By Luis Lima, Senior Advisor, CenterPoint

The rollout of the Common Core State Standards (CCSS) and similar college- and career-ready academic standards for mathematics, has introduced the idea of “shifts” or changes that distinguish these standards from their predecessors.  Despite having figured prominently in the lexicon of the education world, much about the term is not understood deeply by enough educators.

The most significant of these shifts, arguably, is the expectation that all students will encounter greater rigor in their mathematical learning experience.  Greater rigor means a move toward students developing deeper conceptual understanding of mathematics.  Prior to the CCSS and similar college- and career-ready standards, instructional practice seemed to concentrate on having students do procedures and use the standard algorithm efficiently.  It was not uncommon for teachers to introduce a new idea by routinizing the process and by providing students with catchy mnemonic devices to help them remember the routine (i.e.: Keep – Change – Change for subtraction of integers and Keep – Change – Flip for the division of fractions are two examples).

In this way, teachers were essentially training on the steps for solving equations, for instance, without really spending time on the concepts and reasoning underlying it.  What does it mean to solve an equation?  Is there more than way to solve an equation?  If there is more than one way, are those methods equivalent?  These are the kinds of questions that should drive instruction if the objective is to build conceptual understanding – to make the concepts “stick” and strengthen students’ ability to apply mathematical knowledge in new contexts.

In order to build students’ conceptual understanding, however, the teacher needs to understand how that mathematical idea develops towards the standard algorithm.  The teacher needs to help build the concept for students so they can experiment and manipulate the math in a meaningful way, eventually arriving at the standard algorithm as the most efficient way to solve the problem.

Through my experience in the classroom, as an administrator, and working with schools and districts to implement more rigorous college- and career-ready math standards, I have found several challenges in helping educators develop the capacity to teach math in this way:

  • First, most teachers didn’t learn math conceptually. Additionally, in a significant number of teacher preparatory courses the preparation of new teachers to teach mathematics conceptually has taken time. And building the depth of content knowledge necessary to promote student development of math concepts is not the norm of professional development practice.
  • Second, this approach to teaching mathematics requires educators to let go of some of their tried and true practices because they may not serve the purpose of helping kids develop conceptual understanding. Teachers may inadvertently still be doing all of the thinking, and not giving students the opportunity to experiment and to reason.
  • Third, research shows (Jamil, Larse, & Hamre, 2018; NCTM, 2014) that teacher expectation shapes student outcomes in the long run – particularly the impact of positive expectations for students. If a teacher doesn’t believe that every child can do the higher-level math that’s proposed by the CCSS, then it’s unlikely that teacher will engage them to the conceptual phase of learning math. In this way, it’s a significant equity issue, as evidenced so clearly by TNTP’s recent report, The Opportunity Myth [].

What are the implications for professional learning?  Fundamentally, the challenge is that professional learning must not only help teachers understand the math at a deeper level but also understand the pedagogy that comes with the expectation of the standard.  Teachers have to explore and understand concepts much more deeply than the standard algorithm for that grade level content. They have to be able to break down that standard and build it back up using mathematical understanding in different dimensions.  They need to learn how to give students enough experiences experimenting with a given concept.  It’s about purposeful engagement of students, and that’s not a simple task.

Strong curriculum – tied to professional learning that truly builds capacity – helps set the foundation for moving math instruction in this direction.  The professional learning, grounded in the curriculum, must include experiences where educators can break down the expectation of the standards, and see how that translates into sequences of events that are organized in the curriculum and aligned, embedded assessments.

Supervisory capacity is also critical.  Principals and other school instructional leaders need to understand the expectations for math instruction aligned to the CCSS/college- and career-ready standards so they can support teachers.  Instructional leaders need to understand that instructional practices do not fit every discipline – for example, modeling as part of instruction is great for many content areas, but not in mathematics where kids need to engage with the math and make connections.

If teachers have the content knowledge and pedagogical understanding, they can help build students’ procedural fluency in the context of math application.  It is critical to get this right:  Math success correlates with social mobility, particularly for students from lower socioeconomic backgrounds (Bureau of Labor Statistics, 2017; Fryer, 2010).  Closing achievement gaps and providing equitable access to educational opportunity is one shift we must master.


Dr. Luis Lima is Senior Advisor for Curriculum Assessment, and Professional Services in STEM at CenterPoint.  He has several decades of experience as a K-12 mathematics teacher, a college instructor, and a school and district leader.


Jamil, F. M., R. A. Larsen, & B. K. Hamre (2018). Exploring Longitudinal Changes in Teacher Expectancy Effects on Children’s Mathematics Achievement. Journal for Research in Mathematics Education, 49(1), 57-90

NCTM (2014). Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: NCTM. p. 11

Household Data Annual Averages – Bureau of Labor Statistics (2017). Available online at

Roland G. Fryer, Jr. (2010). Racial Inequality in the 21st Century: The Declining Significance of Discrimination. NBER Working Paper No. 16256