Planting the Seeds of Algebra, PreK–2

Explorations for the Early Grades

Monica M. Neagoy

• Publisher: Corwin
• Year: 2012
• ISBN:
  9781452283630
• Edition: 1
• Topic: Éducation
• Subtopic: Enseignement des mathématiques

Help young minds explore algebraic concepts

This book shows teachers how to create a strong foundation in algebra for very young children. Using in-depth math “explorations,” the author unpacks—step by step—the hidden connections to higher algebra. Each exploration contains an elegantly simple grade-banded lesson (on addition, subtraction, patterns, and odd and even numbers), followed by a discussion of the lesson's algebra connections, as well as suggestions for additional problems to explore. Throughout, readers will find:

• Clear explanations of algebraic connections
• Specific strategies for teaching the key ideas of algebra
• Lesson modifications for older or younger students
• An array of age-appropriate problems and games

1 Making 7—The Lesson: (Grades PreK-K)

For much of the 20th century, learning single-digit addition and subtraction has been synonymous with memorizing math facts. The predominant learning theory supporting this view was stimulus-response: A card with “3 + 4” flashed before children's eyes was the stimulus, and the memorized, isolated answer of “7” was the expected response. Despite all we now know about how the brain learns mathematics (Dehaene, 1997; Sousa, 2008), in many schools, memorizing mindless math facts is still stressed by some parents … and teachers.

Making 7 explores alternatives to rote memorization and fosters sense making of single-digit addition combinations. The lesson described below took place in the classroom of Ms. Brady, an extraordinary kindergarten teacher. Ms. Brady had split up her class into two groups: Group 1 was her “regular students,” Group 2 her “high flyers.” I worked with each group separately using different explorations into the same mathematics. The lessons were filmed and discussions transcribed.

Box 1.1 Reflect …

Suppose a parent told you, “My daughter Ping already knows her addition facts for making 10. I'm afraid she's not being challenged in math.” What would you devise for Ping and other students like her?

Group 1

Setup

Ms. Brady's ever-growing love for and knowledge of mathematics were contagious. The children's excitement was palpable. Seated in a circle on the colorful learning rug, they were eagerly waiting for Dr. Monica. The manipulatives requested—two large dice, unifix cubes sorted by color, and a white board—were ready, as was a beautiful drawing by Ms. Brady on a magnetic green board. A large, leafless tree was drawn on the right of the board and a small dense bush on the left. The artwork had intrigued the children since their arrival that morning, but as requested, their teacher had kept its purpose a secret. I like adding suspense and theatrics to my math classes. It comes from my theatre experience.

Dr. M: You know I'm a mathemagician, right?

Some (politely): Yes.

Dr. M: Would you like me to start our lesson this morning with a trick from my MathMagic show?

All (excitedly): Yes!/Yeah!

Dr. M (picking up the large dice, one in each hand): Before I perform my trick, I have a question for you: What are these?

Some kids called out “cubes”; others said “dice.” I addressed both:

Dr. M: Yes, a cube is the name of their shape. But dice is also correct. Why?

Chloe: Because they have numbers on them.

Dr. M: Absolutely! How many different numbers do you think are on these dice?

Many hands went up. Answers included a couple of fours, some fives, but mostly sixes, so we went with six.

Dr. M: So can anyone tell me what the six numbers are?

Silence fell upon the classroom. My question was too ambitious. So I modified it:

Dr. M: What's the smallest number?

Many children (in unison): One!

Dr. M: What's the biggest number?

Many children (again in unison): Six!

Holding up just one die and rotating it to reveal the cube faces in random order, I asked the students to call out the number of dots I was pointing to. They correctly identified the dot numbers in random order, then in ascending order calling out, “one, two, three, four, five, six.” Revisiting their first answer, “cubes,” I asked one more question, nudging them toward the one-to-one correspondence between the six dot numbers on a die and the six faces of a cube:

Dr. M (placing my entire palm against one face): This is called a face of the cube. Some of you called it a side. How many faces, or flat “sides,” are on this cube?

Once again, silence. So I placed the red die in the middle of the floor for all to observe, as I retreated from the center to take a seat among the children, in the circle.

Jacob: Six.

Tyler: No, five!

Elizabeth: It's six because there's one on the bottom.

One girl said “four.” Keeping a neutral face so the students would persevere, I broke down the number of faces into three parts, calling them top, sides, and bottom. Students successfully called out the corresponding numbers for these three subgroups, namely, 1 (for the top), 4 (for the lateral sides), and 1 (for the bottom). We then added them together in two stages as I simultaneously gestured to indicate each addend of the sum: “How much is one plus four?” I asked. “Five.” “How much is five plus one?” “Six.”

Then, before I could utter my closing question, an amazing thing happened: Juliana leaped from her spot to the center of the circle, grabbed the cube, and said, “I know! There's like … um … six sides on it and each side has a number. So there's six numbers!” This was a significant insight into the one-to-one correspondence I was probing into. Notice Juliana's answer: She had connected the number of “sides” to the number of dot-numbers, so much so that she actually answered “So there's six numbers” to my question about “How many faces?” Thrilled, I exclaimed, “You just got what I was hoping you'd figure out!” To reinforce this connection that only a few students had grasped, we concluded by singing, “Six numbers for six sides, six sides for six numbers.”

Discussion

1. The Math-Magic Trick

We were all ready for the magic trick. I rolled the two large dice into the middle of the circle and had students identify the dot numbers “on top” (Figure 1.1). Five for the red die and four for the black one they said, accurately. Mesmerizing the class with a trembling gravity in my voice, I intoned, “Abracadabra, using my mathemagician eyes, I'm looking through these dice and I see that under the four, there is a three; and under the five, there is a two.”

Figure 1.1 Math-Magic Trick: Seeing through the Dice

A girl's voice (breaking the silence): On the bottom?

Dr. M: (clarifying): Yes, on the bottom of the black die there's a three (I picked up the black die to reveal the 3), and on the bottom of the red die there's a two (I picked up the red die to reveal the 2).

Noah: I think you remembered them.

Emma: I don't really think you're looking through the cubes … I think … I think that …

Sadie (interrupting, and pointing to the 4): … that you see around the cube that there's not that number.

Emma (pursuing her own idea, also pointing to the 4): So you know it's a three on the bottom because it's the opposite of four … kinda.

Sadie (insisting on theory, and clarifying): No, you look around it [the 4] and there's no three around, so you know on the bottom there's a three.

Emma was onto something. I encouraged her to think further about her theory of “opposites.” But to the class, Sadie's theory was convincing: When asked how I knew there was a 2 under the red 5, the class had adopted her idea. I acknowledged that looking around to find the missing number was a nice explanation, but that there was more math to my magic. To convince them, I turned my back and assigned two quieter students to roll the dice and then call out the numbers on top. Without looking, I had to identify the numbers on the bottom. They obliged me. Ethan called out a “one” for the red die and Abigail a “three” for the black ...