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In this text, we present the structuring of a study and research path (SRP) carried out within the scope of the initial education of pre-service mathematics teachers attending the curricular supervised teaching practice at a public university in Brazil. One of the research objectives was the constitution of a theoretical-methodological device capable of promoting teachers' research and continuing education on plane analytic geometry (PAG) topics. The work built around the generative question of the SRP, "How to teach the analytic geometry of the point and the line?", brought the possibility of answering the general question of the research promoting a process of professional formation from a theoretical-methodological device with specific characteristics. The planning moments identified in the activities executed by the pre-service teachers were quite significant. However, we could not identify in the recordings the possible praxeologies thought for the teaching of the PAG. Overall, the SRP-PF developed proved to be a theoretical-methodological device with the potential for scientific research in the didactics of mathematics and the professional formation of pre-service teachers. [ABSTRACT FROM AUTHOR]

How well do traditional and analytical hydraulic geometry hypotheses predict discharge? We tested whether Dingman's analytical expressions can be validated with field data by comparing it with the traditional Leopold-Maddock at-a-station hydraulic geometry (AHG). Data from a small basin (66 km2) in southern Brazil were used. The correlations for all the coefficients and for the exponents of depth and velocity of AHG relations were strong. Other results were confirmed: (i) there is a strong linear correlation between bankfull width and the AHG width coefficient; (ii) the hydraulic depth exponent and coefficient κ from generalized hydraulic equation are well predicted using relations between exponents and coefficients of AHG. The test of the hypotheses of the Leopold-Maddock and Dingman hydraulic geometry shows similar results and can be used to estimate discharge. However, the Leopold-Maddock method is simpler to apply from field measurements. [ABSTRACT FROM AUTHOR]