“How many gas stations do you think we’ll pass on the way home?”" “You mean on the side of the road, or at exits?”" “Whichever – places where we’d consider buying gas.”" “I don’t know, six, eight why?”" “Well, if we pass the first 1/e gas stations and then stop at the first place that offers cheaper gas, then we’ll maximize the chance that we’ll end up buying the cheapest gas. I mean, assuming that gas prices are random and don’t follow any trend with regards to location, which obviously isn’t the case.”" “Oh. Ha! But that’s not the algorithm we want – that one maximizes the probability of getting the cheapest gas; it doesn’t minimize the expected price of the gas we buy, which is what we want.”" “Oh, yeah, you’re right: if we apply that first algorithm, then in the likely event that we don’t get the very cheapest gas, there’s still a reasonable chance that we’ll end up with pretty expensive gas.”" “Right.” “So what’s the strategy for minimizing the expected price of gas?” “I don’t know.” “So, is there a probabilist in the house?
Driving with mathematicians
Mar 12
When I was a kid, my father told me about a strange phenomenon, which was later explained to him, that he’d observed while eating on airplanes. Back in the seventies and early eighties, Dad travelled a lot on business, and he noticed that the slices of hard-boiled eggs he received never included any ends. Each egg slice that made its way onto his tray consisted of yellow and white concentric circles – and, even more curiously, they all appeared to be congruent. At first, my father chalked this up to coincidence – maybe he just always happened to get the middles of the eggs? – but soon, he noticed that his seatmates also received only egg middles. It was nearly a statistical impossibility that such a large random sample of egg slices would never contain the end pieces. Were the airline chefs throwing out the yolkless ends? It didn’t make any sense.”Dad’s question was answered some time later, when he had the opportunity to watch a video on the production of airline food. Included was a segment specifically devoted to the preparation of eggs. At last, a chance to settle this question!”The documentary showed a huge, industrial-sized kitchen, where several chefs were diligently cracking eggs, separating the whites from the yolks. One large mixer processed hundreds of egg yolks; a second contained the whites. The chefs then poured the yolks into a long, hollow tube, where they were boiled. Then they formed the whites into a sheet half an inch or so thick, where they were hardened just enough to be moved without collapsing. Then, the whites were positioned to surround the stick of yolk. The entire apparatus was cooked one final time, resulting in a symmetric, several-foot-long cylinder of hard-boiled egg – which a machine then sliced into the discs that would be served to airline passengers.”This is why my father never got the ends of eggs.”In related news, friend and reader oxeador sent me a link to The Muddle Machine, a textbook editor’s firsthand account of why elementary and high school texts used in the United States are bland, incoherent, expensive, and updated with every new phase of the moon – even as they offer little new content with each edition. Tamim Ansary has written an appalling exposition of how a lot of bureaucracy, a lot of money, a lot of politics, and hardly any pedagogical or subject expertise has given rise to books that serve their creators and their financial backers rather than the students and teachers that use them.” I got a hint of things to come when I overheard my boss lamenting, “The books are done and we still don’t have an author! I must sign someone today!”" Every time a friend with kids in school tells me textbooks are too generic, I think back to that moment. “Who writes these things?” people ask me. I have to tell them, without a hint of irony, “No one.”"Last year, I did some contract work writing and editing a textbook – elementary school math for college students, more or less. “Writing” and “editing” such a text consists of taking perfectly functional texts and guidelines, and processing them in a manner that is disturbingly similar to the way that our airline chefs of yore mass-produced hard-boiled egg slices. My result, arrived at after hours of poring through specs and sources, differed from the existing texts about as substantially as the egg tube different from its (hard-boiled) constituent parts. Ansary’s experience is similar:” [A]t each grade level, the editors distill their notes into detailed outlines [later], they divide the outline into theoretically manageable parts and assign these to writers to flesh into sentences.” What comes back isn’t even close to being the book. The first project I worked on was at this stage when I arrived. My assignment was to reduce a stack of pages 17 inches high, supplied by 40 writers, to a 3-inch stack that would sound as if it had all come from one source. The original text was just ore. A few of the original words survived, I suppose, but no whole sentences.” To avoid the unwelcome appearance of originality at this stage, editors send their writers voluminous guidelines. I am one of these writers, and this summer I wrote a 10-page story for a reading program. The guideline for the assignment, delivered to me in a three-ring binder, was 300 pages long.”There’s so little I can add to this piece; it’s a bit long, but it’s an easy read. And Asmary does provide some constructive suggestions at the end, the second of which I especially support:” Reduce basals to reference books
The adage that addresses the issue of judging books by their covers counsels unambiguously against, but I’ve always rejected it on the grounds that it assumes, generally incorrectly, that authors have no editorial control over the presentation of their work. I unabashedly judge books by their covers in the figurative sense; but at the moment I’m being quite literal. Specifically, I am judging Rethinking Mathematics by its cover:
The authors of this tome aim to “provide examples of how to weave social justice issues throughout the mathematics curriculum and how to integrate mathematics into other curricular areas”, and if the cover is one such example, then we can safely conclude that the integral of SOCIAL JUSTICE ISSUES + MATHEMATICS CURRICULUM equals GIBBERISH. It’s an equation! No, it’s an inequality! No, wait…it’s bullshit! Seriously -what are the units of MULTICULTURALISM * POVERTY / INEQUALITY?
I haven’t read the book. My local library doesn’t carry it, and if I were to pay the $16 cover price to purchase it from pair of white guys who write about economic racism, well, I fear that that would make me part of the problem.
Nevertheless, even those of us who haven’t read the book can find plenty to criticize (cf “critical thinking”) in the introduction alone, which decries the “unfortunate scarcity of social justice connections” that are to be found in conventional high school mathematics curricula. Sadly, some old fuddy-duddies think that math classes should deal with stuff like trigonometry, as opposed to, say, the War on Iraq Boondocks Cartoon. Those naysayers, however, are just party poopers who totally don’t get it, but they can easily be convinced that a a social justice approach is consistent with your (yes, your) state’s mathematics standards:
Occasionally, a teacher needs to defend this kind of curriculum to supervisors, colleagues, or parents. One approach is to survey your state’s math standards (or the national standards) and to find references to “critical thinking” or “problem solving” and use those to explain your curriculum. Also, the NCTM clearly states that “mathematical connections” between curriculum and students’ lives are important.
There’s a valuable lesson to be learned here, actually, and it is this: if the description of your curriculum is impenetrably vague and long-winded, then people can use it to justify anything. For instance, the paid-by-the-word folks behind the Illinois math curriclum talk about a goal, within the math class, for students to “express and interpret information and ideas”, from which one could argue that it follows that we should be teaching interpretive dance in lieu of geometry.
This book pisses me off. It pisses me off a lot. Not because I think that math and life should be disjoint – quite the contrary, as I’ve frequently argued in this space. Hell – during the first ten minutes of the statistics-for-social-scientists course I taught last year, I stated, in so many words, that no one can accurately claim to be a fully-functioning member of a democratic society if they can’t interpret quantitative data. (Nor am I the first to make this argument; John Allen Paulos, author of the marvellous – and bestselling – book Innumeracy, says as much himself. The fact that the authors of Rethinking Mathematics are so unfamiliar with the literature that exists on the topic of mathematical literacy that they claim that theirs is the only resource of its kind, does not speak well to their expertise on the subject.) I also don’t think that the social-justice-based math curriculum is the dominant force behind students’ appalling inability to work with quantities. Sure, it’s not helping, but to claim that students think that 2/3+5/7=7/10 because their teachers are ideologues who have rethought mathematics is to diminish the roles played by innumerate elementary school teachers, innumerate curriculum developers, absent fathers, working mothers, fucking graphing calculators, sugary breakfast cereals, sex on TV, and shock rock in bringing about that sorry state of affairs. Hell, a good many of the topics in this book look quite worthwhile: the section on how unemployment figures are reported, for instance, seems like a nice topic for the “how you present data impacts what people think” section that appears in every single statistics chapter in every single high school math text, not just this “first of its kind” resource, but anyway! No, I am not opposed to mathematical literacy, and I wish that folks who are more politically-inclined than I would invoke it more often.
No, what bothers is this: is anyone familiar with a movement among social studies educators in secondary schools to use math in their courses, or does the movement toward interdisciplinary studies of social justice only go in the other direction? I am aware of none. Why are the educators who are motivated by political issues – and who see numeracy as a means to that end – injecting those issues into the math curriculum, rather than injecting math into social studies classes – which seems more natural to me? If I think that potters would improve their craft by learning some elementary Newtonian mechanics, I’d sooner give impromptu physics lessons at the pottery wheel than drag my physics classmates to the studio.
Is the overall effect to the high school curriculum, a net reduction of mathematical content?
The authors of Rethinking Mathematics are unabashedly politically-driven, and from the table of contents it is apparent that the math they present in their book leads students, none too subtly, to such conclusions as the one that capitalism is a fundamentally damaging economic system. Leaving aside for the moment the validity of this conclusion – I personally dispute it – let’s consider just how very involved a topic economics is. To come to any conclusion about capitalism requires one of two things: 1) a great deal of in-depth studies of economics and related issues, issues that Ph.D. students have written theses about; or 2) some superficial examination of pre-selected data (is this the Global Capitalist Economy Cartoon mentioned in the book’s table of contents?) that leads directly to the desired conclusion. In the context of a high school math class, (1) entails a huge use of the mathematics class’s time to teach and learn economics, while (2) constitutes brainwashing.
Given how ill-prepared the majority of high school students are to either do mathematics or think (let alone “think critically”, and the first person to point out case of that phrase being used by anyone who doesn’t have an ideological axe to grind, gets a cookie), you’ll forgive me if I can’t get on board with either of those two options.
This book, if used more than very sparingly, will give innumerate high school students highly skewed foundations on a wealth of complicated topics, and direct them to predetermined conclusions. Judging from the table of contents, it might prepare students for jobs preparing statistical expositions of positions espoused by lefty think-tanks. And, hell, that’s more than a lot of high school classes prepare students for, so I can’t even find fault with that; the problem is that while grooming students for that path, the social-justice math class will inevitably omit, because of time constraints, some other topics that might prepare students for further study in other areas. Will students whose teachers are motivated by social justice concerns learn enough trigonometry to hold their own in a university engineering course, should they wish to pursue that path? Will they learn enough algebra to succeed in the chemistry courses required by every medical school? The authors of this text talk about using mathematics to “potentially change the world”, which is hardly the exclusive domain of the social justice activists: anyone who thinks that engineers and doctors haven’t used math to change the world, has spent too long at rallies and is brainwashed beyond salvation. Engineers and doctors have changed the world for the better, even if measured in social justice terms. A robust, demanding, contentful high school mathematics curriculum, even one that suffers from an “unfortunate scarcity of social justice connections” (yes, they did use that phrase to describe the standard high school math curriculum, because you know that when I see the “how to use your fucking graphing calculator to plot a straight line” unit, the first thing I think is “but where’s the social justice?”) will leave the door open for students to acquire the tools they will need to use math to change the world – whether or not they later choose to become social justice crusaders. A curriclum designed to “guide students towards a social justice orientation” will cripple them if they choose any path other than that one.
And given how deluded high school students seem to be about the nature of equations , it can’t be a good idea to let them anywhere near that horrible cover.
This just in: I come across as blunt, abrasive, aloof, distant, and ostensibly averse to small talk – and damned if some people don’t much care for that.”Pardon me, allow me to clarify: this just in from my supervisor, who took five minutes to get [above] across to me. I knew exactly what was coming halfway into the “Well, I’m not quite sure how to tell you this, but sometimes in large groups ” prelude, but coaxing it out of him any faster would have required me to bypass the requisite small talk and cut straight to bluntness, and never let it be said that I can’t take a hint.”Anyway, this is apparently a problem. Not just to my supervisor, but to the anonymous chorus of indeterminate size (”some people”) that has approached him with concerns about my demeanor. Oh, they all know I mean well, but would it kill me to smile a bit more? Spend more of my lunch breaks indoors with others in the lunchroom, instead of running errands or relaxing at the park? Interrupt my work (which I’m apparently doing quite well, thankyouverymuch), whenever the circumstances demand it, with multi-word commentary about how by gosh, it is raining again, whenever will the madness end?”It’s not that I’m not aware of all of this, mind you; it’s that what others see as friendly banter, I see as distractions from my work – work that no one else in the office can do. I am a task-oriented person, dammit, not a people-oriented person! It’s that, much as I like my coworkers, I don’t come to work to better my social life. It’s that in my metric, it’s better to approach people, possibly bluntly, than it is to mediate your concerns through a third party, so that the offending individual is left suspecting all of the friendly chit-chatters of filing complaints with that party and leaving her to guess whether this modification of her behaviour is enough, because it’s not like she’s had the chance to speak directly (that word again!) with anyone who actually wanted her to modify it. So help me, this all strikes me as remarkably inefficient, not to mention, highly inconducive to creating a pleasant working environment, at least for me. I’m just saying.”In summary: damned if I know what precisely I need to change (goddamn, it’s hard to get straight answers from the directness-averse HR set), and damned if I could make whatever changes are necessary without driving myself insane even if I did know exactly what they were. The good news is, I continue to provide my employer with a sort of specialized expertise that they’ve been seeking for years; and by all accounts, I do a very good job of what I was hired to do.”Call me old-fashioned, but right now, I plan to just continue to do my job well, and I reckon that’ll be enough.
Doppelgangers
Feb 8
On the buisiness trip:” * Guy who tells me that it’s “nice seeing [me] again.” Again? I say. Not that I doubted that I’d run into him before; I remember having seen people far less often than they remember having seen me, and I often feel tremendouly guilty about this. However, the last time this guy supposedly saw me was “at the conference in Edmonton”, and I was never at the conference in Edmonton. But he swears he saw me, or at least, someone who looked just like me.” People tell me this disturbingly often, and I can only assume that not all of them think that “I know I’ve seen you before” is a functional, if not especially clever, pick-up line. Apparently there are plenty of people who look just like me floating around, but I’ve never met any of them personally.” * Speaker who looks exactly like a ten-to-fifteen-years-younger version of my father. Same height. Same build. Same colouring. Same gait. Same haircut. Same glasses. Same style of dress. This is slightly jarring.” More jarring, though, is the fact that this dude could not possibly sound less like my father. No, whenever this guy opens his mouth, he speaks in a lispy, slightly breathless, gay-guy-from-crappy-sitcom tenor.” And file under “just plain strange” (or “race is a social construct”, if you’re so inclined): though I’m a mutt and I can’t reliably enumerate all of the nationalities that feature in my bloodline, I am fairly certain that there is no genetic explanation for my parent looking like some guy named Sanchez.”The customer is always something”" Know Thyself, Welcome To The Occupation.”The room in the somewhat swank hotel where I am staying on business was not ready when I arrived two minutes after check-in time.”Now, I didn’t mind one whit, because I mostly just think about hotels as places where someone else changes your sheets and cleans your bathroom, and I pretty much figure that once you’ve got that, it really doesn’t make much sense to gripe about the timing of the sheet-changing and bathroom-cleaning, you know? Gift horses and misdirected gazes, is all. But I am clearly not a member of the class of hotel patrons that the staff at the somewhat swank hotel has been trained to serve, because the woman at the check-in counter immediately shifted into damage control mode as soon as she had discerned the state of my room-to-be: We’re so sorry! Your room was not available at three-oh-two, so here’s a gift card for a free shoe shine! And another that will allow you two hours at our luxury spa, free of charge! And here, take an exercise kit that we usually provide only to our preferred customers, of which you were not one five minutes ago, but you are now! Because you have been wronged! By us! Oh, and while you wait, go get yourself a drink at the bar – at no cost to yourself! And if you want to enjoy one of the pay-per-view movies that we offer this week, you will not see the charge on your hotel bill!”Eyes widened in anxious anticipation. Expectant stare. Is there anything else we could do to pursuade you to forget this unpleasantness?”At this point I was so disoriented by this wholly unnecessary song and dance, that my ability to adjust my behaviour to my surroundings took leave of me completely. “Well, to be honest,” I deadpanned, “short of giving me a free upgrade to one of your luxury suites – no, I’m afraid there really isn’t.”"Without a word, the clerk nodded, pursed her lips, and briskly entered some data into her computer. A minute later, she handed me a key and dispatched me. It was only after I arrived at the door to my temporary residence that I realized that there was no reason for my room to be ready so soon, and oh my god there are two bathrooms in this place and I need a pair of binoculars to watch TV in here.”It did come to my attention, however, that even the preferred customers who spend, I swear it says this right on the door, seven hundred dollars a night to stay here, still have to pay for their own high speed internet connection on top of all that, ten goddamned dollars a day.”I considered calling the front desk and asking for that charge to be waived, but I thought that might be pushing it.
“Applicant must be willing to travel”, read the job ad, and they weren’t kidding: I’ve got five trips scheduled this quarter alone, and they range in purpose from the interesting to the Dilbertesque. Next month, for example, I am to attend a conference that will teach me the ins and outs of effectively managing one’s employees, and I leave the taxonomy of that one as an exercise to the reader.”I’ll give the reader a hint: I do not, myself, have any employees. I am the managee in this relationship, not the manager.”And another: preparation for said conference involved taking a personality test, which revealed such things as – try to contain your shock – “subject is highly independent”, “subject is task-oriented” and “subject scored in the lowest decile on the
Business sense
Jan 27
1. At the local Sears, women’s jeans, unlike men’s, are indexed not by pairs of numbers that denote waist size and inseam, but by single numbers that denote nothing. Yeah, I know, I’ve been over this before, but that’s what you get when you keep reading the same blog for over a year. Anyway: a few weeks ago I subjected a (middle-aged, male) coworker of mine, nevermind why, to a passionate tirade about how I will not even try on jeans whose manufacturers can’t even be bothered to provide a two-parameter description of them. Why waste my time? Except that some of the Sears jeans are labelled with waist size and inseams: one pair, filed under the marker “8?, sported a tag that read “30/32?. Seems the dolt in charge of the women’s jeans section decided that that information should be hidden from immediate view. (Aside: you know those women in fiction who describe themselves as “I’m a size n”? Does anyone actually do that? Because if I were to give the single-number pants size, I’d need to provide a margin of error as well.)” Credit where it’s due, however: I did manage to pick up some great pyjama bottoms at Sears. They came from the men’s section – and were labelled with waist and inseam. Which is kind of weird, but goddamn, do those pyjamas ever fit.” 2. My employer sprung a new business trip on me after I’d already booked tickets for another one. I called the airline to cancel my original flight, had some conversation involving the words “non-refundable” and “thirty dollars”, and I agreed to a bunch of stuff, and then, a few weeks later, saw my credit card bill, and -” “Excuse me, but did you people charge me thirty dollars to change a flight from return trip to Edmonton to return trip to nowhere?”" “Yes, we did.”" “That doesn’t make any sense.” “It’s our policy. We explained it over the phone to you.”" And, in all fairness, they did: they said that my tickets were nonrefundable, and that I could cancel the flight “for thirty dollars”, which I (understandably) parsed as ” but we’ll refund you $30.” You know how I always complain about students not reading the damned question, and instead just doing whatever they want with the numbers in their word problems? Feel free to point me to this post next time that happens.” Nevertheless, “But if I just didn’t show up to the airport, it wouldn’t cost me anything to fly to nowhere. I thought that the airline would prefer to know that I wouldn’t be flying so that they could sell my seats to someone else.”" “Yes, we do appreciate it, thank you for notifying us.”" “But you just showed your appreciation by charging me thirty dollars.”" To be fair, they were charging my employer thirty dollars, and perhaps I should have shown my appreciation for my employer by not spending more than thirty dollars’ worth of my time debating this issue with the airline. However, it wasn’t about the money; it was the principle of the thing.” “That’s our policy.” “May I speak to your supervisor?”" Yes, I may! And let me skip the ensuing thirty-minute conversation and go directly to the coda, which is this: supervisor agreed that why yes, now that I mentioned it, this was ridiculous from the perspectives of both company and customer, and we’ll credit your account thirty dollars, have a nice day.” 3. The hotel where I last stayed on business had a pizza place on the first floor. “Available in the restaurant on the first floor, and in room service!” boasted the menu on the desk. Also: “15% gratuity extra for room service.” Why not? I’m sure that lots of folks who pay their own money to stay in places like this also pay their own money to avoid walking to the lobby.” “I’d like to order a pizza,” I said.” “Room number?”" “Oh, I’ll pick it up myself.”" “Okay, but we still need your room number so that we can call you when it’s ready.”" Fair enough.” Twenty minutes later, a knock on my door, along with man holding a pizza.” “Oh,” I said, “I told them I’d pick it up in the lobby.”" The fellow nodded, and walked over to the end of the hall. I followed him into the elevator and into the restaurant. “That’ll be twelve dollars,” he said, reading from the receipt he’d carried up to my room and back down again. “Price of the pizza plus tax.”
What do you call it when a person deliberately seeks out psychologically unhealthy attachments?
Jan 26
I often forget just how dysfunctional a relationship many of my weaker students have with mathematics.”It’s been a long time since I’ve been surprised at the extent to which such students harbour an unproductive and damaging belief that mathematics is nothing more than a mishmash of symbols and voodoo procedures. After all, this is understandable, what with students being taught that memorizing templates of questions and plugging memorized formulas into their fucking graphing calculators is homologous with “doing mathematics”.”What surprised me for a long time after – at least the first fifty times I encountered the phenomenon – was how resistent these same students are to seeing mathematics as anything other than a collection of disconnected formulas and calculator algorithms.”The other week, I found myself teaching introductory graphing to a handful of students. Partway through a lesson, one student asked me – how do I graph the line in this question? Do I find two points and join them, or should I just find one point and the slope and then graph it that way?”Giddy with delight at this hint of outside-the-box thinking, I replied: you can do it either way you want! It’s your choice! Both of these options are totally valid methods of graphing the line! Two points, point slope, it’s up to you! In fact, you can graph it one way, and then if you want to check your work, you can graph it the other way, and ISN’T MATHEMATICS SUPER?”Pregnant pause. Hesitation. The barely-perceptible tremours of a worldview beginning to collapse unto itself.There are two ways to do this question?”Yes! Not one, but two (2) ways to achieve the goal of graphing a straight line! Pick one! It’s entirely up to you!”But which way should WE do it?”EITHER way! The easy way! The quick way! Try
Oh, Canada, did you just vote in an even more unstable government than we had last time? You did, didn’t you? Whatever are we going to do with you?”I have an idea: haul in some cameras and a reality TV crew, because The Real World: House of Commons damn near writes itself. Think about it: we’ve got the most right-wing prime minister this country has had in, well, ever, and the man’s got no history of consensus-building to speak of; but never before has it been so vital to a Canadian prime minister’s political survival that he compromise with the other parties in the House. In this minority government – which is weaker than most polls predicted – Harper is going to have to compromise with some party on every issue. And, oh, the possibilities:”Behind door number one, we have the Liberals, whom Harper slammed as corrupt at every turn! Will compromising with the Liberals mean compromising his integrity?”Alternatively, Harper could open door number two, behind which we find the separatist Bloc. How cozy can Harper get with them before alienating his western base, who have long complained about the government being determined by Ontario and who would therefore probably not take terribly well to being at the mercy of Quebec?”Fortunately, there’s a third option: Harper can go with door number three and deal with the Satanic NDP! Oh – wait – that won’t quite give him a majority of seats. He’d have to get the radio shock jock in on that one!”I predict high ratings. Wonder how long it’ll be till the third season
On the agenda at work today: coordinate a meeting that as many as possible of two dozen-odd clients would be able to attend. Inane administrative duty? No: opportunity for field research into alternative voting systems!”Dear clients, I wrote, I’d like to hold a meeting with you during the first week of March. Could you please email me a list of times when you will be available during that week (eg, “Monday morning”, “Thursday afternoon”, etc)? If you have some times that are better than others, feel free to send that information as well (eg, “I am free Monday and Tuesday mornings, but the best times are Tuesday, Wednesday, and Friday afternoons”). I’m going to do my best to accommodate as many of you as possible, but I can’t guarantee anything.”And I got responses, some of which provided me with nice, STV-compliant rankings. But, oh, the others – for instance, the dozen that went something like this:” dear moebius stripper i am free tuesday mornings”Thank you for your quick response, I wrote back, But you’re a retired grandmother and like hell that’s the only time slot you have free Are there any other times you have available? I am trying to accommodate as many people as possible, and while I can’t guarantee that everyone will get their first choice, I will have a better chance of arranging a meeting that most people can attend if I have more options.” dear moebius stripper” yes there are other times i could come but tuesday mornings are the best, i do not want the meeting any other time.”Fully half of my clients did not want to provide me with their second choice because they were afraid that doing so would weaken their first choice vote. I’m wondering now how widespread this attitude is – does it account for a significant proportion of votes against electoral reform? If so, it’s both selfish and irrational, because we’re probably going to end up with a meeting date that hardly accommodates anyone.”Kind of like the government we’re going to end up with on Monday.