Author Archives: David Pearce

What’s in a 10?

The number 10 is dominant in our mathematics because the numeration system used is base 10. The understanding and using of tens is essential when working with place value, addition, subtraction, multiplication, and many other areas. How do children think about and use 10 in their mathematics?

Baron

Baron, a 1st-grader, was asked to show three fingers. He slowly put up one finger at a time while whispering “1, 2, 3,” then tentatively extended his hand to show the teacher his three fingers. “Good job! Can you show me six fingers?” Once again he extended his fingers one at a time while quietly counting. When he said six, he looked at his fingers and then touched each one while counting again, “1, 2, 3, 4, 5, 6.” Finally, he extended his hands toward the teacher to show his fingers. “Nice! Can you show me ten fingers?” Baron quickly threw both hands in front of him with all his fingers flailing out saying, “10!.”

AJ

“1-2-3-4-5-6-7-8-9-10-20-30-40. What?”. AJ stopped and stared down at his blocks with a puzzled look on his face. He started counting again, “1-2-3-4-5-6-7-8-9-10-20-30.” He stopped a second time and said, “Wait.” Now he looked confused. Counting is something he has done a lot. He has counted beyond 100 multiple times and this time he had just 13 blocks sitting on the ground in front of him. It has been a few months since we have done anything like this, but this was unexpected. He has never counted by tens in previous attempts, so it surprised me, and it also seemed to surprise him. My guess is that his kindergarten class has been practicing counting by tens.

Reno

Reno, a 2nd-grader, excitedly asked me if I knew what 10 and 10 are, then quickly said, “It’s 20!” He did the same for 20 and 20 along with 50 and 50, eagerly letting me know the answer for each. His questions led right into a line of questioning I had planned for him. I place a card in front of him with the numeral 14 on it and then a second card with the numeral 10 and asked for the total. He sat for a while staring at the cards before beginning to count while extending fingers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30…” He continued counting until he reached 52 and realized he was lost in his counting.

Nelson

Another time while working in a 2nd-grade classroom, I asked a student to count out 23 snap cubes. Once he had done so, he was asked to connect cubes to makes stacks of 10. He made two stacks of 10 and had three loose cubes remaining that he pushed to the side. These were covered, and then he was told to count out ten more cubes and make another stack of 10 from them. Finally, he was asked how many cubes he has all together. He begins counting “23”, then extending his fingers sequentially counted “24-25-26-27-28-29-30-31-32-33” until he had ten fingers sticking up. “There’s 33!” Given the opportunity to check the total number of cubes, he lifted the cloth and pointed to each stack saying “10-20-30.” Then pointed to each of the individual cubes and said, “31-32-33.”

Keifer

Keifer is a 2nd-grade student with whom I was continuing to explore ideas of tens in math situations. On one occasion I placed two cards in front of him with the numerals 33 and 10. I then asked for the total. Keifer quickly responded, “43.” When asked how he knew, he told me that “there’s 30 here”, pointing to the card with 33 printed on it, and then he added “and a ten here, which makes 40.” He then counted three more saying, “41-42-43” while extending his fingers. I gave him 24 and 21 more. He began counting “24”, then sequentially extended his fingers and said, “25-26-27-28-29-30-31-32-33-34-35…45” until he had extended 21 fingers while counting.

In each of the stories, what is the child’s understanding of 10, and what does that mean for their mathematical understanding? How could we as teachers use their understanding to help guide their learning?

Rubik’s Cube Insanity

I remember getting a Rubik’s cube when I was younger. I sat down immediately scrambled it up, put in the Def Leppard cassette on my Walkman and set out to solve the puzzle. It soon became apparent to me that it was not going to easy or quick. I would play with it for a… Continue Reading

Mathematical Strategies: The Chicken or the Egg?

Which came first: the chicken or the egg? This is an age-old question based on the observation that all chickens hatch from eggs, and all chicken eggs are laid by chickens. The problem is it’s difficult to answer because it is not clear which of the two events is the cause and which is the… Continue Reading

Creating Centers in the Classroom – Part 4

This blog is the fourth part of a multi-part series titled “Creating Centers in the Classroom.” If you’ve missed the previous installments, you can read part 1 HERE, part 2 HERE, and part 3 HERE. In our continuing series of blog posts on creating math centers in the classroom, I began thinking about the idea… Continue Reading

Who’s Prior Knowledge?

“To teach mathematics well, you need to be an expert who is able to recover what it’s like to think like a novice” – Brent Davis, 2017 When students enter our classrooms, they come to us with a wide range of pre-existing knowledge, skills, beliefs, and attitudes, often referred to as prior knowledge. In the… Continue Reading

Down the Rabbit Hole

“The rabbit-hole went straight on like a tunnel for some way, and then dipped suddenly down, so suddenly that Alice had not a moment to think about stopping herself before she found herself falling down a very deep well.” ― Lewis Carroll, Alice’s Adventures in Wonderland Alice had it easy. In a recent colleague’s blog,… Continue Reading

Pointing Towards Finger Usage

“Sixteen!” My grandson shouts, sitting on the floor smiling, fingers extended. We had been playing with his toy cars. He counted out nine, gave them to me, and I put them behind me out of his sight. He then counted out seven more, gave them to me also, and I hid them under a nearby… Continue Reading

The Future’s So Bright, I Gotta Wear Shades

I am always impressed with the passion teachers have for their work. Education in general, and teachers specifically, have received a lot of negative press. The story line seems to be that the system is broken and failing our kids, so let’s throw those bums out. But if you listen to teachers talk about their… Continue Reading

I Didn’t Know What I Didn’t Know

I’m coming up on the two year anniversary at my current job and I’ve been reflecting on how my thinking has changed over this time. Before this job, I felt I had a good grasp on teaching math. I had just spent 20 years teaching junior high math. I faced down the torch and pitchfork… Continue Reading

Teaching Children Fractions

“Fractions. Ugh! I’ve never been good with fractions.” I can’t count how many times I’ve heard this statement. Every teacher knows that working with fractions is an area where many kids struggle. As a middle school teacher, I saw these struggles and how they can lead to further struggles in math. In fact, mastery of… Continue Reading