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We present the first flexion-focused gravitational lensing analysis of the first of the strong-lensing "cosmic telescope" galaxy clusters, observed as part of the Hubble Frontier Fields initiative. Using HST observations of Abell 2744 (z = 0.308), we apply a modified Analytic Image Model (AIM) technique to measure source galaxy flexion and shear values at a final number density of 82 arcmin$^{-2}$. By using flexion data alone we are able to identify the primary mass structure aligned along the heart of the cluster in addition to a major substructure peak offset 1.43' from the cluster core. We generate two types of nonparametric reconstructions: a flexion aperture mass map, which identifies the central potential and substructure peak with mass signal-to-noise of 3.5$\sigma$ and 2.3$\sigma$ respectively; and a convergence map derived directly from the smoothed flexion field. For the primary peak we find a mass of $1.93\times10^{14}\,h^{-1}\,M_{\odot}$ within a 45" (145h$^{-1}$ kpc) aperture, and for the western substructure we find a mass of $7.12\times10^{13}\,h^{-1}\,M_{\odot}$ within a 25" (80h$^{-1}$ kpc) aperture. The associated peak velocity dispersions were determined to be $\sigma_v$ = 1630 km/s and $\sigma_v$ = 766 km/s, respectively, by fitting nonsingular isothermal sphere profiles to the flexion data. Additionally, we use simultaneous shear measurements to independently reconstruct the broader cluster mass structure, and find that it is unable to reproduce the small-scale structure associated with the flexion reconstructions. Finally, we perform the same analysis on the Abell 2744 parallel sky field, and find no strong phantom signals in the noise reconstructions., Comment: 12 pages, 5 figures. Submitted to MNRAS