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/

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.

Project PAR: Promoting Algebra Readiness

By Dawn Woods, RME Elementary Mathematics Coordinator

Many students appear to be on-track for mathematics achievement in 4th grade, but exit 8th grade without having developed critical skills in the area of rational numbers (National Center for Education Statistics, 2009). Conceptual understanding of rational numbers, as well as, their symbolic representations is a critical component for understanding everyday situations in algebra. Students must master these foundational skills and concepts at the elementary and middle school levels. Project PAR: Promoting Algebra Readiness is an intervention curriculum designed to build this rational number understanding.

Project PAR is a three-year, Institute of Educational Sciences (IES) funded research study that is working to develop a strategic intervention on rational number concepts that use evidence based strategies. The purpose of this project is to promote algebra readiness for sixth grade students by developing students’ conceptual understanding of rational numbers on a number line. The project team consists of researchers and curriculum experts from the University of Oregon (UO) and Research in Mathematics Education (RME) at Southern Methodist University who have extensive experience designing math interventions for a range of student learners as well as vast teaching experience in the mathematics classroom.

Project PAR is completing the development phase of this study where curriculum writers from the UO and RME have designed the scope and sequence for the intervention, developed approximately 100 print-based lessons, and conducted preliminary feasibility testing of individual lessons. The project is now moving into the implementation phase where classroom intervention teachers in Texas and Oregon are teaching the lessons. At this time curriculum writers and researchers will determine if the lessons have realistic expectations and goals for classroom use as the teachers use the lessons and provide critical feedback. This summer, curriculum writers will revise the curriculum based on the results of the feasibility study in preparation for the pilot study scheduled for the 2014-2015 school year. During the pilot study, the potential promise of the intervention increasing student achievement will be examined.

RME would like to thank the sixth grade math teachers Bush Middle School in Carrollton-Farmers Branch ISD who opened up their classrooms for the preliminary feasibility testing as well as the sixth grade math teachers at Fowler Middle School in Frisco ISD who are implementing the PAR curriculum during the feasibility study. We could not do our work with out the support of great teachers at great schools who are putting evidence- based strategies into practice!

National Center for Education Statistics (2009). The nation’s report card: Mathematics 2009. Washington, DC: National Center for Education Statistics

What Makes a Pre-AP Math Course Pre-AP?

By Sharri Zachary, RME Mathematics Research Coordinator

Pre-AP courses are designed to prepare students for college. According to The College Board (2014), Pre-AP courses are based on the following premises:

• All students can perform at rigorous academic levels
• Every student can engage in higher levels of learning when they are prepared as early as possible

As we transition into implementation of the revised Texas Essentials of Knowledge and Skills (TEKS), we have to ensure that Pre-AP courses still fulfill the purpose for which they are intended. The revised TEKS have added a level of academic rigor for ALL students in the general education classroom. Students are expected to deepen their conceptual understanding of math concepts, including reasoning and justifying their solution. This means that students in Pre-AP courses have to be met with challenges that expand their knowledge and skills and push them a notch above, toward the next level. We have to be cautious to avoid students receiving Pre-AP credit for course work that is not Pre-AP.

Pre-AP Math Course Goals:

• Teach on grade level but at a higher level of academic rigor
• Assess students at a level similar to what is offered in an AP course (rigorous multiple-choice and free-response formats)
• Promote student development in skills, habits, and concepts necessary for college success
• Encourage students to develop their communication skills in mathematics to interpret problem situations and explain solutions both orally and written
• Incorporate technology as a tool for help in solving problems, experimenting, interpreting results, and verifying solutions

This is just a small list of goals for Pre-AP math courses. The College Board has official Pre-AP courses in mathematics (and English language arts) for middle and high school students offered through their SpringBoard program (The College Board, 2014). These courses offer rigorous curriculum and formative assessments consistent with their beliefs and expectations.

The College Board. (2014). Pre-AP. Retrieved from http://apcentral.collegeboard.com/apc/public/preap/index.html

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.

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.

The Pythagorean Relationship

By Saler Axel, RME Research Assistant 

Math has a reputation of being dull. Luckily, there are some fun math holidays that exist throughout the year. Two popular ones are Pi Day (3/14) and Mole Day (10/23). Last week was 5-12-13 Triangle Day! What makes the 5-12-13 right triangle worth celebrating?

Let’s spend time considering special right triangles, which are some of geometry’s extraordinary shapes. A right triangle contains sides lengths that can be calculated using the Pythagorean Theory, a² + b²= c². We will spend time discussing right triangles like the 5-12-13 right triangle.

Side-based special triangles, such as a 5-12-13 right triangle, contain proportionate side lengths that make computing easier. Called Pythagorean Triples, these triangles contain angles with degrees that are never rational numbers. If students understand the relationships of a special right triangle’s side lengths, they can calculate other side lengths in geometric problems without having to employ difficult strategies.

An easy way to calculate Pythagorean Triples: a = m²– n², b = 2mn, c = m²+ n². where m and n are relatively prime positive integers and m>n.

Below are some things that you can do in your classroom to celebrate this extraordinary shape.

• Challenge students to calculate scaled examples of 5-12-13 triangles.

• Draw a 5-12-13 right triangle on grid paper. (An example of a 3-4-5 triangle is below.) Have your students make a square from each side. The diagram should have a 5•5 square on the left, a 12•12 square on the bottom, and a 13•13 square off of the hypotenuse. Encourage your students to measure the number of square units. They will discover that 5²+ 12²= 13². Then ask your students to try the same activity with an isosceles triangle (or any other type of triangle except a right triangle). This will help them understand that if they measure the squares, the sides will not make a right triangle.
• Here, the two squares together are a "proof without words." Here we see that:
  
  a² + 2ab+b²= c²+ 2ab

a²+ b²= c²

Other common Pythagorean Triples include those with side length ratios of: 3-4-5, 8-15-13, 7-24-25, and 9-40-41, though the possibilities are endless using the formula (3n)²+ (4n)²= (5n)². For an extensive list of Pythagorean Triples, visit www.mathisfun.com/numbers/pythagorean-triples.html.

How can you tailor these and other classroom lessons to expand your students’ thinking about special right triangles and their importance in geometric calculations?

Teaching Fractions Beyond Pizza and Pie Charts

By Sharri Zachary, Mathematics Research Coordinator

Research shows that “algebra is the [career] gateway to success for many students” (Williams, 2011, pg. 1). For students to understand the concepts and symbolic representations for everyday situations in algebra, students must master certain foundational skills and concepts at the elementary and middle school levels. According to the U.S. Department of Education (2006), a solid foundation in math in early grades will assist in the development of critical thinking skills necessary to pass algebra. The National Mathematics Advisory Panel (2008) identifies the skills that demonstrate algebra readiness as: (1) whole number computation, (2) fraction and decimal proficiency, (3) number concepts (including percents), (4) general concepts, proportions, and geometry, (5) problem-solving, and (6) basic understanding of integers, variables, and simple equations.

Fraction and decimal proficiency is often developed with the use of pizza and pie charts as the introductory piece. Visually, this allows students to see one “whole” partitioned into equal parts, which in turn, helps students understand the part-whole idea of fractions. However, this leaves out the idea that fractions are numbers with magnitude that can be compared (Siegler et al, 2010).

If students are to be truly algebra-ready, visual representations must extend to model fraction concepts that teach the part-whole relationship, as well as, fractions as a distance and magnitude. The IES practice guide recommends the use of number lines as the “central representational tool” to help students recognize fractions as numbers and expand student thinking beyond whole numbers (Siegler et al, 2010, p. 19).

The use of number lines can help students visualize and understand the magnitude of fractions, the relationship between fractions and whole numbers, and the relationship between fractions, decimals, and percents (Siegler et al, 2010). This conceptual understanding is foundational to understanding algebra (Siegler et al, 2010). Based on the recommendations put forth in the IES practice guide, teachers should center fraction instruction around the number line model and support that instruction with other models that include (but are not limited to) fraction circles and strip diagrams. If number lines are recommended as the “central representational tool” for fraction and decimal proficiency, then teachers must consider moving beyond pizza and pie charts in their instruction and assessments to prepare students for mastery of this component of algebra readiness.

National Mathematics Advisory Panel (Spring 2008). Final report, Washington, D.C. 

Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., & Wray, J. (2010). Developing effective fractions instruction for kindergarten through 8th grade (NCEE 2010-4039). 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/practiceguide.aspx?sid=15 

U.S. Department of Education. (2006). Math now: Advancing math education in elementary and middle school. February 2006. Retrieved from: http://www.ed.gov/about/inits/ed/competitiveness/math-now.html 

Williams, T. G. (2011). Reaching algebra readiness (RAR): Preparing middle school students to succeed in algebra – the gateway to career success. Rotterdam, The Netherlands: Sense Publishers.

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.

RTI in a Middle School Mathematics Classroom

By Lindsey Perry, RME Research Assistant

Are you looking for tools and resources to help you reach all students, including those who are struggling in mathematics? Are you seeking out professional development to help you grow in your teaching? The Middle-school Students in Texas: Algebra Ready (MSTAR) initiative can help you learn instructional strategies to assist students struggling with mathematics, assess student understanding, and meet the needs of all learners.

The MSTAR initiative, funded by the Texas Legislature and developed by the Texas Education Agency, is a comprehensive project that provides teachers and administrators with assessments, professional development, and intervention lessons to improve grades 5–8 mathematics achievement in Texas and to sustain the implementation of Response to Intervention (RTI).

An important step in the RTI process is assessing student understanding. To do just that, the MSTAR initiative provides teachers with screening and diagnostic instruments, the MSTAR Universal Screener and the MSTAR Diagnostic. The MSTAR Universal Screener assists teachers in determining if a student is at-risk or on-track for meeting grade level algebra-readiness expectations and the level of support the student may need in order to be successful. The MSTAR Universal Screener is administered three times per year in order to monitor student progress and is administered online at mstar.epsilen.com. The spring administration window is April 8 – May 10, 2013. To find out more, visit http://www.txar.org/assessment/mstar_screener.htm or email universalscreener@region10.org.

The MSTAR Diagnostic Assessment is currently in development. The MSTAR Diagnostic should be administered to students who have been identified by the MSTAR Universal Screener as at-risk for meeting algebra-readiness expectations. This instrument provides teachers with information about why students are struggling and the misconceptions students may have. We are currently seeking a small set of classrooms to participate in the MSTAR Diagnostic Beta test. These classes must have already taken MSTAR Universal Screener at least once this year. While this is a beta test, teachers will receive data on how their students performed. If you are interested, please email us at rme@smu.edu.

The MSTAR Initiative also includes numerous online and face-to-face professional development opportunities. Trainings are available that focus on providing all students with quality Tier I instruction (MSTAR Academy I), strategies for Tier II instruction (Academy II), and data-driven decision making (Implementation Tools). Trainings on topics such as addressing the needs of English language learners, addressing the College and Career Readiness Standards, and teaching fraction/decimal relationships are also available, among many others. Many of the trainings are now available online at www.projectsharetexas.org. For more information, contact your Education Service Center or search the Project Share course catalog at http://projectsharetexas.org/about.

The MSTAR Initiative can help you improve your teaching and help you better understand your students’ needs and how to meet those needs. We encourage you to check out the MSTAR assessments and professional development offerings!

For detailed information about the initiative and the Response to Intervention framework, we invite you to click the link for a copy of “Supporting Students’ Algebra Readiness: A Response to Intervention Approach” in Texas Mathematics Teacher.

Screener vs. Diagnostic

By Savannah Hill, RME Professional Development Coordinator

One project we are involved with at RME is an initiative with the Texas Education Agency and Education Service Center, Region 13 called Middle School Students in Texas Algebra Ready (MSTAR). It began in the summer of 2010 with the goals of (1) improving overall mathematics instruction, and (2) impacting student achievement. MSTAR is comprised of three lead components structured and integrated to support students and teachers in grades five through eight to achieve mathematics success: the MSTAR Universal Screener, MSTAR Diagnostic Assessment, and MSTAR Professional Development.

After talking with many teachers, we have found there is some confusion on the different ways to utilize the MSTAR Universal Screener and the MSTAR Diagnostic Assessment. The intent of this blog is provide a short description of each of these components and how they should be implemented.

MSTAR Universal Screener 
The MSTAR Universal Screener is designed to be administered to all students and identifies studentsʼ level of risk for not being ready for algebra.  The Universal Screener helps teachers make two important decisions within the Response to Intervention (RTI) process:

• Identify students on-track or at-risk for meeting expectations in algebra and algebra-readiness.
• Determine the degree of intensity of instructional support or supplemental intervention needed for students who are at-risk for not meeting expectations in algebra.

Teachers monitor studentsʼ risk status by administering comparable forms of the MSTAR Universal Screener in fall, winter, and early spring.

MSTAR Diagnostic Assessment
The MSTAR Diagnostic Assessment is designed to address those students identified as struggling in Tiers 2 and 3. The diagnostic assessment is given after the MSTAR Universal Screener to those students in Tiers 2 and 3. Its purpose is to:

• Inform educators where a student is on a learning progression.
• Identify the underlying misconception(s) that caused the student to answer incorrectly.
• Identify students current understanding of algebra-related content.

None of the diagnostic assessments are tied to a particular grade level because there may be a 7th grade student who is struggling from misconceptions about 5th grade content. However, when the teacher decides which assessment the student will take, there will be some direction about which assessment may be better for each grade. The reports given will provide information that can be used to plan supplemental instruction. This assessment is not intended to provide screening information.

MSTAR Professional Development
The MSTAR Professional Development provides tools for delivering instruction to all students in achieving algebra readiness and supports informed decision-making based on the results of the MSTAR assessments. The MSTAR Professional Development academies were created to support teachings in preparing students for success in algebra. Trainings are available in face-to-face sessions and/or online. RME researchers, along with TEA, delivered Professional Development in three training sessions for the MSTAR project for the Texas Education Agency in spring and summer 2011 and 2012. The trainings were replicated across the state by certified trainers.

The MSTAR Universal Screener can be accessed through the Project Share Gateway at www.projectsharetexas.org. It can also be accessed directly at http://mstar.epsilen.com. This option will allow you to bypass the Project Share site entirely. Users will see an MSTAR icon after logging in. The same username and password is used for either option. For more information, you can also contact your local Educational Service Center.

Why is Factoring So Important? Factoring Series Part I

By Beth Richardson, RME High School Math Coordinator

“Of all pre-college curriculum, the highest level of mathematics one studies in secondary school has the strongest continuing influence on bachelor’s degree completion. Finishing a course beyond Algebra 2 more than doubles the odds that a student who enters postsecondary education will complete a bachelor’s degree (Adelman, 1999).”

Picture taken from
http://www.aplusalgebra.com/algebra-tiles.htm

The United States is boosting the rigor of its mathematics curriculum. The state of Texas requires students to receive 4 math credits, 3 of which must be Algebra 1, Geometry, and Algebra 2 in order to graduate. Another highly recognized initiative is the Common Core State Standards (CCS) adopted by most states. The traditional pathway of the nation-wide CCS requires high school students to take Algebra 1, Geometry, Algebra 2, and, in most cases, one other math course.

Based on the above information, completing Algebra 2 is a target for our students. Therefore, we need to provide our students with the math skills and knowledge that will lead them towards success. As a former Algebra 2 teacher, I strongly believe a crucial concept for success in Algebra 2 is factoring. Students are required to use factoring as a problem solving strategy and a method of simplification with multiple parent functions and equations (ex: quadratic, rational, and all 4 conic sections) throughout the curriculum.

Many students struggle with factoring. At its core, is an application of the distributive property in which students should begin developing in elementary school through arithmetic. In Algebra 2, factoring of polynomials with variables is a complex, abstract idea. Few students are provided experiences to connect the application of the distributive property with their prior learning. We need to make this clear to them. It is important to take time and explain the underlying mathematical properties at play.

Concrete and visual models are great tools to use when new material is introduced. They help give students a real-world context to connect abstract algebraic properties. Below are some samples of models that are used with the distributive property when students are first learning the concept and when students apply the concept to factor in algebra.

A typical model used to demonstrate
the distributive property in 4th grade.

A typical model used to teach factoring
(opposite of distribution) in Algebra.

Summing it All Up
So, how do we help students’ transition between the different uses of the distributive property (including factoring) while understanding that the property always remains the same? In the next parts of this factoring series, we will explore specific strategies that can be used in elementary, middle, and high school to help cement factoring and the underlying distributive property.

Teachers of all grade levels, we’d like to know strategies you use (types of diagrams, models, vocabulary) that you’ve found help your students be successful with the distributive property and/or factoring. Your comments will help shape our next blogs in this series.

Texas Education Code: Chapter 74: Curriculum Requirements 
Common core state standards for mathematics: Designing high school mathematics courses based on the common core state standards. Appendix A
Common Core Standards Writing Team. (2011). Progressions for the common core state standards in mathematics (draft). Retrieved from http://www.isbe.net/common_core/pdf/math_progression052911.pdf
Adelman, C. (1999). Answers in the Tool Box: Academic Intensity, Attendance Pattern, and Bachelor’s Degree Attainment  http://www2.ed.gov/pubs/Toolbox/Part1.html

Planning in the Problem-Based Classroom

By Saler Axel, RME Research Assistant

Problem-based classrooms provide natural learning opportunities for students by giving them the latitude to explore at their own pace. This type of student centered learning helps encourage exploration and in turn enhance comprehension. Many teachers though struggle with successfully implementing problem-based learning into their teaching because the approach is often very different from what they were taught.

Implementing a problem-based curriculum takes time and patience. When teachers plan lessons within this framework, it is imperative to remember that pre-planned lessons may not always follow a formal time table. Lessons need to be tailored to the students’ needs and fulfill the curriculum objectives, which doesn’t always allow for the rigidity of a schedule.

John Van de Walle offers teachers nine steps to successfully planning problem-based lessons in his book Elementary and Middle School Mathematics: Teaching Developmentally (2013).

Step 1: Begin with the Math! Consider what you want your students to learn by thinking in terms of mathematical concepts instead of skills. Students will better comprehend and retain new information when you approach your teaching in this manner.

Step 2: Consider Your Students. Begin by thinking about what your students already know. Consider what background knowledge they need and whether they have enough to begin or whether they will require a review. What do you expect may cause your students to struggle? How can you best present mathematical concepts to match your students’ prior knowledge base?

Step 3: Decide on a Task. Use Van de Walle’s book to help you compose a task that will best match the lesson and concept you plan to teach. Remember, not all tasks need to be complex or elaborate; simple can be better!

Step 4: Predict What Will Happen. Predict what your students will do with the presented task. Make sure that each student has the opportunity to participate and benefit from your lesson. Students may approach tasks differently, but it’s important that each student learns new skills. If you feel unsure about whether your task will benefit everyone, reconsider. Does the task help accomplish teaching the concepts you set out to teach?

Step 5: Articulate Student Responsibilities. For almost all tasks, students should be able to tell you:

• What they did to get the answer. 
• Why they did it that way. 
• Why they think the solution is correct. 

Consider how you expect students to share the information above. Consider asking students to answer in several different formats throughout the year. Be clear that everyone will be expected to provide this information when their tasks are complete.

Step 6: Plan the Before Portion of the Lesson. It is important to prepare students for the task at hand by first encouraging them to quickly work through easier, related tasks. This can better familiarize students with your expectations of each task and refresh their memories of past-presented information.

Step 7: Think about the During Portion of the Lesson. Consider your predictions. What types of accommodations or modifications can you provide in advance for students that will likely need extra help? What types of extensions or challenges can you offer students who finish before their peers?

Step 8: Think about the After Portion of the Lesson. Determine how your students will present their. Consider the best way to assess your students’ learning. How will you be assured of their comprehension and ability to retain any new material?

Step 9: Write Your Lesson Plan. Now that you have considered your lesson in such detail, this step should come easily! Below is a possible lesson plan outline format:

• The mathematics or goals. 
• The task and expectations. 
• Materials needed and necessary preparation. 
• The before activities. 
• The during hints and extensions for early finishers. 
• The after-lesson discussion format. 
• Assessment notes (whom you want to assess and how)

Summing It All Up
When planning lessons in a student-centered, problem-based classroom, remember that your students’ needs and learning styles should heavily influence what tasks you implement in the classroom. Take some time this week to plan a mathematics lesson using these nine steps.

What tasks will best meet your students’ needs?

Van de Walle, J. A., Karp, K. S., Bay-Williams, J. M. (2013). Elementary and middle school mathematics: Teaching developmentally. Upper Saddle River, NJ: Pearson Education, Inc.

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Don’t Tell the Answers, Just Ask the Right Questions

By Sharri Zachary, RME Mathematics Research Coordinator

I recently came across an interesting newsletter that highlighted 8 tips for asking effective questions in the mathematics classroom. In this newsletter, teachers are encouraged to never say what a student can say. Rather than tell students what to do, the practice should be to stimulate thinking and deepen students’ conceptual understanding of mathematical concepts. To be able to effectively do this, teachers have to make a commitment to continually develop their mathematics knowledge so that they can ask questions that will help their students make connections between concepts.

In order to improve at asking effective questions, it is suggested that math teachers consider the following tips:

1. Anticipate Student Thinking. Consider different ways or approaches students may take to solve problems, including possible errors or misconceptions, and form questions that will prompt students to be reflective about their problem solving.

2. Link to Learning Goals. Ask questions that connect back to previously taught concepts. Consider the following example:

Learning goal –Understand that a unit fraction can represent a point on the number line, a distance between two points, and magnitude.

Student problem: Represent the fraction ⅕ on a number line. 
Possible questions:

• Why is it possible for ⅕ to be represented as both a point and a distance between two numbers?
• How would you describe the unit interval?

3. Ask Open Questions. Open questions allow for differentiation because student responses may vary depending on the level of understanding a student has about the topic.

Closed question: Is ²/₇+²/₇+²/₇ equivalent to ⁶/₇ ?
Open question: What other equivalent expressions can be written to represent ⁶/₇ ?

4. Ask Questions that Actually Need to be Answered. Avoid asking rhetorical questions because they provide students with an answer without allowing them to engage in reasoning about their answer (“Asking Effective Questions,” 2011).

5. Incorporate Verbs that Elicit Higher Levels of Bloom’s Taxonomy. Include verbs from Bloom’s list in your questions because they require students to tap into specific cognitive processes that engage thinking (“Asking Effective Questions,” 2011).

6. Ask Questions that Lead to Conversations with Others. Students often benefit from mathematical conversations held with their peers. However, it may be difficult for students to initiate this conversation. This allows for students to discuss the “big ideas” within a given topic.

7. Keep Questions Neutral. Avoid prefacing questions with qualifiers such as easy or hard, or offering verbal and non-verbal clues, facial expressions, or gestures. Qualifying questions may prevent a student from thinking through the question or may deter them from answering.

8. Provide Wait Time. Wait time encourages thinking and provides students with an opportunity to formulate their thoughts into words.

Summing it All Up
The goal, in asking effective questions, should be to help students focus their thinking about problems rather than lead to a solution. By stimulating their thinking, they will gain a deeper understanding about a concept that will lead them to make connections to other mathematical concepts. Don’t tell students what to do but ask questions that will lead them to the right answer.

To read the entire article:
Asking effective questions. (2011, July). Capacity Building Series. Retrieved from http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/CBS_AskingEffectiveQuestions.pdf

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