Discovery learning in math: Exercises versus problems Part II

Barry Garelick - July 26, 2009
Columnist EducationNews.org

Making Meaningful Discoveries

               To better explain the effectiveness (and efficiency) of carefully sequenced, well-posed problems, let me use my favorite example: my daughter. When she was 10, I tutored her on addition and subtraction word problems, and I found that her understanding deepened as we worked on her procedural fluency with problems that demanded new, harder applications. Using the Saxon math series, Math 65, (Hake, et. al., 2001) we were on a chapter that explained how to do certain types of word problems.  Specifically, these were called “Some and Some More” and “Some Went Away.”  Some and Some More was defined by way of an example: “Before he went to work, Tom had $24.50.  He earned $12.50 more putting up a fence.  Then Tom had $37.00.”  Some Went Away was defined similarly: “Tom took $37.00 to the music store.  He bought a pair of headphones for $26.17.  Then Tom had $10.83.”  

                I asked my daughter if she understood what was going on in each type.  “Yeah, you add the numbers together for the Some and Some More and you subtract for the Some Went Away.”  She seemed very pleased with this and added, “Seems pretty easy.”

                The next few practice problems, however, weren’t quite as easy as she thought. They weren’t exactly like the sample problems; she needed to combine the new procedures with prior knowledge in order to solve them. Along the way, her understanding deepened substantially.  For example, one problem was “After losing 234 pounds, Jumbo weighed 4,368 pounds.  How much did Jumbo weigh before he lost the weight?”  Another was “The price went up from $26 to $32.  By how much did the price increase?”

                We read the first problem.  I asked her what type of problem it was.  Silence.  “Where are you stuck?” I asked. “It doesn’t tell me how much the elephant weighs,” she said.  This was correct; it was different than the initial presentation. “Does the elephant lose some weight?”  I asked.  She nodded.  “So is it a Some and Some More?” I asked. She replied, “No, it’s a Some Went Away because the weight goes away.”

                Because this was an example problem, the solution was given in the text.  It was represented as follows:

                                Before:   Jumbo weighed…              W pounds

                                Then:     Jumbo lost…                      –234 pounds

                                Now:    Jumbo weighs… 4,368 pounds

                I showed her the set-up, and she knew immediately what to do. “Oh, you add 4,368 and 234 to get what he weighed before.”   She knew this because in previous lessons she had solved problems with missing numbers in subtraction (e.g. F – 15 = 24).  With these carefully sequenced lessons, she had been well prepared; she knew how to solve a missing number in a subtraction problem. Now, she discovered how the Some Went Away scheme worked in conjunction with her prior knowledge and was able to make the connection.   Similarly with the second problem, now knowing how the schemes connected with prior knowledge, she readily identified the problem as falling in the Some and Some More category. From previous experience she knew how to solve such problems. “You just subtract: $32 minus $26 is $6.”

                I asked if she could write it out as a “missing number in addition” problem using a letter for the missing number.  She reluctantly complied: X + $26 = $32.  I noticed that she resisted putting the problems in this form.  When I asked her why, she said “Why bother? I can just add or subtract and get the answer.  Besides, that’s algebra and I’m not ready for it.”  I decided not to argue that point given that I was tired, things were going well, and the steps for solving simple equations in algebra were clearly falling into place.

                By contrast, if all the subtraction word problems had been in the form of “John had $30 and spent $15; how much did he have left?” there would have been no discovery.  (This is originally what my daughter was expecting—and hoping—would happen).  If an assignment is constructed properly, as it was in the case of these problems, students should have an unscript-like understanding of the material by the time they are done—whether conducted in class or as homework They should also have some level of “aha” experience brought about by procedural fluency.  With my daughter, I’ve seen that the procedural fluency resulting from the exercises helps clarify the concept, even if it wasn’t fully understood before starting the problem set. 

Scaffolding

This process of extending previously learned material to slightly more difficult problems, in the jargon of education, is called “scaffolding”.  When done properly—whether in a set of homework problems or in a classroom—it is an extremely effective method to bring about meaningful discovery.  I have had the pleasure of observing a teacher who does this very well. Let me tell you about a lesson I observed from the teacher’s honors geometry class. Having already taught the students the basics of trigonometry and how to use it to find the lengths of the sides of a right triangle in a previous lesson, she drew the following figure on the board:

She asked the class to find the perimeter of the triangle (which is not a right triangle) given the two angles, and length of one side as shown.  She told them that they had all the knowledge necessary to solve it. 

The key to solving this problem is to draw in the altitude and then, using trigonometry, solve for the lengths of various sides. 

Some students drew in the altitude and solved it.  Others were stymied and said they couldn’t solve it “because it isn’t a right triangle.”  The teacher provided a hint: “Well maybe you have to create a right triangle.”  They thought about this and understood that they needed to draw the altitude.  I asked one girl who had drawn in the altitude without any hint from the teacher her how she knew what to do.  She explained that drawing in the altitude of a triangle was something she did automatically any time she had a triangle—she didn’t really know why. In this case an automatic procedure helped her to solve the problem—something I’ll come back to in the next section on critical thinking.   

 Algorithmic Procedures and the Development of Critical Thinking 

                 Mathematics demands mastery of foundational steps in order to build upon them. As such, it is relentlessly linear. Without such mastery or foundation students will not be prepared to solve new and complex problems. Yet some practitioners believe that the above examples are “inauthentic work”—mere exercises—and as such, they do not lead students to learn how to do the “authentic work” of solving problems.

                In fact, the application of learned and mastered material in new, off-the-script context does not happen immediately—nor is it brought about by giving students problems which they‘re not equipped to solve. Daniel Willingham, a cognitive scientist who teaches at the University of Virginia, maintains that it takes time and effort for knowledge to accumulate to the point that connections between learned material and new and difficult problems can be made. (Willingham, 2002).  Willingham refers to the difficulty that novices have with thinking critically as “inflexible thinking,” which he characterizes as perfectly normal and to be expected among students.  Specifically, the critical thinking that allows for the application of prior knowledge to new, unfamiliar type problems requires recognition of an underlying principle that can be used to solve the problem.  The underlying principle can be found in any number of simpler problems to which a person has been exposed—the challenge is in looking beyond what Willingham calls the surface structure to see the deep structure.  For example, the following two problems are based on the same underlying principle—rate—and in fact the same equation is used to solve both of them:   

1)            John can mow the lawn in 20 minutes while his brother Bob can mow the same lawn in 30 minutes.  If both mow at a constant rate, how long does it take for both of them to mow the lawn together?

2)            It takes John 20 minutes to walk to school from home, while it takes his sister 30 minutes, both walking at constant rates of speed.  John starts walking from school to home at the same time that his sister starts walking from home to school.  How long will it take for them to meet?

                (1/20 + 1/30)X = 1 For the first problem the rate is the portion of the job completed per minute.  Specifically, John's rate is 1/20 of the job per minute and Bob’s rate is 1/30 of the job per minute.  The amount accomplished in X minutes is (1/20)X for John and (1/30)X  for Bob.  Therefore, the portion of the lawn mowed (or job completed) in X minutes by both John and Bob working together is (1/20 + 1/30)X.  In setting it up this way, we are adding up their accomplishments in terms of how much of the job is completed in X minutes.  Since the problem asks how long it takes to mow the lawn, we are interested in exactly one job, and want to find what value of X is needed to get this done.  To do this, we set the above expression equal to 1 and solve for X: (1/20 + 1/30)X = 1

                The second problem is solved using the same reasoning as the first.  The key to solving it is to see the connection between distance and time, and that this is still a rate problem.  In this problem, however, the rate is the portion of the total distance walked per minute.  In one minute John walks 1/20 of the way between school and home, while his sister walks 1/30 of the way.  The solution follows as described above for the lawn mowing problem, and results in the same equation.           

                Beginning algebra students may understand how to solve the first problem but may not make the connection that the same concept of rate underlies the second problem as well.  In fact, as Willingham explains, it is unlikely that students will make such connections readily until they have developed true expertise. Only experts see beyond the surface level of a problem to its deeper structure.

                So how do you teach students to make such connections; i.e., to think critically?  Is it a failure of the math program—or the teacher—if they do not?  Willingham (2002) argues that understanding the deep structures of a discipline such as mathematics is an important goal of education, “but if students fall short of this, it certainly doesn’t mean that they have acquired mere rote knowledge and are little better than parrots.” Rather, they are making the small steps necessary to develop better mathematical thinking. Simply put, no one leaps directly from novice to expert.

                While there is no direct path to learning the thinking skills necessary to apply one’s knowledge and skills to unfamiliar territory, Willingham (2002) argues that one way to build a path from inflexible to flexible thinking is to use examples.  This approach could be used for rate problems such as the two problems just described.  In fact, this is what was done in the two well crafted lessons I observed.  Students extended their knowledge along scaffolding built from examples—examples that fit on the underlying structure.

                Although it does not necessarily happen automatically, thinking becomes more flexible as more knowledge and experience are acquired. Think of the girl in the high school geometry class who solved the triangle problem. What had become a mechanical or algorithmic habit for her—drawing in the altitudes of triangles—ultimately led to the solution?  Many problems in mathematics involve evaluating their form through algebraic manipulation, or in the case of geometric figures, adding supplementary lines.  Such analysis leads to insights about a problem’s underlying structure. The girl’s habit, which some might consider algorithmic thinking (and therefore “inauthentic”) was part and parcel of her flexibility in thinking and applying a previously learned principle in a novel way to a new problem.

Students should certainly be given increasingly challenging problems—but problems that draw upon what they have learned.  As discussed above, applying what has been learned to new problems one hasn't seen before does not simply happen. It comes about from practice and acquiring tools and experience using them.  If you look at an expert "problem solver", someone who is a mathematician, engineer, or scientist, say, you'll find someone who has command over many procedures and skills and has memorized many things.  Such experts have so much extensive background knowledge that many problems—for them—are like exercises.  People who know math can look at a problem and based on their knowledge and experience will break the problem down and apply a straightforward, mathematical solution. For people who haven’t mastered what they need to know in math, just about everything is a problem that is not easily broken down. 

Students given well-defined problems that draw upon prior knowledge, as described in this article, are doing much more than simply memorizing algorithmic procedures. They are developing the procedural fluency and understanding that are so essential to mathematics; and they are developing the habits of mind that will continue to serve them well in more advanced, college level mathematics courses. Poorly-posed problems with multiple “right” answers turn mathematics into a frustrating guessing game.  Similarly, problems for which students are expected to discover what they need to know in the process of solving it do little more than confuse. But well-posed problems that lead students in manageable steps not only provide them the confidence and ability to succeed in math; they also reveal the logical, hierarchical nature of this powerful and rewarding discipline.

References:

Hake, S. et. al , Math 65: An Incremental Development, 2nd ed. (Norman, OK: Saxon Publishers, 2001).

Interactive Mathematics Program Year 1; Key Curriculum Press (Emeryville, CA 1997)

Ma, Liping, Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States (Mahwah, New Jersey: Lawrence Erlbaum Associates, 1999), 115.)

Primary Mathematics Standards Edition: Textbook 4A (Singapore: Marshall Cavendish Education, 2008).

Russell, S, et. al. , Investigations in Number, Data, and Space: Implementing Investigation in Grade 4 (Printed in the United States of America: Pearson Education, 2008).

Willingham, D.,  “Inflexible Knowledge: The First Step to Expertise,” American Educator 26, no. 4 (2002): 31–33, 48–49.

Discovery learning in math: Exercises versus problems Part 1