The use of visuals does not advance understanding in any way because most times the children to not transfer from the concrete to the abstract. And the problem that most students face in math is working in the abstract. I have received most of my education outside of North America and I am no researcher but early in my learning there was rote, and now I understand because of all the procedures that I have learned to automaticity. I remember before we were taught fractions we were taught to find the LCM of numbers by factoring out primes, and I was wondering why we were spending so much time learning that. When we started to do fractions, it was like drinking milk because I realized we were creating equivalent fractions all with a common denominator when we computed an LCM.
The point about this debate over rote or discovery, I view as mathematical innovation-consistent with views of Andrew Nikiforuk author of the book” If Learning so Natural, Why Am I Going to School”. Nikiforuk in his book, talks about Innovations versus Reforms” He defines innovations as ideas that administrators implement in the hope that they may do some good. He goes on to say that ” innovations are implemented with no foreknowledge or subsequent measurable proof of their effectiveness”. He posits that genuine educational reforms yields measurable improvement in student learning. There is some research by Jennifer Kaminski and Vladimir Sloutsky that says that manipulative or visuals impedes the ability of students to work in the abstract. The proponents of discovery math sell the idea that if people can visualize procedures they will understand them. I have a daughter who is the victim of discovery math. Every time I show her how to change the subject formula to solve an equation she is flummoxed. That is what you get from learning by problem solving rather than learning a defined technique that can be generalized. I remember reading that the late mathematician Herb Wilf said that after examining a slew of math research that they are not robust enough to be relied on. He described some of them as being simplistic.

The use of visuals do not advance understanding in any way because most times the recipients do not transfer from the concrete to the abstract. They never make the link between the concrete and the abstract. And the problem that most students face is working in the abstract. I have received most of my education outside of North America and I am no researcher but early in my learning there was rote, and now I understand because I of all the automatic procedures I developed through rote learning. to me, It forms the foundation of my understanding. I remember before we were taught fractions we were taught to find the LCM of numbers by factoring out primes, and I was wondering why we were spending so much time learning that. When we started to do fractions it was like drinking milk because I realized we were creating equivalent fractions all with a common denominator when we do an LCM. The point about this debate over rote or discovery, I view as mathematical innovation-consistent with views of Andrew Nikiforuk author of the book” If Learning so Natural, Why Am I Going to School”. Nikiforuk in his book talks about Innovations versus Reforms” He defines innovations as ideas that administrators implement in the hope that they may do some good. He goes on to say that ” innovations are implemented when no foreknowledge of subsequent measurable proof of their effectiveness”. There is some research by Jennifer Kaminski and Vladimir Sloutsky that says that manipulative or visuals impedes the ability of students to work in the abstract. I support this theory because it was always my belief. I remember my daughter using counting squares to find area and perimeter and then given the question of finding the dimensions of a figure that has an area of 20 and a perimeter of 18. I showed her how to do it using the formula for area and perimeter, but she said her teacher did not teach them about using formulas. The proponents of discovery math sell the idea that if people can visualize procedures they will understand them. I have a daughter who is the victim of discovery math. I remember reading that the late mathematician Herb Wilf said that after examining a slew of math research that they are not robust enough to be relied on. He described some of them as being simplistic.

Michael C,
I want to know if creative mathematics works in every situation if you are doing a trig proof, a differentiation or a composition of functions, I do not think that creative math can help you there. You must know the standard procedure for solving these problems. In my mind creative math is an attempt to create a way for everyone to be successful in math. A grade 3 girl in my spelling bee class challenged me to solve the problem 3953/5 i did the traditional division and I returned the answer 690 remainder 3. She told me that was incorrect and set about solving it by deducting 500 each time until she reached to 453 then she deducted 450 and ended up with the remainder 3. she divided each of the deductions by 5 and and added up the quotient of the division by 5. You can do a division like that, but using the traditional algorithm is a more efficient way, and where time is of the essence efficiency counts; it also help you to use the concept of division in practical applications. I remember when we had word problems my mother would demonstrate it to us. she would show me that if she had 30 oranges to share between my siblings and I she would divide by 5 to see how many each of us would receive. In those algebra questions where you divide both sides by the coefficient of x to find the value of x, I follow how she demonstrated it to me-if you bought 4 oranges and you spent 40 cents how much would each cost; she explained that you had to share the cost among the oranges by dividing by the number of oranges to find the cost of each just as she divided to find how to share the oranges. In math you need to know the traditional way of doing things, and learn how they are relevant to everyday problems.

Good skills in addition, multiplication and division is the key. nothing more nothing less. it is skirted in the earlier grades, but when students encounter higher level math, the absence of the basic foundation becomes a serious impediment to learning.

In response to Dick. Repetition is a part of the learning process. As someone who had learned math by the old method of repetition, I swear by it because, I do not think that concepts help children very much in understanding. Recently, I witnessed a lesson where children were shown the concept of area through counting squares, and the concept of the perimeter through counting the squares on the edge of the figure. They were then taught how to calculate area and perimeter using dimensions, and asked if they knew what they were doing with the expectation that they would associate this new way of computing area and perimeter with the concrete way they had previously learned. Unfortunately they did not.

I will now talk about my daughter who was educated in North America. She recently had to do a standardized test to obtain a job. The math questions were easier than the language questions, but she was unable to work them out mentally. she had to rely on the language problems. Let us hope they were enough to gain a pass. The questions were simple, any student with fair arithmetic skills should have been able to do it. But her problem is that the way she was taught in Canada, she never did repetition, she also never did interleaved practice, and interleaved practice is not new. i am 62, and that is what I was exposed to in my little third world country. I do not blame her for her situation, although she should not really be in the position she is now because I had always predicted that the way they were being taught in school, only the very best will be competent in math. I offered to teach her arithmetic, algebra and some geometry, but since that idid not correspond with the school syllabus it seemed like a different kind of math. What they did at school is work out problems without being taught the methods that will allow them to generalize or transfer their knowledge. They were not taught changing the subject of a formula. They applied it in solving an equation, but were not aware that if you have one unknown in any equation, you can solve it by changing the subject of the formula. Thy were not taught a general method for calculating the Lowest Common Multiple, so she is unable to do fractions fluently. They were taught to find out by doing iterations until you get the same product. I can name many more instances of what should have been taught but never was. She is now trying do undo the damage. Recently, she told me that she realized that you must know your tables to master factorization. I think that it is unfair to students when the authorities who control what they learn indulge in educational innovations for which there is no robust evidence that they will work. At the end of the day, the bells a whistles never stay, and what should have been learned to form the foundation for higher math was never learned. My daughter is a victim of the education system like so many other children are victims. I have done many math courses that involved application of math, and learning by repetition never hampered my conceptual understanding instead it fostered it. It provided me with a good foundation for higher math. By the way repetition alone is not responsible for making one remember multiplication tables; using them all the time is.

That is exactly what needs to happen. But remember how it goes-a student does not understand something, and asks for help. The teacher asks what don’t you understand. The student cannot articulate. As I am older, and I revisit some math that i did in school without the pressure to perform; I realize that an effective teacher should recognize a student’s problem. I learned this by sometimes looking at a demonstration, and thinking I do not understand it. I would usually reflect on it , and then an ‘aha’ moment arrives, and I would ask myself why my teacher did not recognize my problem when I was at school.

Most state testing is not consistent with the curriculum; that is why it is difficult and futile. The people creating the test make assumptions about what students should know without actually knowing what they know. I came from a system where we did exit exams a form of standard testing. But the testing was based on our syllabus. Standard testing are designed by people who pay no attention to what students are actually studying. So it is reduced to the teacher teaching to the test and neglecting the syllabus, or following the syllabus and neglecting the testing. Testing needs practice on what is to be tested.

Well said Erik,
You articulated well my thoughts. I can cook fairly well and I know that feeling. But there are some times that you need to follow recipes. I am a good artisan baker, and I am always researching how to make the perfect bread. One day a friend saw my well-leavened golden loaf, and decided to come for some lessons on making bread, but after seeing the rigmarole that was required, she said it was too difficult. But becoming proficient in math is something like that chef, but the self discovery is really self realization. individuals must realize that they know all the techniques required for problem solving. Sometimes they know it without knowing that they know it. I have looked at some of Salman Khan’s videos, and I see the same problems that most teachers make -they assume that everyone could follow their applications. The videos are good for me, but not for my daughter. Sometimes you know something, but you do not know how to use it. And it would help if someone would remind you when solving a problem why they do what they do.
I think the worse situation is sometimes you get it and sometimes you do not. I remember sometimes getting all my factorization correct, and sometimes getting it wrong. Then, I realized I was patterning without understanding- although I always wondered why I sometimes seemed fluent and other time not. When I got it correct was when I regurgitated something I did before (standard recipe), when I saw a problem that I had never done I was stuck, but all along I had the requisite knowledge, but did not know the technique. The moment I realized how to work out the middle term ‘bingo’. I knew my tables very well, but I missed the explanation of the technique-find two factors of the last term that will return the middle term. The chef has flexibility, and a person who can only pattern has no flexibility. A chef has flexibility because he can bring to bear all his knowledge of cooking to create new dishes, our artisan cook may not have that flexibility. The chef easily transfers his skills to create new recipes. Students need to have skills, and to know how to transfer those skills, but to transfer those skills, they must have them. So it is in cooking so it is in math.

I agree with you about one thing spending reams of time on rounding and estimating is wasting valuable instruction time, and students seem to have a difficult time understanding rounding. However, on the use of pen and paper math, I think wunderlic testing which many employers use as a means of elimination. in wundelic participants are not allowed to use a calculator, and they have 50 questions to complete in 12 minutes and they need to make between 20 and 25 to pass.
This is one of the questions four men entered into a partnership contributing capital in the ratio of 50, 15, 15 and 20 percent. They agree to share the profits equally. In the first year they make 120 000. How much more the partner contributing the highest amount of capital would have received if profit was shared in proportion to the capital contributed. This must be done in 14.4 seconds without a calculator. You may think that students need to know pen and paper math, but they do need to know it, because employers know that they may not be fluent in it and use it as a means of elimination. This test also contain a lot of vocabulary and interpretation of proverbs. Given the dismal state of language arts it could be a chalennge in all departments.