ELEMENTARY
MATH
This section of the site gives access to constructivist resources for elementary mathematics. The Constructivist Learning section abovee has conceptual papers on constructivist concepts. The link immediately below takes you to my books entitled Understanding Fractions for Adults. A few other links to resources are given also below. Some resources are written as HTML files. MS Word versions are available on request.
Attribute Games and Venn Diagrams To bridge to the concept of a Venn diagram we start with the basic attribute game and focus on how it can be used with elementary school children. Similar attribute games can also be used at a higher level to bridge to additional concepts. Below are pictures of the tokens that are used for this game. Items vary by the attributes types {size, color, shape}. Sizes are large and small. Colors are blue and red. Shapes are circle and diamond. A larger set of attribute items can also be used, but even college students can benefit by starting with the basic game.
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Aviation Headings This resource compares describing angles in aviation to describing them in the standard coordinate system. One of the building blocks of navigation is the concept of describing the direction that a plane is flying. Navigation uses the concept of a heading for this purpose. In the first set of activities students are given, a heading and asked to convert it to a standard angle description. The next set reverses the procedure. In both cases students are asked to explain in words how to do the procedure. The third set of activities relates the concept of heading to pairs of cities.
Bridging from Color Pieces to Fractions This is one of several brief papers for educators illustrating the use of bridging. Bridging involves connecting concepts that a person finds remote to anything that seem manifest to that person. More about the concept of bridging is discussed near the end of this paper. To bridge to equivalent fractions we focus on what we will call the trading principle. This is a choice principle for making exchanges having equivalent value, when we have some other reason to make such trades. We begin with some familiar monetary examples. We then consider an imaginary monetary example that we use to directly bridge to trading equivalent fractions. Once equivalent fractions seems manifest to a person and is integrated with other concepts, then bridging to other fraction concepts can also be integrated into a person’s routine network of concepts. This approach in this paper is primarily a simplified version of what is done in my book Visualizing Fractions with Color Pieces.
Coin Flipping & Voting Probabilities Section 1 is intended both to help students appreciate some of the mathematical concepts involved in probability and to obtain a feeling for how mathematical probabilities relate to what actually happens in coins flipping trials. This section can be skipped or skimmed by anyone who has a functional mastery of the concepts involved. However, running the trials and formulating the dialog actually expanded my perspective. Section 2 presupposes basic probability concepts and applies them to simple voting situations and on how voting probabilities might relate to reasons for voting. Altho an expanded perspective will suggest more live options, I am not advocating anything other than a flexible perspective that will make for a better discussion of the reasons different people might have for voting. Of course, I not disinterested. I hope that this will result in a less simplistic attitude towards voting and a greater openness to options that might otherwise be dismissed by adopting a rigid or narrower attitude.
Explorable Cluster for Probability This resource contains the specific example of an explorable cluster on elementary probability concepts that is taken from the paper Explorable Clusters. I want to emphasize that an explorable cluster is merely one type of resource that can be designed for constructivist learning. It is a type that can be used in an extremely flexible an open-ended manner by a mentor to both assist students in immediate concept construction and provides a basis for the construction of fairly remote concepts at a later time. It is also a type of resource that easily lends itself to being redesigned.
Understanding Division. Section 1 focuses on the concept of division. Section 2 focuses on algorithms for doing what has been called long division. The last part of this section discusses some purposes of learning about both a standard and a modified algorithm for long division. It also relates this to a constructivist perspective on learning mathematics. Section 2 can be read without reading Section 1, as long as concept of division has been understood in relation to subtraction.