This module will explore math content integration. You will examine comprehension strategies as they relate to mathematics instruction, problem-solving, and writing connections.
Math content integration is blending math with other areas of the curriculum in order to enhance the meaning of the program and to allow each discipline to both reinforce and be reinforced by other subject areas. The focus of math content integration is to enhance students’ comprehension of facts and concepts while stimulating their interest in math. The purpose of content integration is to enable students to practice making predictions, drawing conclusions, observing, and classifying. It also serves as a springboard for meaningful content-related, interactive, and journal-writing experiences.
Math content integration requires using literacy to enhance and extend mathematical thinking in daily instruction. Content integration allows students the time to process their reading skills throughout all subject areas and allows time for practical, real-world applications.
In her book About Teaching Mathematics: A K–8 Resource, Marilyn Burns discusses Piaget’s theory of disequilibrium. This involves the realization of a mental misperception that produces confusion. This confusion is positive for meaningful math instruction. When students are faced with a confusing scenario, we want them to problem-solve, using many of the same techniques we use when looking at nonfiction text structures. Students need to address and attack these problems as an active process. With strong literacy instruction, these links can be made across the math curriculum. This time of disequilibrium is the prime “teachable moment” in mathematics. It requires students to step outside of their comfortable, independent area and move into the uncomfortable and frightening instructional zone. They then have the opportunity to reorganize their thinking, and they need tools to do that. This is when nonfiction text features become important. The graphics and organization of information can help students begin to make sense of mathematical situations. Writing will also begin to play a key role in the organization of ideas. As learners, they can better understand a key concept when they are responsible for explaining it using the writing process. This is an excellent form of assessment in determining true understanding of a concept. Many students are quite capable of the basic computation in early mathematics, but if they do not have the concept of “why,” they will be lost at a higher level.
Content integration gives students time to process information using their reading and writing skills. It will move students to the next level of understanding and allow them to make reading and writing connections to the real world.
Math is part of our everyday lives. Many times the actual math involves simple computations, but we still have to know when to apply each operation. In order to apply the appropriate strategy, we must understand what the task is asking. Just as a reader of a traditional text must understand the author’s purpose, a mathematician must understand the purpose of the problem. This is why before, during, and after strategy instruction is crucial in mathematics.
Students should learn to dissect problems before embarking on the search for the solution. This allows students the chance to find out what the problem is really asking. Once they understand this, they are able to find the correct operation or an alternate way to reach the solution (pictures, numbers, words).
During problem-solving, students should be encouraged to use pictures, numbers, and words to describe their answers. Math is a tough subject for many students because teachers often move from the concrete (manipulatives) to the abstract (numerical representations) before students are ready. For this reason, it is imperative that we model how to use manipulatives and have them available for students to use when they need the support. Students should be encouraged to draw pictures. Pictures allow them to utilize a concrete representation while moving into abstract thinking. They can no longer touch and feel an object, but they have a mental image to represent the objects involved in the problem. Pictures allow students an important opportunity to bridge the concrete to the abstract.
After solving the problem, students should write. This writing is a critical step in the problem-solving process because this, more than the solution to the problem, will help you, the teacher, gauge your students’ understanding. Writing is the most powerful means of assessment for students of mathematics. Not only should they be able to solve the problem, but they should be able to provide a detailed explanation for their thinking. This allows the teacher to examine thinking processes and fill in gaps where they are found.
Problem-solving is not specific to the traditional word problems. Problem-solving is embedded in all mathematical situations. Students should have direct instruction in different problem-solving strategies. For example, students should understand exactly what it means to “use a friendly number.” This means to round to a number they can readily use without difficult computation—for example, 10 instead of 9. Rounding will not lead students to the exact answer, but often the exact answer is not needed. Even if the exact answer is needed, this can still be a valuable strategy, because it provides a range of possible answers.
Problem-solving should be a daily experience and should be integrated throughout the content areas. Teachers who take advantage of opportunities to integrate vocabulary, strategy, and comprehension instruction into their mathematics lessons provide their students with a strong academic base. Once you begin to open up your thinking to reading as deciphering any symbol and understanding it, mathematics is an obvious link. This is certainly the most difficult stretch to make, but it will provide the links your students need to be successful in all subject areas.
“Computing has to do with thinking, not writing with paper and pencil. Requiring children to learn paper-and–pencil computation first is putting the cart before the horse. It’s like expecting children to learn to write before they can tell their own stories.”
- Marilyn Burns, Math: Facing an American Phobia
Although the recent focus in standardized testing has been on problem-solving, computation has and always will play a large role in mathematics instruction. What schools often do not consider is the time children need to develop the number sense that forms the foundation for computation. The quote above reminds us that our first job in the primary classroom is to allow our students opportunities to think about and work with numbers in their heads. They need time to process the information, ask questions, and have concrete representations.
Manipulatives are an integral part of mathematics instruction at all levels. Numbers are abstract. The fact that the numeral 2 represents 2 objects is a complex notion. Learning these representations requires modeling in many forms. Students must have an opportunity to see what 2 looks like in multiple situations. If we always use 2 beans to make 2, our students will only know numbers in the form of beans. To move from this concrete level to the abstract, modeling is the key. Show your students all different forms of 2. For example: 2 leaves, 2 dogs, 2 pencils, 2 pieces of paper, 2 houses, etc. After all of these examples of 2, students can generalize the meaning of 2. The concept is now firmly in place. Your students have prior experiences with the number and can recognize and work with it in a variety of formats. Developing number sense is not difficult; it just requires modeling and time. As your students build on this foundation, they begin to move effortlessly between the concrete and abstract. They have learned when to work with concrete representations through the use of manipulatives. They begin to build confidence to work with abstract representations. This bridge between the concrete and the abstract is necessary for students to internalize the symbolic representation of numbers.
Problem-solving is often introduced and discussed before students are actually presented with a problem to solve. Students need the opportunity to confront, figure out, find order, reason, guess, and test the solutions they find. Real problem-solving activities have three ingredients: desire, blockage, and effort.
For a problem to be worth solving, the need for the solution should have real-life implications.
For a problem to be a problem, there must be something in the way of directly arriving at the answer. If a student is simply asked to compute two numbers, there is no problem — just a computation. When students must understand the action in the problem and apply a problem-solving strategy, such as guess and check, there is a hurdle to cross. Blockage is a key ingredient for a rich problem-solving experience. The academic abilities of your students determine the difficulty of the blockage.
The final ingredient for a quality problem-solving experience is effort. The student has to put forth the effort to solve the problem. Many of the students in your classroom will eagerly tackle these problems without any prodding, but there are also those that will attempt one strategy, arrive at an unreasonable solution, and quit. Your classroom environment will determine the culture of problem-solving in your classroom. For your students to believe in themselves as problem solvers, they must have daily opportunities to solve problems on their own. When they fail, you should be there to provide support through whole-group, small-group, and individualized instruction. These are perfect opportunities to provide strategy reinforcement for the group of students with which you are working.
Gradual Release of Responsibility
In a previous subtopic we discussed the idea of the gradual release of responsibility in relation to reading instruction. The same information can easily be applied to mathematics instruction. When we work with students we want to take them through a shared experience. During these experiences, the role of model and coach falls heavily on the teacher. The student is learning how to process the information. It is during this stage that you will explicitly teach the problem-solving strategies. Students will have an opportunity to see the application of the strategies. They will also have access to you as a mathematician. This is a perfect opportunity to model the metacognitive process behind the strategic thinking.
After explicitly modeling these strategies and talking through them with students, you can begin to work with each student at their own level. This might look like small-group instruction, individual conferences, peer conferences, or student-led discussion groups. These informal practice sessions provide a chance for students to work with problems with support available if needed. This is a time to make mistakes and try new strategies.
Math is comprised of an internal code. Just as there is a letter-sound relationship with the alphabet, there is a number-quantity relationship with our number system. This is a simple, yet extremely complex and abstract relationship. When students are provided opportunities to see, practice together, and apply their learning independently, we see an increase in academic retention.
When we ask our students to add, subtract, multiply, or divide, we must expect that they understand all of the words in the problem that prompt that operation. By utilizing proven reading comprehension strategies, our students will meet this challenge. When a student is able to visualize the problem or draw a picture, the abstract nature of the algorithm instantly has a concrete quality. When students question the intent of the problem, they are dissecting the requested action. Each comprehension strategy has an applicable place in the math classroom. Click on the link below to study the file Comprehension Strategies for Math.
Metacognition means being aware of how and why you are doing something. In simple terms it means thinking about your thinking. There is a strong correlation between problem-solving success and instruction that has cognitive monitoring practices in conjunction with strong strategy instruction (Van De Walle, 2001). Students with strong problem-solving abilities regularly monitor their actions and actively seek additional strategies to clear up their confusion. Metacognitive strategies can be taught through modeling. Click on the link below for a student checklist to help your students internalize metacognitive strategies.
“Exploring, investigating, describing, and explaining mathematical ideas promote communication. Teachers facilitate this process when they pose probing questions and invite children to explain their thinking.”
-National Council of Teachers of Mathematics
When students can explain their thinking, we truly have an accurate assessment of their understanding. Reading what your students write is an opportunity to learn a wealth of information about them. Mathematics assessments are generally formulated as arithmetic problems to be solved and a few word problems. Rarely do these assessments ask students to explain their thinking. If a student can explain exactly what it means to multiply or divide certain numbers, you can be certain the concept is firmly in place. Student writing samples allow you to confidently ascertain the abilities of your students and gear your instruction accordingly.
How do you get your students to write about math? Many times students will try to write a description of how they solved the problem. With modeled and shared writing experiences, students will understand that you want to hear the voice inside their head. You want to know exactly what they were thinking when they decided to utilize a certain strategy.