Mark W. Ellis, Ph.D., NBCT
New content standards for mathematics, such as the Common Core State Standards (CCSS), are designed to help today’s children build a solid foundation of knowledge and skills in preparation for the world they will enter and eventually lead.
The new standards emphasize the concepts behind the calculations and how to reason about complex problems. It’s no longer enough for students just to know basic algorithms and facts, given that the machines around us do millions of calculations every second. Instead, today’s young people need to reason and analyze in order to make data-based decisions and develop creative approaches to the non-routine problems of the 21st century.
This series of blog entries examines the four most challenging areas of math standards—measurement, modeling, fractions, and statistics. When educators focus on the challenging standards first, they can maximize their time, accelerate progress, and create learning environments in which all students can succeed with mathematics.
Q1: What are the biggest changes to the CCSS math standards and how are they affecting the math classroom?
The biggest change is that the new standards emphasize both knowledge and practice. They recognize that students must be able to use math as a tool for understanding the world in order to leave school ready for college or career.
In today’s world, it’s unlikely that students will encounter many neatly ordered problems they can solve with predetermined rules, since these are routinely taken care of by computers. Instead, today’s young people will be called on to tackle complex situations that require reasoning, sense-making, and perseverance.
As a result, learning environments must change to focus on discourse, reasoning, and problem solving so that students can develop deep, coherent content knowledge and proficiency with rich practices of mathematical thinking. These skills are essential for the 21st century and I would argue should begin even in kindergarten.
Q2: All the math standards are more rigorous and challenging. Which are the most challenging and why is it important to identify and focus on them?
- Geometric measurement
- Modeling problem situations
These are the key concepts educators must pay attention to, focusing their time where it matters most. The most difficult standards require students to develop new conceptual understandings or to recognize new relationships among mathematical ideas, tasks that become more difficult when new content is not well connected to prior knowledge or experiences.
For example, when students are learning to use standard units of linear measure, they must understand no fewer than six important concepts. If they lack an understanding of these concepts, they will carry with them shallow, incomplete, and often incorrect ideas as they move on to more advanced content. It is essential that students develop their own understanding through meaningful activities and discourse that allow them to own the concepts and use them as building blocks for future learning.
Research confirms that students are more likely to use their skills accurately and flexibly when they understand the concepts behind the computations, becoming more efficient learners who are better able to recall and retain knowledge over time. In my own classroom teaching experience, when I taught in this way, I found it less necessary to review prior content over and over because students really owned it.
Q3: How can we better prepare students to tackle difficult math standards?
As educators we benefit from extensive research on how to support meaningful learning for all students. Carol Dweck’s work on the impact of fostering a “growth mindset” has shown that when learners are given the message that greater effort leads to greater learning, they develop more productive behaviors.
We must promote this concept among teachers, administrators, parents, and students to let them know they can make sense of math! The Growth Mindset Maths website from Helen Hindle offers some concrete strategies to get folks started.
Students need learning environments that emphasize coherence, reasoning, sense-making, and the relationship among mathematics concepts and procedures. They need classrooms in which there is time and space to work individually, in small groups, and as a whole class, engaged in mathematical practices that support meaningful learning. They also need resources and tools to help them explore mathematical ideas.
New content standards for mathematics, including the Common Core, require that students learn with meaning and develop proficiency with practices (e.g., the Standards for Mathematical Practice; see here for how these are unpacked for students in K–5).
In the past, teachers generally presented math examples that students mimicked as best they could, with little or no discussion of how procedures worked, few opportunities to engage in non-routine problems requiring time and thought, and limited resources beyond a calculator with which to explore mathematics.
Teachers must be able to revisit their own mathematical knowledge while at the same time designing and implementing new learning activities for students. I wrote about my own experience addressing the changes as a high school math teacher in an article for Mathematics Teacher titled, “Constructing a Personal Understanding of Mathematics: Making the Pieces Fit.”
Teachers and administrators must recognize that students are not the only ones learning and that students’ input can help teachers deepen their own understanding of mathematics. And administrators are called upon to facilitate professional development driven by teacher input, organize schedules that promote collaboration, and model intellectual curiosity by increasing their own understanding of mathematics.
Q4: What makes a standard difficult to teach and difficult to learn?
The most difficult standards require students to develop new conceptual understandings or to recognize new relationships among mathematical ideas, tasks that become more difficult when new content is not well connected to prior knowledge or experiences.
For example, when learning about fractions students are faced with new and sometimes counterintuitive ideas about numbers. Attention must be given to building new knowledge in a coherent way where ideas they already know can be leveraged, extended, and refined. With fractions, this means building from student understanding of whole numbers and informal knowledge of the ideas of halves, thirds, and fourths. The Common Core standards do this by bridging from whole number units (ones, tens, etc.) to the idea of unit fractions and by using visual models of number lines and rectangular area diagrams to represent the relationship between fractions and whole numbers.
This blog is the first in a series from Curriculum Associates about the most challenging math standards that complements they’re recently released white paper – “Mastering the Most Challenging Math Standards with Rigorous Instruction“.
For more see:
- Spotlight on Math: Strategies for Addressing the Most Challenging Math Standards – Measurement
- Spotlight on Math: Strategies for Addressing the Most Challenging Math Standards – Modeling
- Spotlight on Math: Strategies for Addressing the Most Challenging Math Standards – Fractions
- Spotlight on Math: Strategies for Addressing the Most Challenging Math Standards – Statistics