December 3, 2025Cognitive demands of mathematical tasks, higher-order thinking (HOT), Neurodevelopmental Functionsby Math for All

Math for All and Supporting Students’ Higher-Order Thinking

Over the past few decades, researchers, educators, and policymakers have agreed upon the importance of fostering students’ higher-order thinking (HOT) in mathematics, which has led to reforming mathematics curriculum, instruction, assessments, and teacher training (National Council of Teachers of Mathematics, 2000; National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). Developing higher-order thinking is essential for students’ success in learning mathematics. Math For All employs a neurodevelopmental framework (NDF) to analyze the cognitive demands of mathematical tasks. This framework identifies eight key areas of cognition that can come into play when learning: language, sequential ordering, spatial ordering, motor functions, psychosocial functions, attention, memory, and higher-order thinking.

Engaging all Students: Rich Tasks and a Strengths-Based Approach

Consider the following 6th-grade task from Illustrative Mathematics. Take a few moments to solve it using any method you like.

Jessica gets her favorite shade of purple paint by mixing 2 cups of blue paint with 3 cups of red paint. How many cups of blue and red paint does Jessica need to make 20 cups of her favorite purple paint?

Did you read the problem and hear yourself mutter “Hmmm…”? Were there multiple options to consider for working out a solution? Did you find yourself mulling over why one strategy was stronger than another? Could you solve this question quickly or did you need to take your time? Did you arrive at an answer?

Manipulatives: Not Just for Kids, Not Just for Strugglers

By 1953, scientists knew much about DNA, including its chemical components and how they related to each other. There was even X-ray crystallography of the molecule. However, the structure of DNA in three-dimensional space remained a mystery. James Watson and Francis Crick experimented with building wooden models, creating several until they found the one that fit all of what was already known. This work, which relied heavily on the use of physical models, earned them and their colleague, Maurice Wilkins, a Nobel Prize. Their discovery of the physical structure of the DNA molecule was a breakthrough that lead to massive advances in the understanding of genetics and in medicine.

AI: A Coaching Thought Partner

My husband is a data scientist, so I’ve been hearing about AI and ChatGPT a lot over the past year. When I listened to him share his experiences using this generative AI tool, I—like many other educators—wasn’t sure how we should be using it. Educator and researcher Donna Shrum was encouraged to try out AI in her teaching practice by her colleagues. She wrote about her positive experience using it to give immediate feedback during a writing workshop in an opinion piece in Education Week (Ferlazzo, 2025).

Strategies for Math Intervention Can Benefit All Students

Every math classroom is filled with students who have different strengths, experiences, and learning needs. While research-based math intervention strategies are designed to support students who may struggle, these same practices can also strengthen learning for all students.

This blog post explores two powerful practices that can help students deepen their understanding of mathematical concepts: building mathematical language and using models and manipulatives (also known as representations). Language and representation might seem distinct at first, but in reality, they are constantly interacting. When students are explaining their ideas aloud or showing their thinking by using models, language and representations work together to help students make sense of math.

Supporting Students with Limited or Interrupted Formal Education

Years ago, I worked in classrooms and in after-school settings with refugee youth at a school that was established specifically for students newly arrived in the U.S. The teachers who founded the school in the Bronx promoted literacy as a means to help students succeed in their world. They were skilled and reflective practitioners. They designed project-based curricula that allowed students to draw on their personal backgrounds, and understood how to integrate language and content in ways that were engaging for adolescents. Yet as rich and thoughtful as the curriculum was, a subset of students challenged their competence. These Multilingual Learners (MLLs) were designated as Students with Limited or Interrupted Formal Education (SLIFE).

Fostering Spatial Thinking in Young Children

Spatial thinking—the ability to visualize, manipulate, and reason about objects in space—is essential for success in mathematics and other subjects such as science, engineering, and technology. Spatial thinking plays a role not only in learning about basic shapes, but it is also important in higher mathematics such as geometry and algebra. For instance, spatial thinking can help students to visualize word problems (e.g., using arrays to think about multiplication problems), picture equations and graphs, and to solve problems related to area, volume, and angles. In today’s world, where more time is spent indoors with increased passive screen time and limited opportunities for exploration and play, spatial skills are often underdeveloped in young children. However, research shows that young children can improve their spatial thinking by working with teachers (Casey et al., 2008) and parents (Borriello & Liben, 2018; Hall et al., 2023; Verdine et al., 2019).

Mathematical Discourse & Our Students

[T]here is frequently more to be learned from the unexpected questions of a child than the discourses of men, who talk in a road, according to the notions they have borrowed and the prejudices of their education.

—John Locke, Some Thoughts Concerning Education, 1693

When John Locke wrote these words at the close of the seventeenth century, the world was in the midst of the Enlightenment and change was in the air. Mary and Edward Clarke, Locke’s friends and fellow aristocrats, began seeking his advice on educating their eldest son, who was not having much success with what was then considered a typical education for boys of his class. Locke’s advice was long and specific, but he elevated virtue, a love of learning, and practicality above all. He warned the Clarkes that any tutor they found for their son should “not so much to teach him all that is knowable, as to raise in him a love and esteem of knowledge.” Locke also spoke of a child’s curiosity and how to “keep it active and vigorous” through acknowledging and answering their questions and taking seriously that which interests them. By the end of the eighteenth century, Locke’s influence on educational theory was well known and well regarded.

Putting Students at the Center: Helping Teachers Make Decisions About Instructional Strategies to Ensure the Success of Diverse Learners in Math

As mathematics classrooms are becoming more and more diverse, teachers aiming to help all of their students learn are increasingly challenged. Pandemic-related learning gaps, the arrival of newcomers who may have experienced interrupted schooling, and the increase in the number of students with disabilities in general education classrooms have intensified performance differences among students. It is widely recognized that teachers require help and support to meet the varied needs of the different students they serve. There is less certainty, however, about the kind of support that will help teachers to make this happen.

Author: Math for All