Mat 521, Fall, 2002

Responses to "The art of teaching is the art of assisting discovery."

Richard E. Ferro

Mark Van Doren, a literary critic and English professor, was quoted in Stewart's Calculus as saying, "The art of teaching is the art of assisting discovery." Not only is this comment relevant for teaching mathematics at a secondary school level, but it is relevant for all levels. Not matter what the level the student has attained, he or she will be learning new concepts as they progress with their education. The teacher must somehow convey the principle to the student. If the student never really grasps what the teacher is explaining, he or she will never truly understand the concept. When he finally does grasp the concept, it is literally like a light bulb going off in his head. As a teacher it is truly gratifying to see students 'discover' new principles. I have had the opportunity to teach many different levels of mathematics from arithmetic through calculus and differential equations. I have seen the above scenario happen many times and from students at different points in their education. With the beginning algebra student, the concept of negative numbers can be extremely hard to grasp. The teacher will try to explain negative numbers using like borrowing money or temperatures, etc. Hopefully, the student does finally understand. With calculus students conceptual understanding of limits is rather difficulty. I have seen students that have A averages in every previous mathematics course struggle with this topic. Furthermore, the number of teachers who introduce limits with epsilon - delta definition is few and far between, which might magnify the difficulty in the complexity.

Van Doren's comment is also relevant for teaching graduate level courses in Mathematics, especially Mat-521, titled Algebra for Teachers. In this class everybody either is or wants to be a mathematics teacher. So, if one can relate learning to act of discovering a new concepts, then one certainly relate teaching to the act of assisting discovery. For me, I have learned little things, like to tell if a polynomial is irreducible, and introduced the simpler methods into the math classes that I teach.

Finally to sum up, I believe that teaching any course can be said to be an act of assisting discovery. Teach an introductory course in a programming language, say C++, where there are no prerequisites, you will see many students struggle until one day they program better than you do (i.e. a very steep learning curve).

Josh Roberts

I believe that Mark Van Doren's comment about teaching is at the essence of what teaching at the secondary school level should be. I have often thought about what it means to teach high school students, and I have an idea about it that I like to call "the light bulb principle." To me, this means that you guide students in a way so that they aren't fed the material, but are allowed to make the connections for what is applicable in Mathematics. I have accomplished this in small ways when tutoring peers in subjects in the past, and I can recall from my experience that as far as teaching is concerned, there are few greater joys when that "light bulb" goes on in your peer's mind. In my opinion, the word discovery probably isn't found in those texts you perused for on simple reason: Some people who write texts have a very detached way of looking at things. I liken this to how a doctor refers to a patient as "the patient" instead of a human being with an illness. Anyway, the ability for the student to discover something is their right and is the greatest gift of learning. To simply feed information and expect regurgitation when it comes test time is a tragedy and a disservice to the child's education, as well as to the teacher, since they will not evolve as a teacher in such an environment.

There are obstacles which are not always overcome at the secondary level regarding discovery in learning. The student may fail to make the connection, either not having the desire nor the ability to do so.

Is this comment about discovery relevant to our class? Although I feel incapable of making a genuine discovery myself, I would say yes. It is difficult for me personally to discover the connections between related ideas, although I believe this is primarily due to the amount of time I have to focus on the material. In past Mathematics courses, I have made these discoveries, and I am sure that I felt just as my peer did when I successfully guided them towards their own realization. However, 1 believe that discovery is very important throughout learning, and our class is no exception.

I close by remarking that an unfortunate thing about education at this time is that so many students don't see how wonderful it is to discover new things with minimal guidance until they lack the ability to do so. I had to work very hard to get back to a position where I was discovering new things, instead of just following along with the group work or teacher and regurgitating material back when necessary.

Naomi Carmichael

I highly doubt that anyone researching the NCTM or NYS standards will ever come across the word discovery. These sets of achievement levels are not about the best interests of the students. They are really about the great need for the United States to compete with other countries on standardized testing, and for New York to be one of the states producing pupils who can rise to that challenge.

I hope my sarcastic tone is evident because I really think that the grand drive for success is squashing its competitors. Do the ends really justify the means? Will we ever have satisfactory ends? I do not care for the answers to these questions if answered by the people creating the standards.

In a perfect world, learning, and therefore teaching, would be all about discovery. Isn't that what it really is? Isn't learning the task of being interested and motivated enough in a topic to want to know more about it? I would really say that it is. Unfortunately, this is not how we teach at the secondary level today.

High school is all about meeting requirements and getting the job done no matter what it takes. This means long periods of teaching materials that the kids don't like or understand, or even care to try to understand. We take the beauty of sharing knowledge and facilitating, and turn it into menial tasks where no one wins. This was not my vision of teaching, but, hey... we live in the wonderful world of standards, now, don't we?!

James Colton

There are certainly pros and cons to discovery learning in the current educational system. Ideally, it would be wonderful to be able to assist students in discovering mathematical concepts and algorithms. I think that there is a small amount of discovery that can be done by the student with the assistance of the teacher, since so much of our high school mathematics is algorithmically based. For example, just yesterday, I was teaching my high school sophomores how to solve equations, which included radicals. I don't find the algorithm to be necessarily difficult (to isolate the radical and then square both sides), so I did give them the opportunity to try to solve the equation without first giving them direct instructions. Nobody was able to come up with the algorithm that I was going to teach, but I did get some other interesting methods that were used by the students. The benefit of assisted discovery learning is that it gives the student ownership over the material, and then the student will have a better chance of remembering.

But there are some cons to this type of instruction as well. The biggest being that it's terribly inefficient under our working conditions. When there is a ratio of one teacher to twenty-five students, and those twenty-five students have twenty-five different ideas, it's difficult for that one teacher to get around and make sure that every one in that classroom in on the right page. Plus, a forty-two minute period is not enough time, and with a curriculum that you have to make sure is taught. Under the conditions of our current educational system, the most efficient way to teach math is by direct instruction. The problem then becomes keeping the students attention.

Discovery might not be an important issue for the NCTM, but problem solving currently is. As teachers, our goal now is to give students the math that they need to apply it to given situations. This is where discovery takes place. No matter how many word problems you do in class, you ran never really teach students how to problem solve. You can only give them the opportunity to discover how.

The word discovery may not appear in the NCTM standards, but every pre- service teacher in this state is taking classes and learning that there are alternatives to direct instruction, where discovery is implied. The reason that you do not see the word in the NCTM publication is because every standard the NCTM writes has to be assessed, and how do you assess students on what they discover?

Jim Davies

Mark Van Doren's statement, "The art of teaching is the art of assisting discovery," can be interpreted in many ways. One way I interpret the phrase is that good teaching can increase opportunities for the student. For example, teaching a language assists students in discovering a different culture and might encourage them to travel, something they might not have desired to do before taking the class. Teaching art or music might allow a student to appreciate the subject more or even unveil an unknown talent.

Another way this statement can be interpreted is that good teaching engrains in the student the skills needed to successfully learn and understand the subject matter at a deeper level. These skills will allow the student to further his or her study in the subject, and potentially add their own discoveries to the knowledge of the subject.

This second interpretation is one I think is more appropriate for teaching mathematics at the secondary level. A skilled teacher has the opportunity to introduce to his or her students the main concepts of mathematics, not merely the algorithms and techniques commonly used. Knowledge of these concepts allows a student to think beyond the material at hand, and perhaps "discover" some mathematics oh his or her own. This discovered mathematics might be something that is practical and already discovered or just a unique observation that might not be of any use, as of yet.

Teaching to assist discovery also comes into play in Mat 521. This course is unique in that it allows teachers and future teachers to study high school mathematics at a higher level. Having been exposed to this advanced material we, as teachers, have an added perspective to teach our students from. Some of Mat 521's material is suitable for investigation in the secondary school. For example, encouraging students to create new algorithms or find other ways to solve problems allows students to think creatively about a subject that is all too often taught one-dimensionally. Even if we choose not to teach the material learned in Mat 521, we may show our students that the subject matter they are learning is practical and they can use it and their own discoveries to answer questions they encounter throughout their lives.


There are two major pieces to the education puzzle; what should be taught and therefore tested and how should those topics be taught: Content vs. Pedagogy. Whether you think it is correct or not the New York State Education Department selects the content that they feel students should have learned to be given credit for achieving a certain proficiency level; whether that be Math A or Math B. That is out of the hands of teachers in the classroom. What isn't out of our hands is how we teach the material. Creative and hardworking teachers can take material and present it in such a way that it builds on prior knowledge and allows the students to learn in the best way possible for that individual student. The whole idea of NCTM as I understand it has nothing to do with this part of the teaching process, at least not in as straightforward ways as they direct content that should be taught....

Limiting how ideas can be taught limits the true strength of many teachers, that being their creativity and adaptability. Not every example reaches every student, so being locked into one or two examples being the only ones that they can see makes it difficult for them. How this applies into Math 521 is that when trying to do a problem either from the text or from the homework sheet, it can become difficult if the explanation given in the text and notes and the previous material doesn't lead right to the questions being asked. If how the prior knowledge enters into the new material isn't clear, then the new material isn't building on anything. That makes for difficult problems. Not that I'm saying this is happening, I'm just drawing the line from the quote into Math 521.

Constructionism in teaching and learning is a great way to work. I try to teach that way, and when I have time to dig and figure things out I try to learn that way as well.

Jenn Amyot

The comment made by Mark Van Doren is relevant to teaching at the secondary level; "The art of teaching is the art of assisting discovery." While the standards may not specifically use the word discovery that is in essence what learning is about. When a student learns mathematics, or any subject for that matter, they are always discovering new things. The job of a teacher is to assist this process, so the comment made was very well stated.

Learning mathematics is sometimes referred to as sense making. As a student tries to make sense of the new things they are learning the teacher wants to help but also allow the student to try and discover some things on their own. Instead of just giving a formula to students, if they are first given the opportunity to come up with one on their own it will help them to better understand it. Allowing for this discovery opportunity will give students a feeling of ownership of their learning, and hopefully further engage them. Along with ownership, discovery also allows students to be creative and try various ways of solving a problem. Solving problems in various ways fosters better understanding, as the student will most likely be able to see more connections in the material they are learning.

In learning there must be an opportunity for discovery. However, students are not expected to discover the mathematical world on their own. Teachers provide necessary background information and helpful hints along the way. They guide their students toward the discoveries they hope they make and in the process may come across some discoveries they did not anticipate. Unanticipated discoveries were mentioned in the teaching principle of the NCTM principles and standards book. Once students have learned a new topic in math they may then discover new ways to solve previous problems as well. Teaching students is to help them understand and discovering new connections and solutions in math can only deepen understanding.

The comment is relevant for Mat 521 as well. While we are learning new methods and topics we may also discover new connections or solutions. When doing homework problems for example some of them require using new methods but some can also be done using ways, which we already know or come up with on our own. However, we may never have known the connections that can be made without being taught the new methods. The more connections and thorough understanding of math that we have the better teachers we can be.

Jon Zehnder

Mark Van Doren's comment is extremely relevant for teaching mathematics at the secondary level. The best type of teacher in any discipline is the teacher who facilitates discovery. The learning experience is entirely more meaningful if it is just that: meaningful. If a student can make a discovery on their own, it will be so much more meaningful than having the same thing dictated to them. Although the word discovery may not be present in NCTM's Principles and Standards for School Mathematics, the spirit of discovery can be found at several points throughout the document. In an excerpt entitled "Standards for Grades 9-12", the following expectations are outlined for mathematics education:

"teach students to think and reason mathematically, not just to perform routine operations....

emphasize modeling the real world and develop problem-solving skills.

promote experimentation and conjecturing.

provide a solid foundation in mathematics that prepares students to read and learn mathematical material at a comparable level an their own. "

These expectations all embrace the idea of discovery being a fundamental part of mathematics education. The goal is to produce students who are ready for the real world. We want to educate students to think for themselves and attack new problems using their own resources. In order to do this we must let them discover mathematics concepts on their own by setting up meaningful problems and scenarios. If teaching were just spoonfeeding information to students, they wouldn't be able to develop the higher level thinking skills which are necessary to be able to solve problems creatively. This isn't to say that all mathematics can be discovered; much of what must be learned includes universally agreed upon notation, or rather the language of mathematics. The best teachers, however, are the ones who let their students discover as much as is possible.

Erica Greene

"The art of teaching, Mark Van Doren said, is the art of assisting discovery." Although the word discovery cannot be found in the NCTM's standards, it is a relevant comment for the ideal high school. The good teacher simply passes on knowledge, but the great teacher will help the students discover the concepts for themselves. This is because the great teacher is aware that the student who obtains the knowledge himself will have greater ownership over the ideas. That student will know and remember the process that brought him or her to their end goal. If a student truly understands the process, he or she is far more likely to remember it for the long term.

However, today's society and today's standards essentially dictate that teachers cannot provide time for discovery because of the volume of concepts that must be taught in order for our students to pass the exams. This year's freshmen and sophomores are mandated to take three years of math in order to graduate. They also need to pass the Math A Regents. Currently I have several sophomores taking the first half of Math A, which means they need to pass this year and every following year in order to graduate on time. These students are struggling to learn concepts that have no basis in the life they envision just so they can pass an exam to graduate and begin that life.

This country is determined to raise standards. New York State determined that the way to do this is to eliminate the local diploma and mandate certain Regents exams. This has certainly not raised standards. More students will simply drop out of school because of their inability to complete the math requirements (or other stringent requirements). The Regents itself is not full of difficult concepts. The problem students have is deciding when to apply what concept. There is far more reading and interpreting on the Math A Regents than in the old Course 1, Course 11, and Course III exams.

Students would probably do better if they were given time to discover when to apply the concepts they are learning, how to use different skills, and how to build upon prior knowledge. However, the new standards have eliminated that possibility. With the amount of reading and additional concepts in these standards, there is simply a lack of time for student discovery.

As for Mat 521, discovery is certainly essential. I don't learn abstract concepts, and neither do the majority of people. We learn during the homework to try to take the abstract presented in class and turn it into the concrete required for homework. Sometimes we learn from each other, sometimes from the book, and other times from people outside this class. Our students would throw a fit if all we did was present abstract concepts and expect them to complete the homework without finishing several examples in class.

Nancy L. Van Oort

The quote, "The art of teaching, Mark Van Doren said, is the art of assisting discovery.", from Stewart's Calculus is very thought provoking. It creates images of teachers that are providing pieces of information, like a box of puzzle pieces, and students that are motivated enough and engaged enough to want to put these pieces together, as in completing the puzzle to see the picture. Is this vision realistic for the teaching of mathematics?

As I explore my beliefs about teaching and "discovery", I realize that I want this theory to work at the secondary level. I believe that this approach would enable students to more firmly grasp concepts and to retain them for longer periods of time. By "discovering" their knowledge, students would be happier to come to class and more willing to participate in the leaming process. In other words, I believe that students would enjoy mathematics.

This "discovery" theory does have some flaws for teaching mathematics at the secondary level. Many of the ninth grade students that I work with have come to me with the belief that mathematics is boring and difficult. They think that they cannot understand it so they do not try. When faced with their opinion, I have difficulty fitting the "puzzle image" into my classroom. Teaching Math A to the students that have difficulty with mathematics has caused me to begin to feel like a cook at Thanksgiving who is stuffing a turkey. How much information can you really stuff into a student's brain?

Due to this feeling, I have spent many hours reflecting on my teaching style and the teaching style of my peers. I have been trying to combine the student "discovery" theory with the students with which I currently work. I think that this theory will only work if the student is motivated enough to want to "discover' something. If there is no desire, the pieces of information and the joy of discovery will fall on deaf ears.

As a student progresses through to higher-level mathematics, I believe that a student becomes more motivated to seek out information. In upper secondary grade levels and college courses, including Mat521, I believe that the students want to learn about mathematics. These students have the desire to search for information and try to put it into logical thoughts and theories. I truly find that the more that I try to use the facts that a teacher has provided for me, the more I am able to understand the material that is being discussed. It is truly a wonderful feeling to believe that I understand a topic that is being discussed in enough depth to be able to apply the concept. Awesome!

In essence, I think that teachers can provide the students with interesting information that will engage the students if the student wants to learn. The students need to have the maturity and the understanding that learning is important. I believe that I will continue to strive to encourage my students that mathematics is a wonderful world of fascinating facts and unlimited options. I will support them in their travels of learning mathematics and encourage their belief in themselves and their ability. Furthermore, I will continue my goal of trying to instill in my students my love and excitement for mathematics.

Natalie Perillo

"The art of teaching is the art of assisting discovery" I feel that Mark Van Doren's comment is very relevant for teaching at the secondary school level. My main reason is because learning is in fact discovery. These students are able to open up their minds to new ideas presented by the teacher and not only apply them in the classroom, but also in the real world. The first concept that I thought of when reading this comment was Vygotsky's theory of zone of proximal development, This is when a range of tasks is too difficult for a student to master alone, but can be learned through the assistance of an instructor. This is an important concept to me because there are times when a student will took at certain material and think that they are not able to accomplish it. When a teacher is there to assist the student and influence them, it is very encouraging for the student. This is the same situation that happens to me in MAT 521. When first looking at the material, it looks very challenging. Once the professor explains it and demonstrates how it works, it is much more comprehensible. I also feel that the tables could be turned on Van Doren's comment. There have been many times when the students have opened my eyes to new discoveries. I have learned new ways and techniques to do things inside and outside the classroom. Keeping the students self-esteem high and their willingness to try harder is what can lead them all to the great discovery of learning.

Jennifer Cervoni

"The art of teaching, Mark Van Doren said, is the art of assisting discovery." This comment is relevant for teaching mathematics at the secondary level and for math 521. In teaching students you don't want to give them all the answers, you want them to come up with answers on their own by playing around with the problem. In doing this students are teaming through discovery. When given a problem they try to work it out with prior knowledge and use problem solving strategies to solve the problem. This is done or at least should be done at the secondary level in teaching mathematics. If you "spoon feed" students and give them all the answers, you are not teaching. In teaching students I try to guide them through the problem by giving them some hints, but I want them to do most of the work to give me an answer and I want them to be able to explain how they came up with their answer. This is how they will learn. Leaming is done by discovery. When you discover things on your own they have more meaning to you. I know for me this is true. You are more Rely to remember how to do the problem if you did problems similar to that one on your own. If you are just given the answer, you won't be able to do similar problems because you won't know how to go about doing the problem in the first place. The Math A exam, which all student must pass in order to graduate forces students to explain their work and show their work through problem solving techniques. They can't just write down an answer. It's very comprehensive. In math 521, we are all learning by discovery in the homework that we are given each week. Dr. Childs teaches us a lesson for the night and then gives us problems for homework that deal with what we did in class so we can practice the concepts. In doing these assignments, I am always trying to take what I was taught and apply it and when I do this I am teaming through discovery. The following week if we have problems that we just can't do, Dr. Childs gives us hints on how to go about the problem. We can then take those hints and try the problems again and then hand them in the next week. In conclusion, I always want my students to learn by discovering things on their own. I try to give them hints to get them on the right track, but I want them to take those hints and try different things to come up with an answer. The object is to give students problems in which once given a hint they will be able to solve it by themselves. If you don't help students to get them on the right track, they will become frustrated easily and tend to give up. Hence the "art of teaching is the art of assisting discovery."

Anne Battaglia

Van Doran's comment, "The art of teaching is the art of assisting discovery", is extremely relevant for teaching mathematics at the secondary level. It is in agreement with standards based reformed mathematics and the Principles of the NCTM. NCTM Principles advocate high school students building on their prior knowledge and learning more varied and higher level problem solving techniques. In addition, the NCTM Principles and Standards stress the importance of making connections. They believe students develop a much richer understanding of mathematics when they look at a problem from many different mathematical perspectives. This is best done by teaching through problem solving, where students are led to discover the math. Teaching in this manner, a teacher introduces a problem to the students and they work together building on their prior knowledge to discover the answer on their own. They apply what they know in new and different situations. By learning in this manner they are better able to solve other unique and different problems. In addition, the connections students make when discovering math on their own are stronger and enable them to make more sense of the math. The NCTM also advocates integrated content where students have to draw on different aspects of mathematics, and where problems are solvable in many different ways. Again this stresses the importance of the connections students make when solving problems. It also emphasizes the importance of graduates having the confidence and willingness to take on new and difficult tasks as they enter the workforce and higher education. Van Doran's comment is also relevant in Mat 521. However, many times in order to complete our assignments we have to be familiar with the algorithms or methods of the particular problems taught in class. For example, Fast Fourier Transformation or Horner's Expansion method. And I know for a fact I would not be led to discover any of these methods on my own! On the other hand, there are times when I work for hours and finally get an answer that I know I was led to discover!