I feel for both these kids and their parents.

They are having problems with what teachers call number sense. The answer, according to the professionals, is number talks and number sense activities. They want to teach number sense.

The nice side of me says, "Sure. Right, of course, that is what you should do." And that is sort of what I would do with our young lady who doesn't get missing numbers, and that is sort of what I will do with my Algebra 1 high school student. 

The snarky side of me says:

How about we don't create these problems in the first place?

How about instead of fixing the issue with number sense activities, we teach in such a way that the student is able to discover how to break apart numbers and put them back together, shift symbols, and become expert manipulators?

What if we didn't ADD number sense activities to a mind-numbing, soul-sucking, boring curriculum and instead ALLOWED children to discover the fun and amazing world of quantity? 

What if we stopped teaching in a way that traumatizes our kids?

How about just teaching math efficiently? Good Stinking Grief.

You couldn't see me stabbing at the keys on my computer, but I that is what I was doing. 

What is number sense? According to Gersten & Chard, 2001, number sense basically means a student's “fluidity and flexibility with numbers." Students can break numbers apart, put them back together, they understand the relationships and the symbols and can use them flexibly and, I say, creatively. 

Caleb Gattegno calls this symbolic mastery. He doesn't give attention to understanding numbers. You will not see in Gattegno's textbooks the kind of work that ties the symbol "2" to two balls, two stars, or two dots, for which the student counts 1 then 2 touching each object. 

Numbers are not objects and treating numbers as though they are sets the student up to develop a faulty intuition about how numbers work, especially when it comes to fractions, which is where most kids get stuck. How does 3/4 of an apple divided by 3/4 of an apple give you one whole apple?

Rather, you will find an emphasis on supporting gains in symbolic mastery and allowing students to make what understanding they will, over time, of what numbers actually mean. Dick Tatha once summarized Gattegno's approach (this is a bit of a paraphrase, as I don't know where I read it) "teacher's look after the symbols, the understanding looks after itself."

Gattegno realized what is most important for mathematicians: each symbol's connection to other symbols, in other words - algebra. It is the student's ability to play with the symbols, including numerical symbols, that determines a student's success in mathematics. 

Math educator, Jo Boaler, explains what researchers have discovered about number sense in a recent video. Interestingly, Gattegno was preaching about this 60 years ago.

What Boaler is describing in the video is algebra: a general understanding of how the symbols work and how to manipulate numbers/symbols. What if there were a curriculum geared towards developing number sense instead of merely adding numbers sense activities on top of a traditional curriculum?

There is! Have I said lately that I love Caleb Gattegno? Thank goodness for Gattegno.

So what can we do to foster number sense in our children? Well, we can add number sense talks and numbers sense activities or we can allow our students to learn math in such a way that students become master manipulators of mathematical symbols. Oddly enough, students enjoy this more and it's much more fun for mom. 

You will note, in Gattegno, the repeated use of the same structures over and over again, as well as the same types of activities. Each time we approach a structure, we expand the possibilities for discussing that structure and how we write about it. 

If successful math students, and mathematicians, develop rich and varied connections between numerical symbols, doesn't it make sense to begin by showing students this way of working with symbols?

If you are using Gattegno, don't skip ahead to get to working with real numbers too quickly. Remember my high school algebra 1 student? Guess where we will begin work? Chapter 3 of book 1. Why? She is having trouble with substitutions; she doesn't understand what is happening. 

What do I notice? I notice that I have 4, 5, and 6 year old children who create more complex substitution problems than this young woman has encountered yet. And they think it is fun.