Discovery, authenticity and understanding, Part I.

(Originally published at Nonpartisan Education Review in different form. This is an update and revision of same).

By way of introduction, I am a math teacher, but not a mathematician.  I majored in math and have used it throughout my life including my pre-teaching career in which I worked in the field of environmental protection. My facility with math is due to good teaching and good textbooks. The teachers I had in primary and secondary school provided explicit instruction and answered students’ questions; they also posed challenging problems that required us to apply what we had learned. The textbooks I used also contained explanations of the material with examples that showed every step of the problem solving process.

I fully expected the same for my daughter, but after seeing what passed for mathematics in her elementary school, I became increasingly distressed over how math is currently taught in many schools.

Optimistically believing that I could make a difference in at least a few students’ lives, I decided to teach math when I retired. I enrolled in education school and obtained my certification in secondary math teaching, which spans grades 6-12. Although I had a fairly good idea of what I was in for with respect to educational theories, I was still dismayed at what I found in my mathematics education courses.

In class after class, I have heard that when students discover material for themselves, they supposedly learn it more deeply than when it is taught directly. Similarly, I have heard that although direct instruction is effective in helping students learn and use algorithms, it is allegedly ineffective in helping students develop mathematical thinking. Throughout these courses, a general belief has prevailed that answering students’ questions and providing explicit instruction are “handing it to the student” and preventing them from “constructing their own knowledge”—to use the appropriate terminology. Overall, however, I have found that there is general confusion about what “discovery learning” actually means. I hope to make clear in this article what it means, and to identify effective and ineffective methods to foster learning through discovery.

To set this in context, it is important to understand an underlying belief espoused in my school of education: i.e., there is a difference between problem solving and exercises. This view holds that “exercises” are what students do when applying algorithms or routines they know and the term can apply even to word problems. Problem solving, which is preferred, occurs when students are not able to apply a mechanical, memorized response, but rather have to figure out what to do in a new situation. Moreover, we future teachers are told that students’ difficulty in solving problems in new contexts is evidence that the use of “mere exercises” or “procedures” is ineffective and they are overused in classrooms.

As someone who learned math largely though mere exercises and who has creatively applied math in my work, I have to question this thinking. I believe that students’ difficulty in solving new problems is more likely to be because they are novices, not experts.  They have neither the experience, nor the requisite knowledge and/or mastery of skills to allow solving widely varying problems—not because they were given explicit instruction and homework exercises.

Those who make such a differentiation and champion “true” problem solving espouse a belief in having students construct their own knowledge by forcing them to make connections with skills and concepts that they may not have mastered. But, with skills and concepts still gelling students are not likely to be able to apply them to new and unknown situations. Nevertheless, the belief prevails that having students work on such problems fosters a discovery process which the purveyors of this theory view as “authentic work” and the key to “real learning.” ( This concern about “authentic” versus “inauthentic” work comes from progressive education reformers who believe that it’s best for students’ school work to be as realistic as possible, that is, for it to be focused on
learning about and trying to solve “real world” problems. )

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’?” 

Those are important questions, but I will argue in this article the following points: 1) “Aha” experiences and discoveries can and do occur when students are given explicit instructions as well as when working exercises; and 2) Procedural fluency does not exclude conceptual knowledge—it leads ultimately to conceptual understanding and the two are key for applying mathematics to complex problems.

      I’m not against asking students to discover solutions to novel and challenging problems—the experience can be quite powerful, but only under the right conditions. A quick analogy may be useful here. Suppose a person who knows how to drive automatic transmission cars travels to a city and is forced to rent a car with a standard transmission—stick shift with clutch. The person in charge of rentals gives our hero a basic 15 minute course, but he has no opportunity to practice before heading out. In addition to this lack of skill in driving a standard transmission, the city is new to him, so he needs to rely on a map to get to where he needs to go. The attention he must pay to street names and road signs is now eclipsed by the more immediate task of learning how to operate the vehicle. In fact, he would be wise to take a taxi in order to avoid a serious accident. But now suppose that prior to his trip he is told that he will need to drive a standard transmission because where he is going, rental car companies don’t rent out automatic transmission cars. With proper training and guidance, he can start off on quiet streets to get the feel of how to coordinate clutch with shifting, working up to more challenging situations like stopping and starting on hills. Over time, as he accumulates the necessary knowledge, and practice, he’ll need less and less support and will be able to drive solo. There will still be problems that he has to figure out, like driving in traffic jams that require starting, slowing, downshifting, and so forth, but eventually, he will be able to handle new situations with ease. Now, having already achieved driving mastery of the vehicle that will take him where he needs to, the task of driving in a strange city although challenging is more manageable.  He will be able to focus all of his attention on navigating through new streets.

      Whether in driving, math, or any other undertaking that requires knowledge and skill, the more expertise one accumulates, the more one can depart from the script and successfully take on novel problems. It’s essential that at each step, students have the tools, guidance, and opportunities to practice what they learn. It is also essential that problems be well posed. Open-ended, vague, and/or ill-posed problems do not lend themselves to any particular mathematical approach or solution, nor do they generalize to other, future problems. As a result, the challenge is in figuring out what they mean—not in figuring out the math. Well-posed problems that push students to apply their knowledge to novel situations would do much more to develop their mathematical thinking.


To be continued.


NCSM/NCTM Annual Conf, Dept.

Every year the National Council of Teachers of Mathematics has its annual conference, complete with celeb speakers, vendor booths, instructional seminars, and the usual array of topics that pass for effective practices.

From what I hear from a friend who teaches high school math, this year’s was no different.  Her report follows below:

I signed up for a pre-conference workshop on teaching math for social justice.  They made the accusation that colleges of education exacerbate the problem of achievement gap for minorities.  I asked an ed-school professor (maybe from Connecticut) what her school did to alleviate this problem.  Her answer sounded awfully general, so I asked her to give me one, explicit example of a topic they teach that would work toward alleviating social injustice in schools. Her example was that, oh, you can teach students that a comma can mean the same thing as a decimal point in other countries. Of course, this is not what they are talking about at all, so she missed the point. 
They made us do this activity where three siblings were going to give a party for their father’s 70th birthday.  One made something like $20,000 a month; another made $6,000 a month, and the third — a single mother with two children — made $3000 a month.  The sibling making $20,000 a month thought they should split the $4500 cost of the party equally.  Another sibling suggested amounts that are proportional to their salaries.  We were supposed to converse (in groups, of course!!) what is “fair.”  Of course, it launched into a huge discussion about missing information, such as the one who makes $20,000 a month may have a spouse with a disease that requires  a $5000 shot each month, so in other words, were weren’t told about their disposable incomes or other circumstances.  After about 20 minutes, we still hadn’t settled in on anything other than the proportional one.
My issue is this:  I think that MANY kids don’t know how to compute what would be proportional to the incomes in the first place, so why impose all of that drama on it? 
I also attended a session of which Phil Daro was a co-presenter, but had to leave right before he was on stage.  His partner, Kyle Pearce, didn’t know beans about math. Their big thing was about this photo of 5 reams of copy paper stacked against a concrete block wall, and how many reams would it take to reach the ceiling.  They gave the height of the ceiling and the height of the stack of 5 reams of paper.  I divided that height by 5, and then divided the height of the ceiling by that quotient.  I did not set up a proportion at all.  One can say that I used proportional reasoning, but I didn’t need to formally set up a proportion.  I didn’t like the way they presented the solution of the problem.  Clearly, it was designed for teachers who are at middle school or lower level.
The bottom line is that it was all pretty bad.  My school district paid a few thousand dollars to send my colleague and me to this stuff.  We were there for the last day of NCSM and the first day of NCTM.  Needless to say, I was glad to get home!!