Strict partial order is a mathematical structure commonly seen in relational data. One obstacle to extracting such type of relations at scale is the lack of large scale labels for building effective data-driven solutions. We develop an active learning framework for mining such relations subject to a strict order. Our approach incorporates…

Alternating series have the simplest of sign patterns. What about series with more complicated patterns? By inspecting the alternating series test closely, we find a theorem that applies to more complicated sign patterns, and beyond.

If a series of real numbers converges absolutely, then it converges. The usual proof requires completeness in the form of the Cauchy criterion. Failing completeness, the result is false. We provide examples of rational series that illustrate this point. The Cantor set appears in connection with one of the examples.

Within the limited field of research on teachers' probabilistic knowledge, incorrect, inconsistent and even inexplicable responses to probabilistic tasks are most often accounted for by utilizing theories, frameworks and models, which are based upon heuristic and informal reasoning. More recently, the emergence of new research based upon informal…

Size and scale are crosscutting ideas integral to scientific understanding. However, research shows that students have little understanding of the size of objects, particularly objects too small to see with the unaided eye. Using a cross-sectional study with 101 middle-school through undergraduate students, a teaching experiment with 24 middle…

Lists six basic principles for building numeracy. Presents variety of activities in four categories of number concepts: spatial relationship, classifying, patterns, and correspondence. Suggests music, movement, and book ideas for each category of activity. Gives directions for making a geoboard and a reusable graph board. (DLH)

Linear measurement is more than just learning how to use a ruler. In the early grades, measurement activities develop students' understanding of the properties of objects as well as what it means to measure objects. Hands-on activities can enable students to explore such measurable properties as height and length, and direct comparisons of various…

This 19-page workbook presents the elementary student with topological concepts through sequences of pictures with related questions. Generally the questions ask, "How are the pictures different?" and "How are they the same?" Several topological concepts are presented in this manner: connectivity, number of holes, closed and open curves, networks,…

The author's secondary school mathematics students have often reported to her that quadratic relations are one of the most conceptually challenging aspects of the high school curriculum. From her own classroom experiences there seemed to be several aspects to the students' challenges. Many students, even in their early secondary education, have…

First- and second-grade students given instruction on length performed better on tests of seriation than students not given this instruction. (SD)

The purposes of this study were to: (1) determine the influence of a series of experiences involving the equivalence relation "same length as" and the asymmetric transitive relations "longer than" and "shorter than" on the ability of first and second grade children to classify and seriate objects on the basis of length; (2) investigate the…

Through seriation, verbal seriation, and conservation tasks, investigated blind, partially sighted, and sighted children's understanding of quantity. Subjects were 81 children equally dispersed through these 3 groups. Age range was 4 to 17 years. Found similarity in concept acquisition among three groups that extended beyond quantity conservation…

Noticing relationships between similar things and developing rules for treating things the same or differently, based on characteristics like color, size, shape, and texture provides the basis of beginning mathematics learning. Based on the view that teachers and parents of young children should provide children with age-appropriate words,…