Transferring- using knowledge in a new context or novel situation

In a traditional classroom, the teacher’s primary role is to convey facts and procedures. The students’ roles are to memorize the facts and practice the procedures by working skill drill exercises and, sometimes, word problems. Students who can recall and repeat the appropriate facts and procedures score well on the end-of-unit or end-of-semester test. By contrast, in a constructivist or contextual classroom, the teacher’s role is expanded to include creating a variety of learning experiences with a focus on understanding rather than memorization. Contextual teachers use the strategies discussed above (relating, experiencing, applying, and cooperating) and they assign a wide variety of tasks to facilitate learning for understanding. In addition to skill drill and word problems, they assign experiential, hands-on activities and realistic problems through which students gain initial understanding and deepen their understanding of concepts. Students who learn with understanding can also learn to transfer knowledge. Transferring is a teaching strategy that we define as using knowledge in a new context or novel situation—one that has not been covered in class.

“If students are expected to apply ideas in novel situations,

then they must practice applying them in novel

situations.”47

American Association for the Advancement of Science

Project 2061

Research shows that, when teachers design tasks for novelty and variety, student interest, motivation, engagement, and mastery of mathematics goals can increase. Excellent teachers seem to have a natural ability to introduce novel ideas that motivate students intrinsically by invoking curiosity or emotions. As an example of invoking emotion, a mathematics teacher with 16- and 17-year-old students could distribute a magazine article that uses statistics to argue that young people should not be allowed to obtain drivers’ licenses until they are 18 years old. Predictably, many students will react emotionally to this argument. The resulting energy can be directed to engage the students in a discussion or debate, followed by a class assignment to work in groups to write critiques of the article. Critiques will include analysis of the mathematics. Were statistics misused? Were facts or assumptions misrepresented or omitted? Was the argument logical? If the critiques are persuasive, the teacher can even encourage students to submit them to the editor of the magazine as rebuttals.

Students also have natural curiosity about unfamiliar situations. A teacher can capitalize on student curiosity with problem-solving exercises such as the following.

 A sheet of notebook paper is approximately 2 mils thick. (A mil is one-thousandth of an inch.) If you fold a sheet of notebook paper in half, the total thickness is 4 mils. If you fold it in half again, the thickness becomes 8 mils. Suppose you could fold a sheet of notebook paper 50 times. Which of the following best describes the total thickness?

a. Less than ten feet

b. More than ten feet but less than a ten-story building

c. More than a ten-story building but less than Mount Everest

d. More than Mount Everest but less than the distance to the moon

e. More than the distance to the moon

While folding a sheet of paper is not novel, students cannot be familiar with 50 folds because it is impossible to fold the paper that many times. The teacher encourages students in small groups to discuss the possible choices of thickness and then vote as groups for the choice they predict to be true. A spokesperson for each group explains the rationale for the prediction. After the votes are tallied on the board, students have “bought

in” to the problem, and are eager to know the right answer. At this point, the teacher can have each student group find the thickness, without giving them a formula. The problem solution involves sequences, patterns, mathematical modeling, exponential functions, conversion factors, powers, and scientific notation. The solution is surprising, and the teacher can lead a class discussion of reasons most predictions are wrong, other situations where many doublings can occur, and the need for mathematics in these situations.

“. . . A major goal of high school mathematics is to equip

students with knowledge and tools that enable them to

formulate, approach, and solve problems beyond those that

they have studied.”-----National Council of Teachers of Mathematics

Excellent teachers use exercises like these to invoke curiosity and emotion as motivators in transferring mathematics ideas from one context to another. In addition, sensed meaning created by relating, experiencing, applying, cooperating, and transferring engages students’ emotions. One of Caine and Caine’s 12 Principles of Brain-Based Learning says, “Emotions and cognition cannot be separated and the conjunction of the two is at the heart of learning.” Although they did not use the term constructivism, their ideas about felt meaning, emotions, and cognition clearly paved the way: “. . . The brain needs to create its own meanings. Meaningful learning is built on creativity and is the source of much of the joy that students could experience in education."