I teach at a smallish school, and share an office with the entire 5th-12th math department. Of the many reasons why this is fantastic, one is that I have a much better sense of what our students are doing at different grade levels. Teaching one year of sixth grade (and one was enough) was immensely valuable for my teaching of older grades.
In the younger grades we use Singapore Math, and the last few years we have been emphasizing a distinctive of it called Model Drawing. We have an awesome 5th-6th grade math teacher who has really embraced this, and is taking it to the next level. She showed me this the other day.
AJ has $15 more dollars than RJ. Together they have $55. How much does each have?
This is how a sixth grader answered this:
In the sixth grade this student is solving equations, not by blind use of rules, but by reasoning from diagram. My older students ask why you have to subtract before you divide – this student is grasping intuitively that you must. And everyone in the class is answering this way.
So who wants a cookie?
Not that the end all of math education should be getting kids doing algebra, but model drawing is the single greatest tool I have seen for giving younger kids something to attach mathematical thoughts to. This is just a simple problem too, more for practicing the method. Wait until you see the kids doing ratios – way easier and more powerful.