High-Yield Math Learning For Longhorns

Why High Yield Instructional Strategies for Math?

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The principal reason we use

Marzano's strategies is simply because they WORK!

Drawing Pictures & Pictographs

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Drawing Pictures and Pictographs. Drawing pictures or pictographs (i.e., symbolic pictures) to represent knowledge is a powerful way to generate nonlinguistic representations in the mind. For example, most students have either drawn or colored the human skeletal system or have seen a picture of one in the classroom. Similarly, most students have drawn or colored a representation of the solar system. A variation of a picture is the pictograph, which is a drawing that uses symbols or symbolic pictures to represent information. The above example shows how a 1st grade teacher uses symbolic pictures in a geography lesson.

Math Concept Pattern Organizers

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Concept Patterns. Concept patterns, the most general of all patterns, organize information around a word or phrase that represents entire classes or categories of persons, places, things, and events. The characteristics or attributes of the concept, along with examples of each, should be included in this pattern. For example, students could use a graphic like the one in Figure 6.7 to organize the concept of fractions, along with examples and characteristics.

Generalization/Principle Pattern Organizers

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Generalization/Principle Patterns. Generalization/principle patterns organize information into general statements with supporting examples. For instance, for the statement, “A mathematics function is a relationship where the value of one variable depends on the value of another variable,” students can provide and represent examples in a graphic like that shown in Figure 6.6.

Episode Patterns as Math NLR Strategy

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Episode Patterns. Episode patterns organize information about specific events, including (1) a setting (time and place), (2) specific people, (3) a specific duration, (4) a specific sequence of events, and (5) a particular cause and effect. For example, students might organize information about the French Revolution into an episode pattern using a graphic like that shown in Figure 6.5.

Descriptive Pattern Organizers for Math

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Creating Graphic Organizers
Graphic organizers are perhaps the most common way to help students generate nonlinguistic representations. One of the most comprehensive treatments of the use of graphic organizers can be found in the book Visual Tools for Constructing Knowledge by David Hyerle (1996). Actually, graphic organizers combine the linguistic mode in that they use words and phrases, and the nonlinguistic mode in that they use symbols and arrows to represent relationships. The following six graphic organizers have great utility in the classroom because they correspond to six common patterns into which most information can be organized: descriptive patterns, time-sequence patterns, process/cause-effect patterns, episode patterns, generalization/principle patterns, and concept patterns.
Descriptive Patterns. Descriptive patterns can be used to represent facts about specific persons, places, things, and events. The information organized into a descriptive pattern does not need to be in any particular order. Figure 6.2 shows how teachers and students can graphically represent a descriptive pattern.

Time-Sequence Patterns as NLR

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Time-Sequence Patterns. Time-sequence patterns organize events in a specific chronological order. For example, information about the development of the Apollo space program can be organized as a sequence pattern. Figure 6.3 above shows how you might represent a time-sequence pattern graphically.

Math Process Cause and Effect Strategies

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Process/Cause-Effect Patterns. Process/cause-effect patterns organize information into a causal network leading to a specific outcome or into a sequence of steps leading to a specific product. For example, information about the factors that typically lead to the development of a healthy body might be organized as a process/cause-effect pattern. Figure 6.4 shows a graphic representation of a process/cause-effect pattern.

Non-Linguistic Representations

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Research and Theory on Nonlinguistic Representations
Many psychologists adhere to what has been called the “dual-coding” theory of information storage (see Paivio, 1969, 1971, 1990). This theory postulates that knowledge is stored in two forms—a linguistic form and an imagery form. The linguistic mode is semantic in nature. As a metaphor, one might think of the linguistic mode as containing actual statements in long-term memory. The imagery mode, in contrast, is expressed as mental pictures or even physical sensations, such as smell, taste, touch, kinesthetic association, and sound (Richardson, 1983).
In this book, the imagery mode of representation is referred to as a nonlinguistic representation. The more we use both systems of representation—linguistic and nonlinguistic—the better we are able to think about and recall knowledge. This is particularly relevant to the classroom, because studies have consistently shown that the primary way we present new knowledge to students is linguistic. We either talk to them about the new content or have them read about the new content (see Flanders, 1970). This means that students are commonly left to their own devices to generate nonlinguistic representations. When teachers help students in this kind of work, however, the effects on achievement are strong. It has even been shown that explicitly engaging students in the creation of nonlinguistic representations stimulates and increases activity in the brain (see Gerlic & Jausovec, 1999).

Using Kinesthetic Methods To Represent Math

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Engaging in Kinesthetic Activity. Kinesthetic activities are those that involve physical movement. By definition, physical movement associated with specific knowledge generates a mental image of the knowledge in the mind of the learner. (Recall from the previous discussion that mental images include physical sensations.) Most children find this both a natural and enjoyable way to express their knowledge. The following example below illustrates this in the context of a math class.
Often, to take a brief pause in math class, Ms. Jenkins asks her 4th grade students to think of ways they can represent what they are learning. For example, during the lesson on radius, diameter, and circumference of circles, Barry uses his left arm outstretched to show radius, both arms outstretched to show diameter, and both arms forming a circle to show circumference. During a different lesson on angles, Devon depicts obtuse and acute angles by making wide and not-so-wide “Vs” with her arms as the children yell out the degrees. They even have ways to show fractions, mixed numbers, and turning fractions into their simplest forms.
Ms. Jenkins started the activity she called Body Math just to give the students a break from the routine of doing math drills, but then realized that it was a powerful way for students to show whether or not they understood the concept behind the problems. Once the word got around, other students could be seen peeking in the classroom to see what they were doing that day with body math.

Generating Mental Pictures of Math

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Generating Mental Pictures. The most direct way to generate nonlinguistic representations is to simply construct (i.e., imagine) a mental picture of knowledge being learned. For abstract content, these mental pictures might be highly symbolic. To illustrate, psychologist John Hayes (1981) provides an example of how a student might generate a mental picture for the following equation from physics:

Making Physical Models (Manipulatives) For Math Representation

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Using Other Nonlinguistic Representations
Making Physical Models. As the name implies, physical models are concrete representations of the knowledge that is being learned. Mathematics and science teachers commonly refer to the use of concrete representations as “manipulatives.” The very act of generating a concrete representation establishes an “image” of the knowledge in students' minds. The following example illustrates this process in the context of a science class.
Mrs. Allison helped her 4th grade class to understand why we see different phases of the moon by presenting a concrete representation of the moon's monthly journey around the earth and its relationship to the sun. For the moon, Mrs. Allison gave each student a white Styrofoam ball and had them stick it on the end of a pencil. For the sun, she used a lamp with the shade removed. She told her students each of them would be the earth.
Mrs. Allison placed the lamp in the middle of the room, pulled down the window shades, and turned off the lights. Then she had each student place the ball at arm's length between the bulb and their eyes, simulating a total solar eclipse, which, she explained, is quite rare. Because the moon usually passes above or below the sun as viewed from Earth, Mrs. Allison then had her students move their moon up or down a bit so that they were looking into the Sun. From this position the students could observe that all the sunlight was shining on the far side of the moon, opposite the side they were viewing, simulating a new moon.