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'Same' and 'Equal' Are Not Congruent Terms

Our respect for equality has a long and storied history in America. From the very beginnings of our nation, we declared the concept central to our principles with the bold-for-its-time proclamation that

2013-08-24 all men are created equal.jpgIn mathematics, the term "equal" is used when things are "equal in value," and being equal in value is what was intended with those daring strokes of 1776. It meant that the man sweeping the floor and the merchant running the business were of equal value - equal in the eyes of the law, equal in their value as humans, equal in their right and opportunity to pursue their happiness. This radical-for-its-time idea had essentially never before been implemented, and the "grand experiment" that followed has wrought massive struggles and remarkable successes. Some of those struggles have found us grappling with (what we thought at the time were) big questions. Are black men equal, too? What about women? Of course they are. "Men" means all of mankind, not just the ones who are actual [white] men. Clarifying this point took a lot out of us, but we reached a just conclusion: that each of us is equal in our value as a being.

(Frustratingly, in many ways and in many places across our globe, each one of us being of equal value is still a bold statement even today...)

Yet in spite of our reverence for all humans being of equal value, I think we are terribly confused about what that actually means in practice, notably in the world of Education.

Broadly speaking, teachers desire to treat their students equally. It is a core principle we've all grown up on; it is ingrained in our way of thinking. But in practice, this comes out as essentially treating all students the same, i.e. identically, whether justified or not, whether necessary or not, whether appropriate or not. Everyone gets the same number of minutes of reading instruction utilizing the same book (whether they need it or not, or whether they are ready for it or not). Everyone is taught their times tables the same month (whether they are ready for them or not, or whether they had already mastered them or not). Everyone is taught the letter "A" the same week, even the ones who figured it out on their own two years before.

But treating all learners as the same does not acknowledge their equal value as learners. "Same" and "equal" are not congruent terms.

Each student is of equal value, therefore each is worthy of an education, of learning, growing, being educated. The learner's value as a learner lies in his or her right to an opportunity for and fulfillment of the pursuit of an education.

But one does not become educated through an ill-fitting education. One does not become educated when nothing new is taught to him.

In the following photos, all of the items in each image are equal. But they are not the same. They are equal in value (i.e. area, in this case), but they are expressed in ways that are not the same as each other.
2013-08-24 equal but not the same 1.jpgThe above two items are equal in value to one another, but they are not the same.


2013-08-24 equal but not the same 2.jpgThe above two items are equal in value to one another, but they are not the same.


2013-08-24 equal but not the same 3.jpgThe above three items are equal in value to one another, but they are not the same.


And the concept holds for our students as well:
2013-08-24 equal but not the same 4.jpgThey are equal in value, but not the same.


Treating kids "the same" and treating kids "as equals" are two totally different things. Treating students the same means giving them identical amounts of instruction, identical lessons, identical learning materials, an identical education. Treating students as equals means acknowledging each one has equal value as a learner, which in turn means giving them each what they need to fulfill their value as a learner. They are not the same! They do not all need the same things. But they are equal. They are all equal in their right to pursue becoming educated, their right to receive an education, their right to fulfill their educable abilities. They are equal in their right to and need for tailored opportunities to become what they can become. Because they are equal in value, we should show each of them dignity as a learner. And showing a child dignity as a learner means providing the opportunities that that learner needs and is ready for.

Denying some students what they need as learners simply because they are not the same as other learners robs them of their inherent state of being equal by ignoring their value as a learner. Denying a learner an opportunity to actually learn disregards their dignity and value as a learner.

These concepts are even imbedded in our standards, and always have been imbedded in what we teach. The latest version, the Common Core, notes it in the following ways (among others):
* CCSS.Math.Content.4.NF.A.1 "Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions."
* CCSS.Math.Content.1.OA.D.7 "Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2."
* CCSS.Math.Content.6.EE.A.4 "Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for."

We need to practice what we teach!

Students who have already mastered concepts and content shouldn't have to sit through further lessons on those but rather they need the next level so they can continue to learn and grow (as we should want ALL students learning and growing). In American schools, we seem to think treating all students equally means treating all students the same way or treating them as if all students were the same. On one hand we celebrate differences (i.e. "diversity") but on the other we ignore (even deny!) differences (i.e. intellectual diversity) because it is neither comfortable nor convenient to do what is necessary to accommodate those differences. The mathematical concepts of "same" vs. "equal" help to explain our innately-human differences in a new way. We are all equal (e.g. in the eyes of law, as created, in our humanness), but we are not all the same. Equal opportunity does not mean the same thing for every kid. Rather, as Mary Slade says, "Every student deserves an equal opportunity to struggle." And what is a struggle for one is not necessarily a struggle for another, and vice versa.

NOT giving a learner what he or she needs as a learner results in inequality of opportunity to become educated to one's full potential. NOT giving a learner what he or she needs as a learner sends that child the message that "I don't value you as a learner as much as I value everyone else as learners." What the child hears is, "You have less value as a learner to me, so I'm going to choose to not put as much effort into stretching you, I'm not going to provide you the challenge you actually need, I'm not going to educate you to your full educable potential." Yes. It's true. When we don't give our more advanced learners what they need as learners, the message they interpret and receive is that they are of less value as learners. Don't believe me? Ask the brightest students in your class or school if they feel equally valued as learners. If they have been getting what they need as learners, they will say, "yes." But if they haven't been getting what they need as learners, they will say, "no."

As teachers, we should be saying and showing to all students: "I value each of you enough to give each of you what you need as a learner."

Every single child deserves to hear, "You have equal value as a learner. You have equal value as a being becoming. Therefore, I will give you what you need to do just that."


(All photos by Tamara Fisher.)

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