Rules that Expire: "Just add a zero!"

By Cassandra Hatfield, RME Assessment Coordinator

Many tips and tricks that we teach our elementary students as rules of mathematics, are introduced as a way to help students recall a procedure rather than truly promote their conceptual understanding of the content. However, many of these rules learned early on don’t hold true as students start to learn more advanced content in middle and high school.

An article in Teaching Children Mathematics, 13 Rules that Expire, by Karp, Bush and Dougherty addresses some of these common misconceptions. Let us know if you see these rules that expire in your classroom, and how you address them.

The first rule we are going to talk about is, "Just add a zero!"

When you multiply 4 by 30 what strategy do you use?

Consider these possible strategies for solving this problem:

Strategy A	Strategy B
4 times 3 is 12. Then add a zero and you get 120.	4 times 3 is 12.  12 times 10 is 120.

At first glance one may think both of these strategies are appropriate. However, use the same strategies to multiply 0.4 by 30:

Strategy A	Strategy B
0.4 times 3 is 1.2. Then add a zero, so 1.20.	0.4 times 3 is 1.2.  1.2 times 10 is 12.

The strategy of adding a zero to the right of the number when multiplying by a multiple of 10 only applies to whole numbers, and can’t be generalized. Additionally, utilizing this trick of “adding a zero” isn’t mathematically sound, and does not support students in reasoning and justifying their answer.

Let’s take a look at the mathematics behind Strategy B for each of the above problems.

4×30	0.4×30
4×3×10	0.4×3×10	Decomposition or Partitioning into Factors
(4×3)×10	(.04×3)×10	Associative Property of Multiplication
12×10=120	1.2×10=12

Elementary students can and do use the properties of operations when computing; it’s our job as teachers to help students see and understand the value of the mathematics behind each strategy.

Cluster problems are one way to support students with using facts and combinations they likely already know in order to solve more complex computations (Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M., 2016). Here’s a set of cluster problems that lead to 34 x 50. Consider how these problems are related and the rich discussion you can have with students about the properties of operations they used to get their final answer.

4×5

3×5

3×50

30×50

34×50

Karp, K.S., Bush, S.B., & Dougherty, B.J. (2014). 13 Rules that Expire. Teaching Children Mathematics, 21 (1), 18-25.

Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2016). Elementary and middle school mathematics: Teaching developmentally (9th ed.). Boston: Pearson.

Creating New Learning Opportunities with FAQs

By Brea Ratliff, RME Secondary Mathematics Coordinator

One of the most important components of any informational resource is a section labeled “Frequently Asked Questions” or “FAQs”. The FAQ section is often easy to find, can be a very helpful tool when you need to quickly find the answer to a question you might have, or you need a reminder of a process or an idea you had seen previously and need a refresher. Whether you are a novice or an expert, the FAQs are helpful for everyone.

So, where are the FAQs in the classrooms? Would students know how to access answers to the pertinent and relevant questions they have about whatever concept they are learning? More important, as teachers, are we aware of some of the questions students might have which could be included in a FAQ section about our classroom?

Here are a few strategies for helping you establish an FAQ space in your math classroom.

1.  Have a clear understanding of your expectations. If we anticipate our students will rise to our expectations, we must be clear about what the expectations are. Many of our expectations are outlined in a syllabus or a letter that goes home to parents at the beginning of the year, but what about our expectations for learning mathematics in the classroom? Here are a few questions to consider:

• What are my expectations for collaboration in the classroom?
• What techniques will I use to ensure my students comprehend what they are learning?
• What opportunities can I provide for students to communicate with me when they have questions about the math?

For the concept you are teaching, identify the major misconceptions or misunderstanding students might have. Understand the background knowledge necessary for being successful with this topic, as well as why the topic is foundational for future studies. Emphasize content vocabulary and mathematical processes. The FAQ is not only a resource, but can be used as an evaluative tool to help you identify what students do and don’t understand about a concept or unit of study.

Here are some example questions for a lesson or unit focused on dividing fractions:

2.  Get to know your audience. Most – if not all – people want to succeed in whatever they do, and can sometimes feel embarrassed to ask questions. I know many middle and high school students who would never raise their hands and tell their teachers, or the entire class, they don’t understand math concepts taught to them years earlier. I also know several highly educated adults who would rather “play it safe” and not ask questions, out of the fear of looking as if they don’t know something. An FAQ space can make learning accessible for everyone.

3.  Find your medium. So, now you’ve developed your FAQs, but where will you keep it? As I mentioned earlier, a syllabus can be a great starting point, but let your creative juices flow when selecting your medium. Try creating an FAQ bulletin board in the classroom, or maybe adding an FAQ section to your classroom website. Use Twitter as an FAQ space or create posters throughout the school so students can see and be reminded of these ideas outside of your classroom.

4.  Make it collaborative. One of the greatest rewards of being an educator is the gift of being a teacher and a student at the same time. To quote science fiction author Robert Heinlien,“When one teaches, two learn.” As you teach concepts, allow your students to draft and share questions to be added to your FAQs.

Please share some of your FAQ space examples with us on Twitter at @RME_SMU

Quote by Robert Heinlein. Think Exist http://thinkexist.com/quotation/when_one_teaches-two_learn/149371.html retrieved 23 September 2014

RME at CAMT - June 24-26

Looking for a good conference this summer? Come join us at CAMT - the Conference for the Advancement of Mathematics Teaching in Houston on June 24-26. CAMT is an annual Texas conference for K-12 mathematics teachers. The conference is sponsored jointly by the Texas Council of Teachers of Mathematics, the Texas Association of Supervisors of Mathematics, and the Texas Section of the Mathematical Association of America.

If you have never heard of the CAMT Conference, visit their website to learn more.

We have several members of our team presenting this summer. Come join us at one of the following sessions!

ESTAR and MSTAR: Supporting RtI in Texas, Wednesday, 10:00: This session will inform teachers about ESTAR (Elementary School Students in Texas: Algebra Ready) and MSTAR (Middle School Students in Texas: Algebra Ready), a TEA initiative that is available at no cost to all Texas public school districts. ESTAR and MSTAR support grades 2 to 8 by improving overall mathematics instruction and impacting student achievement.

Interpreting MSTAR Universal Screener Reports, Wednesday, 1:00: Universal screening is a step in the RtI process to identify students who may be at risk for success in mathematics. This session will provide a brief overview of the MSTAR (Middle School Students in Texas: Algebra Ready) Universal Screener and describe how to interpret the results.

Interpreting ESTAR Universal Screener Reports, Thursday, 10:00: Universal screening is a step in the RtI process to identify students who may be at risk for success in mathematics. This session will provide a brief overview of the ESTAR (Elementary School Students in Texas: Algebra Ready) Universal Screener and describe how to interpret the results.

The Anatomy of High-Quality Multiple Choice Assessment Items, Thursday at 1:00 and Friday at 8:30: In this session, participants will learn the different purposes for giving students assessment items, how to develop high-quality items that adhere to best practices in assessment development, how items can be crafted to target increasingly sophisticated levels of understanding, and how to use data obtained from multiple-choice items to inform instruction.

Interpreting MSTAR Diagnostic Assessment Reports, Friday, 8:30: In the RtI process, diagnostic assessments are given to students in order to determine what areas and specific misconceptions a student might hold. This session will provide a brief overview of the MSTAR (Middle School Students in Texas: Algebra Ready) Diagnostic Assessment and describe how to interpret the results.

Interpreting ESTAR Diagnostic Assessment Reports, Friday, 10:00: In the RtI process, diagnostic assessments are given to students in order to determine what areas and specific misconceptions a student might hold. This session will provide a brief overview of the ESTAR (Elementary School Students in Texas: Algebra Ready) Diagnostic Assessment and describe how to interpret the results.

RtI Guidance at Your Fingertips, Friday, 10:00:This session will inform teachers and administrators about an ongoing initiative by the Texas Education Agency to support educators’ understanding of Response to Intervention (RtI). The RtI iOS project delivers best practices in RtI through a mobile application and complementary website.

RME Conference Morning Breakout Summaries

Our RME Conference was held at the end of February. Below are summaries of the morning breakout sessions.

Morning Breakout 1 – Solving Word Problems Using Schemas

Presented by Dr. Sarah Powell and facilitated by Cassandra Hatfield

In this session, Dr. Sarah Powell, presented problem solving strategies teachers can use to help
elementary students organize their thinking when approaching word problems. Dr. Powell emphasized the importance of teaching students to recognize schemas, specifically additive and multiplicative problem types. The example word problems used in Dr. Powell’s presentation highlight the importance of teachers moving beyond problem solving strategies that place emphasis on the identification of “key words”, and suggested students should instead focus on understanding the context and meaning of the language used in word problems. Dr. Powell also suggested students should have a strategic plan for solving word problems that is used regardless of the problem type. In order to ensure all students are familiar with the same problem solving processes, Dr. Powell suggests educators adopt a problem solving strategy for their entire school.

• Students need an “attack strategy” anytime they solve a word problem. Regardless of the problem type, students should know what process they will use to solve a given word problem. Many attack strategies involve reading the word problem, paraphrasing the question, developing a hypothesis, using a diagram or equation to represent a process, estimating or computing an answer, and checking your work. These strategies could be considered an algorithm for solving a word problem. Examples include R.I.D.G.E.S., S.T.A.R., D.R.A.W., S.I.G.N.S., and S.O.L.V.E.
• Students should not be encouraged to identify “key words” as a strategy for solving word problems. Students should understand the context and meaning of all language within a word problem.
• When using strategies, it is important to help students identify the three problem types for addition/subtraction (additive schemas) and four problem types for multiplication/division (multiplicative schemas). Additive schemas include part-part whole, difference, and change (join/separate). Multiplicative schemas include

Morning Breakout 2 – Mathematical Problem Solving in Real World Situations

Presented by Dr. Candace Walkington and facilitated by Megan Hancock

At the 2015 RME conference, Dr. Walkington spoke about personalization matters! Specifically in mathematics, it is important that students feel personally connected to what they are studying. This is central to helping some students feel more comfortable and be more successful. Personalization means that instruction is tailored to the specific interests of different learners and problems are introduced using different topics that can be implemented efficiently through technology systems. Students have rich engagement with their interest areas. It is important that instructors incorporate students’ passions into what they are learning.

Personalization interventions should seek to include depth, grain size, ownership, and richness. Depth means to make deep meaningful connections to the ways students’ use quantitative reasoning. Grain size refers to knowing the interests of individual learners. Ownership allows students to control the connections made to their interests. Lastly, richness means to balance rich problem solving with explicit connections to abstractions afterwards. If instructors can implement these important personalization interventions in their mathematics teaching, students will feel more connected to their learning and likely be more successful as well.

• The TEKS Process Standards should be interpreted through real-world situations. Students should be introduced to a topic they can relate to, then, the specific mathematics topics should be brought in after they have a firm understanding of the context.
• Studies show that students learn best from concrete thinking to abstract thinking. The teacher teaches the content using concrete scenarios and then moves to abstract thinking after the students understand the math content.
• When mathematics is connected to students’ interests, they can gain a better understanding of the content being taught. Students with little exposure to algebra can reason about and write a linear function in the context of their interests without realizing they are using algebra. This peaks their interest, then the teacher can follow up with the concrete mathematics topics.

Morning Breakout 3 – Fostering Small-Group, Student-to-Student Discourse: Discoveries from a Practitioner Action Research Project

Presented by Dr. Sarah Quebec Fuentes and facilitated by Becky Brown

This session focused on the use of small group peer discussions to increase student understanding with an emphasis on communication. Three of the math process standards include communication, quality communication with reasoning, explaining, and justifying. By asking the students to communicate, you are effectively changing the way they approach mathematics. When you put kids into a group they will communicate but the communication is not always of quality. The teacher’s role is to facilitate the discussion, not to set a rubric or tell them exactly what to do. Students gain process help through their peer interaction, which aids their problem solving abilities by increasing their adaptive qualities. This type of meaningful communication is achieved
through the Action Research Cycle: planning, acting, observing, and reflecting.

• You can improve student communication in your own classrooms in three phases. Stage 1 is to evaluate student communication and just get them to communication. Stage 2 is to evaluate group communication. Which point on the action cycle is this group? Stage 3 is to evaluate your communication. Are you effectively facilitating meaningful discussion? Lastly Stage 4 is to try a customized intervention.
• There is no blanket intervention strategy because each team interacts differently and operates in different phases of the action cycle.
• This practice can be scaled to an entire math department as long as it is scaled down and adjusted for the time needs of the professional.

Supporting English Learners in the Mathematics Classroom

By Dr. Deni Basaraba, RME Assessment Coordinator

The number of English Learners (ELs) in the United States is growing at an unprecedented rate that shows no signs of slowing. As of 2013, for example, over 60.6 million people (21%) spoke a language other than English in the home and, of those, 37.6 million (62%) spoke Spanish in the home (Ryan, 2013). Moreover, the National Center for Educational Statistics (2011) reported that the number of ELs attending public schools has increased in the last three decades, from 4.7 million to 11.2 million. In Texas specifically, the percentage of students classified as ELs increased from 15.3% to 17.5% from 2003 to 2013 and the percentage of students receiving bilingual or English as a second language services grew from 14% to 17.1% (Texas Education Agency [TEA], 2014). This steady increase in the number of ELs attending our schools, combined with a persistent achievement gap in mathematics on both state and national assessments on which ELs exhibit consistently lower levels of proficiency than their non-EL peers, (NCES, 2013), underscore the need to ensure that our mathematics instruction incorporates evidence-based principles of instructional design and delivery to support the development of ELs mathematics understanding and proficiency.

Listed below are three research-based recommendations for supporting ELs mathematics understanding and proficiency.

Situate mathematics problems in contexts that are familiar to students. One of the primary goals of education is to provide students with instruction and practice in skills that they can generalize outside of the classroom to real-world contexts. Consequently, situating mathematics problems for students to solve in contexts that are familiar to them is important not only because it increases their likelihood of engaging in meaning-making actions that rely on conceptual understanding (as opposed to carrying out rote procedures) (Domínguez, 2011) but also because it increases students’ engagement in the problem-solving process (Brenner, 2002; Domínguez, LópezLeiva, & Khisty, 2014). Examples might include: grocery shopping, preparing meals, playing video games, reading books aloud to siblings and/or adults, or eating meals in the school cafeteria.

Focus explicitly on mathematical vocabulary. Although proficiency in mathematics requires students to think in terms of abstract ideas, concepts, and symbols that may be similar across languages, this does not support the common misconception that mathematics is “culture free” (Garrison & Mora, 1999). Rather, it could be argued that explicit instruction of mathematics vocabulary may be critical for some ELs because some mathematical words such as odd, times, table, or line may have specific mathematical definitions that are different than their meaning in everyday conversation (Fang, 2012; Garrison & Mora, 1999; Schleppegrell, 2007)

Strategically incorporate visual representations and manipulatives. One means of fulfilling the recommendation for developmental mathematics instruction put forth by the National Council of Teachers of Mathematics (NCTM, 2000) is to scaffold students’ understanding of abstract mathematical concepts with concrete and visual representations. Concrete representations, or manipulatives such as tangrams, for example, can be used to provide students with tangible experience with mathematical concepts such as greater than and less than, larger and smaller, or concepts of size (e.g., small, smaller, smallest) (Garrison & Mora, 1999). Visual representations, such as graphs or tables, may be useful methods for helping ELs to communicate their preliminary understanding of complex mathematical concepts such as multiplication or division that can be represented graphically more easily than they can verbally or with written words. Not only do these representations provide ELs with opportunity to see and touch while simultaneously being exposed to new mathematical vocabulary, but they also provide ELs with access to the key mathematical concepts in formats that don’t require dependence on language (Cirillo, Bruna, & Herbel-Eisenmann, 2010).

References

Brenner, M. (2002). Everyday problem solving and curriculum implementation: An invitation to try pizza. In M. E. Brenner & J. N. Moschkovich (Eds.) Journal for research in mathematics education. Monograph (Vol. 11): Everyday and academic mathematics in the classroom (pp. 63-92). Reston, VA: National Council of Teachers of Mathematics.

Cirillo, M., Bruna, K. R., & Herbel-Eisenmann, B. (2010). Acquisition of mathematical language: Suggestions and activities for English language learners. Multicultural Perspectives, 12, 34-41.

Domínguez, H., LópezLeiva, C. A., & Khisty, L. L. (2014). Relational engagement: Proportional reasoning with bilingual Latino/a students. Educational Studies in Mathematics, 85, 143-160.

Domínguez, H. (2011). Using what matters to students in bilingual mathematics problems. Educational Studies in Mathematics, 76, 305-328.

Fang, Z. (2012). Language correlates of disciplinary literacy. Topics in Language Disorders, 32, 19-34.

Garrison, L., & Mora, J. K. (1999). Adapting mathematics instruction for English-language learners: The language-concept connection. Changing the Faces of Mathematics: Perspectives on Latinos, 35-48.

National Center for Educational Statistics. (2013). NAEP data explorer [Data file].Washington, DC: U.S. Department of Education. Retreived from http://nces.ed.gov/nationsreportcard/naepdata/report.aspx.

Ryan, C. (2013). Language use in the United States: 2011. American Community Survey report (ACS-22). U.S. Census Bureau; U.S. Department of Commerce. Retrieved 02/26/14 from http://www.census.gov/prod/2013pubs/acs-22.pdf

Schleppegrell, M. J. (2007). Linguistic challenges of mathematics teaching and learning: A research review. Reading & Writing Quarterly, 23, 139-159.

Texas Education Agency (2014). Enrollment in Texas public schools: 2013-2014. (Document No. GE15 601 03). Austin, TX: Author.

RISD Enterprise City - Tackling Financial Literacy

By Brea Ratliff, RME Secondary Mathematics Coordinator


The revised mathematics TEKS for grades K-8 include a strand addressing financial literacy. The student expectations within this strand were developed to ensure students have a fundamental understanding of economics, and the skills connected to being a consumer and investor (TEA, 2012).

While many schools and school districts are for the first time investigating ways to implement these standards, a program created by the Richardson Independent School District could serve as a prototype for educators looking to cultivate students’ understanding of financial literacy using a real-world model.

Enterprise City is a miniaturized representation of an actual city, and is housed on the campus of Canyon Creek Elementary in Richardson, Texas. Almost 30 years ago, Richardson ISD developed Enterprise City to promote students’ understanding of economics through the development of an innovative interdisciplinary curriculum. While students begin ascertaining the essentials of business and financial literacy through classroom experiences, their knowledge is put into action when given the opportunity to manage the operations of the city for one day.

In addition to gaining an appreciation for the fiscal responsibilities of maintaining a city, students are utilizing critical thinking skills and being engaged citizens as they work collaboratively with other students to support businesses and organizations within the city. I had a chance to interview Jodi Freeman, Enterprise City Program Coordinator, to learn more about the program and how it has impacted their district.

How was the Enterprise City curriculum developed?

In 1983 one of our superintendents visited a program called Exchange City in Kansas City. The curriculum was developed based on that program and has been revised several times since then. It teaches basic economic concepts and personal finance.

Several financial literacy concepts have been embedded within the Enterprise City curriculum since it was established in 1985. How has the adoption of the new mathematics TEKS impacted the curriculum?

Our curriculum has focused on PFL and free enterprise from the start. Students complete job applications, practice money management by using a checkbook, maintain a register and understand the purpose of a debit card. They also formulate advertisements for their business, secure a bank loan, and budget for their business as well as develop an understanding of their role as a good citizen.

How has Enterprise City impacted the community (both RISD and the city of Richardson)?

Our program has won several awards, most recently the Magna Award - a national recognition program that honors innovative programs that advance student learning. Hundreds of businesses have generously donated to our program over the past twenty years, city and state leaders and representatives have visited our facility in support of our program and goal of teaching the free enterprise system and we receive several emails/calls per year from prior “citizens” of Enterprise City who want to participate again as a teacher/parent volunteer or have chosen their career path based on what they learned at Enterprise City as a child.

Does the district provide any programs similar to Enterprise City that are available for students in other grades? How has the district addressed the study of financial literacy and economics in other grade levels?

There isn’t another district-wide program in RISD; Enterprise City is the only program offered for our 6th graders. However, some of our high school business classes have used our facility and revised parts of the curriculum to meet their objectives. We also have our high school LOTE (Languages Other Than English) classes attend. The purpose is to give the students a simulation of life in this environment and to use their language for more than just one class period. The rules of the day in the city require that the students speak only their language at all times. Currently, every RISD 6th grade class attends Enterprise City; however, our district has moved the program over the years from one grade level to another. Teachers in various grade levels at different schools have implemented their own economic activities/programs such as classroom economies. Many students learn how to use check registers, apply for jobs and earn money as incentives.

Can students and teachers outside of RISD participate in the Enterprise City program?

Yes, we offer our program to non-RISD schools for a fee. Currently, we have 8 districts, which send several schools within each district and about 20 private schools that participate.

What advice would you give to teachers and instructional leaders looking to implement a similar program in their district?

Do exactly what we did- secure the funding from their community/school district to build a city and implement a similar program and recruit a core group of teachers to develop the curriculum. Anyone interested is welcome to come visit our facility to see the “best city in Texas” first-hand!

To learn more about Enterprise City, visit http://www.richardson.k12.tx.us/enterprisecity/index2.html

References
(J. Freeman, personal communication, October 23, 2014).

Welcome to Enterprise City. (n.d.). Retrieved November 21, 2014, from http://www.richardson.k12.tx.us/enterprisecity/index2.html

Analyzing Assessment Items

By Dr. Pooja Shivraj, RME Educational Assessment Researcher

Much of the work we do at Research in Mathematics Education involves the development of assessments used by educators to identify students who may be struggling with algebra-readiness knowledge and skills, so that teachers can provide additional instructional support. The research process we use is rigorous and begins with an assessment blueprint, then item writing, internal reviews and external expert reviews, followed by a pilot test and finally the development of the test forms. The pilot test is given to a large number of students in order to determine the validity of the assessment items. Our researchers receive the results of the pilot and perform an extensive statistical analysis to determine if an item is good, psychometrically speaking.

The point of obtaining item statistics is to develop a pool of items that function well from which future tests can be designed. There are two kinds of analyses that can be performed: a Classical Test Theory (CTT) analysis, which is sample-dependent and non-model based, or an Item Response Theory (IRT) analysis, which is sample-independent and model-based. Regardless of the type of analysis performed, three primary statistics are used to determine if an item is psychometrically good. The ranges listed below are the acceptable norms found in the literature.

(1) The item should have a strong correlation between each item score and the total score. In other words, the correlation should show that the test-takers choosing the correct answer on the item are likely to receive a higher score. This statistic is measured by the point-biserial correlation (CTT) or the point-measure correlation (IRT). A good item would have a point-biserial correlation of >0.2 or a point-measure correlation of >0.25.
(2) The difficulty of the item, measured by the proportion of students answering the item correctly (CTT), should be between 30% to 80% of the test-takers. In IRT, the difficulty parameter, b, should be between -4 and +4.

An item characteristic curve depicting the discrimination parameter
(a) and the difficulty parameter (b) in an IRT model

(3) The discrimination of the item, also measured by the point-biserial correlation (CTT) should be higher for the correct response than the distractors. In IRT, the discrimination parameter, a, should be between 0.5 and 1.5. The greater the discrimination, the better the item discriminates between lower ability and higher ability students.

What can you do with items that don't function well? For the items that don't function well, reviewing the data would be the first step. Are the items functioning poorly because the majority of students are choosing the correct answer? Is one distractor not being chosen at all? Are the majority of students choosing a single distractor more often than other options? These data would all be red flags. The next step would be to review the content of all the items that don't function well, especially the items that were flagged in the previous step. What about the content led students to choose or not choose a particular response choice?

Using this process of analyzing data, reviewing items, and adjusting the content of the items, a pool of items that function well can be developed for use in the future.

Note: Many other statistics (e.g., fit statistics in IRT like Chi square, infit, outfit, etc.) could be used to determine if an item functions well in addition to the ones described above that could also provide information at the test level. Please feel free to email me if you would like more information at pshivraj@smu.edu.

Benjamin Banneker Week

By Brea Ratliff, RME Secondary Mathematics Coordinator

For many students, mathematics is viewed as a faceless, and sometimes meaningless, course of study, but learning more about the fascinating and prodigious minds who have shaped the subject can be inspiring. In the face of doubt, criticism, failure, and even seemingly impossible circumstances, many great men and women have been intellectual trailblazers whose extraordinary contributions to society are a testament to the power and importance of teaching mathematical processes and critical thinking. One such individual was African-American mathematician, author, scientist, agriculturalist, astronomer, publisher, and urban planner, Benjamin Banneker.

Image from
http://www.bnl.gov/bera/activities
/globe/banneker.htm

Benjamin Banneker was born outside of Baltimore, Maryland on November 9, 1731. He was born a free black, and was generally self-taught through most of his young adult life. Banneker began to display his brilliance as an engineer while he was a young man; first through his often noted affinity toward solving puzzles, and later through his mathematically-perfect creation of the first clock made entirely of hand carved wooden parts and pinions (Washington Interdependence Council, 2014). This clock, which Banneker built after carefully studying a borrowed pocket watch, accurately kept time for decades.

Benjamin Banneker’s love for learning encouraged him to begin studying astronomy and advanced mathematics from sets of books loaned to him by a neighbor. As a result of these studies, he was able to accurately predict solar and lunar eclipses, and became the author of an internationally published almanac, which contained his many scientific and mathematical calculations. The international recognition of his almanac also served as a springboard for Banneker to become a recognized proponent for the abolishment of slavery. He famously composed a letter addressed to Thomas Jefferson, in which he insisted black Americans possess the same intellectual ability and should be afforded the same opportunities as white Americans (Chamberlain, 2012). This letter led to an ongoing correspondence between the two men, and led to Banneker receiving a considerable amount of support by abolitionist groups in Maryland and Pennsylvania (Biography, 2014).

Banneker was also selected to assist Major Pierre L’Enfant to survey and develop the city plans for our nation’s capital, which was later named the District of Columbia. After L’Enfant abruptly quit the project, Benjamin Banneker was able to reproduce the plans – from memory - for the entire city in just 2 days. These plans provided the layout for the streets, buildings, and monuments that still exist in Washington D.C. (Chamberlain, 2012).

During the week of November 9th through the 15th, individuals and groups across the nation will honor the many contributions of this great mathematician by celebrating “Benjamin Banneker Week”. The Benjamin Banneker Association, an organization dedicated to mathematics education advocacy by providing support and leadership for educators and students in order to ensure equity exists for all students, is sponsoring a mathematical task competition to continue his legacy.

Schools, libraries, community and professional organizations, or interested citizens are urged to make mathematics a significant part of children’s lives by coordinating a Benjamin Banneker Celebration event in their communities. Visit the Benjamin Banneker Day website (www.benjaminbannekerday.weebly.com) to learn more about Benjamin Banneker, and how you and your community can participate in this year’s celebration.

Benjamin Banneker: A Memorial to America’s First Black Man of Science (2014). Retrieved Oct 13, 2014 from http://www.bannekermemorial.org/history.htm

Benjamin Banneker. (2014). The Biography.com website. Retrieved Oct 13, 2014, from http://www.biography.com/people/benjamin-banneker-9198038.

Chamberlain, G. (2012) Benjamin Banneker – The Black Inventor Online Museum. Retrieved Oct 13, 2014 from http://blackinventor.com/benjamin-banneker/

Bringing the Associative Property of Multiplication to Life

By Cassandra Hatfield, RME Assessment Coordinator, and Megan Hancock, Graduate Research Assistant

The Institute of Education Science (IES) Practice Guide for Improving Mathematical Problem Solving in Grades 4 through 8 Recommendation five states that it is important to “help students recognize and articulate mathematical concepts and notation” (Woodward et al., 2012). One way to carry out this recommendation is to “ask students to explain each step used to solve a problem in a worked example” and “help students make sense of algebraic notation” (Woodward et al., 2012).

The Associative Property of Multiplication will illustrate this recommendation by going beyond a procedural skill and making connections conceptually that support the symbolic notation. Our goal is to give evidence that the Associative Property of Multiplication can be taught through multiple representations. Through our research we found that some representations are mathematically accurate, but may not provide students with a compelling reason to use this property.

When developing the concept of volume of rectangular prisms, decomposing the rectangular prism into layers allows students to make the connection with content they are already familiar with, arrays and area. This decomposition also exemplifies the Associative Property of Multiplication. Here are some examples of how the rectangular prism shown above can be decomposed in different ways.

 

• A: 2 × (6 × 4)
• B: (2 × 6) × 4
• C: Supports commutative property of multiplication too
• 2 × 6 × 4; 2 × 4 × 6; (2 × 4) × 6

By designing activities and lessons that support the decomposition of rectangular prisms into different layers, teachers can support students in making sense of the notation of Associative Property of Multiplication, A x (B x C) = (A x B) x C, and finding the volume of rectangular prisms. Explorations like this also support teachers in holding students accountable for understanding the notation because students can use the different models to support their explanation of their understanding.

Woodward, J., Beckmann, S., Driscoll, M., Franke, M., Herzig, P., Jitendra, A., Koedinger, K. R., & Ogbuehi, P. (2012). Improving mathematical problem solving in grades 4 through 8: A practice guide (NCEE 2012-4055). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http:// ies.ed.gov/ncee/wwc/publications_reviews.aspx#pubsearch/.

Closing the Learning Gaps: Strategies to ensure your students will be successful with the new TEKS

By Brea Ratliff, RME Secondary Math Research Coordinator

The Texas Essential Knowledge and Skills (TEKS) are the state standards that identify the information students should learn and the academic proficiencies they should demonstrate in each grade level or course. The newly adopted math TEKS are evidence of increased expectations for mathematics education in the state of Texas. Although several changes have been incorporated into the math TEKS, our students do not have to enter the next grade or course with gaps in their understanding of mathematics. As educators, we are charged with the difficult task of meeting students where they are through our reflective practice, which includes the development of instructional techniques designed to support students as they learn mathematics. The biggest, and perhaps the most important step in this process, is for educators and administrators alike to analyze and become familiar with the new math TEKS. In this blog series, we will examine strategies to help teachers and students experience success with the new math TEKS.

Strategy 1: Identify your resources

The Texas Education Agency (TEA) has published several resources for math teachers and administrators to help them transition into teaching the new math TEKS. These resources can be used to plan lessons, develop an understanding of the knowledge and skills addressed in a particular grade level / course, and foster conversations with parents and other stakeholders in your community about the changes in the state math standards.

• Side-by-Side TEKS Comparison - this document compares the revised TEKS (adopted in 2012) to the previously adopted TEKS (revised in 2006) and allows the reader to see all of the major changes and shifts made to the math content and mathematical process standards. Documents for grades K through 8, Algebra 1, Geometry, and Algebra 2 are available on Project Share. (Side-by-Side TEKS)
  
• Vertical alignment charts - TEA has published four vertical alignment documents, which organize the TEKS by major concepts and show how these ideas are connected across grade levels / courses. These charts can also be accessed on Project Share. (Vertical Alignment)
• STAAR Mathematics Resources - changes in the math standards have also impacted the state mathematics assessments. The State of Texas Assessments of Academic Readiness (STAAR®) assessments Assessed Curriculum, Blueprints, and Reference Materials documents have been updated to reflect these changes. (STAAR Math Resources)
• Texas Response to Curriculum Focal Points - Revised in 2013, this document guides mathematics teachers in understanding the topics within each grade level that require the most emphasis, and can be used to inform instructional pacing and lesson development. This document is also available on Project Share. (TXRCFP)

After reviewing these resources, please share any questions or comments you might have with us via email or on Twitter (@RME_SMU). In the next blog, we will examine how these documents can be used to impact math instruction, with a specific emphasis on Number and Operation.

CAMT is Coming Up!

By Savannah Hill, RME Professional Development Coordinator

Looking for a good conference this summer? Come join us at CAMT - the Conference for the Advancement of Mathematics Teaching in Fort Worth on July 21 - 23. CAMT is an annual Texas conference for K-12 mathematics teachers. The conference is sponsored jointly by the Texas Council of Teachers of Mathematics, the Texas Association of Supervisors of Mathematics, and the Texas Section of the Mathematical Association of America.

If you have never heard of the CAMT Conference, here is a little bit about them from their website:

The CAMT program consists of approximately 750 sessions, ranging from 90 minutes to three hours. The workshops and sessions consist of mathematics content and pedagogy appropriate for K-12 mathematics teachers. Outstanding speakers from all over the state and nation submit proposals to present sessions, and a program committee selects sessions most appropriate for the conference. Also, a large number of presenters are invited from all over the country to present featured sessions. Most session presenters are outstanding, practicing mathematics teachers, who present ideas for teaching that they have found effective in their classrooms. Many sessions involve hands-on learning activities that teachers can use in the classroom to address various topics in the mathematics curriculum. Effective use of manipulative materials in the classroom is an important component of the conference, and a number of sessions regarding manipulative use occur at each conference. Use of technology in classroom instruction is also an important component of the CAMT conference.

We have several members of our team presenting this summer. Come join us at one of the following sessions!

• Engaging Models and Activities to Support Fraction Instruction - Monday, July 21, 10:00 - 11:00 and 11:30 - 12:30, CC 203A 
• BYOD: RtI at Your Fingertips - Monday, July 21, 10:00 - 11:00, CC 120 
• MSTAR: Understanding the Value of an Assessment Plan - Monday, July 21, 1:00 - 2:00, Omni FW 5 
• PreCal 911: Engaging Activities to Save the Day! - Tuesday, July 22, 10:00 - 11:00, CC 201C 
• Teacher T.O.M. - A Strategy for Reflective Practice - Tuesday, July 22, 11:30 - 12:30 and 1:00 - 2:00 Omni Stockyards 3 
• ESTAR: Understanding the Value of an Assessment Plan - Tuesday, July 22, 1:00 - 2:00, Omni FW 5 
• Implementing the NEW TEKS with Best Practices - Tuesday July 22, 1:00 - 2:00, CC 114 
• Money Management: Developing Appreciation Through Mathematics - Tuesday, July 22, 1:00 - 2:00, CC 204AB 
• Spaghetti & Meatballs and Algebraic Reasoning - Wednesday, July 23, 10:00 - 11:00, Omni Sundance 2 
• ESTAR: Understanding the Value of an Assessment Plan - Wednesday, July 23, 2:30 - 3:30, Omni FW 5

Hope to see you there! 

More than Just Fun and Games: Using Apps for STEM Learning

By: Elisa Farrell, Guest Blogger, SMU Undergrad

Dr. Candace Walkington, math education professor at Southern Methodist University, teaches a course on STEM integration for pre-service elementary teachers. As part of the class, her students author a series of blogs where they discuss issues related to the integration of science, technology, engineering, and mathematics in elementary school. In this set of blogs, her students were discussing how they can use educational “apps” that are related to STEM in their classroom, focusing particularly on math. They were encouraged to take a critical stance towards the use of apps and to give clear guidelines for how teachers can find and evaluate high-quality apps for math learning.

How much time did you spend watching TV when you were in elementary school? When did you first get a cell-phone? My personal technology milestones were a laptop in eighth grade, a cellphone as a high school freshman, a smartphone as a college first-year, an e-reader as a college sophomore, and a tablet as a college junior. The following chart shows that in 2009, the average age for receiving a personal mobile phone was 9.7. That’s fourth or fifth grade!

Technology is becoming increasingly prevalent in today’s society. As Collins & Halverson (2009) point out in their book Rethinking Education in the Age of Technology, “the world is changing and we will need to adapt schooling to prepare students for the changing world they are entering” (p. 9). The following graph shows the time spent per week on certain media and devices by age, including video games, computers, tablets, e-readers, mobile phones, and more.

You can see that in 2010, at least one-fifth of children ages four to five are using media for 5-9 hours per week. Using the waking hours per day as the maximum of 12 (Pantley, 2009) for a total of 84 waking hours in a week, 9 hours of media is 10% of waking hours. That’s a large chunk of time in front of screen. As I mentioned in my earlier blog post about the use of technology in education, I am highly in favor of its proper integration. Collins & Halverson (2009) also point out that “technology gives us enhanced capabilities for educating learners” (p. 9). One of the best examples of this enhanced capability incorporated into an educational setting is mobile applications, or apps. A research study funded by Nickelodeon found that gaming is the primary use of electronic devices, where “96% of kids say they use their computer for gaming, compared to 88% on the tablet and 86% on the smartphone” (2013).

Educational apps combine the natural inclination of children to enjoy games with the new technology and media that has become ubiquitous to our daily lives. In our course on Integrated STEM Studies through the education department, we had the chance to test several STEM apps, specifically focusing on math apps. Here are some of my favorites from class and from my own personal exploration:

Pizza! by Motion Math
This app combines a favorite childhood food with business decision-making to teach children important math skills. For example, they learn division by calculating the unit price for ingredients to make sure they’re getting the best deal. Multiplication and addition are used to find the total bill of a customer’s order, and rapid computation is necessary to keep customers satisfied and sales high. In later rounds, pricing decisions require number comparison – is the cost of producing pizzas offset by the money customers will pay? How much is too much to charge for the oft-requested “Sardine Special?” Finally, students must keep track of ingredient inventory to maximize sales and avoid the angry speech bubbles, “You ran out of pineapple!” From my criteria for what makes a good app this app pretty much fulfills them all.

Hungry Fish by Motion Math
In this game, children feed number bubbles to an insatiable fish. The easiest level is simply number recognition and matching – if the fish says “1” then the correct bubble is that which also says “1.” In higher levels, bubbles must be combined, adding or subtracting them to create the appropriate feeding value. This game is simpler in concept than Pizza! but still highly engaging, at least in my personal experience. I had the opportunity to observe a group of second-grade students play this app together in my field experience.

Overall, using educational apps can help make learning fun and provide good individual or small-group reinforcement activities. However, since most are formatted as games, the STEM skills and knowledge must be central to success in the game or it is just a game with numbers. I saw Hungry Fish being used in a second grade classroom without teacher supervision, and the student in charge of combining bubbles was simply dragging adjacent bubbles together without trying to reach the correct number to feed the fish. It is important to remember with education apps that education needs to be the focus, not the app. In addition, as whenever technology is used, there are concerns about sharing and taking turns, theft or other damage, and if the technology is functional when it’s needed.

Educationally, apps need to be standards-based, STEM focused (for this blog), and challenging but not frustrating.

Standards-based apps are preferable, since they pull their goal concepts and skills from an already created and approved list. For example, Hungry Fish, the bubble-eating game I mentioned earlier, includes a list of the Common Core Standards addressed in the game on the company’s website.

Without a basis in standards, apps may help students practice math or other STEM skills, but fall short of helping them stay on track with the actual curriculum. The curriculum and standards are developed to provide a framework of logically sequenced knowledge and skills acquisition, and following them creates a uniform education system across the state (and the nation).

STEM focused apps use STEM concepts and processes as part of the core mechanics of app, not a side benefit. For example, in Pizza! , math knowledge and skills are necessary for success. In contrast, Hungry Fish could simply be an amusing way to combine bubbles and see numbers change.

Finally, apps need to be educationally challenging, but not frustrating. Vygotsky’s Zone of Proximal Development posits that students learn best when they are working in the area where they need guidance but can still accomplish a task. This is illustrated as the middle circle in the diagram.

Even though you use your own criteria to evaluate apps yourself, some of the work has already been done for you. My favorite place to look for app suggestions is on websites that focus on reviewing media and apps. Here’s a list of a couple that focus specifically on apps for children, and some that even focus in on educational apps for children as reviewed by parents and teachers.

• Children’s Technology Review's goal is “to provide objective reviews of children’s interactive media products,” and there are many apps reviewed. However, the search feature not as polished as some of the other sites.
• Appysmarts finds the “best apps for young brains.” You may need to create an account to access some features of this site.
• Common Sense Media: “We rate, educate, and advocate for kids, families, and schools.”
• Graphite is “a platform we created to make it easier for educators to find the best apps, games, and websites for the classroom” (by Common Sense Media).

Overall, educational apps can be highly useful for extending already-popular technology and gaming into the classroom. Educators should be careful to keep learning goals in mind when choosing apps, and can make use of some of the above resources when choosing apps for the classroom.

An 'App'etite for STEM Education

By Nate White, Guest Blogger, SMU Undergrad

Dr. Candace Walkington, math education professor at Southern Methodist University, teaches a course on STEM integration for pre-service elementary teachers. As part of the class, her students author a series of blogs where they discuss issues related to the integration of science, technology, engineering, and mathematics in elementary school. In this set of blogs, her students were discussing how they can use educational “apps” that are related to STEM in their classroom, focusing particularly on math. They were encouraged to take a critical stance towards the use of apps and to give clear guidelines for how teachers can find and evaluate high-quality apps for math learning.

There is a multitude of iPad applications, or "apps," out there for elementary education that can be utilized effectively in the classroom. There are nearly 100 new "education apps" added to the Apple App Store every day! Therefore, the number of educational applications is so broad that it encompasses both "good" and "bad" apps. When looking at science, engineering and mathematics, there is quite a variety in apps out there, even free ones! Let's look at some examples of free apps in the mentioned STEM subjects, with a focus on mathematics.

Engineering: Pettson's Inventions
This app balances fun and entertainment beautifully. Basically, there are numerous different scenarios (like the one in the photograph below) where the player must design and build some solution to an engineering problem. You are provided objects on the left side of the screen to drag over to the incomplete engineering design on the right, and each "part" is only used once in a specific manner (see the picture at left). It may not even seem like you are learning engineering problem solving skills when playing this game, but the game requires so much creativity, analysis and trial-and-error that you will develop engineering skills. This app seems appropriate for both younger and older elementary students. A couple plausible weaknesses of this app is the lack of engineering vocabulary, as well the lack of support or directions in difficult problems in particular, which could deter some students from persevering to solve the problem.

Science: Bill Nye The Science Guy
This science app is engaging and versatile. There are several different content areas within the app, which the user can choose to pursue. You can explore out solar system and learn about each planet (as seen in the picture below where you have missions to go to each planet and take pictures, place satellites and learn several interesting facts); you can learn about geology (layers of the Earth); or you can even learn about optical illusions. All of these topics are taught by interactive games with informative narration by Bill Nye himself! This app also includes video episodes (which you must purchase) and science experiments with elaborate directions. While this app is very engaging and informative, it could pose problems for English Language Learners because of the app's reliance on (English) narration.

Mathematics: Oh No Fractions! 
This app is composed of various fraction problems in which you must manipulate a visual representation in order to show the answer of the given problem (as seen in the picture at left). You can choose between addition, subtraction, multiplication, division and compare problems, all of which exclusively use fractions. This game is simple and explicitly educational without too much "fun and games," but I think it is a good tool for students in upper level elementary to practice manipulating fractions.

Math Champ
This app is categorized by grade and difficulty (grades 4-7 and five levels of difficulty). The app s free, however, it requires you to buy the uprated version to access the majority of its content. As for the elementary grades (4th and 5th) in the app, there are questions about geometry, fractions, decimals, multiplications etc. All of the questions are multiple choice (as you can see in the picture). The app does a good job of keeping it educational while adding a fun and engaging design. For example, you can unlock alien characters to play as.

Sushi Monster This is perhaps my personal favorite because of how challenging and fun the app is. This app assesses addition and multiplication of positive integers from zero up to the thousands place. First, you choose either addition or multiplication, and then pick a level of difficulty (1-5). In each level there is a different monster who is inside of a circular sushi bar. The sushi chef places several dishes of sushi, each with a number on it, and the monster will have a number on him. The goal is to add or multiply the correct sushi dishes to equal the number on the monster, and you do this by dragging the desired sushi from the outer sushi bar to the monster sitting at the center. You accumulate points and strive for a low time of finishing each stage. It is engaging because of the focus it requires due to the fast paced nature of the game, plus the fact that there is fun Japanese music playing while a monster is eating sushi you serve.

Lobster Diver The last math app I will discuss is a number line game where you play as a scuba diver who is fishing for lobsters. At the bottom of the screen there is a sea floor with a number line on it (which could be counting numbers, negative numbers, fractions, etc.). Each level is timed and you are told the point on the number line where you must dive to get the lobster, however, most of the number line is not marked, so the user must be able to count and decipher the number line in order to dive at the right place. In addition, there are eels constantly swimming by that you must avoid. I love this game because it provides the positives of a recreational game (e.g. eels and interactive scuba diver) yet challenges the player to understand a number line and thus correctly compare fractions and negative numbers. This could be a great app in the classroom.

Strengths and Weaknesses Because of the highly diverse array of apps in the STEM subjects, there are strength and weaknesses to different types of apps. Apps like Oh No Fractions! and Math Champ do not offer anything exclusive to technology, i.e., those problems can be done on the board or on a piece of paper in class, so you can definitely make a case for unnecessarily using technology in those examples. They do, however, enable students to test they knowledge of a wide variety and quantity of math problems efficiently. Games like Lobster Diver risk students doing poorly on the app, not because of their math skills/knowledge, but because of their lack of game playing skills, such as not avoiding the eel and losing all three of his/her lives thus having to start over. Math games like Sushi Monster are great for gifted students because you can always get faster and faster at the game and work with big numbers in the higher levels.

Criteria for Evaluating STEM Apps
As mentioned earlier, because of the quantity of education apps in existence currently as well as those being created everyday, there are plenty of STEM apps that are a waste of money and class time even if they are free. So how does an elementary school teacher evaluate whether or not he/she should purchase/download an app for STEM learning in the classroom? There are several variables that go into this, not to mention the context of the school (such as kids coming to school from impoverished homes where technology is a rarity). But generally speaking, there is certainly some criteria that should be used when evaluating apps.

Educational Technology and Mobile Learning offers some insight as well as rubrics and evaluation questionnaires for iPad educational apps. One example of a rubric I found to be quite effective is seen in the following figure:

I love how this rubric weighs things like engagement, levels of difficulty, various modes of play, and randomly presented content because it is important for students to want to play an app as well as be able to replay it without easily mastering it. As seen in this rubric, it is important to ask yourself if an app meets the students' needs and is aligned to your state standards. At the end of the day, your class time should be used to further your students knowledge to meet, if not exceed, state standards (TEKS, Common Core Standards e.g.). It is easy to find an extremely creative and engaging app with some educational ties and instinctively have your students play it, but careful analysis must take place where the teacher looks for concrete content in the app that assesses or instructs standards-based content. Other practical things should be considered as well, such as the availability of technology to use the app, the presence and extent of feedback given by the app to students. Cost is another factor, which is rather obvious in my opinion: A mediocre or even good app is not worth speeding money when there are so many great apps out there for free!

Another resource teachers can use to find ideas for free mathematics apps is found at TCEA.

More STEM apps for elementary are listed here at Imagination Soup.

Teachers should not rely on the Apple App Store ratings, or even comments, for evaluating and finding apps. It is wise to speak with other teachers, especially trying their apps out. Additionally, simply searching Google for Elementary iPad apps in whatever subject or specific content can provide good results as long as you carefully consider the given pros and cons.

Subitizing and Decomposing Numbers for Early Math

By Dr. Deni Basaraba and Cassandra Hatfield, 
RME Assessment Coordinators

In 2013, the National Center for Education Evaluation and Regional Assistance (NCEE), in partnership with the Institute of Education Science (IES) released an educator’s practice guide focused on Teaching Math to Young Children. The intent of this Guide (and all similar Practice Guides) is to provide educators with evidence-based practices they can incorporate into their own instruction to support students in their classrooms. In this ongoing blog series we will focus on specific recommendations put forth in the Practice Guide Teaching Math to Young Children and provide practical suggestions for incorporating these recommendations into your classroom instruction.

Recommendation 1 in this Practice Guide is to teach number and operations using a developmental progression. Using and understanding a developmental progression for number serves as the foundation for later mathematics skill development. As noted in the work documenting the development of learning trajectories for mathematics (Clements & Sarama, 2004; Daro, Mosher, & Corcoran, 2011) as well as in our own work in the development of diagnostic assessments using learning progressions, developmental progressions can provide teachers with valuable information regarding students’ knowledge and skill development by providing a “road map for developmentally appropriate instruction for learning different skills” (Frye et al., 2013).

Specifically, the research recommends that teachers first provide students with multiple opportunities to practice subitizing, or recognizing the total number of objects in a small set and labeling them with a number name without needing to count them. According to Clements (1999), two types of subitizing exist:

• Perceptual subitizing: The ability to recognize a number without using other mathematical processes (e.g., counting).
• Conceptual subitizing: The ability to recognize numbers and number patterns as units of units (e.g., viewing the number eight as “two groups of four”).

The role of subitizing as it relates to numeracy (Kroesbergen et al., 2009) and procedural calculation (Fuchs et al., 2010) has been documented in the literature. Kroesbergen et al., (2009), for example, not only found that subitzing was moderately correlated to the early numeracy skills of kindergarten students, but that it also explained 22% of the overall variance observed n counting skills and 4% of the variance in early numeracy skills after controlling for language and intelligence. Moreover, research also indicates that instruction designed using a developmental progression can support students’ ability to subitize (Clements & Sarama, 2007), as evidenced by relatively large gains in the pretest to posttest gain scores observed for students receiving this type of instruction compared to a “business as usual” comparison condition.

To support students with subitizing and decomposing numbers, flash images of arrangements of dots visually for students for about 3 seconds. Then give students an opportunity to share what they saw. Over time, student’s verbal descriptions can transition to writing equations. For younger children, subitizing may be fast and efficient only when the number of objects is less than four (Sarama & Clements, 2009); numbers larger than this may require decomposition into smaller parts.. For students learning multiplication arrangements of multiple groups of dots can be shown to support visualizing equal groups.

How do you see this image?

5 and 5, minus 1 

4 and 4, plus 1 

2 and 2, doubled, plus 1 

2 groups of 4, plus 1

Print these dot cards or 10 frame cards on cardstock and put them on a ring. They can be used in various ways:

1. Hang them in places throughout the hallway of your school. Working on subitizing is a great way to keep students engaged during transition times.
2. Place them as a center for partners to flash the images and ask “How many?”
3. Independent think time: Students can be given an arrangement and write all the different ways they see the arrangement.
4. Warm-up activity to get students thinking prior to small group instruction

Hyperlink for dot cards: http://www.k-5mathteachingresources.com/support-files/dotcards1-12.pdf

Link for 10 frame cards: http://www.k-5mathteachingresources.com/support-files/large10frames1-10.pdf

References

Clements, D. H. (1999). Subitizing: What is it? Why teach it? Teaching Children Mathematics, 5, 400-405.

Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical thinking and learning, 6, 81-89.

Clements, D. H., & Sarama, J. (2007). Effects of a preschool mathematics curriculum: Summative research on the Building Blocks project. Journal for Research in Mathematics Education, 38, 136-173.

Daro, P., Mosher, F. A., & Corcoran, T. (2011). Learning trajectories in mathematics: A foundation for standards, curriculum, assessment, and instruction. (CPRE RR-68). New York, NY: Center on Continuous Instructional Improvement.

Frye, D., Baroody, A. J., Burchinal, M., Carver, S. M., Jordan, N. C., & McDowell, J. (2013). Teaching mathematics to young children: A practice guide (NCEE 2014-4005). Washington, DC: National Center for Education Evaluation and Regional Assistance (NCEE), Institute of Education Sciences, U.S. Department of Education Sciences. Retrieved from the NCEE website: http://whatworks.ed.gov.

Fuchs, L. S., Geary, D. C., Compton, D. L., Fuchs, D., Hamlett, C. L., Seethaler, P. M., Bryant, J. D., & Schatschneider, C. (2010). Do different types of school mathematics development depend on different constellations of numerical versus general cognitive abilities? Developmental Psychology, 46, 1731-1746.

Krosebergen, E. H., Van Luit, J. E. H., Van Lieshout, E. C. D. M, Van Loosbroek, E., & Van de Rijt, B. A. M. (2009). Individual differences in early numeracy: The role of executive functions and subitizing. Journal of Psychoeducational Assessment, 27, 226-236.

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

Math STAAR: Strategies for Success

By Dawn Woods, RME Elementary Mathematics Coordinator

As every math teacher across the state of Texas knows, the State of Texas Assessments of Academic Readiness (STAAR) testing window is upon us. You have worked diligently, teaching vocabulary, concepts & skills, through the lens of mathematical process standards thereby empowering your students to implement mathematics in everyday life, as well as perform on this assessment. The strategies listed in this blog are suggestions that could enhance your students’ success.

1. Teach goal setting. Research suggests that when students are taught to set specific academic goals they make progress in learning skills and content, discover how to self-regulate learning, and improve their self-efficacy and interest in the task (Bandura & Schunk, 1981). Through this goal setting and self-assessment process, students are enabled to monitor and evaluate their performance during a lesson, unit of instruction, or review of course material thereby increasing student performance and instilling responsibility for their learning. An example of goal setting for STAAR could look like:
   
   
2. Teach “timed” test strategies. A few strategies include:
• Listen to the test proctor’s directions.
• Budget time appropriately. Work quickly but do not rush.
• Work the problems in the test book, not in your head!  Double check if you copied numbers correctly, if the units are similar, and if you applied the appropriate formulas. Use good handwriting so you do not misread your answer.
• Do not be too happy to see your computed answer as one of the answer choices!  Test makers know what wrong choices could be made and include them in the answers. So check your answer before marking it on the answer sheet!
• Do not panic. If the question is difficult, return to it later. Maybe another question will job your memory on how to answer the difficult question.
• Position the answer sheet next to the test booklet so that you can mark answers quickly while checking that the number next to the circle on your answer sheet is the SAME as the number next to the question you are answering.
• Before turning in your test, double-check your answers.
• Make sure you bubbled in the answers correctly on your answer sheet.
•  Don’t be disturbed by other students finishing before you. Extra points are not given for finishing early!
3. Communicate with parents and students to encourage healthy pre-test behaviors. A few pre-test behaviors include:
• Relaxing for a few hours before bedtime.
• Getting enough sleep the night before a test.
• Eating a healthy breakfast and avoiding foods that could make you groggy or hyper.
• Don’t stress!  You’ve worked hard and are prepared for the test.

Works Cited:
Bandura, A., & Schunk, D.H. (1981). Cultivating competence, self-efficacy, and intrinsic interest through proximal self-motivation. Journal of Personality and Social Psychology, 41(3), 586-598.

Open Ended Assessments: Part 2 - Grade 8 Math

By Brea Ratliff, Secondary Mathematics Research Coordinator

 With the assessment season upon us, many teachers and administrators are looking for strategies to ensure their students are successful with all of the concepts being assessed. This blog describes a few ideas for open-ended assessments that build on this student expectation:

8.7(C) – The student is expected to use pictures or models to demonstrate the Pythagorean theorem.

Level 1 – Assessments designed to develop proficiency in 1 student expectation. Assessments build around one particular skill are often helpful after when introducing a concept, or providing targeted intervention.

Level 2 – Assessments designed to develop proficiency in 2 or more student expectations. The assessments for this level can vary in degree. While some may be designed to assess a combination of content knowledge, others may be written to include the process skills. This assessment addresses a wealth of knowledge and skills, and could possibly be used for several class periods.

8.2(D) - use multiplication by a given constant factor (including unit rate) to represent and solve problems involving proportional relationships including conversions between measurement systems. 

8.6(B) - graph dilations, reflections, and translations on a coordinate plane. 

8.7(C) – The student is expected to use pictures or models to demonstrate the Pythagorean Theorem. 

8.7(D) - The student is expected to locate and name points on a coordinate plane using ordered pairs of rational numbers. 

8.15(all) - The student communicates about Grade 8 mathematics through informal and mathematical language, representations, and models.

Discovering Pi (π), 3.14159265359…

By Sharri Zachary, RME Mathematics Research Coordinator and RME Collaborator Patti Hebert, Garland ISD

As presented in the opening session of our RME conference, there are three key components that we, as educators, should maintain as we transition into the new math TEKS: (1) balance, where the emphasis is on students’ conceptual understanding and procedural knowledge (2) focus, where we centralize instruction around the “big” ideas, and (3) coherence, where the instruction is aligned within and across grade levels.

Consider this standard from the revised math TEKS for grade 7:
The student applies mathematical process standards to use geometry to describe or solve problems involving proportional relationships.

The student is expected to: 
5(B) –describe π as the ratio of the circumference of a circle to its diameter

In honor of Pi Day, I would like to share an activity that you may want to consider for use with your students, Sidewalk Circles. At the end of this activity, the students should be able to explain that:

1. The distance from the center to the edge of a circle is "1" /"2" the distance from one side of the circle to the other side of the circle through the center (OR the distance from one side of the circle to the other side of the circle through the center is about 2 times the distance from the center out to the edge of the circle).
2. The distance all the way around the outside of the circle is about 3 times the distance from one side of the circle to the other side of the circle through the center (discovery of Pi).

You will need the following materials for each team:

• One (1) center tool (we used small funnels with shoelaces strung through the end of the funnel to keep the shoelace from coming through)
• Chalk
• One (1) pre-marked ribbon piece (with indicated measures, 10 cm, 23 cm, and 36 cm) 
• One (1) 2.5 m piece of string 
• One (1) tape measure

Take the class outside to an unused pavement area. (If raining, let students use large pieces of butcher paper to complete this activity.)

Student teams directions:

1.  Pick a center point and mark it with a clear mark so that you will know where it needs to be every time you are creating a circle.
2. Stretch your pre-marked ribbon out tight and wrap it around the piece of chalk so that the first mark on your ribbon is at the edge of the chalk.
   
   
   
   
   
3. HOLD THE RIBBON TIGHT as you move the chalk around the center drawing a circle on the sidewalk. Work as a team and do not let the center move.
4. You must take 3 measurements for EACH circle. Use the 2.5 m string. Lay it out, then take it to a measuring tape to find the actual measurement: a) From the center to the edge of the circle b) From one side of the circle to the other side of the circle THROUGH the center c) Around the outside of the circle.
5. Repeat this process for the other 2 tape marks on your ribbon.

The general premise is that each group of students will create sidewalk circles using the pre-marked lengths of the ribbon piece (one each: 10 cm, 23 cm, 36cm), a center point, and chalk. They will use string and a tape measure to find the distance from the center to the edge of the circle (radius) and the distance around the entire circle (circumference). They will repeat these processes for all three measurements until they have drawn one circle for each measurement. The group should discuss their measurements and use reasoning skills to analyze the relationships among the measurements.

Developing Numerical Magnitude Understanding in Young Children

By Dawn Woods, RME Elementary Mathematics Coordinator

Research shows general math achievement is closely related to children’s understanding of numerical magnitudes, or the amount of a quantity (Gersten, Jordan, & Flojo, 2005; Siegler & Booth, 2004). Children develop the ability to quantify and order numbers through subitizing and counting (Clements and Sarama, 2009) and many children can answer the question “which is more, 5 or 3?” by five years of age. However, some children may be unable to tell which of two numbers is bigger or which number is closer to another number and may not have developed the “mental number line” representation of numbers (Gersten, Jordan, & Flojo, 2005; Griffin, Case, & Sigler, 1994; Clements and Sarama, 2009).

This concept of numerical magnitude is a core component of number sense, which is widely viewed as crucial to success in mathematics (National Council of Teachers of Mathematics, 2006). Furthermore, existing data on the relationship between mathematical proficiency and understanding of magnitudes are consistent with the view that helping young children develop a better understanding of numerical magnitudes may lead to improved performance on mathematics tasks (Laski & Siegler, 2007).

So with this research in mind, how can teachers and parents help young children develop a better understanding of numerical magnitudes? One way is to use a clothesline as number line in order to build understanding of numerical relationships (Suh, 2014). The list of activities below can help young children develop flexible thinking with numbers.

• Encourage young children to equally space and hang number cards on the number line, using benchmark numbers such as 0, 5, and 10 as a guide. As the child masters this range of numbers, expand or change the range. Encourage the child, as he/she hangs the number cards, to reason and talk about mathematical ideas such as: 
•  Is your number card closer to 0 or 5? How do you know? 
•  Is your number card closer to 5 or 10? How do you know? 
•  How far is 4 from 10? How do you know? 
•  Support children’s reasoning about comparing and ordering numbers by having them justify solutions. For example, 
•  Which number is bigger, 4 or 5? Why? 
•  Why is 225 smaller than 250? 
• Discussing placement of fractions and decimals highlights equivalency concepts. As children work with these number cards ask questions such as, 
•  Which fraction is equivalent to 1/2? 
•  Are 0.09 and 0.90 the same or different number? How do you know?

References
Clements, D. & Sarama, J. (2009). Learning and teaching early math: the learning trajectories approach. New York: Routlege. 

Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38, 293 – 304. 

Griffin, S., Case, R., & Siegler, R. (1994). Rightstart: Providing the central conceptual prerequisites for first formal learning of arithmetic to students at risk for school failure. In K. McGilly (Ed.), Classroom lessons: Integrating cognitive theory and classroom practice (pp. 24–49). Cambridge, MA: MIT Press. 

Laski, E., & Siegler, R. (2007). Is 27 a big number? Correlational and casual connections among numerical categorization, number line estimation, and numerical magnitude comparison. Child Development 78(6), 1723-1743. 

National Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics. Washington, DC: National Council of Teachers of Mathematics. 

Siegler, R. S., & Booth, J. (2004). Development of numerical estimation in young children. Child Development, 75, 428 – 444. Suh, J. (2014). 

Line ‘em up! Teaching Children Mathematics, 20(5), 336.