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Mathematical reasoning does not lie in the connection of mathematics to the real world, but in the connection within mathematics.
How mathematics is applied is its connection to your world.
4-15-2007
making connections
help desk
Monday, April 16, 2007
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20 comments:
"Mathematical reasoning does not lie in the connection of mathematics to the real world, but in the connection within mathematics".
One of the greatest Russian scientists of all times, a real genius in math, physics, chemistry, and geology, M. Lomonosov, used to say, "Mathematics is a calistetics for the brain".
Every math classroom used to have his words posted on the wall.
After some recent indirect involvement (because they don't allow direct involvement) with my son's school regarding a potential change from Everyday Math, I have a few comments. Remember I said that I thought they were going to change to Sadlier-Oxford Progress in Mathematics? Well, there is a bit of resistance.
This is not a philosophical discussion. It is one of competence. This is not a debate between understanding and mastery. The teachers at my son's school think it is, but they are wrong. It's a matter of competence. The teachers and curriculum advisors are not competent in math. They have no basis for understanding and deciding on what has to be done.
The discussion at the school is hovering at or below a mere superficial level of debate. They may be genuinely trying to do what is best for the kids, but they are not competent to do so.
Real world problems and understanding are just non-issues that mask their lack of mathematical understanding. The blind are leading the blind. A parent told me that one curriculum advisor seemed to think that Singapore math had a good balance between practice and understanding, even though she doesn't have copies of the textbooks and workbooks. They have been at this for months and they don't even get the books. If they did so, they might be struck by how far ahead Singapore Math is over Everyday Math in fifth and sixth grades.
Philosophical talk about understanding versus practice just hides the fact that they really don't know what they are talking about. There is nothing in Everyday Math that is meaningful in terms of mathematical understanding. At the very best, it trades off depth and mastery for superficial (not mathematical) explanations that do not provide any support as the problems become more complicated.
I don't want a philosophical discussion of real world problems or understanding because the teachers who are pushing these ideas don't know what they are talking about. Philosophy cannot be used to hide ignorance.
By the way, tomorrow night is science fair night, where all of the kids test the sailboats on wheels they built to see which can go the furthest and the fastest when pushed by a box fan. Real world design with absolutely no foundational knowledge. Feel-good play learning.
They may be genuinely trying to do what is best for the kids, but they are not competent to do so.
Steve, I'm with you on this one. However, I don't expect every 3rd grade teacher out there to have her MS in math and to know better. They are just as much suckers for marketing as the rest of us. What I want to know is, who is it at the state level that is approving these books for adoption? That list is what the school's choice is going to be limited to, isn't it? Is it a committee? Are their names known to the public? What are their qualifications?
Is there the math equivalent of the Gablers? ;-)
I know where Steve is coming from. There are well paid people on the staff of our schools that are supposed to know better. We have curriculum directors, math consultants, and curriculum review teams. All of these people are charged with knowing something about math and the teaching of math. The fact that they are incapable or unwilling to look beyond the marketing materials they pick up at a conference demonstrates an appalling lack of competence.
"What I want to know is, who is it at the state level that is approving these books for adoption?"
Our schools (public and private) don't just select from a list. I suppose they could, but they don't. They discuss the issues and choices locally. I never hear them complain that they don't have enough choices and a list won't stop them from picking something else. A third grade teacher might be clueless, but schools have upper grade teachers and curriculum advisors who are supposed to know something about math. A masters degree in math is not required.
All they have to do is look at the high school curriculum and make sure that the lower grades bring the kids up to that level. They can't even do that! Why? I don't know? I guess they see enough (?) students get into honors math in high school that they think that everything is OK. Schools might provide tougher options for math in seventh and eighth grade, but they really don't know why or how kids get into those classes in the first place. Our schools don't even do that. There is a known curriculum gap between eighth grade and high school. I call it a curriculum wall. High school and lower schools are completely different worlds and they don't talk to each other. One high school math teacher told me that all they can do is advise the lower schools, and it seems that the lower schools don't like it when they do that.
It is philosophically driven and complicated by a lack of competence in math. I do see some conflict with the seventh and eighth grade math teachers. They seem to know more about what it takes to prepare kids for high school math, but they are still steeped in fuzzy ed school ideas. They also have to deal with the very unprepared students who walk into their classes.
At my son's school, lower grade teachers love Everyday Math and argue that the students get good marks on SSAT testing and do well in high school. Of course, they don't know whether this is because of the school or in spite of the school. They don't want to know about outside tutoring. If they hear about it, they just complain about pushy parents who want "superkids".
Then there is the competency issue that they don't get the job done, no matter what curriculum they use. I don't think Everyday Math is so bad that it produces kids in fifth grade who don't know their basic math facts. That's what happens when schools just go through the motions and don't set high enough expectations.
"However, I don't expect every 3rd grade teacher out there to have her MS in math and to know better."
I feel the same way about my son's third grade teacher. I think she knew the TERC program the school adopted would not benefit the kids, but she was powerless to do anything about it.
--PaulaV
Our school district is in the process of choosing a new elementary math curriculum to replace MathLand. The school board has adopted a 6-page evaluation rubric to choose which programs to pilot. The rubric is driven by a clear constructivist bent, and includes items like the following:
"The program is designed with the expectation that calculators are available to all students at all times and that all students have access to a computer for individual and group work."
"Students are encouraged to explore and investigate mathematical ideas. They are expected to read, write, and discuss mathematics. The program asks students to conjecture, test, and build arguments about a conjecture’s validity. Students are asked to reason about different strategies and outcomes, weigh the pros and cons of alternatives, and pursue varied paths when working on tasks. Students are expected to work on group and individual projects and assignments."
and, my very favorite:
"Students are expected to reflect on, make judgments about, and report on their own behavior, performance, and feelings. Students are asked to do self-assessment on selected aspects of their experiences as one method for evaluating students performance and disposition."
Not a word anywhere in the six pages about "learning to mastery," "distributed practice," or "preparing for algebra by 8th grade."
I have the opportunity to comment on the rubric, or on anything else about the curriculum evaluation process, but the school board's view of what they want is so different from what I think would be valuable that I just don't have any idea where to start.
--DaleA
I would suggest, at a bare minimum, that a math program be logically sequenced, with mastery expected.
When we implemented EM about 5 years ago, no one from the MS or HS were asked to participate in the decision making process. This was a k-6 math curriculum. The parents and teachers on the committee had no math background. Strangely enough, we have some high competent teachers with a passion for math, and the collegiate level math experience to prove it. They were not included in the search committee.
The committee began by looking at what was available and what other school systems were using in the area. At no point did the committee consider the "ultimate" objective - preparing kids for the next step.
After they narrowed their choice to two options (no indication how they narrowed it), Saxon and EM, they asked the state math consultant for advice. They never asked people who would be dealing with these kids in later grades.
Steve is right, they could have picked anything. No list exists, school systems can implement whatever they want. EM provided extensive "assistance" during this process. Everything the committee relied on in making their decision came directly from EM.
Now that EM kids are starting to reach the upper grades, MS teachers are starting to make some noise about the lack of preparation these kids have had. As long as our CMT scores remain high (due largely to extreme cramming just before the tests are taken), EM will stay entrenched.
"When we implemented EM about 5 years ago, no one from the MS or HS were asked to participate in the decision making process."
This is exactly what I see. This is their world and they don't want others telling them what to do, especially about philosophical assumptions.
"They were not included in the search committee."
Philosophy trumps knowledge. Competence is not a necessary ingredient.
"As long as our CMT scores remain high (due largely to extreme cramming just before the tests are taken), EM will stay entrenched."
This is what's happening with my son's school. Enough kids do well that they think everything is just fine. They don't document what that means and they don't want to know.
"... but the school board's view of what they want is so different from what I think would be valuable that I just don't have any idea where to start."
Start with a solid algebra course in eighth grade and work backwards. Get the textbooks for the honors track in high school and demand that the lower school offer the honors-track algebra class in eighth grade.
This is just the beginning. They have to work backwards to see what math curriculum best leads to that algebra course. Many students may not be ready for that course in eighth grade, so the school has to provide a path that will get the students to that same algebra course in ninth grade, not track them off to checkbook math.
This will be difficult to do when they get stuck on superficial discussions of understanding versus practice, or different methods of multiplication. Perhaps one way is to say that you don't care what they do about "understanding" if they achieve specific mastery levels in a list of mathematical skills. Use sample tests from Saxon or Singapore. Show them what you expect kids to be able to do in each grade. They can talk about understanding all they want, but if the kids can't do the problems, then what does understanding mean?
By the way, if they are just getting rid of MathLand now, after it has disappeared off the face of the earth for several years, then you have a big uphill fight.
"By the way, if they are just getting rid of MathLand now, after it has disappeared off the face of the earth for several years, then you have a big uphill fight."
Well, yes, not to mention that the only reason that they're getting rid of MathLand now is that the publisher no longer supports it!
Unfortunately, I don't think there's any possibility that they will listen to what I think should be covered each year . . . or what is covered in Singapore, or anywhere else for that matter.
The state standards (and state tests) completely dominate that aspect of the discussion. "Mile wide and inch deep" has been mandated from above, and I think we're stuck with it. I don't think I could even get them to consider the slightly narrower focal points from the newer NCTM document.
I'd like to argue that our kids might do better on the state tests if we just skipped some of the "mile wide" stuff and got a lot deeper than an inch on what matters, but I know that the conclusion will be that I don't have the credentials to opine on that (which is probably true -- I'm going on nothing more than common sense).
--DaleA
but I know that the conclusion will be that I don't have the credentials to opine on that,
Having an opinion different than theirs disqualifies you, not your background. Jim Milgram, a mathematician from Stanford, told me that he has had school board and other administrative personnel tell him that he "doesn't know what he's talking about" when it comes to math education. He is one of the top mathematicians in the country! But apparently that doesn't hold water if your opinion is such that it opposes fuzzy math.
It's possible that the ony creditials that matter are a degree from an ed school and the ability to use rhetoric that puts child centered guidance above actual content.
I've gotten to the point where I don't want to discuss things like "understanding", "constructivism", "mile wide", and mastery. Discussing these things makes it seem like it's all a zero sum trade-off; that schools and teachers are doing all they can already; that they come to the table with the ability to even understand the trade-offs.
I've also noticed that it is really difficult to get to the required level of detail in discussions, even with parents. One parent, who has a degree in Applied Math, talked about "understanding" and real world applications. I don't think he has yet figured out that he is talking about something quite different than the school. He says that we should not try to emulate people in Japan or Singapore. I'm not sure where that came from. I didn't ask.
Another parent I know has a PhD. in science. We were talking about a piano competition my son played in. He commented on the high percentage of Asian kids who were playing in the competition. He remarked on how they all push their kids too much. Suicide rate and all that stuff, you know. I'm smart enough to know when to move on to the weather.
It seems that you have to fight with generalities to get to a point where you can even begin to argue the details. Schools like to argue with generalities (who doesn't like balance) to get you to go away, and then they get to decide all of the details.
Real world problems and understanding are just non-issues that mask their lack of mathematical understanding. The blind are leading the blind. A parent told me that one curriculum advisor seemed to think that Singapore math had a good balance between practice and understanding, even though she doesn't have copies of the textbooks and workbooks. They have been at this for months and they don't even get the books.
We are HAMMERING AWAY at the idea that all committees must henceforth include real subject matter experts.
Of course, no such thing has occurred as of yet, BUT this may be one item on which the community has consensus.
We may be able to push it through; we may be able to set a standard that when educators and mathematicians disagree, parents are within their rights to insist that we use the curricula mathematicians would choose.
The fact that they are incapable or unwilling to look beyond the marketing materials they pick up at a conference demonstrates an appalling lack of competence.
It's worse than that.
In my own experience school personnel have acted as advocates and salespeople for the curricula they choose.
They don't want to know about outside tutoring. If they hear about it, they just complain about pushy parents who want "superkids".
Great way of putting it!
Superkids -- I'm going to remember that one.
Jim Milgram, a mathematician from Stanford, told me that he has had school board and other administrative personnel tell him that he "doesn't know what he's talking about" when it comes to math education.
This has long been one of my favorite edu-stories.
Another parent I know has a PhD. in science. We were talking about a piano competition my son played in. He commented on the high percentage of Asian kids who were playing in the competition. He remarked on how they all push their kids too much. Suicide rate and all that stuff, you know.
OK, I have the answer to that.
Our suicide rate is higher.
A lot higher.
That's in The Learning Gap, Stevenson & Stigler
oh heck
I don't see my copy at the moment, or I'd look it up
I'm finding that some lines of argument are highly persuasive with other parents, namely:
* subject matter experts ARE DIFFERENT FROM teaching experts and must always be included and consulted in curricular decisions. When the two are in conflict we should go with the judgment of the subject matter expert - or, at a minimum, offer choice so that parents who wish to go with the judgment of the subject matter expert may do so
* choice within the public schools - "schools within schools" - etc. Most parents don't instantly think choice within a school is a good idea, but I've found that they also feel less less concerned that I'm out to take things away from their children in order to impose something new they didn't choose themselves. It's **very** important, when talking about choice within the public schools, to point out that this is not a new idea -- and to point to instances in which choice has been offered in your own school. (You can also observe that teachers would probably like some choice as well.)
* international comparisons - Steve's experience with the scientist is a bit hairraising ...however I've consistently found that parents with whom I speak feel that their kids should be on the same track as kids in Europe and Asia - and certainly as kids in the Bronx.
* the professional model: I don't think I've **ever** had a parent object to the idea of moving to a professional model similar to the physician/patient model. When you think about it, that's remarkable. How many proposals for radical change make instant sense to lots of people? When I say, "Educators could be in the position of physicians, offering professional counsel and expertise. Parents would be in the position of decisionmaker, holding the final authority. Most of the time you would take the advice of the professional with whom you're working, but some of the time you would not - or you would seek a second opinion before doing so." Everyone gets this immediately. I also point out that when you have a good working relationship with a child psychiatrist your doctor typically wants to know about the resources & articles you've seen. (I assume this is less true of fields with a more solid research & practice base. But psychiatry is more analogous to teaching than cardiology, say.)
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