Number fluency, problem solving skills.
When parents, teachers, administrators, or other concerned citizens ask what any random elementary-aged student needs to work on in math, a good nine times out of ten the answer is … number fluency and problem solving skills.
Number fluency is generally the more familiar territory to navigate. It entails practicing and understanding basic math facts, fact families, number combinations, skip counting, doubling and halving, and so on as you move through arithmetic. We almost always teach and learn number fluency — which is the speed and ease of building, breaking, and sharing quantities — with very concrete examples. Blocks. Beans. Buttons. Bears. Bricks.
Problem solving is another story. We sometimes begin with concrete materials, but we quickly move on assuming mastery of number fluency to more abstract ideas where pure numeracy meets the intricacies of language. This leap is where many students get left behind. Our job as teachers is to equip students with a diversity of tools that will enable them to journey into those unfamiliar waters.
This 2009 US Department of Education study by The Institute of Education Services (IES) makes eight research-based recommendations for struggling elementary aged students in mathematics instruction. Two of the recommendations, showing evidence of strong and moderate success respectively, were teaching the common underlying structures in problem solving and using visual representations of mathematical ideas — including strip diagrams, or bar models.
This video gives a brief (ahem) overview of using bar models. It’s not a silver bullet and it’s not for everyone and it’s not for every problem, but it’s a concrete jumping-off point for students to organize their information and reason their way through a problem. And like most math skills, it can’t be shown on the board on Monday and tested on Friday. It needs consistent modeling and implementation by teachers and students, until it becomes a skill students naturally conceptualize in their head and have no need of drawing it all out! It’s easy to forget as an adult how hard these abstract ideas can be when you are first learning them.
Anyway, here it is. I need to augment this video with a newer version that talks about comparing problems a bit more, in addition to changing problems. But the basics are all there. Have fun!