Math
Deep Dive
How are our students doing?
The percentage of highachieving math students in the United States  and most of its individual states  continue to rank around the middle of the pack, and shockingly below those of many of the world’s leading industrialized nations. The most recent results (from 2015) place the U.S. at an unimpressive 38th out of 71 countries in math. Among the 35 members of the Organization for Economic Cooperation and Development, the U.S. ranked 30th [www.pewresearch.org].
Looking at scores across the United States, Calfiornia does not fare well. The most recent scores from the National Center for Educational Statistics indicate that in 2019, the average score of fourthgrade students in California was 235 or 5 points lower than the average score nationwide.
The percentage of students in California who performed at or above the NAEP Proficient level was 34 percent and above the Basic Level was 75 percent. What is more concerning is the continuing achievement gap. Black students perform on average 26 points lower and Hispanic students perform on average 25 points lower than White students. Student’s from low socioeconomic backgrounds score on average 28 points lower.
Why is this so important?
The United States has always prided itself (and our economic wellbeing depends) on a welleducated workforce. In the early part of the 20th Century, the large investments made in education (particularly secondary and higher education) resulted in unparalleled growth which made us a dominant economic power and leading innovator across the globe. Our long, slow slide into mediocrity when it comes to the performance of our students, at the time when technological innovations continue to intensify, is certainly cause for alarm. While we have stagnated, other countries have advanced rapidly. [For an indepth look at this problem, check out this great read from the Harvard Program on Education Policy and Governance “U.S. Math Performance in Global Perspective."] This policy brief strongly recommends the following three important targets:
bring ALL students up to a minimum level of achievement
support high performers in becoming our next generation of highly qualified scientists & engineers
lift MORE students to higher levels of educational achievement
But it’s more than just competing in a 21st Century, global economy. Building mathematical skills has life long implications for students, particularly students with disabilities, that can be easily overlooked if our focus is solely on creating the next generation of scientists and engineers.
Math Makes Your Life Add Up!
Basic life skills, such as paying bills, balancing a checkbook, creating budgets, and arriving at work on time can be the “make it or break it” point for a student to move out of the house and live independently.
More advanced skills may determine the type and pay of a student’s future employment. Skills such as measuring in the construction trade, estimating the amount needed in inventories, budgeting expenses, and reading charts and graphs all require math skills. As teachers, we need to be helping our students prepare marketable skills for an increasingly technical workforce across the board. [Summarized from the chapter on “Assessing Student’s Needs for AT: Assistive Technology for Mathematics”, the Wisconsin Assistive Technology Initiation.]
Things we know about our students who struggle: [from PBS  Misunderstood Minds: Difficulties with Mathematics]

At nine years old, students with math disabilities have, on average, a 1st grade level of math knowledge.

At seventeen years old, students with math disabilities have, on average, a 5th grade level of math knowledge.

Experts estimate that for every two years of school, children with math disabilities acquire about one year of mathematical proficiency.
Let's dive deeper...
Children with math disabilities often reach a learning plateau in seventh grade, and acquire only one year’s worth of mathematical proficiency in grades seven through twelve.

Incomplete mastery of number facts: Recalling these facts efficiently is critical because it allows a student to approach more advanced mathematical thinking without being bogged down by simple calculations.

Computational weakness: Many students, despite a good understanding of mathematical concepts, are inconsistent at computing. They make errors because they misread signs or carry numbers incorrectly, or may not write numerals clearly enough or in the correct column. These students often struggle, especially in primary school, where basic computation and "right answers" are stressed. Often they end up in remedial classes, even though they might have a high level of potential for higherlevel mathematical thinking.

Difficulty transferring knowledge: One fairly common difficulty experienced by people with math problems is the inability to easily connect the abstract or conceptual aspects of math with reality. Understanding what symbols represent in the physical world is important to how well and how easily a child will remember a concept.

Making connections: Some students have difficulty making meaningful connections within and across mathematical experiences. For instance, a student may not readily comprehend the relation between numbers and the quantities they represent. If this kind of connection is not made, math skills may be not anchored in any meaningful or relevant manner. This makes them harder to recall and apply in new situations.

Incomplete understanding of the language of math: For some students, a math disability is driven by problems with language. These children may also experience difficulty with reading, writing, and speaking. In math, however, their language problem is confounded by the inherently difficult terminology, some of which they hear nowhere outside of the math classroom. These students have difficulty understanding written or verbal directions or explanations, and find word problems especially difficult to translate.

Difficulty comprehending the visual and spatial aspects and perceptual difficulties: A far less common problem  and probably the most severe  is the inability to effectively visualize math concepts. Students who have this problem may be unable to judge the relative size among three dissimilar objects. This disorder has obvious disadvantages, as it requires that a student rely almost entirely on rote memorization of verbal or written descriptions of math concepts that most people take for granted. Some mathematical problems also require students to combine higherorder cognition with perceptual skills, for instance, to determine what shape will result when a complex 3D figure is rotated.
What can stand in the way of a student’s math development? Research that went in to the PBS documentary “Misunderstood Minds” identified the following major skill set areas that can be impacted in a student struggling with math:
Explore the red flags and signs of math difficulties from the PBS “Misunderstood Minds  Math Difficulties” summary.
It is important to consider that for some students, challenges with math go beyond poor instruction, or math anxiety, but are based on neurologicallybased learning differences. This is often referred to as “dyscalculia”.
So what do we need to do differently?
“We need to stop expecting every student to learn the same way at the same speed. We need to go more in depth and show how different subjects of math are related and used in real life...we have cars that can drive themselves and phones that recognize our voice and facial structure  surely we can teach students math.” [Elle Venezky, “Why U.S. Students are Bad at Math”]
What good math teaching looks like...
Take a look at these change ideas proposed by Chris Ozarka in “Teaching Math in the 21st Century: Changing the Focus from Calculations to Critical Thinking."
Differentiate (options!)
Provide a collaborative environment
Be inspiring and informative with feedback
Make sure students are making mistakes along the way
Ditch the inclass lecture and focus on selfpaced, collaborative learning
Focus on students actively creating their own understanding collaboratively
Inform students on “ how learning works” (our goal is to develop “expert learners”)
Provide clear instructions (what is the goal? what are your options for getting there?)
Jo Boaler, a Professor of Mathematics Education at Stanford University, describes two ways we can engage students in learning mathematics...
We can show students methods and then they repeat them. This approach is used in most schools, but the methods often lack meaning, and students reasonably ask: when are we going to use this? Additionally, students only ever get to use what they were shown, not select a method themselves, one of the most important mathematical acts.
We can engage students in rich, open, visual and creative tasks. They use their intuition and thinking, and choose methods that can be useful in the task. When they need to learn new methods, teachers teach them inside the task. Students can see how important they are and learn more deeply. They engage in the important acts of choosing and making connections between ideas.
OR
Watch Dr. Boaler contrasting these two instructional paradigms in the video below.
Consider Universal Design for Learning as a framework that can help teachers intentionally make critical instructional design changes. Based on the work of Neil Albero, CAST Implementation Specialist, designing mathematical instruction with UDL in mind can be scaffolded with some really good questions.
Here are some questions Neil uses to help teachers reflect on their mathematical instructional design:
How about UDL?
How do I ...

design for relevance?

design for connecting representations?

design for student discourse?

design for math progressions in a lesson?

design for rigor?
Here are some questions Neil uses to help teachers reflect on their UDL instructional design:
How do I...

design the environment?

design for goals to be posted, shared, and reflected on?

design for learners to reflect on progress and selfassess their understanding to inform their next steps?

design for choice to recruit learners’ interest and help learners sustain their effort and persistence?

design collaboration so learners share mathematical thinking?

design for tools and resources to be leveraged so learners use them strategically?

design for learners to monitor their own progress through the process of learning?

design multiple representations for students to learn new information?

design flexible assessments so students can demonstrate understanding?
How do I...

promote high expectations for all and the belief that all are capable of reaching those expectations?
How do learners...

seek and receive feedback during the process of learning to make adjustments to their thinking?
Strategies & Tools
for Diverse Learners
We have organized the tools and strategies in our Open Access AT FlipKit based on the types of supports a teacher might be looking for to support students who struggle with math. The flipKit includes tools and strategies to support legibility, building concepts, no/low tech tools, and tools built into devices.
The following are some additional resources that you can explore to go deeper in making the connection between UDL and strategies that can support the designing of math instruction from a UDL lens:
In the book “Universal Design for Learning in the Classroom: Practical Applications” (Hall, Meyer and Rose, 2012), the authors connect the FIVE ATTRIBUTES associated with proficiency in mathematics (for the National Research Council  “Adding It Up”), and took them a step further by by relating them to the UDL networks for learning (affective, recognition, engagement), identifying specific barriers to math learning and potential UDL supports or strategies.
In the book “What Really Works with Universal Design for Learning” (Murawski and Scott, 2019), the authors identify a variety of researchbased practices, within a UDL framework that includes multiple modes of presentation, expression, and engagement, that can be used to teach math effectively.
In the book “Teaching in Today’s Inclusive Classrooms: A Universal Design for Learning Approach” (Gargiulo and Metcalf, 2017), the authors dedicate a chapter to Developing an Understanding of Mathematics in All Learners, with numerous examples of evidencebased methods, strategies, materials and resources that can be used in constructing universally desgined lesson plans for mathematics.