# Common Core Motivation Math

Mathematics is a universal subject. All individuals must know basic mathematics. The demand to understand and be able to apply mathematics in everyday life and in the workforce is an ever present need and will continue to grow. The National Council of Teachers of Mathematics (NCTM, 2000) agrees that mathematics is for everyone. Thus, “all students should have the opportunity and support to learn significant mathematics with depth and understanding.”

The National Assessment of Educational Progress (NAEP) reports “positive trends” of scores in mathematics for Grades 4 and 8. However, research studies indicate that American students are outperformed in mathematics when compared to students from other countries. Other results noted by NAEP (National Mathematics Advisory Panel, 2008) show that mathematics education in the United States needs improving. For example, “32% of students perform at or above the proficient level in Grade 8, while 23% are proficient at Grade 12.” Four-year colleges and community colleges have determined the need for remedial mathematics for entry-level students. The United States Department of Education (2004) shared that “the recent National Assessment of Educational Progress (NAEP, the Nation’s Report Card) showed that 27% of eighth-graders could not correctly shade 1/3 of a rectangle and 45% could not solve a word problem that required dividing fractions.” Philips (2007) offered statistics that indicated adults have difficulties with everyday applications of mathematics in the real world. Other research indicates that students and adults experience problems in foundational mathematical skills (Hecht, Bagi, & Torgeson, 2007). Based on these findings, the evidence clearly shows mathematics literacy is a serious problem in the United States. Therefore, it is understandable why the National Mathematics Advisory Council (NMAC) expresses concern for mathematics education in the United States. While some data indicates progress, there continues to be a need for the United States to focus on improvement in mathematics education.

The international assessment, Trends in Mathematics and Science Study (TIMSS), showed students in the United States perform poorer in eighth grade than fourth grade and 1995 data for twelfth grade results were even lower than eighth (Evan et al., 2006). Newly released data from the Third International Mathematics and Science Study for 41 countries highlight the continuing problems in science and mathematics education in the United States. While scores for fourth graders in both content areas reflected top and middle rankings respectively, the performance of eighth graders was just above the median in science and below the median in mathematics. Although fourth graders performed better than eighth and twelfth, only 7% of U.S. fourth-graders scored at the advanced level in TIMSS compared to 38% of fourth-graders from Singapore. Throughout the world, Singapore is recognized as a leader in mathematics. The 2007 Programme for International Student Assessment (PISA), points out similar findings in that the United States ranked 25^{th} out of 30 in math literacy and problem solving (Baldi, Jin, Shemer, Green, Hergert, & Xie, 2007).

The aforementioned findings document that the United States must increase its focus and become more coherent in order to improve mathematics performance. In the past numerous standards for mathematics existed, but the curriculum contained little or no depth. The Common Core Standards (CCS) in Mathematics addresses the absence of focused and coherent standards, the concerns identified by the National Mathematics Panel, and the challenges noted in international benchmarking studies.

Standards are the skills and knowledge students need to be academically successful and be prepared for college and careers. High standards, but also reachable standards, are essential in order to develop the abilities to reason and think mathematically and to acquire the necessary knowledge and skills in order to function in life. Standards ensure that students are learning and understanding the critical information needed to succeed at higher levels. The Common Core Standards provide the foundation from which students can achieve mathematical competence.

Students must develop depth of understanding as well as the ability to apply mathematics. Obviously, teachers can benefit from quality educational resources that align with the CCS in Mathematics and also provide additional information that can guide them in the implementation of the standards. *Motivation Math Common Core Aligned* is a student and teacher resource that focuses on teaching and learning of mathematics.

Students’ mathematical achievement, however, is ultimately determined and limited by the opportunities they have had to learn. Mathematics is not restricted to a select group of students. “All students must learn to think mathematically, and they must think mathematically to learn” (Kilpatrick, Swafford, and Findell, 2001). The RAND Mathematics Study Panel (2003) also emphasized the importance of mathematics. Furthermore, the panel declared that it is essential that students develop math proficiency. No Child Left Behind Act (NCLB, 2001) suggested research about effective practices be a guide in changing the way mathematics is taught so that high standards are met, leading to an improvement in achievement. The CCS for Mathematics, released in June 2010, define what students should understand and be able to do in their study of mathematics. The goal is for students to graduate high school prepared to succeed in college and in a modern workforce.

*Motivation Math Common Core Aligned *incorporates research-based strategies and pedagogically sound principles for teaching and learning. This product is designed to support and enhance the best practices for teaching the common standards in mathematics. *Motivation Math Common Core Aligned* is founded on the modeling of "Active Teaching," which is teacher-directed instruction that proceeds in small steps. Research indicates that this approach is associated with higher levels of student achievement. Students are guided through the learning process and are afforded multiple, varied opportunities for mastery of testable student expectations. Student-centered activities are also an essential part of daily mathematics instruction. The National Mathematics Advisory Panel (2008) indicates that research supports both teacher-directed and student-centered activities as integral to student achievement as opposed to merely one or the other.

*Motivation Math Common Core Aligned* is written to reflect the depth, rigor, and complexity of the Common Core Standards (CCS) and to help students gain a deep understanding of mathematics. The 2010 Mathematics Standards provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them. The standards are focused, coherent, and relevant to the real world, describing the knowledge and skills that students need for success in college and careers. This resource complements existing mathematics curricular and can serve as a reinforcement or intervention program. *Motivation Math Common Core Aligned* aligns to the Common Core Standards and addresses National Council of Teachers of Mathematics process standards.

While the Common Core Standards in Mathematics establish grade-specific standards, they do not describe intervention methods or materials for supporting students who exceed or function below grade-level expectations. Educators are asked to continue appropriate accommodations in order to engage students as much as possible. *Motivation Math Common Core Aligned* does include a section in the Teacher Edition titled Interventions. The small group or partner interventions define learning experiences that parallel the unit’s standards and include formative assessment.

The Common Core State Standards for Mathematics are comprised of two sets of standards: Content Standards and Standards for Mathematical Practice. Content Standards are what students should know and be able to do at each of the designated grade levels. The Content Standards focus on** ***what *students are learning. Standards for Mathematical Practice are the eight mathematical standards that remain the same for K-12. The Practice Standards focus on *how *students are learning math. More specifically, these practices describe how students should engage with the mathematical content as they grow in mathematical maturity and expertise throughout the elementary, middle, and high school years. Thus, it is imperative that mathematical programs and resources address how these standards intertwine. *Motivation Math Common Core Aligned* identifies the eight Practice Standards in a condensed version. This shortened version describes what the teacher and the student would be doing for each of these practices. The Standards for Mathematical Practice should be embedded within the teaching and learning of the Content Standards at all grade levels. Within the introductory display of each *Motivation Math Common Core Aligned* unit in the Teacher Edition, there are several headings that relate to the Standards: Domain, Cluster, Standard, Other Standards Addressed in the Unit, Common Core Mathematical Practice, and Unpacking the Standards. This displayed information enables teachers to readily identify essential elements of each unit. The Standards for Mathematical Practice serve as a reminder for teachers to incorporate these practices on a daily basis.

PARCC (2011) and Smarter Balanced Consortium (2011) also support the consistent incorporation of the Standards for Mathematical Practice in classroom instruction, discussions, and learning activities. Both consortia advocate the implementation of processes and proficiencies embedded within the Mathematical Practices. PARCC and Smarter Balanced stress mathematical environments that integrate these practices with students participating in carefully designed tasks that align with the standards and vary in difficulty, context, and type. Teachers are encouraged to carefully designed standards-based mathematical tasks that assess content knowledge and elicit evidence of mathematical practices. *Motivation Math Common Core Aligned* connects mathematical practices to mathematical content in mathematics instruction. These connections are identified within the initial part of each unit in the Teacher Edition. This educator resource includes a balance of instructional and assessment activities that provide students with numerous opportunities to develop and demonstrate their expertise as described in the eight practices.

The National Association of Elementary School Principals (2012) attests to the importance of principals in the successful implementation of the Common Core Standards. A Common Core Implementation Checklist was designed by this group to guide administrators in determining their strengths and assisting in the development of successful plans for implementation. One of the areas noted on the checklist was professional development. More specifically, the checklist identifies indicators from which administrators must facilitate teachers’ understanding of the curriculum changes for Mathematics. With *Motivation Math Common Core Aligned, *administrators can direct teachers to be cognizant* *of the wide range of information that will assist them in understanding the changes in the way the Student and Teacher Editions are designed. Unpacking the Standards is one component that will specify the meaning behind the standards addressed in the unit and aid teachers in Conceptual Understanding. Focus is addressed by identifying “what” to teach and “where” to begin and end. The Content Standard and the Mathematical Practices are also identified for teachers. The arrangement of the Student Edition increases the probability of maintaining the focus so that the depth of the standards can be addressed. Reasoning is promoted through the activities, critical thinking opportunities, and journal prompts. The background information, provided for the teachers, guides the flow of instruction. Mastery of the standards is easily addressed through formative assessment. Whereas, in the past, assessment was more summative in nature, *Motivation Math Common Core Aligned* includes an array of instructional activities, vocabulary suggestions, interventions, and guiding questions that increase the probability of leading students to mastery.

Earlier in 2012, the United States Department of Education and the Federal Communications Commission announced a blueprint to invite schools to transition to digital textbooks by the end of the next five years. While not mandated, the initiative encouraged schools to make the switch from print-to-digital materials based on the projected cost-savings and the academic improvement. These benefits are due to the expense of printed textbooks and the personalization of digital content. *Motivation Math Common Core Aligned* also features a print-to-digital transition. Campuses will have digital access to all the Student and Teacher Edition pages if using Internet-connected computers. Using the same aligned content as *Motivation Math Common Core Aligned*, educators have access to an interactive delivery method for their students and classrooms. This new dimension of flexibility, *Motivation Math Online*, offers an engaging learning environment, not only for educators, but also for students. Tools such as online progress monitoring, automatic tracking, and reporting are built into this innovative program. With the appropriate use of technology, students can develop deeper understanding of mathematics.

Hiebert and Wearne (1992; 1993; 1996) reported that a critical attribute in regards to student learning in mathematics, is the nature of the learning task in which to engage students. Students need mental engagement in challenging and worthwhile mathematical tasks that emphasize the conceptual aspects of the topic and promote the formation of mathematical connections. This type engagement is a prerequisite to learning skills with meaning and being able to apply those skills to solve problems. Students must receive direct encouragement to think and persist with the mathematical task at hand.

Grouws and Cebulla (2000) state that teaching mathematics with a focus on number sense encourages students to become problem solvers in a wide variety of situations and to view mathematics as a discipline in which thinking is important. “Number sense” is an intuitive feel for number size and combinations, as well as the ability to work flexibly with numbers in problem situations in order to make reasonable judgments. The processes of mentally computing, estimating, sensing number magnitudes, moving between representation systems for numbers, and judging the reasonableness of numerical results must be flexibly used. This type of instruction requires teachers to have a deep understanding of mathematics and how students learn mathematics. More specifically, teachers will encounter difficulty in teaching number sense without a working knowledge of number sense themselves.

The experiences, discussions, and review of the literature convinced the Mentoring Minds Product Development Team that resources for mathematics needed a change. Thus, the format for the Student Edition is designed to move mathematics forward so that teachers can incorporate standards-based instruction in mathematics on a higher level, developing within students the confidence needed for success.

Numerous studies** **indicate that increasing the amount of time spent in mathematics instruction is positively correlated with student achievement in mathematics. The 2001 National Research Council publication, “Adding It Up: Helping Children Learn Mathematics” states that significant time should be devoted to daily mathematics instruction in every grade of elementary and middle school. In addition, the 1999 Handbook of Research on Improving Student Achievement (Cawelti, 1999) states that a favorable relationship exists between total time allocated to mathematics and general student performance. A finding in The Nations’ Report Card: Mathematics 2000, NAEP showed that the average scores of fourth and eighth graders generally increased as the amount of instructional time for mathematics increased. Grouws and Cebulla (2000) also concluded in their work that a positive relationship existed between total time allocated to mathematics and general mathematics achievement.

Furthermore, the way in which time is utilized in mathematics class is paramount to the degree of student achievement. Additional time is recommended to be spent in direct instruction as opposed to seatwork or written drill. The 1999 TIMMS video study indicated that nations scoring higher than the United States on tests of mathematics achievement at grade 8 devoted more time on the average to studying new content (a range of 56 to 76% of lesson time) than reviewing previous content. In the United States, there was no observable difference between the average percent of lesson time devoted to reviewing previous content and studying new content (53 and 48% of lesson time respectively).

The levels of each Student Edition are comprised of all the identified Common Core Standards for Mathematics for the designated grades. Each unit reflects a central focus standard integrated with other Content Standards whose expectations support the focus standard. The components in each unit are Introduction, Partner Practice, Independent Practice, Assessment, Critical Thinking, Journal Prompt, and Motivation Station with Parent Activities components. According to Grouws and Cebulla (2000), students need to be given both an opportunity to discover and invent new knowledge and an opportunity to practice what they have learned to improve student achievement. *Motivation Math Common Core Aligned* presents multiple learning experiences in which to apply both findings. The Introduction features open-response questions from which teachers can diagnose and prescribe issues as students work in whole group settings. Partner Practice provides guidance practice opportunities for students to work in pairs or small groups with the selected-response questions. Research recognizes the importance of social and intellectual support from peersand teachers. Support is essential for African-American and Hispanic students. These relationships, evidenced with continuous support, can lead to higher mathematics performance for students. With Independent Practice, students are provided independent opportunities with selected-response questions. Thus, many learning opportunities are given within the Teacher and Student Editions of *Motivation Math Common Core* *Aligned* to build proficiency and conceptual knowledge with obtrusive and intrusive assessment opportunities.

Beyer (1991) advocates teachers activate students’ relevant prior knowledge. When teachers link new learning to everyday and academic experiences, students seem to better comprehend new information. Studies acknowledge the necessity of connecting classroom activities with topics familiar to students. Kujawa and Huske (1995) confirm the importance of prior knowledge. Therefore, students appear to best learn and remember newly presented information when the content is linked to their cultures and experiences. *Motivation Math Common Core Aligned* provides an array of teaching and learning opportunities within the Teacher Edition from which teachers can implement learning experiences that build on students’ prior knowledge, forming meaningful connections. For example, when activating prior learning, teachers might employ graphic organizers and small group or class discussions. Using dialogue or visualizations, teachers can determine the extent of future instruction. Necessary adjustments can be made so that incomplete or inaccurate prior knowledge or connections are corrected. The Unpacking the Standards component within *Motivation Math Common Core Aligned* Teacher Edition also clarifies prior learning so that teachers have a firm understanding of the prior learning. This information then helps to strengthen the connections between the new learning and prior learning, as well as understand the meaning of the standard(s) addressed in the unit.

“Assessment plays a critical role in all aspects of teaching and learning mathematics” (NCTM, 2000). In the publication compiled from numerous writers, “What We Know About Mathematics, Teaching, and Learning,” Nancy Kober (1996) from North Central Regional Educational Laboratory (NCREL) reported that evaluation tools which closely align with the objectives are usually more beneficial for diagnosing and revising instructional needs. No Child Left Behind Act (NCLB, 2001) stated, “Beginning no later than the 2005-2006 school year, each State must administer annual assessments in … and math in each of grades 3 through 8 and at least once in grades 10 through 12.” When assessment is an integral part of mathematics instruction, it contributes significantly to students’ mathematical learning (Stecker et al., 2005). Assessment should inform and guide teachers as they make instructional decisions. The tasks that teachers select for assessment convey messages to students about what kinds of mathematical knowledge and performance are valued. Formative assessments are interwoven throughout each unit in the Teacher Edition. In the Student Edition, a page of four-to-five selected- response items is followed by one short constructed-response item. Feedback from the variety of formative assessment tasks in *Motivation Math Common Core Aligned* will help students know how to improve and what next steps to take. Other benefits will be seen as students play prominent roles in setting goals, assume responsibility for their own learning, and become independent learners. Therefore, teachers can gather timely student information or data to readily and continuously maintain accountability for academic achievement standards in mathematics. Assessment for learning is a common occurrence within both Student and Teacher Editions so that teaching can be adjusted and learning can improve and grow.

The National Mathematics Advisory Panel (2008) recommends formative assessments are used on an ongoing basis for students in the elementary grades. The Panel reports that in classrooms where teachers regularly employ formative assessment, student learning appears to improve. When teachers have guidance on the use of assessment to individualize instruction, the formative assessment process is strengthened. In the Suggested Formative Assessment section throughout each unit in the Teacher Edition, detailed information is provided in order to guide teachers in assessing for learning during the introduction, vocabulary, instructional, reflection/closure, and intervention activities. Being able to gauge student learning relative to standards is essential in developing mathematical proficiency in all students.

The Chart Your Success chart is included at all levels and is located in the back of the Student Edition for each student to visually record and observe individual progress on an ongoing basis. The involvement of students in assessment promotes student engagement in individual learning targets. Students need to know what learning targets they are responsible for mastering, and at what level. Marzano (2005) states, “students who can identify what they are learning significantly outscore those who cannot.” Research on formative assessment suggests that students should be aware of the learning target, their present status, and the next steps in reaching that goal or closing any gaps (Atkin, Black, & Coffey, 2001). Such knowledge helps students keep track of their achievements, know how close they are to their learning targets, and determine future steps to advance their learning. When students are aware of their achievement gaps and teachers motivate students with continuous feedback linked to the expected outcomes and criteria for success, students are able to surge ahead and close performance gaps in mathematics. Black and Wiliam (1998) note there is evidence to support a strong relationship between interactive feedback and student achievement. Therefore, *Motivation Math Common Core Aligned* reflects formative assessments mingled with interactive student/teacher conversation.

Studies support the use of several measures from which to gauge student achievement. PARCC (2012) and Smarter Balanced (2012) consortia have both released specifications regarding the general structure of performance tasks in mathematics. Those specifications provided guidance from which to develop the performance tasks for *Motivation Math Common Core Aligned*. All tasks are relative to real-world application. Typically, four or five performance tasks for each grade level are presented in the Teacher Edition. The design for the student task includes the following descriptors: Problem Stimulus, Task Overview, and Performance Task. In addition, a section Teacher Information is featured. It consists of Primary Claim, Secondary Claim, Content Standards, Standards for Mathematical Practice, DOK, Rigor/Relevance Quadrant, Scoring Criteria (for inclusion into the design of rubrics), and Answers. Due to the accountability issue for schools, Mentoring Minds encourages teachers to maintain accurate and useful data as well as employ a variety of assessment opportunities in order to form a more valid insight on where a campus, classroom, or student stands in mathematical performance. The teacher is provided a Standards Checklist to document which of the eight Standards for Mathematical Practice are addressed during instruction. In addition, teachers can designate when Common Core Standards are introduced, retaught, or mastered. A Class Performance Chart is available for teachers to maintain a running accountability document with cumulative scores for the standards of students in the class as well as running scores of student performance. The student assessment page used in conjunction with the aforementioned measures provides crucial information to the teacher for use in improving performance.

Evidence from research demonstrates that a successful mathematics program must include time for students to practice what they are learning and experiences to perform the tasks for which they are to demonstrate competence. The additional Introduction Activity, Suggested Instructional Activities, and Interventions in the *Motivation Math Common Core Aligned *Teacher Edition support students in their quest for mastery of the standards. The Introduction Activity engages the students with a whole class or small group learning experience facilitated by the teacher. It is interactive and may consist of a game, or a literature based activity, or a similar experience that introduces the students to the focus standard. Suggested Instructional Activities are hands-on, offering several in order to meet student needs in skill acquisition and conceptual understanding. Formative assessments are suggested for use in between and during these varied experiences so that needs of learners are determined and misconceptions are corrected. Assessment data inform teachers and students as to the next steps and if Intervention Activities are needed. In the Student Edition, the organization of the unit components benefits students in acquiring skills and concepts. The components are arranged by Introduction, Partner Practice, Independent Practice, Assessment, Critical Thinking, and conclude with Motivation Station and Parent Activities. Practice is repeatedly an integral part of each unit in order to strengthen mathematical understandings.

Critical thinking is an important issue in education today. Attention is focused on quality thinking as an important element of life success (Huitt, 1998; Thomas and Smoot, 1994). In the 1950s, Bloom found that 95% of the test questions developed to assess student learning required them only to think at the lowest level of learning, the recall of information. Similar findings indicated an overemphasis on lower-level questions and activities with little emphasis on the development of students’ thinking skills (Risner, Skeel, and Nicholson, 1992). “Now, a considerable amount of attention is given to students’ abilities to think critically about what they do” (Hobgood, Thibault, and Walberg, 2005). It is imperative for students to communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

When solving mathematical equations, it is crucial to invite students to explain their thought processes. If the results are inaccurate, teachers can identify the precise point at which students deviated from using critical thinking. Thus, it is essential that classrooms promote critical thinking as part of the learning experiences in mathematics. The literature notes that when students use their critical thinking abilities integrated with content instruction, depth of knowledge can result. Teachers are encouraged to refrain from limiting instruction to lectures, rote memorization, and other strategies that exercise only lower levels of thought as opposed to incorporating those that build conceptual understanding** **(Bransford, Brown, and Cocking, 2000).

The models used to develop critical thinking throughout the Student and Teacher Editions are: Bloom's Taxonomy(1956), Revised Bloom’s Taxonomy (Anderson et al, 2001), Webb’s Depth of Knowledge (2002a; 2002b), Hess’ Cognitive Rigor Matrix in Mathematics (2010; 2009; Hess & Petit, 2006), and the International Center for Leadership in Education (2012) Rigor/Relevance Framework®. These cognitive rigor and complexity models were used by the product developers to stimulate and develop students' higher order thinking skills and make extensions to the real world. Critical thinking is integrated into each component of the unit through higher-order questions and complex problematic situations. Students are invited to shift to new levels of increased awareness when calculating, analyzing, problem solving, and evaluating. In the Student Edition, one page is dedicated to the component Critical Thinking. This opportunity is presented to entice students to think critically and move them beyond basic comprehension and rote memorization. This page typically offers two open-ended questions that are coded to higher levels of Bloom’s and Revised Bloom’s Taxonomies. While students are applying and using higher order thinking skills in real-life situations, they are also learning to question the accuracy of their solutions.

Nonroutine problems (those not familiar to the problem solver) and transfer of problem solving require high level transfer, which is effortful and conscious (Salomon & Perkins, 1989), whereas routine problems involves less conscious attention and rely more on low level transfer. Routine problems are those in which the learner knows a correct solution method based on past experience and is able to reproduce it and apply it. Experts advise teachers that students can lose the ability to articulate and reflect on their use of reasoning in solving problems if they are exposed to mostly routine problems. A basic characteristic needed to become a proficient problem solver is flexibility. Flexibility develops through the expansion of knowledge required for solving nonroutine problems rather than just routine problems. Nonroutine problems require the learner to use productive thinking to create a way to understand and solve the problem since an immediate solution method is not known. A balance is needed between the time students spend practicing routine procedures and the time they devote to discovering new method solutions for nonroutine problems. There is no need for teachers to make a choice between which of these two type problems to use if students are to develop mathematical thinking power.

Assessing the work of students in a problem-solving situation differs from a traditional method of determining the accuracy of computational skills. Open-ended problems can be solved using a variety of methods or the problems can have multiple responses. *Motivation Math Common Core Aligned* utilizes a variety of formative assessment opportunities in the Teacher Edition and offers a page dedicated to Assessment in the Student Edition. Typically, three-to-five selected- response questions and one constructed or extended response are presented. These opportunities provide the teacher with informal data about student understanding of the concept or skill. The Assessment page in the Student Edition is designed for independent completion, with the student recording mastery of the skill or concept on the Chart Your Success page.** **

Students must learn to read, write, speak, listen, and use language effectively in a variety of content areas. The Common Core State Standards advocate content areas supplement English Language Arts standards. Journal prompts are an important element in each mathematics unit for the purpose of incorporating writing into mathematics. Journal prompts are used in the Student Edition to provide authentic writing opportunities and, as promoted by research, serve as a valuable instructional learning experience for concept application to real-world settings. Open-ended problems are presented for students to solve using words, numbers, or pictures, and to follow up with written explanations. This writing prompt applies a mathematics concept to the student’s own life, thus making a real-world connection with a cross curricular topic. Mathematical concept prompts allow students to reflect and communicate their knowledge of mathematics. The journal prompts in *Motivation Math Common Core Aligned* serve as another formative assessment opportunity for students to express their thoughts and reasoning abilities as they transfer a mathematical concept across the disciplines.

Motivation Station, another unit component, is typically an activity or game that is designed to be completed with a partner or independently, or as a class. This section offers an opportunity to students with which to reinforce one or more Content Standards within the existing unit.

The section Parent Activities is located at the conclusion of each unit in the Student Edition.** ***Motivation Math Common Core Aligned* includes these activities to invite and encourage parent engagement in mathematics education. Product developers recognize that teachers must support and encourage parent collaboration with students regarding mathematics. Teachers are provided activities per unit with which to cultivate parent involvement with their children by reinforcing previously introduced skills. Research concludes that productive collaboration and interaction with parents have a favorable impact on attitudes towards mathematics and student achievement (Calabrese Barton et al, 2004). Parents can be significant contributors to the learning process. Opportunities for parents to be involved in their students’ learning allow parents to show an interest in the students’ work. Parent involvement helps parents become familiar with the content and the way students are learning (National Council of Teachers of Mathematics, 2000). When parents take time to provide home encouragement, students have another opportunity to apply and practice the mathematical concepts previously learned. Teachers and other educational leaders should consistently help students and parents to understand that an increased emphasis on the importance of effort is related to improved mathematics performance (The National Mathematics Advisory Panel, 2008).

Research indicates that the more parents are involved and excited in the learning of their children, the more successful a child can be academically. When schools cultivate partnerships and engage families in their children’s education, author Constantino (2008) stated that student achievement can increase. In addition, Constantino noted that schools must continuously nurture relationships with parents by providing them with resources to help their children succeed in school. Constant attention in strengthening relationships lays the foundation for high-quality engagement. West (1985) and Weller (1999) indicate there are parent behaviors that can lead to effective schools. When parents show support, interest, and become involved the success rate of students can rise. Students in at-risk situations show an increase in grades, test scores, and academics when their parents become involved in instructional programs (Dolan, 1996). The activities for parents in *Motivation Math Common Core Aligned* offer opportunities within each unit to reach and engage parents.

Bagin and Gallagher (2001) note that communicating on a regular basis with parents can promote student learning and reduce attendance problems. Weller (1999) advocates that when schools and teachers treat parents with genuine concern and make them feel important, welcome, and needed, parents are more apt to take active roles in supporting their children in academic achievement. Findings from an extensive research review on parent/family involvement programs are shared by Henderson and Mapp (2002) in the report A* New Wave of Evidence: The Impact of School, Family, and Community Connections on Student Achievement*. Henderson and Mapp concur with other researchers that a favorable and substantiated relationship exists between family involvement and student success, regardless of race/ethnicity, class, or parents’ level of education. A key finding is that children of parents who are involved in home and in school settings show improved performance in school. Thus, the section Parent Activities is provided to help parents support their children with meaningful and relevant applications to the previously taught concepts. The information given helps the parent and child build oral language through informal conversation. Simply written, the text invites parents to support the mathematical learning by asking questions, making relevant comments, or setting up other home learning activities to reinforce previously introduced concepts. Assignments, intended to be completed in class or at home, enhance students' understanding, skills, and proficiency in mathematics. *Motivation Math Common Core Aligned* reflects the careful planning taken by the Mentoring Minds Product Development Team so that the primary focus of Motivation Station and Parent Activities are meaningful extensions of the skills/concepts taught in the unit; however, these two components may incorporate skills/concepts from previous units.

Teachers cannot teach what they do not know. When reviewing the literature regarding the relationship between teachers’ mathematical knowledge and students’ achievement, the research indicated that teachers’ content knowledge is an important element. *Motivation Math Common Core Aligned* designed the Teacher Edition in recognition and support of this finding. Teachers are provided the guidance they need as they prepare and deliver high-quality instruction to improve mathematical performance for all students. Each** ****Teacher Edition **lists the Common Core Standards for mathematical content by domain, cluster, and the standard that is the focus of the unit. Other standards addressed within each unit are also listed. In addition, the Common Core Standards for Mathematical Practice are identified based on those practices most closely linked with the learning activities. These practices appear at the beginning of each unit as a reminder for teachers to incorporate the mathematical practices into instruction on a daily basis. High-quality research does not appear to support that instruction be either student-centered or teacher-directed. The findings seem to confirm that neither approach should be used exclusively. Thus, *Motivation Math Common Core Aligned* is designed to incorporate both. The Unpacking the Standards component identifies the grade expectation supported by prior learning. The description links mathematical ideas and builds upon previous ones in order that student understanding and knowledge deepen and student application of concepts increase. This component provides teachers a tool with which to deepen their knowledge and conceptual understanding of the standards within the mathematical unit of study. * *

Research findings indicate that certain teaching strategies and methods are worth careful consideration as teachers strive to improve their mathematics instruction. Stigler and Hiebert (2004) advocate when the improvement of teaching methods becomes the focus, student performance will show more positive results. Teacher and student interaction is key to improvement. Many students learn mathematical concepts best through the manipulation of concrete materials because it helps them to build a mental representation of the concept. Manipulatives provide concrete introductions to abstract ideas. Every student should have an opportunity to have adequate "hands on" experiences with appropriate manipulatives before engaging in pencil-and-paper activities. Textbooks and other printed resources, show the pictorial and symbolic representations of mathematical concepts.

According to several studies, the use of manipulatives can enhance the cognitive process. Suydam and Higgins (1977) researched activity-based teaching approaches, including the use of manipulatives, in kindergarten through eighth grades. The conclusion reported was “…lessons using manipulative materials have a higher probability of producing greater mathematical achievement than do non-manipulative lessons.” Findings revealed that manipulatives are effective no matter the achievement, ability, or socioeconomic levels of students. Manipulatives and pictorial representations produce higher achievement as opposed to only symbolic representations, as students can construct models to show their understanding of mathematical ideas or processes. This allows teachers to observe how students think or reason so that misconceptions can be corrected in a timely manner. It also offers students opportunities to demonstrate their learning other than with paper and pencil. The relationship between longevity and the use of manipulatives indicate positive findings of enjoyment, interest, understanding, thus, increasing student engagement in mathematics (Sowell, 1989; Ruzic and O’Connell, 2001). When students’ interest grows, mathematical ability is affected and attitudes towards mathematics improve. Sutton and Krueger (2002) report that long-term usage of concrete materials seems to be positively related to increases in mathematical ability. Research by Grouws and Cebulla (2000) suggests that teachers use manipulative materials regularly in order to give students hands-on experiences in order to construct meaning for the mathematical ideas they are learning. A major benefit for students would be to use multiple types of manipulatives when learning mathematical concepts to ensure broader comprehension.

In* Curriculum and Evaluation Standards for School Mathematics*, the National Council of Teachers of Mathematics (NCTM, 1989) recommends the use of manipulatives in math education especially for elementary levels. In the revised document, *Principles and Standards for School Mathematics*, NCTM (2000) continues to place emphasis on the importance of manipulatives and supports the use of manipulatives in mathematics instruction. Although studies report there is no one way to best teach mathematics, the use of manipulatives combined with other strategies can cultivate depth and understanding of abstract concepts. *Motivation Math Common Core Aligned* does not limit the use of concrete materials to demonstrations, but suggests ways that encourage students to think and verbalize their thoughts.

It is highly recommended that every classroom have an assortment of manipulatives for student accessibility at all times. If the same materials to teach multiple ideas can be used during each school year then the amount of time to introduce the manipulatives can be shortened and students are helped to visualize and establish connections between ideas. This does not preclude a teacher from introducing other manipulatives but provides consistency with essential manipulatives utilized at more than one grade level. Thus, research and mathematical experts agree that the one essential component in a mathematics program should be the appropriate use of manipulatives. Thus, *Motivation Math Common Core Aligned* supports the use of manipulatives and identifies manipulative-based activities throughout the Teacher Edition.

Literature can stimulate a variety of creative and critical thinking responses from the students, such as performing a skit from the story followed by mathematics-related problems. Problem-solving strategies, including acting it out, drawing a picture, and constructing a model using manipulatives, materialize quite readily as a result of literature. Evidence shows literature promotes thinking and reasoning in mathematics when questions are presented on higher thinking levels. Discussions are encouraged to build conceptual understanding. Thaiss (1986) advocates mathematics and literature connections to strengthen student motivation and increase higher levels of engagement. *Becoming a Nation of Readers: The Report of the Commission on Reading *(1985) stresses the importance of the integration of reading. Mathematics lends itself easily as a communication tool and thus, works directly with reading to help students become successful learners. Suggested literature that can be used for integrating lessons across the curriculum is noted in the Teacher Edition for each unit in *Motivation Math Common Core Aligned*. Children's literature offers excellent resources for connecting literature to mathematics instruction for students. “Through the use of books, students see mathematics as a form of communication. It has been proven that children learn best when they can apply their learned knowledge from one subject to another” (National Council of Teachers of Mathematics, 1989). Problems that emerge from books make the mathematics relevant, are highly motivational, and present meaningful contexts for establishing mathematical thinking.

Active instruction includes a wide range of instructional approaches: small groups, class discussion, concrete objects, hands-on experiences, reading, and writing. Research does not favor all of one way to instruct, either student-centered or teacher-directed. A combination of both seems to be what is recommended (NMAP, 2008). In *Motivation Math Common Core Aligned*, teachers can ask students to: think aloud, consider different options for solving problems, show evidence for the solution reached, and put their thoughts in writing. All of these techniques help students organize their thinking and assist teachers in determining the level of understanding of mathematical concepts. Studies indicate that instruction which emphasizes active student engagement in hands-on opportunities improves attitudes toward math and indicates a positive effect on mathematics achievement.

The National Mathematics Advisory Panel (2008) shared results about the use of explicit instruction with struggling students. A recommendation was made that these students receive explicit mathematics instruction on a regular basis, but not explicitly deliver all mathematics instruction for the struggling learners. When students are provided models that clarify procedural steps such as in problem solving, followed by extensive practice of the newly learned strategies or skills, when students participate in thinking aloud, and when they are given specific feedback, their performance improves. Thus, *Motivation Math Common Core Aligned* provides several intervention activities per unit for students exhibiting mathematical difficulties in order to build the foundational skills and conceptual knowledge identified for that grade level. Students may work in pairs or in small groups. A formative assessment is planned for these intervention activities so that instruction may be adjusted.

Students have to understand vocabulary to understand the academic content they encounter in school. In the Teacher Edition of *Motivation Math Common Core Aligned* is the section Vocabulary Activity. This area entails graphic organizers, games, or similar opportunities to actively engage students in learning with vocabulary essential for the unit and concludes with a formative vocabulary assessment. Stahl and Fairbanks (1986) revealed when specific vocabulary from academic subject areas is selected as the focus of instruction, the result was a 33 percent increase. Therefore, it appears when students are taught specific content vocabulary in each subject area at each grade level, students have an excellent opportunity to acquire the academic background knowledge they need to understand the subject area content. Teaching content vocabulary using a systematic approach appears be a powerful tool for student success (Marzano & Pickering, 2005). Furthermore, research firmly documents that academic background knowledge has an effect on academic achievement. Any intervention for the achievement of students should identify increasing students’ content vocabulary knowledge through direct instruction as a leading priority (Marzano, 2004). In earlier research, Becker (1977) concluded that the implementation of systematic vocabulary programs appeared essential in order to close gaps between students from economically disadvantaged backgrounds and those who were not.

Mentoring Minds seeks to understand the issues involved in teaching and learning mathematics. The National Research Council (2001) asserted that the performance of students in both reading and math at the conclusion of elementary school is an important predictor of their educational success. The review of literature notes that students who have not mastered certain basic skills can expect to encounter problems in mathematics throughout their schooling and later. Summary statements similar to these, a review of mathematical literature combined with recommendations from studies, and observations from classroom experiences yield much knowledge about what works. With this wealth of information, *Motivation Math Common Core Aligned* was developed as a complement to an existing mathematics program for any grade or campus. The Mentoring Minds Product Development Team embraces the goal that all students receive a quality mathematics education.

To ensure student success in mathematics, all five strands of mathematical proficiency must be addressed (Kilpatrick, Swafford & Findell, 2001). Research strongly supports the importance of student acquisition of conceptual understanding of mathematics. The literature indicates when factual knowledge and procedural proficiency are aligned with conceptual knowledge, students can achieve mathematical proficiency. This integration is found in *Motivation Math Common Core Aligned*, a mathematics resource that provides access to high-quality mathematics for all students, regardless of their characteristics, backgrounds, or challenges. *Motivation Math Common Core Aligned* addresses the five strands; thus, as students progress from grade to grade, students should become increasingly proficient in mathematics. Proficiency in mathematics will prepare students to meet challenges they may face in college, in the work force, and in life.* *

The Mathematics Product Development Team is comprised of educators who have served as administrators, teachers, and mathematics coordinators. The developers of *Motivation Math Common Core Aligned *reviewed research-based evidence on how students learn, gathered input from a wide array of mathematics’ experts and educators, attended NCTM and NCSM conferences, collaborated with practitioners in the field, studied released documents from Smarter Balanced and PARCC, unpacked the Standards, and employed individual expertise and collective judgment as they designed a mathematics resource to lead students into the 21^{st} century. The contents of *Motivation Math Common Core Aligned* focus on the Common Core Standards in Mathematics and the Standards for Mathematical Practice, ensuring that the product is appropriate, high-quality, and up-to-date. Bloom's Taxonomy, Depth of Knowledge, Hess’ Cognitive Rigor Matrix, and Rigor/Relevance Framework® are incorporated to stimulate and develop students' higher-order thinking skills, encouraging rigor and depth in thinking. Examples of evidence-based techniques found in *Motivation Math Common Core Aligned **are many**,* including standards-based instruction, active teaching, cooperative learning, hands-on/minds-on learning activities, critical thinking, formative assessments, and real-world applications. *The contents of** Motivation Math Common Core Aligned* are aligned with these principles for improving student performance. The literature on improving student performance in mathematics concludes that effective mathematics programs provide specific information on individual student performance for teachers, parents, and students; peer feedback and support; direct or explicit instruction; and real-world problems.* Motivation Math Common Core Aligned* meets these criteria for improving student performance.

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