Mathematical ability rests on two kinds of skill- problem solving and arithmetic. As with the development of many psychological processes, these two skills emerge somewhat separately in early childhood. But they become intertwined over time. During the elementary school years children need to become comfortable and adept at arithmetic- addition, subtraction, multiplication, and division. They need to grasp what those operations refer to in the real world (i.e. the number sentence 1+1=2 is a representation of what happens when you have an orange in your lunch box, put another one in there and then have two oranges).  But children also need to become so quick and comfortable with arithmetic that it is almost automatic and can be used in the service of more complex and interesting problems. The youngest children at Hayground work on this by grouping objects, and performing arithmetic with real things. We want them to acquire an understanding of how written and spoken numbers correspond to the concrete world of objects. Most of the time children learn about the connection between the abstract and concrete worlds of numbers through simple hands on activities and games. Once they have a firm grasp of what those operations are in real terms, they can focus on practicing arithmetic in the abstract (solving written number problems) – again, this is sometimes done with simple activities and games, but also through straightforward repetition.  There is nothing mystical or fancy about this, but since it is necessary for more sophisticated mathematical work, not to mention for every day life, we stick with it.

    Meanwhile, the second strand of numeracy requires comfort with and ability to solve a wide range of problems. The numerate child is willing to try out different solutions until he or she finds the one that works. Some of these problems involve numbers: The Profit.  A dealer bought an article for $7, sold it for $8, bought it back for $9, and sold it for $10.  How much profit did she make? And some have little to do with arithmetic, and lots to do with thinking about different ways to solve a puzzle: Getting a Pole on a Bus.  For his thirteenth birthday, Adam was allowed to travel down to Sarah's Sporting Goods store to purchase a brand new fishing pole.  With great excitement and anticipation, Adam boarded the bus on his own and arrived at Sarah's store.  Although the collection of fishing poles was tremendous, there was only one pole for Adam and he bought it: a five foot, one piece fiberglass "Trout Troller 570" fishing pole.
    When Adam's return bus arrived, the driver reported that Adam could not board the bus with the fishing pole.  Objects longer than four feet were not allowed on the bus.  Adam remained at the bus stop holding his beautiful five foot Trout Troller.  Sarah, seeing the whole ordeal, rushed out and said, "We'll get your fishing pole on the bus!"  Sure enough, when the same bus and the same driver returned, Adam boarded the bus with his fishing pole and the driver welcomed him aboard with a smile.  How was Sarah able to have Adam board the bus with his five foot fishing pole without breaking or bending the bus line rules or the pole?*   
    In order to work on a range of both arithmetic and mathematical problems, we group children who display the same level of mathematical expertise, though they may or may not be the same age as one another. The teachers use this math block time

to get children working and thinking in the world of numbers; they present math problems or arithmetic tasks and ask children to solve and/or complete them. At other times teachers want children to use problem solving and arithmetic skills within a more complex and meaningful context eg: children might be creating maps for a larger project and need to learn to scale things down or up for the map.  All children at Hayground spend part of every day doing work that involves both arithmetic and mathematics. Some of this time is spent with those with similar skills, focusing explicitly on numbers, while some of the time is spent in mixed age groups working on larger projects within which number tasks are embedded.

* Thanks to Edward Burger, author of The Heart Of Mathematics, Dept of Mathematics at Williams College, for providing these examples.