Teaching Practices to Improve Math Learning

150 150 DC Special Education Cooperative

During SY14-15, the Co-op piloted a new professional development program, Project SOAR, for special education math teachers in 3 member schools. Through Project SOAR, the Co-op was able to bring in an expert math coach to help charter school staff implement math interventions for students with disabilities in grades 4-8. Our coach, Angela Stoltz, is a Doctoral Fellow at the University of Maryland’s Center for Mathematics Education with extensive teaching experience in middle school math working with struggling learners. We asked Angela to share some of the best practices she introduced to teachers as the Co-op’s math intervention coach. Read on to learn more from Angela about key teaching practices to improve mathematics learning!

In 2014, the National Council of Teachers of Mathematics published Principles to Actions: Ensuring mathematical success for all (2014). In this document, NCTM identified eight mathematics teaching practices as a framework for improving mathematics teaching and learning. These practices included:

  1. Establishing mathematics goals to focus learning
  2. Implementing tasks that promote reasoning and problem solving
  3. Using and connecting mathematical representations
  4. Facilitating meaningful discourse
  5. Posing purposeful questions
  6. Building procedural fluency from conceptual understanding
  7. Supporting productive struggle in the learning of mathematics
  8. Eliciting and using evidence of student thinking (NCTM, 2014, p. 10).

3As educators, we are accustomed to developing mathematics goals to focus our students’ learning; however, practices two through eight require a significant shift from traditional mathematics teaching and, on the surface, seem to conflict with the explicit instructional practices that are often the focus of special education teacher training. The question is, as special education instructors who teach mathematics, is it possible to meet the unique learning needs of our special education students and shift our mathematics teaching from direct instruction toward the mathematics teaching practices outlined in Principles to Actions (NCTM, 2014)? I believe research has indicated that it is possible, but we must accept that shifting our teaching practices takes time and is more effective when we focus on one or two teaching goals deeply and have a community of peers and administrators to support us in achieving these goals.

By setting a goal to implement tasks that promote reasoning and problem solving and incorporating a plan to examine potential tasks using a framework like the Stein & Smith Task Analysis Framework (1998), we can work to identify mathematical tasks that: develop students’ conceptual understanding; are represented in multiple ways and support students’ connections among multiple representations; support students’ self-monitoring and self-regulation; and require students to draw on their own knowledge and experiences. A high quality mathematics curricula will contain tasks that meet Stein & Smith’s definition of cognitively demanding tasks. Two middle grades curricula which we have examined in Project SOAR that incorporate these types of tasks are Connected Mathematics and Do the Math.

The benefit of incorporating a curricula like Connected Mathematics is that the teacher’s guides map the conceptual framework of the units and explain connections among concepts that are found across the units, so teachers can visualize the scope and sequence of the content and concepts and see the connections among them. In addition, the teacher’s guides outline specific questions for teachers to use to elicit evidence of student thinking and use that evidence to help students identify misconceptions and make connections among concepts themselves. Thus, a curriculum like this can provide us with tasks that promote reasoning and problem solving as well as the strategies we need to implement the tasks in ways that allow our students to use and connect mathematical representations, engage in meaningful discourse and productive struggle, and build procedural fluency from conceptual understanding. However, while Connected Mathematics incorporates strategies for self-monitoring, we may find we need to build additional supports into the pre-existing lessons such as adding tables, graphs or questions to provide greater access and support students’ self-monitoring as they engage in problem solving.2

Do the Math, although broken into short, thirty minute lessons, also supports conceptual understanding, connecting mathematical representations, building procedural fluency from conceptual understanding, and implementing tasks that promote reasoning and problem solving. However, when incorporating Do the Math, we need to carefully plan our questioning and task implementation to promote meaningful discourse and monitor student thinking in order to identify misconceptions and use student thinking to support connections across strategies and concepts.

A key component of both Connected Mathematics and Do the Math is the incorporation of CRA (Concrete-Representational-Abstract). CRA is explicit and systematic instruction which guides students from concrete manipulation to mathematical representations and abstraction. Also known as EIR, these instructional strategies have been found to significantly improve MLD student achievement in mathematics (Scheuermann et al., 2009). CRA instruction is the convergence of the reform mathematics teaching practices outlined by Principles to Actions (NCTM, 2014) and the scaffolded/explicit instruction that are highlighted in special education teaching. By incorporating CRA into mathematics curricula, we special educators can meet both NCTM’s recommended teaching practices and the individual learning needs of our students.

Finally, Principles to Actions (NCTM, 2014) contained a chapter on access and equity which stated that:

attending to access and equity requires being responsive to students’ backgrounds, experiences and knowledge when designing, implementing, and assessing the effectiveness of a mathematics program… (it) also means recognizing that mathematics programs that have served some groups of students, in effect privileging some students over others, must be critically examined and enhanced, if needed, to ensure that they meet the needs of all students. That is, they must serve students who are black, Latino/a, American Indian, or members of other minorities, as well as those who are considered to be white; students who are female as well as those who are male; students of poverty as well as those of wealth; students who are English language learners as well as those for whom English is their first language; students who have not been successful in school and in mathematics as well as those who have succeeded; and students whose parents have had limited access to educational opportunities as well as those whose parents have had ample educational opportunities (p. 60)

NCTM outlined unproductive beliefs that create obstacles for student access to challenging mathematics and equitable mathematics teaching. These unproductive beliefs were: students (or particular groups) possess different innate mathematics abilities that cannot be improved by instruction; equality and equity are the same-all students having the same opportunities will have the same outcomes; equity issues only impact high minority/high poverty  schools; math learning is independent of race/culture/gender/language/socioeconomic status; high poverty students lack characteristics that support mathematics achievement; tracking supports student learning; only advanced students can reason mathematically (p. 63-64). On the contrary, productive beliefs outlined by NCTM were: math ability is not innate, but is the confluence of opportunity, experience and effort; equity requires differentiation to support individual student needs; equity is essential in all schools; English Language Learners can learn grade level mathematics as they develop their language skills; race/culture/gender/language and socioeconomic status must be leveraged in mathematics instruction; effective mathematics teaching can increase opportunities and achievement for poor students; tracking should be eliminated; all students are capable of reasoning mathematically (p. 63-64).

Not surprisingly, issues pertaining to access and equity were the focal point of NCTM’s National Conference this April (Ball et al., 2015; White et al., 2015; Aguirre et al., 2015; Cuoco et al., 2015). As special educators, we know firsthand the negative effects that tracking and fixed mindsets can have on our students’ beliefs about themselves and their beliefs about their mathematics abilities. While many structural obstacles impacting our students’ access to higher level mathematics is out of our control, we do have the power to provide mathematics instruction that is equitable by: learning about our students and drawing on their prior knowledge and experiences; reflecting on our teaching practices and the micro-messages we send (do we reinforce fixed or growth mindsets, do we provide male and female students and students of different racial/ethnic backgrounds equal opportunities to share their knowledge and ask questions, do we highlight the mathematical contributions of females and non-western European cultures/ethnic groups); providing access to challenging mathematics by incorporating CRA into our curricula; advocating for our students to be de-tracked; and seeking professional development that increases our own mathematics content and pedagogical knowledge. Shifting our society from the historically imposed “fixed” mindset that has sent the message that some people are mathematics people, and some are not, to a “growth” mindset that sends the message that all of us are capable of learning mathematics, is a much greater challenge than incorporating CRA. While CRA helps our students gain access to challenging mathematics, awareness of our students’ knowledge, beliefs and experiences and awareness of our own teaching practices and beliefs, as well as awareness of the structural inequities that create obstacles to access and equity will help us identify ways we can provide students with greater access and equity in mathematics.


Aguirre, J., Bartell, T., Drake, C., Foote, M., McDuffie, A., and Turner, E. (2015). Proceeding from the NCTM Research Conference: Preparing Culturally Responsive Mathematics Teachers; The TEACH MATH Project.

Ball, D., Martin, D., Meyer, D., and Leinwand, S. (2015). Proceedings from the NCTM Research Conference: Turning the Common Core into Reality in Every Math Class.

Cuoco, A., Munter, C., Badertscher, E., Boston, M., Johns, T., Fisher, C., Murdock, T., Schmitt, R., Martin, D., and Nasir, N. (2015). Proceeding from the NCTM Research Conference: Co-designing for Equity: Tensions That Arise in Formative Intervention Research.

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

Stein, M. K., & Smith, M. S. (1998). Mathematical tasks as a framework for reflection. Mathematics Teaching in the Middle School, 3, 268-275.

Scheuermann, A. M., Deshler, D. D., & Schumaker, J. B. (June 06, 2009). The Effects of the Explicit Inquiry Routine on the Performance of Students with Learning Disabilities on One-Variable Equations. Learning Disability Quarterly, 32, 2, 103-120.

White, D., Crespo, S., Bieda, K., Aguirre, J., Goffney, I., and Civil, M. (2015). Proceeding from NCTM Research Conference: Mathematics Methods Classrooms as Sites for Courageous Conversations about Equity.