By John A. Van de Walle
Ten years after NCTM’s release of the Curriculum and Evaluation Standards for School Mathematics, this country is having a wrenching debate about what should be taught in mathematics and how it should be taught. Debate has degenerated to “math wars.” On one side are those who fervently believe children need to learn “the basics.” On the other side are those who believe or think they believe in the message of the Standards. These are the educators who believe in “reform mathematics.” Objectivity often gets lost in rhetoric and what either side believes is frequently vague at best.
Understanding the War
In order to understand how we arrived at this state of war it seems useful to offer some definitions of basics and reform. First the basics. This is the mathematics that parents and legislators recognize as the mathematics that they attempted to learn when they were in school. It consists primarily of arithmetic or computation. It is finding answers to questions such as “30 is what percent of 87?” It is “solving for x” and memorizing formulas. A list of items important to the basics position would certainly include the following:
- Counting accurately to 100 or more.
- Mastery of basic facts for all four operations.
- Pencil and paper computation skills with whole numbers, decimals and fractions.
- Solving percent problems.
- Knowing and using formulas for area and perimeter of basic shapes.
- Mastery of measurement conversions (2 pints in a quart, 3 feet in a yard, etc.).
Of course the above is an obvious over simplification. Regardless of your specific list, the position of the basics camp is that kids should know some “basic stuff.” Basics are about what children should know – content. It is trivial pursuit mathematics.
Reform mathematics is a bit more difficult to pin down but the principal positions can be found in the 1989 Standards document. Reformers want the five goals for students as outlined in that document: students should 1) value mathematics, 2) be confident in the ability to do mathematics, 3) become mathematical problem solvers, 4) learn to communicate mathematically, and 5) learn to reason mathematically. Those who believe in reform mathematics talk about mathematical power; the ability to reason and solve unique problems. The four process standards (problem solving, communication, reasoning, and connections) have almost become a mantra over the past ten years. Reform is about children and thinking.
These two positions, reform and the basics, are not opposite ends of the same continuum. On one hand, the basics tend to be about content, specifically about the content that was common when today’s adults were in school. On the other hand, reform is much more about how children learn and how to achieve the content goals one desires.
Again, this is an oversimplification. The basics camp has also taken some extreme positions. In some states, California being a prime example, the skills recommended are not always appropriate for the grade levels suggested nor are they reflective of today’s societal needs. Many skills such as multiplication by a 3-digit decimal number are archaic in a world of omnipresent calculators. And most onerous is the demand in some quarters for nothing other than direct instruction, what Tom O’Brien (1999) so aptly labeled “parrot math.” This approach suggests that children mimic mindlessly what teachers model with the hope that somehow the mimicry will lead to learning. Do parrots understand? Ernst von Glassersfeld (1995) talks of 50 years of domination by behaviorists. “The behaviorists succeeded in eliminating the distinction between training (for performance) andteaching that aims at the generation of understanding” (p. 4).
The reformers have also made some mistakes and are guilty of misguided emphases. By praising the values of calculators and continually harping about tedious computations, we in the reform camp failed to emphasize many of the valid content objectives in mom and dad’s typical list of basics. We became obsessed with all sorts of interesting problems that ate time in the mathematics period. (Nineteen pigs and chickens with a total of 54 legs. How many pigs and how many chickens?) Detailed applications and “alternative assessments” became so involved with the context that often the mathematics was difficult to find (Rainforest math). Manipulatives, cooperative groups and calculators became hallmarks of reform no matter how superficially or inappropriately these materials and methods were used. In short, we forgot that there is much of what is basic that is still important. Basic facts are essential! All children must be able to compute!
Mom and dad think we are pushing “fuzzy math” and have forgotten what is important. The reformers feel as though they are being returned to the dark ages by state mandates.
Children and Learning
We simply cannot go backwards! The behaviorists’ parrot math, direct-instruction approaches are doomed to failure. Evidence of this can be found in any gathering of adults, even educated adults, where most will gladly proclaim that to them mathematics is a mystery. Most adults believe that mathematics is composed of mindless rules and while it is certainly important, they were “never any good at math.” Direct, rule-oriented methods failed them miserably and will fail today’s children in exactly the same manner.
Today we know a lot about how children learn and how they learn mathematics. There is abundant evidence that instruction reflective of a constructivist theory learning is more effective than teaching by telling. The NSF supported reform curricula all reflect, though somewhat imperfectly, a constructivist view of learning. Most reform-movement advocates espouse constructivism as the best explanation for how children learn. Unfortunately, we have not moved very far in this direction if measured by the typical classroom in the United States.
The basic tenet of constructivism is simply this: Children construct their own knowledge. Construction requires tools. The tools children use to construct knowledge are the ideas they already have. To use ideas to construct new ideas means that children must be mentally engaged in the act of learning. They must call up those ideas that are relevant and use them to give meaning to the new or emerging or changing ideas that they are developing.
Since ideas are used to develop new knowledge, those existing ideas will necessarily be connected to the new idea. The result is a network of meaningful, related, useful ideas. The more ideas and the more interconnected they are, the better all of the ideas are understood. Things make more sense when they can be related to a lot of the ideas already understood.
Constructivism is a theory about how we learn. If it is correct then that is how all learning takes place, even rote learning and that which is a result of direct instruction and parrot math. But what tools are used to construct rote learning? To what is the new knowledge connected? Typically they are not mathematical ideas nor will they build to form a mathematical network of ideas. Children searching for a way to remember 7×8=56 might note that the numbers 5, 6 and 7, 8 go in order. Or they may connect the number 56 to “that hard fact” since 56 is unique in the multiplication table. But then so is 54. Repetition of a procedure may be connected to some mantra-type recitation of the rule. “Divide, multiply, subtract then bring down.” This sequence has even been related to the mnemonic “Dirty Monkeys Smell Bad.” Such connections will never form a useful network of ideas or develop into understanding.
On the other hand, when mathematical ideas are used to create new mathematical ideas, useful cognitive networks are formed. Return to 7×8 and imagine a class where children discuss and share clever ways to figure out the product. One child might think of 5 eight’s and then 2 more eight’s. Another may have learned 7×7 and noted that its just one more seven. Still another might look at a list of 8 seven’s and take half of them (4×7) and double that. This may lead to the notion that double 7 is 14, and double that is 28 and double that is 56. Most of these methods suggest ways to think about other facts where doubling or five’s can be useful. The result is a meaningful network of ideas.
Meaningful networks of ideas mean fewer details to retain, easier recall of ideas after extended periods of time, better application of ideas to novel problems, and a feeling that mathematics makes sense. Apply this same idea to any area of study in your experience. When you are able to or helped to connect new ideas to those you already have, the new information seems to make sense; you understand it and most likely will remember it easily. Conversely, when facts do not connect with things you know they seem foreign or mysterious and they are almost certainly forgotten or confused.
An Idea to Consider
The main thesis of this paper is the following:
Most if not all important mathematical ideas can be taught via problem solving.
The term problem in this statement should be taken to mean any task or exploration
· for which the solution has not been explained,
· that begins where kids are (with their ideas),
· that is challenging mathematically, and
· for which justification and explanations for answers, methods, and results are understood to be the responsibility of the students.
· may be posed for individual students, pairs, or groups,
· may involve hands on materials or drawings, or involve only mental work, and
· may involve calculators or not.
A problem should always include the expectation of a report be it oral, written or some other product such as a poster or graph.
The intent of the above definition is to broaden the concept of problem to include any task that causes children to wrestle or struggle with the mathematics that we want children to learn. It is important to understand that mathematics is to be taught through problem solving. That is, problem solving is the vehicle by which the desired curriculum is developed.
A few examples are in order:
Take seven counters. Find out how many ways the seven counters can be separated into two groups. Draw pictures and use numbers to tell what you find out.
Make the number 346 in one pile. In another pile, make the number 282. What is a good way to figure out how much you have all together? Try it out! Test your plan on these two numbers: 165 and 473. Be prepared to explain your plan to the class.
Find four fractions that are more than 1/2 and less than 1. Put your fractions in order from least to most. Write an explanation to convince me you are correct. Pictures are always a good idea but you must also use some other argument other than a picture to convince me. Challenge: Make each of your fractions have a different denominator.
It is important to notice that these examples involve very traditional content. Children engaged in these problems are likely to “bump into” and develop or construct the desired ideas. It is not a matter of discovery but rather the personal development of the expected curriculum, based on the students’ existing ideas. In the first example above, students will find ways to think about a number in two parts. Some children will be very systematic. Others will be quite haphazard, starting each combination anew. When these ideas are shared, the thoughts from the class as a whole will aid in everyone’s development. Part/whole concepts are at the very foundation of addition and subtraction facts.
In the third example, students have to wrestle with the size of fractions. They will need to use what they understand about the meanings of top and bottom number and the relative size of fractional parts. Some may learn from others in their group since all students in a class rarely master basic fraction concepts. Some may use a physical model but the explanations that develop at the end of the lesson will go beyond simple showing of fraction pieces. Computation and estimation are built much more easily on a foundation of good fraction concepts such as these.
Admittedly, teaching through problem solving is a difficult task when compared with teacher-directed instruction guided by a traditional basal textbook. Why should anyone want to go to this trouble? Here are some good reasons:
1. Problem solving places the focus of the students’ attention on ideas and sense making rather than on following the directions of the teacher.
2. Solving problems develops the belief in students that they are capable of doing mathematics and that mathematics makes sense.
3. Problem solving provides ongoing assessment data that can be used to make instructional decisions, help students succeed, and inform parents.
4. It is a lot of fun!
A Three-Part Lesson Format
A problem-solving lesson should always have three components: before, during and after.
Before: Your task in this part of the lesson is to get students mentally prepared to work on the problem. You want to be sure they understand the task. You want to get them thinking about the kinds of ideas that will help them the most. You want to be sure they understand their responsibilities beyond getting an answer.
Depending on the task you might do a related or simpler task. For computational situations you may want them to do the task mentally or make and share estimates before working with paper and pencil.
During: Many teachers find this step the hardest. Your first task here is to LET GO! Have faith in your students. Give them a chance to work without your constant guidance so that they will use their ideas and not simply follow directions. Be an active listener. Find out how different children or groups are thinking, what ideas they are using, how they are approaching the problem. This is a time for assessment. Take notes. Do not be afraid to offer hints but don’t guide students to the point that you slip back into directed teaching.
After: Have different teams or individuals share solutions, approaches and justifications. DO NOT EVALUATE. Be an attentive listener to both good and not so good ideas. Require all students to listen and have the class assess other teams’ ideas. “Terrell, what do you think of Sarah’s group’s method? Could you solve another problem using their method? Do you think it works all the time?” Support effort but expect students to do good work. Praise of individuals suggests they did something unusual and is negative feedback for those who do not get praise. Doing good mathematics is intrinsically rewarding.
Be certain to plan ample time for this portion. Often this is when the best learning will take place. Twenty minutes or more is not at all unreasonable for a good class discussion and sharing of ideas.
In the Before portion of the lesson, have students supply the missing part of 100 after you supply one part. Try numbers like 80 or 30 at first then try 47 or 62. You might also ask students if the answer to the problem is more or less than 300. Now let go. Students here will develop an add-on approach to subtraction.
How many small squares (ones) will fit in a rectangle that is 54 units long and 36 units wide? Use base ten pieces to help you with your solution. Make a plan for figuring out the total number of pieces without doing too much counting. Explain how your plan would work on a rectangle that is 27 units by 42 units.
This last example is designed to help students develop a method for multiplying two 2-digit numbers. Earlier tasks may have been rectangles such as 30 by 8 or 40 by 60. The discussion or after portion of the lesson may take place the second day with as much as a whole period devoted to developing written schemes for solving this problem.
If you didn’t know the answer to 12-7, what are some ways you could find the answer?
Here students working alone or in pairs will come up with a wide variety of strategies for this fact. In the after portion of the lesson, focus on those strategies that are most efficient. If no one uses a make-ten strategy (3 on to 7 is 10 and 2 more is 5), you will know to plan a subsequent lesson involving ten frames and adding on to make numbers in the teens.
Tips and Suggestions from Teachers
The following ideas have been learned from elementary teachers who have been working hard at developing a problem-solving approach in their classrooms.
1. Predict! Don’t hope. When planning a task it is not sufficient to think about how it will work if everything goes well. “Hopefully the kids will …” Rather, think about your students – all of them. What are they most likely to do. Predict as many responses as possible. Be prepared to deal with them. You may find yourself looking for a different task.
2. Tasks come in all sizes. A good lesson can be built around a single task. However, many tasks require only ten minutes. Others may take two days or more.
3. Be clear in your own mind about the purpose of the task. Don’t select problems simply because they are interesting. Always ask “What ideas will students be working with or develop by working on this task.”
4. There is much more to a problem than the answer. Often the most learning goes on in the discussion about the answer, the various paths to the solution, and most importantly deciding why an answer is right or wrong. When you the teacher determine for students whether an answer is right or wrong, you eliminate discussion. When children find out they can determine the correctness of answers they learn that mathematics makes sense.
5. Do not confuse open-ended problem solving with encouraging creativity. Good problems often have multiple paths to a solution and these lead to rich discussions. However, open-ended solutions are not the same as encouraging creative stories, artistic drawings, or having students make up their own numbers for a problem. These latter approaches often detract from the mathematics of the task as students spend time coloring or writing colorful stories. Student-invented numbers often add little value and can create real difficulties if inappropriate numbers are selected.
6. Distinguish between conventions and conceptual ideas. Many things in mathematics are the result of common conventions that have developed over many years. Tens are recorded to the left of the ones place. The numerator counts how many parts in a fraction are being considered. These conventions must simply be told to students. Concepts or mathematical ideas can be developed through problems or with minimal “judicious telling.”
FAQ’s Concerning Teaching Through Problem Solving
The following are typical of the questions teachers have about this student-centered approach to teaching mathematics.
1. How can I teach all those basic skills I have to teach?
Number concepts, basic facts and even computational procedures can be developed as examples in this paper have demonstrated. Even a standard algorithm can be explored as a problem after children have invented a variety of meaningful methods of their own. Show how a standard procedure works and pose the problem of figuring out why it works. Even traditional mathematics makes sense and children can be expected to make sense of it.
2. If I am not supposed to “tell” or “explain” why is it OK for one of my students to explain?
First, students will question their peers when an explanation does not make sense to them. However, from the teacher, the explanation is usually taken on faith – it comes from the Teacher! Second, When students explain, the class develops a sense of pride and confidence that they can figure things out and make. They have power and ability.
3. Where can I find the time to cover everything? This approach takes a lot of time.
First, teach with a goal of developing the “big ideas,” the main concepts in a unit or chapter. Most of the little skills and ideas in your list of objectives will be covered that way. If you focus on the little bits on the list, big ideas and connections are unlikely to develop. Second, We spend far too much time reteaching in this country because students don’t retain ideas. By spending time up front to help students develop meaningful networks of ideas, time spent in review and reteaching is drastically reduced.
4. Where can I find enough tasks? I don’t have time to search for problems.
There never will be a perfect set of tasks for your class; every class is different. Here are places to begin: 1) Look at the lessons in your textbook. Many of the teach or explain sections on the pupil pages can be transformed into good problems. Try it! 2) Read the NCTM journals regularly! Copy and file articles to go with each chapter in your text or each section of your curriculum. Don’t wait until you need an idea to begin flipping through journals. 3) Check out NCTM publications. The Addenda Series books are excellent and fit perfectly with the ideas in this paper. 4) Check out the reform curricula at your grade level. Buy a single module to get your feet wet. See these Implementation Centers. For elementary: Alternatives for Rebuilding Curricula (http://www.comap.com/arc). For middle school: Show-Me Center (http://showmecenter.missouri.edu).
5. Is there any place for drill and practice?
Absolutely! The error is to believe that drill is a method of developing ideas. Drill is appropriate when a) the desired concepts have been meaningfully developed, b) flexible and useful procedures have been developed, and c) when there is a real need for speed and accuracy. Not very many items in the curriculum meet all of these criteria today.
6. What do I do when a task bombs?
Do not give in to the temptation to “tell ‘em.” Set it aside for the moment. Ask yourself why it bombed. Did the students have the ideas they needed? If not, that tells you where to go next. Was the task too advanced? Often we need to regroup and offer students a simpler related task that gets them prepared for the one that proved difficult. Was the task getting at the ideas you wanted to develop? When you sense a task is not going anywhere, regroup! Don’t spend days just hoping that something wonderful might happen. If you listen to your students you will know where to go next.
What is Basic in Mathematics?
The most fundamental or basic thing to keep in mind is this: MATH MAKES SENSE! Children learning through problem solving will soon understand this and more importantly will come to believe they are capable of making sense of mathematics. We can get all of our students to this level of confidence – all students! To begin we must first believe in our students.
Commission on Standards for School Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: NCTM.
O’Brien, T. C. (1999). Parrot math. Phi Delta Kappan, 80, 434-443.
von Glassersfeld, E. (1995). A constructivist approach to teaching. In L. P. Steffe & J. Gale (Eds.), Constructivism in education (pp. 3-15). Hillsdale, NJ: Erlbaum.
Battista, M. T. (1999). The mathematical miseducation of America’s youth: Ignoring research and scientific study in education. Phi Delta Kappan, 80, 424-433..
Hiebert, J. (1999). Relationships between research and the NCTM Standards. Journal for Research in Mathematics Education, 30, 3-19.
Schoen, H. L., Fey, J. T., Hirsch, C. R., & Coxford, A. F. (1999). Issues and options in the math wars. Phi Delta Kappan, 80, 444-453.
Presentation for the 77th Annual Meeting of NCTM, April 23, 1999
John A. Van De Walle passed away in 2006. His passion for mathematics education is missed.