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Like I said, I'm fairly new to pre-university education, though I see the results, and all my colleagues agree that we've been getting students with poorer and poorer math skills over the years. Having said that, we don't want to get into how I ended up tutoring (it's a long story). My tutee -- I'll call him Ricky -- is an 8th grader. His parents both have PhDs (before you ask why they aren't helping him, I do not know, and there really is no tactful way to ask). I'm being intentionally vague about details because, well, for personal and political reasons.
Ricky is extremely sharp, by the way. He immediately understands what I'm saying. I never have to explain anything more than once (he has the attention span of a gnat, though he is only in the 8th grade). Ricky enjoys learning, and is naturally curious. He's like a knowledge sponge. When I started working with him, he couldn't tell me what 16-9 was without reaching for his calculator. He had no mastery of basic operations, so that's what we did. He resisted at first (why do I have to do this when I can use my calculator?) but now that he has that information mastered, he can do things like basic algebra much more quickly and effortlessly, and he understand now.
Back to the course.
Okay, I may not have an ed PhD (thank God for that!) and I may not have pre-university classroom experience, but I've been teaching, writing exams, producing materials, and creating (and fiddling with) curricula for a long time. So even though I may not have any primary or secondary school teaching experience, the basic principles are the same.
They use worksheets (in-class worksheets -- the really frightening ones, with substance-free discussion and "food for thought" questions and "try this!" exercises -- as well as the take-home worksheets -- traditional math problems -- she gives them so they'll get decent scores on the state exams.
We have a problem right there. This is an admission that what they're doing in class -- the discussion and "food for thought" questions, and the "try this! Create three ways to calculate the area of a circle!" exercises -- are not teaching students the skills they need to get decent scores on the state math exams. If I were teaching the class and my students weren't learning the material, I'd change what I was doing. But this has not occurred to the teacher.
What strikes me, however, is how disorganized the course is.
They hop from topic to topic weekly, with no logical progression from topic to topic (for example, going from simple linear equations to probability -- and why they're doing probability in the 8th grade, I do not understand). They don't spend enough time on one topic to actually learn it, nor do they cover any topic to any depth, which makes no sense to me. The first thing on the list when they go back after break is limits -- why would 8th graders be doing limits when they don't understand basic fractions? Why would 8th graders be doing limits even if they did understand basic fractions?
The teacher also gives them "review" sheets (the sneer quotes are there because "review" implies a topic that's already been covered, and they don't cover a topic in any sense of the word). One was on long division. Ricky couldn't do it. He'd never seen anybody do long division, nor did he understand factoring. I asked him if his teacher had shown them how to do this, and he said no -- which normally I would take with a grain of salt, had I not seen so much disorganized nonsense already. He said she sent it home with this -- and he dug out another worksheet, a "how to" sheet on long division. He's in the 8th grade.
Forget the silly in-class worksheets, and forget the fact that the teacher sends them home with worksheets (traditional problems) so they'll get decent grades on the state exams. Here's my question: How can you send work home with your students and ask them to do something they've never been shown how to do? How can you justify that? If I tried to get away with that, I'd be in the Dean's office -- and it wouldn't be pretty.
The biggest battle has been teaching him to approach a problem, take it apart, and figure out how to solve it. It's been as much tutoring logic as math. (Does that mean I've been teaching "higher-order thinking" skills?) I had to be sort of ruthless at first because he didn't understand why he had to learn all this, but I've got him not only solving the problem, but checking it by finding a second way to do it and using that to work backwards. Like when we were doing probability (I still think that's weird). There was a basic problem on the worksheet, one of those standard balls problems (there are 30 red balls in the basket and the rest are blue. The probability of choosing a red ball is 0.75. How many balls are there total?) So I showed him how to calculate the total number of balls with the equation 0.75x=30. Then we calculated the number of blue balls, and checked it by first calculating the probability of choosing a blue ball, then subtracting that from one to find the probability of choosing a red ball. Once he started to catch onto the logic, he started picking it up fast. What's sad is that a kid that bright shouldn't need a tutor.
Another few months, and I'll turn Ricky into a card-carrying math geek.
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36 comments:
Here's my question: How can you send work home with your students and ask them to do something they've never been shown how to do? How can you justify that?
All I can tell you is that it begins on day one of the first grade Everyday Math curriculum. Nothing that's come home has been taught in class as of yet.
What is the curriculum his school uses?
Incredible post - thanks so much for taking the time to write it.
I have to get back in touch with the Ed Wonks folks. This should go in the carnival.
In our middle school many of the brightest math students have tutors. When they don't have tutors they have parent reteachers.
This has become par for the course. The district doesn't care, isn't interested, etc.
Many of the tutors, in fact, are teachers working for the district. A couple of years ago the administration was apparently planning to put a stop to this, but I believe parents objected (though I don't know).
I can't endorse ending the tutoring business for Irvington teachers myself. A parent's first priority has to be his child's education; reforming the district comes second.
The question of why his parents aren't teaching him is simple: 8th graders loathe being taught by their parents.
Loathe it.
This makes it extremely difficult to teach.
If Ricky were my son and I had access to a tutor as knowledgeable as you I'd go the tutor route, too.
Ricky is pretty fast. Imagine how the slow learners are doing.
By the way, they introduce probability as early as third grade now. Why? Because it's on the state test.
"In our middle school many of the brightest math students have tutors. When they don't have tutors they have parent reteachers.
This has become par for the course. The district doesn't care, isn't interested, etc."
That's just plain sad.
"Many of the tutors, in fact, are teachers working for the district. A couple of years ago the administration was apparently planning to put a stop to this, but I believe parents objected (though I don't know)."
Interesting. Most universities at least discourage faculty from tutoring university students in their spare time, if not have rules against it. It's seen as a conflict of interest.
"The question of why his parents aren't teaching him is simple: 8th graders loathe being taught by their parents."
That's a very good point. I hadn't thought about that.
"By the way, they introduce probability as early as third grade now. Why? Because it's on the state test."
That's ... bizarre. Especially since what Ricky got as probability didn't get any more complex than "what is the probability of choosing a green ball?" and that could be argued really isn't probability -- it's simple algebra. So Ricky knew (when we did it) that the probability of tossing heads was 1:2. But he didn't know about the independence of trials.
Like I said, they don't cover anything to any depth. They just touch on something, then touch on something else next week. It's really bizarre. And there's no logical progression from topic to topic, which is even more bizarre, so they students never get the opportunity to build knowledge based on what little they do know already.
In New York state Statistics & Probability is one of five “content strands” that make up the core curriculum standards. These apply from kindergarten on up.
I suspect that one source of motivation for introducing statistics throughout K-12 math was to move from "hard," boy-type math, i.e. the kind of math one uses in chemistry, engineering, and physics, to "softer" girl-type math, i.e. the kind of math one uses in the social sciences.
The topic of statistics is also attractive from a "critical thinking" perspective because people use statistics so badly. A teacher can ask students to criticize a newspaper's or politician's use of statistics.
Here's a passage from David Klein's new paper
Through the 1990s, funding for textbooks aligned to the Standards flowed from the National Science Foundation (NSF) and corporate foundations.[10] Parents were the first to object, especially in California,[11] and they formed grassroots organizations[12] to pressure schools to use other textbooks, or allow parental choice. NCTM aligned books and programmes were criticized for diminished content and lack of attention to basic skills. The elementary school programmes required students to use their own invented arithmetic algorithms in place of the standard algorithms of arithmetic. Calculator use was encouraged to excess and integrated even into kindergarten lessons. Student discovery group work, at all grade levels, was the preferred pedagogy, but in most cases, projects were aimless or inefficient. Statistics and data analysis were overemphasized repetitiously at all grade levels at the expense of algebra and more advanced topics.
When you put this passage next to a passage he quotes from Hung-Hsi Wu the "hard-soft" opposition jumps out:
This reform once again raises questions about the values of a mathematics education ... by redefining what constitutes mathematics and by advocating pedagogical practices based on opinions rather than research data of large-scale studies from cognitive psychology.
The reform has the potential to change completely the undergraduate mathematics curriculum and to throttle the normal process of producing a competent corps of scientists, engineers, and mathematicians.[19]
My guess is that the authors of the NSF-funded textbooks are slightly at odds with the notion that we teach mathematics in order to produce a competent corps of scientists, engineers, and mathematicians.
Here's another interesting passage:
An Agenda for Action also argued that 'emerging programs that prepare users of mathematics in non-traditional areas of application may no longer demand the centrality of calculus. . .' The de-emphasis of calculus would later support the move away from the systematic development of its prerequisites: algebra, geometry, and trigonometry. The 'integrated' high school mathematics books of the 1990s contributed to this tendency. While those books contained parts of algebra, geometry, and trigonometry, these traditional subjects were not developed systematically, and often depended on student 'discoveries' that were incidental to solving 'real world problems'.
I just sent this post to the Carnival of Education.
It's those 21st century skills they keep talking about. Unfortunately, they're having problems with the other century skills that feed into them.
My LD son used to come home with multi-step problems involving graphs. No one had ever explicitly explained that problems can have steps. He was (with severe LD issues and a lower IQ) supposed to "figure it out" or something.
I told him that the problem was a multi-step one where he needed to gather info from other steps before he could finally solve it, like we did in Singapore and Saxon math. That helped him a lot. Then I could lead him to the answer.
If we followed up with 50 more examples that were very similar, he might have been able to understand how to approach problems with more than one step.
But no one does this with him (except me), so the schools come to the conclusion that he just CAN'T.
Even gradually practicing multi-steps with basic operations would have been a helpful prior sequence that could have lead to him understanding how to approach analyzing graphs, but the foundation wasn't there.
The goofiness goes on...
Susan
I'm only lately coming to see how profoundly unfair our standard curriculum and pedagogy are to almost anyone who actually needs to be taught.
At the same time, I think I'm becoming aware of the hidden deficits in the education of speedy learners like me.
I read an interesting short article at... Lisa Van Damme's site (Capitalism Something or other...)
She contrasted "pattern recognition" with "comprehension."
I have no idea whether this is a sound idea, but it struck me as potentially true of my own learning.
She said that there are students who can pick things up via pattern recognition, which I assume is a form of incidental learning.
I also assume that I've always done quite a lot of "pattern recognition"-type learning.
For instance, I can spell quite well in various languages, including languages like German. I may not know the meaning of a single German word, but I can spell in German.
She said that these kids basically have no idea what they're doing.
That's often true of me (uh...no irony intended...)
For instance, I probably learned quite a lot of grammar through pattern recognition. I "picked it up."
But I have no idea what the rules are, which means I can't access them when I need to use formal knowledge of grammar to interpret a passage of poetic or archaic prose.
It's better to be a fast learner than a slow learner in our schools, but I now think the fast learners have hidden gaps and deficits they aren't even aware of.
From what I've seen, students may be covering a topic called statistics, but they don't have a handle on the basic concepts, much less actually being able to run statistical tests or interpret the results.
And I've been saying for years that one requirement for voting should be that you've passed a statistics class, but I don't think teaching watered-down baby stats solves the problem.
Teaching watered-down baby stats definitely does not cut it!
It's not just eighth grade that jumps from topic to topic (triangles to probability to graphing to early algebra). That starts in 6th grade. I defy anyone to look at the tables of contents from 6th, 7th, and 8th grade text books and tell me which is which. I suspect that the same would be true if you inspected the actual lesson content. The plan is apparently to cycle through every topic fast. If the kids don't get it, that's okay; they'll see it again next year. Only the kids who score well--whether they're inherent math heads or well tutored outside school--get to escape the cycle. They get to start Algebra early.
On the whole, I find only a few of the middle school math topics really objectionable (too much emphasis on mean, median, and mode that began in 3rd grade, stem-and-leaf, box-and-whisker, etc.) The problem is that none of the good topics are taught to any depth or in any logical order.
Five years ago, I learned that math tutors in Palo Alto -- generally retired teachers -- were charging $75 an hour and up. I'm not sure about tutors in other subjects: Math -- especially pre-algebra and algebra -- seems to be the biggest hurdle for the most students.
Catherine,
I thought of your comment about boy math and girl math when I read this summary of an episode of The Simpsons:
"Springfield Elementary School Principal Skinner is ousted after casually remarking that girls aren't much good at math. Skinner's female replacement divides the boys and girls into separate schools since, she says, girls can't learn math around "aggressive, obnoxious" boys.
Brainy 8-year-old Lisa Simpson is delighted until she attends the girls' math class. "How do numbers make you feel?" the teacher begins. "What does a plus sign smell like? Is the number 7 odd or just different?" Aghast, Lisa poses as a boy to attend the ghettolike boys' school, where real math is being taught."
Found here:
http://www.sciencenews.org/articles/20060610/bob8.asp
Hi, Dan!
Your comment reminds me of Mike Feinberg at KIPP:
You know, talk about curriculum, if I put in front of you a fifth, sixth, seventh, and eighth grade textbook in math and opened up to page 200 and I jumbled them up, and said, “order them from fifth through eighth grade in order,” you'd have a very tough time because they all look the same. That's because, unfortunately, we have this national strategy of “we're not really going to teach to master, we're going to teach to exposure and over lots and lots of years of kids seeing page 200 in the math book, eventually somehow they're going to learn it. We're going to teach them how to reduce fractions in fifth grade, in sixth grade, in seventh grade, in eighth grade, in ninth grade and continue until finally somehow magically they're going to get it.” Instead of thinking, “let's teach the kids how to reduce fractions at a mastery level in fifth grade, maybe spend a little time reviewing it in sixth grade but let's move on to pre-algebra and let's move on to algebra then.” And that's been our take and so it's not that we have a different math curriculum as much as we have a different math strategy and a different math philosophy.
The plan is apparently to cycle through every topic fast. If the kids don't get it, that's okay; they'll see it again next year. Only the kids who score well--whether they're inherent math heads or well tutored outside school--get to escape the cycle.
Exactly.
In our middle school we've gone one better; the "accelerated" kids are in the same boat as the non-accelerated kids, churning topics so fast they forget most of what they saw the day they move on. The entire Phsae 4 course, all 3 years of it apparently, is one long cram-fest.
Actually, our accelerated kids may be in worse shape than the non-accelerated kids. A parent whose older child dropped out of the accelerated course a few years back told me that a lot of the non-accelerated kids had better knowledge & fared better on state tests.
I'm sure that's true.
At this point I'd drop Christopher down to "Phase 3" in a heartbeat (at least I think I would), but for the fact that Christopher doesn't want to drop.
Of course I say that, but I didn't do it last year when Christopher did want to make the change. The whole idea of dropping down to Phase 3 when I know he's capable of learning the material wakes up my Stubborn gene.
We've gotten into a head-butting match with the school that no one's going to win.
otoh, when it comes to head-butting I have few equals
My mom once said, "I can be led, but I can't be pushed."
No!
That wasn't it.
She said, "I can be led, but don't push me."
Sadly, there appear to be few Scots-Irish types in these parts.
ONE MORE THING!
I'm happy that Christopher refuses to drop out of Phase 4.
He's one of the youngest kids in his class; he has no typical siblings to knock heads with; he's by nature a cautious and sensitive boy.
The fact that he's developed the determination to stick with a dreadful course - a course that will keep him off the Honor Roll all school year - is great.
His life would be a whole lot easier if he ditched this class.
He's choosing to stick with 'hard.'
That will stand him in good stead.
"My LD son used to come home with multi-step problems involving graphs. No one had ever explicitly explained that problems can have steps."
Yes, yes, that's exactly what I've run into with this tutoring thing!
But the problem is they spend so little time on any given topic and cover it so superficially that there is very little for students to learn (not to mention just incorrect information, like the definition of a circle my tutee was given, that it is a shape without corners).
The real problem with all this jumping about and covering things so superfically is that even if students were learning any substantial amount of material, they wouldn't be making connections between topics. Way back when I was taking my teaching practicum in grad school, one thing they beat into our heads was make connections for students and create continuity from class to class. I guess that idea has fallen out of favor.
My theory about why all this emphasis on data analysis and what the alleged curriculum designers call statistics is because of the emphasis on "real world" problem solving.
It's considered boring and "rote" to have give students ordered pairs of numbers to plot on a graph, or to give them ready-made textbook problems with "nice" answers that conform to the y = mx + b linear graph. The fear is that students, rather than learning and mastering the basic skills and concepts of linearity, will grow up thinking that that's what the real world is like, and won't be able to handle real-world problems.
So instead, they skimp on the basics, and have them collect real world data, such as ratio of hand length to arm length for a sample of students, or weight vs height, and then use their calculators to generate a best fit line to the collected data.
Math then becomes empirical and statistical rather than a logical sequence of basic skills and concepts that they will later be able to generalize to solve the real world problems. Deduction based on logical principles has not been on the agenda for some time.
Fuzzies believe that the "traditional" word problems in algebra texts are just extensions of exercises in algorithmic skills, and solving them does not involve the alleged "higher order thinking skills". Learning how to break things down into smaller problems (i.e, multi-step problems), however, is an application of higher order thinking skills, just as a complicated dance is a series of smaller pieces/steps that the dancer can perform because he/she has mastered the various forms.
My theory about why all this emphasis on data analysis and what the alleged curriculum designers call statistics is because of the emphasis on "real world" problem solving.
That makes sense.
I remember the first page of Trailblazers for grade 5 has a whole long narrative about a Hispanic girl on her first day of school wanting to wear sneakers or something while her mom wants her to wear nice shoes.
She goes to school and takes data on what kind of shoes the other girls are wearing.
Students are supposed to "solve" this problem by saying what the mom is going to do when she finds out all the other girls are wearing sneakers.
oy
the definition of a circle my tutee was given, that it is a shape without corners)
good lord
one thing they beat into our heads was make connections for students and create continuity from class to class. I guess that idea has fallen out of favor
WE HAVE A NEWBIE!
oh no, no, no, no, no, no, no
MAKING CONNECTIONS IS IT
MAKING CONNECTIONS IS EVERYTHING
MAKING CONNECTIONS IS THE WHOLE ENTIRE POINT OF CONSTRUCTIVIST MATH
the difference being that today it's up to the student to do it
the student is to:
a) activate prior knowledge
b) make connections
c'est tout!
I got all excited at one point last year because Ms. K started giving the kids pre-tests.
I thought YAY!
FORMATIVE ASSESSMENT!
Turned out they were just supposed to be assessing students' prior knowledge so the students could make connections.
That's ed school today.
Assess the prior knowledge, not the post knowledge.
Math then becomes empirical and statistical rather than a logical sequence of basic skills and concepts that they will later be able to generalize to solve the real world problems. Deduction based on logical principles has not been on the agenda for some time.
oh!
good point
right
the real-world emphasis leads logically to watered down statistics
Ben learned the range, maximum, minimum, mode, and median every year from 1st grade through 4th grade. In the 5th year they learned to divide and so they taught them to calculate the mean. This is still going on in middle school -- but in the curriculum they are using, they do seem to give it less emphasis.
You brought back my Everyday Math experiences with Ben in grade school -- every day he'd come home with something in homework that they hadn't been taught. I remember the day he came home with division problems, when he hadn't been taught long division. It turned out they were intended to do the divisions with calculators -- that left them more time for learning higher-level skills like calculating the max, min, median, etc..
In NYC we use the Prentice Hall book for our 9th grade algebra students. This is a book written without any sense at all. The topics are randomly scattered through the book with a few days of probability followed by some other topic, back to probablility, etc. We are supposed to teach scatter plots to weak 9th graders. It doesn't get much more ridiculous than that. We teach graphing parabolas before we teach graphing straight lines and solving literal equations before we teach addition of polynomials. I could go on and on about the silliness of this course. The seasoned teachers have devised their own way of teaching so the course is not so scattered. The new teachers really don't know how to do anything but follow the curriculum. It's sad. To make matters worse, kids are supposed to spend lots of time doing group work--they teach each other. I don't have to go in to how sad this is.
I'm one of those $$$$ tutors and it's a shame that kids can't get a proper education in school.
Sometimes new homeschoolers ask, "How can I get my husband on board with homeschooling?" I advise the technique I used with my husband, who studied math and ended up as an analytic philosopher: I got hold of a math textbook used by our local public schools and asked him to look through it. That was it.
So many of the dads where I live are engineers that a look at the ps math (or science) curriculum alone is usually sufficient to make them determined to teach their own kids.
I don't have any school-aged children and I'm not a teacher, but I've already encountered this...
Two years ago I went to visit my parents in Florida. My brother, sister in law and her daughter (then age 9) were living with my parents to get my niece into a better school system (although very bright, she was repeating 2nd grade) during my SIL's last year of PharmD. Anyway, my brother worked odd hours and was usually gone afternoons and evenings, and my SIL had classes/work when her daughter's (henceforth "my niece") school let out, so my mom (who also works, but got off earlier at the time) would go pick her up and spend time with her after school. My mom helped with a lot of my niece's schoolwork during the year they all lived together.
Anyway, while I was there 2 years ago, my niece really wanted me to help her with her math. No problem, I thought. Yikes, was I in for a surprise! They had word problems that required simple ALGEBRA (2nd grade!!!), and what's more, my niece had NOT been taught the fundamentals for algebra. I looked at her book for guidance on what they were learning, and it was COMPLETELY unrelated to the homework. WTH??? I was stumped by the instructions given and both she and I were frustrated that I had to start from scratch and teach her things NOT in her book. She was receptive, but it was tedious for both of us.
I do remember the "soft" math concepts in her book, though, now that I think back. I remember being appalled by the whole scenario, but until today when I read this post I thought it was just her class. Silly me!
By the way, I majored in a science (didn't use it--went into the military instead) and tutored trig for a semester or two before I graduated. I am not a math whiz by any stretch of anyone's imagination, but even I was flabbergasted by what I saw with my niece. She was not supposed to use a calculator at home on her mom's orders (I was not apprised of this prior to my session with her--she was so lost she was sneaking in the use of the calculator as a crutch; I really had to start at ground zero to teach her anything), but apparently they are required classroom items. Again, 2nd grade, WTH??? I didn't know how to use a calculator until 7th grade or later, and then only used it sparingly (usually to check my manual computations). This trend in math is absolutely moronic.
Stephanie
Opinionated Homeschooler,
Thank you for that suggestion. I was having exactly that problem. My husband and I both are scientists/engineers, and he thinks that my homeschooling interest is just me being anti-establishment on principle, rather than because it's *really that bad*.
He did have Prof. Wu has a prof, though, and Prof. Wu's thoughts on lack of current preparation might encourage him to home schooling as well.
The "pattern recognizers" find that their knowledge has no depth when they reach college--esp. if they go into science or engineering. It hurts like hell to realize you don't know how to learn, and you're 18, and because of your patterning, people expect quite a bit more of you. Being bright and untrained is a hazard just as being not-so-bright and untrained is.
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