The last 20 years has seen a global increase in studies investigating various aspects of Home Education (HE) and the physical location learning relation to schooling; outside schooling remains under researched. This paper provides review some, but not all, existing literature, leading us see where there are potentially gaps research as well opportunities for creative methodologies. starts with ...

Proof. Suppose that π ∈ Sn is such that the minimal j ∈ {1, . . . , n} for which aπ(j) ∈ BX(x, r+dX(x, y)) actually satisfies aπ(j) ∈ BX(x, r−dX(x, y)). Hence j r (x) = j and therefore aπ(j) = ar (x). Also, dX(aπ(j), y) 6 dX(aπ(j), x) + dX(x, y) 6 r, so j r (y) 6 j. But dX(x, ajπ r (y)) 6 dX(y, ajπ r (y)) + dX(x, y) 6 r+ dX(x, y), so by the definition of j we must have j r (y) > j. Thus j π r (...