"The depressing thing about arithmetic badly taught is that it destroys a child's intellect and, to some extent, his integrity. Before they are taught arithmetic, children will not give their assent to utter nonsense; afterwards they will."
-- W.W. Sawyer, Mathematician's Delight
"Ours is not to reason why; just invert and multiply."
-- rhyme used to remember the algorithm for dividing fractions
Teachers see it all the time: otherwise intelligent and curious children who check their brain at the door as math time begins. "Just give me the rule" they demand, when asked to solve a problem. These children have learned, from previous school experience or from home, that mathematics is just following a bunch of rules that don't often make sense. Fortunately, teachers using Investigations are seeing fewer and fewer students with this behavior. Most students who have had Investigations since the early grades expect math to make sense and look for reasons behind steps and procedures they encounter.
Even so, many Investigations teachers still feel anxiety about what to teach their students regarding algorithms. Investigations advocates treating the conventional American algorithm for each operation as just one more way to perform the operation. As with any other strategy, students using a conventional algorithm should be able to explain why it works. It's not that there's something wrong with the conventional algorithms. But our ultimate goal is computational fluency and there are other efficient, accurate algorithms that students understand better and that can be notated in a way that doesn't obscure the meaning of the numbers. Using these methods, students more easily achieve computational fluency rather than blindly applying poorly understood rules.
Given this goal of computational fluency, it is critical that by the upper elementary grades, students have at their fingertips whatever algorithms they have chosen to use for the four operations. According to Susan Jo Russell, principal investigator for Investigations, students at this age need to consolidate one or two approaches that they use most often for each operation, including having a clear way of keeping track of multiple steps in a problem. Teachers need to help students with this process of consolidation and generalization so that their strategies become fluent. If teachers don't do this, they end up wanting to fall back on the conventional algorithms, because it doesn't look like their students have efficient strategies.
Some teachers, however, feel compelled to deviate from what the Investigations program recommends, by teaching and drilling the familiar (to us) conventional algorithms. There seem to be three reasons why teachers feel a need to explicitly teach the conventional American algorithms:
- in response to pressure from parents who only know one method or from teachers at the next grade level
- to prepare students for a standardized test that assesses for knowledge of specific algorithms
- because they feel it is important for students' education
The first reason reflects that we live in a real world where, alas, not everyone appreciates the Investigations approach. In these situations, you may have to make concessions and explicitly teach the conventional algorithms to some degree. But be sure to ask yourself: is this a real need from parents and teachers or just a perceived one? Talk to parents and other teachers to find out the real nature of their concern. Perhaps they just want students to be able to solve computation problems accurately and quickly, and they are unaware that other algorithms can accomplish this too. As for standardized tests, your students should do well on these tests without knowing the conventional algorithms. Perhaps all they need is some familiarity with particular notation and vocabulary.
For some teachers, though, it's not just external circumstances, such as parents and middle school teachers, that are leading teachers to supplement Investigations with teaching the conventional American algorithms. Deep down they believe that students should know and value these time-honored methods for performing addition, subtraction, multiplication, and division. Like their students' parents, almost all teachers grew up using only these methods for computation. Unlike most parents, they also have invested huge amounts of time and energy teaching these methods over their teaching careers. So naturally it is hard to let go of the conventional algorithms overnight. Teachers should give themselves some time to appreciate the idea that there is not one best algorithm for each operation. One way to do this is by practicing using alternative procedures when in real life problem situations, such as mentally determining the change from $10 when the bill comes to $3.59. Many Investigations teachers find themselves supplementing with teaching the conventional algorithms a little less each year, as they feel more comfortable that this is the right thing to do with their students.
As teachers gain more experience with student-generated strategies that make sense, they start to see how often the "making sense" part is lost on students using just the conventional American algorithms. As an example, let's look at multiplying 36 x 3 with the conventional method. First you do 3 x 6 = 18, where just the 8 is written down and the 1 (ten) is carried over and written above the 3 in 36 (in the ten's column). Then 3 x 3(0)=9(0), then 9(0) + 1(0)= 10(0), which gets written to the left of the 8 already written, to give an answer of 108. The problem here is that usually students do not think in terms of the numbers written in parentheses, and when pressed to explain what they're doing often are unable to do so. Using a different algorithm, a student may think of 36 as 30 + 6, find 30 x 3 = 90, then 6 x 3 = 18, and then 90 + 18 = 108. Both algorithms actually do the same thing; they just find the partial products in a different order. The difficulty is that students who ONLY know the traditional algorithm usually cannot find the answer without pencil and paper, i.e., mentally. And more important, when they get to algebra, they are unable to make the connections between the arithmetic 36 x 3 and the same problem using variables and the distributive property: (a+b)c = ac + bc, or (30+6) x 3 = 30 x 3 + 6 x 3.
While the developers of Investigations don't recommend forcing students to use the historically taught American algorithms, they don't recommend hiding these algorithms either. There is no reason why these algorithms shouldn't come into the classroom in the later grades, as algorithms that students will see many people in our society use. Again, the criteria for using them are the same as for any other algorithm: the student must be able to explain it and must have more than one approach available to match a solution strategy flexibly to the problem situation.
To summarize, what do we want our students to learn about algorithms? The list below is adapted from a conversation with Dr. Christine Moynihan, mathematics and science specialist in Wayland, Massachusetts and from the 1998 Draft of the NCTM's Principles and Standards for School Mathematics.
Goals for Learning about Algorithms:
- Each student is provided with opportunities to develop and use their own algorithms. This is doing mathematics, instead of following a recipe.
- Each student understands the algorithm s/he uses and can explain it, whether it's the conventional American algorithm or some other algorithm.
- Each student realizes that there is no one "right" algorithm for addition, subtraction, multiplication, and division. Certain algorithms are better for certain situations.
- Each student needs to be provided with ample opportunities to use his/her algorithms in problem situations that are meaningful and varied in context. This will allow for decision-making in terms of which algorithm may fit the situtation better than another.
- Each student uses efficient and accurate algorithms for computing.
The fifth goal stated above raises another issue about algorithms: how does an Investigations teacher help all students to develop efficient computational algorithms on their own? We hope to start a conversation about this common question with our readers. Please send in your ideas by clicking on "contribute a comment" below.
In closing, keep in mind that if you teach the conventional American algorithms too soon, students will abandon their invented algorithms. It's almost better to not let students develop their own algorithms at all than to send the message that what they have developed is not worthy of being "real" mathematics, and that they should instead use the "real" algorithm that the teacher has presented. With Investigations, teachers have an opportunity to make mathematics the active discipline it is meant to be, a discipline that recognizes that, although there may be one correct answer, there are often many ways to get there.
CESAME Support Site for Investigations Users
December 1999
This information was reprinted with permission of CESAME, Northeastern Univ., and the Educational Alliance, Brown University.
