Entries from AssessmentTests

(Well, maybe it is; I don't know. If you come across other sites being built along these lines, could you let us know? I'd love to see how other people handle the information architecture challenges we're facing here at ktm.)

Yesterday I came up with the image that Kitchen Table Math is an 'office.'

It's a new office, one that hasn't existed before.

It's an office for parents, teachers, therapists, and, I hope, eventually students, too.

So far, that image is working for me.

As a parent, I need colleagues.

And I don't have them.

I do have parent-colleagues for the standard issues of child-rearing: behavior, discipline, friends, moral values, chores, summer camp, siblings, allowances--all of that good stuff.

I also have parent-colleagues, to a limited degree, when it comes to education. I can talk to other parents about the various doings and goings-on in our schools.

But I don't really have parent-colleagues with whom I can discuss supporting and supplementing and, in some instances, replacing my son's curriculum and teaching.

collaborating with teachers

Schools simply aren't set up to promote teacher-teacher collaboration or teacher-parent collaboration. Every minute of a teacher's day is spent in the classroom, teaching.

We need release time! We do!

This year Christopher's 5th grade teacher, Mrs. D'Arcy, spent a huge amount of time just sitting me down and telling me how she teaches math.

She could do this because she's young (no kids yet), lives close to the school, and just so happened to have a classroom on the first floor close to where I was running my Singapore Math class in the after-school program.

So we'd run into each other, and she'd give me advice.

I believe strongly that we need formal mechanisms to create, promote, and sustain parent-teacher collaboration (and not the public-diplomacy-masked-as-collaboration event-oids that TRAILBLAZERS advises. Uggh.)

So, at the moment, I'm thinking that's what ktm is, and will become.

It's an office for parents, teachers, therapist, kids and all other interested parties.

So today, thanks to Carolyn's heavy lifting, we're one step closer to that reality.

I see you started with quotes. I have a favorite from Samuel Clements: There are lies, there are damn lies, and then there are statistics! I see that Catherine Johnson has a similar line in her July 2 info. Does she know if it came from a book character of Mark Twain's or if it was in a speech by Samuel Clements? I've never known the context. I've used it frequently in my own speeches about our local School Board in Thousand Oaks, California regarding their exaggerated claims of greatness.

The Conejo Valley Unified School District spent quite a bit of money distributing 26 pages in the newspaper about how "great" the 29 schools were doing. They bragged that the 3 high schools had 30% of the students at the California level of Advanced or Proficient. The public didn't understand the inverse. That performance was pathetic for our "excellent" district. That meant that 70% of those teenagers were in the 3 lower groups of the 5 levels: Advanced, Proficient, Average, Basic, Far Below Basic. 7 out of 10 high school students here were NOT even Proficient.

The District fools the public by this omission. One of your writers today also had some cogent remarks on statistics that are omitted.

Another interesting statistic here is that a couple of years ago a third of the CVUSD schools failed to make the minimum target of 800 points, which is only 75% of the API (Academic Performance Index.) The API starts at 200 and goes to 1000, so there are 800 points available, not 1000. Every 80 points translates to 10%. This further confuses the public. Many do not understand when I explain that the true top 10% is really 920 points. It is the empirical 10% that is important, not the artificial 10th decile. In our state the kids' scores are so bad that in the first few years of the API there were high schools that scored only 726 but were ranked in the "top ten". Yeah, decile, not empirical. Every year ONE OUT OF EVERY TEN California schools gets to brag that they are a "10", often with scores more than 150 points below 920, the true cut off for 90%, because the cutoffs for the deciles continue to be down in the basement.

This really isn't a math wars issue, precisely; it's just good marketing in the face of bad statistics. It's amazing that while mathematics, including statistics, is a discipline with very clean edges that would not appear to admit much potential for fudging, nevertheless it's so easy to mislead people using statistical language.

Not to lie, though; because it's definitely true that, every year, one out of every ten high schools is in the top ten percent of high schools. But what if 90% of high schools are failing miserably? That remaining 10% could lie anywhere in the range from excellence down to barely-crawling-along. So the fact that a school is in the top ten percent tells you very little.

In an academic world that is benchmarked with standardized tests such as the California API (and the Colorado CSAP), the ability to Lie with Statistics is more valuable than ever. That doesn't mean that standardized tests should go away -- quite the contrary. It just means that we'll continue to need watchdog groups like Cathy's to keep pointing out the real meaning behind the marketing.

When I give my tutoring clients an assessment test, I give them mostly calculation problems.

I usually give four word problems:

• one problem is slightly below what they should be able to do. It is easy to read and very straight forward.

• One problem is at their grade level. It uses straight foward numbers but is multistep.

• If they are above 3rd grade but still in elementary school, the third word problem involves fractions. It is usually a problem from Singapore math that has several steps clearly deliniated with an a, b, and c. They should be able to get at least part of the problem.

• The fourth problem is a multistep problem that requires that the student have some logical way (ie bar model or equations) of keeping everything straight.

don't rely on state tests

In practice, I'm still in favor of them, but I don't rely on them. High-stakes testing is subject to enormous political pressure from all concerned. Years ago Ed worked on the California History Social Science Frameworks. He helped the CA Department of Ed develop assessments for the Frameworks, evaluating off the shelf tests, which were, in his words, 'insanely easy.' 12th graders were evaluated at a 9th grade reading level.

The Dept of Ed developed its own tests, & tried them out. (They didn't test the entire state, and he doesn't remember which groups took them.) Two political groups objected: some conservative Christians objected to the critical thinking portion of the tests, and some minority groups objected that their children's scores would go down (which they probably would have, at first). These two groups put enough pressure on their respective representatives that the new tests were scotched before they were ever rolled out. CA went back to using off-the-shelf tests.

No state test will survive a high failure rate in my opinion. That's why I view the current situation in NYC, where Mayor Bloomberg's campaign is based on a sudden, monster increase in student scores, as being far from ideal. I'm fine with the idea of a mayor campaigning on improving student scores. And now that I've seen what can happen to one child's scores thanks to simple, hard work, I believe that you could have a sudden, monster increase in student scores on a broad scale. It's possible.

But I want to see independent audits of those scores. I want to see the test items, and I want to see an audit. Sunshine laws are a good thing. Let's have sunshine laws for state & local testing.

I once read a Diane Ravitch essay on this issue (if I find it again, I'll drop in the reference). She argued that the solution is to establish different levels of 'Pass,' as they do in British universities. Students could pass exit exams with high honors, honors, no honors, and so on. That would probably allow states to maintain rigorous testing in the face of parent opposition.

You might still have an inflated pass rate, but then again, maybe not. Competition spurs people on to higher achievement, and not just because people are naturally competitive, which I believe we are. Seeing someone you know & like do well implies that you can do well, too.

Given the pressures on state testing, I don't rely on New York state tests to tell me how well Christopher is doing. At the end of 4th grade, when Christopher had flunked fully one-third of his year's math course, he earned a '4' on the state math test. 'Exceeds state standards.'

I'm sorry, but a 68 on Unit 5, a 39 on Unit 6, and a 4 on the state exam don't square.

(This is kind of funny. A couple of months later I called one of the guidance counselors at the Middle School to ask about Christopher's chances of moving to Phase 4 when he entered 6th grade. The counselor said nobody ever moves to Phase 4 from Phase 3, so the chances were slim to none. I said, 'But he got a 4 on the state test!' He said, 'That doesn't matter.' I was outraged at the time, but even in the midst of my outrage I knew exactly what he was saying. He was saying Don't rely on state tests.)

So today I'm reminding everyone about these Practice Problems for the California Mathematics Standards Grades 1-8 for the Los Angeles County Board of Education, which David Klein developed for the Los Angeles County Board of Education.

The state of California has the best math standards in the country, according to the Thomas B. Fordham Foundation assessment of state math standards. David's problems will tell you whether your child meets CA standards--and, if not, which topics he or she needs to work on.

I count 85 questions on the 5th grade test in all, divided into 4 areas:

• number sense
• algebra & functions
• measurement and geometry
• statistics

The test isn't as time-consuming as it sounds, since often there are 4 separate questions in one larger question (such as identifying several points on a graph). Answers are included.

If giving the test seems like a lot to do in the face of Massive Pre-teen Resistance, just divide it up across a few days' time. That's what I did.


related posts:
Assess Your Child for Free Part 2
Assess Your Child for Free
and
David Klein at the AEI

OnlineTIMSSTest 27 Jul 2005 - 23:40 CatherineJohnson



This is a terrific resource. You can give your child 10, 15, or 20 questions from the 1995 & 1999 TIMSS tests. The web site scores them for you.




Explore Your Knowledge

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SampleEighthGradeTIMSSProblems 27 Jul 2005 - 23:50 CatherineJohnson




10 items

OK, I'm going to take this test.

I assume everyone can link to the same sample test, but I don't know for sure. The first question is about Penny & her bag of marbles.

oh, yay

I got all ten right, and my results around the world are just peachy. Penny and her marbles stumped 59% of U.S. students, 56% of international students (this is all intl students, I believe, including kids from very poor countries who've just started taking the TIMSS' test). Obviously, fractions are impossible. Although the Singapore Challenging Word Problems Grade 3 book made all the difference. That and Russian Math.

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OnImplementingNCLB 11 Aug 2005 - 22:56 CarolynJohnston


When I went looking at Education Next for the Caroline Hoxby article that Catherine recommended here, I found another article, by the same author, on implementing NCLB.

NCLB (the No Child Left Behind Act) was implemented in 2001, and is an ambitious bit of legislation to ensure that every school child will be proficient in reading and math by the year 2014. Hoxby actually gives a nice summary of it:

A core principle of NCLB is that every student must reach the desired level of performance: no group of students -- minority, disabled, poor, limited English proficient, mobile -- should be left behind.

Another core principle of NCLB is that every child is capable of attaining proficiency, defined in an appropriate way. Thus, while progress is important, NCLB deliberately emphasizes reaching proficiency, not making gains each year, regardless of past performance. NCLB provides no special recognition to students or schools that exceed the minimum. This is not a good thing or a bad thing, but it clearly demonstrates that the focus of NCLB is on bringing low-achieving students to a sound level of academic achievement.

A third principle of NCLB is that it works through the states, long the workhorses of the country's education system. States and localities provide more than 90 percent of funding for schools, so it makes sense for them to exercise control. Furthermore, with fewer schools to watch, states are in a much better position than the federal government to monitor multiple targets. Thus, even though NCLB monitors only proficiency, it encourages states, in their own accountability systems, to reward schools that make gains along the entire spectrum of achievement.

NCLB doesn't offer answers to the tough questions about the problem with American education: it just requires that schools improve, or suffer the consequences. That's a good thing. There is no roadmap for improvement in NCLB, because noone has one. There is a requirement for standardized assessment, which I consider a positive step -- although I think that high-stakes testing has to be handled very carefully in order to ensure that the incentives they create are the ones that we want people to respond to. I'm not concerned about time spent 'teaching to a test', which is a huge complaint of educators -- I have the feeling that in many places, teaching to a well-designed test might actually be a good thing, ensuring a base performance level, and some degree of consistency of curriculum from school to school. If the stakes are high enough, especially for an individual teacher or a school, the temptation to cheat will be there.

I think that NCLB has many elements of what's needed to improve our schools. But I have my doubts that it can succeed in its goals as it currently is implemented, and I'm afraid a big failure to 'get there' by 2014 will do a lot of harm.

One big problem is that absolute "No child left behind" language. Everybody who knows anything about quality assurance knows that there are always failures in manufacturing or software production -- the objective in QA is to drive the failure rate arbitrarily close to zero, not to set a deadline by which perfection will be achieved. Demands for absolute perfection are impossible to meet, and everyone knows it; so schools with big problems will not be thinking of ways to improve -- they'll know the goal is impossible, and they'll be looking for an out from the very beginning.

And here's their out: there is a world of trouble in the phrase "every child is capable of attaining proficiency, defined in an appropriate way." Anyone with a child who has special needs, and is involved in advocating for their child in the highly individualized and labor-intensive process dictated by the IDEA (individuals with disabilities education act), knows that the potential of a child is something that everyone assesses differently. I think that the 'defined in an appropriate way' phrase is going to be used as a way around the requirement of proficiency for individual children, plain and simple. The most likely scenario is that more low-performing children will be identified with special needs, so that 'appropriate levels of proficiency' can be defined for each of these children without harming a school's score. The wording about requiring each child to make 'adequate yearly progress' reminds me strongly of similar language I hear all the time in special education.

The poor academic performance of American students as a whole is something of a mystery. Are our expectations too low? Are our math curricula too boring and tedious, or too touchy-feely? Are our teachers too incompetent, are they undertrained, are they overtrained in the wrong areas, are we spending too little or too much money, are we expecting too little of minority students?

One good thing about NCLB is that it doesn't try to find the answers to these questions; it lets the states try to do that for themselves. Accountability is a big step forward - even if the metrics for success need retooling.

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BestPerformingStudentsPartThree 14 Nov 2005 - 02:32 CatherineJohnson


The question of how our top students compare to everyone else's top students has made me realize I need to be paying attention to this. My goal as a homeschooler-on-the-side is for Christopher to be able to major in a math-related subject in college if he chooses, which apparently means he should be able to score a 625 or higher on TIMSS.

So I'm going to start scouting information on all ranges of student achievement, and posting it here.

Here's my first:

Researchers determined which items students who achieved at the various levels on the total test were likely to get right. Then they placed the items on a scale from 200 to 750. So we have a pretty good idea of what the best students know that others have difficulty with.

Only the top 10 percent of 9-year-olds were likely to get this math item right. Students had to explain their answers verbally, symbolically or pictorially.

In the first part they had to indicate that 20 is twice as large as 10 or that 10 is half of 20. 10 percent of third graders and 21percent of fourth graders did this. A small number of students (less than 1 percent in any country ) received credit for satisfactory explanations even though they did not give a yes or no response to whether Julia was right.

U.S. percentages were 13 percent at third grade and 25 percent at fourth grade.

For the second part, only 6 percent of third graders and 15 percent of fourth graders responded correctly. 6 percent of U.S. third graders and 17 percent of U.S. 4th graders got credit. However, 30 percent or more got credit in Japan, Korea and Singapore.

I'm going to spring this one on Christopher tomorrow. I really can't tell whether he could have gotten this item right at age 9. If you showed him 10 girls and 20 boys he would have known instantly that boys and girls weren't half and half.

But I tend to think he would have been thrown by the sight of the numbers '10' and '20.'

As well, I'd say this problem imposes a high cognitive load. You have to keep Juanita and Amanda straight in your mind, unless you've developed seriously good informal chart-making skills, which Christopher has not done now and certainly had not done in 4th grade.

update: Christopher's answer

Christopher turned 11 yesterday (boo hoo).

His first impulse, as I feared, was to say 'yes,' Amanda is right.

He obviously had the 'environmental dependency' effect of seeing the numbers '10' and '20' and thinking: 1/2.

But then he corrected himself, and said, confidently, that Juanita is right and Amanda is wrong. (Nice to see that the Designated Stupid Person concept has spread to TIMSS, too.)

His explanation was a bit strangled, but it was right. He said, 'Well, if there's 1 girl for every 2 boys, then there's 1 girl and 2 boys, then 2 girls and 4 boys, then 3 girls and 6 boys...'

This is pretty interesting, because I think he had a 'number sense' or 'pattern' way of getting this answer. In other words, I think he got the answer without really knowing why or how he got it. He just knew it. Juanita's correct statement of the problem instantly became his statement of the problem; he didn't have to do any adding or subtracting or logical reasoning to test Juanita's statement.

Then, when I asked him to explain why Juanita was right, he explained how her answer would work as a kind of Fancy Skip Counting Mechanism. If you kept counting up by 2-to-1 ratios, eventually you'd hit 30 kids, and your ratio would be 10 girls, 20 boys.

After he gave this illustration I asked him, 'how many girls and how many boys would there be in the class' (forgetting that in fact THE PROBLEM TELLS YOU THIS UP FRONT) and Christopher said, instantly, '10 girls and 20 boys.'

When I asked him how he knew (TIMSS should just have 'Catherine' be the Designated Stupid Person) he said, 'I just knew it.'

Apparently he had forgotten the fact that we'd been given this information, too. Like mother like son.

In any case.....this is something I was talking to Carolyn about the other night: what is the relationship of implicit knowledge to expertise when you're talking about math?

Certainly in every other field (I think) implicit knowledge is a sign that you're getting good at what you do, because you don't have to think about it. You 'just know it.'

But math has been confusing for me in this realm.....our friend Fred was here a few weekends ago, and I asked him to take a look at a RUSSIAN MATH problem that was stumping me. Fred is a Big Brain; he went to Yale undergrad, then got a Ph.D. in experimental psychology at Stanford, I think it was; then got a law degree at Yale; then clerked for the Supreme Court.

So I hope you're impressed.

Anyway, Fred was keenly interested in math when he went to college, but pretty quickly found out that pure mathematics wasn't going to be for him.

anti-constructivist digression

"I always loved finding the right answer," he said.

This is SO important; it's one of the core pleasures of math. Finding the right answer. Radical constructivists gleefully snatch this pleasure this pleasure away, the drips.

back on topic

Anyway, once he realized that pure mathematics was beyond him, Fred moved to statistics. Looking at the Russian Math problem, he instantly knew how to do it. But he didn't know why He knew.

This was yet another Problem Involving Reciprocals, and Fred said, 'I don't know why I knew to use the reciprocal there.'

So......

This is where I get confused.

Fred is a super-smart person with, I would say, high expertise in elementary math & in applied math. On the other hand, he isn't doing a math-related job as a career, so maybe he's no longer in the 'expert' category after all these years. I don't know where to put him.

So I don't know what to think about the fact that he could instantly solve the RUSSIAN MATH problem, but didn't know why his solution worked. Is that a sign that he has advanced knowledge (because people with advanced knowledge often 'just know' things they can't explain), or a sign that he doesn't?

This brings me back to Christopher.

Watching and listening, I felt like the fact that he instantly knew Juanita was right was a sign he's developing expertise. It was as if math is starting to be 'in his bones.'

On the other hand, I don't think he could show me how to do the problem, if the problem were too advanced to do just by eyeballing it. (If the numbers weren't 'friendly.')

Actually, that's a good question. In the next day or two I'll find out what he would do with a more complicated version of this question.


How good are our best?
BestPerformingStudentsPartTwo
a word problem only the top 10% of 9 year olds solve
England vs America vs Singapore

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SingaporeMathPlacementExam 05 Sep 2005 - 13:33 CarolynJohnston


The last two nights, I've been giving Ben the Singapore Math 4A placement exam (all the Singapore Math placement exams can be found here). I had a look at the Singapore Math 3A and 3B tests, and decided that Ben can probably do them fairly easily; but I wasn't so sure at all about Singapore Math 4A.

I've been giving the test to him in little chunks. The first day I did it -- it was several days after school had started, and I hadn't tutored him at all, and he was having an easy time of it since all they were doing was factoring numbers into primes -- he howled as though I were slipping bamboo shoots under his fingernails. That was to be expected. We always get the worst resistance after he's had a break.

At this point, I've gone as far with him in these placement tests as I plan to go -- 4A is definitely the place for him to start. What I'm finding is that in the first part of the placement exam, where the problems are computational, he is doing fine; I've taught him well in that regard (using mostly Saxon math, with some Prentice-Hall). However, after the first ten or so problems, the placement exam starts to test a kid's problem-solving ability. In Ben's case things got ugly quickly. He fell apart emotionally in the face of these problems, of a type he'd never seen before.

The first two problems involved analyzing a figure for parallel and perpendicular lines, and determining the area of a rectangle that had had a couple of rectangular pieces removed. That last is a real-world problem, by my lights, if there ever was one.

These two problems were on the placement exam as well:

A rectangular swimming pool measures 24m by 16m. A concrete path 2m wide is paved around it. What is the area of the path?

Mary bought 1m of ribbon. She used 2/5m to tie a package, and 2/7m to make a bow. How much ribbon had she left?

Ben's reaction to the second one was especially interesting. By the time he got to that problem, he was frazzled by having had to skip a few of the earlier ones. He shouted:

"What do you expect me to do, add 2/5 and 2/7?"

"Yes," I said. "Oh," he said.

Ben's confidence crumbled fast with this placement exam. I tried to assure him that it was just a pretest, and that he should skip problems he can't do; but he's just frail these days. Perhaps all kids are.

I think the Singapore math curriculum may work for us. It's challenging, but we can do it; it's not impossible. And at least the evidence says we're on the right track with it.

And the books are cheap, to boot (check them out here).

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SaxonItWillBe 23 Sep 2005 - 19:17 CarolynJohnston


I had my meeting this morning with B's special ed teacher, his math teacher, and an unexpected guest -- the principal. Perhaps they were a little nervous because of this letter I had sent them, in which I mentioned that I have a math Ph.D. and I'm a Powerful Math Ed Blogger (be afraid: be very afraid).

I asked them if they would have a teacher's aide work with Ben on his math, one-on-one, using the Saxon Math curriculum. The special ed teacher, bless him, said that he could make it work; that he thought he could spare a teacher's aide during that last period of the school day, and it would just be an (easier) matter of finding them a quiet place to work. I was so relieved I could have hugged him.

It's been two years of struggle for me and Ben, supplementing from Saxon and trying to work around the vagaries and inconsistencies of Everyday Math; and here we were, once again, facing another year of it, after having worked so hard last year to find a school that offered a traditional math class, and then fighting the open enrollment system to get him into it, and then committing to the 45-minute-per-morning commute that it entails. I wanted so much for this year to be the end of it. I never really wanted Ben to have to do two math curricula, especially when one of them seemed to be a total waste of time for him.

And then I found on the first day of school that Ben's math class would be using Connected Math after all. I just about despaired. I've had to give up my dream of having Ben mainstreamed in math -- I always thought it was the one class in which he could hope to really hold his own and have a Typical Kid Experience. But I don't care any more -- math education is a mess in this country, and we're perversely fortunate to be able to opt out.

I got some insight into why Ben's new middle school had chosen to go 50-50 with Prentice Hall and Connected Math this year, following many years during which they had a reputation for doing solid traditional math classes (and for having the best math department in the city). It's not ideology; it's fear.

The special ed teacher told me that if I wanted Ben to be taught from a traditional math class, that I would have to just 'ignore the CSAP' (the CSAP is Colorado's assessment test for students, given in compliance with NCLB). "He'll do badly on it," he told me. "The test is very applications-oriented. You can't hold us responsible for that."

"If he does poorly on the CSAP," I told him, "I'll hold myself entirely responsible." No way will he do poorly on the CSAP. He didn't this last year -- except in those sections, data representation and probability, that I chose not to supplement.

Apparently, on the CSAP, kids are frequently asked to give verbal explanations for what they did on a problem. Math CSAP scores for students at Ben's school have been getting worse and worse over the last few years, and the teachers and principal don't know why, and don't know what to do about it. This adoption of Connected Math is therefore, I conclude, their attempt to grasp at straws. There is no way for them to know in advance whether Connected Math is going to solve their problem; I doubt they even know what the cause of the problem is.

An even deeper question is whether the CSAP itself -- or any other state assessment -- is worth a hoot. Who's vetting the CSAP to check whether kids who do well on it in 5th grade have the skills, on average, to go into calculus in college?

I believe in the value of assessment -- it provides a minimal benchmark of proficiency and keeps people accountable. But the assessment has to be good, and we have to know what to do about the weaknesses it reveals. If it leads good schools astray, I call that backfiring in a big way. I've been assuming that the metrics, at least, are good; now I wonder. The more deeply I look at the problem of math education in our country, the more I realize that there are "unknown unknowns" all the way down to its foundations.

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FormativeAssessnent 19 Dec 2005 - 01:30 CatherineJohnson



Doug's comment reminded me that I'd pulled an OECD article on formative assessment to post:

Formative assessment – the frequent assessments of student progress to identify learning needs and shape teaching – has become a prominent issue in education reform. In fact, Studies have shown it to be among the most effective educational interventions ever reported.

Between 2002 and 2004, CERI examined exemplary practice of teaching and formative assessment in secondary schools in eight OECD countries – Australia (Queensland), Canada, Denmark, England, Finland, Italy, New Zealand and Scotland – and brought together literature reviews from English, French and German research traditions, relating all this to the broader current policy environment.

The resulting publication, Formative Assessment: Improving Learning in Secondary Classrooms, combines those elements to clarify the concept of, and approaches to, formative assessment and its relation to teaching strategies. The culmination of this study was a major international conference organised by CERI in Paris, on 2-4 February 2005. The conference highlighted international research and case study evidence from the CERI study.

CERI will co-sponsor a regional conference on formative assessment in Budapest, on 29 – 30 September 2005....

Beginning in 2005, the project has just started to look at assessment strategies for adult learners. The study will highlight the issues of why, what and how institutions should assess adult students, and implications for policy.

I think this may be the web site that assured me 'adult learners' don't remotely learn the way young learners do, a fact I decided not to learn.

Being an adult learner, not learning that I can't learn was easy.

update

ah-hah

yes, indeed, I have done a bang-up job of not learning the bit about adult learners not learning, because the CERI web site, far from being the bearer of bad tidings about adult learners, is in fact the bearer of the Certain-To-Be-Correct observation that one can learn at any age. (pdf file)

In recent years, brain science has captured the interest of policymakers and educators. Many believe that new discoveries about the brain yield new insights into early childhood and adolescent learning. However, most of the brain science policymakers and educators cite is not new and even this “old” brain science tends to be oversimplified and misinterpreted in policy and educational contexts. Contrary to popular understandings about the brain, most learning is not limited to early critical periods in development. Furthermore, there is no simple relation between the number of neural connections in the brain and rate or ease of learning. What we do know, from psychological studies of the mind, is that rate and ease of learning depend critically on what one already knows, not on one’s age. We should attempt to use what we do know about learning across the lifespan to provide optimal learning environments for all our citizens.

Does that sound like domain knowledge to anyone else?

oops

Nope, wrong again.

This is the web site with the bad news about adult learners, a fact I seem to have learned in spite of the many obstacles created by my advanced age.

Here's the Good Word from Manfred Spitzer, Psychiatric Hospital, University of Ulm, Germany (pdf file):

You cannot train 15 year olds and 50-year olds in the same way, as the younger ones will perform better.

I'm going to forget that now.

what does this mean?

Spitzer recently attended a meeting on the retraining of employees where he said he noted that the official dogma of every learning institute for retraining of employees stated emphatically that age does not matter. However, he says you cannot train 15-year olds and 50-year olds in the same way as the younger ones will perform better, and that this causes anxiety in the older subjects. But this is not officially recognised, and so when Spitzer told them about the declining learning rate and what the consequences should be for educational programmes it was evident that they were doing exactly the opposite. He explained his theory of a more cost-benefit effect: if this type of retraining was more focussed on split groups according to age decline, it would ultimately produce a curve effect, and in turn produce a cost benefit effect. He says when you start to think about such issues it becomes evident that there is an endless list of possibilities of things you can do, and this is what he will now be exploring in his new Transfer Center.

I wonder if the author of this passage is too old to learn to express himself clearly?

Surely not.


KUMON & formative assessment

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DanOnFractionPreTest 28 Nov 2005 - 17:27 CatherineJohnson



Last week sometime I was asking people whether this online fraction pre-test was OK.

The website has some glitches, so the question became: would this test be OK if the website worked?

Here's Dan's response, which I'm filing in the Book-style index:

I think the test is too brief. If they ask two fraction addition questions and you get one wrong, was it a careless error or do you fail to grasp the concept? Four questions of each type would be a little more telling.

It should clearly state whether it wants improper fractions or mixed numbers as answers, or if it doesn't matter.

I also think the fraction addition problems are too easy. The denominators are too similar; there's no difficulty in finding a common denominator. Also, you could solve these by simply drawing a square cut into eighths. I want a problem that requires you to convert the fractions to a meaningfully different common denominator.

A problem with a negative answer might also be nice.

You could argue that it belongs in a decimals or percents section instead, but it might be good to ask for 25% to be written as a fraction, or 0.4 written as a fraction in lowest terms.

And I can't resist singing my favorite note: they should include problems that explicitly ask for cancelation of common factors in the numerator and denominator to prove that the student gets it. Then, this should be extended to units, i.e. dimensional analysis! (I couldn't resist)

trust but verify redux

I bring this up because Christopher missed the fraction pre-test his teacher gave last week, and I had a gnawing suspicion he's not remotely where he should be on the subject.

But more than that, especially in the wake of reading Engelmann, I think we parents need our own set of assessment tools. (The link above includes Lone Ranger's advice on using the Iowa Test of Basic Skills at home.)

Ideally, I would like to see every curricula used in the schools publish pre- and post- unit tests parents can administer at home if they choose; I would also like to see the results of any and all such tests the school administers. (The middle school—and, I assume, the other schools as well—administered pre-tests in every subject this fall. I think that's excellent, but I'd like to know how the kids fared. Another question for the TEAM MEETING.)

Given the fact that in my experience schools aren't especially forthcoming on these questions, I want access to such tests myself.

Beyond that, I would like to be able to administer the TIMSS test, or a valid TIMSS equivalent, to my own children. (You can administer a small portion of it.)

And because TIMSS is given only to 4th, 8th, and 12th graders, I need a valid, norm-referenced standardized test to use each year if I so choose.

I don't know what an 'A' or a 'B' means in the larger world, and I certainly don't know what a canned comment like 'Making satisfactory progress' indicates.

We need to introduce some checks and balances into the system—or more than we have now, at any rate.


pattern training redux

I gave the online test to Christopher. It was a disaster.

He answered only 6 out of 10 questions, and 1 of his answers was wrong.

I went on a frenzy of workbook/worksheet acquisition before I realized that I might be looking at pattern training. The online fractions aren't written with fraction bars, and the run-on word 'dividedby' was used instead of a division sign.

So I wrote out the problems in standard form. Christopher did every problem correctly.

Pattern training lives.

I think I have a fairly good sense of where he is with fractions at this point.

His knowledge is higly inflexible, and shaky to boot. He's nowhere near procedural fluency, although he does have the basic procedures down.

He does not know how to add and subtract with borrowing. That information has gone missing.

Ed said, 'Can you talk Mr. Liu into giving you the fraction worksheets out of sequence?'

I'm thinking that one over.

In the meantime, I have 3 very good workbooks, all purchased at Lakeshore Learning:

• Kelley Wingate Publication Pre-Algebra (Illinois Loop likes this series)

• if Pre-Algebra Math Grades 5-8

• Spectrum Math Grades 6, 7, & 8 (separate books)

The Spectrum series is organized by chapters, and includes a pre-test and a chapter test for each one.


another fraction pre-test

Cure Your Math Anxiety: Basic Math Skills-fractions

This site includes 8 lessons plus a fraction pre-test:

Fractions Pretest and Terminology

answers to fractions Pretest and Terminology

---

NewAirReportOnEducationalResearch 19 Dec 2005 - 01:30 CatherineJohnson



Ken just pointed me to a new AIR report on School Reform Models (pdf file) that I think should be a boost for DI (as for Success for All):

A new guide using strict scientific criteria to evaluate the quality and effectiveness of 22 widely adopted comprehensive elementary school reform models rates 15 as “limited” to “moderately strong” in demonstrating positive effects on student achievement.

Direct Instruction and Success for All are the only two 'reform models' withi moderately strong evidence of effectiveness.


from the full report:

Goals/Rationale

The Full Immersion Model of Direct Instruction has two foundational principles: all students are capable of learning if taught using proper techniques, and all teachers can be effective if provided with researchbased strategies and materials. Thus, the model seeks to accelerate learning for all students and provide teachers with appropriate strategies by targeting factors that are within a school’s control. These factors include assessment, instruction, grouping, scheduling, professional development, and resource allocation.

Notably, the model does not rely on parental involvement or technology; NIFDI believes that school leaders often cannot control these factors or use them efficiently.

The main component of the Full Immersion Model of Direct Instruction is Engelmann’s curricular program. Engelmann asserts that an implementation plan, such as DI, seeking to accelerate student achievement should include

• A scientifically research-based instructional program;

• Homogeneous and flexible grouping;

• Appropriate student placement within the instructional sequence;

• Daily practice and application of skills and strategies;

• Scheduling that allows for cross-classroom grouping and provides sufficient daily instructional time;

• Instructional activities that motivate, engage, and interest students; and

• Ongoing data collection for instructional decision making.

Ken's taken the time to type up some passages from Engelmann's War Against the Schools' Academic Child Abuse, and I'll be copying more myself.

With Engelmann, the game is: did the students learn what you taught? Period. Constant formative assessment, the purpose of which is not to grade and categorize kids, but to find out what they've learned.

When they aren't learning, you don't ship them off to the school psychologist to assess learning failure.

You revise your curriculum and/or teaching methods.

I'm thinking......schools should have some writers on staff. (They've got everyone else! Why not writers!)

When you write for a living, you never, ever, get to send the folks who don't like and don't get your books to the school psychologist.

You get to go back to your computer and revise.


here's more:

Curriculum and Instruction

The DI approach is based on the belief that learning is affected by the sequential development of skills, instructional approaches, amount of skill practice and application, ongoing feedback given to students, and continuous monitoring of student progress. Four basic principles guide the DI curriculum and instruction:

• The programs should develop specific skills through continuous practice and later combine these skills to form higher order thinking skills.

• Lessons should emphasize reviewing and practicing already learned skills and integrate new skills as they are mastered.

• Scripted and predictable lessons ensure daily assessment of student progress.

• Field-tested instructional practices should be revised, adjusted, or maintained based on student progress and responses.

The central element of The Full Immersion Model of DI is the scripted curricular program. The curriculum materials include highly interactive yet fast-paced lessons. Each lesson builds on the previous lessons; therefore, the lessons gradually introduce new skills. The lessons require teachers to adopt specific instructional strategies such as directing choral responses and signaling. NIFDI suggests that schools phase in the implementation of the model typically by implementing the reading and language curricula during the 1st year, spelling and math during the 2nd year, and handwriting during the 3rd year.

Prior to implementing the model, NIFDI mandates that schools discontinue their use of other instructional programs that may compete for time and resources unless the programs are approved by NIFDI. For example, schools should not continue to implement programs that take students out of the classroom during instructional blocks.

Schools purchase the DI instructional materials from SRA (http://www.sraonline.com). Reading and language materials include Reading Mastery I–VI; Horizons A–D; Corrective Reading (Decoding and Comprehension); Language for Learning; Language for Thinking, Reasoning and Writing; and Expressive Writing. DI math and spelling materials include Distar Arithmetic 1, Connecting Math Concepts Levels A–F, Corrective Mathematics, and Spelling Mastery. These materials include scripted lesson plans, workbooks, and materials for assessing student performance. NIFDI provides teacher training for implementing these materials in the classroom.


I wonder if I can get hold of Expressive Writing.


I love this part:

The implementation manager places all students, including most students with special needs, in instructional groups; for this reason, the model does not generally accommodate pull-out programs.

DI doesn't allow a school to do pull-outs!

Class time in America is utterly fractured by constant, chronic, unceasing interruptions. Therapists coming in to pull-out kids; enrichment teachers 'pushing-in' to enrich kids; band teachers collecting kids for band; it goes on and on and on. And then there's the PA system.

A friend of mine has been helping out at the K-3 school. She said it feels like you don't get more than a few minutes' uninterrupted classtime there.


just keeps getting better

Technology

NIFDI feels that technology is peripheral to the mission of accelerating student achievement.

I'll say.


formative assessment

Monitoring Student Progress and Performance

During the initial stage of implementation, students take a placement test that determines their instructional grouping. Throughout implementation, teachers monitor student progress and grouping using daily assessment of student performance on lessons. Teachers are taught techniques to analyze and interpret data from these assessments; the techniques help with reflecting on their instructional practices, evaluating students’ responses to instruction, and identifying students who do not demonstrate mastery.

Each week teachers provide the school principal with a summary report of student performance, which notes any students that do not make adequate progress during that week. In return, the principal submits a summary report of student performance to NIFDI. During weekly conference calls, the school management team (principal, building coordinator, and peer coaches) discusses the progress of instructional groups and individual students with the implementation manager and project director.

If individual students do not make adequate progress for 3 consecutive weeks, the management team establishes a plan for remediation. The principal or building coordinator continues to monitor the students’ performance on a weekly basis. The model requires the management team to ensure that teachers receive feedback, coaching, and appropriate instructional materials to meet the needs of students requiring remediation.

If individual students do not make adequate progress for 3 consecutive weeks, the management team establishes a plan for remediation.

This is what I like about DI.

If a student isn't learning, they don't let years go by before anyone notices something's amiss.

Three weeks of no learning, and you're On The List.

Back when Christopher failed two unit tests in a row, amounting to a full 1/3 of his year's work in 4th grade math, I heard nothing from the school. I was working under intense deadline pressure, and I came close to missing what had happened.

Today he routinely says, "I didn't learn anything in 4th grade (math)," and I'm inclined to agree. But at the time, I had no idea. And the school didn't jump into a principal-managed remediation plan.

I did know that his dad was reteaching every concept at night, but I didn't know that wasn't good enough.

Christopher's partner in 4th grade math failure is still behind today. He's never closed the gap.


Executive Summary (pdf file)

---

IepsForEveryChild 19 May 2006 - 21:47 CatherineJohnson



Rereading Parent Pundit's post about her daughter's experience with Everyday Math and ALEKS, this passage caught my eye:

...they give a pretest and a posttest for the curriculum. In other words, they give the final at the beginning of the year and at the end of the year to track the learning. My daughter received a 25 at the beginning of her 5th grade year in math, but she only received a 69 at the end of the year....

Clearly, intervention was needed. In the summer at the end of 5th grade, I had her try the Aleks computer program in math, www.aleks.com. The Charter School in my town uses it, and I decided to try it for my own daughter. A tutor would have been expensive and less than optimal in this situation because my daughter does get concepts, she just needs more drill (how can most kids hone their number sense if they aren’t ever asked to multiply and divide numbers continuously), and she needs algorithms that have fewer steps so there is less possibility of error (everything that Everyday Math does not provide.)

I give Parent Pundit's school—and the authors of Everyday Math—credit for the pre- and post-testing.

My problem is: what comes next?

They give this child a pre-test and she scores 29; they give her a post-test and she scores 69.

And then......nothing.

"Clearly intervention was needed."

I'll say.

Why is intervention the parent's responsbility?

The school has failed to teach this child 5th grade math. When she takes the ALEKS test, the program tells her she knows only 21% of a typical 5th grade curriculum. (I'm wondering whether ALEKS allows people just to take the grade-level tests, and if so, how much they charge. I'll check.)

If this child were classified as having special needs, she would be entitled to be taught the content that is listed on her 'IEP,' which stands for Individualized Education Program.

Of course, in my experience the content on the IEPS doesn't get taught, either, but still.....it's there; the parent has a leg to stand on. (And in my own children's case, in fact it's extremely difficult to know what they are and are not able to learn, though I suspect Engelmann would make short work of some of the IEP meetings we've had.)

But with a typical child with normal intelligence, there's no mystery. She can learn 5th grade math in 5th grade. It's the school's job to teach it to her—and to reteach it if they failed the first time around. If that means providing tutoring or summer classes, so be it. It's the school's failure; the school needs to fix it.

This mother was in the same position I was in at the end of 4th grade. My child was failing; the problem was the school's, not his or mine. (In his case the problem was almost certainly the teacher, who I liked very much, but who apparently just could not teach math at that early stage of her career. The school didn't give her tenure, which was the right move. But children who lost a year of math in 4th grade weren't given any help or remediation. No one came to parents of these children and said: Your child failed to learn math this year, because his teacher was inexperienced and didn't manage to teach the subject to mastery. Here's what we're going to do to re-teach the material he missed.

American schools, by and large, teach for coverage.

Not for mastery.


free assessment at ALEKS?

It looks like ALEKS offers a free assessment. (I haven't tried to use it, because I'm not sure I can run the test twice on one computer, and I'm most interested to see where Christopher scores.)

If this assessment really is free, and is easy to use, it could be a useful tool in talking to teachers and administrators.

What we really need is our own simple-to-administer, at-home assessment, 'rolling' assessment tools.

I'd like to be able to send my school a report each month on where Christopher is in the curriculum.

Of course, that's another project.

report cards for the school

---

SmartestTractorsAssessmentForm 19 May 2006 - 21:54 CatherineJohnson







"Attached is a page from our Guide to the Provincial Report Card. It is not required we use it in our classrooms, but I find it helpful in focusing some students. At worst, it is an alternative to the page you have been handed."


thank you





my contract to improve Christopher's grades
a Grade Contract that makes sense
the book
Grade Contract for married people
climb down
Smartest Tractor saves the day
KIPP Academy contract

---

IfTheStudentHasntLearned 23 Dec 2005 - 22:16 CatherineJohnson









revision

From Catherine:

Our new pretend-shirt specifically says "If the student hasn't learned, the school hasn't taught," not 'the teacher hasn't taught'.

No more thoughtless (and unintended) teacher-bashing.

Seriously. I'm the last person to want to make teachers feel blamed and bashed, seeing as how half my relatives have been or are currently teachers. I'm sure I'll be one again at some point, too.

The problem is that, when you talk about schools, it's the teachers who are visible. They're in the trenches, so they get the blame. (I realize I'm not telling teachers anything they don't know.) I know better than that, but I've been sounding like I don't.

Time for a course correction.

From Carolyn:

Hey, my entire family on my mother's side were also teachers, every man and woman Jack of them. I've been a teacher too; so has Catherine.

My observation is that policy flows downhill in a school, and the buck stops with the teachers. They get the responsibility, but not the authority; policy changes really have to start with upper management.

We're here to put the pressure on upper management, and support the teachers in doing what they know how to do.

---

FormativeAssessmentSummary 19 May 2006 - 22:01 CatherineJohnson



the OECD weighs in

The educational gains associated with formative assessment have been described as “among the largest ever reported for educational interventions.”
source:
Organisation for Economic Co-operation and Development

summary of Black & Wiliam

(full passage quoted below)

• formative assessment: all activities schools, teachers, and students undertake to collect information that can be used diagnostically to alter curriculum, teaching, and learning

• information gleaned from formative assessment allows teachers to make necessary instructional adjustments: reteaching, trying alternative instructional approaches, or offering students more opportunities for practice. Formative assessment allows schools to make necessary curricular adjustments.

• Black and Wiliam literature review of 250 journal articles and book chapters: formative assessment produces significant learning gains, with effect sizes ranging between .4 and .7

• students need specific comments about errors and specific suggestions for improvement; formative assessment is designed to provide this information

• formative assessment allows teachers and students to identify gaps in students' skills and understanding and guides them through the process of remediating those gaps

• formative assessment instills confidence in teachers, parents, and students that all students can 'learn to high levels'

• formative assessment in the form of self-assessment and self-monitoring improves student learning when students understand the assessment criteria

• specific feedback from formative assessment "emphasizes that students can improve as a result of effort rather than be doomed to low achievement due to some presumed lack of innate ability"

• Black and Wiliam: low-achieving students, including students diagnosed with LD, improve most
  

source:
The Concept of Formative Assessment by Carol Boston

Purpose and Benefits of Formative Assessment

Black and Wiliam (1998b) define assessment broadly to include all activities that teachers and students undertake to get information that can be used diagnostically to alter teaching and learning. Under this definition, assessment encompasses teacher observation, classroom discussion, and analysis of student work, including homework and tests. Assessments become formative when the information is used to adapt teaching and learning to meet student needs.

When teachers know how students are progressing and where they are having trouble, they can use this information to make necessary instructional adjustments, such as reteaching, trying alternative instructional approaches, or offering more opportunities for practice. These activities can lead to improved student success.

Black and Wiliam (1998a) conducted an extensive research review of 250 journal articles and book chapters winnowed from a much larger pool to determine whether formative assessment raises academic standards in the classroom. They concluded that efforts to strengthen formative assessment produce significant learning gains as measured by comparing the average improvements in the test scores of the students involved in the innovation with the range of scores found for typical groups of students on the same tests. Effect sizes ranged between .4 and .7, with formative assessment apparently helping low-achieving students, including students with learning disabilities, even more than it helped other students (Black and Wiliam, 1998b).

Feedback given as part of formative assessment helps learners become aware of any gaps that exist between their desired goal and their current knowledge, understanding, or skill and guides them through actions necessary to obtain the goal (Ramaprasad, 1983; Sadler, 1989). The most helpful type of feedback on tests and homework provides specific comments about errors and specific suggestions for improvement and encourages students to focus their attention thoughtfully on the task rather than on simply getting the right answer (Bangert-Drowns, Kulick, & Morgan, 1991; Elawar & Corno, 1985). This type of feedback may be particularly helpful to lower achieving students because it emphasizes that students can improve as a result of effort rather than be doomed to low achievement due to some presumed lack of innate ability. Formative assessment helps support the expectation that all children can learn to high levels and counteracts the cycle in which students attribute poor performance to lack of ability and therefore become discouraged and unwilling to invest in further learning (Ames, 1992; Vispoel & Austin, 1995).

While feedback generally originates from a teacher, learners can also play an important role in formative assessment through self-evaluation. Two experimental research studies have shown that students who understand the learning objectives and assessment criteria and have opportunities to reflect on their work show greater improvement than those who do not (Fontana & Fernandes, 1994; Frederikson & White, 1997). Students with learning disabilities who are taught to use self-monitoring strategies related to their understanding of reading and writing tasks also show performance gains (McCurdy & Shapiro, 1992; Sawyer, Graham, & Harris, 1992).

key worsd: gapology
James Milgram on long division & time
can you cram math: learning a year of math in 2 months
overlearning
remediating Los Angeles algebra students
Inflexible Knowledge: The First Step to Expertise by Daniel Willingham
Matt Goff & Susan S on remediating gaps
Anne Dwyer on diagnosing gaps & request for 'gap' stories
failing algebra in Los Angeles
formative assessment
formative assessment in a nutshell

---

NewYorkStateMathTestGrade6 19 May 2006 - 21:52 CatherineJohnson



New York state Sample Test Mathematics Grade 6 Book 1 (pdf file)

New York state Sample Test Mathematics Grade 6 Book 2 (pdf file)


I haven't looked through them yet, but I trust your opinion more than mine.




boy

Already, on page 6, I'm having doubts about how well Christopher will do.

His super-duper, accelerated Phase 4 Math Class has ZERO word problems.

I'll see if he can do this problem tonight, but I'd put money on it that he can't. And he's just finished the chapter on fractions. We're going to have to get back to the bar models big-time.

Obviously I'm going to have to print out these tests, and start seeing to it he can do the problems.

This is just great.

Now I'm going to be teaching to the test.





This is the kind of problem bar models were invented to solve.



update: Christopher can do this problem

He did it in no time flat. I was shocked. Ed said he could do it, and he was right. Ed remembers the two of us working on these problems in Saxon Math.

In fact, he remembers us working on these problems a lot.

I must have been in a trance at the time (a math trance!) because I have no recollection of teaching Christopher how to do such problems.

Have I mentioned that cortisol is bad for your memory?

Well, it is.

Cortisol is a stress hormone, and I've been pumping out a lot of stress hormones ever since I discovered that:

a) Christopher was flunking 4th grade math

b) U.S. students are 1 to 2 years behind their peers in high-achieving countries

c) the only children in Irvington who are on grade level with their peers in high-achieving countries are the so-called gifted children in Phase 4 Math

d) Irvington was adopting TRAILBLAZERS



update: what Singapore children can do at the end of 6th grade

Here's the placement test (not a pdf file) for New Elementary Mathematics 1, which is the 7th grade book in the 'Singapore Math' series. [note: If these links are bad, go to singaporemath.com and search for placement tests and New Elementary Math.]

Here's a fun question:




I always loved this kind of thing.

And—I can still solve one. (At least, I can still solve one if, while copying the problem onto a nice, crisp, clean, brand-new piece of scratch paper, I write '1/12' as '1/12,' not '1/2.')

That's good news, especially seeing as how I have never in my life attempted to solve—or been taught to solve—a problem like this one:

a) A hole with a diameter of 3.5 cm is drilled through a square metal nut of thickness 4 cm and length 6 cm. What is the mass of this nut if the density of the metal is 6 g/cm3? (Take pi = 22/7)

b) What is the surface area?



word problems Singapore children can do at the end of 7th grade

3. The HCF (highest common factor) and LCM (lowest common multiple) of 2 numbers are 8 and 408 respectively. If one of the numbers is 24, find the other number.

4. 6 men, working together, can finish a job in 2 h 20 min. If 3 men leave after one hour, how long will it take the remaining men to complete the job?

5. John spent $4 less than 60% of his money on a book and $3 more than 75% of his remaining money on another book. He still has $2 left. What percentage of his original money did he spend?

8. How many liters of 60% acid solution must be mixed with a 75% acid solution to get 20 liters of a 72% solution?

9. A man bought 450 books for $1,350. He sold half of them at a profit of 20%, 150 of them at a profit of 10%, and the rest at a loss of 4%. What was his gain percent, to the nearest percent?

13. A man has just enough money to buy 60 apples or 40 oranges. If he wants to buy an equal number of apples and oranges, how many of each type can he buy with the money?

16. Water flows at 4.5 m per second through a pipe. The water is collected in an empty cylindrical tank of an internal diameter 10 times the internal diameter of the pipe. Find the height of the water after 2 minutes.



word problems some New York state children can do at the end of 7th grade

26. On Friday and Saturday, there were a total of 200 cars in the parking lot of a movie theater. On Friday, 120 cars were in the parking lot.

Part A

What percent of the total number of cars were in the parking lot on Friday?

Show your work.


Part B

What percent of the total number of cars were in the parking lot on Saturday?

Show your work.



28. Mr. Roberts asked his students to solve the three equations below.

784 ÷ 2 =       125 x 6 =       14 x 28 = 

Which equations have the same solution?

Show your work.



31. Simplify the expression below.

6 x 4 ÷ 2 + 3³

Show your work.







NY State Grade 6 multiple choice questions

forget I asked

I obviously didn't need a professional opinion on the level of math achievement being tested here.

I wonder how many New York state kids score 3s and 4s? I'll see if I can track that information down quickly.

I'm going to give Christopher both of these tests, and see where we are now.

---

PretendAlgebraInMaryland 23 Dec 2005 - 02:38 CatherineJohnson


from Jerome Dancis's website:

Nice Problem A tube of tooth paste costs 90 cents to make, and sells for $2.50. The company has "fixed costs" (machinery or rent or whatever). of $3000. How many tubes of toothpaste does the company need to sell to cover/balance-out the fixed costs?

The profit on the sale of each tube is $2.50 - 0.90 = $1.60. Hence, the company will need to sell 3000/1.60 = 1875 tubes. (O.K. to use a calculator for the division only.)

This Nice Problem was not on the sample MD Algebra test; — well not until all the conceptual understandings, and problem solving had been removed and after it had been rewritten in a long-winded and pretentious manner. ( I suggest that you read the first paragraph of the problem, then jump to the Pedagogical Analysis, below.):

Problem #32

A Pedagogical Analysis of Problem #32

A crucial part of problem solving is "setting-up" the equations for a "word problem". Also know as "modeling and interpreting real-world situations". This problem does not test this skill because the equations are provided. In sharp contrast, read the mis-claimed stated-expectation for this problem, on the state's website.:

Expectation 1.2: "The student will model and interpret real-world situations, using the language of mathematics and appropriate technology."

(Click, on view Core Learning Goal, Expectation and Indicator this item tested)

In fact, I counted only one of the 49 problems on the sample MD Algebra test, which actually required the student to set up the equations.

Solving simple equations both by hand and with a graphing calculator, is an important part of real Algebra. Here the equation 2.5x = 0.9x + 3000 needs to be solved. But the students do not need to do the simple calculations; they are encouraged to use their graphing calculators (which provide graphs of the functions). In fact, I counted only two problems on the sample MD Algebra test, which required students to solve equations, none, which required students to solve equations without a graphing calculators.

Here, the thinking part was reduced to choosing the correct "window" to view on the graphing calculator. Even that was deemed too hard as suggested "window" ranges are supplied.

Economists use q for quantity and c for cost. Never the cryptic x for quantity and y for cost as in this problem. A needed skill, in setting up a problem, is to choose names of variables that assist in understanding the problem and the equations. But then graphing c = 0.9q + 3000 on a graphing calculator requires some conceptual understanding unlike y = 0.9x + 3000 which does not.




Another solution, which received the highest possible score when graded....Here the student typed the two given equations into the calculator and had the calculator list their table of values. The student then "scolled through the table until [the numbers for both Y's] were the same." Precious little [Grade 6] conceptual understanding and problem solving involved.

This avoidance of conceptual understandings, and problem solving is in sharp contrast to the Maryland State Dept. of Education statement:

"In all mathematics content standards, the emphasis is on achieving a balance among memorization of facts, proficiency with paper and pencil skills, appropriate use of technology, conceptual understandings, and problem solving" (Underline added). On the web at here.

A big No-No in real Algebra is never using the same variable to mean two different things in the same problem. This problem violates this rule, having y representing both "income" and "cost". This type of ambiguity often confuses students. This suggests a problem writer, with little understanding of the very basic algebraic concept of "variables" (the x's and y's) in algebra. Of course, problem writers, who actually understand Algebra would require more pay for each problem. This would reduce the profits of the profit-making, test-writing company.

The following was added to the webpage for this problem between June 2001 and March, 2002 (I informed them of this and all the errors listed above on Oct. 30, 2001):

"The variable y is used to represent both the income for selling x tubes of toothpaste and the production cost for x tubes of toothpaste. This is an error in the use of a variable."

The first version of this problem was easy.

The second version was utterly mystifying, not least because X and Y were used to represent different values.

Worse yet, I have no clue what goes on with these graphing calculator thingies.

I'm going to have to take a whole course just on calculators.

I guess I can do that while I'm teaching to the test.

---

NCLBAndGiftedProgramming 03 Jan 2006 - 03:24 CarolynJohnston


My friend Jen sent me a link to an op-ed at the Washington Post on NCLB. The author is concerned that NCLB will cause schools to have to struggle so hard to pull up their low achievers to proficiency that the educational needs of gifted kids for accelerated classes and special programming will be neglected. From the article:

Perhaps these schools, along with the drafters of NCLB, labor under the misconception that gifted students will fare well academically regardless of whether their special learning needs are met. Ironically, included in the huge body of evidence disproving this notion are my state's standardized test scores -- the very test scores at the heart of the No Child Left Behind Act. Reflecting the schools' inattention to high performers, they show that students achieving "advanced" math scores early in elementary school all too frequently regress to merely "proficient" scores by the end. In recent years the percentage of California students scoring in the "advanced" math range has declined by as much as half between second and fifth grade.

I don't know how to interpret that last statistic, actually -- "In recent years" and "as much as half" aren't specific enough. Here's what I want to know: In the last 4 years, after holding steady at 10% for many years before NCLB, has the percentage of advanced scorers fallen from 10% to 5%? Tell me something like that, and I'll start to worry about the gifted issue in particular. As it is, though, I'm going to subsume this worry into the pile of other worries I already had about NCLB.

NCLB is in many ways, I think, good legislation (for an unfunded federal mandate). I approve of the notion of assessing kids as they move through school, and holding schools accountable; and I like the way NCLB is set up to keep jurisdiction local.

My concern with NCLB is that 100% proficiency goal. I don't think 100% proficiency is attainable, so in the next 8 years until 2014, I fear that we'll see schools falling off the cliff at an accelerating rate. By that I mean that at some point, all schools will be failing to make adequate yearly progress ('AYP' in the edubuzz). How will we deal with this -- by dumbing down the tests until everyone can pass them, or by backing off of that impossible 100% goal? (My guess is that a percentage somewhere in the 90s is actually attainable with earnest work, and would represent a significant improvement in the public schools).

Anything less than 100% may not be politically feasible (think of the slogan: "Only a few kids left behind"). So getting an actual usable policy out of this may be an impossible dream. I fear that a lot of teachers and administrators are going to get burnt out in the next few years, fighting a battle they know from the beginning is unwinnable. And I am afraid the failure is going to set us back in the fight to improve standards in public schools -- an unintended consequence of demanding an unrealistic goal.

As long as I'm airing my darkest fears about NCLB, here's another one: that not only gifted programming, but other 'non-core competency' classes (such as art, music, etc.) are going to get short shrift as more and more money goes into struggling vainly to reach the 100% goal. These classes may not be as 'core' as reading and math -- but it's activities like this that keep many kids in school, and it would be very sad to leave them behind.

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ValueAddedTesting 09 Jan 2006 - 17:33 CarolynJohnston


The link my friend Jen sent me on NCLB and gifted programming also led to this link on value-added testing. Value-added testing tests with an eye toward ensuring that all students are making the progress they need to be making in their education, focusing not only on the low-performers but also on average and high-performing students.

As Catherine would say if she were here, "Read and Discuss!"

The value-added methodology, by contrast, doesn't create such incentives to focus on a handful of students. Under the system, every child's improvement counts the same towards the school's overall rating. And the methodology itself is widely seen by those who use it as fairer and more accurate. Value added should thus make it easier for teachers to accept the idea of higher pay for outstanding performance and for working in the toughest schools—changes many see as important next steps in reforming education. Indeed, Dallas is already doing this. Teachers at schools with high value-added scores get financial bonuses.

Many of the obstacles to the widespread use of value-added ratings have been overcome in recent years. Thanks in part to the passage of No Child Left Behind, schools all over the country are on their way to testing every year from 3rd to 8th grade—a prerequisite for the value-added methodology. States are also beefing up their computer and statistical resources. Researchers are still working to address some of the toughest technical issues raised by the value-added method, such as how to measure students who move from school to school and how to compare scores on a subject year-to-year when the curriculum changes. But enough progress has been made that more and more states are looking at the value-added idea.

afterthought

I can imagine that if you had a testing mechanism in place that really did measure the improvement each individual kid was making, and reward for it, you'd have a lot more teachers who'd be willing to take on the toughest schools, because the failingest kids have the biggest potential for improvement on the margin.

So this idea -- provided it really measures what it proposes to measure -- is growing on me.

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NoGradeInflationInTheSuburbs 16 Sep 2006 - 21:07 CatherineJohnson



I say we get rid of middle schools altogether.

Ed just called.

On the train he had a chat with a distinguished academic, a Brit.

Her daughter is in middle school, and is doing badly. As the mom put it, 'my very bright daughter who is getting bad grades.'

The mom just wrote a paper, start to finish, for her daughter.

The grade?

C-

Ed said, "Very few Brits who've become distinguished professors can't write."

update: Ed now says it was a C+, not a C-. He also talked to the professor again, and learned that the only reason she'd written the paper was that her daughter was completely overwhelmed with work that night. There was no way she could finish everything, so the mother wrote the paper and the daughter did everything else.



Ed gets a B-

So Christopher just handed in his first paper to his new English teacher.

Ed worked closely with him on it.

He didn't write it. He read Christopher's rough draft and made comments, as a teacher would do, and as this teacher does.^*

Then Christopher revised.

Ed checked grammar, punctuation, paragraph structure, and topic sentences.

The paper came back yesterday with a grade of 80.

I better try my hand on the next one. See if we can get that baby up to 83 or 84.

[update: ok, bad idea ]



my Secret Plan

This reminds me of my Secret Plan.

Back when Christopher got his two Ds from she-who-shall-be-nameless and was asked, in front of the class, 'Are you trying to do the work at all?' I mentioned that Christopher would not be writing any more papers for thi