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.

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.

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/.

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.

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.

Open-ended Assessments: Part 1

By Brea Ratliff, Secondary Mathematics Research Coordinator

As schools prepare for standardized testing this spring, many educators often wonder which instructional strategies will be most effective in terms of ensuring student success.

Beyond the scope of content, state mathematics assessments are measuring students’ ability to problem solve, recognize appropriate conjectures, communicate and analyze knowledge, and understand how mathematical ideas connect.

In short, these tests evaluate whether or not students are able to demonstrate the appropriate processing skills necessary for mathematics at each grade level.

On the STAAR Mathematics assessment for grades 3-8, mathematical processing skills are “incorporated into at least 75% of the test questions…and [are] identified along with content standards” (TEA, 2013). In order for students to perform well on an assessment designed this way, they should show success on formative assessments where they are challenged to apply their content knowledge while confidently using these skills. Over the next several weeks, we will explore a variety of open-ended formative assessments for students in grades 5 and 8, and students who are taking Algebra 1. These assessments can be implemented in a variety of ways. Depending on the makeup of your classroom, they could be a bell ringer, or the performance indicator during or following direct instruction. Many teachers also use them to start meaningful discussions with their students, as well as for an exit ticket or homework assignment. The possibilities are endless.

This week, let’s begin by looking at a few open-ended assessment ideas for 5th grade – all of which build upon student expectation 5.10(C) from the TEKS:

5.10(C) – The student is expected to select and use appropriate units and formulas to measure length, perimeter, area, and volume.

Level 1 – Assessments designed to develop proficiency in one 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 two or more student expectations. The assessments for this level can vary in degree. While some may be designed to assess a combination of content skills, others may be written to include process skills. In this next example, we continue to look at 5.10(C), but also student expectations 5.3(A) and 5.3(B):

5.3(A) – The student is expected to use addition and subtraction to solve problems involving whole numbers and decimals.

5.3(B) – The student is expected to use multiplication to solve problems involving whole numbers (no more than three digits times two digits without technology).

For this next example, we assess students’ logical reasoning while addressing student expectation 5.16(A):

5.16(A) – The student is expected to make generalizations from patterns or sets of examples and nonexamples.

This assessment covers two content standards we have already addressed [5.3(A), 5.10(C)], and introduces two standards from Probability and Statistics and Underlying Processes and Mathematical Tools:

5.13(B) – The student is expected to describe characteristics of data presented in tables and graphs including median, mode, and range.

5.14(B) – The student is expected to solve problems that incorporate understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness.

These are just a few examples to use with your students, but they are guaranteed to challenge your students to apply what they know about length, perimeter, area and volume in a new way.

Texas Education Agency. (2013). STAAR Assessed Curriculum, Grade 5. Retrieved from http://www.tea.state.tx.us/WorkArea/linkit.aspx?LinkIdentifier=id&ItemID=2147488330&libID=2147488329

Success With Elapsed Time: Part 2

By Cassandra Hatfield, RME Assessment Coordinator

Using the number line to facilitate students understanding of elapsed time can support understanding because it is a familiar model they have used with the base 10 system. Initially, in Success with Elapsed Time: Part 1 I discussed ways to support students in thinking about elapsed time out of context and support the transition to thinking about the base 60 system of time. In this part, I will focus on three basic underlying types of contextual situations that student’s encounter with elapsed time and how to use the structure of those problems to facilitate further use of the number line.

Read through these three problems, and consider what the problems have in common and what is different about them.

1	Sam’s school starts at 7:50 am. He goes to lunch is at 12:20 pm. How much time elapses between when school starts and when he goes to lunch?
2	Jessie has soccer practice at 4:15pm. Practice lasts for 1 hour and 30 minutes. What time will practice end?
3	Michelle’s mom needs her turkey to be done for dinner at 6:30 pm. It will take the turkey 4 hours and 15 minutes to bake. What time does the need to put the turkey in the oven?

Using this model, it is clear to see that each problem has 2 of the 3 pieces of information:

Basic Structure of Elapsed Time Problems

1
2
3

In working in classrooms on this topic, I found that it was effective to give students the opportunity to brainstorm in groups and then discuss the similarities and differences between the problems as a class. The students were able to realize the structure of the problems and that one part was missing without me providing the overarching model. By giving the students the opportunity to develop the model, I became the facilitator of the learning.

As you are planning for lessons on elapsed time, plan to give students a variety of different problem types. Many traditional textbooks only offer problems with a start time and an elapsed time.

Dixon, J. (2008). Tracking time: Representing elapsed time on an open timeline. Teaching Children Mathematics, 15(1), 18-24.

Success with Elapsed Time: Part 1

By Cassandra Hatfield, RME Assessment Coordinator

One of the challenges many teachers face is how to teach students to calculate elapsed time. In fact, "on the 2003 NAEP assessment, only 26 percent of fourth graders and 55 percent of eighth graders could solve a problem involving the conversion of one measure of time to another" (Blume et al., 2007).

This blog will focus on a strategy for computing the elapsed time, given a start and end time. The second blog of this series will focus on the three types of elapsed time solving story problems and how to support students in understanding the structure of those problems.

Using a procedure similar to the standard algorithm to calculate elapsed time can be challenging for students because time is in a base 60 system and depending on the times given, students have to calculate considering the change from AM and PM.

An open number line is an great tool that supports students in calculating elapsed time mentally. Before making the transition to the open number line, in a whole class setting have students count around the class by benchmarks of time and record the times on an anchor chart.

When students understand the benchmarks of time it supports them in being flexible in which strategy they use. Some students will gravitate towards one strategy while other students will select the strategy that is most efficient for the times given.

Calculating by benchmarks of time

Calculating to benchmarks of time

Some students will find it difficult to combine the minutes and hours when calculating to benchmarks of time. It is also important to focus your classroom discussions on how to combine benchmarks of time. An anchor chart to support this can also is beneficial for your students. Students will come up with many different ways. Here are just a few.

We would love to get some feedback on transitioning to a number line for calculating elapsed time. Let us know how it goes!

Blume, G., Gilindo, E., & Walcott, C. (2007). Performance in measurement and geometry from the viewpoint of Principles and Standards of School Mathematics. In P. Kloosterman & F.Lester, Jr. (Eds.), Results and interpretations of the 2003 mathematics assessment of the National Assessment of Educational Progress, 95-138. Reston, VA: NCTM.

Insight to ‘Teaching Math to Young Children’

By Sharri Zachary, RME Mathematics Research Coordinator

The National Research Council (NRC) and the National Council for Teachers in Mathematics (NCTM) describe two fundamental areas of mathematics for young children: 1) Number and Operations, and 2) Geometry and Measurement. According to the NRC (2009), conceptual development within number and operations should focus on students’ development of the list of counting numbers and the use of counting numbers to describe total objects in a given set. It is recommended that teachers provide students with opportunities to “subitize small collections [of objects], practice counting, compare the magnitude [size] of collections, and use numerals to quantify collections” (Frye et al., 2013). Conceptual development in geometry and measurement should support the idea that geometric shapes have different parts that can be described and include activities that model composition and decomposition of geometric shapes.
The Institute of Educational Sciences (IES) released a practice guide recently on Teaching Math to Young Children. The recommendations put forth in the IES practice guide are:

1. Teach number and operations using a developmental progression
2. Teach geometry, patterns, measurement, and data analysis using a developmental learning progression
3. Use progress monitoring to ensure that math instruction builds on what each child knows
4. Teach children to view and describe their world mathematically
5. Dedicate time each day to teaching math, and integrate math instruction throughout the school day

These recommendations are intended to:

• Guide teacher preparation that will result in later math success for students
• Provide descriptions of early content areas to be integrated into classroom instructional practices
• Assist in the development of curriculum for students in early grades

To access/download the full IES practice guide, please visit http://ies.ed.gov/ncee/wwc/PracticeGuide.aspx?sid=18

Frye, D., Baroody, A. J., Burchinal, M., Carver, S. M., Jordan, N. C., & McDowell, J. (2013). Teaching math 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 Educa-tion. Retrieved from the NCEE website: http://whatworks.ed.gov

National Research Council. (2009). Mathematics learning in early childhood: Paths toward excellence and equity. Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: The National Academies Press

Mastering Explicit Instruction - Part 4

By Dr. Deni Basaraba, RME Assessment Coordinator

We have come to an end on our series on explicit instruction (Please see our previous posts for more information!). Van de Walle (2013) characterizes explicit instruction as highly-structured, teacher-led instruction on a specific strategy. He explains how this approach can help uncover or make overt the thinking strategies that support mathematical problem solving for students with disabilities.

Here are our final thoughts on the elements necessary to carry out explicit instruction in the classroom.

Monitoring the performance of all students closely to verify students’ mastery of the content and help determine whether adjustments to instruction need to be made in response to student errors. Closely monitoring student performance can also help you determine which students are confident enough in their knowledge of the content to respond when signaled and which students are cueing off their peers by waiting for others to respond first.

Providing specific and immediate affirmative and corrective feedback as quickly as possible after the students’ response to help ensure high rates of success and reduce the likelihood of practicing errors. Feedback should be specific to the response students provided and briefly descriptive so that students can use the information provided in the feedback to further inform their learning (e.g., “Yes, we can use the written numeral 2 to show that we have two pencils”).

Delivering instruction at a “perky” pace to optimize instructional time, the amount of content that can be presented, the number of opportunities students have to practice the skill or strategy, and student engagement and on-task behavior. The pacing of content delivery should be sufficiently brisk to keep students engaged while simultaneously providing a reasonable amount of “think-time,” particularly when students are learning new material.

Providing distributed and cumulative practice opportunities to ensure that students have multiple opportunities to practice skills over time. Cumulative practice provides an opportunity to include practice opportunities that address both previously and newly learned content, skills, and strategies, facilitating the integration of newly learned knowledge and skills with previously learned knowledge and skills and supporting retention and fostering automaticity.

Thanks for joining us on our journey with explicit instruction! Please feel free to leave comments below with your thoughts or questions!

Baker, S. K., Gersten, R., & Lee, D. (2002). A synthesis of empirical research on teaching mathematics to low-achieving students. The Elementary School Journal, 103, 51-73.

Carnine, D. W. (1997). Instructional design in mathematics for students with learning disabilities. Journal of Learning Disabilities, 30, 130-141.

Doabler, C. T., Cary, M. S., Jungjohann, K., Clarke, B., Fien, H., Baker, S., Smolkowski, K., & Chard, D. (2012). Enhancing core mathematics instruction for students at risk for mathematics disabilities. Teaching Exceptional Children, 44, 48-57.

Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and middle school mathematics: Teaching developmentally, 8th edition. Boston: Pearson.

Witzel, B. S., Mercer, C. D., & Miller, M. D. (2003). Teaching algebra to students with learning difficulties: An investigation of an explicit instruction model. Learning Disabilities Research & Practice, 18, 121-131.

Mastering Explict Instruction - Part 3

By Dr. Deni Basaraba, RME Assessment Coordinator

We have been doing a series on how all students, particularly those who are struggling, benefit from the provision of explicit instruction (Baker, Gersten, & Lee, 2002; Carnine, 1997; Doabler et al., 2012; Witzel, Mercer, & Miller, 2003). Van de Walle (2013) characterizes explicit instruction as highly-structured, teacher-led instruction on a specific strategy. He explains how this approach can help uncover or make overt the thinking strategies that support mathematical problem solving for students with disabilities.

We started a list of the methods and elements of explicit instruction (see the previous two posts!). To continue this series of blogs, here are some additional elements of explicit instruction as well as examples and/or steps to carrying out these recommendations.

Using clear, concise, and consistent language that is not ambiguous, does not introduce irrelevant or unnecessary information into instruction, and is consistent across teacher models, whole group practice, and to elicit student responses so that students know what they are expected to do. Here are some suggestions to help with this element: Train interventionists to explain math content Include math concepts, vocabulary, formulas, procedures, reasoning and methods Use clear language understandable to students

Providing a range of examples and non-examples to help students determine when and when not to apply a skill, strategy, concept, or rule. Including a wide-range of positive examples during instruction provides students with multiple opportunities to practice the using the target strategy or skill while including non-examples can help reduce the possibility that students will use the skill inappropriately or over-generalize it to inappropriate mathematics problems. For example, here are some examples and non-examples of an algebraic expression: Example: Suppose we wanted to write an expression to find the total cost of movie tickets for different groups of people, given that tickets cost $7 each. We can define the variable "p" to represent the number of people in a group. Then the algebraic expression 7 x p or 7p represents the total cost of movie tickets for a group of p people. Non-example:The expression 550 + 20 is not an algebraic expression because it does not contain at least one variable.

Requiring frequent responses by relying minimally on individual student turns (e.g., asking students to raise their hand if they have the correct response) and relying more on the use of questioning to elicit various types of responses such as oral responses (e.g., having pre-identified partners or groups of students such as boys/girls, 1s/2s, apples/oranges respond), written responses, and/or action responses (e.g., thumbs up/thumbs down). Increasing the number of opportunities students have to respond during the lesson promotes a high level of student-teacher interaction, helps keep students engaged with the content, provides multiple opportunities for student practice with and elaboration on the newly learned content, and can help you monitor student understanding.

We have four more suggestions for mastering explicit instruction! Stay tuned!!

Baker, S. K., Gersten, R., & Lee, D. (2002). A synthesis of empirical research on teaching mathematics to low-achieving students. The Elementary School Journal, 103, 51-73.

Carnine, D. W. (1997). Instructional design in mathematics for students with learning disabilities. Journal of Learning Disabilities, 30, 130-141.

Doabler, C. T., Cary, M. S., Jungjohann, K., Clarke, B., Fien, H., Baker, S., Smolkowski, K., & Chard, D. (2012). Enhancing core mathematics instruction for students at risk for mathematics disabilities. Teaching Exceptional Children, 44, 48-57.

Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and middle school mathematics: Teaching developmentally, 8th edition. Boston: Pearson.

Witzel, B. S., Mercer, C. D., & Miller, M. D. (2003). Teaching algebra to students with learning difficulties: An investigation of an explicit instruction model. Learning Disabilities Research & Practice, 18, 121-131.

Mastering Explicit Instruction - Part 2

By Dr. Deni Basaraba, RME Assessment Coordinator

A couple of weeks ago, we discussed how all students, particularly those who are struggling, benefit from the provision of explicit instruction (Baker, Gersten, & Lee, 2002; Carnine, 1997; Doabler et al., 2012; Witzel, Mercer, & Miller, 2003). Van de Walle (2013) characterizes explicit instruction as highly-structured, teacher-led instruction on a specific strategy. He explains how this approach can help uncover or make overt the thinking strategies that support mathematical problem solving for students with disabilities. 

We started a list of the methods and elements of explicit instruction. To continue this series of blogs, here are some additional elements of explicit instruction as well as examples and/or steps to carrying out these recommendations.

Beginning each lesson with a clear statement of the objectives and goals for that lesson so that students know not only what they are going to learn but why it is important and how it relates to other skills and strategies they have already learned. For example: Today, we are going to continue our work with variables and expressions. You are going to learn how parentheses are used in expressions. You need this knowledge in order to solve algebraic equations.

Reviewing prerequisite knowledge and skills prior to starting instruction that includes a review of relevant information not only to ensure that students’ have the prior knowledge and skills needed to learn the skill being taught in the lesson but also to provide students an opportunity to link the new skills with other related skills and increase their sense of success. For example: First, let’s review. Tell your partner what a variable is. (A variable is a symbol that represents a number.) Yes, a variable is a symbol that represents a number. Look at these expressions (5 + x; t – 10). In the first expression, what is the variable? (X) Yes, x is a symbol that represents a number. Everyone, what is the variable in the second expression? (T) Let’s look again at the definition of an expression...

Modeling the desired skill or strategy clearly to demonstrate for students what they will be expected to do so and can see first-hand what a model of proficient performance looks like. When completing these models, think aloud for students so that each step of the strategy or skill you perform is clear and to clarify the decision-making processes needed to complete the procedure or solve the problem. For example: Look at this expression: 3 × ( 4 - 2). When an expression contains more than one operation, parentheses can be used to show which computation should be done first. So, when we have an expression, we first look for parentheses and do the operation inside the parentheses. In this problem, 4 - 2 is inside the parentheses, so I will do that operation first...

Providing guided and supported practice by regulating the difficulty of practice opportunities from easier to more challenging and providing higher levels of guidance and support initially that is gradually decreased as students demonstrate success. For example: Let’s do some problems together. Please stay with me so we can do these items correctly. Write this expression on your paper, but don’t solve it. Do we do the operations inside or outside of the parentheses first...  Write this expression on your paper. Do we do the operations inside or outside of the parentheses first? Inside. Find the value of the expression. (After several examples where the teacher slowly gives less help) Copy this expression and find the value of the expression. Don’t forget . . . parentheses first...

Baker, S. K., Gersten, R., & Lee, D. (2002). A synthesis of empirical research on teaching mathematics to low-achieving students. The Elementary School Journal, 103, 51-73.

Carnine, D. W. (1997). Instructional design in mathematics for students with learning disabilities. Journal of Learning Disabilities, 30, 130-141.

Doabler, C. T., Cary, M. S., Jungjohann, K., Clarke, B., Fien, H., Baker, S., Smolkowski, K., & Chard, D. (2012). Enhancing core mathematics instruction for students at risk for mathematics disabilities. Teaching Exceptional Children, 44, 48-57.

Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and middle school mathematics: Teaching developmentally, 8th edition. Boston: Pearson.

Witzel, B. S., Mercer, C. D., & Miller, M. D. (2003). Teaching algebra to students with learning difficulties: An investigation of an explicit instruction model. Learning Disabilities Research & Practice, 18, 121-131.

Mastering Explicit Instruction - Part 1

By Deni Basaraba, RME Assessment Coordinator

Research (Baker, Gersten, & Lee, 2002; Carnine, 1997; Doabler et al., 2012; Witzel, Mercer, & Miller, 2003) indicates that all students, but particularly struggling learners, benefit from the provision of explicit instruction, which can be defined as a direct an unambiguous approach to instruction that incorporates instructional design and instructional delivery procedures (Archer & Hughes, 2011).

Moreover, students benefit from having strategies for learning content made conspicuous for them but only when these strategies are designed to support the transfer of knowledge from the specific content being taught (i.e., the mathematics problems that are the focus of instruction during a particular class period) to a broader, more generalizable context (e.g., similar mathematics problems with similar or more complex types of numbers, similar mathematics problems presented during whole group instruction to be solved collaboratively, in pairs, or independently as part of homework) (Kame’enui, Carnine, Dixon, & Burns, 2011).

What is explicit instruction?

As noted previously, the idea of explicit instruction includes both instructional design and instructional delivery features, and requires that we carefully consider not only the content we plan to teach and the order in which that content is sequenced but also the methods we used to deliver that instruction to our students. In this series of blogs, we will provide you with elements of explicit instruction as well as examples and/or steps to carrying out these recommendations.

Focusing instruction on critical content (e.g., skills, strategies, concepts, vocabulary) that is essential for promoting student understanding of the target content as well as future, related content. The amount of time designated each day to explicit vocabulary instruction will depend upon the vocabulary load of the text to be read as well as the students’ general prior knowledge of the new vocabulary.  Computer-assisted instruction can be another effective way to provide practice on newly-introduced vocabulary words.  Other methods include using graphic organizers and semantic maps to teach the relationships among words and concepts.

Organizing instruction around “big ideas” that can help students see how particular skills and concepts fit together. This will provide students with a clearer understanding of how content, skills, and strategies are related and can help students organize that information in their minds, making it easier for them to retrieve that information and integrate it with new material. Example: Fractions are numbers that can be represented in different ways. Modeling part/whole relationship Writing fraction numbers Comparing fractions Measuring fractions on a number line

Sequencing skills in a logical fashion so that (a) students learn all prerequisite skills before being asked to perform the target skill, (b) high-frequency skills are taught before those needed less frequently, (c) skills and strategies that are similar and may be confusing to students are not taught in close proximity within the scope and sequence, and (d) easier skills are taught before more difficult skills.

Breaking down complex skills and strategies into smaller units in an effort to minimize cognitive overload and the demands placed on students’ working memory. Teach skills in small, discrete steps that can be added to one another to form the target skill. Present new information in small steps will give students enough to have success on the topic.  Once students have mastered that step, reinforce the topic and add to it.

Archer, A. L, & Hughes, C. A. (2011). Explicit instruction: Effective and efficient teaching. New York NY: Guilford.

Baker, S. K., Gersten, R., & Lee, D. (2002). A synthesis of empirical research on teaching mathematics to low-achieving students. The Elementary School Journal, 103, 51-73.

Carnine, D. W. (1997). Instructional design in mathematics for students with learning disabilities. Journal of Learning Disabilities, 30, 130-141.

Doabler, C. T., Cary, M. S., Jungjohann, K., Clarke, B., Fien, H., Baker, S., Smolkowski, K., & Chard, D. (2012). Enhancing core mathematics instruction for students at risk for mathematics disabilities. Teaching Exceptional Children, 44, 48-57.

Kame’enui, E. J., Carnine, D. W., Dixon, R. C., & Burns, D. (2011). Introduction. In M. D. Coyne, E. J. Kame’enui, & D. W. Carnine (Eds.) Effective teaching strategies that accommodate diverse learners (4th ed.) (pp. 7-23).

Witzel, B. S., Mercer, C. D., & Miller, M. D. (2003). Teaching algebra to students with learning difficulties: An investigation of an explicit instruction model. Learning Disabilities Research & Practice, 18, 121-131.

Connecting the Area Model to the Standard Algorithm

By Cassandra Hatfield, RME Assessment Coordinator

Using the area model for multiplication and using the standard algorithm for multiplication are often put in two separate and unrelated categories. Often times textbooks spend very little time developing the conceptual understanding and focus on the procedure of the standard algorithm.

However, “as much time as necessary should be devoted to the conceptual development of the algorithm with the recording or written part coming later.” (Van De Walle, Karp, Bay-Williams, 2013). Students are more successful when they can relate their prior knowledge with a new concept. Designing lessons that connect the area model and partial products can then lead to the understanding of the standard algorithm. This powerful transition allows students to visually see the why behind the standard algorithm.

The model below uses color to amplify the connection between the area model, the partial products strategy and the standard algorithm with 2-digit multipliers. Notice that the area model was drawn proportionally, not as a “window pane.” The importance of drawing area models proportionally was discussed in one of my previous posts, It's Not a Window Pane... It's an Area Model.

It is important to consider the value of the digits rather than the digits themselves when using partial products or the standard algorithm. For example, when multiplying 20 x 20, use the base 10 language 2 tens times 2 tens is 4 hundreds or 20 times 20 is 400. Try to avoid “two times two.”

Students can use the partial products strategy just as effectively as the standard algorithm. In fact, it is of utmost importance to give students the opportunity to explore, explain, and demonstrating their understanding of the value of the digits over the digits themselves.

Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and middle school mathematics: Teaching developmentally, 8th edition. Boston: Pearson.

Back to School with a Home Run!

By Cassandra Hatfield, RME Assessment Coordinator

This summer I attended two trainings. Each training was aligned to different standards: (1) the newly revised Texas TEKS to be implemented in the 2014-2015 school year and (2) the Common Core State Standards. Although aligned to different standards, both of these trainings highlighted the math Process Standards in Principles and Standards for School Mathematics (NCTM, 2000) and the National Research Council’s Strands of Mathematical Proficiency discussed in Adding It Up. A common theme exemplified through the trainings was that it is essential for mathematics instruction to move beyond rote procedural knowledge and for instruction to be grounded in conceptual understanding and mathematical reasoning.

While attending these trainings I heard teachers, specialists, and administrators grappling with when and how to make this shift in instruction. Sherry Parrish’s resource Number Talks: Helping Children Build Mental Math and Computation Strategies is an incredible learning adventure that enables an individual teacher, a team of teachers, or a teacher leader to make changes in classroom instruction and build students repertoire of computational strategies in just five to fifteen minutes a day! By using this resource teachers are given the opportunity to “hit it out of the park” by:

1. Promoting environment and community
2. Facilitating classroom discussion
3. Developing the role of mental math
4. Providing purposeful computation problems

I’ve implemented this resource in more than 25 classrooms over the past three years and have seen the joy from students as they move from being told how to compute to telling how they computed and seen the confidence they have build as they have moved from counting on their fingers to using flexible mental math strategies. In addition, many teachers I’ve worked with have used this resource to help them shift their teaching style into being a facilitator. In the beginning it can be overwhelming, but start with the basic fact problem sets and enjoy the journey with your students.

Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for mathematical practice. Washington, DC: National Governors Association Center for Best Practices and Council of Chief State School Officers (CCSSO). http://www.corestandards.org.

National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: NCTM.

Kilpatrick, J., Swafford, J. & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.

Parrish, Sherry D (2010). Number talks: Helping children build mental math and computations strategies. Sausalito, CA: Math Solutions.

Parrish, Sherry D (2011). Number talks build numerical reasoning. Teaching Children Mathematics, 18(3). 198–206.

Teacher Time Allocation in the Classroom

By Saler Axel, RME Research Assistant 

Research has shown that a strong positive relationship exists between on-task learning time and student achievement (Codding & Smyth, 2008). The basic components of learning time include time allocated to instruction, time engaged in the learning process, and academic productivity (Codding & Smyth, 2008). In many classrooms, lost time occurs despite the fact that most school professionals are aware of the strong relationship that exists between on task learning and academic success (Codding & Smyth, 2008). As educators, we often worry that there is not enough time to complete all of our goals. With the new, heightened standards in mathematics, it is even more important that we take advantage of instructional time. Just think: If you reallocate the time you already have in your classroom toward more mathematics instruction, students’ academic success and understanding of mathematics concepts may increase!

As suggested by Engelmann and Carnine (1982), it is important to consider your instructional environment when planning ways to improve students’ academic performance. Gagne and Dick (1983) assert that it is necessary to separate the external and internal influences on instruction when trying to change a behavior (or in our case change how time is appropriated in the classroom). This means that, when you try to reallocate time in your classroom for additional mathematics instruction, concentrate on the things you can see and do (such as enhancing your classroom management skills or timing how long it takes to transitioning from one activity to another).

Studies have shown that students can spend up to one-half of instructional time engaged in tasks not related to learning (Codding & Smyth, 2008). Information like this highlights the importance of making goals to increase instructional time and discourage tasks unrelated to instruction. Lee (2006) suggests that decreasing transitions between activities is a primary way to increase instructional time (and in our case, increase mathematics instruction!). Lee (2006) identifies two types of transitions that can be decreased within the classroom: (1) transitions between programs and (2) transitions between classroom routines.

Consider your most recent mathematics lesson. How did it end? Did you have the opportunity to conclude as you had planned? Did you run out of time? How was your instructional time allocated? Were students engaged in instructional activities for most of the time dedicated to your lesson? As many of us have experienced during an instructional “groove,” when transitions happen, the classroom momentum can be stymied. By reducing the transitions between subjects or programs, we can increase academic learning time (Lee, 2006). This may help increase the potential of accomplishing increased academic success. Furthermore, by reducing transitions between classroom routines, potential opportunities for students to misbehave and waste educational time are lessened (Codding & Smyth, 2008). If you use classroom time gained in a positive way, just think how much you can favorably impact your students’ academic outcomes.

So, what are some successes you can look forward to if transitions in your classroom are reduced and classroom time is allocated more toward additional mathematics instruction? If you enhance your ability to allocate more classroom time toward more mathematics instruction, you can further

• utilize gained time effectively, 
• utilize gained time to implement more detailed lessons, 
• utilize gained time to increase interactions with students, and 
• utilize gained time to improve instructional quality. 

The good news? When teachers enhance these things, studies have shown that teachers can increase academic outcomes in students (Codding & Smyth, 2008; Lee, 2006). In addition, if you help enhance your students ability to

• utilize gained time effectively, 
• utilize gained time to increase interactions with teachers, and 
• utilize gained time to complete assignments, studies have demonstrated that their academic successes can also grow (Codding & Smyth, 2008; Lee, 2006).

Think about how you approach mathematics teaching. How do you allocate instructional time in your classroom? What are some ways you can enhance your transitions and increase the opportunity to provide more instruction to your students and enhance the likelihood of furthering the goals above? Share your thoughts by responding to this blog.

Codding, R. S. & Smyth, C. A. (2008). Using performance feedback to decrease classroom transition time and examine collateral effects on academic engagement. Journal of Educational and Psychological Consultation, 18, 325-345. 

Engelmann, S. & Carnine, D. (1982). Theory of instruction: Principles and applications. Manchester, NH: Irvington Publishers. 

Gagne, R. M. & Dick, W. (1983). Instructional psychology. Annual Review of Psychology, 34, 261-295. 

Lee, D. L. (2006). Facilitating transitions between and within academic tasks: An application of behavioral momentum. Remedial and Special Education, 27(5), 312-317.

It's Not a Window Pane... It's an Area Model

By Cassandra Hatfield, RME Assessment Coordinator

As students develop an understanding of multiplication, instruction often moves from an equal sets into an area model. In my experience as a math specialist, sometimes an area model was called a “window pane.” In this blog we will focus on the conceptual understanding of an area model and the need to shy away from calling it a “window pane.”

Just last month, I was visiting my nephew who was very proud that he had memorized several multiplication combinations and he asked me to quiz him. He was giving me answers at a rapid pace until I got to 8 • 7. His response, “I don’t know that one yet!” He had just told me what 8 • 5 was and what 8 • 2 was. However, my nephew was simply memorizing combinations and did not have a strategy to compose the combination with other combinations he knows. The area model is a powerful way to assist students in composing combinations.

When students know and understand combinations through 10 • 10 then they can decompose numbers to find any other combination.

I’ve been in classrooms and seen anchor charts titled “Window Pane Strategy” with an example of finding the product of a 2 digit number by a 2 digit number. However, notice that prior to this area models did not look like window panes. The models shown are proportional. The units are squares, so in the problem 12 • 8 the factor 12 is longer than the factor 8. When teaching students to multiply 2 digit by 2 digit numbers before transitioning to an open area model (without the grid lines) it is extremely important for students to make the connection to their prior understanding by using a model with grid lines. Using an area model is not a procedure for finding an answer; it’s a conceptual understanding of the distributive property.

Even after moving to an open area model, while students do not need to measure to make the parts perfectly proportional, it is important that students still draw the open area models to show that each of the parts is a different area. The open area model can continue to be used to support students in multiplying larger numbers.

In another post next month, I’ll share how the area model connects to the standard algorithm and can be a powerful model to support student understanding of the standard algorithm beyond a procedure.

Help Your Students Experience Fractions Conceptually

By Dawn Woods, RME Elementary Mathematics Coordinator

Many students find fraction concepts difficult to understand yet the understanding of fractions is essential for learning algebra and advanced mathematics (National Mathematics Advisory Panel, 2008). As an elementary mathematics educator, I noticed that many of my students struggled with fraction concepts across the curriculum. I wondered how I could help my students experience and understand fraction concepts conceptually so they could succeed not only in my classroom but also in advanced mathematics.

My search for answers began with research. I discovered that to understand fractions means to recognize the multiple meanings and interpretations of fractions. Furthermore, I needed to explicitly present these different constructs in a contextual way to build understanding. Mathematics educators generally agree that there are five main fraction constructs and that they are developmental in nature.

The first construct presents fractions as parts of wholes or parts of sets. Research suggests that this construct is an effective starting point for building fractions (Cramer & Whitney, 2010). However, it is important to realize that the part-whole relationship goes way beyond the shading of a region. For example, it could be part of a group of animals such as (¼ of the animals are dogs), or be part of a length, (we ran 1 ½ miles) (Van De Walle, Karp, Bay-Williams, 2013).

Researchers such as Cramer, Wyberg, and Leavitt suggest that the fraction circle manipulative is a powerful concrete representation since it helps to build understanding of the part-whole relationship as wells as the meaning of the relative size of fractions (2008). Here, they use fraction circle models to help build mental images that aid in the ability to judge relative sizes of fractions. It is also important to remember that the fractional parts do not need to be identical in shape and size, but must be equivalent in some other attribute such as area, volume, or number (Chapin & Johnson, 2006). However, it is important to teach beyond this first construct to include other fraction representations and models.

The second construct presents fractions as measures. Measurement (Van De Walle, Karp, Bay- Williams, 2013) involves identifying a length and then uses that length to determine a length of an object. The number line plays an important role in this construct by partitioning units into as many subunits that one is willing to create (Chapin & Johnson, 2006). For example, in the fraction ¾, you can use the unit fraction ¼ as the selected length and then measure to show that it takes three of those to reach ¾ (Van De Walle, Karp, Bay-Williams, 2013). Research suggests that students who develop an initial understanding of rational numbers as measures, develop ideas of unit, partitioning, order, addition and subtraction (Cramer & Whitney, 2010) while using the number line as a model. Essentially, this powerful construct illustrates that there are an infinite number of rational numbers on the number line as it focuses on how much rather than parts of a whole.

Fractions can also result from dividing two numbers. This construct is often called the quotient meaning, since the quotient is the answer to a division problem (Chapin & Johnson, 2006). Think about the number of cookies each person receives when 15 cookies are shared between 3 people. This problem is not a part-whole scenario (Van De Walle, Karp, Bay-Williams, 2013) but it still means that each person will receive one-third of the cookies expressed as ¹⁵⁄₃, ⁵⁄₁, or 5. Connecting division to fractions enables students to feel comfortable with seeing division expressed in multiple ways such as 16 ÷ 3, ¹⁶⁄₃, and 5¹⁄₃ and is important for continued success in advanced mathematics.

The fourth construct presents fractions as operators. In this construct, a fraction is a number that acts (or operates) on another number to stretch or shrink the magnitude of the number (Chapin & Johnson, 2006). For an example, a model of a car may be 1/16 the size of the original or a cell maybe magnified under a microscope to 400 times the actual size demonstrating a multiplicative relationship between the quantities. This construct takes fractions beyond representation to a place where students know how to use fractions to solve problems across the curriculum.

The fifth and final construct characterizes fractions as the ratio or comparison of two quantities. A ratio such as 1/3 can mean that the probably of an event is one in three (Van De Walle, Karp, Bay-Williams, 2013). Or a ratio can also represent part-whole relationships such as 11 children at the park compared to the total number of 18 people. We could write this part-to-whole relationship as the fraction ¹¹⁄₁₈. However, it is important that realize that all fractions are ratios but not all ratios are fractions (Chapin & Johnson, 2006). Part-to-part comparisons such as the number of children to the number of people at the park, 11:18, is not a fraction because this comparison does not name a rational number but presents a comparison of two numbers.

Fraction understanding, although a challenge to students, is a critical mathematics concept. For students to really understand fractions, they need to experience fractions across all five constructs in meaningful ways that build conceptual understanding. This conceptual understanding, in turn, provides students with mental representations that enable students to connect meaning to fractions across a variety of contexts.

Chapin, S.H., & Johnson, A. (2006). Math matters, 2nd edition. Sausalito: Math Solutions Publications.

Cramer, K., & Whitney, S. (2010). Learning rational number concepts and skills in elementary school classrooms. In D.V. Lambdin & F.K. Lester, Jr. (Eds.), Teaching and learning mathematics: Translating research for elementary school teachers (pp. 11-22). Reston, VA: NCTM

Cramer, K., Wyberg, T., & Leavitt, S. (2008). The role of representations in fraction addition and subtraction. Mathematics Teaching in the Middle School 13(8), 490-496.

National Mathematics Advisory Panel (2008). The final report of the national mathematics advisory panel. Jessup, MD: Education Publications Center. Retrieved from http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf.

Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and middle school mathematics: Teaching developmentally, 8th edition. Boston: Pearson.