Author Archives: David Pearce

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 effect. We are currently asking similar questions in math education. Do we 1) help children make sense of their math and then expect them to develop their own strategies OR do we 2) teach strategies and then expect children to make sense of their math?

Current standards in mathematics education grew out of the realization that children were learning mathematics through the memorization of algorithms, but were unable to generalize math concepts and apply them to increasingly sophisticated mathematics. This is often expressed as a lack of “number sense,” which leads to an inability to solve problems strategically. As children advance through their math education, an increasing number of them become frustrated with math and give up. As a child’s ability to make sense of the teacher’s behaviors diminishes, so does their ability to solve using those behaviors.

While developing the current standards, those involved in the process realized that people who do well in mathematics understand and use a variety of strategies. From this realization, the language of “strategy” was included in the Common Core Math Standards with the purpose of improving math education for all. Some specific examples of the strategies found in the first-grade math standards are counting-on, making ten, decomposing a number leading to a ten, and using the relationship between addition and subtraction (CCSS.Math.Content.1.OA.C.6). Although these strategies are written into the standards, the standards do not require teaching any specific instructional strategy. What is called for is that children learn multiple ways to solve problems, use strategic thinking and be creative in problem-solving.

Textbooks have lesson specifically teaching these strategies. Is this helping improve children’s mathematics?

I was recently sitting with a second-grade student while the teacher asked him how many total flowers were picked if one child picked 8 and another picked 4. The student gave answers of “16” and then “maybe 15.” I asked if he could use counting to determine the total. He suddenly said, “I could put the big number in my head (placing his hand on his forehead) and count on.” But he struggled in his attempt to use this strategy. He became confused as he tried to count 9, 10, 11, 12 and keep track of how many times he had counted beyond the 8. In a later conversation with the teacher, I was told that “He started by just throwing out numbers. He has zero number sense. He doesn’t know how to use the strategy, but at least he knows the strategy. He just needs to practice more to be able to use it.”

Can number sense be developed just by practicing a strategy?

What happens when a teacher intentionally puts a child into situations to develop number sense which creates the desire to stop counting all and begin counting on? In the following video a 1st-grade student, who has not been taught the counting on strategy, is given a series of increasingly large addition situations.

Notice that in the final problem he begins counting all but moves to the more efficient method of counting on. Does this make sense to him because he initiated the idea of counting on?

Which comes first in children’s mathematical development: making sense of the math and then using strategies or learning strategies and then making sense of the math by using them?


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

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

Takeaways from Subtraction (Part 3): The Struggle

Every elementary school teacher has seen children struggle with subtraction. From these struggles, attitudes of “I’m not good at math” emerge. Our team recently worked with students on the concept of subtraction. We presented situations in which students would count out 23 cubes, hide them, and then remove some of the cubes. The students were… Continue Reading

Takeaways from Subtraction part 2: To Take Away or Not to Take Away

In my last blog entry, I described three goals suggested by Dr. Les Steffe which support introducing subtraction as take away. Yet, there is a belief among some math teachers that thinking of subtraction as take away interferes with future mathematical development. They argue that using the words “take away” should be eliminated completely from… Continue Reading