## Posts

### Learning Assessment: Choosing Goals

Introduction I recently received a copy of a new book on assessment, one that was highlighted at this year's big assessment conference: Fulcher, K. H., & Prendergast, C. (2021). Improving Student Learning at Scale: A How-to Guide for Higher Education . Stylus Publishing, LLC.   This is not a review of that book; I just want to highlight an important idea I came across, from pages 60-63 of the paperback edition. The authors contrast two methods, described as deductive or inductive,for selecting learning goals that form the basis for program reporting and (ideally) improvement.Here's how they name the two methods, along with my suggested associations (objective, subjective) in italics: Deductive ( Objective ): "A learning area is targeted for improvement because assessment data indicate that students are not performing as expected." (p. 60). Inductive ( Subjective) : "[P]otential areas for improvement are identified through the experiences of the faculty (or th
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Last time I showed how we can think of matrix multiplication as (1) a matrix of inner products, or (2) a sum of outer products, and used the latter to unpack the "positive definite" part of the product $$X^tX$$. At the end I mentioned that the outer-product version of matrix multiplication makes it easy to show that  trace $$AB$$ = trace $$BA$$.  In the late 1980s I took a final exam that asked for the proof, and totally blanked. I mucked around with the sums for a while, and then time ran out. A little embarrassing, but also an indication that I hadn't really grasped the concepts. One can get quite a ways in math by just pushing symbols around, without developing intuition.  Some Sums The matrix multiplication of $$AB = C$$ is usually defined as an array of inner products, so the $$i$$th row and $$j$$th column of the product is  $$c_{ij} = \sum_{k=1}^q a_{ik} b_{kj}$$ where $$q$$ is the number of columns of $$A$$, which must be the same as the number of rows of \(B\