Understanding and Addressing the Challenges of Teaching Middle Grades Mathematics Conceptually

Research Summary

By: Holly Henderson Pinter


Understanding students' academic needs and developing curricula to address them has always been a challenge for middle level educators. Competing perspectives on academic rigor and external demands for accountability have often created additional layers of stress for teachers and administrators. Many try to find the best balance they can for their particular students in their particular subject. With recent controversies over the Common Core (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010), these tensions have become more pronounced, especially in the area of mathematics. At the same time that the popular press calls for higher levels of achievement, politicians and parents express confusion about the "new" standards. At the heart of this turmoil is an emphasis on deep conceptual understanding. Everyone wants students to be able to "think mathematically."

Tenets of This We Believe addressed:

  • Curriculum that is challenging, exploratory, integrative, and relevant
  • Educators use multiple learning and teaching approaches

Decades of research on mathematical learning have identified "conceptual," or standards-based approaches that nurture reasoning that goes beyond surface level calculation and memorization. The stronger emphasis on conceptual understanding and deep knowledge of mathematics closely adheres to the NCTM principles and standards (NCTM, 2000) and the teaching practices outlined by NCTM Principles to Actions (NCTM, 2014). There are several implications of this for middle level teachers: First, there is a shift of content objectives within grade levels, and, second, there is a need for a deeper understanding of how standards follow a learning progression, or trajectory, through the grade levels.

Prior to the implementation of Common Core State Standards for Mathematics (CCSSM) across much of the United States, states typically structured mathematics standards and curricula in a spiraled format. Each year, the five content strands outlined by the National Council of Teachers of Mathematics (NCTM) were covered, with the complexity of tasks increasing with each grade. Students would explore units focused on numbers and operations, geometry, measurement, data analysis and probability, and algebra. With this structure, teachers were able to focus their teaching using a very small scope and sequence of specific content standards for their grade level. The curriculum would then spiral to teach overlapping concepts each year with more depth. Since Common Core has come down the pipeline however, the scope and sequence is shifting to incorporate a stronger emphasis on coherence and focus (Porter, McMaken, Hwang, & Yang, 2011). The content in each grade level is more rigorous and covered more deeply within a grade level and with less overlap across grade levels.

This research summary will operationally define learning trajectories in reference to middle grades Common Core mathematics implementation, address the changes in content across the middle grades, and make recommendations to appropriately emphasize vertical alignment across grade levels. This We Believe: Keys to Educating Young Adolescents (National Middle School Association [NMSA], 2010), denotes the importance of offering curriculum that is challenging, exploratory, integrative, and relevant; and that educators use multiple learning and teaching approaches. The widespread implementation of CCSSM has challenged middle level educators to reframe and refine the teaching of mathematics.

Conceptual Understanding and the Common Core

The Common Core was initially conceptualized by a collaboration of the National Governors Association Center for Best Practices (NGA Center) and the Council of Chief State School Officers (CCSSO) in partnership with teachers and experts in the field of mathematics education. The final standards were released the summer of 2010 after a three-year progression of development. Porter, McMaken, Hwang, and Yang (2011) compared the CCSSM standards to the previous state standards for all U.S. states and found that the CCSSM standards tended towards higher cognitive demand than previous standards. This is one of the many reasons the National Council of Teachers of Mathematics supports the CCSSM in their position statement, "The Common Core State Standards offer a foundation for the development of more rigorous, focused, and coherent mathematics curricula, instruction, and assessments that promote conceptual understanding and reasoning as well as skill fluency" (NCTM, 2013, para 1). Implementation of CCSSM has had an impact on students, teachers, and parents alike. Teachers have had to alter their teaching practices to accommodate higher levels of demand, students have had to adapt their approach to mathematics from a more conceptual standpoint utilizing the standards for mathematical practice, and parents have had to familiarize themselves with an approach to mathematics that feels drastically different from their own learning experiences due to less focus on memorization and procedures. The CCSSM was designed as a learning progression across K-12 and because of this parents, students, and teachers may struggle to see the bigger picture of how the standards connect and produce complete conceptual understanding. The curriculum materials being used to implement CCSSM based on conceptual understanding are referred to as standards-based curriculum. These standards-based materials were originally constructed with strong connections to the NCTM documents (1989, 1991, 2000, 2014) and have been updated to reflect CCSSM standards and teaching practices.

Standards-Based Teaching Practices and Curricula

All of these changes came in response to an ever-growing body of research that supported the impact of reforms on teaching practices and student achievement. Weiss & Miller (2006) found that implementing the central tenets of reform (problem solving, communication, connections, reasoning, and use of representations) required a fundamental shift from teacher-centered, direct instruction toward a student-centered, more collaborative environment, a growth process that takes time. To be successful, teachers must "develop a deeper understanding and broader view of mathematics, strengthen their pedagogical knowledge, explore assessments aligned with inquiry-based instructional strategies, and foster and sustain collaboration for continued growth" (Weiss & Miller, 2006, para 5). When teachers accomplish these changes, student achievement increases. In a study of more than 2,000 middle grades students who participated in standards-based mathematics for two years or more, students equaled or outperformed achievement of students in comparison (Reys, Reys, Lapan, Holliday, & Wasman, 2003). In Slavin, Lake, and Groff’s (2009) systematic investigation of effective programs for middle and high school mathematics, the authors noted a considerable impact of the use of cooperative learning environments (which use problem solving and discourse practices) situated in the context of standards-based curriculum and teaching, towards achievement of students.

Common Core Content

While not all states had similar content standards in each grade level prior to the CCSSM, the adjustment of the coherence and focus of CCSSM standards certainly shifted objectives slightly, if not dramatically, in many state sets of standards. This section will outline the general progression of content in each grade level according to CCSSM progressions for each strand, and note any major changes from prior sets of standards.

Statistics and Probability
Statistics and Probability is the strand of mathematics that potentially has shifted the greatest in the middle grades. Previous standards emphasized a spiral where students began working with measures of central tendency in sixth grade and then worked towards applications of these measures towards statistical displays such as box plots in seventh and eighth grade. In contrast, the Common Core standards delve more deeply into statistical analysis through understanding variability and bivariate data. In grade 6, students engage with vocabulary that describes variability (cluster, symmetry, outlier, etc.), and then they apply the vocabulary to graphical representations of histograms and box plots. The application of vocabulary within one school year allows students to understand why they are learning terms and how those terms can be used in a more coherent and connected manner than is done with traditional models of scope and sequence. The focus in seventh grade is for students to start to apply the skills of analyzing data toward producing data. At this point the study of probability is developed—a concept that was previously spread out across multiple grades in elementary and middle school. Students integrate this learning trajectory with the trajectory of algebra in eighth grade as they can begin to study data in relation to functional relationships by studying scatterplots and bivariate data.

Ratio and Proportions
The study of ratios and proportional relationships in middle grades represents a pivotal point in a learning trajectory. Students’ extensive work with measurement, multiplication, and division is extended and applied in the middle grades to working with unit rates, slope, and percentages. This foundational use of ratio and proportion then extends towards students work with geometry and trigonometry in high school where ratios are embedded in concepts such as the use of sine, cosine, and tangent. This is one of many areas of CCSSM where the middle level content links foundational knowledge with applications in the high school years. According to Briggs and Peck (2015), the progression of proportional reasoning is perhaps the most important strand in elementary mathematics as NCTM identifies it as one of the five major foundations in mathematics.

Algebra: Expressions and Equations
Under the CCSSM, algebra has been re-framed to the study of expressions and equations. In previous sets of standards, the use of a variable for an unknown in an expression or an equation would not be introduced until the middle level years. Under CCSSM however, students have been using this skill since grade 3. Additionally, students have been writing expressions since grades 4 and 5. By grade 6, students begin working with expressions in a more sophisticated way. They can recognize variables in contextual situations and write corresponding expressions and equations to match. Additionally they begin to analyze relationships between variables, which connect to the study of linear and non-linear functions. The connection amongst graphical representations, equations, and tables is made explicit in seventh and eighth grades. As previously mentioned, the learning trajectory of proportional reasoning intersects here as students begin to connect ratios to slope and to graphs of linear functions.

The Number System
The number system CCSSM standards really highlight is the vertical alignment of building mathematical ideas and the natural progression through learning trajectories. There are two primary concepts of rational numbers that build on students’ foundational K-5 experience: representing fractions and whole numbers on the number line and the properties of operations using whole numbers and fractions. This is an area where teacher knowledge of and use of learning trajectories is particularly important. The consistency in use of tools and representations to depict mathematical ideas is what helps students make conceptual connections and properly build on their elementary experience.

Geometry
The beginning of middle grades geometry content hinges on geometric measurement. Students work with finding area, surface area, and volume. At grades 7 and 8 the sophistication of the shapes learned about increases at each grade level. The concept of decomposing and composing complex shapes is a key component of the geometry strand. Students build formulas from recognizing relationships and making conjectures based on those relationships. Grade 7 geometry standards closely connect to ratios and proportional reasoning through the exploration of scale drawings. There is also emphasis on the construction of geometric shapes using appropriate tools such as protractors and compasses. The middle level geometry experience is rounded out by exploring transformations and the Pythagorean Theorem. This is a strand that may be similar to previous state standards because Van Hiele levels of geometric development have been used to guide curriculum decisions. The main difference is the alignment of geometric objectives to connect more explicitly to the other strands within each grade level in the CCSSM.

Learning Trajectories

The term learning trajectory, first used in the field of mathematics education by Simon (1995), refers to how teachers frame their lesson planning based on anticipation of how students will reach mathematical understanding based on several variables. These variables include not only the activities that students will engage with during instruction, but also how the teacher anticipates the multiple solution pathways toward understanding a mathematical objective. This definition was expanded by Simon and Tzur (2004) with the term hypothetical learning trajectory (HLT), which includes the existing knowledge of students, a medium for planning the learning of concepts, the chosen tasks that promote learning of mathematical concepts as an influential part of instruction, and regular adaptions made by teachers depending on student progress due to the hypothetical nature of the trajectory. According to Daro, Mosher, and Corcoran (2011), the roots of learning trajectories can be traced to Piaget’s (1970) stages of development and Vygotsky’s (1978) zone of proximal development, which both examine the pathways to a child’s understanding of a concept. Clements and Samara (2004), define learning trajectories as descriptions of a student’s thinking contained in an isolated mathematical concept through exploration of tasks designed to direct students through a progression of thinking with the end goal of student understanding of the mathematical concept.

By analyzing learning trajectories in mathematics, teachers can spend time deeply analyzing clusters of knowledge that can help them adequately anticipate student thinking and plan appropriate assessments. Briggs and Peck (2015) argue that using a learning progression approach to evaluate student growth has the potential to link the use of formative and summative assessments in a way that can help teachers make data driven decisions.

In a recent study of learning trajectories in mathematics related to instructional practice, Wilson, Sztajn, Edgington, and Myers (2015) noted that the progression of thinking in a learning trajectory is not necessarily linear, and functions more as a web in which students progress in understanding via multiple pathways—some much more complex than others. In general, Wilson et al. (2015) argue that teachers help students learn through learning trajectories by addressing the following components of instruction:

anticipating, where teachers have given thought to the potential strategies and knowledge that students bring with them;

attending to proficiency levels, where teachers know their students well enough to anticipate their likely level of proficiency with any given content objective;

attending to strategies, where teachers are active listeners in their classrooms to take notice of the multiple pathways students use to solve problems and progress towards understanding; and

attending to misconceptions, where teachers have anticipated potential misconceptions students might encounter in the midst of their work on a mathematical task.

Here is an example with a lesson objective to analyze proportional relationships and use them to solve real-world and mathematical problems. A teacher chooses a task where students are given four different "recipes" for mixing juice using water and concentrate. The four different mixtures give varying degrees of flavor and students are to determine which mixture is the most and least concentrated. A teacher attuned to learning trajectories would spend quite a bit of effort anticipating potential strategies based on student prior knowledge. In this case the teacher will think through the students’ experiences with fractions, decimals, and percents all while keeping in mind the end goal of this unit, which is for students to think flexibly about proportional relationships.

As students explore the task, the teacher would take note of the strategies and processes students use. In this case students may compare the amount of concentrate to the water, or maybe the amount of concentrate to the entire amount in the mixture. This would tell the teacher if students were using part-to-part or part-to-whole relationships. Students might also have misconceptions when they solve the problem and may use subtraction or some other arbitrary method. When this happens, the teacher can address those misconceptions and give recommendations for students to get on a correct pathway to a solution. A teacher with a strong knowledge of learning trajectory can build on students’ previous learning and create a natural pathway towards understanding new concepts that can help them make mathematical connections down the road.

What Can Teachers Do to Promote Higher Levels of Conceptual Understanding Using Learning Trajectories?

As is often true of middle school, the middle level years in mathematics teaching and learning are serving several purposes. One purpose is to build on the foundations of concepts that start to flourish in earnest in upper elementary school, and the second purpose is to build a new foundation of concepts to set students up for success in high school. Where these two foundations intersect is key, and it is up to middle level mathematics teachers to provide the consistency needed for students to make the necessary connections to understand mathematics in a deep and conceptual way.

One immediate way to strengthen this foundation is to study the standards for the grades above and below the grade level they instruct. Doing so enables teachers to better anticipate and attend to strategies and misconceptions they encounter during instruction. Studying the standards across the grade levels becomes even more effective when teachers have regular opportunities for conversations about content and scaffolding. Grounding dialogue sessions with assessment results and student work samples provides a process for identifying and filling content gaps. A series of iterative conversations over time can help teachers refine their understanding of learning trajectories and have greater impact.

Another way middle level teachers can potentially have an immense impact on helping students stay on target with their learning trajectories is to implement a whole school agreement. According to Karp, Bush, and Dougherty (in press), a whole school agreement functions much like a school-wide discipline policy, except the focus is consistency in the use of language, symbols, models, and rules in mathematics across all grade levels. In each of these domains, teachers may unintentionally encourage use of language, models, etc. in ways that only have a very limited use. For example, instead of using outdated terms like "borrow" or "reduce," all teachers across the school would agree to use mathematically accurate language such as "regroup" or "simplify." This aligns with the CCSSM emphasis on attending to precision in the standards for mathematical practice. Using mathematically accurate language, symbols, models, and rules allow students to see the connection and build on concepts from the elementary years through high school.

Several recommended resources are listed below to help teachers get started in exploring learning trajectories.


References

Briggs, D.C., & Peck, F.A. (2015). Using learning progressions to design vertical scales that support coherent inferences about student growth, measurement. Interdisciplinary Research and Perspectives, 13(2), 75-99. doi: 10.1080/15366367.2015.1042814

Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6(2), 81–89. doi:10.1207/s15327833mtl0602_1

Daro, P., Mosher, F.A., & Corcoran, T. (2011). Learning trajectories in mathematics: A foundation for standards, curriculum, assessment, and instruction. CPRE Research Report# RR-68. Consortium for Policy Research in Education.

Karp, K.S., Bush, S.B., & Dougherty, B. J. (2015). 12 math rules that expire in the middle grades. Mathematics Teaching in the Middle School, 21(4), 208-215.

Karp, K. S., Bush, S. B., & Dougherty, B. J. (in press). Establishing a mathematics "whole school agreement". Teaching Children Mathematics, accepted for publication.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics (1991). Professional standards for teaching mathematics. Reston, VA: Author.

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

National Council of Teachers of Mathematics (2013). Supporting the Common Core State Standards for Mathematics (Position Statement). Retrieved from http://www.nctm.org/uploadedFiles
/Standards_and_Positions/Position_Statements/
Common%20Core%20State%20Standards.pdf.

National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.

National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards. Washington, DC: Authors.

National Middle School Association. (2010). This we believe: Keys to educating young adolescents. Westerville, OH: Author.

Piaget, J. (1970). Science of education and the psychology of the child. New York: Orion.

Porter, A., McMaken, J., Hwang, J., & Yang, R. (2011). Common Core Standards the new US intended curriculum. Educational Researcher, 40(3), 103-116.

Reys, R., Reys, B., Lapan, R., Holliday, G., & Wasman, D. (2003). Assessing the impact of" standards"-based middle grades mathematics curriculum materials on student achievement. Journal for Research in Mathematics Education, 74-95.

Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114–145. doi:10.2307/749205

Simon, M., & Tzur, R. (2004). Explicating the role of mathematical tasks in conceptual learning: An elaboration of the hypothetical learning trajectory. Mathematical Thinking and Learning, 6(2), 91–104. doi:10.1207/s15327833mtl0602_2.

Slavin, R.E., Lake, C., & Groff, C. (2009). Effective programs in middle and high school mathematics: A best evidence synthesis. Review of Educational Research, 79, 839–911.

Vygotsky, L. (1978). Interaction between learning and development. Readings on the Development of Children, 23(3), 34-41.

Weiss, I. R., & Miller, B. (2006). Developing strategic leadership for district-wide improvement of mathematics education. National Council of Supervisors of Mathematics. Lakewood, CO.

Wilson, P. H., Sztajn, P., Edgington, C., & Myers, M. (2015). Teachers’ uses of a learning trajectory in student-centered instructional practices. Journal of Teacher Education, 66(3), 227–244. doi: 10.1177/0022487115574104.


Selected Annotations

Karp, K. S., Bush, S. B., & Dougherty, B. J. (2015). 12 math rules that expire in the middle grades. Mathematics Teaching in the Middle School, 21(4), 208-215.

In this article, Karp, Bush, and Dougherty tackled the set of "rules" or rather language, notation, and models that expire when students reach middle school. The researchers asked students to choose the equation they think would be easiest to solve out of a set of four equations with varying degrees of complexity. They found that students were more inclined to choose the equations that looked "normal," written in the following format: variable, operation, constant, equal sign, and solution. From this the authors began to investigate other generalizations and potential misconceptions that teachers potentially convey during instruction. The authors present 12 different examples of ways that teachers may use language, symbols, or models that may lead to misconceptions or confusions as students progress through their mathematical career. One example is the use of language in describing absolute value. Students are often told that the absolute value of a number is just the number as a positive. The lack of connection to the concept (that the absolute value is the distance a number is from zero on the number line) can create confusion for students when they are then asked to manipulate absolute value with a negative sign outside the absolute value symbols. This and the other 11 examples outlined in this article help teachers to see the vertical alignment of language, symbol, and model use as students progress through mathematics.


Weber, E., Walkington, C., & McGalliard, W. (2015). Expanding notions of "learning trajectories" in mathematics education. Mathematical Thinking and Learning, 17(4), 253-272. doi: 10.1080/10986065.2015.1083836.

Weber, Walkington, and McGalliard address distinctions in operational definitions of learning progressions versus learning trajectories in this theoretical piece. The authors define learning trajectories to be the way students explore and reflect on tasks and the development of knowledge based on those tasks. They argue that learning progressions are distinctly different as they rely on predetermined benchmarks. The authors highlight the CCSSM as a learning progression across K-12 noting that there are learning trajectories embedded within the larger progression. Relying on Simon’s (1995, 2004) seminal work regarding learning trajectories, authors explore the different ways that researchers conceptualize learning and make suggestions to the field of mathematics education for future research to help more clearly delineate operational definitions of learning trajectory and learning progression.


Wilson, P. H., Sztajn, P., Edgington, C., & Myers, M. (2015). Teachers’ uses of a learning trajectory in student-centered instructional practices. Journal of Teacher Education, 66(3), 227–244. doi: 10.1177/0022487115574104.

This study investigated how knowledge of learning progressions impacted the enactment of student-centered teaching practices. Blending together research on teacher professional development, student learning trajectories, and student-centered instruction, the authors used a design experiment to examine the predictions that teachers would use learning trajectories to choose lesson objectives and build on students’ prior knowledge; that teachers would use learning trajectories to anticipate student responses and monitor student progress; and that teachers would use learning trajectories to sequence and connect mathematical ideas. The findings of this study suggest that learning trajectories have strong potential to inform how teachers plan and implement instruction. By planning conceptual lessons based on anticipating students’ prior knowledge and potential solution pathways, teachers were able to provide high quality, student-centered instruction. This study, while focused on math, has potential to transfer to other domains of instruction as well.


Suggested Resources for Teachers

Achieve the Core: Find the progression documents, lessons, assessments, and other resources http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8.

National Council of Teachers of Mathematics:

Collins, A. (2011). Using classroom assessment to improve student learning: Math problems aligned with NCTM and Common Core State Standards. National Council of Teachers of Mathematics. Reston, VA.

Koestler, C., Felton, M.D., Bieda, K., & Otten, S. (2013). Connecting the NCTM process standards and the CCSSM practices. Reston, VA.

Lannin, J.K., Elliott, R., & Ellis, A.B. (2011). Developing essential understanding of mathematical reasoning for teaching mathematics in prekindergarten-grade 8. National Council of Teachers of Mathematics. Reston, VA.

Lloyd, G.M., Herbel-Eisenmann, B.A., & Star, J.R. (2011). Developing essential understanding of expressions, equations, and functions for teaching mathematics in grades 6-8. National Council of Teachers of Mathematics. Reston, VA.

Lobato, J., Ellis, A., & Zbiek, R. M. (2010). Developing essential understanding of ratios, proportions, and proportional reasoning for teaching mathematics: Grades 6-8. National Council of Teachers of Mathematics. Reston, VA.

Small, M. (2013). Uncomplicating fractions to meet Common Core Standards in math, K-7. New York: Teachers College Press.


Author Information

Holly Henderson Pinter, PhD, is an assistant professor of elementary and middle grades education at Western Carolina University. She holds a master’s degree in middle grades mathematics and language arts education from Western Carolina University and a PhD in mathematics education from the University of Virginia. Her specific research interests are quality of standards-based mathematics teaching practices, professional development, and pre-service education.


Citation

Pinter, H. (2016). Research summary: Understanding and addressing the challenges of teaching middle grades mathematics conceptually. Retrieved [date] from http://www.amle.org//TabId/622/artmid/2112/articleid/618/Understanding-and-Addressing-the-Challenges-of-Teaching-Middle-Grades-Mathematics-Conceptually.aspx


Published March 2016.


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