The Separate Path and the Well Travelled Road

3.31.10 – Barry Garelick – Much has been written about the debate on how best to teach math to students in K-12—a debate often referred to as the “math wars”. I have written much about it myself, and since the debate shows no signs of easing

The Separate Path and the Well Travelled Road

Much has been written about the debate on how best to teach math to students in K-12—a debate often referred to as the “math wars”.  I have written much about it myself, and since the debate shows no signs of easing, I continue to have reasons to keep writing about it. While the debate is complex, the following two math problems provide a glimpse of two opposing sides:

Problem 1:  How many boxes would be needed to pack and ship one million books collected in a school-based book drive?  In this problem the size of the books is unknown and varied, and the size of the boxes is not stated.

Problem 2:  Two boys canoeing on a lake hit a rock where the lake joins a river.  One boy is injured and it is critical to get a doctor to him as quickly as possible.  Two doctors live nearby:  one up-river and the other across the lake, both equidistant from the boys. The unhurt boy has to fetch a doctor and return to the spot.  Is it quicker for him to row up the river and back, or go across the lake and back, assuming he rows at the same constant rate of speed in both cases?

The first problem is representative of a thought-world inhabited by education schools and much of the education establishment.  The second problem is held in disdain by the same, but favored by a group of educators and math oriented people who for lack of a better term are called “traditionalists”. 

In the spirit of full disclosure, I fit in with the latter group. I also plan to teach mathematics when I retire soon.  I have a more than brushing acquaintance with the thinking of the education establishment side of the war from classes I have taken over the last few years at education school

I have heard various education school professors distinguish between “exercises” and “solving problems”.  Textbook problems are thought of as “exercises” rather than “problems” because they are not real world and therefore, in their view, not relevant to most students.  Opponents of such problems contend that they contain the quantities students need to solve the problems and therefore do not require students to make value judgments.  Such criticisms of traditional approaches in mathematics have led to an approach that I call “The Separate Path” because it generally takes students on a path they have never seen before. 

         The first problem presented above is an example of a “separate path” type problem which appeared in a paper called Teacher as designer: A framework for teacher analysis of mathematical model-eliciting activities, by Hjalmarson and Dufies Dux.  The problem is called “The Million Book Challenge”.  While it may be engaging, students will generally lack the skills required to solve such a problem, such as knowledge of proper experimental approaches, systematic and random errors, organizational skills, and validation and verification. 

         Problems such as the million book challenge are predicated on the idea that by repeatedly confronting students with new situations as well as problems with which they have little to no experience, they will develop the “habits of mind” that lead to mathematical reasoning. The open-endedness of the problem is seen as a means to engage students in the “process” of critical thinking.  In my opinion, however, such approach is like learning German by practicing particular sentence constructions in English (e.g., “I know that he the book read has”) in the hopes of building up a structure that then only needs vocabulary to complete the learning process. 

         Let’s turn now to the second problem—a problem appropriate for an honors algebra 1, or regular algebra 2 class.  Unlike the million book challenge, it allows students to rely on prior knowledge, it is well-defined, and has specific mathematical goals.  I think of problems such as the canoe problem as going down a well travelled road—you know where it is taking you even though it may have a few twists and turns and detours in unfamiliar territory.  

While problems such as the million book challenge are touted as showing how the real world describes what’s going on more than mathematics alone can do, the canoe problem challenges that view. It demonstrates that mathematics is needed to describe what is going on—and contradicts what one intuitively believes.

Many students will assume that either route will take the same amount of time.  They reason that if the boy travels upstream and downstream for the same distance at the same rate of speed, the amount the canoe is slowed by the current when travelling upstream is cancelled by the additional speed the current imparts when travelling downstream.  But the mathematics shows otherwise.

Letting D be the distance to each doctor, R the rate of speed of the canoe and C the speed of the river current, we know that since distance = rate x time, the time to go across the lake and back is 2D/R.  The time to go up the river and back is D/(R-C) + D/(R+C).  If the time to cross the lake is equal to the time to go up river and back, then 2D/R would equal 2DR/(R2 – C2) . .

Algebraic exploration shows, however, that 2D/R is less than 2DR/(R2 – C2); thus it takes less time to go across the lake and back than to go up the river and back. Note that this problem does not provide any values for distance, speed of boat or speed of current.  Students must model the situation using symbols, and apply their knowledge of expressing time as a function of rate and speed. It is also interesting that this problem does not lend itself to a “plug-in” solution.

 Also, the proof that 2D/R is less than 2DR/(R2 – C2) is not easy and it may well be that students will have some difficulty with this part of the problem.  Proofs of inequalities do not easily lend themselves to algorithmic solutions–in fact, they require the critical thinking and analytical skills that many believe problems like the Million Book Challenge develop.  To do this problem in the later part of an Algebra 1 course or in Algebra 2 would not introduce anything students haven’t learned.  The student has the tools with which to do it, though this doesn’t mean that all students will be able to solve it.  Some students may find such problems very difficult.  But according to Vern Williams, a middle school math teacher, and who served as a member of the President’s National Mathematics Advisory Panel, “By taking math that has been taught to them and attempting to solve difficult problems, they will discover relationships between content and methods that they already have in their arsenal even if they don’t solve the problem or arrive at the correct answer.”   

The issue of separate path type problems versus the well travelled path has particular significance in light of the recent interest in developing national assessments for math.  Critics of the well-travelled road approach to math tend to believe that assessments should evaluate the “critical thinking” skills of students rather than having students solve “exercises” that lend themselves to applications of previously learned procedures.  Many such critics also believe that students in other countries that surpass the US in math on international tests are being taught only how to take tests.  Ironically, the problem with a test that emphasizes the separate path type of problems is that it accommodates students’ learning how to answer open-ended questions.  The U.S. may then achieve high test scores, but when process trumps content, what mathematics are students really learning?  In the end the problems which students in Singapore, Hong Kong and Japan excel at solving are still likely to be off the script for many US students.

In a world in which problems that have a unique answer obtained through systematic application of mathematical skills and principles are deemed “mere exercises”, students are heading down a separate path approach to learning that leads at best to math appreciation, and at worst, a turn-off to the subject.  If, however, students are taught the skills and concepts necessary to solve well-defined and challenging problems, they will learn to surmount what a disheartening number of U.S. students now consider to be insurmountable.

Barry Garelick is an analyst for the Environmental Protection Agency in Washington DC.  He majored in mathematics at the University of Michigan, and is a co-founder of the US Coalition for World Class Math, a group of mathematically literate parents and educators who want excellence in mathematics education in the U.S.


  1. Matthew

    Fascinating article. There must be more emphasis on algorithms and sufficient practice.

  2. Niki Hayes

    Interestingly, Barry's article echoes another leader's view about U.S. math education. In a 1984 American School Board Journal, John Saxon, the maverick math teacher, author, and publisher who was immensely disliked by the math establishment as supporters of "discovery" learning, wrote the following:

    “I contend that our job is to teach rewarding responses to mathematical stimuli, to teach thought patterns that have been found to lead to the solutions, to allow the students to practice reacting to the stimuli with these thought patterns and to be rewarded with the warm feeling of pride that accompanies the correct answer… Mathematics classes can become warm sanctuaries towards which students gravitate because there they are asked to solve puzzles by using familiar thought patterns.”

    Further, Saxon said that students must be taught that many problems can be classified according to type and can be solved by using particular thought processes. If the authors carefully select for emphasis the important skills, the student also develops a firm foundation of basics on which it is possible to stand comfortably and reach for understanding of advanced concepts.

    He also said that mathematics students who learn the standard “coin problems, time-to-do-a-job problems and trains-leaving-Detroit-at-midnight-problems” would benefit from these because they represent the ancient mathematicians’ gifts of concepts and skills, which are the foundation of modern problems in engineering, physics, and advanced math classes. He accused the reformists/constructivists of dismissing this approach because they wanted to prepare students not for the hard sciences but to make them feel comfortable with mathematics in their everyday lives; hence, their focus is on "open ended" questions that have no "defined" answers.

    More on the philosophy and methodology of John Saxon and his revolutionary textbook curriculum that proved unbelievably successful can be found at

  3. Kirthi Raman

    A very interesting analysis. I used to wonder why students from USA are falling back in Math Olympiads. I also agree that if a student learns the concepts,understand them and learns how to model real world problems then they tend to do much better in their career path.

  4. MRM

    I disagree with you about the difference between “exercises” and “problems”. It’s not a case of “textbook” vs. “real world”.

    In your example above, the problem with the million books is ill-defined and fits neither category as far as math education is concerned (although problems like that–where the input is unknown and variable–are very useful for computational geometry, a topic in computer science, and have very real applications in manufacturing, etc.)

    An “exercise” is an activity where the students know up front which algorithm to use to find the solution. Someone has already shown the students how to do a similar example, and they are basically just walking through the same steps. (“How do you divide fractions?” “Invert and multiply.” “I’ll do an example.” “Let’s do a million examples together.” “Now you do another load of examples at your seat.” Now “How many 1/4-pounders can we make out of 3 pounds of meat?”)

    A problem is a (well-defined) scenario where students are required to determine the steps to find the solution themselves. They are not just digging through the recipe box in their memory to find the determine what the steps are.

    Exercises are important for students to learn mathematical procedures; but they are like playing scales in music, phonics in reading, or calisthenics in P.E. At some point, if you want students to learn how to THINK mathematically, you need to have them practice thinking.

    Real problems–whether they are real-life scenarios or mathematical proofs–get the students to think, not just recall “Where have I seen this before?” Obviously, different students are able to stretch different amounts to make connections of concepts in order to find a solution. But ultimately, they need to be able to convince you using a logical argument that their solution is mathematically correct.

  5. Barry Garelick

    Thank you for your comment.

    Your points are well taken, but I maintain that the prevailing thoughtworld in education schools is that even problems that require thinking through what steps are necessary (as opposed to digging through the “recipe box”) are viewed as “exercises”. One teacher I had in ed school said that Asian students do well on TIMSS exams because they have seen these types of problems before and therefore they are merely exercises. The point is, that as one progresses in math and becomes more versatile at solving problems, the difficult problems become a series of exercise-like steps. In essence, they HAVE seen these problems before, in different parts, but now have to put all these together. So solving a problem is like doing a lot of different exercises in a multi-step fasion. For example, this problem from an AMC 10 competition synthesizes many different mathematical ideas:

    “For a particular peculiar pair of dice, the prbabilities of rolling 1, 2, 3, 4, 5 and 6 on each die are in the ratio 1:2:3:4:5:6. What is the probability of rolling a total of 7 on the two dice?”

    Solving such problem requires a knowledge that the probability of getting at least one of the numbers from a roll of a die is 1. This can be expressed as the sum of the probabilities of rolling a 1, 2, 3…6. One also has to know the formula for summing consecutive integers, as well as how to compute the probability of getting a 7 on a single throw of two dice. All of these taken by themselves can be considered exercises.

    I would agree with you that the million book problem is ill-posed, but there are many who are thinking that providing such problems to students builds a path for problem solving, whereas more traditional problems do not. One teacher with whom I spoke summed up this philosophy with the following questions: “What happens when students are placed in a totally unfamiliar situation that requires a more complex solution? Do they know how to generate a procedure? How do we teach students to apply mathematical thinking in creative ways to solve complex, novel problems? What happens when we get off the ‘script’?”

    I agree with you that to do that we need to give students problems for which they have not seen the “worked example”. But there is some amount of scaffolding and preparation to get students to that step, and the ed school approach is to skip a lot of the scaffolding in the belief that students will learn the information and tools they need in order to solve the problem in a “just in time” sort of way.

    As a post script, some people think the canoe problem in the article is not good because it does not reflect what would happen in the real world–it’s too contrived. This lack of seeing the forest for the trees is all too prevalent, unfortunately,

  6. J. D. Salinger

    “if you want students to learn how to THINK mathematically, you need to have them practice thinking.”

    Yes, and exercises are a way to lead to that, because it gives them the tools with which to do such thinking, much as language allows you to express ideas.

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March 30th, 2010

Barry Garelick Contributor

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