always true. However, AngLegs is a static way of exploring the concepts because it is not interactive in nature. Different types of AngLegs can be used to experiment and verify the triangle similarity concept in a multiple ways, but this may be time consuming. Additionally, the number of different sets of similar triangles that can be made is limited because

we only have six type of given AngLegs. Therefore, AngLegs might not be enough to make various types of triangles with different measurements in order to explore the triangle similarity concept.

GeoGebra provides opportunity to explore and experiment using the triangle similarity concept effectively and efficiently in multiple ways, which is not easy with AngLegs. Students can explore and experiment using the concept by manipulating sides as well as angles of similar triangles in an interactive way. Thus, integrating GeoGebra would further enhance and reinforce the concept in a more interactive and dynamic way, which fosters deeper conceptual understanding with multiple representations. Moreover, utilizing GeoGebra in conjunction with AngLegs helps to bridge the gap in students’ knowledge, if any, during AngLegs activity exploration. The hands-on learning experiences that students acquire during AngLegs exploration is transformed into more numeric and dynamic representation of the triangle similarity concept during the GeoGebra activity. Connecting AngLegs exploration with GeoGebra applet also fosters the different stages of learning mathematics as explained by Bruner’s (1966) three modes of mental representations.

ReferencesBruner, J. S. (1966). Toward a theory of instruction.

Cambridge, MA: Belknap Press of Harvard University Press.

National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Author.

Figure 6. GeoGebra applet side-ratio relation.


Hohenwarter, J., Hohenwarter, M., & Lavicza, Z. (2008). Introducing dynamic mathematics software to secondary school teachers: The case of GeoGebra. Journal of Computers in Mathematics and Science Teaching, 28(2), 135–146.

Tall, D. (1994, July). A versatile theory of visualisation and symbolisation in mathematics. Plenary presentation at the Commission Internationale pour l’Étude et l’Amélioration de l’Ensignement des Mathématiques, Toulouse, France.

Suh, J. M., Johnston, C. J., & Douds, J. (2008). Enhancing mathematical learning in a technology-rich environment. Teaching Children Mathematics, 15(4), 235–241.

Wisconsin Teacher of Mathematics | 2018, Issue 2 8

Using Number Talks to Develop Student Understanding of Equality and the Equal Sign

Jill Leffler and Mindi Stoneman, School Districts of Greenfield and South Milwaukee

Mrs. Stoneman noted, yet again, that her 7th grade students were struggling to solve basic algebraic equations. For example, when asked to solve 4 + 2a = 10, one student wanted to begin by subtracting 4 from 2a. Another student wanted to add 4 to 10. Another student wanted to subtract 10 from both sides then asked what to put on the right side. She thought, “What the heck am I doing wrong? It’s like they don’t understand what the equal sign means.”

These types of experiences have led us to search for a way to help students gain an understanding of equality and the equal sign. This understanding would give them a solid foundation for solving algebraic equations and eliminate scenarios such as the one described in Mrs. Stoneman’s class. The purpose of this article is to share how number talks can be an effective and efficient vehicle to develop student understanding of equality. To begin, we will discuss the importance of students having an understanding of equality and provide a progression showing four levels of student reasoning for equality. Second, we will define and describe a number talk. Finally, we will give specific strategies for implementing number talks that focus on developing student understanding of equality and the equal sign.

Why Focus on Equality?Since the release of the Common Core State

Standards for Mathematics (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010), students are expected to work with operations and algebraic thinking in Kindergarten and throughout elementary school. By sixth grade, the Operations and Algebraic Thinking domain narrows to the Expressions and Equations

domain. In sixth grade, students spend quite a bit of time learning to solve one-variable equations and inequalities and work with independent and dependent variables.

To get a sense of how our students were thinking about the equal sign, we conducted an informal survey. We asked them to solve an open number sentence, similar to tasks in research on the equal sign (Falkner, Levi, & Carpenter, 1999). In our survey, we asked students to put the number in the blank that would make the following statement true: 8 + 4 = __ + 3. We have summarized our results in Table 1. It is worth noting that one third of our sixth-grade students did not think nine would make the equation true. This is a problem considering that important foundational work with equations is done in sixth grade.

Why do some students think that the solution is 12 or 15? Students often interpret the equal sign as a signal to compute (Falkner et al., 1999; Knuth, Stephens, McNeil, & Alibali, 2004). It appears that our students do make progress in their understanding of equality from sixth to eighth grade. This is also helpful information. Our instructional work with eighth graders is to develop a more sophisticated understanding of equality known as relational thinking.

So, where do we begin? Matthews, Rittle-Johnson, McEldoon, and Taylor (2012) identified four levels for analyzing student understanding of the equal sign, as shown in Table 2. This progression is a useful reference tool when doing a formative assessment of student reasoning. For example, using the progression, it appears that one third of our sixth graders most likely use Operational reasoning about equality that is considered

Table 1Results of Student Survey

Solution 6th grade 7th grade 8th grade

9 66.7% 77.3% 88.9%Other (12, 15, 6, 13, 2) 33.3% 22.7% 11.1%


to be Level 1. We would want to make instructional decisions that would get our students to grow to at least Basic Relational reasoning, as represented by Level 3.

It is important to note the warning made by Matthews et al. (2012) when interpreting these levels: We should not expect students’ progress through these stages of reasoning to follow a strictly linear trajectory. Learning is a fluid process that may occur at multiple levels simultaneously (Matthews et al., 2012). After assessing current levels of reasoning, instruction and practice can be done through a number talk.

What Is a Number Talk?Humphreys and Parker (2015) define a number talk

as “a brief daily practice where students mentally solve computation problems and talk about their strategies” (p. 5). A number talk only needs a small amount of class time, 15 minutes at most. During the number talk, the teacher presents students with a problem to solve. It is helpful if the problem can be solved in a number of different ways to promote discussion among students. After students have had an opportunity to arrive at a solution, the teacher allows students to share their answers with the teacher recording each student’s thinking. Students are given a chance to defend their own answers and to question the thinking of others. A well-designed number talk may consist of a sequence of problems that will help students’ recognize a pattern

or help thinking progress.Although number talks are frequently used as a tool

to build computational fluency, teachers can also use number talks to help increase students’ number sense and reasoning skills. In addition to the mathematical learning achieved through number talks, students’ also gain confidence and learn the skills necessary to participate in meaningful discourse in the classroom. Humphreys and Parker (2015) outline the benefits of number talks as, “helping students learn to work flexibly with numbers and arithmetic properties; and helping them build a solid foundation and confident dispositions for future mathematics learning” (p. 6). Number talks are often thought of as something that is done in an elementary setting, but we have found them to be helpful in the middle grades as well. We have found that number talks can be impactful when done at least two times per week.

Suggestions for Developing Student Understanding of Equality Through a Number Talk

Because a number talk should not take more than 15 minutes, it is an example of getting a big bang for your buck. The overarching goal of a number talk on equality is to help students see the equal sign as a signal that we have two equivalent expressions. True–false questions and open number sentences are a place to start because we can get an idea of what students already know about

Table 2Levels of Reasoning

Level Reasoning Equation structure(s) understood by student

Level 1 Operational• equal sign means operate• only successful when operation is on the left

4 + 8 = __

Level 2 Flexible Operational• equal sign means operate• can solve when operation is on either side

8 = 6 + __

Level 3 Basic Relational• solves with operation on either side• solves by operating on both sides

3 + 4 = __ + 57 + 6 + 4 = 7 + __6 - 4 + 3 = __ + 3

Level 4 Comparative Relational• compares the expressions on both sides• uses compensation strategies

898 + 29 = 896 + __

Note. Adapted from Matthews, Rittle-Johnson, McEldoon, and Taylor (2012, p. 320).


equality and what gaps they have in their understanding. Following the four levels of reasoning shown in Table 2, instruction would begin where student reasoning is and progress toward Comparative Relational thinking. When working toward the Comparative Relational level of reasoning, the number talk should include equations that cannot be easily calculated. Encourage the habit of taking a moment to compare the expressions on each side of the equal sign, looking for a relationship before attempting to calculate a solution (Basic Relational). In addition to equations with large numbers, also use problem strings that apply properties of operations, an understanding of the meaning of multiplication, and the order of operations. We want our students to look for, notice, and use these mathematical ideas when working with equations.

In developing our number talks, we referred to the Mathematics Routine Bank (San Diego City Schools, 2004) and Carpenter, Franke, and Levi (2003) for guidance in thinking about equality and how to develop problem strings for the purpose of advancing our students’ conceptual understanding of equality. For example, Carpenter et al. (2003) provided some guidelines for a number talk such as leaving each equation from the problem string on the board because the first one is usually more obvious and is a source for comparison with the unfamiliar ones.

The Appendix shows the sequence of problem strings that we developed along with a focus, structure, and instructional notes. This is just a sampling of problem strings; it is not a complete sequence. The sequence includes many of the valuable areas of focus when working with equality, such as properties of operations, the meaning of multiplication, the order of operations, and analysis of the equation through comparing each side. Although some of the equations seem very simple, much of the student learning that occurs in number talks is due to student discourse related to proving and justifying.

A typical number talk would flow as follows: Write an open number sentence such as 3 x 6 = __ x 2 on the whiteboard. Read the equation to the students as “three times six is the same quantity as some number times two” and ask the students to decide what number goes in the blank (or box) to make the equation true. Wait until all students indicate a decision by giving a “quiet thumbs-up.” If some students give a thumbs-up right away, the teacher can ask students to think of two or

three other ways to get the same answer. If a student struggles, the teacher can walk over to the student and support him or her in entering into the problem. After all students indicate that they have a number, the teacher asks some or all students what number they got, writing all ideas on the board. There will be some incorrect answers. Next, the teacher asks any of the students to prove or justify that their answer is correct. The teacher faithfully records student thinking without adding teacher input other than clarifying questions. Thoughtful questioning can help the students understand the mathematical thinking being recorded. Allow a student with an incorrect answer to attempt to prove their answer. This is where the magic happens as they start to realize their number does not work and see where they made an error. After the student thinking is recorded and the correct solution is determined, the teacher can lead a discussion. This process is engaging for all students. Of course, the teacher needs to set the tone for understanding the value in not being afraid to be wrong and in learning from our mistakes. In the Appendix, the first problem string includes the equation 9 = 9. The fact that 9 = 9 is an equation may seem so obvious that it is not worth including it in a number talk. After talking to some elementary teachers who told us that their students do not believe that 9 = 9 is an equation, one of the authors decided to see what her middle school math intervention students thought. She put 9 = 9 on the whiteboard with the question: "Is this an equation?" She told her students to decide yes or no and defend their answer. Of those students, 58% of the eighth graders (19 students), 61% of the seventh graders (18 students), and 67% of the sixth graders (9 students) said that 9 = 9 was not an equation. The reasons they provided were that it is not an equation because: there is no operation, there are no actions to be taken, and no rules to be followed. We had a rich discussion proving that 9 = 9 is an equation, and we arrived at a working definition of an equation as two equivalent expressions.

The true value of a number talk lies in the discussions that occur. One way to improve the discourse is through the use of talk moves such as revoicing, repeating, adding on, wait time, and using turn and talks. Through these rich discussions, student misconceptions can be surfaced and resolved.

Reflecting on Number Talks in ActionOne of the authors recently conducted a math


intervention with 15 sixth-grade students. The focus of the intervention was to facilitate student growth in algebraic thinking, specifically their understanding of equality and the equal sign. Within the context of equality, the intervention also focused on some properties of operations that are key to the mastery of operations in algebra. As noted by Carpenter et al. (2003), “The fundamental properties that children use in carrying out arithmetic calculations provide the basis for most of the symbolic manipulation in algebra” (p. 2). The main instructional practice was number talks, which were done four times per week for 6 weeks. The results of the intervention are in Figure 1. The equations were from a set of formative assessments published by Tobey and Fagan (2014).

After administering the preintervention probe, there was compelling evidence that the students did not demonstrate understanding of equality, the equal sign, or some properties of operations. The data showed that they interpreted the equal sign as a signal to compute and disregarded any other quantities in the equation. Based on these results, the learning goals for the intervention were as follows.• I engage in discussions where I share and justify my

solution and listen and respond to others.• I use the language “is the same quantity as” when

reading the equal sign.• I apply my understanding of properties of operations

(distributive property for multiplication over addition, the associative property for addition and multiplication, and the commutative property for addition and multiplication) to analyze an equation.

• I understand that an equation is composed of two equivalent expressions.

• I take time to analyze the entire equation to see if I notice anything on both sides that will help me be more accurate and efficient.Number talks brought student misconceptions

and understanding to the surface. In one number talk (at the beginning stage of working with the concepts of equality and the equal sign), much discussion and inquiry occurred around the solution to 19 + 8 = __ + 20. Some students said, “I don’t know, do I add 19, 8, and 20? What do you want me to add? One of the students responded saying that, “Well, you have to add what’s on the left side first.” Another student said, “Oh I have another way. Can I come up and show you?” Normally in number talks, the teacher records student thinking, but on occasion we allow students to come up to the whiteboard and show their thinking themselves. The student whose work is shown in Figure 2 said, “Well, since 19 is only one away from 20, then I took one from the 8 and gave it to the 19, so obviously the answer is 7 cuz both sides have 20 already.” This student showed a clear understanding of equality and utilized relational thinking by looking at the relationship between the two equivalent expressions.

Figure 1. Results of the intervention. The y-axis is percent correct, and the x-axis shows the equations used both pre- and post-intervention.


Successful progress through this intervention helped students improve their understanding of equality and confidence in handling equations. However, it takes more than 6 weeks for all students to gain proficiency with equality. They need consistent opportunities over the entire school year in order to demonstrate understanding in a variety of contexts.

In conclusion, when we work with our students in sixth through eighth grade on the Expressions and Equations state standards, we cannot assume that they

have the foundational understanding of equality and the equal sign that is necessary for success. The levels of understanding and the questions and tasks presented by Matthews et al. (2012) gave us a good place to start planning our instruction. Through number talks, we can incorporate targeted work so that all of our students can develop a foundational understanding of equality and the equal sign. With these tools in hand, our students have a much better chance to be successful in their work with equations in middle school and beyond.

Figure 2. Student work showing relational thinking during a number talk.

ReferencesCarpenter, T. P., Franke, M. L., & Levi, L. (2003).

Thinking mathematically: Integrating arithmetic & algebra in elementary school. Portsmouth, NH: Heinemann.

Falkner, K. P., Levi, L., & Carpenter, T. P. (1999). Children’s understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6(4), 232–237.

Humphreys, C., & Parker, R. (2015). Making number talks matter: Developing mathematical practices and deepening understanding, Grades 4-10. Portland, ME: Stenhouse.

Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37(4), 297–312.

Matthews, P., Rittle-Johnson, B., McEldoon, K., & Taylor, R. (2012). Measure for measure: What combining diverse measures reveals about children’s understanding of the equal sign as an indicator of mathematical equality. Journal for Research in Mathematics Education, 43(3), 316–350. doi:10.5951/jresematheduc.43.3.0316


Parrish, S. D. (2011). Number talks build numerical reasoning. Teaching Children Mathematics, 18(3), 198–206. doi:10.5951/teacchilmath.18.3.0198

San Diego City Schools. (2004). Mathematics routine bank: Secondary number sense and algebraic thinking routines. Retrieved from http://www.svmimac.org/images/Cristo_Rey_-_Middle_Level_Bank.pdf

Tobey, C. R., & Fagan, E. R. (2014). Uncovering student thinking about mathematics in the Common Core: Grades 3–5. Thousand Oaks, CA: Corwin.

http://www.svmimac.org/images/Cristo_Rey_-_Middle_Level_Bank.pdf




APPENDIX

Table A1Sequence of Problem Strings With a Focus, Structure, and Instructional Notes

Focus Structure Problem strings Notes

What defines an equation?

True/false equations

9 = 4 + 59 = 9

Ask: “Is the equation true or false?”Elicit current conceptions of the equal sign with: “Is this an equation?”, “What does the equal sign mean?”, “Can you define the equal sign without using the word “equal”? Model use of the language “is the same quantity as” for the equal sign when reading an equation.

Different ways to write an equation.

Open number sentences

7 = �9 = � + 1

Ask: “What number can you put in the blank to make this equation true?”

Compare both sides of the equal sign


6 + 2 = � + 615 + 16 = 15 + �

Can students start to “notice” something on both sides of the equal sign that allows them to not have to calculate to solve?



4 + 6 = � + 64 + 6 = � + 44 + 6 = � + 5

What is the relationship amongst the numbers on the two sides of the equal sign?



12 + � = 10 + 1383 + 29 = � + 30

Some students will use a compensation strategy for addition such as taking one from 13 and adding it to 10.

Compare both sides of the equal sign; multiplication as groups of


7 x 4 = 8 x 4 - 4 In the discussion, ask questions such as, “What is the difference between seven groups of 4 and 8 groups of 4?”



9 + 3 - 3 = 9 In the discussion, ask, “What is the same on both sides?” and “What does 3 - 3 mean?”

Identity property of zero for addition


9 + 5 = 149 + 5 = 14 + 09 + 5 = 14 + 1

Distributive property for multiplication


5(2 + 3) = 10 + 157(2 + 1) = 14

The second equation gets at the idea that some students forget to multiply the seven to both the 2 and the 1.



10(2 + 4) = 20 + �� x (3 + 4) = 5 x 3 + 5 x 4�(2 + 4) = 10 + 20(7 x 3) + (7 x 5) = � x (3 + 5)

Students need a lot of practice with seeing the distributive property worked out in different ways.


(3 x 2) + (3 x 1) = �(2 + 1)14 x 4 = (10 x 4) + (4 x �)13 x 9 = (10 x 9) = (� x 9)6p - � = 3(2p - 1)


(8 x 2) + (8 x 1) = (2 + 1)� Discussion: “Does it matter that the 8 is on the right side of the parentheses?”


5(x + 3) = (5 x 3) � (5 x 3) What symbol goes in the box?

Recognizing what is not the distributive property for multiplication


5 + (2 + 3) = (5 x 2) + (5 x 3) Did anyone get fooled into thinking the left side is the distributive property?


Multiplication as repeated addition


3 x 7 = 7 + 7 + 73 x 7 = 14 + 7

Multiplication as groups of


3 x 7 + 7 = � x 7� x 13 = 7 x 13 + 13� x 26 = 19 x 26 + 26

Multiplication as groups of


7 x 4 = 8 x 4 - 4

Commutative property for multiplication


8 x 3 x 5 = 3 x 8 x 55 - 3 = 3–512 ÷ 3 = 3 ÷ 12

Commutative properties for addition and multiplication


12 + 5 = � + 12� x 4 = 8 x 4

Associative property for addition

12 + 2 + 3 = 14 + �(24 + 5) + 10 = � + 15

Associative property for multiplication


3 x 4 = 3 x 2 x �6 x 4 = 2 x � x 6

Some students will notice that both sides have something the same such as multiplying with 6. In the discussion it may be helpful to rearrange the equation to 4 x 6 = 2 x � x 6 so that all students can see that both sides have x 6.



50 = 2 x 5 x 54 x 5 = 2 x 5



3 x 6 = � x 23 x 3 x 5 = � x 5(2 x 3) x 5 = 2 x (3 x �)



5 x 3 x 12 = � x 1225 x � x 5 = 5 x 13 x 255 x � x � x 2 = 10 x �



9 x � = 3 x 126 x 5 = 5 x 3 x �� x 5 x 2 = 5 x 4

Bringing meaning to mathematical symbols


½(�) = 8½(�) = ½(6 - 2)1 = ½ + �2 + ¼ + � = 3

Read the first equation as: “Half of a number is the same quantity as 8”

Identity property for one


1 x � = 41 x = 1 x �1 x � = x 4

Bringing meaning to mathematical symbols


1/2x = x/23 x 8 = 2 x 8 + 8

Order of operations Open number sentences

�(4 + 2) = 3 x 4 + 3 x 2(16 + �) ÷ 2 x 5 = 20 ÷ 2 x 5

Do we need parentheses on the right side of the equation?

Order of operations Open number sentences

3 x � ÷ 5 = 30 ÷ 5

Order of operations True/false equations

2 x 6 ÷ 3 = 2 x 2

Order of operations True/false equations

5 x 4 ÷ 2 = 20 ÷ 2


The Struggle is Real: Anticipating Student Misconceptions

Katelyn Albright and Derek Pipkorn, Mequon-Thiensville School District

If we want students to be making mistakes, we need to give them challenging work that will be difficult for them, that will prompt disequilibrium. This work should be accompanied by positive messages about mistakes, messages that enable students to feel comfortable working on harder problems, making mistakes, and continuing on. (Boaler, 2016, p. 19)

As a middle school math specialist and a seventh-grade teacher in a suburban community, we strive to foster a classroom environment in which students are comfortable sharing mathematical strategies with their peers. Although some middle school students immediately embrace the opportunity to engage in productive struggle through challenging tasks and the sharing of ideas, it takes a supportive environment for other students to continue building their mathematical confidence. Regardless, we find that students’ willingness to take mathematical risks increases when they understand that mistakes are valued in our classroom. As students become more comfortable identifying and sharing their mistakes, it is important that we are there to support and prompt them with questions that continue their thinking rather than feeding them hints or clues toward the correct answer.

Anticipating student misconceptions is a strategy that we use to ensure we are asking our students questions to promote their own thinking. This strategy has become an important part of professional practice because it gives teachers the opportunity to collaboratively discuss the strategies that we expect to see from students as well as the mistakes that we predict will surface. While implementing tasks in the past, we often provided struggling students with a clue to guide them in the right direction. However, we now prompt them with a question that continues their thinking and gives all students the opportunity to find success from their own ideas.

In this article, we delve further into the importance of anticipating student misconceptions and planning teacher responses prior to implementing a rich task. We begin by establishing the need for teachers to plan for student misconceptions. We then share an example of how two middle school teachers prepared

and implemented a seventh-grade lesson that engaged students in productive struggle. Finally, we reflect on our experiences and provide major takeaways for educators to think about for their own classrooms.

The Importance of AnticipatingIn order for students to have the opportunity to

engage in productive struggle, it is critical that educators are equipped with knowledge and understanding of best teaching practices. As stated in Principles to Actions (National Council of Teachers of Mathematics, 2014),

For students to learn mathematics with under-standing, they must have opportunities to engage on a regular basis with tasks that focus on reasoning and problem solving and make possible multiple entry points and varied solution strategies. (p. 23)

One best practice that teachers can utilize in preparing to administer a rigorous task is to anticipate what misconceptions students might have during the task. As Smith and Stein stated (2011),

Anticipating students’ responses involves developing considered expectations about how students might mathematically interpret a problem, the array of strategies—both in correct and incorrect—that they might use to tackle it, and how those strategies and interpretations might relate to the mathematical concepts, representations, procedures, and practices that the teacher would like his or her students to learn. (p. 8)

The importance of the work put in prior to task implementation sets the stage for purposeful student collaboration, discourse, and understanding.

Preparing for the TaskThe artifacts in this article come from a task

implemented in three sections of seventh grade math in a suburban community near Milwaukee, Wisconsin. When selecting a task, it was important for us to pick one that complemented the ratio and proportional relationships content that students were already learning. We chose the task “7.RP Sale!” from Illustrative Mathematics (https://www.illustrativemathematics.org/content-standards/tasks/114) because it focused


on students’ understanding of markup and markdown problems, which align with CCSSM 7.RP.A.3 (National Governors Association Center for Best Practices [NGA] & Council of Chief State School Officers [CCSSO], 2010). The task was:

Four different stores are having a sale. The discounts for each of the stores are in the surrounding boxes. Which store gives the biggest price reduction? Which store gives the smallest price reduction?• Store A: Two for the price of one.• Store B: Buy one and get 25% off the second.• Store C: Buy two and get 50%off the second

one.• Store D: Three for the price of two.

This task also incorporated Mathematical Practice 4, “Model with mathematics” (NGA & CCSSO, 2010), which supported our work with tape diagrams. Our expectation was that students would communicate their thinking by creating models, writing their explanations, and sharing with their peers.In planning for student discourse, we chose to make groups of three students. We felt this was the best fit for our students because two students might limit the quality of conversation, and four students might allow some students to take a “back seat” during the conversation. We also wanted to be explicit that we expected students to record their work and justify their reasoning. Thus, we chose to use a place mat (see

Figure 1) as an organizer for the students because it placed the problem in the middle of the page with room for students to record their work surrounding it.

In preparation for task implementation, we anticipated potential student misconceptions that might appear while students worked collaboratively on the given task (see Table 1). By working together, we were able to come up with more student misconceptions than if we were to just brainstorm alone. Some misconceptions that we anticipated may surface with our students were not understanding how a sale works, feeling uncomfortable because they were not given an exact item or price, and not delineating between whether to use the percent off of an item or the percent paid for the item. We realized through brainstorming that most of the misconceptions would likely occur towards the beginning of the problem-solving process. We agreed that, although we still wanted to introduce the problem to the students before they began their work, it was important to not guide students toward entering the problem in a certain way.

We additionally came up with a set of prompts to respond to students who were stuck (listed in Table 1). In Hattie, Fisher, & Frey’s (2017) Visual Learning for Mathematics, prompts are defined as “questions or statements used to remind students to leverage what they already know in order to think

Figure 1. Place mat provided for student groups to record their work.


further” (p. 93). Hattie et al. also noted that “prompts should challenge students rather than do the thinking for them” (p. 93). By working together, it allowed us to accomplish this step in a small amount of time. Regardless of the time spent, we knew that we were now well equipped to present the task to the students and were also aligned in our understanding of the task.

Implementation of Task

As we implemented the task, we made certain that we did not overwhelm students with directions. Students transitioned into their groups of three, reread the task, and began their collaborative work (see Figure 2). While students worked, we circulated with our chart of student misconceptions. It was not long before

student questions arose or we observed some of the anticipated struggles. By having our chart of anticipated misconceptions ready, we were able to ask students probing questions that continued their conversations without revealing any strategies or answers.

During the student work time, we found it helpful to tally the number of times that we heard each student misconception in addition to recording new misconceptions (see Table 2 for additional misconceptions). We also recorded additional questions students brought forth. As we visited each group, we made notes of which groups we wanted to share during our whole class debrief. We were able to connect toward the end of student work time to quickly create an order for the groups to present based on the work, strategies, and misconceptions that we witnessed. For those students who finished early, we had an extension question ready to ask them: “Will your answers remain true for items of any value?”

Debriefing the Task

Using the order of student work that we selected and ordered during group work, it became an efficient process for us to create a student-lead debrief of the task. When we noticed that most students had come to a close on their work and many had transitioned to the extension question, we called everyone back together. Through the conversation, students were able to share their strategies using the SMARTBoard and document

Table 1Anticipated Student Struggles (Done Prior to Task Implementation)

Anticipated student struggle What might students do / say? What will we do if / when this happens?

Not having a set price for the item “How much does the item cost? Ask student if it matters and have them try one or two prices to compare

Not understanding how a sale works “Do I have to buy multiple items to get a discount?”

Discuss the process of a sale and how it works

Inability to find percent off “I’m confused by the 50%. What do I do with that?”

Ask students to consider a visual - connect 50% and 1/2

Not understanding that we are comparing the amount of savings

Students might think that the cheapest deal is the biggest amount of savings

Refer back to question - what does “price reduction” mean in terms of the problem?

Not sure of an entry point to the problem

Ask “How do we start?”Not engage/ side conversations

MP4 is the target - so we can ask How could you show the discounts for each of the stores using a model?

Figure 2. Students working collaboratively with place mat organizer.


camera (see Figure 3). They were able to ask questions to peers where they were still struggling, and they were able to make connections amongst their representations.

Reflection and Takeaways

This activity challenged our thinking on what we thought we knew about our students and how they actually responded. After planning, implementing, and reflecting on the task implemented, we determined five key takeaways from carrying out this process.

Anticipating student misconceptions should be collaborative work. Collaborating on a task allows for different points of view on student misconceptions. Teachers can role-play student responses and how to address them before actually implementing the task. This allows for classroom discussions to flow naturally while keeping the momentum of the task.

It is okay to not anticipate every student misconception. With a classroom full of young, bright minds, it is nearly impossible to anticipate every misconception. Don’t sweat it; this just leads to richer discussions in the debrief. Praise students for catching something that you didn’t! (Just remember to document that response for future use.)

Record or take notes of conversations to help you select or order student presenters for debrief. As you move from group to group, keep a clipboard and take notes on student responses. Put a check next to each anticipated misconception that you heard. Begin by letting students know that you would love to have them share their response or strategy with the whole group.

Debriefing at the end of a task is a not negotiable. Ensure that you have at least 5 minutes to debrief as a whole class. After selecting students to share responses, just facilitate and let your students lead the discussion. They just completed the task, and you already anticipated and documented responses, so let this flow naturally, and ask probing questions when appropriate.

The entire process takes time, but ensuring student success is worth it. Don’t get overwhelmed by doing this process for every rich task, but the habit will build naturally. Recruit a partner to support you because it will take time. Your students will reap the benefits of increased productive struggle opportunities with a teacher who is prepared to and comfortable with supporting them.

As Boaler (2016) suggested, tasks implemented in the classroom must prompt disequilibrium and mistakes. The small amount of time that we spent anticipating student misconceptions on our task

Table 2Student Misconceptions Identified During Task Implementation

Student struggle that we did not anticipate What students said or did What we said / how we responded

Not understanding that buying different amounts of items can offer the same price reduction.

“Store D can’t have the same reduction as Store A, because we are buying three items instead of two.”

Remind students about the overall goal of the activity.

Attempted to solve the problem algebraically.

Students assigned variables to full priced items and struggled to identify discounted variables.

Drew a visual representation to assist students in assigning algebraic terms for percent discounts.

Figure 3. Student-lead debriefing of the task.


had a long-lasting impact on the depth of student understanding. Conversation remained productive, and students worked together to achieve an outcome on the

task. Through the professional process of collaboration, implementation, and reflection, we as educators ensure the success of all our students.

ReferencesBoaler, J. (2016). Mathematical mindsets: Unleashing

students’ potential through creative math, inspiring messages and innovative teaching. San Francisco, CA: Jossey-Bass.

Hattie, J., Fisher, D., & Frey, N. (with Gojak, L. M., Moore, & Mellman, W.). (2017). Visible learning for mathematics, Grades K-12: What works best to optimize student learning. Thousand Oaks, CA: Corwin.

National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.

National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Authors.

Smith, M. S., & Stein, M. K. (2011). 5 practices for orchestrating productive mathematics discussions. Reston, VA: National Council of Teachers of Mathematics; Thousand Oaks, CA: Corwin.


Stop Counting By Ones: Seeing Twos and Threes as a Foundation for Kindergarten Mathematics

Laura McNelly, Milwaukee Public Schools

At the start of the school year, I began holding number talks with my Kindergarten students using dot patterns. To start a number talk, I would quickly flash my students a dot pattern and then ask my students to use their newly learned strategy of finding twos and threes in the pattern. Figure 1 shows one of the patterns and the response from one of my students. (The pattern in my classroom had two red dots, shown as black in the figure, and three yellow dots, shown as gray circles).

Before this year, I was never a teacher that encouraged my students to focus on the parts of a number; I was a teacher who wanted my students to find the whole amount. It was not until I immersed myself in a Master’s program at the University of Wisconsin-Milwaukee this past summer focusing on mathematics in early childhood that I realized I was putting the cart before the horse, meaning that I was expecting my students to see the whole amount before they could see parts within the whole. Our Kindergartners need to learn to subitize, which is the ability to see groups (those twos and threes) so that they can understand relationships between numbers, flexibly and meaningfully, giving them a strong foundation in number sense. For me, this meant having my students stop counting by ones all the time and asking them to start seeing twos and threes.

This article examines why it is critical for our youngest students to see smaller groups and how it will help them later in their mathematical learning. First, I will explain what subitizing is, what it looks like for your

students, and the difference between perceptual and conceptual subitizing. Second, I will explain how this concept connects to the Common Core State Standards for Mathematics (CCSSM; National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010) and learning trajectories. Lastly, I will explore some activities and ideas for instruction you can do with your students. If you are finding your Kindergartner whispering “1, 2, 3, 4, 5” and pointing to each dot, cube, counter, or cheerio, then this article will give you ideas for building a stronger mathematical foundation among your young learners.

Subitizing: What Is It? Let us break down subitizing. What is it? As

mentioned above, it is the ability to instantly know how many objects are in a group. Seeing and thinking in groups begins at a very young age, and it is important to encourage in both pre-Kindergarten and Kindergarten. The following developmental guidelines for subitizing were identified by Clements (2004) as• Ages 2–3: “Collections of 1 to 3” objects;• Ages 4–5: “Collections of 1 to 5” objects; and• Ages 5–6: “Collections of 1 to 5” objects, and

“patterns up to 10” objects (p. 28).Children learn two types of subitizing, perceptual

and conceptual. Perceptual subitizing is being able to see smaller quantities, such as the twos and threes, at

Student: I see 5 dots.Teacher: Tell me how you saw 5 dots.Student: I saw 2 red dots and 3 yellow dots so I put 2 fingers up and then 3

fingers up and then I counted all of my fingers and that was 5.

Figure 1. Dot pattern discussion for a number talk.


a glance without having to count, whereas conceptual subitizing focuses on working with larger numbers by seeing groups of smaller quantities within it. For example, using Figure 1 from before as a dot pattern, a student who recognizes the two red dots as "2" and does not need to count each dot is a perceptual subitizer. A student who sees the group of two red dots as well as the group of three yellow dots (without counting by ones) and can put the two subgroups together to see the whole group as the quantity of five, is a conceptual subitizer. Huinker (2011) notes that it is important to distinguish between perceptual and conceptual subitizing and explicitly help students develop both types.

In the operations and algebraic thinking progressions document for the CCSSM, the authors explain that “focusing attention on small groups in adding and subtracting situations can help students move from perceptual subitizing to conceptual subitizing in which they see and say the addends and the total” (Common Core Standards Writing Team, 2011, p. 8). Last year,

this is where I failed to overtly help my students find and name the parts (or as the progressions document says, the addends) and the whole (or total). Currently, I ask my students to name the parts and the whole. For example, for the number five, I would ask my students to tell me a way we can make five, some would say “2 and 3,” “4 and 1,” or maybe even “5 and 0.” By doing so, my students are explaining to me that they know when they have two objects and three objects, and if put together, that would be five objects all together; this is the foundation of addition. Thus, I now recognize the value of subitizing to focus attention on small groups to develop students’ addition and subtraction fluency and future mathematical skills.

The Why: Learning Trajectories and the CCSSMLearning trajectories help guide educator’s

instruction based on the student’s developmental progression in a specific math area. Each trajectory describes a path of student learning and how it grows

Table 1Learning Trajectory Developmental Levels for “Recognizing Number and Subitizing”

Level Level name Description

1 Small Collection Namer

The first sign of a child’s ability to subitize occurs when the child can name groups of one to two, sometimes three. For example, when shown a pair of shoes, this young child says, “Two shoes.

2 Nonverbal Subitizier

The child can name the value of a small collection (one to four objects) only briefly, the child can put out a matching group nonverbally, but cannot necessarily give the number name telling how many. For example, when four objects are shown for only two seconds, then hidden, child makes a set of four objects to “match.

3 Maker of Small Collections

The child can nonverbally make a small collection (no more than five, usually one to three) with the same number as another collection. For example, when shown a collection of three, makes another collection of three.

4 Perceptual Subitizer to 4

Progress is made when a child instantly recognizes collections up to four when briefly shown and verbally names the number of items. For example, when shown four objects briefly, says “four.

5 Perceptual Subitizer to 5

The child instantly recognizes briefly shown collections up to five and verbally names the number of items. For example, when shown five objects briefly, says “five.”

6 Conceptual Subitizer to 5+

The child can verbally label all arrangements to five shown only briefly. For example, a child at this level would say, “I saw 2 and 2 and so I saw 4.”

7 Conceptual Subitizer to 10

The child can verbally label most briefly shown arrangements to six, then up to ten, using groups. For example, a child at this level might say, “In my mind, I made two groups of 3 and one more, so 7.

Note. This table is modified from Sarama and Clements (2009, pp. 48–50).


along a number of levels. The trajectories help teachers answer questions about their students’ academic success, such as: What areas are my students proficient in, and in what areas do my students need more support?

The learning trajectory for “Recognizing Number and Subitizing” is shown in Table 1. You can see that the development path goes from a child being a perceptual subitizer to a conceptual subitizer. At the first three levels, the child does not call the group of objects by a number; instead, they are simply matching and making replicas of what they see. At level four children can correspond the number name to how many objects they see, such as visually perceiving and naming twos and threes. As students move further along the trajectory, they can take these foundational skills of seeing small groups of twos and threes to then build up to seeing larger groups of fours, fives, and sixes.

It is important for students to have the ability to recognize number values because it is a foundational part of number sense and develops over the course of several years. Beginning at about age two, children begin to name groups of objects, whereas the ability to instantly know how many are in a group begins at about age three. By age eight, with instruction and number experience, most children can identify groups of items and use place values and multiplication skills to count them (Sarama & Clements, 2009).

So, how does this learning trajectory tie in with the CCSSM? In the Operations and Algebraic Thinking (OA) domain, Kindergarten students are to “understand addition as putting together and adding to, and understand subtraction as taking apart and taking from” (National Governors Association Center for Best Practices [NGA] & Council of Chief State School Officers [CCSSO], 2010, p. 10). Within this cluster, one of the standards, K.OA.3, states that students are to “decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1)” (NGA & CCSSO, 2010, p. 11). The ability to decompose and compose numbers stems from the knowledge of subitizing.

In the first-grade standard 1.OA.6 in the OA domain, students are to “add and subtract within 20, demonstrating fluency for addition and subtraction within 10” (NGA & CCSSO, 2010, p. 15). Use strategies

such as counting on, making ten, decomposing a number leading to a ten, using the relationship between addition and subtraction, and creating equivalent but easier or known sums. Many of these strategies support the developmental levels of the learning trajectory of recognizing number and subitizing.

The How: Implementing Activities With Your Kindergartner

Here are a few activities that I use to support my students in seeing twos and threes. I use these activities as warm ups or as number talks in my classroom, especially at the beginning of the year, to familiarize them with seeing groups of twos and threes.

Environmentally seeing twos and threes. Start introducing children to identifying twos and threes in their daily life, such as two hands, two feet, two eyes, three cookies, or three cats. Ask your students to look for twos and threes in real-life objects. For example, I ask them how many characters they see in books or television shows or, when grocery shopping, to help their parents grab two apples or three bananas. Make it into a fun game for students to see how many twos and threes they can find at school in the classroom and hallways, and have them draw in a journal what they found.

Bunny ears. By using both hands, have students make fists on top of their head. Ask students to use their fingers to show you “3” by using both hands. You may see 2 fingers on one hand and one finger on the other hand, or students might show three as zero fingers and three fingers. Increase the quantity to four, five, six, and so on throughout the school year. The goal is for students to be able to put up the total number of fingers without having to count by ones.

Dot patterns. Making dot patterns on plates or

Figure 2. Sample dot patterns.


index cards is a valuable tool to help Kindergartners learn to subitize. Some samples of dot patterns are shown in Figure 2. When using dot patterns, make sure you quickly “flash” it to the students so they can visualize the group and not count by ones. You may show a pattern for a second flash, if needed, then ask the students how many dots they see and how they see it. If students are having trouble seeing the quantity that fast and “holding it in their heads,” have them draw what they see on a whiteboard or give them counters so they can make the pattern they see. To help reinforce my students to subitize and not count by ones, I explicitly ask my students to look for the groups of twos and threes, and then describe to me where in the pattern they see it (this also addresses student’s language skills of using positional words such as above, below, “next to,” or “in the middle”).

Closing ThoughtsIn closing, I am now finding that my students

this year have an immensely stronger foundation in number sense then my students in the past because of my instruction with these activities and asking my students to see twos and threes. Of course, I still have students that count by ones, particularly when showing more than five objects, but my instructional goal is to help all students learn to see twos and threes. As an educator, I have significantly transformed my teaching in early mathematics through the research and activities presented in this article, which has benefited my students’ foundational skills in being stronger math students. This has made a huge impact for on young learners by teaching them to stop counting by ones and to start seeing twos and threes.

ReferencesClements, D. H. (2004). Major themes and

recommendations. In D. H. Clements & J. Sarama (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics education (pp. 7–72). Mahwah, NJ: Erlbaum.

Common Core Standards Writing Team. (2011). Progressions for the common core state standards in mathematics (draft): K–5 progression on counting and cardinality and operations and algebraic thinking. Tucson: University of Arizona. Retrieved from https://commoncoretools.files.wordpress.com/2011/05/ccss_progression_cc_oa_k5_2011_05_302.pdf

Huinker, D. (2011). Beyond counting by ones: "Thinking groups" as a foundation for number and operation sense. Wisconsin Teacher of Mathematics, 63(1), 7–11.


Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. New York, NY: Routledge.


Engaging Our Students in Upper-Level Mathematics: Finance Applications

Dave Ebert, Oregon High School, Oregon, WI

This is the third article in a continuing series of articles on engaging our students in upper-level mathematics. Although these activities are intended to be used in an upper-level mathematics class such as Algebra 2, Precalculus, or Integrated Mathematics 3, they can be revised to be used in any high school mathematics course.

There are state and nationwide calls for all students to have an increased awareness and understanding of financial literacy. The National Council of Teachers of Mathematics’ (2014) publication Principles to Actions: Ensuring Mathematical Success for All urges teachers to implement tasks that promote reasoning and problem solving. This call applies to teachers of all levels of K–12 mathematics. At all levels, it is often difficult to balance the need to engage our students with relevant tasks and the need to prepare our students for their

future mathematical studies. Ideally, teachers are able to do both.

In upper-level mathematics classes, students need to learn about exponential and logarithmic functions. It is clear from the Common Core State Standards for Mathematics (CCSSM; National Governors Association for Best Practices [NGA] & Council of Chief State School Officers [CCSSO], 2010) that there is a need for students to be proficient with exponential and logarithmic functions in the context of financial literacy. Relevant standards appear in three separate high school categories within the CCSSM, and multiple times within those categories. Table 1 lists a number of these standards that relate to financial literacy.

In 2006, the Wisconsin Department of Public Instruction released Wisconsin’s Model Academic Standards for Personal Financial Literacy. These

Table 1Standards Relating to Financial Literacy (NGA & CCSSO, 2010)

Category Example Page

Algebra “Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret as the product of P and a factor not depending on P.”

64

“Use the properties of exponents to transform expressions for exponential functions. For example

the expression can be rewritten as to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.”

64

Functions “The return on $10,000 invested at an annualized percentage rate of 4.25% is a function of the length of time the money is invested.”

67

“Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as

, and classify them as representing exponential growth or decay.”

69

“Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.”

70

“Express as a logarithm the solution to where a, c, and d are numbers and the base b is 2, 10, or e.”

71

Modeling “Modeling savings account balance … or investment growth.” 72


standards are intended to be integrated across the curriculum. “Teachers in every class should expect and encourage the development of these shared applications, both to promote the learning of the subject content and to extend learning across the curriculum” (p. xvi). They also include many connections to the mathematical standards in the CCSSM (NGA & CCSSO, 2010).

Some standards identified in Wisconsin’s Model Academic Standards for Personal Financial Literacy (Wisconsin Department of Public Instruction, 2006) include the following. Students will:• “Apply various money management strategies to

authentic situations and predict results over time” (p. 7),

• “Identify and evaluate interest rates, fees, and other credit charges” (p. 8),

• “Calculate the cost of borrowing” (p. 8),• “Compute and assess the accumulating effect of

interest paid over time when using a variety of sources of credit” (p. 10),

• “Calculate and compare the total cost of borrowing for various amounts and types of purchases” (p. 10),

• “Apply strategies for creating wealth/ building assets” (p. 12), and

• “Evaluate the effect of ‘compounding’ earned interest” (p. 13).The standards go on to state that “Educators from

all grade levels can use the financial literacy standards to align instruction and create curriculum and activities designed to instill within students a desire to be financially literate” (Wisconsin Department of Public Instruction, 2006, p. 1).

Thus, it is clear from these various standards that there is a need to teach our students about exponential and logarithmic functions as well as to have them apply their knowledge in the context of a variety of financial applications. What follows is an example of an engaging, real-world project that has been used with precalculus students with a great deal of success.

The Exponential Equations and Finance Project (see Appendix) is divided into two main sections. In the first section, Problems 1–6, all of the students solve the same finance problems. The second section, Problems 7–11, has students explore their own unique situations. In the first section, the problems are ordered similar to how the students will encounter situations in their lives, starting with credit card debt, new car financing, and auto depreciation. It then continues with saving

for a down payment on a house, retirement saving, and home financing. All students work on the same realistic problems and get the same answers. Students are surprised to learn how much compounding interest can work against them on a credit card and how much it can work for them when saving for retirement. Students are also surprised to learn that saving a small amount of money for a longer time is much more beneficial than saving a large amount of money for a shorter time. Additionally, students are shocked to learn that the total amount paid to finance a house is often over double the actual cost of the home.

In the second section of the project, students need to research the cost of items that they wish to buy, such as computers, cars, and houses. They also research the interest rates of credit cards, savings accounts, and home mortgages and then apply what was learned in the previous problems to their own unique situations. In this section, many students use their own personal savings account rates and use their family’s car and house as examples. This often leads to discussions with their parents about finances and is very eye opening to the students.

Students are given 3 days of class time to work on this project and usually need to devote some additional time outside of class to complete the project. Day 1 in class is devoted to completing problems in the first section, with students using Days 2 and 3 to research and complete problems in the second section. Students may use a computer, tablet, or phone to find all of the information needed. Use of a calculator or Desmos is encouraged to compute their solutions.

The final question asks students to reflect on what they have learned in this project. Students often share that their learning extends beyond what is typically learned in a mathematics class and will hopefully lead to a longer impact on the students’ lives. Some of the responses from my students who completed this project last semester include the following.

I learned that you should start saving early for retirement savings because the interest will start gaining more and more. I also learned that you should pay off your charges every month from your credit card because if you wait the interest will gain more and more so you will owe more than if you pay it off at the end of the month.I learned that you should start saving money for retirement early so it has more time to collect interest.


ReferencesNational Council of Teachers of Mathematics. (2014).

Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.

National Governors Association for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Author.

Wisconsin Department of Public Instruction. (2006). Wisconsin's model academic standards for personal financial literacy. Madison, WI: Author.

You should also either try to get a low interest rate on a loan or pay if off as soon as possible so you don’t pay a high amount of interest.This was actually a really interesting project for me. It really helped show me how much interest can affect the prices of things over long periods of time. I also realized how affordable some things can be depending on the amount of time you plan on paying something. I really liked this project!

An additional benefit of this project is that, although the questions are the same for every student, the problems that the students solve are personalized around their interests. I gain valuable insights into their lives, having individual conversations with them about their collegiate plans, material desires, dream cars, and activities outside of school. For example, I recently learned that one of my students has a passion for high-end sports cars and aspires to be a collector someday.

I also learned that a different student works in the fashion industry, buying and selling merchandise for a local retail store.

A final benefit of this project was recently shared with me. I ran into a parent of a student, and she told me that her daughter had set up an investment account because of what she had learned through working on this project. Ideally, every student applies what they learned about finances in their future, and the benefits of completing this project extend far beyond what is learned in class.

Teaching upper-level students about exponential and logarithmic functions in the context of finance gives students a relevant, engaging learning experience. By making these connections, the students’ learning extends beyond the classroom and into their lives, leading to increased financial literacy and, ideally, to some wise long-term financial decisions.


APPENDIX

Exponential Equations and Finance Project

Section 1

1. At the beginning of your freshman year of college, you buy a pizza for $25 and charge it to your credit card. The credit card charges 18% interest compounded continuously.

• How much money do you owe when you graduate from college after five years?

• Let’s say you forget about your credit card bill and decide to pay it off when you retire. How much money do you owe after 40 years?

2. After you get your first “real job”, you decide to buy a new car. You decide on a new 2016 Honda Accord for $22,225. The financing the dealer offers you is for $471 per month for five years. What is the total amount that you will have paid for your car?

• How much of this amount is interest?

3. A new SUV that sold for $30,788 has a book value of $24,000 after 2 years. Find an exponential depreciation model for the SUV.

• What will be the value of this SUV after 5 years?

4. You want to start saving money to use as a down payment on a house. You decide to put $200 each month into a savings account that pays 3% interest.

• The formula for this type of account is:

• Where A = final amount, P = amount deposited each month, r = interest rate, and n = number of months.

• How much money is in this account after one year?

• How much money is in this account after five years?

• How long will it be until you have saved $25,000?

5. After you get your first job, you start to save money for retirement (it’s never too early to start). The formula for this type of account is the same as the one used in problem #4.

• If you start early and save $100 per month in an account with an interest rate of 8%, what is the total amount after 40 years?

• If you start later in life and save $200 per month in an account with an interest rate of 8%, what is the total amount after 20 years?

• Compare the answers to the two questions above. Write what you should do to save money for retirement.

6. After a few more years, you decide to buy a house. The house you select has a price of $250,000. You have saved $25,000 to use as a down payment.

• The formula for monthly payments is:

• Where P = monthly payment, L = loan amount, r = interest rate, and n = number of months.

• If you get a 30-year loan with an interest rate of 7%, what is your monthly payment?

• After 30 years, what is the total amount you will have paid?









Section 27. Research the cost of some item you’d like to buy and a credit card rate.

• What are you buying?

• What is the cost?

• Where did you find this price?

• What is the credit card rate?

• What kind of credit card is it?

• Where did you find this credit card rate?

• Use the above cost and credit card rate to find out how much you would owe after five years.

• Do you think that you’ll still be using this item after five years? Explain.

8. Research the cost of a new car.

• What kind of car are you getting?


• Where did you find this car price?

• Us the formula in problem #6 to find the monthly payment for a five year loan at 6% interest.

• What is the total amount you will have paid for this car?

• How much of this is interest?

9. Research the value of a 2-year old model of the same car you bought in problem #8.

• What is the value of this car after two years?

• Where did you find this information?

• Write an exponential depreciation model for the value of this car.

• What will be the value of this car after five years (when you’re done paying it off)?

10. You want to start saving money. Realistically, how much money could you save each month right now? Put this amount in the first three rows of the table on the next page.

• Research the interest rate for a savings account at a bank. What is the interest rate?

• What bank is this at?


• Research the interest rate for a Certificate of Deposit (CD). What is the interest rate?

• What bank is this at?


• Historically, the long-term interest rate paid on money invested in the stock market is about 8%.

• Complete the table below using the interest rates you have found.


Type of account Interest rateAmount invested

monthly Time (years) Final amount

Savings 10

CD 10

Stock market 10

Savings 100 10

CD 100 10

Stock market .08 100 10

Savings 200 10

CD 200 10


Savings 200 40

CD 200 40


• If you saved $200 each month at a rate of 8%, how long would it take you to have $1,000,000?

11. Research the cost of buying a house and the mortgage interest rate.


• How many bedrooms and bathrooms does this house have?


• What is the interest rate for a 30 year fixed loan?


• Calculate the monthly payment for this house.



12. Please write a paragraph about what you have learned in this project.

The Wisconsin Mathematics Education Foundation (WMEF) was founded in 2010 in order to support math education in the State of Wisconsin. WMEF's mission is to financially support excellence in mathematics education for students and teachers in the State of Wisconsin by providing educational opportunities so that all Wisconsin students will acquire the mathematics skills necessary to be productive citizens ready to globally compete in the 21st century.

WMEF is growing in the amount of donations each year, increasing the number of ways we invest in math educators, and expanding the number of ways that fellow educators can get involved in the organization. The following are some of the grant and scholarship opportunities available from WMEF.

WMEF Grant Opportunities• Henry Kepner Long Term Professional Development Grants: The purpose of this grant

is to provide financial assistance to teachers, grade level groups, or school districts for long-term professional development pertaining to mathematics education.

• Julie Stafford Professional Development Grants: WMEF awards grants of up to $1,500 each to encourage and support the efforts of an individual or teams of mathematics educators to take course work or to attend conferences or workshops.

• Material/Resources Grants: The purpose of this grant is to provide financial assistance to teachers, grade level groups and school districts for the purchase of classroom material to support in the development and implementation of innovative teaching strategies or projects in the field of mathematics.

• Student Activity Grants: WMEF awards up to $500 to encourage the involvement of students in mathematics activities beyond their regular classrooms.

WMEF Scholarship Awards• Sister Mary Petronia Van Straten and Jane Howell Scholarships: WMEF offers two

$2,000 scholarship opportunities for Elementary and Secondary Education Majors that honor Sister Mary Petronia Van Straten and Jane Howell.

• Arne Engebretsen Memorial Scholarship: WMEF offers one $2,000 scholarship opportunity for a high school senior who plans to major in mathematics education or to have a mathematics concentration at the elementary or middle level.

For more information, please visit:http://wmefonline.org/

Wisconsin Mathematics Council, Inc. 2019 State Mathematics Contests

The WMC State Math Contests for both middle and high school students will be held the week of March 4-8, 2019. Schools participate at their home sites, and each school chooses the day the team will participate in the contest. The entry fee is $25 for WMC members and $50 for non-members.

Registration and information available at http://wismath.org/WMC-Math-Contest.

The school team advisor corrects the individual events and all of the items on the team event, and tests and results are returned to WMC. Medals are given to individuals from top-scoring high school teams, ribbons to high scoring middle school teams, plaques to top scoring schools. Top team and individual scores are published in the WMC Newsletter, and high school students with perfect scores are awarded $50, middle school students receive a medal.

The meet consists of five sets of problems. For each of the first four problem sets, each individual student will attempt to solve problems worth 1, 3, and 5 points, respectively. For the team problem set, all team members will work cooperatively to solve 6 problems, each worth 10 points.

Each school may enter one team. A team consists of eight students (all from the same school) consisting of no more than four seniors for high school teams and four 8th grade students for middle school teams.

School enrollment is used to determine the appropriate class for each school. A school may compete in a higher classification upon request. The classes/divisions are determined according to the following:

Class Grades 6-8 Grades 9-12

AA 901 & above 1201 & above

A 501-900 751-1200

B 301-500 501-750

C 300 & below 500 & below

For more information, please visit www.wismath.org.

May 1 -3, 2019

Green Lake

Conference Center

KEYNOTE SPEAKERS

Graham

Fletcher

Ron

Lancaster

Jennifer Lempp

Sponsored by HMH

RunningHorse

Livingston

Fawn

Nguyen

Andrew

Stadel

Tracy

Zager

Registration and Conference Information

WISCONSINMATHEMATICS COUNCIL, INC.

PO Box 130Holmen, WI 54636

2018 | issue 2 teacher of mathematicswismath.org/resources/documents/wtm issues/wmt_2018_issue...the...

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