Rewriting Equalities to Build Numeracy - Part 2

In our last post, we wrote about the work of Dr. Barbara Dougherty, the Director of the Curriculum Research and Development Group at the University of Hawaii. Dr. Dougherty's work emphasizes the importance of of instilling a solid sense of numeracy (being able to reason with numbers and other mathematical concepts) and operations (recognition of the relationships among addition, subtraction, multiplication, and division) to provide students with a true understanding of math.

One way to do this is to teach students to "Rewrite Equalities", a process that begins with understanding the meaning of = .

How do we do this?

Step 1: Establish an understanding of =

= does not represent “the answer” (as most kids think it does). It means that whatever is on one side has the same value as whatever is on the other side.

This seems simple, but it’s a big intellectual leap for kids. One of the biggest revelations will probably be the insight that it’s possible to have multiple numbers on each side of the =.

Try modeling this with manipulatives and sticky notes. Start with this:

  

Then show students that you can also do this:

Or this:

Or even this:

In all of the examples above, there are still a total of seven pennies on each side of the =. Challenge them to build their own models.

Step 2: Entry-Level Equalities

When it’s time to move from manipulatives to numbers, the teacher should demonstrate how to rewrite an equality in a few different ways. Begin by writing and explaining a series like this:

3 + 5 = 8

8 = 3 + 5

8 = 4 + 4

7 + 1 = 8

Now, invite students to take over with their own ideas. How many can they come up with? 

The student who remembers the commutative property doubles the number of expressions s/he can write, but don’t tell them this! Wait for someone to figure it out.

In my classroom, after they’ve worked for a while, students pick their two favorite expressions to write on the board and explain to the class.

Questions to Push Thinking:

• Can you rewrite the equality using a different operator, like a subtraction sign?
• Can you rewrite the equality by putting two or more numbers on each side of the equal sign?
• Can you rewrite the expression using decimals or fractions? What about negative numbers?

Step 3: More Advanced Equalities: Focusing on Operations

Challenge students to rewrite an equality using only the numbers in the inequality. They may use any operators and any format they’d like.

So a simple expression like this:

4 + 6 = 10

could be rewritten like this:

10 – 6 = 4

A more complex expression, like this:

3 x 4 x 2 = 24

could be rewritten like this:

2÷24 = 4 x 3

Students will notice that more complicated expressions can be rewritten in more ways.

I’m my classroom, we’ve learned that there is tremendous power in simple activities and procedures. I hope other educators find this activity to be as valuable as we have!

Rewriting Equalities to Build Numeracy - Part 1

Recently, I was fortunate to attend a two-day seminar on teaching math to struggling learners in the middle and upper grades, led by Dr. Barbara Dougherty. Dr. Dougherty, the Director of the Curriculum Research and Development Group at the University of Hawaii, is an insightful and innovative educator with a passion for helping all students develop of deep appreciation for math. If I could sum up the most important lesson I learned during my two days with Dr. Dougherty, it would be with a statement she repeated many times over the course of the conference: Pay now, or pay later.

Dr. Dougherty was referring to the time it takes to build numeracy skills and operational sense. These two critical components of math education are often given a mere nod in the classroom, if not overlooked completely by teachers scrambling to cram as many procedures into students’ memories as possible before exam time. This rushed approach, while understandable, is a mistake, according to Dougherty. A solid sense of numeracy (being able to reason with numbers and other mathematical concepts) and operations (recognition of the relationships among addition, subtraction, multiplication, and division) underlies a deep understanding of math. Happily, these competencies are available even to students who struggle with numbers and even those with dyscalculia. But learners need time and the right platforms for exploring the relationships between numbers.

There are lots of ways to set the stage for this kind of inquiry, but one of the simplest is an exercise in which students rewrite equalities. This powerful strategy is deceptively simple: give students an equation and ask them rewrite it, keeping both sides equal. So,

6 + 4 = 10

might be rewritten as

10 = 6 + 4

9 + 1 = 10

15 – 5 = 10

5 x 2 = 10

7 + 3 = 20 – 10

etc.

My fifth graders have loved this activity, and even those who will swear they hate math cheer when I announce that we’ll be warming up for class with a new number sentence to rewrite. There are lots of reasons to try this in a classroom setting or with an individual student. 

Rewriting equalities:

• It's fun! – My students, without any guidance from me, treat these challenges like riddles, and they are tickled when they come up with novel solutions.  
• Builds operation sense – Crunching numbers with a real goal in mind makes practice relevant and motivating. 
• Builds numeracy – With practice, students develop a sense for how to point the magnitude of a quantity in the direction they want; for example, they discover that most of the time, multiplying by a positive number will make a quantity much larger than adding by the same number. 
• Accommodates a wide range of skill levels – More advanced students can use fractions, roots, exponents, etc., while those still building basic skills can gain comfort with simpler operations like subtraction or multiplication. 
• Sets the stage for algebraic thinking – Many teachers take for granted that students who have only ever seen a mathematical expression with a single number to the right of the =, structured like this:  5 + 3 + 1 = 9 will be able to transition to algebra, when problems may have multiple numbers and variables!) to the right of the =, like this: 5 + 3 = 9 – x.

A misunderstanding of = is often behind the confusion because students haven’t been taught to think flexibly about an equality. (Check out Part 2 of this post for examples of how to implement this process in the classroom and for more about the “equals” sign.)

Specific Learning Disabilities in College

As we have written before, students who have graduated high school are no longer covered by the Individuals with Disabilities Education Act (IDEA), which comes to an end either upon graduation or when students who will not be graduating with a standard diploma "age out" of their eligibility for public education under their state's laws (usually age 21). Instead, students in college are protected by the Americans with Disabilities Act (ADA). The protections afforded by the ADA are broad, but there is one particular area where it differs from the IDEA that can make college entrance and college completion a problem for students who have specific areas of learning challenge.

The IDEA permits modifications to curriculum. So, for example, a student with a language disability may not be required to take a foreign language in high school, or a student with a disability of mathematics may not be required to take the same math courses as their classmates; the curriculum for each of these students can be modified in their area of specific learning disability. Note that these students may not be eligible for certain academic diplomas if their curriculum is significantly different than that of their classmates, but they are usually still eligible for a "local" diploma, one that is accepted by colleges, employers, and the military.

The ADA does not require colleges to make any modifications to curriculum. It requires accessibility, in both a physical and academic sense, but a student must be "otherwise qualified" for admission and graduation. What that means is that colleges can set standards for qualifications for students they admit, and they need not admit students who are lacking certain courses (staying with our example, this could be students who are lacking advanced math or foreign language courses), and even if such students are admitted, they may not be permitted to graduate without taking required courses or obtaining a waiver of such requirements from a college committee charged with granting such waivers.

This issue is of particular concern to some students in community colleges. These colleges are generally "open enrollment" in that they accept all high school graduates from their community. Because they do not rely on the stricter standards of selective colleges, they need a way to make sure students are prepared for college work. For students who have taken SAT or ACT exams, good scores in these exams are a way to demonstrate such preparation. Students who cannot submit strong scores on standardized tests, often because their learning disabilities impact their performance, must take placement tests to demonstrate their proficiency in areas such as reading, writing, and math. Those who cannot pass these tests are required to take remedial coursework and to pass such remedial courses in order to fully matriculate. Then, like all college students, they must also take the courses required for their major in order to graduate. Furthermore, students have to pay for remedial coursework just like they pay for regular courses, but do not receive credits towards their diplomas for these. And, often, they are limited in how many times they can take a particular course -- a "three strikes and you're out" approach.

The issue requires a balancing between the legal right of a college to set standards for admission and graduation and the rights of students with specific learning or other disabilities to be able to access a college education. There is no easy answer and it raises some fundamental questions that we expect courts to deal with at some point in the future.

Estimation Skills and Dyscalculia

Just about everyone is familiar with dyslexia, a learning disability that can make reading a difficult process because of an inherent difficulty in connecting sounds to symbols like letters. Fewer people, however, have heard of dyscalculia, a learning disability specific to mathematics. Like dyslexia, true dyscalculia affects only a small portion of the population (estimates range from 5-8%), yet it poses a tremendous difficulty to those affected by it.

Studies indicate that children who demonstrate difficulty differentiating between different sounds during their early years of life are at risk for dyslexia. Similarly, preschool-aged children who demonstrate difficulty with estimation tasks may be showing early signs of dyscalculia. Several recent studies have found that young children’s innate estimation capabilities are predictive of their later performance in math. Kennedy Krieger Institute research scientist Michele Mazzocco, Ph.D. says that children with dyscalculia lack “number sense,” an instinct for quantity that cannot be simply memorized.

Emerging research in this field was considered as we redesigned some of the assessment tools we use at The Yellin Center to evaluate children’s math capabilities. For students who demonstrate difficulty in math, we now administer a brief, proprietary dot estimation task in which children are given brief exposures to cards with varying numbers of dots, then directed to complete different estimation tasks. The results are often illuminating, and we appreciate the increased insight we can gain by better assessing our young students' estimation skills.

Image used under Creative Commons by Shawn Campbell