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/

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.

All Resources are Not Created Equal! A Closer Look at Algebra I Textbooks

By Dr. Candace Walkington, Assistant Professor of Mathematics Education & Elizabeth Howell, RME Research Assistant

Quite often teachers inherit the textbooks that will be used as a primary teaching tool and resource in their classrooms. Textbook choices may be district-level decisions or may be decided in a multi-year cycle. As a classroom teacher, the decision may be out of your immediate control. However, as the ultimate instructional leader in your classroom, it IS possible to use current research about textbooks in order to improve outcomes for your students, regardless of the required curriculum.

Background
Experts agree that algebra is a crucial course in students’ mathematical trajectory, and success in this course has been identified as important to college and career readiness (Stein, Kaufman, Sherman, & Hillen, 2011; Cogan, Schmidt, & Wiley, 2001; Kaput, 2000; Moses & Cobb, 2001). A key learning goal in algebra is the use of symbols to represent and analyze situations and problems, and many textbooks are written with a heavy emphasis on symbolic usage and presentation.

However, working with symbolic representations is challenging for students (Walkington et al., 2012), and evidence suggests that students actually have improved learning when they first learn about a new concept using concrete and familiar formats (Goldstone & Son, 2005) - like verbally presented algebra story problems. By giving students story scenarios first, instead of symbols alone, we can draw on the things they already know and understand. Over time, these verbal supports can be faded, as students begin to understand and work with symbols more.

Textbook Classifications
Current Algebra I textbooks are classified as being traditional or reformed. Traditional texts have the lion's share of the textbook market (Holt, Pearson, Saxon, etc.), and introduce concepts by showing definitions and worked examples, and then presenting problem sets. These texts are typically expected to go along with a teacher-directed approach to instruction. Reformed textbooks are more rare in Algebra I, and often follow NCTM standards for reformed teaching, taking a student-centered approach. They may present students with more complex, open-ended problems or mathematical investigations, and accentuate the use of problem-based learning. Differences have emerged in the way that traditional and reformed curricula introduce the use of symbols in Algebra I, and many reformed textbooks in mathematics have taken less of a symbolic approach and adopted a more verbal presentation style.

Example of problem presented in VERBAL format:

• Maria just got a new cell phone, and on her phone plan each text message she send costs $.10. Write an algebraic expression that relates the number of texts Maria sends to the cost in dollars. How much will it cost to send 7 texts?

Example of problem set presented in SYMBOLIC format:

• Solve for y when x = 7:
  y = 3x + 5
  y = 0.25x
  y = 2x - 3

Current Research
In a recent study, researchers found that the presentation format of the examples and homework problem sets in commonly used Algebra I textbooks varied depending on the type of textbook, traditional or reformed (Sherman, Walkington, Howell, 2013). Reformed texts favored a verbal presentation first, and this verbal first approach was faded over time in the text. Thus in reformed texts, when students are first learning about algebra, they get a lot of verbal problems, but as their expertise develops, they get more symbolic problems.

Traditional texts favored symbols first - in each section, symbolic problems were presented to students before verbal problems. Traditional texts also had fewer single format only sections - there were fewer sections that had only verbal problems, or only symbolic problems. Most traditional texts contained a mixture of symbolic and verbally presented problems in the homework, yet the instructional examples provided in the text trended toward symbolic. Other research has also suggested that in traditional texts, the student recommended exercises in the teacher’s edition sometimes excluded the verbal problems from the students’ assignment.

Implications
Because many schools are using traditional textbooks, it is likely that your district adopted text has a heavy prevalence of symbol-first presentation. Be aware of the challenges that this approach may present to your beginning Algebra I students, and use verbal presentations in your initial classroom examples whenever possible. When assigning homework, be aware that the recommended exercises may exclude all of the rich verbal contextual problems, and add a few back in to your homework assignment. Better yet, choose a few to discuss and work on together as a class.

The choice of an Algebra I textbook may not be a decision that you can determine, but how to best use the examples and problems presented in the book is always your choice as the teacher. What type of textbook are you currently using? Take a closer look, and be aware of the presentation style that the book favors. If verbal scaffolding is not prevalent, it is easy to add those supports back in to help your students to succeed. A closer look will help you to see those places in the curriculum where a verbal presentation could be beneficial.

Resources:
Texas State Adopted Textbooks: http://www.tea.state.tx.us/index2.aspx?id=2147499935

References:
Cogan, L.S., Schmidt, W.H., & Wiley, D.E. (2001). Who takes what math and in which track? Using TIMMS to characterize U.S. students’ eighth-grade mathematics learning opportunities. Educational Evaluation and Policy Analysis, 23, 323-341.

Goldstone, R., & Son, J. (2005). The transfer of scientific principles using concrete and idealized simulations. Journal of the Learning Sciences, 14(1), 69-110.

Kaput, J. J. (2000). Teaching and learning a new algebra with understanding. U.S.; Massachusetts: National Center for Improving Student Learning and Achievement.

Moses, R., & Cobb, C. (2001). Radical Equations: Math Literacy and Civil Rights. Boston: Beacon Press.

Sherman, M., Walkington, C., & Howell, E. (April, 2013). A comparison of presentation format in algebra curricula. Presented at National Council of Teachers of Mathematics Research Pre-session, Denver, CO.

Stein, M.K., Kaufman, J. H., Sherman, M., Hillen, A.F. (2011). Algebra: A Challenge at the Crossroads of Policy and Practice. Review of Educational Research, 81(4), 453-492.

Walkington, C., Sherman, M., & Petrosino, A. (2012). ‘Playing the game’ of story problems: Coordinating situation-based reasoning with algebraic representation. Journal of Mathematical Behavior, 31(2), 174-195.

After High School: What's Next?

By Elizabeth Howell, RME Research Assistant

High school graduation is certainly cause for celebration! All of the hard work, studying, and preparation have paid off. Students have taken mathematics courses and learned advanced algebra skills, geometry, and maybe even more. So what’s next?

In Texas, students pursuing a postsecondary degree require a stamp of college readiness. There are many ways that a student can demonstrate college readiness: SAT/ACT scores, TAKS scores, math dual credit courses, to name a few.

But what if a student's SAT/ACT or TAKS score in math wasn’t so hot, and the student never took a dual credit math class in high school? Without one of the many approved exemptions, a student will be required to take a placement test upon enrolling in college. This test will be used to determine if a student is indeed college ready in reading, writing, and mathematics. And the results will determine which classes a student can enroll in for their first semester.

Most placement tests will be administered via a computer, and there is typically a fee associated with taking the test. Common placement tests are Accuplacer, Compass, and THEA. Each of these tests will have a reading, writing, and mathematics component in order to assess a student's skill level before enrolling in classes. Each test has a pre-set cut off score. Any student that does not meet the cut off score will be required by state mandate to enroll in remedial or developmental classes.

Approximately forty-one percent of students in Texas higher education require remediation upon entering college (THECB, 2013). Remedial classes have homework, tests, and grades, just like any other class. They cost tuition dollars, just like any other class. BUT…they do not count toward any degree! Remedial or developmental classes are designed to reteach the material that high school mathematics courses should have taught, and they are required if a student's math placement score is not passing. Completion of the remediation specified by a student's test score is required to enroll in credit mathematics courses.

Sadly, students that require developmental education are far less likely to graduate from a college or university (Morales-Vale, 2012). Developmental courses delay degree completion, cost tuition dollars, and can be a major roadblock to a student’s academic goals.

So how can student's avoid developmental courses?

1. Take high school courses seriously. The reading, writing, and mathematics skills a student learn sin high school is critical to college success.
2. Take the placement test seriously. If a student is required to take a placement test for college, emphasize  that reviewing notes and looking at practice questions is critical. The importance of the test cannot be overstated. The placement test will determine the academic path a student will start on, and being on the right path is crucial.
3. If remediation is needed, take the remedial classes seriously. These classes are designed to improve academic skills, but sadly many students do not realize the importance of these classes because they think that they do not count. In a sense that is true, these classes do not transfer or count toward a degree. But, remedial classes can be the gatekeeper between a student and the degree they want -  because not completing them successfully means that a student cannot move on to the classes that DO count toward their desired degree.

In Texas colleges and universities, far too many students end up in developmental coursework. College readiness is a demonstrated skill, and students' need to take the initiative to brush up on skills before taking a placement test. Practice versions of many tests are available online for free. Make sure students talk to a high school counselor or a college advisor if they have concerns. In addition, have them visit websites dedicated to college readiness such as http://gentx.org for resources, hints, and checklists to help transition successfully from high school to college.


Morales-Vale, S. (2012). TSI and developmental education updates. Presented at CRLA/CASP Convention, November 8, 2012, Austin TX. 

Texas Higher Education Coordinating Board (THECB). (2013). Developmental education/Texas success initiatives. Retrieved May 10, 2013 from http://www.thecb.state.tx.us/index.cfm?objectid=233A17D9-F3D3-BFAD-D5A76CDD8AADD1E3.

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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Converting Fractions to Percentages

By Beth Richardson, RME High School Mathematics Coordinator

As a high school math teacher, I taught a wide range of students from ESL Algebra 1 and regular Geometry to Pre-AP Algebra 2. The resounding similarity I saw between all of my students was that, for some reason, students cringe when they see rational numbers. They feel like rational numbers automatically make the problem “hard”. I was amazed that by high school, students were still struggling with something as simple as converting from a fraction to a percent. Perhaps this is because, as teachers, we sometimes teach our students shortcuts that leave out the logic behind the scenes of the procedures they learn.

The IES Practice Guide, which is supported by research evidence, recommends that teachers “'help students understand why procedures for computations with fractions make sense’ and ‘develop students’ conceptual understanding of strategies for solving ratio, rate, and proportion problems before exposing them to cross-multiplication…’ (Siegler et al., 2010).”

Some common shortcuts teachers use are changing the fraction to a decimal then multiplying by 100 or changing the fraction to a decimal then moving the decimal to the right twice.

Examples:

Neither method above leads students to a percentage as the final answer, unless the student “remembers” to tag it on at the end. Units are crucial when converting in any context. In order for students to understand why they must multiply by 100% rather than 100 when converting from fraction to percent, units must be used properly.

Instead, students should be taught to set up proportional relationships, including units, between the fraction and unknown out of 100%. It is important that students understand that when the units of the numerator and denominator are the same, they cancel and the fraction is unit-less.

Example:
25 students went on a field trip and 5 wore a hat. What percentage of the students wore a hat?

20% of the students wore a hat on the field trip.

Through the process above, students see why they are multiplying by 100% and why the units in their answer must be a percentage. Also, students can use number sense to reason that x must be a percentage between 5 and 100.

Summing It All Up
Fellow teachers: it’s not safe to assume that our students understand why they are doing a particular procedure, even if it is one they “should” have mastered several grade levels ago. If we take a little more time to illustrate examples with labeled units and explanation, we will hopefully catch any previous misconceptions our students have and steer them on the right path towards math success.

Now it’s your turn. Share with us common misconceptions, similar to what we described above, that you’ve found in your classroom!

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

Siegler, R., Carpenter, T., Fennel, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., & Wray, J. (2010) Developing effective fractions instruction for kindergarten through 8th grade: A practice guide. 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/practiceguides/