## Saturday, October 27, 2007

### notes on Bloom & teaching to mastery

Mastery Learning

Notes from Benjamin Bloom lecture [ACSA, April, 1987]:

With traditional instruction, the correlation of pupil performance from grade-to-grade is 80%+. The variation within each grade is greater each year. The range is double the second grade in the fourth grade, triple in grade 6. Rank order is already fixed by third grade for the next 8 years for 90% of kids. Similarly, self-concept drops grade-by-grade for the bottom 20% while it rises year-by-year for the top 20%. This is true of most countries, not just the US.

[snip]

The mid-point for conventional instruction is the 50th percentile. For individual tutoring, it is the 98th percentile. For whole class mastery learning instruction, it is the 84th percentile.

[snip]

CORE IDEA OF MASTERY LEARNING: aptitude is the length of time it takes a person to learn not how "bright" a person is, i.e., everyone can learn given the right circumstances.

How to instruct for mastery:

1. Major objectives representing the purposes of the course or unit define mastery of the subject.

2. The substance is divided into relatively small learning units, each with their own objectives and assessment.

3. Learning materials and instructional strategies are identified; teaching, modeling, practice, formative evaluation, reteaching, and reinforcement, etc., and summative evaluation

are included.

4. Each unit is preceded by brief diagnostic tests.

5. The results of diagnostic tests are used to provide supplementary instruction to help student(s) overcome problems.

Time to learn must be adjusted to fit aptitude. NO STUDENT IS TO PROCEED TO NEW MATERIAL UNTIL BASIC PREREQUISITE MATERIAL IS MASTERED.

[There is a difference between "80% of students will master the material" and "each student will master at least 80% of the material" before proceeding.]

This explains a lot.

One of the mysteries here & elsewhere is: why are administrators so unconcerned about the amount of tutoring going on in affluent communities?

I've mentioned before the tutor here who estimates that 50% of Scarsdale kids are being tutored. That tells me that Scarsdale schools are overrated.

But that's not what it says to educators.

The 98% figure may explain this. Middle-aged educators are old enough to have studied Bloom in ed school (teachers younger than 35 may never have heard of him - not sure....).

If they're thinking "98%" when they hear tutoring, no wonder they're dismissive.

In fact, I'm 98% positive (that's a joke!) Bloom was talking about tutorials, not tutoring. He was talking about what homeschoolers are doing, which is direct, one-on-one instruction in the whole of a course, start to finish.

Tutors are trying to put out fires, get kids ready for the Big Test, keep the kid from giving up, etc. Hiring a tutor to try to teach a student who is struggling in a class at school is by no means ideal.

In fact, the one family I know whose kids are doing brilliantly well with tutors simply takes it as a given that all schools, public and private, are fatally flawed. They have a "regular" tutor on the tab who works with their kids almost year round.

Preemptive tutoring.

extra credit

Was mastery teaching ever widely adopted in public schools?

## Thursday, October 25, 2007

### 2nd Grade History - Core Knowledge

Me: "So, tell me about the War of 1812."

A.: "Well, it was ended by the Treaty of Ghent ... in Belgium ... in Europe."

Among other facts elicited:

- The Battle of New Orleans happened after the treaty was signed; the commander was future president Andrew Jackson.
- James Madison was President; his wife was Dolly Madison.
- Washington was burned.
- Francis Scott Key was captured by the British.
- The war began because the British were impressing American sailors.

Observations:

- The material is actually being taught by the school, not parents.
- The material is appropriate to the abilities of the students.
- The school is covering core knowledge (and Core Knowledge) of the subject well.
- The framework being taught now should be useful to hang future information on.

### We'll see

### Glencoe critical thinking problem sequence

CRITICAL THINKING For Exercises 41-43, suppose x is an integer.

41. Write an expression for the next integer greater than x.

42. Show that the sum of two consecutive integers, x and the next integer after x, is always odd. (Hint: A number is considered even if it is divisible by 2.)

43. What is the least number of consecutive integers that must be added together to always arrive at an even integer?

Glencoe Algebra, New York edition, page 442

C. is finding algebra manageable at the moment, though I suppose I should wait to see what the grade on his latest test was before I say that.

Nevertheless, when I was helping him study, I could see that he was on track.

He couldn't do this Critical Thinking sequence, but he understood what I had done -- and could explain it to me -- as soon as I did it.

Which I think is probably fine for now.

### Linda Seebach on the whale problem

A whale swims 40 miles in 1 1/4 hour. How far does he swim in 1 hour?

The math mom I mentioned solved this problem mentally in a couple of seconds, then explained that "1 1/4" is 5 parts.

Linda writes:

What is apparent to someone with good number sense is that 1 and 1/4 is 5/4, that is five of something called "one fourth." There's no work to show, though you could explain that if five somethings equal 50, then one something is 10 and so four somethings are 40.

Since the numbers are easy, you can do it in your head; no need for visual models or bar models or algorithms. But then, I was an algebra person, not a geometry person.

It works just as well when the whale swims 37 miles in 16/13 hours except you probably can't compute (13)(37/16) without writing it down or asking the calculator. Knowing what you need to compute, however, is exactly the same.

I have to say....it's pretty sad that I still can't put it this way (or, rather, don't think to put it this way).

Of course Linda is right; 1 1/4 is 5/4. That's what a person with a good number sense "sees" or, more accurately, grasps.

Remembering back to when I first started reteaching myself K-12 math, seeing that 1 1/4 is 5 one-fourths was an obstacle. I had good fraction knowledge for an American, I think; I could create a correct word problem for 1 3/4 ÷ 1/2, the famous challenge given to 22 math elementary school math teachers in Liping Ma's book. I had no math phobias, SAT math scores were good, took statistics in college.... in short, I am a math literate person, certainly for the U.S., and was before I started studying K-12 math.

An aside: hmmm... It's interesting that I keep using the word "see." I'm going to assume that means I personally do need to "see" this as opposed to "grasp" it.

Interestingly, as I've moved into algebra 2, I don't feel this way about many of the topics I've encountered there (practically all of which are new to me). Will have to give this some thought. At some point, the abstractions of math came to seem natural or "real" to me. At least, I think they did.

Back on topic: In spite of the fact that my own understanding of fractions was perfectly appropriate to my needs, I had trouble with the idea that 1 1/4 is five one-fourths. The "one-fourth" is the unit; the 5 is multiplier and 1/4 is the multiplicand. I remember my neighbor explaining it to me one night in the context of another question. (Wish I could remember it now. I may be able to find it. She was trying to explain something about fractions, I think, by pointing out what Linda has just pointed out --- and I stumbled over the explanation.)

I think bar models are a way of teaching the kind of number sense Linda is describing. Which is why C. is going to carry on doing the bar models in 3rd grade Primary Mathematics.

Sybilla Beckmann's textbook for students in education school teachings bar models, as does Parker and Baldridge's text. (I've found Parker and Baldridge much easier to study and understand than Beckmann's book, but that may not be a fair comparison since I began at the beginning of P&B, and only dipped into Beckmann.)

procedural "versus" conceptual knowledge

Given my experience of (relearning) fractions, I now feel strongly that schools must teach the addition, subtraction, multiplication, and division of fractions to mastery in all kids, bar none.

I think, too, that schools should give kids word problems to solve using all four operations -- and a good number of these word problems should be "real world" word problems, by which I mean the actual real world, as opposed to the constructivist real world, which seems to consist mostly of roller coasters and hay balers (see below).

Non-GATE kids should be given fraction word problems about things like measuring a room for tile or cutting fabric from a bolt to sew a dress (I don't care if the kid actually sews or not. Sewing is something he/she might do one day, especially if he/she has kids and sends them to a public school where the curriculum is project-based.)

This approach, which I think is what I probably had, may not give non math-brain kids a good conceptual understanding of fractions. It didn't give me a good conceptual understanding.

But it did give me all the foundation I needed to use fractions in adult life, and to pick up the study of math later on, when I wanted to.

I now believe that non-GATE kids (not sure about the GATE kids) should be taught bar models, too. No matter what curriculum or approach a teacher/school is using, consistent teaching of bar models should be included.

Cassy T has mentioned that 3rd grade is the big year for Singapore Math; that's the year bar models are introduced. (pls correct me if I'm wrong)

I would ideally have U.S. kids work through either the 3rd grade Challenging Word Problems book, or work all the bar model problems in Primary Mathematics 3.

the hay baler problem

A post from Barry a couple of years back:

Here's a problem that appears in IMP for 9th grade It is known as the "Haybaler Problem"“You have five bales of hay. For some reason, instead of being weighed individually, they were weighed in all possible combinations of two: bales 1 and 2, bales 1 and 3, bales 1 and 4, bales 1 and 5, bales 2 and 3, bales 2 and 4 and so on. The weights of each of these combinations were written down and arranged in numerical order, without keeping track of which weight matched which pair of bales. The weights in kilograms were 80, 82, 83, 84, 85, 86, 87, 88, 90 and 91. Find out how much each bale weighs. In particular, you should determine if there is more than one possible set of weights, and explain how you know.”

David Klein, a mathematics professor at California State University at Northridge comments on the problem. “The process of solving this problem made me resentful of the stupidity and pointlessness of it. There is nothing ‘real world’ about it. It is completely inappropriate for kids who likely have not been taught how to solve simultaneous linear equations, or exposed at most to two equations in two unknowns. If I had been given such problems at that age, I think that I would have hated math.”

Consistent with much of the philosophy of “real life math”, the goal of the exercise is to explore strategies and to be able to write about it. This is made apparent by the “student guide” that accompanies the problem. It is essentially a scoring sheet, containing categories, with points awarded for each, such as “Restate the problem in your own words” (4 points); describe all the methods you tried before reaching your solution(s) (4 points); describe the process that lead to your solution(s) (4 points); describe all assistance provided and how it helped you (2 points); state the solution (2 points); describe why your solution(s) is correct, include all supporting data (6 points). Out of a total of 50 points, only 2 are given for the solution. In fact more points are given for describing why the solution is correct.

### The ACT/SAT and Your Middle-Schooler

As a parent, I have often projected my own ambivalent feelings about testing onto my son, J. Even though time and time again, J. has tested well on standardized tests throughout his school career. So, when his scores came back in the 6th grade showing him in the 95th percentile in math when he normally scores in the 99th percentile, I wasn't too concerned. What was odd, however, were that kids who were 2 and 3 levels behind J. showed scores in the 99th percentile.

Even though J. had been involved with Northwestern's CTD program (Center for Talent Development) since kindergarten, I had never investigated their talent search program because, once again, I was anxious about the testing aspect. I figured that J. had been tested and prodded enough.

But then my eyes were opened when I read the book,

**What High Schools Don't Tell You**by Elizabeth Wissner-Gross, and realized that I might be making a mistake. (KTM regulars--I highly recommend this book. It is packed with information and links.) The book lists 300+ secrets for parents. Here is secret #28:

“Four major universities offer talent searches and talent-search summer programs that remain among the best-kept American secrets and parents often learn about them only after it's too late."So, I called Northwestern to get more information and spoke to the nice lady.

The basic gist is that if your child scores in the 95th percentile in a subject area, (This is for 7th grade. The other grades might be slightly different.), they recommend that he/she takes the EXPLORE test. If he/she scores in the 97th+ percentile, they recommend that your child take the ACT or SAT.

I told her my issue with his math scores and she mentioned that taking one of the big tests would give a better idea of his math abilities because there is no “ceiling” for middle-schoolers. Since J. was in Algebra 1 in the 5th grade and it was unlikely that there was any algebra on the 5th grade state test, it could stand to reason that children who had not had pre-algebra or algebra could score higher.

For an additional fee on top of taking the test (around $30) Northwestern then analyzes the scores and provides the parents with ideas about how to approach you kid’s education.

Sooooo, for the last 2 months we have been preparing for the ACTs by using the big prep books and hiring a whiz kid tutor. The experience has been enlightening for me as a parent. I assumed his math skills would be pretty good, but I got a lot of encouragement watching him perform the English and Reading tests. Plus, after his initial panic (“I can’t do that! That’s a test for high-schoolers!!”), he started to see that much of it wasn’t too hard and that he could handle a good deal of it.

The big “if” will definitely be the stamina required for taking the test. It’s clearly more suited to the older teen, so it will be interesting to see how it affects him. My attitude with him is to let him know that this is just for practice and to go for it.

## Wednesday, October 24, 2007

### do GATE kids have bar models inside their heads?

I'm wondering, now, whether GATE people have internal bar models, too.

Remember the 6th grade boy who was given the problem about the whale?

A whale swims 40 miles in 1 1/4 hours. How far does he swim in 1 hour?

This boy, a GATE type kid, instantly knew the answer, but didn't know how he got it -- and didn't know how to set up proportions. (I'm pretty sure his dad said he didn't know how to set up and solve a proportion, at least.)

A couple of weeks later I talked to a GATE mom, and told her about the boy. I rattled off the problem quickly, and she, too, instantly knew the answer. She said, "Well you talk kind of fast [true], but I think that's the answer, right?"

This was over the phone.

I asked her how she knew the answer, and she said, "Well it's 1 1/4 hours, so that's five parts."

Now that is not the way I would have thought of this problem until I did a lot of bar models. (I didn't think of the bar models when I first heard the problem. Had to set it up as a proportion. But if I'd been "in practice," in practice meaning I'd been doing some bar models recently, I a bar model would have popped into my mind's eye.)

This mom also told me that it would take her a lot longer to set the problem up as a proportion and solve than it did just to "know" the answer, which she was pretty sure she knew because 1 1/4 is 5 parts.

I find this fascinating.

Normal human beings, and that category excludes all you math brains, do not think 1 1/4 is 5 parts.

Normal human beings think 1 1/4 is two parts: 1 and 1/4.

It's a well-known fact that normal human beings fall off the math cliff when they hit fractions in 5th grade. Maybe the presence of an internal number line (not proved, btw) and the absence of an internal bar model, tell us something about why normal human beings fall off the math cliff when they hit fractions and not, say, when they hit multiplication.

I'm inclined to believe this, simply because my guess is that Dehaene will turn out to be right; numbers are represented spatially in the brain in some way. Makes sense that fractions would be, too, in people who "know" the answer to the whale problem without having to set it up.

Assuming this is true - which is what I'm going to assume, since I have nothing to lose if I do assume it to be true - I draw three conclusions:

1. Hung Hsi Wu's recommendation that one teach fractions first and foremost as a point on a number line (pdf file) is a fine idea. Hence: lots of practice with rulers!

2. The bar models in Primary Mathematics are indeed one of the core reasons for the effectiveness mathematics instruction in Singapore. (pdf file)

3. I am going to carry on insisting C. do Primary Mathematics 3rd grade bar model problems.

trip down memory lane

Back when I was working my way through Challenging Word Problems Book 3, I could feel my brain changing as I did those bar models.

I loved the bar models so much I started a solution manual for Challenging Word Problems Book 4. Didn't get too far; not sure where it is today.

Writing this post I'm feeling a lure that I must resist back to that project.....

The Number Sense by Stanislas Dehaene

Solving Algebra and Other Story Problems with Simple Diagrams: a Method Demonstrated in Grade 4-6 Texts Used in Singapore by Sybilla Beckmann

Sybilla Beckmann articles

Mathematics for Elementary Teachers by Sybilla Beckmann (1st edition)

Chapter 2: Fractions by Hung Hsi Wu (pdf file)

Cassy T on bar models

Drat These Greeks on bar models

Challenging Word Problems, Book 3

Spatial attention and the internal number line

### And Now For Some Good News

I was bowled over, shocked into silence. Why? Simple.

He clearly, unequivocally took responsibility for student achievement.

I have never heard a school administrator do that before.

He didn't blame the students or the parents.

He didn't imply that this group of kids just wasn't as bright as the previous bunch.

He didn't blame his predecessor (and that would have been understandable).

What he did do was great. The plan is a "laser like focus" (his words) on improving writing skills. The Middle School teachers are analyzing the curriculum; they are using formative assessments; they are identifying struggling students (those without an IEP that are "proficient" but haven't mastered the content); then they are providing instruction based on results of the formative assessments.

Students are given extra help during the school day at times most convenient to the student, not the teacher or the administrator. If a student can come in early, they meet then, if after school or during a free period work better, that's when they schedule extra help. Students are identified, they are not expected to know themselves if they need extra help.

He ends with the statement, "Imagine what a tremendous gift it would be if we could send every student on to high school being able to write well."

I can't believe they hired this guy. In a system filled by an unabashed constructivist superintendent, we get this gem. I feel like I should write him a letter of support (even though I don't even have a kid in the middle school at the moment).

## Tuesday, October 23, 2007

### another district in NY

Our cliff is between buildings, going from 5th to 6th. In 5th the philosophy is that 'all learners' must succeed, so the classes are not deep in content and different learning styles are included in the pedagogy (with the exception of math). Inclusion & SPED are grouped with high achievers, so much that the Regent's suggest for the grade level curriculum is omitted. In 6th, content rules. Inclusion is still grouped with high achievers, but inclusion is given study guides, help from a teacher, and the IEP is used to reduce the h.w. load and usually extra time and quiet on the test is given. There is a lot of h.w. in 6th - well over 2.5 hrs nightly for those not on IEP in the 'high' reading class, about 30 min. less for those on grade level in reading. It is difficult because the 5th grade curriculum omitted all the writing skills and about 1/3 of the necessary background in mathematics.

Students that do well in the transition are either highly verbal or those that had to develop study skills in elementary to stay in the high achiever group. Those that stumble are usually smart visual-spatial boys who had not developed study skills in the elementary or the ability to translate a verbal sequential lecture into good notes; sometimes they are the highly capable verbal boys who don't care to do the h.w. that doesn't lead them to mastery of the material. The former are usually not placed in honors in subsequent grades unless they develop the study and note-taking skills on their own by Feb of 6th grade; the latter are in honors until they reach the point that they can't keep up on the tests from the material they hear in class (usually 7th grade honors English does them in by mid-terms, if not, 8th grade earth science is the wash out).

No surprise, the top 10% academic group based on grades earned exiting high school will include 1 or occasionally 2 boys out of a starting 9th grade population of over 600 students. Enrollment in advanced science and math is very low compared to similar schools. National Merit, SAT/ACT results and college diplomas earned tell a different story about academic excellence for those whose v-s boys stayed and persisted despite the lack of encouragement from the school. This is a large suburban average needs school, not title 1 or urban. App. 25% of parents are highly educated (meaning B.S./B.A. or beyond).

Well......that was a fun read.

Are middle schools in charge of the gatekeeping? That certainly appears to be the case here.

And, if so, when did this happen?

And why?

They don't seem to mention it anywhere in the NMSA.

### don't look at the naked autistic boy

The title of my memoir, should such occur.

This morning was frantic beyond frantic. Ed's been gone for what seems like a couple of years but has actually been just 10 days, and, today, C. has a math test. This one snuck up on me, although I had it on my calendar; par for the course seeing as how calendars seem frequently to fail me, or I to fail them.

Did I not enter a "Remind me" date?

Probably.

The point is: I can't have math tests sneaking up on me.

I wasn't involved in C's first test of the year. Result: grade of 60.

I taught and retaught like crazy for the second test. Result: grade of 87.

Sneaking up is bad.

C. seems in much better shape this time around, so there's hope. Nevertheless, because I didn't see this one coming, I had to add test prep to the morning's customary festivities, which meant that as I was trying to get Jimmy and Andrew into school-ready condition ("EAT YOUR TOAST" "TAKE YOUR MEDICINE") I was also printing out edhelper worksheets on rational expressions raised to negative powers, stapling worksheets and answer sheets together, discovering I had no idea which answer sheets went with which worksheets, tossing out the original batch, reprinting worksheets in a neat and orderly sequence....then discovering the problems on the worksheets were far too complicated to give C. any real practice in the fundamentals of simplifying a rational expression raised to a negative power, AND THEN concocting simple problems, on the spot, for C. to practice on while I was also spreading peanut butter on toast for Andrew and then discovering that Andrew's toast was actually Jimmy's toast, which meant that, in fact, after all the EAT YOUR TOAST shrieking, Jimmy had not actually eaten any freaking toast.

Down to the wire on toast and rational expressions raised to negative powers, we all had to scramble into the car to make it to the bus, the only problems being that I didn't have my glasses and Andrew didn't have his clothes.

No time for glasses; no time for clothes; pile into car (JIMMY, GET IN THE CAR); roar down to bus stop.

Miss bus.

Onward to school, sans glasses and clothes.

C., being 13, was concerned. After all, we are driving up to the school with his autistic twin and his autistic big brother, so we're already standing out from the crowd, and his autistic twin is not wearing clothes.

I tried to distract him with talk of PEMDAS.

"Will anyone see that Andrew is naked?" he asked.

"No," I said. "They'll see that he isn't wearing a shirt, but they won't see that he isn't wearing any pants."

C. brightened.

"Don't look at the naked autistic boy!"

## Monday, October 22, 2007

### Nonlinear Sixth to Seventh Grade Jump

Schools seem to go from low expectations to sink or swim in just one year. They don't worry about mastery (let alone speed) in K-6 math, and then they decide that they have to ramp up to prepare kids for high school. The onus is on the kids to make this nonlinear jump and the teachers don't seem too concerned. They spend years talking against drill and kill, and then, suddenly, that argument disappears.

Has anyone else seen this sort of nonliner jump? Our schools used to have this jump going into high school, (because of CMP in 7th and 8th grade math.) but after many complaints, they've shifted the jump backwards to 7th grade.

### Alan Greenspan on reform math

A recognition of how poor our mathematics education had become and perhaps some reason for hope was the report in September 2006 by the National Council of Teachers of Mathematics, reversing its ill-chosen advisory of 1989. The earlier report recommended a curriculum that dropped emphasis on basic math skills (multiplication, division, square roots, and so on) and pressed students to seek more free-flowing solutions and to study a range of special math topics. I always wondered how you can learn math unless you have a thorough grounding in the basics and concentrate on a very few subjects at a time. Asking children to use their imagination before they know what they imagining about seemed vacuous to me. It was.

The Age of Turbulence: Adventures in a New World

p. 406

### and now for something completely different

Will post excerpts later on.

Preview: he thinks constructivist math was a big mistake.

Unfortunately, he also thinks constructivist math has left the scene now that the Focal Points have been published.

### Shared Decision Making

What is considered "crucial information", "proper guidance", necessary "knowledge" and/or "inappropriate" is at the discretion of the parent, not the school, nor the state. Parents should certainly have a voice and a choice in what their children are taught particularly in such a personal matter such as this.

This should not be the role of a teacher. Please let them teach academic content instead of using valuable teaching time for something that is clearly a parent's responsibility. The consequences of how they choose to teach their children is their own responsibility. To fail to obtain prior consent, communicate what will be taught with parents beforehand, or establish clear guidelines for the teaching of sexual education, is irresponsible and patently wrong.

Just as it is inappropriate for schools to impose a particular religion or political ideology upon the children in their charge, it is equally inappropriate for schools to teach human sexuality in ... a random and careless manner.

Let me add that in fact IUFSD did not intend for middle school kids to read these brochures. It was a mistake.

I agree strongly with the overall position.

Barry Garelick said...

In Fairfax County, VA, the school district sometimes conducts surveys of students about drug use and sexual activities. Prior to doing so they notify the parents of such survey, state that there will be questions of this nature and asks for parents' permission to allow the student to participate in such survey. I think this is the type of choice Steve H is alluding to and one which was not exercised in the case of the pamphlets. No one was notified, no one saw the pamphlets, and no parent was consulted as to whether they felt such material was appropriate or not.

This is all true.

Yes, the pamphlets were a mistake.

However, if the school had given parents the option of vetting the brochures beforehand, the mistake would not have been made.

Last school year, I made this exact point to the administration of the middle school. I pointed out that, where character education is concerned, you are always going to be dealing with values, which means you will never have consensus.

I suggested they set up a committee of teachers and parents to vet content in these areas, if only to protect themselves.

SteveH said...

"What is considered 'crucial information', 'proper guidance', necessary 'knowledge' and/or 'inappropriate' is at the discretion of the parent, not the school, nor the state. Parents should certainly have a voice and a choice in what their children are taught particularly in such a personal matter such as this."

Very well said.

There is a certain amount of arrogance hidden behind the idea that somebody has to do it, and that they (schools, county) have the knowledge and skills to do it. We see this in math, where schools are supposed to know something about the subject, but only teach math via their own pedagogical opinion. When parents (who are experts in the content) complain, they are ignored.

It's the same idea. Schools 10, Parents 0.

The key sticking point is prior knowledge and choice; not just with sex education, but everything.

The schools have too much power -- all schools, not just my own.

In NY state parents and teachers are entitled, by law, to Shared Decision Making.

Parents don't have it.

Nor do teachers.