I recently reached out to Willem Uittenbogaard. Willem was one of the original collaborators between Math in the City (founded by Cathy Fosnot) and the Freudenthal Institute in the Netherlands. He spent two years in New York City — working with teachers to develop the idea that realistic contexts in mathematics problems help children to build on their understanding of the world. He also taught many New York City teachers how to lead number strings. I was one of those teachers. I was lucky enough to be spend two weeks of the summer of 1999 with Willem, as he challenged me to solve mathematics mentally through number strings. Willem went on to co-author all of the Minilessons Resource books for the Contexts for Learning Mathematics series.
In his response to us, Willem wrote a bit about his thinking about strings. Writing number strings, he explained, was best understood not as following a recipe, but as maatwerk, a Dutch word meaning work that has to be done in that moment, or custom made. That is such an important part of why we wanted to highlight the work going on in classrooms on this site; no book will ever have all the number strings that are needed in those moments, custom made by teachers, with their own students in mind.
He also wrote:
Let us take the problem 71 – 69
Most of the American students see a minus sign and think about subtraction.
For me it has to do with a book. There are in total 71 pages. I am on page 69. How many pages to go?
So, it becomes an addition problem. My idea, as a teacher is: look to the numbers, have a suitable context.
And that can be the beginning of a string:
71 – 69
92 – 4
1072 – 7
These problems don’t have anything to do with a book but with buying something or losing something. I have 71 marbles and I lost 2. How many left?
Support both ways with an open number line.
And don’t forget to pay attention to 71 – 69 as a subtraction problem.
That will help students to see all connections.
The steps you take as a teacher are very dependent of the level and insight of your students. Sometimes you need smaller steps, sometimes you can make bigger leaps.
For me as teacher, when the numbers are close in subtraction: add on. When the numbers are far away: subtract. And use a nice context to support those ideas.