In your case, I don’t know if your students would have flourished even *more* under an alternative program. Nor do I know if your population of students is representative of other populations. If, for example, you teach gifted students in a high-SES district with lots of home resources, that may not generalize well to other students.

I believe your conclusion is an oversimplification of a really complex phenomenon. What you dismissed – videos, practice, and classroom projects – as an impoverished option is just not true. Using exactly this model, my students have thrived. They enjoy math more, their confidence is higher, their standardized test scores are better, their acceptance into accelerated math programs increased, and graduates of this system report great success as they move into the next years of math. The personalized learning system I have been using is Khan Academy, and it is really good. I have seen that it does matter that students can choose the pace or presentation of learning. The thing that sells me most about personalized learning is the confidence boost for kids who are struggling. Your comments (and subsequent others) seem to come from the ivory tower, whereas I have seen actual results. ]]>

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.

]]>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. ]]>

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. ]]>

Sometimes people need modeling to initiate transfer, and this is particularly important in math. I know that I have practiced many procedures to automaticity, and after seeing how it is applied in practice, then I gained a complete understanding of it and could use it generally. I work with students receiving special education, and I am privy to many math lessons, and one of the things I noticed is that after one or two lessons, the students will be given problems to solve using the concepts they have just learned. They usually have problems, because the concepts are not yet cemented in their minds, and having not seen worked examples illustrating its practical use, they are unable to apply them to the problems.

I recently did an online course in trigonometry, and I kept noting all the topics one needed to know to do trig proofs using the different identities: you had to be fluent in the basic math operations-addition, multiplication and subtraction; you had to know exponents, radicals, working with complex fractions, factorization and the use of conjugates to rationalize square roots and some more. It involved a lot of memory to know which techniques were relevant. Knowing concepts are just one part of math, you need to remember them and know when they are to be used. The only way you can remember so many things is through practice, and when deliberate practice is removed from strategies to gain competence in math in favor of knowing concepts alone there is a problem. I am not a researcher, but I know from reflection on my past actions what strategies are necessary for me to gain competence in a subject. ]]>