Author Archives: David Pearce

Christian – Part 2

In my previous blog I introduced Christian. He had a prescribed method for solving addition tasks, but many times his answers were not accurate.

In our second session with Christian our primary goal was for him to use a counting strategy when adding two numbers. We began by presenting cards to him with the numerals 27 and 9 printed on them and asked him how much altogether.

He starts by saying “Let me use my brain,” then sits and stares at the numbers for 45 seconds, then says, “40. I was using my head.”

I turn the first card over revealing 27 worms printed on the back side and ask him to count them. After he’s done, I point to the number 9 and ask, “Now can you continue and count these also?”

Christian reaches out to turn it over.

“Can you count without turning it over?” I ask.

He then begins counting out loud: “28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,” and here he pauses.

“How do you know when to stop?” I ask.

He starts counting again, “40. [pauses] 41.”

“How do you know when to stop?”

He responds, “I count 9 in my head.”

“Okay. Do that for me out loud so I can hear you count the 9. So how many were here?”

Christian says, “27.”

“Okay, count these 9 out loud for me.”

He starts counting, “28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39.”

I ask him, “Is there any way you can count and know when to stop at 9?”

He sits for a few seconds looking up and then says, “I think I’ll use my fingers.”

He then proceeds to count and extend fingers sequentially, “28, 29, 30, 31, 32, 33, 34, 35, 36, 37.”

He stops, stares at his fingers, and realizes all 10 fingers are extended.

“So how many do you have here?” I point to the 9.

He replies, “9.”

“So how could you know with your fingers that you have counted 9?”

He looks at his fingers and pulls back one, leaving nine fingers extended.

Christian begins his count again while extending fingers sequentially, “27, 28, 29, 30, 31, 32, 33, 34, 35,” stopping with 9 fingers extended.

At this point he doesn’t look confident in his results.

I ask, “Do you want to try it again?”

He starts over, “27, 28, 29…(slowly) 35.”

I ask “Are you sure?, Do you want to try again?”

“One more time…28, 29, 30, 31, 32, 33, 34, 35, 36.” Now he looks very confused.

Again, “28, 29, 30, 31, 32, 33, 34, 35, 36.”

“So is it 35 or 36?”

He says, “I think it’s 36,” and counts the images of the worms, ending up with 36.

“Hey, it worked!”

It’s not an exaggeration to say he was surprised. I would say he began constructing a way to use his understanding of number to solve the problem. His journey continues in my next blog. What strategies do your students use to “know when to stop” when adding two numbers?


Christian – Part 1

The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years.—Common Core State Standards for Mathematics, p. 8 CSS.MP1 Make sense of problems and persevere… Continue Reading


In the Common Core State Standards for Math, counting-on is considered “a strategy for finding the number of objects in a group without having to count every member of the group.” Counting-on is an efficient way to add and we want children to count-on. Yet, many young children begin by counting-all. For example: Teacher [placing… Continue Reading

An Excellent Math Program

I recently attended the Annual Conference of the National Council of Teachers of Mathematics (NCTM) in San Antonio and came away invigorated and hopeful about our children’s future in math education. The creativity and passion on exhibit within the many sessions and workshops was impressive. I had numerous conversations with awesome teachers that eagerly shared… Continue Reading

Change Unknown

Recently, while working with students, we offered up a situation where nineteen counters were placed under a cloth.  Seven of the counters were pulled out and the students were asked how many remained under the cloth. One child extended ten fingers, pulled them back, and then re-extended nine. He pulled back seven fingers, one by… Continue Reading

From Borat to The Lone Ranger

The latest results from international testing in mathematics have been released and results show the United States finished behind Kazakhstan. Now I don’t know much about Kazakhstan, except this is the country Borat hails from. I’m sure Kazakhstan is a beautiful country, but it doesn’t jump out with me as a math powerhouse. What does… Continue Reading

The Way I Learned Math

“What’s wrong with the way I learned math?” Seriously, we all made it through school, a few of us even did well, and some even liked math. So, what’s wrong with the way math has been taught for years? The Common Core Standards have focused national attention again on math education. But the truth is… Continue Reading

Show Your Thinking

Recently I was speaking with a parent who was expressing frustration with their child’s work habits in their math class. It was a conversation I have experienced many times throughout my 20 years of teaching. It’s about that child who says they can do the math in their head and they do not need to… Continue Reading