“Science is built up of facts, as a house is built of stones; but an accumulation of facts is no more a science than a heap of stones is a house.” ― Henri Poincaré, Science and Hypothesis
While Poincaré was writing about science he might as well have been writing about mathematics. Mathematics, like science, is not a collection of unrelated facts. “Science is a process of figuring out or making sense, and mathematics is the science of concepts and processes that have a pattern of regularity and logical order. Finding and exploring this regularity or order, and then making sense of it, is what doing mathematics is all about.” (Van de Walle, 2013). In other words, mathematics is an interrelated whole, a set of concepts, processes, and procedures that build and depend on each other.
The Utah Core State Standards in Mathematics have at their heart a focus on the important mathematics that must be learned at each grade. The standards are coherent, meaning that they are connected and build on each other both across grades and within grades. The standards are rigorous. Rigor means that the standards build conceptual understanding, procedural skill and fluency, and application of mathematics to the real world. All three components of rigor must be pursued in the classroom with equal intensity if students are to make sense of and do mathematics. In my last post I discussed procedural fluency. In this post I will take on conceptual understanding.
In Adding it Up the editors state, based on the research they reviewed, “Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kind of contexts in which it is useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know” (Adding it Up, 2001)
Students who understand concepts know why and how mathematical procedures work. For example, when asked to solve a problem involving division of fractions they know more than the old line, “Ours is not to reason why, just invert and multiply”. They know that there are two definitions of division, partitive and measurement (also called quotative), and how each or either definition might explain what the problem means. They know why the quotient of a fraction division problem can be larger than the dividend. They can represent the problem in multiple ways. For example, they may use pictures or objects, fraction circles or bars, the number line, a bar model, or a diagram using sets. They can apply their knowledge to solve actual problems involving dividing fractions and can create an application on their own where fraction division is necessary. They can tell you why the inverting and multiplying algorithm works by citing the relationship of division and multiplication and using properties of those operations (the inverse property, in this case) and can explain or justify their reasoning.
Most of us learned mathematics from a very procedural point of view. Our teacher gave examples of the steps one takes to solve a fraction division problem. Then a few of us came to the board and worked two or three other examples, or we all did the problem on an individual chalkboard (today they use whiteboards or iPads). Then we were given a problem set, usually around 30 to 40 problems to solve. We were told to remember the saying, “just invert and multiply.” We would work alone in silence or take the problems home for homework. Then we would get to the story problems and, if they were assigned at all, they would throw us for a loop. We had no idea how to interpret them or what to do. Then, the next day we would all correct our papers by handing them to the person behind us while the teacher gave the answers. We didn’t even discuss the right answers or the wrong answers. They were just right or wrong and we didn’t know why. Then we might have even called out our scores aloud while the teacher recorded them.
One of the many flaws in the traditional “sit and get” approach is that algorithms were taught without understanding. The emphasis was on getting the right answers. A great deal of research in the last 20 to 30 years, however, has indicated that instructional programs that “emphasize understanding algorithms before using them have been shown to lead to increases in both conceptual and procedural knowledge” (Fuson and Briars, 1990; Fuson, Hiebert, Murray, Human, Olivier, Carpenter, Fennemma, 1997; Hiebert and Wearne, 1996).
A classroom in which conceptual understanding, procedural skill and fluency, and application are the goals looks much different. The students may begin to develop their understanding of the concept through a problem solving approach. Students are given a mathematical task to solve the teacher has carefully selected that has a sound mathematical purpose. A task is a problem in context that has some kind of intriguing or even perplexing quality. Students use mathematical concepts they already know to strategize and find solutions to the task. As they do so they may use concrete objects, pictures, drawing, diagrams, numbers, symbols, and so on. Students are encouraged to explore and to talk with each other about what they are doing, to share strategies and work together toward a solution. They are told that they must be able to explain to other students and to the teacher in mathematical language what they did and why they did it. As they are working the teacher is fully involved and circulates around the class instructing, giving suggestions, and taking notes on how the students are working. She asks questions such as:
- How else could you have…?
- How are these _____________ the same?
- How are these different?
- About how long…? (many, tall, wide, heavy, big, more, less, etc.)
- What would you do if…?
- What would happen if…?
- What else could you have done?
- If I do this, what will happen?
- Is there any other way you could…?
- Why did you…?
- How did you…?
Students may then share what they did and their solution to the problem with others. They tell what they did and why. They show their representations, physical or pictorial or numerical, and defend their reasoning in front of the class or in small groups. The teacher pays close attention to the strategies they use and to how they are leading to understanding the concept. She can then plan for the next step in her instruction.
This process is repeated as the students solidify their understanding of the concept and then practice and become fluent. The end goal is that students will be able to use appropriate mathematical procedures, including standard algorithms, fluently to apply their understanding to real world problems.
We certainly believe in the philosophy that students move from the concrete to the representational to the abstract in their mathematical understanding. That kind of instruction is built into task based teaching. There are also certainly times when a teacher needs to balance problem solving based instruction with direct instruction. Bahr and de Garcia (2010) state, “The word balance in any instructional context means that you should consistently assess the needs of your students and select whatever instructional strategy meets that need.” Good formative assessment is very much a part of teaching for understanding.
Just as we know that the traditional classroom model of sit and get as described above doesn’t lead to mathematical understanding we also know that free wheeling, no holds barred inquiry methods where students simply explore mathematical ideas do not work. Teachers must always have mathematical purposes and objectives in mind for any method they choose.
Of course, this approach to teaching mathematics means that teachers must understand the concepts they are teaching as well as connected concepts both above and below their grade level. Hung-Hsi Wu, an eminent mathematician at the University of California – Berkeley, states, ” Ideally (teachers) should know mathematics in the sense that mathematicians use the word “know”: knowing a concept means knowing its precise definition, its intuitive content, why it is needed, and in what contexts it plays a role, and knowing a skill means knowing precisely what it does, when it is appropriate to apply it, how to prove that it is correct, the motivation for its creation, and, of course, the ability to use it correctly in diverse situations” (Wu, 2010).
It is incredible to see what happens when students are taught in a way that leads to understanding. They are more engaged. They are excited about what they are doing. It is hard to get them to move on to other subjects. They learn more and they feel good about what they learn. This applies to all students, not just those that have had an easier time with mathematics in the past. In fact, it is more fun for teachers as well. I know of one teacher who postponed her retirement because, after learning to teach mathematics for understanding, she was having too much fun in math. That is a great outcome.
Van de Walle, J. A., Karp, K.S., & Bay-Williams, J. M. (2013). Elementary and Middle School Mathematics: Teaching Developmentally, Pearson Education
National Research Council (2001). Adding it Up: Helping Children Learn Mathematics, National Academy Press
Fuson, K. C. & Briars, D. J. (1990). Using a base-ten blocks learning/teaching approach for first and second grade place-value and multi-digit addition and subtraction. Journal for Research in Mathematics Education, 21, 180-206
Fuson, K., Wearne, D., Hiebert, J., Human, P., Murray, H., Olivier, A., Carpenter, T., &
Fennema, E. (1997). Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education, 28, 130-162.
Hiebert, J., & Wearne, D. (1996). Instruction, understanding, and skill in multidigit
addition and subtraction. Cognition and Instruction, 14, 251-283.
Bahr, D.L. & de Garcia. L. A., (2010). Elementary Mathematics is Anything but Elementary. Florence, KY: Cengage Learning.
Wu, Hung-Hsi, (2011) Understanding Numbers in Elementary School Mathematics. Providence, RI: American Mathematical Society