Math, Democracy, Equality, and Classroom Culture

This is a contribution to Sam Shah‘s Virtual Conference on Humanizing Mathematics.

As a secondary matter, it fits into my series of posts exploring the relationship of math to democracy.

One aspect of this exploration has been experimenting with explicitly framing mathematical knowledge building with students as a democratic process, analogous to being part of a democratic polity. In a democracy, at least according to the ideal, the direction of the polity is determined by its members, all having an equal say. In the same way, I’ve been striving to build a way of working with students in which they see the knowledge as determined by themselves engaged in a collective process in which they are all equal participants, substantially inspired by Jason Cushner and Sarah Bertucci’s Consensus Is the Answer Key.[1]

My interest is in having students walk away from mathematical experiences knowing that math is nothing more mysterious than communities of humans trying to figure things out together; that the process that led to all mathematical knowledge is something they, and anyone, can participate in; that they can be the authors of such knowledge. That they are entitled to a say in what the community they are part of believes about math, and that their own sense of what to believe benefits by being part of a community thoughtfully working together to try to figure things out.

I hope these overall goals give you a sense of why I wanted to write about this as part of the Virtual Conference on Humanizing Mathematics: while math is often seen as some kind of disembodied and strangely history-less ancient wisdom handed down by specially-anointed priests (“math teachers”), themselves entrusted with it by an even higher priesthood (“mathematicians”), the truth is that it is nothing but the product of humans trying to figure things out together, and I want this to be what students experience it as.

While I do plan in the future on writing about the specific instructional protocols I’ve been exploring to accomplish this, I’m going to keep the scope limited here, and tell you just one story, about a time when the frame of “democracy” unexpectedly gave me a new resource in handling a situation to do with classroom culture that I priorly would have found challenging.

In 2017 I tried my first experiment building a whole learning experience around the math-knowledge-building-as-a-democratic-process metaphor. (It was a course at BEAM.) On the first day I explained that mathematical knowledge is democratic in character[2] and that they would be working democratically as a community to decide what’s true. They bought in.

They were working through something, I no longer remember what. C asked a question or made an argument and A replied. A’s reply was mathematical and on topic, but his tone was a little condescending. Just a little, but it was there.

This is a type of situation in which I’ve historically found it a little hard to exercise my authority to move the classroom culture in a positive direction. I don’t want a room in which it’s okay for people to be condescending to each other. That’s a recipe for the class to start to feel emotionally unsafe. On the other hand, I’ve often had trouble finding a way of intervening in this type of situation that would have felt fair to A. If I said, “that was disrespectful,” well, perhaps it was, a little, but it was also on topic and advanced the conversation, and, well, “disrespectful” is a powerful word. Furthermore, this intervention would not have been very actionable for A: the thing I didn’t like was not located in his choice of words, but in a subtle tone thing. If he felt defensive at all (and who wouldn’t?), it would be difficult for him even to perceive what he had done that was being criticized; how would he correct it?

I think some teachers deftly handle this type of situation using light-touch humor, but that has always been a difficult tool for me to wield when being corrective. I’m too earnest; it’s hard for me to get that dial just right.

I’ve found myself in situations like this countless times, but this was the first time I had encountered it while teaching a class that had explicitly bought into the idea that they were a democratic community. I found myself, quite to my surprise, with a confident new move:

“A, you’re saying something very interesting, but your tone of voice is a little like you’re the teacher and C is the student. In a democracy, you’re equals. So can you try making that exact same interesting point except from one equal to another?”

And what was beautiful was that he completely, happily, undefensively took it on. In fact, he seemed excited to try. And, he did it! He said the exact same thing, except from one equal to another. The conversation proceeded with a new foundation of safety and mutual respect established.

I knew I wanted to teach them that math was something humans make by coming together as equals and trying to figure stuff out together. I didn’t know this would also give me new moves to support the development of a healthy mathematical culture. Retrospectively, maybe I should have.

Notes

[1] Sarah wrote an essay on this pedagogical principle which unfortunately has never been published, but the link above is a nice description of a session she and Jason and their students led at the Creating Balance conference in 2008.

I’ve been working to develop a community of educators interested in this “math-as-democracy” pedagogy. I facilitated a minicourse and a professional learning team at Math for America this past year on the subject, and James Cleveland, who was part of both, led a session at TMCNYC19. This fall I am co-facilitating another professional learning team on instructional routines, one of which is democracy-focused. If you’re interested in thinking about this circle of ideas with me, get in touch!

[2] I explained the underlying philosophy here, and also see the first minute or so of my TED talk.

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I Just Started a Math Blog!

Hey y’all, I just started a new blog for completely random thoughts about math. I just figured something out yesterday morning that would probably never go in a paper or anything, but I wanted to record it somehow, and why not publicly? So, new blog for that kind of thing:

Every Single Problem

It’s gonna be pretty math-ola. Enjoy!

p.s. No promises it will ever contain more than this single post! But, it might!

Re-invitation

Dylan Kane’s recent post about prerequisite knowledge has me wanting to tell you a story from my very first year in my very first full-time classroom job, which I think I’ve never related on this blog before, although I’ve told it IRL many times.

It was the 2001-2002 school year. I taught four sections of Algebra I. I was creating my whole curriculum from scratch as the school year progressed, because the textbook I had (the UChicago book) wasn’t working in my classes, or really I guess I wasn’t figuring out how to make it work. Late late in the year, end of May/early June, I threw in a 2-week unit on the symmetry group of the equilateral triangle. I had myself only learned this content the prior year, in a graduate abstract algebra course that the liaison from the math department to the ed department had required of me in order to sign off on my teaching degree, since I hadn’t been a math major. (Aside: that course changed my life. I now have a PhD in algebra. But that’s another story for another time.)

Since it was an Algebra I class, the cool tie-in was that you can solve equations in the group, exactly in the way that you solve simple equations with numbers. So, I introduced them to the group, showed them how to construct its Cayley table, and had them solving equations in there. There was also a little art project with tracing paper where they drew something and then acted on it with the group, so that the union of the images under the action had the triangle’s symmetry. Overall, the students found the unit challenging, since the idea of composing transformations is a profound abstraction.

In subsequent years, I mapped out the whole course in more detail beforehand, and once I introduced that level of detail into my planning I never felt I could afford the time to do this barely-curricular-if-awesome unit. But something happened, when I did it that first and only time, that stuck with me ever since.

I had a student, let’s call her J, who was one of the worst-performing (qua academic performace) students I ever taught. Going into the unit on the symmetry group, she had never done any homework and practically never broke 20% on any assessment.

It looked from my angle like she was just choosing not to even try. She was my advisee in addition to my Algebra I student, so I did a lot of pleading with her, and bemoaning the situation to her parents, but nothing changed.

Until my little abstract algebra mini-unit! From the first (daily) homework assignment on the symmetry group, she did everything. Perfectly. There were two quizzes; she aced both of them. Across 4 sections of Algebra I, for that brief two-week period, she was one of the most successful students. Her art project was cool too. As I said, this was work that many students found quite challenging; she ate it up.

Then the unit ended and she went back to the type of performance that had characterized her work all year till then.

I lavished delight and appreciation on her for her work during that two weeks. I could never get a satisfying answer from her about why she couldn’t even try the rest of the time. But my best guess is this:

That unit, on some profound mathematics they don’t even usually tell you about unless you major in math in college, was the single solitary piece of curriculum in the entire school year that did not tap the students’ knowledge of arithmetic. Could it be that J was shut out of the curriculum by arithmetic? And when I presented her with an opportunity to stretch her mind around

  • composition of transformations,
  • formal properties of binary operations,
  • and a deep analogy between transformations and numbers,

but not to

  • do any +,-,*,/ of any numbers bigger than 3,

she jumped on it?

Is mathematics fundamentally sequential, or do we just choose to make it so? I wonder what a school math curriculum would look like if it were designed to minimize the impact of prerequisite knowledge, to help every concept feel accessible to every student. – Dylan Kane

Acknowledgement: I’ve framed this post under the title “Re-invitation”. I’m not 100% sure but I believe I got this word from the Illustrative Mathematics curriculum, which is deliberately structured to allow students to enter and participate in the math of each unit and each lesson without mastery over the “prerequisites”. For example, there is a “preassessment” before every unit, but even if you bomb the preassessment, you will still be able to participate in the unit’s first few lessons.

A Thought on First Days

One of the ideas I’ve encountered in my wanderings that has ultimately been most useful to me in shaping my teaching is about the needs of students on day 1. It’s this:

Students come to the first day of class with a number of important questions. They almost never ask you these questions out loud, and they are often at most barely conscious of them. But how you respond to these questions will have a very significant impact on how the class goes.

The questions are things like:

Do you know a lot about this subject?

Can you teach me effectively?

Will I feel safe and supported here?

Do you believe in me?

Different students have different questions, and it often happens that an effective way to respond to one student’s question is an ineffective answer to another’s. Nonetheless, it’s not hopeless to try to figure out something useful about what questions are dominant in a given class and how to respond to them effectively.

I forget where I first heard this idea. I remember thinking about it a lot during my 4th year in the classroom, in conversation with a particular colleague I’ll call Leslie.

In those long-ago days, I taught:

* Algebra I to 9th graders
* Algebra II to 11th and 12th graders
* AP Calculus AB to mostly 12th graders

I struggled a lot with classroom management with the 9th graders. I almost never had any management problems with the 11th or 12th graders. This was not about “strong” vs. “weak” students: on average, the Algebra II kids were the “weakest.” My in-the-trenches conclusion was that 9th graders are just hard.

Leslie was a history teacher. Like me she taught mostly 9th graders and 12th graders. I was extremely surprised when she told me that she got along great with the 9th graders and was in an epic struggle with the 12th graders.

She eventually resolved it, but I remember being extremely confused and curious when she first told me about the difficulty. Twelfth graders, acting like that? I don’t remember what I asked or what she said. But my takeaway was something like this:

“I like math; I know a lot of math; I work very hard to make lessons clear, creative and engaging. I’m curious about kids and excited about their thoughts, and I will spend a lot of extra time with you to try to understand your mind and help you understand the content. On the other hand, I do not like it when students don’t cooperate with my plans or engage with the lesson I worked so hard on, and I wish they would just cooperate and engage.”

“9th graders are developmentally different from adults. Though they are anxious to be seen as grown-up, they still find it difficult to self-regulate their emotions. In this context, a family of questions they have for their teacher on day 1 is, ‘how will you help me stay focused when I find this difficult? how will you help me self-regulate? will you keep us all safe from undue disruption stemming from ourselves’ and each others’ difficult feelings?’

“I have up to now been bad at responding effectively to this suite of questions. I have resented and wished-would-go-away the part of my job that is about helping students stay in control of themselves. I am sure the 9th graders sense the implied power vacuum. They probably find it terrifying. They want to know class will be happy and productive, and they find out the answer is, ‘only if I, and all of my peers, simultaneously, spontaneously stay focused and positive for the whole period.’ Yeah right.

“Meanwhile, Leslie understands and enjoys this part of her job. Her 9th graders relax quickly as they learn what she is willing to do, happily, to make sure they as a community stay their best, most productive selves.

“On the other hand, 12th graders are much closer to being adults. They self-regulate much more easily. They don’t need you to prove to them what you can do to help them with that. On the other hand, they are anxious to know that you are not on a power trip and that their time won’t be wasted.

“In this context, the deal I was subconsciously offering — I know this stuff really well and I’ll work really hard to help you learn it; I won’t condescend to you about how to act, but I need you to cooperate and engage without much structural help from me — actually probably sounded like a great deal to 12th graders. They were ready to do the self-regulating without me, and I probably implicitly answer the questions ‘do you know your sh*t?’ and ‘can you help me learn it?’ very quickly in the affirmative. That explains why their affect was always like, ‘ok, cool, let’s go.’

“On the other hand, from what Leslie is telling me, she did not successfully reassure her 12th graders that she knows her sh*t early on. She does in fact know her sh*t, but somehow they didn’t get that sense at the beginning, and eventually went into open rebellion. Probably sexism was involved; who knows what the whole story is. But, for whatever reason, that question did not get successfully answered, and it led to a big problem.”

I have no idea if any of that is the truth. But it seemed to explain the puzzle to me, to fit my experience and my colleague’s story, and has shaped my thinking about what needs to happen on day 1 ever since.

All of this was at the front of my mind not too long ago when I started a new class in a new context. It was an advanced college level math course, and I had been told that the students had taken a full sequence of prerequisite courses but that their grounding in that content was uneven. Having been told this, it was hard to plan anything and feel confident it would be appropriate. Would it be too easy and they’d feel condescended to? Too hard and they’d be lost? I was really stuck on this.

I reached out to the students for info: “what do you know about X subject?” My first inquiry went unanswered for weeks. I followed up. One of them said, “To answer your question I’d need to look at my previous syllabi.” I asked an administrator for help with this and they turned up several syllabi. A few more days went by with no word, so I followed up again. A second student wrote back: “Every professor has a different idea about these courses. Maybe if you tell us what you want us to know, we can tell you if we know it.” I replied, listing specific topics. Nothing, for a few more days. With the class beginning the next day, I wrote one last time: “Now’s your last chance to tell me something about what you know before we get going. Can you reply to that list I sent before?”

Another student wrote back to the effect of, “Look, we have taken numerous classes before. Nothing on this list is foreign to us. Our mastery over specifics will vary from topic to topic.”

This email told me so much. I mean, it told me almost nothing in terms of their actual background — how that mastery varies from topic to topic was exactly what I had been asking. And yet, it told me just what I needed to know to make the planning decisions that had been tripping me up.

These folks need to know I stand ready to challenge them!

Underneath that, I supposed they might be anxious to know I planned to take their minds seriously. And my attempts to get some orientation for myself could have exacerbated that anxiety! My “are you familiar with X?” questions had all been about content they were supposed to have seen before! If indeed they were concerned I might not think they were up to a challenge, perhaps these questions had fed that concern. (This would at least be a plausible explanation for their slow and unforthcoming responses.)

So, I felt I knew what question I had to answer on day 1. I put together a lecture full of rich, hard content, outlining a grand sweep for the whole semester. I erred on the side of more and grander content. During class itself, I erred on the side of telling them more stuff, rather than probing what they were making of it. I wanted the experience to say, “I know you are not here to play, and neither am I. We are going to go as far as you’re ready to. Maybe farther.”

At the end of class, I mentioned to the student who’d sent the email that I’d enjoyed its tone of “c’mon now, bring it!” He smiled, like, “yeah, you know it.”

The course is behind us now. In fact, that first day was the fastest, most content-packed day of the class. It is not generally my style to construct class in a way that pushes forward without much information about what sense the students are making of the ideas. Once the students became willing to show me what they actually knew and didn’t know, it was possible to properly tailor the course, and we were able to drill down on key points and really get into what they were thinking. To be clear, it didn’t get any easier — I would say it actually got harder and harder over the course of the semester. By the end we were line-by-line in the thick of intricate, pages-long proofs. But we never again zoomed forward at the breakneck pace of that first day.

That said, with hopefully due respect to the fact that I haven’t had this conversation with the students directly, I do believe it was the right choice for day 1. A different first class might have been a little closer to what the rest of the semester would look like minute by minute, but it wouldn’t have spoken to the question I believed then and still believe that my students really needed answered.

By the same token, for different students, it could have been exactly the wrong choice. If my students’ incoming burning question had been, “are you willing to meet me where I am?,” then that first lesson could have come across like, “no, not even a little bit,” and we might have had a real long semester. And I honestly did not know which question my students had! This is why I’m grateful to the one who emailed me to say, “Look, we’ve done a lot.” That told me what I needed to know.

Math Is Democracy III: A Short Rant about Voting Theory

I have been full tilt exploring the relationship between mathematics and democracy for the last at least year and a half. E.g. check out my TED talk.

And I’ve been thinking about this relationship, in some form, for the last 8 or 9 years, ever since Jay Gillen said to me, “Everything you love about math is what free people love about democracy.” (See this previous post.)

Given this, it may be surprising that until fairly recently, the well-established mathematical subfield that explicitly addresses democracy, voting theory (aka social choice theory), never grabbed my interest. FWIW, this has changed, but I think my journey around it is indicative of something worth keeping in mind for mathematical people who want to use math to think about how society works. (E.g., me, and hopefully you.)

Voting theory asks: given a large number of people forming a collective (e.g. a nation or state!), and a choice the collective is faced with (e.g. electing a political leader!), what are the possible ways one could aggregate individual preferences into a collective decision? (E.g. everybody vote for one candidate, and the candidate with the most votes is elected, aka plurality vote, aka how most political elections work in practice; but other possibilities too, like instant runoff, the Borda count, approval voting, and score voting.) And what are their properties? (E.g. in plurality voting, there is a potential spoiler effect when similar candidates split the vote of the majority, which is mitigated in approval and score voting since they do not force voters to choose only one candidate to support.)

It’s sort of a puzzle: given how much I’ve always cared about math, and how much I’ve always cared about democracy, why wouldn’t I want to understand this developed and beautiful theory exploring this most fundamental democratic operation — how does a collective make a decision?

Well, I’ll tell you.

The first and loudest thing I heard about voting theory, beyond just what it’s about, was Arrow’s impossibility theorem. This is a “foundational” result in the theory. It is often glossed as asserting that “there is no perfect voting system.” (Do not take this interpretation at face value. More below.)

From this, I got the impression that the point of voting theory was for us to brain out about the possibilities, and then conclude (with mathematical certainty!) that “we the people” can’t win no matter what. I’m supposed to be excited about this?

Was the whole thing just an intellectual exercise? Many years ago, I listened to a mathematician excitedly describe the Borda count to some graduate students. I remember thinking, with some confusion, “Where is the excitement coming from?” It seemed to me that the mathematician was excited only intellectually, about the mathematical properties of this system, and was totally disengaged from the question of whether anybody had used it or would use it or should use it to make any collective decision.

In this context, voting theory seemed almost grotesque to me. I love math for math’s sake — my PhD work is in pure algebra. But taking the urgent moral and political problem of building a system of government that adequately reflects the will of the people, and using it as loose inspiration for some math for math’s sake — ick.

Now, I take full responsibility for this impression. Many practitioners of voting theory believe the system can be radically improved, and are much more engaged with the realities of elections than I was giving them credit for. Steven Brams, now a mentor of mine, was involved in the adoption of approval voting for leadership elections in several professional societies. (Steve’s attempts to push something similar with political elections have mostly been frustrated so far, although advocacy by the Center for Election Science, an organization Steve has advised, played a role in the recent adoption of approval voting by Fargo, ND.) And Warren Smith has been developing a website to spread information about score voting since 2005. Just as two examples.

Now that I understand that voting theory is meaningfully engaged with actual elections, I’ve done a 180, and I’ve been reading up on both the foundations and recent research, and am looking for ways to contribute myself.

That said, I retain the feeling (with hopefully due humility that I could be wrong again) that the field would benefit from greater accountability to the problems it’s trying to speak to. The imperatives of mathematical exploration are different than the imperative of improving the mechanisms by which the will of the people expresses itself, and these differing imperatives can be in tension.

I’m reading one of Donald Saari’s books on voting theory. Saari has developed a beautiful, geometric way of looking at election systems that offers great clarity in making sense of some of the counterintuitive things that can happen. This is beautiful and important math. He also tends to portray the Borda count as kind of the best voting system, and I cannot shake the feeling that this is more because of the mathematical beauty of its properties, when viewed through a particular mathematical lens, than a sober assessment of its value to society.

So. I want to see and be part of a voting theory that is responsive to the problem it involves itself in. We can talk about voting systems using the tools of academic discourse, but we must remember the stakes. This is not an academic exercise.

Appendix

I promised above a little explanation of my comment that Arrow’s theorem should not be taken as stating that “there is no perfect voting system”. Lemme get into that for a sec. Arrow’s theorem says a voting system of a very specific type cannot obey a certain very specific list of prima facie desirable properties all at once. My basic objection is that the theorem is too specific for this broad an interpretation. My more fine-grained caveat is that there is nothing God-given about the particular set of desirable features Arrow chose to analyze, so deviation from them is just one definition of “imperfection”, and there are plenty of others.

In either case, I don’t blame Arrow for this, but I definitely thought I was being told that making the system better isn’t on the table. My real motivation here is to disabuse you of that misimpression if you had it too.

Measuring Gerrymandering

I gave a talk at MoMath back in early May, about mathematical efforts to measure gerrymandering, culminating with recent very exciting efforts such as those of the Duke Quantifying Gerrymandering group.

It’s online! Check it out!

Acknowledgement: this talk owes a lot to the conferences organized this past academic year by the Tufts Metric Geometry and Gerrymandering Group, and especially to Mira Bernstein.

Math is Democracy II: Math is Democracy!

I announced a series on math and democracy back in October.

It will deal with a lot of concrete areas. Last time I talked about a case that is before the Supreme Court and will influence voting law throughout the land. In the future I’ll be talking about voting, political participation, technology and who has a say over its development, and of course the classroom.

But I want to properly kick things off with a post that is essentially philosophical. I am here to assert the following proposition:

Math is democracy!

What do I mean?

Democracy — from Greek — literally, “rule by the people.” I am referring to the ideal itself, not any particular system of government. Throughout the world we have various systems attempting to implement this ideal. One can ask questions about the degree of success of these attempts, but that’s not what this post is about. I’m just isolating the ideal — democracy — rule by the people.

Mathematics — from Greek — literally, “learning.” Of all the domains of human inquiry, math occupies a privileged place in terms of our confidence in its conclusions. It is the only field where practitioners regularly express unqualified certainty about its results. We sometimes discuss the wisdom it gives us as some sort of celestial gift (as in Wigner’s classic essay on its applicability to the sciences).

I am about to draw a connection. I expect it is still opaque at this point, but hang on.

If math is a miracle, then there is a second miracle: the divine gift was implanted in each of us, since it springs solely from the universal human capacity for rational thought. The wisdom of mathematics was not given us by way of Mt. Sinai, handed down from on high by somebody with privileged access to The Boss. Although many people think back to childhood and recall inscrutable formulas dispensed by a teacher who mysteriously knew the answer (how did they know??), this memory conceals the real truth, which is that the only place mathematical knowledge comes from is a community of peers reaching some kind of consensus after a period of engaged discussion. Furthermore, at least in principle (if not always in practice), anybody in this community has the right at any time to raise good-faith questions about the logic underlying any of our mathematical knowledge, and the matter is not really settled unless these questions have a good answer.

Thus, the only true source of mathematical authority is the consensus of a community of equals.

The principle of democracy is that this is also the only true source of legitimate political authority.

Broadening further, I offer that the principle of democracy holds that the only source of authority (of any kind) over a community is consensus of that community. So math is literally democracy.

Addendum 3/29/18:

This is edited from the version I posted yesterday, where I used the phrase “functional consensus” instead of “consensus.” This was to acknowledge that in a large-scale community such as a nation, or the international community of mathematics researchers, true consensus is not a viable goal. That said, the “functional” didn’t sit well with me overnight, because I thought it could be taken to suggest some sort of majoritarian principle. To me, majoritarianism is a fatal compromise of the principle of democracy articulated here, and it defeats the purpose of the analogy with math.

The thing about math is that, in principle, if an objection is raised to what is regarded as established fact, then that objection needs to be dealt with. Maybe something was overlooked! In actual practice, it may or may not be, because the question of whether you can get people to pay attention to your objection depends on things like if you’re famous, if you’re well-connected, how much work other people have to do to understand it, etc. But mathematicians’ collective understanding of what we’re doing holds that if somebody raises a new objection to something thought to be well-established, we have to answer it, not ignore it, in order to hold onto the established knowledge. This ideal isn’t attained, but it is still how we think about it.

By the same token, it seems to me that the democratic ideal insists that a minority view has the right to be processed rigorously by the community. I am making a high-level analogy so I’m not getting into what that processing might look like. But the failure of a community to take into account minority constituencies in some way is a failure of democracy.

Addendum 3/31/18:

I want to acknowledge some intellectual debt!

In 2008, I went to the Creating Balance in an Unjust World conference and saw a presentation by Sarah Bertucci, Jason Cushner, and several of their current and former students, entitled Consensus is the Answer Key: Empowerment in the Math Classroom. The presentation was on using consensus as the source of mathematical knowledge in the classroom. Later (in 2009?), I visited the school in Vermont where Jason and Sarah were then teaching, and saw Jason’s class. (Random aside: I also met Jasmine Walker!) The ideas have shaped how I saw both mathematics and the classroom ever since. You can see their clear imprint above (and in many of the things I’ve written on this blog over the years).

In about 2010, I was having a conversation with Jay Gillen of the Baltimore Algebra Project. At the time, I was preparing to apply to graduate school in math. Jay asked me many questions about how I thought about the math classroom and the subject itself. At some point he paused and said, “Everything you love about math is what free people love about democracy.” This comment has been continuously blowing my mind for 8 years, and again you can see its clear imprint in the above.