Math classrooms in countries that score high on international comparisons of mathematics achievement have something in common—a culture of teaching and learning designed to help students make connections and build conceptual understanding. This observation comes from Hiebert and colleagues' 2003 analysis of videotaped lessons of 8th grade teachers collected in the Third International Math and Science Study. The researchers found that teachers in these countries not only assigned their students challenging mathematics problems, but also used active questioning and dialogue to help students see and understand the connections among mathematics concepts as they solved these problems.
In stark contrast, none of the U.S. math teachers in the videotape study used questions or conceptual dialogue that helped students explore mathematical connections. Even when the curriculum included math problems that were conceptually rich, the teachers approached these problems as procedural exercises (for instance, by giving the students a formula and telling them to plug in the numbers). One-third of the time, according to another analysis of the videotape study, the teachers simply told the students the answers (Stigler & Hiebert, 2004).
Unfortunately, teaching by telling, drilling on steps to follow to get the right answer—even taking pencils out of students' hands to show them how to do the problem—are methods embedded in the culture of mathematics teaching in the United States. How can we help mathematics teachers move beyond these ineffective, culturally conditioned teaching routines?
The answer lies in braiding together mathematics, language, and cognition. When we interweave these three strands, we create a rope that is stronger, more durable, and more powerful than any single strand. And we exponentially increase the likelihood that students will develop deep conceptual understanding.
Integrating Reading Strategies and Math Processes
For more than two decades, I have worked with K–8 teachers to investigate various ways to help their students become good mathematical problem solvers. One of the areas we have explored is integrating language—particularly reading comprehension—and mathematics. These two areas, although seemingly disparate, are both based on cognition.
- Making connections,
- Asking questions,
- Visualizing,
- Inferring and predicting,
- Determining importance,
- Synthesizing, and
- Metacognitive monitoring.
- Problem solving,
- Reasoning and proof,
- Communication,
- Connections, and
- Representations.
For several years, a dozen teachers and I investigated what students could accomplish when we juxtaposed the five math processes and the seven reading comprehension strategies. For each reading comprehension strategy, we systematically asked, What principles from cognitive science (Bransford, Brown, & Cocking, 2000) can we use to modify this strategy to help students do each mathematical process?
For example, K-W-L is a specific reading comprehension strategy for asking questions (Blachowicz & Ogle, 2001). The letters stand for the prereading questions, What do I know? and What do I want to learn more about? and the after-reading question, What did I learn? By considering the reading strategy of asking questions in tandem with the math process of problem solving, we experimented with adapting K-W-L to create our math strategy of K-W-C: What do I know for sure? What do I want to find out? and Are there any special conditions that I have to watch out for?
The teachers modeled the K-W-C questions for the whole class and encouraged students to use them to focus as they read story problems. When students met in small groups to discuss additional math problems, these three questions provided a structure for the students' work and helped them connect the problems with their prior knowledge.
For instance, I worked with a 2nd grade teacher, Betty, and her 20 students in a suburb of Chicago. We taped a big chart up on the wall and covered it with another sheet, revealing only the title, The Freight Trains. We asked the class a series of questions: What is a freight train? Have you ever been on a train? What kind of train was it? What is freight? The kids eagerly told us about some personal experiences with trains. As Betty had predicted, most did not know what freight was, but some did, and we asked them to explain.
We continued to slide the paper down to reveal one sentence at a time and asked the students, What do you know for sure? or What new information do you now know? after each sentence appeared. Here is the entire problem.The Freight TrainsAt the train station there are many different freight trains.They carry 3 kinds of freight across the United States: lumber, livestock, and vegetables.Each train has some lumber cars, some livestock cars, and some vegetable cars.Each train always has 18 freight cars.There are never more than 10 cars of one kind.Freight cars that are the same are always connected together.How many different ways of making trains with 18 cars can you find?
We handed out a graphic organizer on a sheet of paper, separated into three sections for the kids to express their ideas under three big headings: K (What do I know for sure?); W (What do I want to do, figure out, find out?); and C (Are there any special conditions, rules, or tricks I have to watch out for?). The students wrote down their thoughts and answers to the questions.
Then Betty gave each group 10 cubes of three different colors (30 cubes in all) to represent the three kinds of freight cars. Each group got one sheet of legal size paper with the outlines of four trains of 18 cars that were the same size as the cubes. The students took out magic markers of the threee colors and began. We encouraged them to make the train by arranging different combinations of colored cubes, check to make sure their arrangement met the conditions of the problem, and then color it in. With this kind of support in using the cognitive strategies of asking questions and visualizing, Betty's students were able to understand and solve a problem that many people might assume was too challenging for 2nd graders.
The teachers and I tried out the adapted reading comprehension strategies with a variety of math activities, from traditional story problems to more open-ended or extended-response tasks. The quality of most students' work, especially their explanations of the concepts, improved dramatically during the school year.
Richer Math Problem Solving
In the past, mathematical problem solving was an application of what students had been taught. Today, most math educators view problem solving asdoing mathematics, a powerful vehicle for building understanding of mathematics concepts.
From our experimentation with adapted comprehension strategies, my colleagues and I developed the Braid Model of Problem Solving. This approach incorporates the seven reading comprehension strategies into the traditional four phases of problem solving familiar to teachers: understanding, planning, carrying out the plan, and looking back. Let's look briefly at how two of the reading comprehension strategies can assist students in becoming skillful mathematics problem solvers.
Making Connections
Making connections is involved in every aspect of reading comprehension. Students make connections when they activate relevant prior knowledge and relate what is in the text to other things they have read, things in the real world, and phenomena around them.
Making connections is also at the heart of doing mathematics—from simple connections (for example, understanding how 1 and 1/10 are related) to major breakthroughs in understanding (for example, realizing that multiplication does not necessarily mean “making something bigger”; it could mean having only part of a group, as in .25 × 84, or having multiple groups of an amount smaller than one, as in 84 × .25).
We should teach students to make a variety of connections as they strive to understand each problem. Adapting the reading connections approach, we ask students to look for connections that are math-to-self (connecting math concepts to prior knowledge and experience); math-to-world (connecting math concepts to real-world situations, science, and social studies); and math-to-math (connecting math concepts within and between branches of mathematics or connecting concepts and procedures). We help students identify these different kinds of connections and build bridges across contexts to help students generalize their understanding.
For instance, a teacher may help students build a specific understanding of the concept of integers—positive and negative numbers—within the particular context of a thermometer, and then later deepen their understanding by working with this concept in other contexts (such as elevation above and below sea level, yardage gained and lost in football plays, or debits and credits in a checking account).
When students tackle challenging mathematics problems, we teach them to create representations that help them see and express meaningful connections and patterns. Representation strategies that students can use include discussing the problem in small groups (language representations using auditory sense); using manipulatives (concrete, physical representations using tactile sense); acting it out (representations of sequential actions using bodily kinesthetic sense); drawing a picture, diagram, or graph (pictorial representations using visual sense); and making a list or table (symbolic representations often requiring abstract reasoning; Hyde, 2006; Zemelman, Daniels, & Hyde, 2005). Each of these five strategies employs a different sensory modality and a different way that humans process information, so teachers can readily differentiate instruction.
In general, making more connections and providing more examples in different but relevant contexts will result in more elaborate networks of ideas and relationships, building a deeper, richer, more generalized understanding of a concept.
Inferring and Predicting
Inferring the deeper meaning of a text is an essential part of reading comprehension. Both inferring and predicting require readers to go beyond the surface information in the text, combining the words before them with their prior knowledge to create connections and build meaning. It is important that teachers help their students become more sophisticated in their inferences and predictions.
In the Braid Model, we ask student groups to engage in the K-W-C process and then look back at the questions they asked and decide what inferences they have made and whether those inferences were accurate. For example, if a problem states, “The car traveled the 90 miles from Philadelphia to Baltimore in two hours,” I often hear one student in a group assert as part of what we knowthat “the car went 45 miles per hour.” When students go back to check whether each statement is a fact or an inference, this statement stirs a lot of discussion. Some think it is a fact; others say “45 mph is an average; they may never have been going at that speed. They may have made a stop for lunch and driven 60 or 70 mph for most of the actual driving time.” This discussion gives me the opportunity to make the point that it is useful to infer, but also important to know when and why you are inferring (Hyde, 2006, p. 108).
An Essential Change
To raise mathematics achievement in the United States to higher levels, it is essential that we infuse language and thought into mathematics. This task is not easy: It requires that teachers help students build the habits of creating representations, asking relevant questions, and seeking patterns and connections. But I believe we can do a far better job of teaching students to understand and love mathematics if we enrich our teaching with practices from reading and language arts adapted by cognitive science.
