The paper is devoted to the study of Toeplitz operators with piecewise continuous and slowly oscillating radial symbols on the Bergman space \({\mathcal A}^2({\mathbb D})\). Let \(PC({\mathbb T})\) be the algebra of all piecewise continuous functions on the unit circle. By \(\mathrm{SO}([0,1))\) denote the algebra of all bounded continuous functions \(a\) on \([0,1)\) satisfying, for every \(\sigma\in(0,1)\), the following condition: \(\lim\limits_{\delta\to 0}\sup_{r,s\in[1-\delta,1-\sigma\delta]}|a(r)-a(s)|=0\). Let \(M_1(\mathrm{SO}[0,1))\) be the fiber of the maximal ideal space of \(\mathrm{SO}([0,1))\) consisting of all multiplicative linear functionals \(g\) such that \(g(a)=0\) whenever \(\lim\limits_{r\to 1}a(r)=0\). Let \({\mathcal K}\) denote the ideal of all compact operators on \({\mathcal A}^2({\mathbb D})\). The main result of the paper says that the Calkin algebra of the \(C^*\)-algebra generated by the Toeplitz operators with symbols in \(\mathrm{SO}([0,1))\otimes PC({\mathbb T})\) is isomorphic and isometric to the algebra of all continuous functions on the compact set \(M_1(\mathrm{SO}[0,1))\times ({\mathbb T}\times[0,1])\). The isomorphism is given by the following mapping of the generators \(T_{ab}+{\mathcal K}\mapsto\widehat{a}(\varphi)[b(t^-)(1-x)+b(t^+)x]\), where \(\varphi\in M_1(\mathrm{SO}([0,1)))\), \((t,x)\in{\mathbb T}\times[0,1]\), \(a\in \mathrm{SO}([0,1))\), \(b\in PC({\mathbb T})\), \(\widehat{a}\) is the Gelfand transform of the function \(a\), and the cylinder \([0,1]\times{\mathbb T}\) is equipped with a special topology first considered by I. Gohberg and N. Krupnik in the study of the algebra of Toeplitz operators with piecewise continuous symbols on the Hardy space \(H^2({\mathbb T})\) [I. Ts. Gokhberg and N. Ya. Krupnik, Funct. Anal. Appl. 3, 119–127 (1969; Zbl 0199.19201); translation from Funkts. Anal. Prilozh. 3, No. 2, 46–56 (1969)]. A local base of neighborhoods at the point \((t,x)\) consists of the sets \(\{t\}\times(x-\varepsilon,x+\varepsilon)\), \(0<\varepsilon<\min\{x,1-x\}\) if \(x\notin\{0,1\}\). For the point \((t,1)\), a local base of neighborhoods is \(([t,te^{i\varepsilon})\times(1-\varepsilon,1])\cup((t,te^{i\varepsilon})\times[0,1-\varepsilon])\) with \(0<\varepsilon<1\). Finally, for the point \((t,0)\), a local base of neighborhoods is \(((te^{-i\varepsilon},t]\times[0,\varepsilon))\cup((te^{-i\varepsilon},t)\times[\varepsilon,1])\) with \(0<\varepsilon<1\).