kitchen table math, the sequel: 3/6/11 - 3/13/11

Saturday, March 12, 2011

the Times on public sector pay

terrific set of charts, graphs, and summaries

2nd SAT

Debbie took the SAT today!

A hundred kids in line waiting for accommodations -- and they separate the twins.

death by data

A teacher in my town posted a link to this story:
The Lab School (Lab School at US News) has selective admissions, and Ms. Isaacson’s students have excelled. Her first year teaching, 65 of 66 scored proficient on the state language arts test, meaning they got 3’s or 4’s; only one scored below grade level with a 2. More than two dozen students from her first two years teaching have gone on to Stuyvesant High School or Bronx High School of Science, the city’s most competitive high schools.

“Definitely one of a kind,” said Isabelle St. Clair, now a sophomore at Bard, another selective high school. “I’ve had lots of good teachers, but she stood out — I learned so much from her.”

You would think the Department of Education would want to replicate Ms. Isaacson — who has degrees from the University of Pennsylvania and Columbia — and sprinkle Ms. Isaacsons all over town. Instead, the department’s accountability experts have developed a complex formula to calculate how much academic progress a teacher’s students make in a year — the teacher’s value-added score — and that formula indicates that Ms. Isaacson is one of the city’s worst teachers.

According to the formula, Ms. Isaacson ranks in the 7th percentile among her teaching peers — meaning 93 per cent are better.

This may seem disconnected from reality, but it has real ramifications. Because of her 7th percentile, Ms. Isaacson was told in February that it was virtually certain that she would not be getting tenure this year. “My principal said that given the opportunity, she would advocate for me,” Ms. Isaacson said. “But she said don’t get your hopes up, with a 7th percentile, there wasn’t much she could do.”

That’s not the only problem Ms. Isaacson’s 7th percentile has caused. If the mayor and governor have their way, and layoffs are no longer based on seniority but instead are based on the city’s formulas that scientifically identify good teachers, Ms. Isaacson is pretty sure she’d be cooked.

She may leave anyway. She is 33 and had a successful career in advertising and finance before taking the teaching job, at half the pay.

“I love teaching,” she said. “I love my principal, I feel so lucky to work for her. But the people at the Department of Education — you feel demoralized.”

How could this happen to Ms. Isaacson? It took a lot of hard work by the accountability experts.

Everyone who teaches math or English has received a teacher data report. On the surface the report seems straightforward. Ms. Isaacson’s students had a prior proficiency score of 3.57. Her students were predicted to get a 3.69 — based on the scores of comparable students around the city. Her students actually scored 3.63. So Ms. Isaacson’s value added is 3.63-3.69.


The calculation for Ms. Isaacson’s 3.69 predicted score is even more daunting. It is based on 32 variables — including whether a student was “retained in grade before pretest year” and whether a student is “new to city in pretest or post-test year.”


In plain English, Ms. Isaacson’s best guess about what the department is trying to tell her is: Even though 65 of her 66 students scored proficient on the state test, more of her 3s should have been 4s.
But that is only a guess.

Moreover, as the city indicates on the data reports, there is a large margin of error. So Ms. Isaacson’s 7th percentile could actually be as low as zero or as high as the 52nd percentile — a score that could have earned her tenure.

Evaluating New York Teachers, Perhaps the Numbers Do Lie
Published: March 6, 2011

I have four reactions.

A 32-variable teacher evaluation scheme does not sit right with me if only because it lacks transparency. This teacher has no idea why her score falls in the bottom 7% of all teachers in NYC, and neither does anyone else including her principal and students.

Is this teacher running afoul of a ceiling effect? Her students were already scoring well above average coming into her class -- isn't it harder to bring above-average students further up than it is to bring below-average students to average? Working on SAT math with C., I'm convinced that the jump from 550 to 600 is a shorter leap than the one from 600 to 650. Whether or not that's true for the SAT specifically, I'm pretty sure people have shown it to be true with other tests.


Yes. It's a well-known effect. *

I flatly reject the assumption that New York state tests are capable of distinguishing between a group of students earning 3.57 on average and a group of students earning 3.69 on average. A few years back, when C., who is a fantastically good reader,** scored a 3 on reading, I got in touch with our then-curriculum director, who told me that NY state tests in some grades have essentially no range of scores in the 4 category at all. That is, if you score a 37 or 38 out of 38 correct, say, you earn a 4; score a 36 and you're a 3. I checked the test and sure enough. She was right. There was no range at all for the 4. I don't know whether David Steiner has changed the tests in the year he's been in office, but even if he has, I reject the idea that the tests are now valid and can accurately assess what the gap between a 3.57 and a 3.69 means (if anything) and whether it is equivalent to the gap between a 3.01 and a 3.13.***

On the other hand, suppose the 7% ranking is right. What might account for that?

One possibility: the Lab School is a constructivist enterprise (here's the Math Department), and this teacher was trained at Columbia Teachers College. She is teaching English and social studies to 7th graders. New York state requires that teachers have a Bachelor's degree in their field of specialty beginning in 7th grade, which means that most 7th grade teachers are teaching English or social studies, not both. One of her students says, "I really liked how she’d incorporate what we were doing in history with what we did in English,” Marya said. “It was much easier to learn.”

Interdisciplinary teaching at the middle school level tends to be shallow because students aren't expert in any of the fields being blurred together (and teachers are expert in just one field), and the only commonalities you can find between disciplines tend to be obvious and current eventsy. e.g.: back when one of our middle school principals explained to us that henceforth character education would be 'embedded' in all subject matter, the best example he could come up with was that the father in The Miracle Worker is an angry patriarchal male who is abusive towards his handicapped child. That "interdisciplinary" reading of The Miracle Worker is anachronistic, simplistic, and wrong.

It's impossible to say whether these scores mean anything.

But if they do, they suggest to me that a 7th grade teacher needs to focus all of her efforts on English or on history, not on both.

English literature and history are very different disciplines.

I wonder how other teachers in the school fared - and, if they did better, how true they were to the middle school model?

* The article doesn't tell us whether the city's statisticians correct for ceiling effects and regression to the mean.

**C. typically missed just 1 or 2 items on SAT reading & writing tests.

*** Of course, given the very wide range for 3s, perhaps a .12 difference is significant. It's impossible to know -- and that's the problem.

parents on schools and time

re: Richard Elmore's observation that family's time is a gift, not an entitlement:
This is exactly how I feel about most of my kids' homework -- it's throwing their childhood away.

One of the reasons we are currently homeschooling the elementary kids, I couldn't take one more scrap booking assignment disguised as literature or math. Not to mention the hours lost to drug awareness, school fundraising pep rallies and watching "Finding Nemo."

help desk - what does this mean?

A problem from Dr. John Chung's SAT Math:
Between 500 and 1000, how many integers are multiples of 5?
page 68
How do I read this?

Are 500 and 1000 excluded?

So that I would be finding how many integers are multiples of 5 starting at 501 and ending with 999?

The solution seems to start with 500 and end with 999 ------


Reading (& re-reading) Chung's solution, it's apparent he means that 500 and 1000 are excluded, but his way of writing the solution is so different from the way I would write it that it looks like he's including 500 but excluding 1000.

His solution also seems to imply that you find the number of integers between 100 and 199 inclusive (i.e. how many integers starting with 100 and ending with 199) is to subtract 100 from 199.

Chung's book seems to be fantastically useful, but I'm wondering whether you need to be already scoring a 700 or so to understand it.

I will soldier on.

Dr. John Chung's SAT Math

wrong answer

C. and I both got question 15 on page 773 in the Blue Book wrong.
15. The Acme Plumbing Company will send a team of 3 plumbers to work on a certain job. The company has 4 experienced plumbers and 4 trainees. If a team consists of 1 experienced plumber and 2 trainees, how many different such teams are possible? 
My friend Debbie Stier got it right -- ! After she explained it to me, C. and I saw what we were doing wrong (double-counting the trainee pairs, basically), too, but we wanted to watch the Khan video to see how he did it.

Turns out Salman Khan did the problem the same way C. and I did.

Which makes me feel somewhat better.

Best explanation for novices on College Confidential:
Best way to approach this:

Fact 1: There are 4 different experienced plumbers.
Fact 2: There are 6 different combinations of 2 trainees in a total group of 4 trainees. If this isn't obvious, let the 4 trainees be A, B, C, and D.

All the possible trainee pairs:


and each of these trainee pairs can go with any of the 4 experienced plumbers, so we multiply 6 X 4 for the total number of possible teams, which gives 24. A common error here would be to erroneously consider AB a different pair from BA, when really order doesn't matter in this problem. That is how you would get a wrong answer of 48. 
The Official SAT Study Guide, 2nd edition

Wednesday, March 9, 2011

a mathematician on math and writing

re: math & writing:
In both math and writing, the core idea that you are trying to express exists somewhere in the aether. In both math and writing, you start out staring at the blank page, trying to figure out how to summon the idea, make it yours, and commit it to the page. In both math and writing, you make false starts (unless you are very lucky) and work hard (unless you are very lucky) to express the idea with precision and clarity. In both math and writing, your familiarity with the idea that you are trying to express and your prior practice at expressing ideas can sometimes give you a head start in knowing in which direction to start.

Math is writing. Most of math is persuasive writing; math is an exquisitely structured argument.

(I am a professional mathematician.)
I love this.

This passage captures what writing is for me.

This one does, too:
Writing is easy. All you do is stare at a blank sheet of paper until drops of blood form on your forehead.
Gene Fowler

Richard Elmore on time

It would be an enormous step forward if adults in schools treated the time that children and their families give to schools as a precious gift rather than an entitlement.

Three Thousand Missing Hours
Where does the instructional time go?
Harvard Education Letter
Volume 22, Number 6
November/December 2006
Somewhere along the line I began to see my middle school with its required "health class" and its required "technology class" and its required PowerPoint assignments for Spanish class (create a menu; pack a suitcase and label the clothing) as scooping up huge, heaping armloads of my son's childhood and simply tossing them in the trash. 

 C. told me a funny story. One of his brainier friends was up late one night, reading stuff for health class. His dad saw that his son was still up and asked what he was doing. When C's friend said, "I'm reading health," his dad barked at him: "Put that away! Get to sleep!"

Dad is a physician.

math museum

Ralph Gardner on math versus writing:
I don't want to say I'm skeptical about the idea of a museum devoted to mathematics. Indeed, I can recall the deep satisfaction I felt on the all-too-rare occasions at school when the concepts or formulas fell into place. It seemed an entirely different discipline from writing, where something arises from a blank page through a combination of hard work and patience, with a sliver of creativity. With math, the experience is more like discovering something that's always existed and finally decided to stop playing hard-to-get.
Making Math Fun (Seriously)
March 9, 2011
Wall Street Journal
I'm curious.

Do people who work in math or math-related fields feel this way?

Or is this more aptly a description of the way it feels to learn math when your expertise lies elsewhere?

Choosing Carnival Junk Food

It’s standardized test time in Connecticut and my child will be busy participating in this painfully drawn-out process over the next couple of weeks.  While I should clarify that I think testing *can* be useful for diagnostic purposes, I consider the following question from the eighth grade CMT Mathematical Applications section to be yet another example of why I find my state's manner of assessing students quite useless:
Sample Item 8-5 (Numerical):  Buying Tickets

The carnival offers you two different options for buying tickets.
                OPTION A:  $2.00 per person plus $0.75 per ride
                OPTION B:  $5.00 per person plus $0.25 per ride

If your uncle gave you $10 for the carnival, which option – A or B – would you choose.  Show the mathematics you used to determine your answer.


This poor excuse for a word problem is just one example of why we started homeschooling.  While I’m assuming the objective of the question is for the student to show mathematically that option B is the better choice because you can go on 20 rides as opposed to only 10 rides with option A, the question does not indicate that the goal is to go on as many rides as possible.  I can easily imagine any of my children (including the 8-year old) coming up with alternate scenarios that could make either option the better one.  Unfortunately, I can just as easily imagine a scenario where the person responsible for grading 200 tests containing these strange open-ended mathematical responses before the end of their shift would mark their mathematically and numerically accurate answer WRONG.
Let's say the student were to choose OPTION A since he really likes to eat junk food at carnivals (just like his favorite uncle who spots him the $10) but hates the rides because they make him dizzy (thereby leaving him with $8 to spend on food instead of $5).  Would that be counted as a correct answer?  I would argue that either option could be the better choice depending on the objective—which was not made clear.  Mathematics is supposed to be clear, precise, and accurate. This question is just silly.

These types of  Everyday Math word problems (I'm being generous here by calling it a word problem) used to make me crazy when my child would come home with them in 4th grade.  Now here we are in 8thgrade running in circles all over again.

Meanwhile back in Singapore children are answering this:
Hooke's law for an elastic spring states that the distance a spring stretches is proportional to the force applied. If a force of 150 newtons stretches a certain spring 8 cm, how much will a force of 400 newtons stretch the spring? (New Elementary Math 2 Placement Test)

People are starting to pay more attention to what is "behind the curtain" in our public schools

I caught a few minutes of a school board budget meeting in the school district next to mine, which happens to be near the one that is the subject of this post.  Residents were complaining about what they considered to be outrageous compensation packages given to school administrators. Some speakers expressed surprise about the way that school employees are able to carry forward so many personal/sick days and bump up their compensation during their last year of employment. Of course, the incentive to do just that lies in the fact that lifetime pensions are based on the last year or so of pay.

A couple of speakers compared their situation to that of Dorothy in The Wizard of Oz, as they were only now learning what was "behind the curtain".

The quote from the movie is, "Pay no attention to that man behind the curtain." Well, people are certainly starting to pay attention now.

(Cross-posted at Education Quick Takes)

Tuesday, March 8, 2011

Bronxville school taxes

A good article in the Times today.

The only problem is that it doesn't mention the Triborough Amendment, which you need to know something about in order to understand why the union would choose to work without a contract.

Also, in the illustration below, substitute the words "since the crash" for "in the last several years."

bubble this

C. and I take sample SAT tests together every day just about.....and I make bubbling errors.

Bubbling errors!

I bubble in the wrong bubble!

Today I bubbled in TWO wrong bubbles! On reading for Pete's sake.

How does that happen?

It's not like I'm racing for time; I finish reading sections several minutes early. (And, yes, I do see that the obvious answer is: stop finishing several minutes early. But I thought I had stopped.)

The good news is that I managed to complete every question in Section 8 (Math), Text 6 on page 729-733 of the Blue Book, and I only missed one: number 14. Number 14 is rated "hard" but I missed it because of a dumb mistake, which I think is a good sign.

Number 14  is one of the problems I'm supposed to skip according to my SAT Game Plan.

The Official SAT Study Guide, 2nd edition

The New Math SAT Game Plan

Anonymous on Professional Learning Communities in Utah

Anonymous left this comment:
Several schools in our school district (Washington County Utah) have started doing this kind of assessment and intervention with individual students. They've had such amazing results that the school district is passing a resolution today requiring all schools and teachers to participate. They call them Professional Learning Communities or PLC's.

From today's newspaper:

"The resolution in support of PLCs would require each educator in the district to participate in the PLC activities, part of a collaborative effort to achieve better results from students by continuously analyzing performance and intervening where necessary with individual students."
School board ponders new background checks
David DeMille • • Published: March 07. 2011 10:29AM

I am a huge fan of "Professional Learning Communities," which I think should probably be seen as an indigenous, American form of Japanese lesson study.

Richard DuFour invented "professional learning communities" at Adlai Stevenson High School in the 1980s, and the school has seen continuous improvement in student achievement since that time. Here's his website: allthingsplc.

The great charter schools use a variant of Richard DuFour's approach, too. Paul Bambrick-Santoyo cites DuFour in his book Driven by Data: A Practical Guide to Improve Instruction which is essentially the companion volume to Doug Lemov's Teaching Like a Champion.

Driven by Data: A Practical Guide to Improve Instruction

Teach Like a Champion: 49 Techniques that Put Students on the Path to College

Arne Duncan supports "Gold Star" teachers

Just put up a post at the Parents Forum.

data in schools

“In education, data has traditionally been used for punitive purposes, not for improvement...”
March 7, 2011, 10:05 pm
Coming Together to Give Schools a Boost
Reading Bornstein's article over breakfast this morning, I had a revelation.

For years now, a number of us have pressed our school district to assess student learning and adjust instruction accordingly -- i.e., to use formative assessment. We've gotten nowhere, and I've tended to see the district as 'allergic' to measurement in general.

But in fact, my district owns cutting-edge software that tracks disciplinary actions. When a student is sent to the office, the principal can instantly pull up his entire computerized disciplinary record and read out loud all of the bad things the student has done going back for years. I don't know whether the current principal of the middle school actually does this, but his predecessor did. He would fire up his computer and take the student on a little trip down memory lane: all the bad memories, none of the good.

Of course, I doubt the district has used its computerized disciplinary records as a means to monitor improvements in student behavior. The district essentially never sets measurable goals of any kind.

But still.

We have cutting-edge software to track misbehavior and disciplinary actions.

We don't have cutting-edge software to track student learning.

Monday, March 7, 2011

Middle School Mathematics Institute updates

Middle School Mathematics Institute (MSMI) is alive and well with lots of progress.

MSMI's mission is to help teachers, schools, and parents ensure students succeed at school algebra. Focused primarily on elementary and middle school grades 4-8, MSMI offers a variety of services that build the Bridge to Algebra.

We've got a nice new website:, we are now officially a 501(c)3 non-profit corporation (so donations to MSMI are federally tax deductible), and we're beginning to get some traction in our mission along all of the three prongs--teacher institutes, school curricular services, and parent outreach.

I'll be posting here in the next few days about MSMI 2011 institutes this summer in St. Paul, MN, one on fractions, and another on rational numbers. I'll also be posting about an upcoming event in April here in St. Paul, a workshop on "How to Use the Best Strategies of Singapore Math to Strengthen your Math Instruction." But for now, I'd like to talk about the parent component.

The parent outreach part is just starting to get underway. I gave a talk the other night on the Math on the SAT, Getting to Mastery to a group of parents. Their kids were mostly in elementary and middle school, and that's the time to start helping them understand just what's happening in their students' math curriculum--before it's triage and remediation. The talk highlighted that the SAT reasoning test math portion is a very good test of mathematical maturity, and that maturity simply isn't going to come from the American curricula and textbooks most students see. The talk gave some ideas on just what mastery looks like (both conceptual development and procedural fluency) and how to help students gain that mastery. I was surprised at how well received the talk was, actually, and I hope to do many more on similar topics.

The talk slides are available on the web site to anyone.

Along with that talk, I hope to develop another talk focused on what is needed to be on a college-prep track in high school, and what is needed for college entrance into a STEM career.

MSMI will also be writing some free pamphlets that help parents understand what's happening in math education. Immediately, I expect to have a pamphlet up explaining "curricula, standards, assessments, and all that" so that parents are familiar with these buzzwords. I hope to also get another pamphlet together talking about the false dichotomy between teaching and learning, a kind of mini-primer on what modern ideas of education mean for their student.

I hope to keep refining MSMI's focus on the parent side to maximize the good it can do. I was surprised after my talk that many parents want to organize to do something to help their students, but simply do not know what to do, and what suggestions on how to get their school to change direction. As you might guess, I didn't have a lot to say on that front; MSMI isn't designed for that kind of advocacy. I think for now, it shouldn't be, but I hope it will facilitate parents coming together to do that for their kids.

Anyway, have a look, download the slides, send suggestions (the web site needs pictures, I know, I know...) and drive traffic to, please! :)

help desk - questions for management

A friend of ours who sits on the board of a private school watched the Terc Investigations video a couple of weeks ago. Until he saw it, he believed boards should leave curriculum decisions up to school administration. Now he wants to write him a list of questions to ask "management" about the school's math curriculum.


what is the effect of the Triborough Amendment over the long-term?

I want to check my reasoning with all of you.

Yesterday the Times carried a long editorial on the subject of New York's public sector unions and the state's fiscal crisis, which included these two factoids:
In April 2009, private sector income was down 9%.
In April 2009, public sector employees were given a 4% raise.


Average salary for New York’s full-time state employees in 2009 (prior to April raises): $63,382
Average personal income in NY state: $46,957

State Workers and N.Y.’s Fiscal Crisis
New York Times
I assume these two facts are connected by the Triborough Amendment, a statute that is apparently unique to New York state. Under the Triborough Amendment, when a public sector union contract expires, its terms remain in effect until a new contract is signed.

This means that raises negotiated in good times must be awarded in bad times, which explains the 4% raises paid out in April 2009. It also means that the union has little incentive to negotiate during an economic downturn.

From the Times:
Last April, in the midst of one of the worst financial crises that New York and the nation have ever faced, the state’s unionized workers got a 4 percent pay raise that cost $400 million. It came on top of 3 percent raises in each of the previous three years. These raises were negotiated long before the recession began, by a Legislature that routinely gave in to unions that remain among the biggest political contributors in Albany.

During the same period, many private-sector workers had their pay or hours cut. Private-sector wages in New York dropped nearly 9 percent in 2008. In 2009, Gov. David Paterson pleaded with the unions to give up the raises to help the state out of its crisis. Union leaders attacked him in corrosive television ads, and Mr. Paterson eventually caved, settling for an agreement that reduced pension payments to new employees. The deal wasn’t enough to address New York’s serious fiscal problems.
The Triborough Amendment was adopted in 1972.

Here is the line of reasoning I want to vet with you.

What is the likely effect of the Triborough Amendment over time?

To me, it seems that over time the relative position of public sector to private sector employees would change, with public sector employees moving ahead and private sector employees falling behind.

Over time, the gap between public and private sector income would grow larger. If private sector employees take income cuts while public sector employees receive raises (or, at a minimum, do not take freezes or cuts) -- then each time the economy recovers, hasn't the ratio of public sector to private sector compensation changed?

Then, when the next recession arrives, doesn't the ratio change again?

If so, is that why we see average salary of full-time public sector workers in New York state at $63,382 while average personal income is $46,957?

I realize this comparison isn't apples to apples, but given that public sector employee incomes are presumably included in the $46,957, the actual ratio of public sector to private sector income must be even larger.

What it looks like to me -- and please tell me if I'm not thinking this through correctly -- is a  decades-long redistribution of wealth from one segment of the middle class to another segment.

Or is that wrong?

Do the incomes of private sector workers somehow bounce back up to where they were before each recession while the incomes of public sector workers 'stand still' long enough for the ratio to return to what it was before the downturn?

Magister Green on guerilla teaching

re: memorization is a dirty word, Magister Green offers this advice:
I have found that using "automaticity" in place of "memorization" lets me get around the stigma of the latter word while retaining its basic meaning. And "automaticity" sounds cool, so people don't press.

Guerrilla teaching forever!
Guerrilla teaching or marketing --- !

Sunday, March 6, 2011

Daniel Ethier on cognitive load theory

on another thread, Daniel Ethier writes:
Cognitive load theory has much to say about the educational implications of our limited working memory on learning.

As I read various papers on cognitive load theory, I keep having aha moments as I find reasons for things I see in the classroom.

The key to getting around our limited working memory is automaticity. If you know something well enough to not have to think about it, it does not take up working memory. And so you are free to think about the problem you're trying to solve.

Use Google Scholar and read some of the many papers about various aspects of cognitive load theory. Well worth the time.


"when A.P. testing began in 1956, memorization was not yet a dirty word"

Rethinking Advanced Placement
Published: January 7, 2011
In theory, the new A.P. courses are going to replace "memorization" with "critical thinking."

In reality, critical thinking depends on memorization: you can't think critically without something to think about, and that something has to be stored in long-term memory. If you're going to think critically, you have to know (i.e. remember) what you're thinking about.

What happens when you try to think critically about a subject without memorizing its terms and concepts first?

What happens is that you can think about 4 items at most. That is the number of new, discrete elements you can hold in conscious, "working memory"* at one time. Four. And four may be pushing it.

Of course, when it comes to critical thinking, 4 is a tiny number. Experts think critically about far more elements at one time; being able to think about a vast amount of complex material is pretty much the definition of an expert, as a matter of fact:
The sine qua non of skilled cognitive performance is the ability to access large amounts of domain specific information [i.e. knowledge]. For example, it is estimated that chess masters have access to as many as 100,000 familiar configurations of chess pieces (Chase & Simon, 1973). As another example, in order to make sense of what he or she is reading, a reader must have access to information gained from previously read text. This is particularly true when reading complex technical material filled with jargon.
summary of Ericsson, K. A., & Kintsch, W. (1995). Long-term working memory. Psychological Review, 102, 211-245.
David Zach Hambrick, 1998,

basal ganglia lollapalooza

Here's an example from my own life.

As a nonfiction writer, I'm essentially a permanent student: I am constantly trying to write interesting articles and books (mostly books) about material that may be brand-new to me. My current project involves the basal ganglia, which I knew nothing about going in. The vocabulary alone is overwhelming: nucleus accumbens, orbitofrontal circuit, putamen, striatum --- and that's just for starters.

So here's the question. How exactly am I to (a) understand and (b) think critically about a passage that contains these four terms if I haven't memorized what these terms mean and how they are related to each other first?

The answer is: I can't.

If I don't memorize vocabulary, I have to look up the definitions and then try to hold the definitions in mind while also reading and trying to think about what I'm reading.

It can't be done, and the reason I know it can't be done is that I've spent a lot of time trying to do it. I always make the same mistake with each new project I tackle. Somehow I think I can just look things up (Google!) and remember them while I read a complex study or article.

But I can't. No one can. Looking up four new words and remembering four new meanings maxes out working memory. There's no capacity left to read and understand a text using those four new words and four new meanings, let alone think.

I don't know why this is. Logically speaking, shouldn't it take just as much working memory to hold 4 memorized terms in mind as it does to hold 4 non-memorized terms in mind?

The answer is no: knowledge - content stored in long-term memory - extends working memory.

When you know a lot about a subject - when you have a great deal of knowledge stored in long-term memory - you can think about more than just 4 things at once.

blackboards vs PowerPoints

* Working memory is essentially consciousness: it's what you're thinking about and/or remembering right now. When you hold a phone number in memory while dialing it, you're using working memory.