Quite often the words "teaching" and "learning" are used synonymously. According to Ackoff and Greenberg, the authors of the book Turning Learning Right Side Up, "Traditional education assumes that for every ounce of teaching there is an ounce of learning by those who are taught."
Is this assumption true? If every ounce of teaching is transferred into an equal amount of learning, there would be no failures. All our children would be "A" students.
The reality is that teaching, as the word implies, is about teachers; learning is about students. We focus so much on teachers and teaching, the word "learning" has lost its relevance. Think about essential life skills, such as walking. A child starts with sitting, turning over, crawling, managing to stand with some support, taking the first step, breaking into running, fumbling, balancing and finally, mastering the art of walking. Learning to walk takes a lot of practice, patience and time. Children do not get frustrated by thousands of failed attempts and decide that they are not going to walk anymore. In fact, each failure seems to motivate them to do better next time. Every child is capable of acquiring this skill on their own by trial and error over a period of time. This is how actual learning takes place -- learning by doing. One can't learn swimming without getting into the water. Learning takes place when the learner is personally engaged and allowed to discover.
This is discovery based learning. The joy of discovery motivates students to learn. Remember your first successful attempt at riding a bicycle and the excitement you experienced? It is not that you were the first person in the world to ride a bicycle, but what mattered was that you discovered that you could do it. It is this process of discovery that we need to recreate for learning in classrooms. People learn through a discovery process. Rote memorization and teaching for exams never help students understand principles.
Pure discovery, however, can take a long time and may lead to frustration. What is needed in a formal education environment is to guide the learners through a discovery process. This is similar to providing a safe environment where children can make mistakes and learn from the failures.
Can we implement a guided discovery approach in our classrooms? Yes, a guided discovery process can be implemented at all levels from K-12 to college. Outstanding educators around the country already use some form of guided discovery approach such as scenario-based, inquiry-based or problem-based learning. Some concerns about discovery based learning that I hear from my colleagues and fellow educators are addressed below:
The guided discovery approach seems good for young children. Will it work for high school and college level students?
The guided discovery approach can be used at all levels and should be used where fundamental concepts are involved. Whether dealing with calculus in high school or engineering mechanics in college, the guided discovery approach is effective.
Is this really practical?
The guided discovery approach is rooted in experiential education, hence it is highly practical.
Won't this approach take too long? How do we cover all required material if we spend so much time in teaching concepts?
The guided discovery approach cannot be rushed -- students must be allowed to make mistakes, pick wrong choices, and face consequences. This requires more time, but will help learners develop a deep understanding of principles; therefore, learning follow-up material is lot easier and faster.
Why is this important?
Quite often we hear people say, "don't reinvent the wheel." From an educator's perspective, this notion is completely wrong and often counterproductive. There is no learning if there is no invention that is personally meaningful to a student. Every leaner should be provided an opportunity to reinvent. Guided discovery approach focuses on helping every student to reinvent important concept in their mind. Rote memorization and figuring out the right answer using blind techniques are not the way to develop understanding. Many concepts in science are not intuitive, even though most people believe in them.
Consider the example, why do all objects fall at the same time? When I ask this question, rephrased "as which object, one heavy and one light, will hit the ground first when dropped from the same height," some students answer the question correctly and others incorrectly. Further probing indicates even the students who answered correctly have no real understanding. They answered correctly not because they know this is a tricky question and they've heard that all objects fall at the same time.
We regularly witness even top performing students (who scored well in AP physics and calculus) show no understanding. This is dangerous. Real understanding is essential for success and it comes from experience. The guided discovery is an effective pedagogical approach that can truly engage learners by providing authentic learning experiences.
It is time to shift the focus from teaching to learning.
Michael Roth: A College Education: It's Not a Product; It's a Platform
Consider the long division algorithm, which has great importance later in their mathematical development. In order to understand the algorithm, students need to have a firm grasp of the decimal system and inequalities, which usually is developed when they begin algebra.
Consider the introduction of the real number system, which requires a solid understanding of basic algebra and logic to show the inadequacy of the rational numbers. How on earth are students supposed to discover by themselves that x^2=2 implies x is not rational, without direct instruction of Hippasus's proof.
Or perhaps the best example, the Fundamental Theorem of Algebra simply can not be "discovered" or understood at the high school level.
There are other examples where certain concepts are counterintuitive, like the differing magnitudes of infinity of the real numbers vs. the rationals; i.e. countable vs. uncountable sets. Another example is the famous Monte Hall problem in probability.
People that think that these are the exceptions are mistaken. If discovering mathematics simply required an instructor to only guide a bunch of kids then I assure you that the entirety of mathematics known today would have been discovered long before Euclid.
It's also more expensive and time consuming.
With standardized tests to cram for, administrators won't take the chance or spend the money. They need a sure thing when it comes to filling in those little NCLB bubbles.
The same is true for heliocentric theory. Aside from genius' like Aristarchus, it required careful observation of data to move from the intuitive observation of the geocentric view to the heliocentric view of the solar system.
So apparently the author and most School of Education professors believe that the average child is a naive replica of Issac Newton. I say naive, because in Newton's famous letter to his contemporary Robert Hooke he wrote, "If I have seen further it is by standing on the shoulders of giants."
The point being that Newton only after prolong study and thorough understanding of the techniques and limitations of his predecessors was he able to expand on their achievements.
For an idea of the type of knowledge base and volume of dead end paths that is required to obtain true discovery read Richard Feynman's outstanding Nobel lecture.
http://nobelprize.org/nobel_prizes/physics/laureates/1965/feynman-lecture.html
You present an intelligent, yet nonetheless false, dilemma.
Constructivist/experiential approaches are particularly important in primary and secondary education. The symbolic representation of reality is not reality. Children learning about the physical universe through active engagement is a necessary precondition for the later symbolic and abstract representations of which you write. As you correctly point out, Galileo and Newton constructed invaluable abstract systems, but as you omit, they were previously steeped in real experience , thereby creating neural pathways that allowed for subsequent and profound abstract understanding. It's not one or the other, but experience must come first. Children in the current educational climate are memorizing algorithms that have no meaning to them and will have no subsequent value, other than reiteration on useless examinations.
Consider this, the top AP Calculus students at your local high school can not even scratch the surface of not only Newton's mathematical knowledge but even Archimedes, who proceeded him by nearly 2 millenia. The notion that the average child should be expected to have Newton-like insight and determination is ludicrous and insulting to the achievements of the most brilliant minds in history.
Constructivist charlatans like to blame the failure of American Science and Math Education on testing without reflecting on the fact that they have dominated curriculum and instruction for the past 20 years. This coincidently coincided with the rapid decline in student achievement in these subjects. Perhaps most telling is that the constructivist ideology is practically unique to US public schooling. It is not taken seriously in the elite prep schools nor in education systems outside the US.
If you want true "discovery mathematics" students would successively read and work through the works of every great mathematician chronologically starting with the Babylonians, Egyptians and Greeks. This would get them in the mindset of those that made the discoveries and introduce them to the notion of what discovery even means.