Summary: We consider an inverse boundary value problem for the heat equation on the interval \((0,1)\), where the heat conductivity \(\gamma(t,x)\) is piecewise constant and the point of discontinuity depends on time: \(\gamma(t,x) = k^2 \;(0 < x < s(t))\), \(\gamma(t,x) = 1\) (\(s(t) < x < 1\)). First, we show that \(k\) and \(s(t)\) on the time interval \([0,T]\) are determined from a partial Dirichlet-to-Neumann map: \(u(t,1) \to \partial_x u(t,1)\), \(0 < t < T\), \(u(t,x)\) being the solution to the heat equation such that \(u(t,0)=0\), independently of the initial data \(u(0,x)\). Second, we show that another partial Dirichlet-to-Neumann map: \(u(t,0) \to \partial_xu(t,1)\), \(0 < t < T\), \(u(t,x)\) being the solution to the heat equation such that \(u(t,1)=0\), restricts the pair \((k,s(t))\) to, at most, two cases on the time interval \([0,T]\), independently of the initial data \(u(0,x)\).