Saturday, December 15, 2007
I'm getting one for me, too. Maybe I'll finally be able to finish The Prince.
- The mathematics knowledge of US future middle school mathematics teachers generally is very weak compared to future teachers in Taiwan and Korea. It is also weak compared to German future teachers in all areas except statistics.
- Taiwanese and Korean future teachers were the top performers in all five areas of mathematics knowledge – including algebra, functions, number, geometry and statistics.
- School algebra (which includes functions) is the topic that, across almost fifty countries studied in TIMSS, was the major focus of instruction at seventh and eighth grade.
- On the algebra and functions tests, US future teachers performed at or near the bottom among the six countries—over a full standard deviation below the performance of future Taiwanesse teachers.
- The results for the statistics test were the only bright spot in future US teachers’ performance. The US future teachers scored enar the mean of the six countries.
- Future middle school teachers prepared by a secondary program performed somewhere between one-half to three-fourths of a standard deviation higher in algebra, functions, geometry and number compared to those prepared in either of the other two programs. The difference was slightly less in statistics.
- On average, the Taiwanese and Korean future teachers reported taking courses that covered around eighty percent or more of the advanced mathematics topics typically covered in undergraduate mathematics programs.
- For analysis (the study of functions) Taiwanese future teachers covered virtually all of the topics (ninety-six percent) while, in Korea, the coverage was seventy-nine percent.
- In the algebra and analysis courses which provide the mathematical background for middle school algebra, the Taiwanese, Korean and Bulgarian future teachers all covered around eighty percent or more of the possible topics while Germany covered around 60 to 70 percent.
- Mexican and US future teachers covered less than half of the analysis topics. The same was true for Mexico on the algebra topics, but US future teachers covered on average 56 percent of those topics.
Why not, everything else is banned in the Military.
The perfect military troop: Non-smoking, non-drinking, church going, Habitat for Humanity voluteering, non-extreme sporting, anti-gambling, feminish supporting, college educated, safety oriented killer.
Thursday, December 13, 2007
This is C. last summer at the picnic table outside our kitchen. We were probably working on percent (scroll down).
I don’t like math.
Why is there 1!?
1 + 1 equals 2!
Why is that?
I don't like math.
Words make sense. If I read “the dog,” I know what “the” means.
Math doesn't make sense.
- CHB summer 2007, 12 years old
I know you will all be impressed by the fact that I did not say, "You know what 'the' means?"
There is more to the "concept" of multiplication than iterative addition. (Try applying iterative addition to 1/8 x 2/5.) Perhaps iterative addition is appropriate for 2nd and 3rd graders learning their multiplication tables (or is it 3 and 4th graders these days?) But "the" concept of multiplication includes the fact that it distributes over addition (and that it's associative as well). The multiplication algorithm invisibly makes use of the distributive "concept," and does not employ an iterative "concept." Perhaps I'm overdoing the disdain quotes but I've been lied to too many times by people telling me that something is the "concept" of a procedure or rule and it turns out not to be.
A child with a conceptual knowledge of multiplication, and a lot of time on his hands, could successfully multiply two digits numbers without the multiplication algorithm:
24 X 86 means that
(20 + 4)(80 + 6) which means/implies that...
Etc. You see where I am going with this. One of the benefits of Singapore is that the kid does end up with a conceptual understanding of multiplication, and can apply his knowledge of concepts to come up with correct answers.
Notwithstanding operations on super hairy numbers, he is capable of doing the algorithm on paper when he needs to and can resort to "concepts" when he needs to do mental calculations.
the multiplication algorithm invisibly makes use of the distributive "concept"
I love that!
I love the whole Comment, in fact. People like me -- people who value liberal arts education in general and mathematics education in particular but who aren't expert in mathematics and probably never will be, have no way to get at these things.
I intuitively grasp the notion that there is some kind of "starter understanding" a person can have without being fluent in procedures. Seeing that 6x4 is the repeated addition of 6 4s or 4 6s as the case may be (I've spent quite a bit of time muddled over that one!) strikes me as superior to not seeing it. (I had no idea multiplication could be called repeated addition until I started reteaching myself math, and then I noticed it on my own.)
But at the same time I am gripped -- and gripped is the correct word -- by the conviction that a starter understanding is not a real understanding.
And yet because I lack a real understanding I have no way to express this and thus no means of combating the forces of reform math when they threaten to overrun my son's education.
I'm logging this post under Greatest Hits so I'll know where it is when I need it.
professor of psychology and education:
WHAT DO WE WANT CHILDREN TO LEARN?
Beyond basic literacy and numeracy, it has become next to impossible to predict what kinds of knowledge people will need to thrive in the mid-21st century....[T]he only defensible answer to the question of what we want schools to accomplish is that they should teach students to use their minds well, in school and beyond (Kuhn, 2005). The two broad sets of skills I identify as best serving this purpose are the skills of inquiry and the skills of argument. These skills are education for life, not simply for more school (Anderson et al., 2000). They are essential preparation to equip a new generation to address the problems of the day.
Deanna Kuhn, Professor of Psychology and Education Teachers College
Is Direct Instruction an Answer to the Right Question?
a response to Why Minimally Guided Instruction Does Not Work (pdf file)
EDUCATIONAL PSYCHOLOGIST, 42(2), 109–113
Apparently Mike Huckabee has not set foot inside an ed school any time within recent memory.
bonus observation: I might actually be willing to pay more taxes to stop the extraordinary professional development and ongoing education teachers "require."
Starting with the workshops on writing to learn in math and science. I would pay to have my district's science and math teachers not attend another one of these things.
Monday, December 10, 2007
Hello all. I am totally blind, but my wife and children are sighted.
My son is nine years old and in the fourth grade, and he is having a little bit of difficulty with long division--Especially when dividing a double-digit number into another number (e.g. 5128 divided by 47).
Can any one give me some pointers on how I might explain and illustrate the concepts of how to perform these types of problems, with emphasis on how to explain how to estimate?
I hope that this question is clear enough and that someone may have some ideas that will help me.
Thank you for your assistance.
This request was posted on a list where most of the members are adult blind mathematicians who are unlikely to know what is currently going on with grade school math and to what extent the son's problem is likely to be related to the educational environment.
I don't think this Blogger interface is very accessible to persons who use screenreaders. However, if anyone has any advice or suggestions, I'm happy forward them to this father.
C. did not recognize this expression as a case of the distributive property:
2x(x+1) + 3(x+1)
Can't say I blame him.
This isn't real obvious when you first come across it. So do this. Let's let x + 1 = T.
Now substitute T in the expression and you get:
2x(T) + 3(T)
Can you factor out the T?
Yes. You get T(2x + 3)
Now substitute the x + 1 back in. You get:
I think it was Ron Aharoni who referred to seeing expressions such as (x+1) as single entities as "chunking". To help students do such "chunking" it helps to do what I did above so they can see that x + 1 represents a number, and as such it can be factored.
I'm amazed by how difficult it is to see that "expression X" is the same as "expression Y." This is an ongoing source of pain in my mental life these days (not to put too fine a point on it), because I came to feel, shortly after Animals in Translation was completed, that Temple's & my thesis concerning hyperspecificity in animals and autistic people is wrong in some important way -- either wrong or perhaps right for the wrong reasons.
We argued that autistic people, children, and animals are hyperspecific compared to typical adults. Autistic people, children, and animals are splitters; nonautistic adults are lumpers; etc. (I know I've said all this before, but feel I must repeat in case newcomers stop by.)
The classic hyperspecificity story re: autistic children is the little boy who was painstakingly taught to spread butter on bread and then had no clue how to spread peanut butter on bread.
Until I began to reteach myself math, this kind of thing seemed to me incontrovertible evidence of the otherness of the autistic brain. But now that I'm factoring trinomials I've discovered I have something in common with that little boy. That's probably why God or the universe decided I should take up math. I needed an object lesson.
Still, the observations Temple has spent a lifetime making of animals' (and autistic people's) hyperspecificity aren't wrong. Normal adult humans aren't hyperspecific in the same way animals and autistic people are hyperspecific.
Sometimes I wonder whether the issue is simply that non-autistic adults pass through the hyperspecific stage of knowledge more quickly or more frequently than autistic people do. When a "typical" adult (typical being the preferred term these days) encounters brand-new material he, too, is hyperspecific, as I am with math. Everyone starts out a splitter.
But I don't think that's quite it, either.
I'm getting the feeling that animals may not be hyperspecific across the board, but perhaps only in certain realms. Maybe animals are more hyperspecific than adult humans when it comes to sensory data? e.g.: To a horse a saddle feels completely different at a walk, a trot, and a canter -- so different that he will buck his rider off when he moves from a trot to a canter if he hasn't been carefully trained to tolerate the saddle at all 3 gates individually.
Better story: Temple's black hat horse.
This was a horse who was terrified of people wearing black hats. He wasn't terrified of people wearing white hats or red hats. Just black hats.
I'm thinking, this morning, that humans may be relatively oblivious to "sensory data," that we're lost in words -- so perhaps words are the place where you'll see us being hyperspecific ? (There's evidence that language masks sensory data, but I don't know it/remember it well enough to summarize.)
Temple complains about this all the time. She'll give a talk and her audience will take away a too-specific meaning from her words; then they'll go out and apply her advice all wrong & bollocks things up. "People get hung up on the specific words," she'll say. (I'll write down the next example of this that crops up - can't think of one offhand.) These conversations have gotten to be quite funny because, after years of reading countless articles on autistic people being literal-minded and "concrete," I am now spending my time listening to an autistic person complain that normal people are literal-minded.
Well, she's right. Looking at an expression like 2x(x+1) + 3(x+1), I'm like the horse with the saddle and so is my 13-year old son.
"x+1" next to 2x is completely different from "x+1" next to 3.
Different enough to make us start pitching our riders into the haystack.
Robert Slavin on transfer of knowledge
rightwingprof on what students don't know
Inflexible Knowledge: The First Step to Expertise
on the 5th grade students she assessed:
I assessed a whole [5th grade] cohort in math (only one was a special education student, and he was no worse than the rest), with similar results.
They scored well in understanding "concepts." Whoop-de-doo. But none could reliably do computation with regrouping, do mental computation of any sort, measure accurately, use a number line, name fractions, or do any operations with fractions or decimals. Surprisingly they could not even count money correctly!! They could not figure out elapsed time, nor read non-digital clocks.
What good is all this great "conceptual understanding" if you can't count coins under $3.00, measure the length of a board, find the perimeter of a triangle or determine whether to add or subtract to compare two numbers? The one thing most were good at was reading graphs -- pictographs, bar graphs and simple tables.
Some of these students were quite bright and had good reasoning ability but what they ALL lacked was knowledge of number facts, facility with algorithms, precise vocabulary (perpendicular, acute angle, numerator, range), an organized approach (if guessing didn't work they were stuck), in short MASTERY at any level. Like [instructivist's] students, all have had nothing but fuzzy blah-blah since entering school. Few to none can afford Kumon and most don't have computers at home or access to them, so even that kind of practice is not available to them.
A recent study of teacher competence in our district found that most teachers in 5-8 grade mathematics did not themselves show mastery of the subject at that level. I think we may have come full circle. The system is now being run by people who are its products, and many are quasi-literate and numerate, however bright and caring they may be. Also, the lack of any kind of intellectual rigor or scientific and statistical training in their preparation has left them vulnerable to every fad that comes down the pipeline.
conceptual understanding w/o procedural knowledge:
The best way to illuminate my point would be to contrast my findings with what might have occurred in "the old days." Time was -- the 50's? 60's? when you might often find students who could proficiently, or at least adequately, perform basic operations -- including, in many cases, operations with fractions and decimals -- but would have been at a loss to explain what they were doing, or why they were (for instance) regrouping in subtraction, or inverting fractions to divide them. It was not considered necessary for students to have a "deep" understanding of the number system per se. Most of us (I remember this myself) "got it" in the process of learning the algorithms and how to apply them, and did (eventually) understand why you had to regroup, what you were doing when inverting fractions (I would have wanted to show how it works, even now it would be hard to explain succinctly in words, but it's easy enough to demonstrate).
The children I was assessing -- with a detailed, widely-used norm-referenced diagnostic math test and also with a locally developed "performance assessment" -- showed the opposite pattern. They understood about place value (haven't played with Base 10 blocks for years for nothing), knew that multiplication is repeated addition, that fractions can name parts of an object, or members of a group, and so on. They could show you (with the ever-present manipulatives), or draw a diagram and explain. They could tell you why you have to rename the ones as tens, why you have to keep the decimal points lined up, what the value of various coins etc. is, what the "big hand" and the "little hand" on a clock indicate, and so forth.
What they could NOT do was reliably apply a procedure to come up with an answer. Given a problem like 24X86, they knew this means you make 24 groups of 86 (or 86 groups of 24), but would get lost trying to build them with blocks or count out the tally marks. If they tried to use the algorithm, typically they got directionality, order of steps, etc. all mixed up, and they didn't know the number facts. The brighter ones would figure out the answer had to be something around 2000, but many did not even get that far. When counting coins, they lacked a strategy such as, counting the quarters first, then the dimes, then the nickels, etc. They randomly counted each one separately and continually lost count. They didn't know how to "count up" to find a difference (or an interval between numbers or clock times).
This is so helpful. It makes sense to me that a student could have some degree of conceptual understanding without procedural knowledge, and yet when I try to think of how that would work I come up with a blank. It's clear to me, for instance, that my "conceptual understanding" of unfamiliar subjects -- economics, say -- is thin at best. God is in the details.
hyperspecificity for conceptual knowledge -- ?
One thing I think I see happening with palisdesk's 5th graders is "hyperspecificity" for conceptual understanding. I'm used to seeing hyperspecificity for concrete knowledge. I hadn't really thought about hyperspecificity for concepts such as the meaning of multiplication. It makes sense, though. I'm pretty sure it happens to me all the time, teaching myself algebra 2. I'll have a basic conceptual understanding of a concept -- logarithms, say -- that doesn't immediately transfer to a problem type I haven't done before. I'll try to come up with an example to post.
As usual, I'm hamstrung by a lack of terminology. Our sturdy workhorse words -- procedural, conceptual -- are failing to give me the distinctions I need within the category of "conceptual understanding."
These 5th graders have for arithmetic what I have for logarithms: some kind of start-up understanding of the concept that won't take them very far when confronted with an actual logarithm problem in the flesh.
thank you palisdesk, pissed-off teacher, instructivist, redkudu, dy/dan, nyc educator, exo, smartest tractor (I'm sure I'm leaving others off ....)
For parents and the broader public schools are a black box. Mike Schmoker says that's by design; the official term for management in schools is "loose-coupling," which means, I gather, that the goal of management is to protect the core functions of the organization from outside scrutiny. (more later...much, much later)
Teachers who share their experiences with outsiders are functioning as the education reporters our country needs but does not have.
Robert Slavin on transfer of knowledge
loose coupling & instrutional leadership
A zillion years ago when my son was in first grade and computers were still a novelty, he decided to write his spelling words on the computer. The little twerp was using copy and paste to write them the required ten times. When I made him stop, he told me "but my teacher said it was okay". As I said, computers were new and she did not know how he was using it.
Kids have always cheated. We just have to be smarter than them to stop them in their tracks.
The middle school's Civil War Museum exhibit last spring put the kids' papers on display. Ed looked at every paper & saw a huge amount of internet text. Same thing the year before at the Jason Project exhibit. Many papers included downloaded text. A couple of papers were the same paper.
Of course most people didn't notice because they were looking at the crafts projects, not the papers.
Crafts projects are to a middle school what misdirection is to a magician.
I once had a 9th grade student who turned in an essay on Greek mythology that was totally plagiarized. When confronted with the computer screen shots of the essay's genesis, he denied it up one side and down the other.
Finally, he broke down, blurting out, "But I didn't even write the essay -- my MOM did!" I spoke that night with the mom, who freely admitted ghost-writing her son's essay. She claimed not to know that cut-and-pasting huge swaths of someone else's writing was wrong. Where to begin?!?
(This happened at a GREAT public high school north of Chicago...)
My favorite cheating story happened at Iowa, where I was teaching the freshman rhetoric course. One of my students plagiarized her entire essay, word for word, from the textbook. She denied having done so and then, when I opened up the textbook and did a side-by-side comparison, said, "I remembered."
Of course, now that I have two autistic kids, it strikes me that she could have remembered....
Sunday, December 9, 2007
I'm amazed at the number of kids who simply copy things off the net, with the labels still there, and expect me not to notice it was written by some professional hack writer instead of an ESL student.
They all do this! Not just the ESL students, everyone.
Ed was talking to a French journalist this week who told him kids in France do the same thing. They think writing means downloading from the internet. The less sophisticated kids cut and paste things whole; the more sophisticated kids cut and paste individual sentences, piecing them together as they go along.
It's as if the basic unit of composition is no longer words but sentences, and you look up the sentences you need on the internet not in the dictionary.
Research Question #2: To the degree that the teachers believe students need to be better prepared, what are the major shortcomings?
The teachers were asked to rate the importance of a “solid foundation” in the each the 15 skill/knowledge areas asked about with respect to their target class students’ background preparation. Since the same background skills and knowledge for which the teachers rated student background as inadequate were also rated as important, the following areas emerge as the major shortcomings: rational numbers, word problems, and study habits.
Final Report on the National Survey of Algebra Teachers for the National Math Panel (pdf file)
National Mathematics Advisory Panel
They're not too keen on parents:
Research Question #12: Do they find more parents helpful in encouraging students in their mathematics studies, or do too many parents make excuses for their children’s lack of accomplishment?
Questionnaire item II.1i asked teachers to rate the extent to which they see “too little parent/family support” as a problem in their school. The responses indicate that about 28% of the algebra teachers feel family participation is a serious problem and another 32% believe lack of family participation is a moderate problem (Figure 13).
It's a bit difficult to cess out what this means exactly. The question itself seems off-base; the handful of direct quotations from teachers have nothing to do with parents "encouraging" their kids versus "making excuses" when their kids do badly:
The written-in “verbatim” responses most often mentioned included handling different skill levels in a single classroom, motivation issues, and student study skills. Some notable responses were:
Walking into a class of 30 students in which 1/3 of them don't have the prerequisite skills necessary to be in the class. Many of whom don't know their basic arithmetic facts and know they aren't going to be successful from day one no matter how hard they try.
Students come to me without a basic understanding of math. I am constantly re-teaching concepts that should have been mastered in the earlier grades.
Parents not letting me do my job as I see fit. (Autonomy in the classroom.)
Getting students and parents to believe that education is important. Students don't do their homework...you call the parents...they say that the student will start doing the work (and coming to tutorials). The students still don't do the h.w. -and still don't come to tutorials.
Engaging students who have come to believe that they are stupid because they are struggling with my state's cognitively inappropriate standards.
The Algebra I teachers generally reported that students were not adequately prepared for their courses. The teachers rated as especially problematic students’ preparation in rational numbers, solving word problems, and basic study skills. A lack of student motivation was by far the most commonly-cited biggest challenge reported by the teachers. The problems the teachers identified with the pre-Algebra I mathematics curriculum and instruction and with the lack of parental support for mathematics were likely to be contributing factors to the lack of adequate student preparation and motivation.
In light of the generally favorable views the teachers report with respect to curriculum and instruction, the issue of unmotivated students implicitly is something the teachers view as more of an “algebra-student problem” than an “algebra-teacher problem”. The generally-negative views expressed by the teachers of parental support for mathematics reinforce that attribution. Taken together with the generally negative ratings of background preparation, the lack of student motivation suggests that careful attention to pre-algebra curriculum and instruction in the elementary grades is needed, both to remedy the specific skill deficiencies as well as to identify ways in which negative attitudes toward mathematics are developed.
Our schools need a paradigm shift. If you've got an "algebra-student problem" then you also have an "algebra-teacher problem" by definition and you need to take action. Institute supervised homework study halls, formal reteaching of foundational skills, whatever.
Motivation as the Magic Key to All Learning is overrated in any case, I think. How motivated were Siegfried Engelmann's students in Project Follow-through? How motivated are the KIPP kids?
My own 8th grade child is not exactly gripped by a coursing desire to learn algebra. However, he is learning algebra, and he's learning it pretty well. That has to do with the teacher.
And, yes, it has to do with "parental support for mathematics," which in our case means that I do every homework assignment so I can check C's work and have him re-do the problems he missed.
You can't leave algebra up to student motivation.
I'd love to help, but as many of you know, I am the resident Math Phobe of KTM. With the husband out of town, you are my only hope to helping him with this one particular problem. I have a feeling it is something really obvious.
Here it is:
"The sum of two numbers is 32, and the product of these two numbers is 48. What is the sum of the reciprocals of the two numbers? Express your answer as a common fraction."
I have the answer sheet, but no explanation on the best way to approach this.
Any and all help would be appreciated. Speak slowly, though....although the kid will probably understand what you're saying.
Of a staff of more than 50, only one other teacher and I have more than 10 years experience, and I'm the only one with much expertise in instructional issues. There is definitely a generation gap in knowledge and skills, no fault of the newer folks. Most had no preparation at all in curriculum, behavior management, instructional design, teaching skills etc. Lots of philosophy instead. I don't dare contemplate what things will look like in another 10-15 years.
My district is hemorrhaging experienced teachers.
In their place we hire novices. More than half of our new hires have never taught; the rest have fewer than 5 years' experience. The administration's stated intention, with all new hires, is to grant tenure. A great deal of time and energy are devoted to mentoring and supporting new teachers who are struggling.
My district has, in the past, refused even to interview experienced math teachers. One person denied an interview was qualified to teach math and physics. He had glowing recommendations. Another experienced NY teacher was told that if he taught here he would have to take a pay cut to the salary of a 5th year teacher. He has retired from the public schools and is now teaching in a private school.
I wonder whether this qualifies as a form of age discrimination? Someone asked the Board about the average age of new hires; the answer was, "We're not allowed to ask age." Wouldn't a policy of interviewing only novices be a problem in terms of age discrimination?
I suppose not. We do interview and hire career changers.
I should add that we've had some terrific young teachers. It's not the case that a young teacher can't be good. She can. (Mostly, these days, it's "she.")
It is the case that he or she isn't as good today as he or she is going to be.
The Why Chromosome
Practice Makes Perfect (the 10-year rule)
This WAS an interesting article. However, there must be tremendous variability -- ours is considered an excellent plan, but I did the math and if I retired now I would have to live on 46% of my current income (doable if I moved to a trailer park or seniors' home -- but I'm not attracted to the former nor eligible for the latter). Even at max pension is 60% of previous income. The coup de grace is that one has to purchase one's own medical and dental plan, and there are no comparable ones available from any providers at any price. This isn't so much a consideration for actual seniors but it is if you're in your 50's. Correspondents from other large districts in the east and Midwest have similar stories.
Fighting the system is something that does definitely lose its appeal over time. Guerrilla instructivism is energy-draining. Can't agree more about the frustration of being confronted with one obstacle after another every time you want to actually teach or help a kid. However I do still get an iconoclastic rush out of defying authority and successfully teaching basic skills: the pedagogical equivalent to manning the lifeboats. One consolation is that if I get REALLY p.o.'d I only have to give 10 days' notice. I could get a pleasant job in a bookstore or something to stay afloat. Of a staff of more than 50, only one other teacher and I have more than 10 years experience, and I'm the only one with much expertise in instructional issues. There is definitely a generation gap in knowledge and skills, no fault of the newer folks. Most had no preparation at all in curriculum, behavior management, instructional design, teaching skills etc. Lots of philosophy instead. I don't dare contemplate what things will look like in another 10-15 years.
That article hits it on the nose--if I retired today I would be bringing home more money than I am working full time.farewell to the baby boomers
I don't work for the money, or for the few extra dollars I will make by staying longer, I am working because I really love what I am doing. Most of us 55 year olds are staying for that same reason. We are leaving, not to get more money but to get away from a system that is abusing us and abusing children. Kids are being forced to learn things they don't understand, and will never understand or need. Classrooms are over crowded. There are no meaningful tutoring programs and the system keeps piling on more and more tests. Older teachers are being harrassed. The schools only want the young ones who are earning a much lower salary and will jump as high as any admin tells them too.
We are just tired of the BS. WE are tired of being fought every time we want to do something to help a kid. We might as well take the money and enjoy our lives.