**I’ve been challenged recently** to provide research which supports "*unassisted discovery*" over more traditional techniques for teaching math. This is not possible, as there are no teachers actually using "*unassisted** discovery*" in their classrooms.

First, it is not possible to engage in the act of "*unassisted discovery*" as a student. Just knowing the language to describe what you are working on is a clear sign that at the very least you have the support of your language and culture in whatever you attempt.

Second, if a teacher has chosen the activity for you, or designed the learning objects you will be using, then they have given you an enormous amount of help by choosing the space in which you will be learning. Even **Seymour Papert’s work with Logo** was assisted discovery, after all, Logo is itself going to direct the inquiry toward what is possible to do with the language.

I can’t give examples of research which supports unassisted discovery, but I can give research which supports discovery learning in general. Without searching too hard, I found the following supportive research:

Bonawitza, Shaftob, Gweonc, Goodmand, Spelkee, Schulzc (2011) **discovered that if you tell children how a toy works**, they are less likely to discover additional capabilities of the toy than if you just give it to them, suggesting that direct instruction is efficient but comes at a cost: "*children are less likely to perform potentially irrelevant actions but also less likely to discover novel information.*"

Chung (2004) **discovered "no statistically signicant differences"** between students who learned with a discovery based approach based on Constructivist learning principles as compared to a more traditionalist approach.

Cobb, Wood, Yackel, Nicholls, Wheatley, Trigatti, and Perlwitz (1991) **discovered that students who learned mathematics through a project based approach** for an entire year had similar computational fluency compared to a more traditional approach, but "*students had higher levels of conceptual understanding in mathematics; held stronger beliefs about the importance of understanding and collaborating; and attributed less importance to conforming to the solution methods of others, competitiveness, and task-extrinsic reasons for success.*"

Downing, Ning, and Shin (2011) **similarly found that a problem based learning approach** to learning was more effective than traditional methods.

Wirkala and Kuhn (2011) **very recently discovered that students who learned via problem based learning** "*showed superior mastery…relative to the lecture condition*."

**In a meta-study of nearly 200 other studies** on student use of calculators in the classroom the NCTM concluded that "*found that the body of research consistently shows that the use of calculators in the teaching and learning of mathematics does not contribute to any negative outcomes for skill development or procedural proficiency, but instead enhances the understanding of mathematics concepts and student orientation toward mathematics.*" (I’ve included this piece of research since many traditionalists oppose the use of calculators in mathematics education.)

Keith Devlin, in his book **The Math Instinct**, **cited research** by **Jean Lave** which found that people had highly accurate algorithms for doing supermarket math which were not at all related to the school math which they learned. In fact, people were able to solve supermarket math problems in the market itself with a 93% success rate, but when face with the exact same mathematics in a more traditional test format only answered 44% of the questions correctly. Later in the same chapter of his book, Devlin revealed more research suggesting that the longer people were out of school, the * more* successful they were at solving supermarket math questions.

It should also be noted that this discussion on what should be done to improve mathematics education shouldn’t be restricted to either traditional mathematics education, or discovery based methods, but that we should look at all of our **possible options**.

Absolutely fascinating. However, what mathematics would be given as the “toy” to play with? How much information should the student get before he or she runs off on his or her own? For instance, with derivatives, would simply giving the students the idea of tangential slope be enough? How much should the student have to rediscover?

In Pursuit of Truth

I think that how much discovery is possible may well depend on the domain of knowledge being explored. In higher level mathematics, for example, it has taken generations to discovery all that we know. It is unreasonable for all of that to be discovered anew by students, even in a guided fashion. However, some small pieces of it can be found through discovery. For example, when I first learned the Power rule in differentiation, my teacher wrote down a bunch of examples of it being used on the board. She then asked us to figure out how these different examples were connected. I still remember the lesson nearly 20 years later because I felt a bit like I’d conquered a difficult piece of mathematics myself when I figured out the pattern. It’s certainly more on the instructivist side between the spectrum between instructivism and constructivism, but it allowed for some exploration of ideas. I had the opportunity to test out hypothesis and reject them when they didn’t work, and then I found a pattern which worked for all of the examples.

I think one of the greatest thing that a childhood spent playing videogames (instead of organized sports) did for me is that it taught me that A) almost all problems are solvable and B) I can reach the solution on my own in almost all cases. These traits — this “confidence” — is a MAJOR asset to me in life.

In the case of videogames, however, there are some very specific parameters designed to encourage the player towards a certain goal. This is similar to a teacher designing an exercise. True, it might not be pure self-discovery, but it is still extremely valuable.

At the same time, not everything needs to be discovered to be properly learned, and there is WAY too much information out there to waste time self-discovering all of it. We would flounder about.

In the end, I think it’s important for students to get the ability to think for themselves and to be able to solve random problems with little to no outside help. But they also need to recognize that not everything needs to be learned on their own (this will hopefully save them a lot of time and frustration as they get older).

I agree with you John. I think most mathematics teachers would agree with you (at least most of the ones I’ve met during my career), and that much of what has fueled the math wars is where to draw the line.