We believe that kids learn mathematics successfully when they can connect three things: their intuitive sense of a problem, fluency with numbers, and understanding the formal representation of numbers — when they can call upon *all* these ways of knowing fluently.

For example, learning to add fractions with unlike denominators is a lot easier (perhaps only possible) when kids first get the idea that they need to split the pieces into common sizes.

We have sixths and halves. If we try to add them, we can’t tell how big the result is…

Now we have sixths and sixths…

We can add them and count up the resulting pieces.

Most students can see how this model works pretty easily. But of course the model is only a beginning. Later, when they work with larger denominators, they need more skills and more ideas — ideas like factors and least common multiples. They need to build these skills — this number sense — through lots of experimentation and even more practice.

But we’re still not done with adding fractions. We throw further difficulties at students when we write the squiggles below. Helping students connect their intuitive understanding and their sense of how numbers work to these formal representations makes a final step towards understanding.

When the models, number sense, and understanding of formal representation are all in sync, we help prevent students from making common errors like this one:

In our tutoring, we think hard about all the places that kids can get confused and all the things they need to know. We help them develop an intuitive knowledge of those things — through pictures, manipulatives, thought-experiments, and many other strategies. At the same time, we help them practice the fundamental skills that support these models, and, finally, to make sense of symbolic representations. We think this approach works, and we think you will, too.