tag:blogger.com,1999:blog-8231784566931768362.post738420398809880079..comments2021-04-19T14:06:43.800-04:00Comments on MathNotations: Another Definitive Report 'Proving' That Discovery Learning Fails in Science/Math!Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comBlogger11125tag:blogger.com,1999:blog-8231784566931768362.post-74833325608364840682007-01-25T15:22:00.000-05:002007-01-25T15:22:00.000-05:00Everyone else I've spoken to understands I am most...<i>Everyone else I've spoken to understands I am most concerned about exposing ALL of our children to the same rich mathematical content.</i><br /><br />Okay, you are most concerned about exposing ALL children ....<br /><br />I'm most concerned that all children are not only exposed, but also learn, the same rich mathematical content. <br /><br />This is why I think I am disagreeing with you. For example, I was puzzled by how you seemed to be only interested in asking one question about a class: "This is all I ever ask!" There are so many other important questions to be asked. <br /><br /><i>First we must agree on the WHAT before the HOW. You didn't address my statements about that.</i> <br /><br />Well you seemed to be quite happy talking about the HOW as well as the WHAT. <br /><br />Anyway, I don't think we need to hold these discussions in sequence. We may as well debate the HOW at the same time as debating the WHAT. Quite possibly, our ideas on HOW will influence our ideas on the WHAT. <br /><br /><i>'Balanced' means a commonsense approach to curriculum and instruction.</i><br /><br />I favour a tested approach to curriculum and instruction. In my experience, commonsense fails too often. The world is not commonsensical. For example, in medicine it seemed to be commonsense that if someone was wounded and losing blood, their fluids should be replaced. Eventually it became apparent that this commonsensical view was wrong - replacing fluids interferred with the blood-clotting process.<br /><br />Successful teaching strikes me as being about as complicated as successful medicine (though not with quite such dire potential consequences if you make a minor slip-up) - I don't see why commonsense would be a reliable guide for teaching. <br /><br /><i>are you talking about a program called "Direct Instruction?"</i><br /><br />Yes - that's where the lesson plan I cited was produced.Tracy Whttps://www.blogger.com/profile/08999246551652981965noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-83614847159791620472007-01-24T23:43:00.000-05:002007-01-24T23:43:00.000-05:00Tracy,
are you talking about a program called "D...Tracy, <br /><br />are you talking about a program called "Direct Instruction?"<br /><br />or using "direct instruction" as a generic opposite of constructivism?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-79722144867662877732007-01-24T20:22:00.000-05:002007-01-24T20:22:00.000-05:00tracy--
I admire your passion and commitment to ex...tracy--<br />I admire your passion and commitment to excellence. Why you perceive I am at odds with your thinking I do not know.<br />Let's clear the air. If you want to legally dissect each of my words then you're missing the forest for the trees. Everyone else I've spoken to understands I am most concerned about exposing ALL of our children to the same rich mathematical content. No one misunderstands that message although they may be very wary of any national approach to anything.<br /><br />To insure that ALL children are exposed to the important concepts, skills, procedures of mathematics we need to come together and agree on what those are. Despite enormous obstacles this can be done. Other countries form math committees and reach conclusions. So can we. This discussion is far more important to me than how you characterize an effective lesson. First we must agree on the WHAT before the HOW. You didn't address my statements about that.<br /><br />'Balanced' means a commonsense approach to curriculum and instruction. Students need to be given the tools to do mathematics: the definitions, terminology, symbols, facts, rules, procedures, theorems, etc. and we need to expect mastery. However, developing conceptual understanding of mathematics and developing problem-solving skill is much more difficult. Mastery of facts is NECESSARY but NOT SUFFICIENT for conceptual understanding. Doing progressively harder and less routine problems is a way of deveoping this. Another is to have students collect data, organize that data and attempt to make interpretations, generalizations and conclusions. The lesson I described did this by having students use the calculator to explore various exponential expressions. The teacher then structured the analysis, asking them to focus on certain examples to look for similarities. I don't care whether you wish to defend this DI or I call it exploration/discovery/pattern recognition etc. Do YOU care what it's called? The point is that the teacher had clear objectives. Here was one:<br />1) Demonstrate understanding of zero and negative exponents<br />After students took the time to explore and discuss their findings, the teacher stated the definitions and rules clearly. Because of the 'exploration', students had a context for these. The definitions did not seem so arbitrary. In the end they were expected to KNOW these definitions and APPLY them. The teacher could have used many other approaches to arrive at the same objective. Some students might have benefited for example from the TABLE approach to developing powers (spacing will be off):<br />N...2^N<br />3...8<br />2...4<br />1...2<br />0...?<br />-1..??<br />This is also a powerful construct to motivate the definitions and further it creates the function mind-set which is so critical. The teacher has to ask the right questions to make it happen, but now I'm getting off content and onto pedagogy (which I also care about!)<br />Now, are we really light years apart, Tracy?<br />I hope we're not, but we're both greedy, aren't we!Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-44731263697427413692007-01-24T17:37:00.000-05:002007-01-24T17:37:00.000-05:00Are you being provocative and attempting to provok...Are you being provocative and attempting to provoke some dialogue again? Or are these actually serious comments? <br /><br /><i>Now let's drop all these labels and ask one simple question: Were these effective lessons? YES! Once we agree that both lessons contain real content and one could assess that students learned, what more could we ask!</i><br /><br />Well, we could ask if the students actually did learn the real content. We could ask which set of students learnt the content most thoroughly and retained it the longest. We could ask if the content covered all the things students would need to know later on in that subject (obviously this would not be asked of one lesson, but if for example a whole elementary school maths curriculum misses out place value, that curriculum is inferior to an otherwise identical curriculum that covers place value). <br /><br /><i>While some 'experts' would argue that both lessons are DI, I would argue that that each is a beautiful blend of exploration, discovery, patterning, generalization, guided practice, and transmission of clear specific information by teachers who are knowledgeable and 'strong' in the classroom. This is all I ever ask!</i><br /><br />Is that all you ever ask? Really? Why is that the only thing you ask? I ask all the questions I outlined above. <br /><br />I'm greedy when it comes to education. I want as many students as possible to learn as much as possible for every hour at school.<br /><br /><i>To summarize, do you believe in a balanced view of instruction and content? I believe you do so in fact there is no difference between us. Of course, I may be in error...</i><br /><br />I don't know what you mean by a "balanced view of instruction and content". I believe schools should use the most effective means of instruction. If that's not a beautiful blend but consists entirely of one method then I still prefer the most effective method. <br /><br />(Of course, there's a lot of detail behind the phrase "most effective means", but I won't go into that now).Tracy Whttps://www.blogger.com/profile/08999246551652981965noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-7376653333986490282007-01-24T09:53:00.000-05:002007-01-24T09:53:00.000-05:00tracy --
you are absolutely right...
my exteme cha...tracy --<br />you are absolutely right...<br />my exteme characterization of DI was just as inane as many critics of discovery learning and exploration. My intent was to be provocative and stimulate some honest dialogue. If you read my other posts you know that I detest labels. They oversimplify and cloud the real issues. <br />Now compare your sample lesson to the one I described in my post. Which one was DI? According to my take on your comments, you would argue that both were. <br />Now let's drop all these labels and ask one simple question: Were these effective lessons? YES! Once we agree that both lessons contain real content and one could assess that students learned, what more could we ask!<br />While some 'experts' would argue that both lessons are DI, I would argue that that each is a beautiful blend of exploration, discovery, patterning, generalization, guided practice, and transmission of clear specific information by teachers who are knowledgeable and 'strong' in the classroom. This is all I ever ask! No K-12 teacher allows students to have 45 minutes of completely unstructured math activity nor do they talk 'AT' students for that length of time. <br />You and I are on the same page, but you seem to take exception to my 'apparent' attack on DI. Everyone justifiably gets defensive if someone attacks their beliefs. I know I do!<br />To summarize, do you believe in a balanced view of instruction and content? I believe you do so in fact there is no difference between us. Of course, I may be in error...<br />Now my real concern is that before we worry about instructional methods we need to agree on CONTENT! Without 'national' consensus on WHAT students should know at each grade level and hs course, the rest of the dialogue is meaningless.<br />daveDave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-62728649337885433582007-01-24T07:02:00.000-05:002007-01-24T07:02:00.000-05:00I think you're doing the same over-simplification ...I think you're doing the same over-simplification you accuse critics of constructivism of doing.<br /><br />Direct Instruction does not consist of expecting students to sit passively while information is hurdled at them. Consider for example a script from a DI maths lesson, available at <a href="http://www.specialconnections.ku.edu/~specconn/page/instruction/di/pdf/math_sample_lesson_a.pdf"> http://www.specialconnections.ku.edu/~specconn/page/instruction/di/pdf/math_sample_lesson_a.pdf </a>. The first exercise in the lesson consists of asking students to add and subtract 10 from various numbers. They are to reply verbally. <br />The rest of the lesson consists of teaching new concepts. Here's one of the longer sections for a teacher to say:<br />"Below the rectangle is a line that shows the total distance around the rectangle. That's the distance something woul dhave to go if it went around all 4 sides of the rectangle. The first part of the line is side A, the next part is side B, the next part is side C, and the last part is side D. Your turn: Measure the whole line and write the number at the end of the line. Raise your hand when you're finished." (page no 35).<br /><br />A teacher would have to be speaking pretty darn slow to stretch that out to a few minutes. <br /><br />And this lesson requires a ruler for each kid, a penny for each pair of kids, and an object that weighs about 1 pound. So kids are being presented with a variety of different instructional techniques, including physical items.Tracy Whttps://www.blogger.com/profile/08999246551652981965noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-82624642624110317852007-01-21T16:12:00.000-05:002007-01-21T16:12:00.000-05:00thanks jonathan and myrtle...
"but it takes a ver...thanks jonathan and myrtle...<br /><br />"but it takes a very skilled teacher to ask just the right question at just the right time in order to lead the student thinking in the right direction without actually giving everything away."<br /><br />that's it, myrtle! the art and practice of teaching! i've seen a few master teachers do exactly this and it's beauty personified! <br />The art of problem solving website is outstanding -- i would recommend it to stimulate and open new vistas for children who have an interest in mathematics and/or a special aptitude; i use it to train my students for the amc and aime math competitions; the best part are the training sessions and jams prior to the contest not ot mention the message board<br /><br />yes, i will give you 100 bonus points but i will give him 200 bonus points for insight and inspiration!Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-76528569004363982892007-01-21T13:48:00.000-05:002007-01-21T13:48:00.000-05:00"Experienced mathematics have something to say, di..."Experienced mathematics have something to say, different from either extreme. And it is not being heard, at least not yet."<br /><br />I agree.<br /><br />The problem is that the term "constructivist" is vague and seldom used in a specific context. <br /><br />I appreciate Dave's blog and hope to see how specific word problems can be presented to stimulate interest in different topics. <br /><br />I can always follow up a word problem with a solution if my son can't figure it out, but it takes a very skilled teacher to ask just the right question at just the right time in order to lead the student thinking in the right direction without actually giving everything away.<br /><br />I just got Art of Problem Solving's "Number Theory" for middle school students and I'm learning a lot about different ways of presenting material. While it's much wordier than our current curriculum it's also much more technical. I'll use this in addition to what we are already doing and hopefully it's going to help my son to make the transition from a heuristic approach to a more formal approach to math.<br /><br />I don't want him to do this for any sort of competition. I just think that there is a lot to be learned by going through the book.<br /><br />By the way, Dave, I gave my 10yo son one of your word problems that involved a bicycle and a car traveling at different speeds. The biggest hurdle he had was the fact that the problem didn't specify the units of time and distance. He thought it was impossible to solve such a problem without expressed units. He invented his own units to make the problem easier for himself and then later complained that he couldn't convert his units to precentage. I want a bonus point for keeping my mouth shut until he had a final answer to show me. <br /><br />The lesson I learned is that while he is really good at solving "real world" word problems, he needs exposure to more abstraction.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-82952102472599192832007-01-21T13:35:00.000-05:002007-01-21T13:35:00.000-05:00You've inspired me to post some old stuff I wrote ...You've inspired me to post some old stuff I wrote about similar issues.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-54277242043341743742007-01-21T13:26:00.000-05:002007-01-21T13:26:00.000-05:00jonathan,
my main purpose in starting this blog wa...jonathan,<br />my main purpose in starting this blog was to address your point #5 -- i have screamed loud and long to hte national math panel knowing that a single lone voice like mine would be easily ignored; i suggested that if the voices of teachers on the frontline were ignored by this panel then education journalists and fellow bloggers would join in a loud chorus of protest. Yes, this is incredibly naive on my part. How can one person make a difference? But they can because word spreads faster than any logistic growth imaginable thanks to this superhighway. The emrgence of my blog from nowhere is testament to this!<br />daveDave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-61591583515380325452007-01-21T12:06:00.000-05:002007-01-21T12:06:00.000-05:00Rant warning
1. Constructivist curricula (not con...Rant warning<br /><br />1. Constructivist curricula (not constructivism per se, but curricula based on regular, repeated use of constructvist techniques) have generally done very poorly.<br /><br />2. Leading "constructivists" push the stuff without regard to the difficulties that make it hard to impleement (or easy to implement badly) in the classroom<br /><br />3. Most critiques of constructivism have come not from those concerned about math education, but from ideologues, as part of a much larger agenda.<br /><br />4. The alternatives offered by leading anti-constructivists are the sorts of instruction that would only reach the strongest mathematics students. Their vision is a grab bag of anti-calculator back-to-basics lectures, where most students do not even attempt topics beyond algebra.<br /><br />5. Experienced mathematics have something to say, different from either extreme. And it is not being heard, at least not yet.Anonymousnoreply@blogger.com