“Pure mathematics is, in its way, the poetry of logical ideas.” — Albert Einstein
The Space Race between the USSR and the United States in the mid-‘60’s gave rise to federal funding for enrichment of math and science education. The Cincinnati Public Schools libraries received impressive instructional materials for the teaching of both math and science.
Our librarian hosted a grand afternoon open house for the school’s staff to preview all the new materials available for us to check out and use in our classrooms. I had read about Cuisenaire rods for math so I selected a classroom set of several dozen bags for my second graders.
Cuisenaire rods are smooth, different from Base Ten blocks which are notched to show number. The rods are wonderful in the hand; silky and beautifully made with a Czech finish, inviting play with them.
The smallest rod, white, is a one centimeter cube. The red rod is two centimeters long. Squares offelt muffled the noise on desks.
With ample numbers of rods in each bag, a list of the exact quantities went in each bag to ensure maintaining the correct number of contents, remembering we’d borrowed the set from the library. My second graders were diligent about checking before and after to ensure the correct numbers, soon having memorized the list.
Students had time to visit several centers as their seatwork was finished but the rods table immediately became the most popular.
I decided to let discovery guide the investigation. Initially, students lined the rods up as stairs, ascending and descending; created pictures, mosaic-like; designed tiny room layouts with living rooms and bedrooms. It could be said that in this urban school, these children had few toys or blocks
at home. The “toy” aspect had a draw of its own. Math curriculum for second grade during that spring semester included introduction to multiplication.
Not surprising, one of the students quite early was using his rods to show multiplication. One day I noticed his classmates sharing their collections of rods with him. When I stopped by his desk, he was beaming. He had covered his desktop with felt squares showing endless combinations of factors. He needed many more ones rods and twos and threes rods to keep up with his findings.
Interested in his discovery, his classmates were adding their bags of rods and enjoying the show. Could he record his findings? Even though he had been dealing with dyslexia, using numbers had never been a problem for him. But, we couldn’t use math symbols for recording 4×3=12 so I quickly decided to create a new form: the numeral 4, a box around the 3, then 12. That way, he could choose to record his findings and use those rods for other factors.
He was finding the commutative property, too. Four 3s equal three 4s. Such additional vocabulary was for a much later time. His classmates quickly replicated his discovery and enjoyed using the new term: factor.
And a bit later, he motioned me over again. He said there were numbers that only had two factors: one and that number. I was so delighted with his work and now he had discovered prime numbers.
He’d had very limited success in school with his dyslexia. He was so proud, justifiably…and he immediately wondered how many prime numbers there were between one and one hundred. New idea and vocabulary: prime number. In an instant he was on the way to finding his answers.
I was in awe of the power of this simple discovery method and its draw on students’ curiosity, especially on that student’s great successes…and the students weren’t finished.
A few days later, a small group of students gathered around a desk. I was summoned over to see what they had found. There were several square shapes made with single color rods. Then they filled the center with ones rods. They were saying that 2×2 squares took four ones, 4×4 took 16 ones…just like the multiplication numbers. Because I had enjoyed the teacher’s manual, I was ready to extend their discovery by saying they had discovered the formula for finding the area of a square; and I identified some of its practical uses. A few students really liked using a formula: W x H = A to record findings, quickly moving on to record more examples.
Yes, someone in the group wondered if the sides were two different colors…and the formula worked for rectangle…and then one of them doubled up the sides and I was ready with the new word Volume and the formula for volume of a cube and cubic forms. We talked very briefly about area and volume, not wanting to compromise their present agenda.
These second graders, seven and eight years old, zipping into much higher mathematical understanding and usage via little bags of rods. Their knowledge of multiplication, driven by their curiosity, significantly beyond the regular second grade curriculum.
Increasingly, the rods were used in regular math lessons.
And the impact on my teaching? No formal course or experience in my own preparation for teaching
could come close.
Appealing materials and approaches, sparkling curiosity, their freedom to investigate, knowing when to stay out of their way, being available to celebrate and to add select information to support the new knowledge and understanding…and seeing all their classmates fascinated and propelled right along with them.
Yes, much to be learned from our students.
Judy Harris is well established in Northern Kentucky life, as a longtime elementary and university educator. A graduate of Thomas More, she began her career there in 1980 where she played a key role in teacher education and introduced students to national and international travel experiences. She has traveled and studied extensively abroad. She enjoys retirement yet stays in daily contact with university students.