#4, Part 1 – The Unreasonable Effectiveness of the Knowledge Graph

是的。

Yeah.

就像是那种

It's like the

特别的能力，你不需要做别人必须做的事——而那些事其实你本来就应该做。

special, and you don't have to do things that other people have to do, which are the things you need to be doing anyway.

这种纯粹技能的两面性。

It's the duality of these pure skills.

一方面，我们可以通过让你更快、更强来提升你的水平，比如在健身房增重，或者单纯地冲刺训练。

It's like, on one hand, like, we can we can up your game by getting you faster, getting you stronger, just like in a pure sense, weight in the weight room or just sprinting.

但如果你只依赖这些，把它们当成全部，那你其实是在限制自己。

But if that's what you that's the only thing that you lean on, if that's your entire game, then you're limiting yourself.

对吧？

Right?

就是这样。

It's like that.

那段时间真的特别难，因为你和我都教过那群六七年级的孩子，教他们最初的代数课。

Well, it it was so tough because you because you and I both taught and worked with these six seventh or sixth graders in the first running algebra.

你知道吗，我儿子科尔比是表现最差的。

And, you know, my son Colby was the all time worst.

天哪。

Oh my god.

是的。

Yeah.

真的没错。

This is true.

原因是他总能蒙混过关。

And the reason is because he could get away with it.

要让他明白道理实在太难了。

It was so hard to convince.

我还有个孩子叫内森，比科尔比小一岁。

I had this other kid, Nathan, who was a year younger than Colby.

我当时就想，兄弟，我太了解你这孩子了。

I was like, dude, I'm like, I know you kid.

我是认真的。

Like, I'm telling you.

我知道你脑子里能一步到位，但总有一天你会碰到靠这套行不通的时候。

Like, I know you can do all these steps in your head, but you will eventually run at a point where it's not gonna work so well.

对他们来说，问题就在于：什么才算管用？

And the problem is for them, they're like, what works?

我记得有一次，嗯……

And I I remember this in okay.

打篮球的时候，我就用这个例子。

In in in basketball, so and I use them.

谁都会听进去。

I got anybody's listens.

我经常用体育方面的比喻。

I do a ton of sports analogies.

抱歉，这其实就是我看待事情的方式。

I'm sorry, but it's just how I think about things.

我不是那种高个子后卫，个子一般，但我靠速度绕开对手，手里有几个拿手动作。

But I would I was not a you know, I'm an average height guard, and I would use my speed to go around people, and I would have a couple of moves.

我记得别人总说：你每次都干这一套。

And I remember people say, oh, you just always do this or that.

我就说：那你来拦我啊。

And I'm like, well, stop me.

对吧？

And I was like, right.

他们老抱怨，因为我总能左右晃过他们，怎么都防不住。

Like, they they would complain because I could just do this left or this right every time, and they couldn't stop it.

但我没好好练外线投篮，因为靠速度就能轻松过人。

But I didn't develop an outside shot like I should have because I could just use my speed to around people.

对吧？

Right?

所以我对这一点是亲身经历的。

So I I know this personally.

我当初犯了错，因为你知道，我们以前聊过那个故事，教练杰森说过，以你的身高想打大学篮球，必须得有个特别特别准的三分球。

Like, I messed up because, you know, and I think just we told talked about the story in the in the past where coach Jason, like, you wanna play college basketball at the college level at your height, you need a really, really good three point shot.

是的。

Yeah.

所以你得开始练了。

So you need to start working on it.

哦，对啊。

I'm like, oh, yeah.

行吧。

Okay.

没错。

So, yeah.

关键是，这种优势其实是天生的。

So the thing is, it's it's the the more of an advantage is inherent.

这份礼物让你明白，通常越难打破的，反而越有价值。

This gift gives you, the harder it is to break them usually.

我的意思是，有时候关键在于他们是否愿意接受指导。

I mean, sometimes it's just a matter of what their coachability is.

对吧？

Right?

但你知道，对孩子来说，如果某种方法有效，你就得说不行。

But, you know, as with kids, if something works, you say, no.

不行。

No.

不行。

No.

不行。

No.

别这么做。

Don't do that.

就这么做。

Do this.

他们就说：这招管用。

And they're like, it works.

既然有效，干嘛不这么做呢？

Like, why would I not do that?

对吧？

Right?

这根本说不通。

It's just this this doesn't make any sense.

你跟我说，我看到的就是事实，每次这么处理都有效。

Like, you're telling me, like, I see the world as it is, and I this happens, and it works every time.

既然有效，干嘛不这么做呢？

Why would I not do that?

因为这其实很简单。

Because it actually it's easy.

没错。

Right.

而且我做起来很自在。

And I'm comfortable doing it.

但一旦这种情况发生，就会固化下来。

But it's especially when it occurs, it locks that in.

对成年人来说，这真的很难打破，他们总说：你看，我这是在帮你啊。

And it's so hard to break for an adult who's like, look, I'm trying to help you, man.

我真的在帮你，求你了。

Like, I'm trying to help you, please.

而且传统观念也说，要抓住有效的方法，对吧？

And conventional knowledge too is like lean into what works, right?

在营销领域，你经常听到这句话。

You hear that in marketing all the time.

到处都能听到：抓住有效的方法。

You hear that like everywhere, lean into what works.

有时候，很难分清到底是顺势而为，还是把当前有效的方法当成了拐杖。

So it's hard to kind of untangle that sometimes between like the difference between leaning into what works versus using what is currently working as a crutch.

有时候这种区别非常微妙。

It's really subtle sometimes.

很难分辨，但从长远来看，这两条路完全是不同的方向。

It's hard to tell the difference, but the in the in the long term, these are two completely different trajectories.

是的。

Yeah.

这真有意思。

It's fascinating.

所以，亚历克斯，我不知道你有没有更多关于这个故事的想法，我们可以继续聊，但我觉得我们还是回到原本该谈的话题吧。

So well, Alex, I don't if you think if you have anything more on the story, I we can we can go on it, but I wanna we can probably pull it back to what we're originally supposed to talk about.

我觉得

I think

这其实也不算意外。

we can kind of unsurprising.

我们还是回到之前聊到的SAT吧。

Let's let's let's go back to where we were talking about the the SAT.

其实我们本来想聊好几件事，比如这个——没错。

So I suppose we wanted to talk there's a number of things we wanted to talk about like so, yeah.

我们刚才花了一些时间讨论所谓的‘中间缺失’部分，就是我们在X上提到过的那个‘中间缺失’。

So we we just spent some time fitting in what's called that the so called missing middle that we've talked a bit about on X about the missing middle.

我觉得这个话题挺有意思的，值得深入聊聊。

And I I think that'd probably be kind of a fun thing to talk about.

是的。

Yeah.

你本来还想讲讲SAT课程的历史，这本来就是我们最初打算谈的内容，只是杰森突然插进来，把话题带偏了。

Thought you'd even go through the history of the SAT course, which is, yeah, what we were originally gonna talk about before Jason just, like, grab the

方向盘和注意力都被抢走了。

wheel and watch.

不过，还是回到历史话题吧。

But, yeah, let's get back to the history.

好，那我先简单梳理一下。

So, okay, so let's just, let me just frame it a little bit.

我们最初是凑合着给Alpha School编了一套SAT课程，这个我可以公开说，因为他们经常提到Math Academy。

And then, so we initially cobbled together an SAT course for, it was Alpha School, which I can say publicly because they always they mention Math Academy a lot.

Alpha School是一家比较前卫的私立学校，主要在德州，也在其他地方有分校。他们当时说，嘿，我们想帮学生备考SAT，你能帮忙弄个课程出来吗？

So Alpha School's a very forward leaning private school, I guess, in Texas and other places, but they were like, hey, you know, we're trying to prepare students to go take the SAT.

你能帮我们编一套课程吗？

Can you put together a course?

我当时也是这么想的。

And I was like, yeah.

对。

Yeah.

我就跟 Alex 说，你帮忙梳理一下 SAT 考试涵盖的那些技能点，把我们现有的内容整合成一门课。因为那些知识点分散在几何、代数、代数二还有综合数学里，太零散了。

And said, Alex, can you just take a look at the, you know, review what is the described skills that are covered in the SAT and take what we have and put it into a single course because they would spread across geometry and algebra and algebra two and algebra, and it's a sort of like or integrated math.

根本没法把它们按紧密程度归类。

It's like there's no way of like, well, how close are they?

哎，我也不太确定。

Like, I don't know.

那时候她就提醒过我。

You know, it's so now at the time, warned me about it.

她说：这门课真得好好做，不能马虎。

She's like, look, I I just really gonna be a good SAT course.

我的意思是，我们确实把所有基础内容都涵盖了。

I mean, if you guys and I said, well, I I mean, I think we have all the the fundamentals there.

应该没问题的。

It should be fine.

她警告我说，这事儿没那么简单，我记得我之前也跟你提过这事。

And she's like, I don't she warned she was gonna ingest and you I think I mentioned I asked you about it too.

我就问她：你不是多年辅导学生准备SAT了吗？

I said, what do because you tutored kids for NSAT for years.

你说过，这课程还不够全面。

And you you were like, it's not it's not totally comprehensive.

是的

Yeah.

这些是学生参加SAT考试所必需的基本能力，无论学校采用的是传统课程还是综合课程，这些内容都包含在标准课程里。

This is are the constituent skills that a student is going to need for the SAT that exists in the standard curriculum, whether it's traditional or integrated.

对

Yeah.

但如果你把SAT的要求直接套用到标准课程上，就会发现陈桑的警告是有道理的。

But if you project the SAT on the standard curriculum Echo Sandy's warning.

这远远不够。

Like, this is not nearly sufficient.

是的

Yeah.

你不能只靠这种课程就直接去参加考试。

To like, you can't just come out of that.

那个纯粹基于标准课程内容的原始课程，根本不够用。

Original course that was just based on the standard curriculum stuff.

学生光靠这些内容，是不可能在SAT数学上考到800分的。

A student can't just take that and then come out like scoring like an 800 on the SAT math.

对。

Right.

我们得额外补充很多标准课程里没有的东西。

We had to do, there's a lot of stuff outside of the standard curriculum that that we were gonna have to build in.

而且在

And at the

那时候，亚历克斯像往常一样，同时在带好几门课。

time so Alex, as usual, has had his he they had we had, like, multiple courses going.

所以我跟他说：亚历克斯，你能把这些内容整合一下吗？

So I like he's like, Alex, can you just can you put this together?

他回答：好的。

And you're like, okay.

那你具体想让我做什么？

What what do you want me to do?

我就说，你看。

I'm like, look.

你干脆就说，行吧。

Can you just you're like, fine.

结果一周之内，你就把内容给分析完了。

And within, like, a week, you just analyze the stuff.

你接着说，好吧。

You're like, alright.

这就是实际情况。

This is this is what it is.

然后你就说，对。

And and you're like, yeah.

我觉得这安排不错，因为你只是在看，没去分析考题。

I think it looks good because you were just looking at you weren't analyzing tests.

你只是看了下各个知识点的描述，觉得咱们该有的都有了，挺好的。

You were just looking at the descriptions of the topics and, like, I think we got all this stuff and I think it's good.

然后，可能过了六个月，甚至还没那么久。

And then I maybe it was six months later, maybe not even that long.

我接到了安迪·蒙哥马利的电话，他是Alpha学校课程方面的负责人之类的。

I got a I a call with Andy Montgomery who's like head of curriculum or something for Alpha School.

人特别好。

Really, really nice guy.

特别聪明。

Really bright guy.

他好像是斯坦福毕业的计算机科学博士。

He's like a PhD in computer science from, like, Stanford or something.

他一直都很出色，以前在数学学院就表现突出，所以人很随和，跟他说起话来很轻松。

And he's always been, like, really impressive Math Academy and, you know, was very so he was he was a supporter, so he's easy to talk to.

他对我说：‘杰森，你们的口碑很好，尤其是在X平台这些地方。’

And he's like, listen, Jason, you you guys have a great reputation, you know, especially on X and things like that.

但要是你们宣传说课程能帮人准备某件事，结果却做不到，那就全毁了。

One way to ruin it is to say you have a course that prepares you for something and doesn't.

我当时就问，你这话是什么意思？

And I was like, you know, what do you mean?

他说，你看，我们让学员做了模拟考试，然后我们的团队做了分析。

He's like, look, there are there are you know, we had our students go through the test and they were taking practice tests and we had our team analyze it.

结果发现有不少知识点和题目，学员们根本没准备好。

And there's just a lot of a lot of there are a number of topics or questions that they just really weren't prepared for.

我就说，听到这个确实不太舒服，但我很感谢你的反馈。

And I was like, and I was like, well, I don't love hearing that, but I appreciate the feedback.

然后我想起来，你之前提醒过，还有桑迪也提醒过。

And then I I remember that, you know, Justin, your warning and Sandy's warning.

挂了电话后，我就去找桑迪聊了聊。

Because then I got off the the call and I talked to Sandy.

她说：我早就跟你说过了。

She's like, I told you.

对吧？

Right?

她说，我就说，唉，他们确实想要点什么。

She's like and I said, well, we they wanted something.

我们只能把最好的东西拿出来给他们。

We had to give them the best thing we had.

但她指出，把这门课公开，多少有点把我们暴露了。

And she's but by making it a public course that that exposed us a little bit.

于是我心想，行吧，这事儿得赶紧改。

And so then I was like, okay, we gotta we gotta fix this.

这实在不太妥当。

This is this is not cool.

对吧？

Right?

这本来就是那种我完全没留意的事情。

And it's just one of those things I just was not paying attention to it.

当时我们手头的事情太多了，课程、技术项目，样样都在忙。

We had too many balls in the air, too many different courses we're working on, too many different tech projects.

我当时就想，哦，对，他们做得不错。

I just was like, oh, yeah, we're they're good.

他们呢，你知道的，不好意思啊，这段有点啰嗦，但其实这挺重要的。

They're, you know, and then and I'm sorry, this wind up is long, but this is a little bit

算是个不错的背景铺垫。

of a it's good background.

对，有点前情提要。

Yeah, a little back.

这事儿吧

It's a

有点复杂。

little complicated story.

然后亚历克斯就说，行了，我们搞定了。

But and then we and then Alex is like, all right, we got it.

我们搞定了。

We got it.

我们得把这个问题理清楚。

We got to sort this out.

我觉得打完那个电话后，我就立刻找你们俩谈了。

I think after that call, I jumped on both you guys.

我说，我们必须马上解决这个问题。

Was like, we got to we got to fix this now.

这已经是个问题了。

This is a problem.

你就是Alex。

And you're Alex.

我们开始对这件事做了一些研究。

We started doing some research into it.

我们请了一位数学家，专门分析过去的考试题，想搞清楚到底缺了什么——哪些知识点是学生根本没准备好应对的。

And, you know, we had a guy we had hired who mathematician who was really analyzing past tests and sort sort of trying to figure out like what what was missing, you know, like what what were topics that questions on these exams that students really would not be adequately prepared for.

这就像是在黑屋子里摸黑找东西一样。

It was sort of like a groping around in a dark room trying to find this stuff.

我们其实找了两个人来帮忙。

And we went through a couple of two guys, really.

工作量太大了，你根本没法全神贯注，毕竟你还要上统计课、机器学习课、街头数学课，还有其他一堆东西。

There's a lot of work and it was there was in getting out, you couldn't put your full attention because you had, you know, I think probably statistics course, machine learning course, the street math course, you know, all those others other things.

所以这门课在你清单上可能排到第四第五了。

So you're just this is just like course number four or five on your list.

整个研究项目就这么拖着慢慢推进。

And it was just sort of this research project was going on.

但到了某个时候，我们就意识到，得系统性地处理这个问题了。

But at a certain point, it was like, okay, we need to get systematic about this.

我们把SAT官方数据库里所有的1500道题都拿过来了。

And we took we took all 1,500 problems on the public database that the SAT what is it?

大学理事会公布的那些题，说这些是代表考试题型的1500道题，我们就全收了。

The the college board publishes say these are the 1,500 problems or whatever they are that are representative of the type of problems that are on the exam.

然后我们开始有条理地梳理这些题目。

And we we got systematic about it.

然后，亚历克斯，那时候我们开始意识到，这些题目到底是什么？这里是SAT里比较难的问题。

And then, Alex, that was in the time we were we started realizing, okay, like what what are these topics like here are the problems here are the harder questions of the SAT.

这些是我们手头材料里已经涵盖的问题。

Here are the here are the here are the questions that are on the in the in the material we have.

那中间缺了些什么呢？

What's what's in that gap?

我们开始把这种缺失的部分叫做‘中间空白’。

We started we started referring to that as the missing middle.

所以，亚历克斯，你和你的团队就开始深入研究这个问题——我们该怎么填补这个空白？

And so Alex, you and your team started really pushing on that, You know, how do we close that gap?

能不能说说这个过程？就是你们是怎么找到这个‘中间空白’的？

I mean, talk a little bit about that process that, you know, finding the missing middle?

是的。

Yeah.

当时我们采取了双管齐下的方法。

So that it was kind of like a double pronged approach.

当我们手头有了所有题库里的题目后，就知道得做个备考课程，一个涵盖SAT可能出现的所有题型的课程。

So the so when we had all the the kind of the the question bank questions, we knew we needed a test prep course, a course that basically had every single type of problem that could come up on SAT.

所以我们想，好吧，得把这些题目和对应的解法都准备好。

So we sort of, okay, we need those questions and we need solutions to those questions.

于是我就着手把符合我们标准的题目和解法一一录入数据库，方方面面都得做到位。

So I sort of set about the process of populating the database with those questions and solutions that are kind of done to our standard and things like that.

然后从这个角度去填补那个缺失的中间部分。

And kind of like fitting out the missing middle from there, from that end.

但只是为了说明一下，

But just to

我们并没有直接用那些题目。

be clear, we didn't use those questions.

我们只是拿它们当参考，来确定我们需要哪些类型的题目。

We just used them as a model for the kind of questions for a class of questions that we would need.

对吧？

Right?

没错。

Correct.

对。

Yeah.

就是这样。

That's that's right.

所以呢，就是说，好吧。

So it's just like, okay.

好吧。

Okay.

没错，就是这样。

This yeah.

完全正确。

Exactly.

而且，这些解法会把课时补全。

So and, you know, the solutions will complete the hours.

我们看了题干部分。

We looked at the question stems.

就是说，好吧。

It's like, okay.

我们通常会把这些题目消化吸收，变成我们自己的东西。

You know, we would we would talk make that our make them our own.

这些答案基本上都是我们团队自己写的。

The solutions were basically completely written by our team.

不过呢，我们开始往系统里填充这类备考题，用这种方式补上中间缺失的部分。

But yeah, so we started sort of populating the system with those kind of like test prep questions and kind of filling out the missing middle that way.

但后来贾斯汀得好好理清楚模拟考试的事，因为作为备考课程的一部分，我们需要高保真的模拟考卷。

But then Justin needed, he needed to get his arms around like the practice test because as part of the test prep course, we need high fidelity exams.

于是他仔细研究了模拟考题，非常细致地找出哪些是缺失的中间知识点，哪些也是缺失的中间知识点。

So he started looking at the practice tests and did a really, really, really thorough job of identifying, okay, this is a missing middle topic, that's a missing middle topic.

所以大概花了多久呢？大概一周左右，还是两周？

So over the course of about, was it about a week or so, two weeks?

你懂的，他就从考试的角度做了个空白点分析，把结果给我，说：好了。

You know, just did like a missing middle analysis from the point of view looking at the the the tests, the the exams, gave that to me and said, okay.

这个内容得加进课程里。

This this needs to be in the curriculum.

所以就这么定了。

So it's okay.

这算是第二阶段，因为第一阶段你只是看了那1500道题。

So That was phase two because you went through phase one because you were just looking at the 1,500 problems.

那时候你就在想，我们是不是快到位了？

You're like, I think we're and I would cure out of glass from like, are we are we close?

我的意思是，你觉得差多少？

I mean, how many you like?

我觉得大概还差十到二十个知识点吧，差不多了。

Well, I think we're maybe 10 more, 20 topics away or whatever.

我们正在逼近目标了。

Like, we're getting in.

我顺口跟贾斯汀提了一句，他却说：我不确定。

And I kind of mentioned it offhand to Justin and he's like, I don't know.

对吧？

Right?

然后呢，

And then that's

就是这样。

what like.

他之前跟我说过，听起来SAT课程下周就能结束了。

Had mentioned to me like, Hey, it sounds like the SAT course is gonna be done next week.

这完全是基于我辅导经验，对这门课该覆盖内容和实际考试之间差距的理解。

And that was just like my conception of like what this course needs to cover versus the actual exam just based on tutoring experience.

我当时觉得，咱们就差一周了。

Was like, we're one week away.

事情没那么简单。

Doesn't something does not.

听起来这方面内容应该再多一点。

It sounds like there should there should be more on it.

这就是我说的意思。

That that's what that's

正是这个让你开始关注我们。

what got you to start looking at us.

我会想要。

I will want.

好吧，咱们先看看一些模拟题，然后你去翻了所有的模拟题，接着又去找了那些公开的旧考题，还有别人凭记忆整理出来的题目，差不多就是这类东西。

Okay, well, we'll go take a look at some practice exams and see like and then you went and looked at all the practice exams and then you dug in and you went and looked for published exams or old exams and the cobbled together by people who remembered certain questions and then just kind of write or something like that.

SAT其实多年来一直发布过官方真题，都可以查到。

They have official, the SAT had released official exams for years past that you can access.

哦，原来如此。

Oh, okay.

就是那个加上模拟题。

It was that plus the practice exam.

所以题目范围非常广。

So give a huge range of questions.

是的。

Oh yeah.

这跟题库里的题目侧重点不一样，题型也不太一样，对吧？

Which sort of was a different emphasis or different types of problems than were just, that were on the question bank, right?

对。

Yeah.

结果发现题库的覆盖范围并不完整。

Turns out the question bank was incomplete in coverage.

没错。

Right.

没错。

Right.

所以，亚历克斯，你觉得漏掉了多少个中间知识点？

So so, Alex, how many topics do you say missing middle topics?

我说的那些缺失的中间内容，可不是只缺了一个知识点那么简单。

And and and when I say missing middle topics, it was not just, like, one topic bridge.

有时候得补上一整串知识点，才能把差距填上，因为很明显，SAT题目里最难的那些，有的地方我们离得近，有的地方离得远。

Sometimes it would be a whole sequence of topics to bridge the gap because it wasn't it was obviously so if you had the questions on the SAT, the hardest questions, it was sort of like we were closer and further on certain topics.

所以你得把这些缺口补起来。

And so you had to bridge that.

有时候只缺一两个知识点。

Sometimes it was one or two topics.

有时候则需要搭建一整套层层递进的支撑内容，学生才有可能在规定时间内解出题目。

Sometimes it was a whole sequence of a cluster of scaffolding that had to be built up for for a student would be able to have a reasonable chance of solving a problem in the time allotted.

对吧？

Right?

是的。

Yeah.

说到底，我们最后加了大约115个新知识点。

I mean, I think in the end, it end up being about 115 new topics is what we added.

很多呢。

A lot of them.

确实很多。

Is a lot.

是的。

Yes.

确实不少。

It is a lot.

最大的一块是描述性统计相关内容。

The biggest area was like the descriptive statistics stuff.

这部分内容挺多的。

It's quite a lot.

SAT考试里对各种数据集的分析要求其实挺高的。

It's not, you know, on the SAT, there's quite a lot of emphasis on, you know, sort of analyzing different types of data sets.

有些题目还挺有难度的。

And some of the problems are quite challenging.

而且我们还得补上大量中间缺失的内容。

And there was a lot of missing middle stuff that we needed to add for that.

我觉得，就算只看常规课程，我们的覆盖范围也不够，更别提那些了。

I don't think we really had enough coverage just even in even just considering the regular curriculum in in let alone the the the

但这个问题是普遍存在的。

But it was across the board.

这确实是我们的薄弱环节。

That was just our weakest area.

统计学部分明显是短板，但其他地方也到处都是问题。

Script of test statistics was a was a was a obvious a clear weak point, but it was, everywhere.

二次函数、分析二次函数、方程组、立体几何。

Quadratics, analyzing quadratics, systems, equations, lepto geometry.

给听众提个参考，115个知识点，对比一下我们代数一课程，大概有200个左右吧？

Just for for listeners, 115 topics, compare that to our our algebra one course has what, like, 200 ish topics?

就是

Like

差不多就是这样。

something like that.

220个。

220.

所以这差不多相当于一整年完整课程的一半内容。

So this is about half of like a full length, full school year course.

没错。

Right.

内容真不少。

That's a lot of stuff.

对。

Yeah.

所以说。

So right.

所以你是说，还有描述性统计。

So you're saying there's descriptive statistics.

还有些什么呢？

What else was

里面还有哪些内容？

in there?

我通常发现，很多内容其实都处在边缘地带。

That was So what I tended to find is that it was A lot of the stuff, it was kind of like on the periphery.

我说的是，描述性统计特别突出，因为那部分内容实在太多了。

I mean, the descriptive statistics stands out because it was just such a huge amount of stuff.

所以我们之前说题库差不多做完了，其实也就做到这儿为止了。

And that's kind of what we So when we said we were kind of done with the question bank, it was like, that's as far as we kind of got.

我们得弄大量这类内容。

It was like, we need a ton of this stuff.

因为涉及图表和类似的东西，我们需要很多不同的主题，那段时间主要精力都放在这些上面，先把它们搭起来。

And because it's graphs and tables and things like that, we need lots and lots of topics, kind of a lot of the attention was kind of focused on that for a little while, just kind of getting that in place.

除此之外，就只是一些我们已有内容的轻微扩展，差不多每样就加一两个小点。

Aside from that, it was kind of just stuff like almost like one or two topic extensions of things we already had.

绝大多数情况都是这样。

That was the vast majority of the cases.

你确实讲到了这部分，但讲得不够深入，让准备SAT的人一看题目根本不会觉得：哦，这个我懂。

Okay, you cover this, but you don't cover it to the point where someone who is using this to prepare for the SAT is gonna see that question go, yeah, oh, yeah.

我知道怎么解决这个问题。

I know how to do this.

你明白吧？

You know?

我有一些例子，是写下来的，我觉得挺有意思的。

I've got I've got some examples which are which are kinda written which are written down, which I thought were quite interesting.

比如系统里有很多关于指数模型的内容，但像时间单位的转换——比如从小时换到天，这会怎么影响模型，还有相关的百分比增减，这些方面就比较少。

So so things like we've a lot of stuff in the system on like exponential models, but things like sort of shifting units of time, you know, going from hours to days and things like how does that affect the model and percentage increase and decrease that are kind of related.

不过我觉得这方面确实是个重点，就这一块。

But that was that was like that was one area that I thought was yeah that was one.

对，这可不是简单的单位换算。

Yeah that's not a straight up unit conversion.

这是因为时间在指数模型的指数位置上。

That's like because the time is in the exponent of exponential model.

对吧？

Right?

所以不能只是简单地乘以365，以为一年就是这么算的。

So you have to it's not just oh just multiply this by like three 65 in a year.

不是的，你得用指数的运算法则来处理，因为这种情况比较复杂。

No, there's, you have to use exponent rules and stuff like that to actually make this conversion because it's in a kind of complicated setting.

对，就是这样。

Yes, exactly.

还有一种题型是SAT特有的，我以前从没见过别的地方有。

And, you know, think, and there's one which, there was one type of topic which was quite exclusive to the SAT.

我以前从没在其他地方见过这种题。

I've never seen it anywhere else before.

还是跟指数函数有关，你刚开始分析的时候管这叫‘谜题’是吧？

And again, exponential functions, you you called it riddles when you first did your analysis?

是啊，因为我当时没想出个好名字。

Yeah, because I didn't have a good name.

我当时就想，这到底是什么鬼东西？

I was like, what, what the hell is this?

谁会这么干啊？

Like, why would anyone okay.

行吧。

Fine.

好吧。

Alright.

这根本就是个脑筋急转弯。

This is just this is basically a riddle.

故意说得特别绕，就是为了让人犯迷糊。

It's just worded intentionally confusingly to try to throw somebody off.

不过话说回来。

But yeah.

所以，没错。

So so yes.

我真的很想保留这个名字，觉得它特别好，但当我们深入核心时，发现其实本质上就是把一个指数模型重新表达一下——比如在指数部分减去一，这会影响常数项。

So I I I really wanted to keep that name because I thought it was great, but I thought when we got kind of down to the crux of it, it's like, okay, what this really is is taking an exponential model and kind of rewriting it so that, you know, you subtract one in the exponent, but that affects the constant.

所以这个常数能反映出函数本身的某些特性。

So the constant can reveal different parts about the function itself.

我们当时确实没有充分探讨过这类问题：如何在不同类型的指数模型之间切换，以及常数是如何与模型本身关联、揭示模型特性的。

So, yeah, we didn't really have that, like, you know, sort of flipping between one type of exponential model with another and how that is related to the very How the constants are related to the model itself, what it reveals.

所以，这确实是个很专门的方向。

So yeah, that's quite niche.

而且考试里还出现了几道特别刁钻的题。

And there was some really nasty questions on the exam.

所以说，其实只是在指数函数的整个内容基础上，硬加了一两个知识点，确保学生能应对这类题。

So again, it was just kind of like, it's just one or two topics that had to be added onto all the exponential function stuff to make sure that they can do that.

但这个内容又必须得有，因为如果你第一次在考试里碰到，或者刚开始做模拟题就遇到这些，根本招架不住——这类题出现的频率太高了。

But it needed to be there because if you see that for the first time in the exam or you start doing practice exams, you're hit with these all the time, it's quite a frequent type of question.

如果你没得到明确的解释，根本搞不清到底在考什么，那这场考试肯定会特别难熬。

It's gonna be pretty brutal if you haven't had like explicit instructions to what the hell is actually going on.

那些二次函数的题目，课程里其实都涵盖了，比如求根、找顶点之类的。

If some of those quadratic ones, I mean, the curriculum had all the stuff about finding roots of quadratics, finding the sort of the vertex and all those kinds of things.

但你想想，又是一类典型题型。

But think about, so almost like, again, like type questions.

这些性质成立，那你能从中推断出系数之间有什么关系吗？

These properties are true, what does that tell you about the relationship between the coefficients?

所以这些内容都得加进去。

And so that all had to be included.

有些题目，说实话，我特别喜欢，比如我们加进来的这一类——我在SAT上见过很多次，题目会用一大段话描述某个物理现象，比如电通量密度之类的。

Some things like, you know, some questions, actually probably one of my favorites, think that we did add was, again, I saw it on the SAT quite a lot, which was like they give you quite a a verbose description of some sort of physical phenomenon like electric flux density or something like that.

但关键是，参加考试的学生未必是物理背景的。

And and, know, bottom line is that the kids taking this aren't necessarily physicists.

他们可能完全听不懂你在说什么。

They might not know have the faintest idea what the hell you're talking about.

所以呢，好吧。

So it's like, okay.

可能连一门物理课都没上过。

May not have even taken a single physics course.

通量？他们根本不知道这是啥。

Flex, they might have no idea what that even is.

没错。

Exactly.

所以我们得找个主题，能把这些物理概念讲给非物理专业的人听。

So it's like, okay, we need a topic that basically explains all this physics stuff to non physicists.

不是去解释什么是电通量密度。

Not the way to explain what electric flux density is.

关键是弄清楚题目想考察学生能做什么。

It's getting to the crux of what's this trying to ascertain the student is able to do.

就是把这段复杂的描述提炼出重点，从中列出公式，再做个单位检验，差不多就是个常识性验证。

It's like take this complex description, sort of highlight the really important parts, use that to write down a formula, do like a unit check, almost like a sanity check.

好，单位平衡了。

Okay, the unit's balanced.

这说明公式看起来没问题了。

That means that the formula looks like it's good to go.

然后稍微做点推导处理。

And then kind of do some kind of manipulation.

我们并不是想把每个人都培养成物理小专家。

So we're not trying to create like mini physics experts here.

就掌握这一个能力。

It's that one skill.

面对那种特别啰嗦的陌生概念，把关键信息提取出来，然后加以运用。

Take an unfamiliar concept that's presented in that very, very wordy kind of form, extract out the useful information and work with it.

其实有意思的是，很多学生根本没学过物理，对吧？

Well, you know, what's interesting about this is because right because because right, a lot of students don't take physics, right?

他们学的是生物和化学，可能上了两年化学，学了基础化学，还选了AP化学。

They take biology and chemistry and maybe take two years of chemistry, take basic chemistry and they take AP chemistry.

他们根本就没学过物理。

They don't even take physics.

所以从很多方面来说，这个问题其实不公平，对没学过物理的学生很不利。

And so it's sort of an unfair, it's unreasonable questions to ask in a lot of ways because it is a disadvantage to anyone who has happened to have taken physics.

但我觉得这更像是通过关联来评估。

And but I feel like it's sort of like a it's like a assessment by correlation.

学过物理、还能记得住的学生，通常能力更强，表现更好，分数也更高。

So students who've taken physics and can remember the physics are probably stronger students and they do better at this and they score.

你明白我的意思吧？

You know what mean?

这其实就是一种不公平的优势，你说对吧？

It's just kind of bullshit, you know, advantage that you're giving, you know.

是的。

Yeah.

而且这已经够了。

And it's enough.

这就像找借口，说他们根本不用懂物理。

It's like, can hide behind the idea that like, oh, well, they don't have to know physics.

关键是能从题目里拆解出一个陌生词，根据描述当场判断哪些信息是关键。

It's about parsing an unfamiliar word out of a problem statement and spinning, like just based on the description, figuring out what's relevant on the fly.

但说实话，如果你以前见过这个词，哪怕只做过一两道题，也不用精通，但只要有过接触，优势就大得多。

But yeah, the truth is, if you have seen this word before, if you have just a couple problems, you don't have to be an expert on it, but if you have exposure to it before, then you're at a big, big advantage.

如果你知道‘通量’这个词，其实就只是指穿过某个几何表面的力线数量。

If you know that flux, it really just comes down to the amount of force field lines going through geometric surface area.

你脑子里马上就会想，哦，这根本就是在考表面积，对吧？

Already have in your head, oh, this is really about surface area, right?

但要是从来没碰过这个词的人，就会想：通量？这跟长度、表面积、体积有什么关系？

Whereas somebody who's never ever seen that before, they're like, flux, What does that have to do with length, surface area, volume?

他们根本搞不清重点在哪里。

They don't even know what's important.

以前见过的话，就能马上反应过来。

Seeing it before kind of tips you off.

这能帮你善用这种抓住重点的策略，因为你脑子里的既有认知已经调整好了。

It helps you leverage this strategy of knowing what's important because your priors in your head are all shifted.

他们会

They're gonna

你如果不熟悉，真正会让你慌神的，就是突然冒出一句：天哪。

be you're not familiar, what's really gonna throw you off, gonna be like, oh crap.

对吧？

Right?

可能会有点措手不及。

Could freak out a little bit.

我是不是该懂什么是flux啊？

Like, am I supposed to know what flex is?

是的。

Yeah.

等等，我根本不知道flux是什么。

Wait, I don't know what flex is.

并不是说，我觉得大多数人会想，哦，这里要用点物理知识。

It's not like, and I don't think most people say, well, they're they're going to use some physics stuff here.

你根本无从知晓。

You're not gonna know.

我们希望你做的，就是跟着描述走，别多想。

And just what we want you to do is just add go with the description and don't think about it.

但我觉得，这并不是大多数学生预期的样子。

But that's I don't think what most students are expecting.

这真的不公平。

It's just it's just unfair.

所以，亚历克斯，你得在数学基础课里，专门设计一套小课程，涵盖这类题目。

So, Alex, you had to build up a whole curriculum, mini curriculum within the within the math with math fundamentals course, SAT math fundamentals course that gave a range of these types of problems.

这类题目其实有很多不同的变体。

There are lots of different versions of this.

对吧？

Right?

所以这不仅仅是密度或者通量这么简单。

So it wasn't just like, well, it's just density or just flux.

而是有各种各样的情况。

There were just different.

所以你得覆盖足够多的类型，让他们熟悉各种不同的词汇、术语、常数这些东西。

So you had to kind of do enough range so they could get comfortable with a lot of different words and and term terminology and constants and stuff like that.

对吧？

Right?

是的。

Yeah.

没错。

Yeah.

当然了。

Absolutely.

我的意思是，整个教学设计得非常细致，层层递进。

I mean, you know, again, like, very, very, very finely scaffolded.

所以，你能根据广泛的情境，看懂描述并写出对应的公式，这种能力本身就很关键。

So, you know, just just the the ability to kind of, like, given a wide range of context, just to do that, you know, read the description and write down appropriate formula.

接下来一步就是，好，你得先写出公式，然后还要会变形它。

And then the next step will be, okay, do that, but then you have to manipulate the formula.

再下一步呢，就是这种涉及变化率的题型。

And then the next step will be, okay, then we've got this one, which is kind of like a rate of change.

你得能一眼认出来，比如胡克定律。

You have to kind of spot that, you know, Hooke's law.

这跟弹簧的原始长度没关系，关键是弹簧的伸长量。

It's not about the length of the spring, it's the extension of the spring.

所以你得明白，这里说的其实是弹簧长度的变化量。

So you have to recognize that that is referring to like some change in the length of the spring.

所以我们当时选了几个主题，我觉得挺有效地填补了这个空缺。

So yeah, so we had a few topics that basically, I think did a pretty good job at kind of like closing the gap.

我觉得‘中间缺失环节’的设计目的就是这个——弥合常规课程和考试冲刺所需技能之间的差距。

And I think that's what the missing middle is designed to do exactly that, is to close the gap between a regular curriculum and like what is actually needed for the test prep, to be successful on the exam.

我本来正想说说这一点。

I was gonna say something about that now.

对，其实有两件事我想提一下。

Yeah, so it's a little bit like, yeah, yeah, oh, there's two things I was gonna say.

我觉得基础课和备考课之间的区别，有点像这么回事。

So the difference between the fundamentals course and the test prep course, I kind of imagine it a little bit like it.

想象你是个职业拳手，虽然没安排比赛，但拳击就是你的饭碗。

Imagine you're like a professional boxer and you know, you haven't got a fight lined up but you're a professional boxer, this is what you do for a living.

你得每天都去健身房，拼命训练，练步伐、练力量、练耐力，练刺拳、练所有出拳技巧，诸如此类。

You know, so you need to be sort of showing up to the gym every single day, training your ass off, you know, making sure you're working on your footwork, on your strength, on your endurance, on your jab, on your all your punches and stuff like that.

就是全面的身体素质和体能训练。

So just general all around conditioning and fitness.

基础课做的就是这个事。

That that's what the that's what the fundamentals course does.

它就是帮你达到那个状态，

It just kinda gets you to the point where you're

就是基本的拳击技巧。

in the fundamental boxing skills.

对吧？

Right?

基本的

The fundamental

拳击？没错。

boxing Absolutely.

基本的拳击技巧。

The fundamental boxing skills.

怎么移动、出拳这些，所有这些东西。

How to move and, you know, in punch and whatever, all that stuff.

他们练的都是这些技能，但并不是说，

These are a lot of skills that they're working on, but they're not they're not, what

你所谓的实战对练。

you call, sparring.

是的。

Yeah.

他们并不是针对某个特定对手进行训练。

They're not exactly training for a specific opponent.

所以呢，就是这么回事。

So it's like, okay.

对，就是这样。

So okay.

那到了第二天，你就会说，好吧。

So then then then, like, next day you go is, okay.

我们安排了一场拳赛。

We've got fight lined up.

有个选手来自X地。

There's a it's it's a guy who's from from X place.

他是左撇子。

He's a southpaw.

他个子很高。

He's tall.

你知道，他身体状态很好。

You know, he's he's he's fit.

他力量很强。

He's strong.

所以当你把基本功都练扎实了，接下来就是针对这个对手做准备了。

So you that that's the point where you kind of got all your fundamentals nailed and you're then preparing for that specific opponent.

这时候你得有针对性地加强某些方面，确保自己为这场对决做好充分准备。

So there's certain things you wanna sort of, like, lean into, focus more on just to make sure you're as prepared as you can be for that opponent.

所以这个对手呢，就像是

So that opponent It's like

策略。

strategy.

对吧？

A Right?

没错，就是这样。

So this is Exactly.

正是如此。

Exactly.

对。

Yeah.

如果你要对阵迈克·泰森，难道你连他以前的比赛录像都不看吗？

So if you're if you're going up against Mike Tyson, are you just gonna not look at his, like, previous fight videos?

你就只按平时的训练流程来，然后直接上场，好像随便谁走进拳台都一样？

You're just gonna do your normal workout routine and then just show up, like like, it could be anyone steps in the ring?

不可能。

Like, no.

你肯定得认真研究一下。

You're gonna actually, like, look.

他平时都怎么打？

What does he do?

他身体是怎么倾斜的？

How does he lean?

他出拳的时候，有没有哪些地方比其他对手更容易暴露弱点？

Where when he throws punches, are there any, like, weak points exposed more often than you would see in other opponents?

你得在进拳击台之前好好研究他，不然肯定会被打得惨不忍睹。

You're kinda studying him before you step into the ring with him because otherwise, you're just gonna get your ass kicked.

没错。

Right.

而且他也知道，你也在对他做同样的事。

And he and you know that he's doing that to you also.

对。

Right.

这在所有运动里都是如此。

This is true of all sports.

不管是大学还是职业比赛，他们都会反复分析录像，每周花好几个小时看对手的比赛录像。

So whether you're, you know, like you especially at college and professional where they break down tape, they spend hours and hours and hours a week watching tape of upcoming opponents.

对吧？

Right?

所以你知道，比如你在NFL打球，面对另一支美式橄榄球队时，就得琢磨清楚：他们的进攻风格是偏向冲球还是传球，有哪些强项和弱项，习惯怎么打。

And so, you know, when when, you know, if they're playing in the NFL, you're playing on another football team, American football, then you're trying to figure, okay, what we okay.

他们通常有特定的防守阵型，有擅长的战术，也有明显的短板。

They have they're a run first offense or they're a pass first offense, so they have these kind of secondary and they have strengths and weaknesses and things that they want to do.

那些曾经赢过他们、或者限制住他们发挥的球队，到底是怎么做到的？

Then teams that have put up either beat them or or taken away some things, how do they do it?

我们能从中学到什么？

What can we learn from that?

所以你说得完全对。

So you're really right.

你得提前为这支队伍做好准备。

You start preparing for that team.

最顶尖的运动员，其实都会自己主动这么做。

Now the the the the best athletes would do that on their own.

比如说，很久以前有个叫丹尼斯·罗德曼的，当年在底特律打球。

Like, there's I think a Dennis Rodman who played for Detroit back many, many years ago.

我不是医生。

I'm not doctor.

我们当年在底特律打过球，后来去了公牛队。

I play we did play for Detroit, then play for the Bulls.

他可是篮球界顶尖的篮板手。

And he was like this incredible rebounder in basketball.

对吧？

Right?

他不仅打NBA，还特别擅长这个。

And and he was that and played the NBA.

我好像看过报道，说他花成百上千个小时，专门研究其他高大球员是怎么抢篮板的。

And he I I think I read something where he would have been hours and hours and hours watching how these other big men from these other teams would rebound.

他还研究那些射手投篮时，球打在篮板上的反弹轨迹。

And he would also have where were certain shooters when they would shoot where their ball would go tend to go off the glass.

所以他就会站在篮下，因为当那个人投篮时，球往往偏短，容易掉在篮筐前面。

So he would position himself under the basket because there's a bigger tendency when this guy shoots, it goes short and it tends to fall short.

所以我要提前站到那个位置，或者球如果偏长的话，我也能抢到。

So I wanna be in that or tends to go long.

球要是打到篮筐侧面弹出来，我也能占到好位置。

It's a bounce at the side.

他就是靠这些细节来提升自己的表现。

So he would do all these things.

他会研究对方其他大个子球员是怎么站位的，这样自己就能占据更有利的位置。

He'd learn how the other big men and the other team would position themselves so he would be in a in a have a more leveraged position.

他还会摸清每个射手的习惯，知道他们投出的球大概会往哪个方向弹，把这些小优势加起来，他就成了史上最强的篮板手之一。

And then he would understand also the tendency of whoever shooting where the ball would tend to And so you add up all these little advantages, and then he was like this unbelievably good rebounder.

不过话说回来，这种道理在所有运动里都一样。

But anyway, so same goes in all sports.

就是这么个道理。

It's like, okay.

先把基本功打扎实。

You get your fundamentals in place.

现在该制定策略、规划安排，开始做那些能让你在比赛当天发挥出最佳水平的事了。

Now it's time to game plan, strategize, come up with, start and doing the things you need to do to maximize your performance on the day of the event.

是的。

Yeah.

所以说，这其实挺有意思的。

So, well, you know, it's interesting.

不过我有件事想说一下。

One thing I wanted to say though.

我觉得这种思维方式真的很棒。

So I think it's a really cool way of thinking about it.

我以前还真没这么想过，但我记得我曾经用过一个比喻，我老喜欢用比喻。

I hadn't really thought about it in that way, but I remember one analogy I used that I sometimes I think of analogies.

我总是会想到类似故事一样的东西。

I I always think of almost like stories.

我其实看了一部电影。

I actually watched a movie.

对吧？

Right?

我当时在谈中间缺失的问题，就说，因为我们先得讲清楚这个。

And so I was talking about the missing middle and I said, imagine because we're first talking about this.

我说，想象一下，你请了一位每小时200美元的SAT家教。

I said, okay, imagine that you hired this really high priced SAT tutor.

价格高得离谱。

$200 an hour, some crazy amount.

而且这人真能出成绩。

And and this guy delivers results.

大家都说，你找这位老师，帮你孩子辅导一下，效果特别好，于是你去见他，心里想，你到底有啥绝活？

And people say, you know, you can hire this guy, work with my son or with my daughter and had this incredible results, you know, and so you and so you you meet this guy and you're like, so what what do you do that's so special?

他却说，唉，我干这行很久了。

And he's like, well, you know, I've done this for a long time.

我分析了所有的考试题。

I've analyzed all the tests.

我先仔细研究了这些学生的情况，给他们做了非常深入的评估。

I analyzed where these students first, I give these kids students have really in-depth assessments.

我真正了解他们的优势和短板。

I really understand their strengths and weaknesses.

所以我们第一步，就是先把这些问题补上来。

And so the first thing we do is remediate all those things.

这当然是他们该做的。

That's obviously what they would do.

但我做的另一件事是，我发现SAT里那些难题和学生在学校里学的内容之间，往往缺了很多关键技能。

And then what I do is I've developed there's typically a lot of missing skills between the harder questions on the SAT and then what students have covered in school.

过去十年里，我自己开发了一套教学材料，专门针对学生需要掌握却没学到位的地方，帮他们一步步提升。

And I've developed my own materials over the past ten years, you know, where I have, you know, that I have teaching students, okay, they need this and they got to do this, I'm gonna build on that.

我把这些关键内容叫做我的‘核心考点’。

And so I call my golden topics.

我有整套方法，你作为家长坐那儿，心里肯定想，这太厉害了。

I have this whole thing, you and know, you're sitting there as a parent.

哇，真是没想到。

You go, wow.

这些所谓的黄金知识点。

The golden topics.

到底什么是这些黄金知识点呢？

Like, what are those golden topics?

他就会拿出一个小册子，说：看，我这儿都整理好了。

And he's like, I know he pulls out his, like, a little and he has, like, a some booklet that he's put together and say, okay.

我们发现，像指数这类内容，还有那些常考的题型，他全都有系统梳理。

We see we cover all this exponential stuff and the scripted that, you know, he's like, just facts what you're talking about.

所以我们得找出这些黄金知识点，就是学校课程里没教到的部分。

And so we gotta come up with this these golden topics that that is sort of this this area of these things are not covered.

这些内容根本不在常规教学范围内。

They're not covered in the normal curriculum.