kitchen table math, the sequel: Ze'ev Wurman on Bill Gates on Common Core

Sunday, March 23, 2014

Ze'ev Wurman on Bill Gates on Common Core

Bill Gates: OK. So what is the Common Core? It’s a very simple thing. It’s a written explanation of what knowledge kids should achieve at very various milestones in their educational career. So it’s writing down in sixth grade which math things should you know, in ninth grade which math things should you know, in twelfth grade which math things should you know.

Ze’ev Wurman: That, indeed, is what content standards are supposed to be.

BG: And you might be surprised to learn how poor those I’ll call those standards, but to be clear, it’s not curriculum. It’s not a textbook. It’s not a way of teaching. It’s just writing down should you know this part of algebra? Should you know trigonometric functions? Should you know be able to recognize a graph of this type?

ZW: Wrong. I wish Mr. Gates would actually read the standards rather than rely on what others tell him. Common Core standards are more than just content standards, they also dictate pedagogy and hence curriculum. Here are a couple of obvious examples.

“Grade 1 standard 1.OA.6: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).”

This standard does not require only knowing addition and subtraction within 20, as a content standard should. It insists on knowing four specific ways to add and subtract. In other words, it dictates pedagogy and curriculum.

Bill Gates Needs an Education on Common Core
I haven't read the math standards start to finish, and I probably won't. (I'm working on the ELA standards, Appendix A in particular.)

Nevertheless, I think it's fair to observe that a "standard" instructing teachers to "use strategies such as counting on..." is more than just a content standard. Quite a bit more.

Unfortunately, in the context of American public schools, the word "strategies" means something quite different from what it does outside of public education.

Within public education strategies is a formal term for guide-on-the-sidery.

I remember being semi-gobsmacked, a decade ago, sitting in on a CSE meeting and hearing the word "strategies" repeatedly used to describe the district's approach to educating children with behavioral and/or learning difficulties. The student in question that day, a 5th grader with emotional and behavioral problems, was going to be given "strategies" he could use to stop acting the way he was acting and start acting the way the other kids were acting.

In another meeting, the same psychologist said a child with dyslexia would be given "strategies" she could use to read.

She wouldn't be taught to read; at least, that wasn't the focus.

Instead, she would be given strategies she could deploy, as needed (the child was to be the judge of that), to help herself read better than she was reading now.

Even then, before I'd delved into all these things, I knew it was all rubbish.

A 10-year old with emotional and behavior problems isn't going to use "strategies" to stop having outbursts, and a 10-year old with dyslexia isn't going to use "strategies" to read at grade level.

29 comments:

SteveH said...

CC doesn't define much of any detail. Everything will be driven by the philosophy and expectations of those who make the tests. There is no incentive to make the tests meaningful to any level over the lowest level one needs to get into the lowest expectation college. The better students, with help from parents, will cover their asses at the above average end.

Jen said...

Gaaaa, just lost my comment clicking on the wrong button!

The distinction being made here between the content standard and a strategy for showing that standard IS the crux of the problem for CC.

The tests could be made to test only for the concept, but are much more likely to be made in such a way that you have to know the specific strategy meant to "scaffold" you to the concept, rather than be secure in the concept itself.

It reminds me of a district level math supervisor when my oldest kids were in elementary. The district was using EDM (I know, I know, it was actually great for my kids and did what it was supposed to do, but I did realize as the years went on that it really didn't work for 50+% of their classmates).

The supervisor was explaining how they were trying to make sure that every teacher was really using the curriculum in exactly the way prescribed. It had clearly come to their attention that some teachers found the program lacking and were modifying it.

Their solution? They were making up district-wide unit tests for the lower grades that would ask the students specific questions about the games and activities that were to be included. That is, besides some questions that actually tested the concepts the kids were to be learning, they'd also have questions that tested whether they knew how the games were played, or what the rules were for playing them.

So, now the students were being tested on...nothing...so that the district could ding the teachers/school for it. And that, in a nutshell, is also the big problem for CC. Instead of asking questions that determine if they know and are fluent with their facts to 20, they are instead going to design questions that try to tease out HOW they were taught them and ding those who didn't use the specified "strategies."

Not proofreading tonight, so I apologize in advance!

Barry Garelick said...

From the US Coalition for World Class Math comments on the final version of the CC math standards:


"Standards should be free of pedagogy as much as possible. Many of the standards in the CCSS contain pedagogy. The “fat standards” often contain pedagogy. Fat standards are ones that 1) could be broken down into multiple standards, 2) lead examples in with “For example” or “e.g.”, or 3) both. While the pedagogy in many cases is used as an example, teachers and administrators will often interpret and apply it as a
part of the standard. Consider this Grade 4 standard.

Compare two fractions with different numerators and different denominators, e.g.,
by creating common denominators or numerators, or by comparing to a
benchmark fraction such as ½. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with
symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.


This standard could effectively be broken into separate standards:

Compare two fractions with different numerators and different denominators.

Recognize that comparisons are valid only when the two fractions refer to the same whole.

Record the results of comparisons with symbols >, =, or <, and justify the conclusions.


The phrases “by creating common denominators or numerators, or by comparing to a benchmark fraction such as ½” and “by using a visual fraction model” are pedagogy.

The standards should be able to stand alone without embedded pedagogical examples. This standard could easily be interpreted as having a choice between creating common denominators or numerators and comparing to a benchmark fraction.

With that being a choice, the CCSS never explicitly require students to find common denominators. "

SteveH said...

My view of CC is that it's really just a rehashed national NCLB. It can only be considered a high standard as a minimal high school graduation requirement. If they now say college is for all high school graduates, then CC is now translated into "college readiness" - the 75 percent probability of passing a college algebra course (PARCC). Of course, many want to hide behind the high expectation sounding "college readiness" label.

This is like the biggest bad secret in K-12 education, but many are trying very hard to make it sound like it's something more than a minimal high school graduation requirement. In our town, it will be business as usual (which still has big problems). In other towns, parents will have to fight back the forces of stupidity.

"The colleges will have to change."


SteveH said...

"the CCSS never explicitly requires students to find common denominators."

I studied the standard once and counted the number of times they used the word "fluent" for all of K-12. I can't find my count, but I think that I just had to use the fingers on my two hands. Besides, they don't define what fluent means.

That's why I think that CCSS will only be defined by the tests each state uses and the scores they use as proficiency cutoffs. Our state is having all towns do "field tests" this year. I want to see a full math test and the score (raw percent correct) needed to achieve their level of 5 "distinguished".

Auntie Ann said...

The long-term effects are going to include a new wave of flight from public schools. Parents are going to be looking for alternative schools, tutoring, online programs, etc., to get their kids what they know the need. Kids with parents who are working multiple jobs and don't have time to check their child's learning, kids whose parents are poorly educated and don't know what they should be learning, and kids of parents who can't afford supplementary education will get screwed.

Colleges, which are already under stress (students aren't studying, a degree means less than it used to, credentialism is threatened by pushes to actually prove a student has done more than sit in classrooms for 4 years,) and facing new alternatives, are going to be in trouble. What do they do? 1) dumb down their curriculum so that those screwed by the CC can go to college, or 2) refuse to take CC-educated kids, who will be largely poor and often minority.

They're going to choose option 1, and degrade their reputations and the value of their degrees even further.

Froggiemama said...

I think these are great standards: "Grade 1 standard 1.OA.6: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).”
Why? Because I don't think it is enough to have simply rote memorized the math facts. I think students should know all those strategies, and be able to use them. It is precisely that kind of fluid "playing around with numbers" that leads to deep understanding. I do these exercises with my own kids all the time because they don't do nearly enough in school, where rote memorization still is the preferred "strategy" taught.It forces them to think about the numbers and how they relate to each other.

I teach computer science and constantly have to deal with students who memorized their way through years of math but are not fluent with relationships between numbers. One of the classic problems they might see in CS1 is the problem where you write a program to sum the digits of a number. If you understand the base 10 number system, it is easy-peasy, as my kids would say. But, you would be amazed at how many college students have NO IDEA how to solve that problem. They can't see how to split the digits up, because they never really understood base 10, and place value. They learned carries, and borrowing, as memorized procedures without a deep understanding. So when they have to look at numbers in a way that is different from their memorized procedures, they can't do it.

I am very hopeful that CC will improve things. It can't get any worse than it is now.




Auntie Ann said...

Unfortunately, they've revised math teaching over and over again for decades, and it does always seem to get worse. Hold on to you hats!

kcab said...

To me, the standard isn't bad in quite the same way, because I read "such as" as meaning "able to use a method to reach their answer, here are some possible methods."

I've only had to deal with standards in engineering. In my experience, if something is required then "shall" is used. "Shall" is what is tested. I don't see ANY shall's here, only an implied one at the start, which wouldn't be acceptable. I guess education standards are more squishy.

Ze'ev Wurman said...

Froggiemama,

I think you miss the point where CC goes astray. Everyone wants kids to understand the *concept* of addition and subtraction. In fact, there is a string evidence that very young children already understand this concept innately. Everyone also wants kids to understand the concept behind addition and subtraction of numbers written in a conventional (decimal) way (as opposed to Roman numerals, for example). That's reasonable, and can be demonstrated to young kids (e.g., in K, 1) using manipulatives, blocks, or the number line ... once or twice.

But there is no particular need to repeatedly "teach" this understanding in multiple ways year in year out, and there is certainly no need to TEST kids (which is what standards effectively define -- what kids will be tested on) that they understand what addition and subtraction are in multiple ways. At most, each kid has to understand what they are in one way.

Further, this particular standard is supposedly focused on fluency and not on understanding. So why does it insists on showing a supposed understanding in 4 different ways? Worse, why does essentially the same standard also show up in grades 2 and 3, just with larger numbers? After all, the understanding doesn't change -- just the the numbers grow so the procedure gets a bit more tangled.

The Common Core went off its rocker in its stupid quest for "understanding" where there is little to understand and a lot to be able to automatically do, the less thinking the better. The results will be kids that are jerked around to show their understanding of trivial things in rather unnatural ways to satisfy the teachers and the test, rather than children that can simply do the arithmetics fluently and with little to no thinking, so they can focus on the mathematics of problems rather than on justifying the steps in a rather mindless calculation.

Anonymous said...

Auntie,

"What do they do? 1) dumb down their curriculum so that those screwed by the CC can go to college, or 2) refuse to take CC-educated kids, who will be largely poor and often minority. "

Another possibility is that they bifurcate.
And maybe that's the goal: CC is the plutonomy pulling the ladder up after itself.

SteveH said...

"The long-term effects are going to include a new wave of flight from public schools."

When I was young, there were only Catholic K-6 schools. Now, there are all sorts. A big trend in our area is to send your kids to a private K-8 school and then bring them back to the public high school. It's still a tough choice because the private schools cost so much, there is a longer drive, and many still use curricula like Everyday Math.

Anonymous said...

You wanna talk about flipped classrooms? When some parents realize that everything their kids are learning in elementary school they're learning at home, they flip their kids' classroom into the circular file and homeschool them.

Expect common core to lead to an increase in homeschooling.

Michael Weiss said...

Nevertheless, I think it's fair to observe that a "standard" instructing teachers to "use strategies such as counting on..." is more than just a content standard. Quite a bit more.

I think you are misunderstanding what the standards are. They do not specify what teachers should do, but rather what students should be able to do. In other words, they say that by the end of 1st grade if you ask a student what 8 + 6 is, they should be able to answer that question, and they should have a toolkit of different ways to figure out the answer. (I would add to this: a student should be able to make good choices about when to use different strategies).

Keep in mind that at the first grade level we are talking largely (not exclusively, of course) about mental and verbal math. It is not reasonable to expect a 1st grader to write "8 + 6 = 8 + 2 + 4 = 10 + 4". But it is totally reasonable to expect that many 1st graders will -- without prompting! -- think some version of "Start at 8, go up 2, then 4 more" (which itself relies on him or her recognizing that 6 can be decomposed into 2 and 4.) It is also reasonable to expect that many 1st graders might think some version of "I know 10 + 6 is 16, so this is two less than that." Teachers should be prepared to see these strategies play out in practice, so that they will understand what their students are doing.

Of course if you just think that every 7 year old should commit 8 + 6 = 14 to memory, the above is moot; but memory is probably our most faulty organ, and the problem with relying on memory is that you have no way of knowing whether you have mis-remembered something.

I think what CCSS is trying to do, at least in this example standard, is capture some of the variety of the informal mathematical heuristics that children naturally and spontaneously use at this grade level, and point out that they are valuable for kids to know and use. All of that is great. Unfortunately the way curriculum and policy-makers tend to respond to calls for greater attention to flexible thinking is by mandating a wide range of specific methods, which ends up undermining the very value flexible thinking that it is supposed to be supporting.

Jen said...

" It is not reasonable to expect a 1st grader to write "8 + 6 = 8 + 2 + 4 = 10 + 4"."

I promise you that as unreasonable as that is, it will be expected. In this case, it's likely to be part of a multiple choice question, though. That just means that they'll have to pick out something like what you wrote from a list of garbled versions of the same.

And if you're a kid who uses doubles plus or minus to understand your math facts as your strategy of choice, well, you're out of luck on that question. (I know 6+6 is 12 so I'll add 2 because I need an 8 for one of those two 6s OR I know 6+6=12 and 8+8=16 so, 8+6 must be in the middle.)

ChemProf said...

Coincidentally, a cousin of mine just started homeschooling her second grader, in part due to this kind of common core-justified math. The final straw? A multiple choice word problem where they needed to add 8 and 6, and the correct choice was 10+4. 14 wasn't given (although if you really knew all your facts and knew both 8+6=14 and 10+4=14, you could figure it out).

allison said...

--It is not reasonable to expect a 1st grader to write "8 + 6 = 8 + 2 + 4 = 10 + 4....

-J:I promise you that as unreasonable as that is, it will be expected.


Well, I am about to give up.

The first graders I know using Singapore math have absolutely *no problem* doing this in their head, or verbally, or even on whiteboards using number bonds.

NONE.

They are all apparently a lot smarter than the complainers.

Honestly folks, if you find learning to think in this way so appalling, stop suggesting you are for better math instruction.

Better math instruction is hard work. You want an easy out? go ahead. But teaching kids numbers for whole numbers to ten works. And teaching them to break apart 6 so they can find a number that, when put with 8 makes 10, works.

And then the kid KNOWS why 8+6 is 14, not 15.

Chemprof, I hope you tell you new-to-homeschooling friend not to bother with Singapore math. Because they ask questions like that.

Because knowing 8+6=10+4 is how THEY teach the steps to the vertical algorithm for regrouping.

By the middle of first grade, kids are solving 34 - 16 in their heads using these regrouping techniques. By the end, 83-57.

Froggiemama said...

Thank you Allison, you stated this much better than I did. I am not a math educator, so I don't always know how to make a good math education argument. I am a mere computer science professor who has to deal daily with the fallout from students who memorized math rather than understood it.

I did Singapore informally with my oldest when he was little. At that time, I had never seen that kind of mental math taught in a curriculum(though I use it all the time myself) and I thought it was great. I continued to do some of that with my next two kids. When our district adopted CC I was thrilled to see mental math coming into our curriculum, too late for my oldest two, but my youngest has benefited. Our district has slowly been improving its math curriculum. When my oldest started, it was an uneasy mix of "draw what 2 + 2 looks like" and "do 59 Mad Minute worksheets".
The school district also claims that math instruction will be mastery-based rather than spiral - sure hope that is the case!

Jen said...

One last time. No one is saying that kids would not know their ten facts, or not know that numbers can be decomposed to 10 + x or related concepts.

It's agreed that first graders should be able to do those facts mentally (and on paper!)

The issue is how you test their knowledge and why. Most of this testing is done as either multiple choice or as "Explain 8 + 6 in words and pictures. Be sure to include how a ten fact is important in your answer."

While a well-taught first grader can do all of those things -- and likely tell you about them as they do them, they are much less likely to be able to pick out the correct answer amongst a thicket of 4 or 5 answer choices full of numbers and operators. They are also unlikely to write an English sentence in legible printing that makes adult sense of the math.

*That's* the point. Not the concepts, but the presentation. It's about not having developmentally inappropriate test questions for developmentally (and academically) appropriate math.

Ze'ev Wurman said...

Allison,

You start with the question whether it is reasonable to expect 1st grader to write "8 + 6 = 8 + 2 + 4 = 10 + 4...." and you answer that first graders using Singapore math "have absolutely *no problem* doing this in their head, or verbally, or even on whiteboards using number bonds."

First, I have no real idea how kids solve the problem "8+6" in their head. Nor, I suspect, have you. What you probably know is that they can answer it correctly, rather than how they actually mentally do it.

Second, I think it is "reasonable" to expect 1st graders to write "8 + 6 = 8 + 2 + 4 = 10 + 4...." -- if one forces them to do write it. (Incidentally, I find it unreasonable for a kindergartner, but CC already expect kids to write formal number sentences in K.) The question I find more interesting is whether regularly writing formal equations in this way, and insisting on mental calculations based on this approach, are important and useful ... for 1st graders.

Specifically, I couldn't find in the Singapore program the almost-maniacal insistence on *exhaustive* decomposition of numbers in multiple ways that I see in CC. Singapore does a bit of number bonds and, after establishing the principle, they just move on to a lot of practice with additions and subtraction problems. I also don't see the obsessive push to make kids find "friendly" numbers like 10 or 20 -- in fact, I barely see it at all. I suspect that's because most 1st graders in Singapore very quickly know -- through memorizing single digit addition facts and without much thinking -- that 8+6 equals 14 rather than go through idiotic gyrations of composing and decomposing numbers to satisfy some know-it-all standard-writers.

My interpretation is that Singapore expects familiarity with numbers, and development of mental tricks (for ease of computation), to evolve naturally when kids have a lot of practice with computing with numbers, within and without the context of word (or picture) problems. The program may provide some suggestions here or there, but it certainly doesn't go overboard insisting that kids not only calculate correctly and fluently both with p&p and mentally, but that they do it in a very specific way as CC insists.


C T said...

Isn't it highly probable that ten-facts are so valued in Singapore Math because in Chinese, they use the word shi (ten) in every two digit number? I don't speak Chinese myself, but that seems to be the way their numbering system works. For instance, in English, we say twelve, while I believe in Chinese, they would say ten-two. For twenty-three, Chinese speakers say two-ten-three. When you have a language like that, it makes perfect sense to constantly break numbers apart to make ten. With English, not so much; we can pick up an understanding of the ten-base number system as we learn about place value and do multi-digit arithmetic. I'm perfectly content for my children to not spend a lot of time on these mental math procedures (make 10, etc.) and just memorize that 8+6=14. Isn't it a little "cargo cult" for us Americans to adopt some Singapore Math practices that stem naturally from the Chinese numbering system?

Glen said...

CT, you're right about the Chinese numbering system (I speak Chinese), but I don't see a lot of emphasis on tens per se in Singapore Math, so I don't see a need to explain it in terms of Chinese. (And I would estimate that most kids in Singapore DON'T speak Chinese.) In SM see the usual regrouping for mental math purposes that I learned as a kid in the US. In other words, I see 8+6 discussed as making a group of ten with four left over in the context of learning carrying and place value, but I don't see 8+6 = 10+4 in SM for mental calculation purposes. You just memorize 8+6=14.

But I do see a lot of 18+56=20+54 and 290+361=300+351 in Sing Math for mental calculation, and this regrouping technique was a standard part of US math class back in my day (when none of us spoke Chinese).

C T said...

While English is the official language in Singapore, around 3/4 of Singapore is Chinese and they are generally bilingual; half the country speaks Mandarin in the home. I wish English were similar to Chinese as far as how numbers were formed.
That's interesting that you did the same mental math long before SM. I don't recall doing it in school, and I went to elementary school in two different states. Despite not doing mental math much, I ended up with a math major and tutoring math for 3 years at my university. I guess I just see mental math as fun for those who like it but not an essential.

C T said...

Wow! I did a quick Google search on "efficacy of mental math" and turned up this study of 6th-graders in Jordan that showed a HUGE benefit (compared to the control and compared to girls) for boys given instruction in mental math. No wonder you're in favor of it and I don't really see a need for it. https://eis.hu.edu.jo/deanshipfiles/pub10376745.pdf But I'll teach my kids some mental math formally anyway on the off-chance it helps them (I only have daughters).

Catherine Johnson said...

Hi Ze'ev!

Great to see you here -- !

Catherine Johnson said...

Haven't read the thread, but for what it's worth, back when I was trying to re-teach math at home I came across research on mental math & became familiar with Singapore Math.

It all struck me, at an intuitive level, as absolutely true and as a huge missing element in my own math education.

I'm trying to remember now...Saxon Math lessons have a brief mental math exercise at the start of each lesson, don't they?

I could be misremembering...

Catherine Johnson said...

I'm going to have to dig up my copy of Liping Ma & re-read the chapter on algorithms and mental math. As I recall, she described very specific mental versus paper and pencil modes of doing **specific** problems in **specific** contexts.

The approach seemed to be highly pragmatic.

Allison said...

I work with kids ever week in grades K, 1st, 2nd, 3rd, and 4th. Often with grades 6-8, and I am in daily contact with grade 6-8 teachers discussing what needs the students have. I work in the classroom and in pull out small groups.

I cannot read their minds, but children, unsurprisingly, are transparent. Kids counting on their fingers count in their fingers. Kids pretending not to do it under the desk. Kids counting often voice or move their lips while doing so. Kids guessing shout out random answers rapidly.

But this is what we *do* when we are together in gr k or 1:
me: " here are 8 blocks; I cover some (cover them. 4 are left. How many did I cover?"
student: 4
me: "why?
student: "because 4 and 4 make 8."

For the kids who don't know that, we don't work on that question, but others, like
"The total is five; I take 2. How many do you take?" And whether ths child is adding, counting, or has the answer memorized already is fairly simple to see by their response time and facial expressions.
in gr 2:
for the kids who don't their facts to 20, this is how they work them.
They are given ten rods and singleton cubes.
me: what's 13-8?
student:...hm..
me: show me 13
student: shows the ten, slowly counts out the 13.
me: we need to take away 8 ones. Do we have 8 ones?
student:
no
me:
no, what?
student: we have 3 ones.
me: so what do we do?
student (after weeks of this): "we take away the 8 from the ten.
me: which leaves?
student: (counting maybe..number bind maybe..) ...2
do now we have 2 and how many?
student: 2 and 3
me: which is?
5.
So 13 - 8 is what?
student : is 2 and 3 makes 5.


In these grades, the teachers use these same questions. Why? Because this is what their teacher's guide tells them to verbalize. That level of detail in the PM standards edition TG.

The PM teacher's guides teach *exactly* these strategies, *exactly* this specificity. I will post some pictures.

Anonymous said...

I don't have a problem with that level of specificity for kids who are struggling with concepts that others have learned in class.

However, that is clearly not the only way to perform that problem mentally, nor is it necessarily a way that makes the most sense to teach.

For instance, 13-8 could also be seen as, oh, 13-3=10 and 10 - the 5 leftover is 5.

Still using tens. But, as mentioned up above, really, truly, the goal is automaticity of the facts to 20.

With 3rd graders, I often find adding the tens and then the ones is easiest for them. Rather than
"18+56=20+54 and 290+361=300+351"

They think 60 + 14 = 74 .

The issue really isn't which accurate strategy kids use. The issue is knowing your basic facts to automaticity and understanding place value well enough to understand the different ways numbers can be broken apart and put back together.

If you test on just one specific strategy, it's a bad test. If you test that the students can see which of several different strategies work, that's a little different.

It doesn't sound like the kids taught using one very specific strategy are getting a better (or any) understanding of the overall concept. They are just being drilled in an intermediate step -- and it's an intermediate step that isn't actually a necessity.