Farrar Posted July 31, 2015 Share Posted July 31, 2015 So, Let's Play Math linked this great little numberless problems book which I think I'm going to use parts of with one of my ds: http://www.schoolinfosystem.org/pdf/2008/10/problemswithoutfigures.pdf It's from 1909. I used OCR on it and cleaned it up then tweaked a few of the problems because they involved things that simply didn't make sense for a kid today or language that was just outdated. So, for example, I changed a problem about boards to build a sidewalk to being boards to build a deck, because, um, I'm not sure where has wooden sidewalks in this day and age. But a few of the problems were just unchangeable and I just chucked them out. Bushels, rods, acres, gills... these aren't really measurements that most kids need to know. Well, acres, okay, but since the vast majority of Americans now live in urban or suburban communities, the need is diminishing even for that one. The real agrarian nature of American society more than a century ago was very evident in the book. There were countless farm related problems but only a couple of urban ones. Anyway, it got dh and I talking. It was really clear that people needed more of a certain sort of math back then. You needed to be able to convert between different measurements more, you needed to be able to make change more, you absolutely needed to be able to apply measurement skills more. While some of that is still true, many things now have standard lengths and a number of the problems in the book (like clocks that lose time) are now laughable. People needed more math to solve things to make up for imperfect technology. Now more things that you measure - like, say, the size of a window or a piece of paper or the like - are standardized. And we now have a much greater separation of labor in regards to these things. Very few people build their own homes - being able to use measurement to the detail that you might have needed a century ago isn't necessary for a large number of people. But... I think we just need different math now. For example, with more choices, I think we use permutations and combinations more often in daily life (how many options do I have for this burrito, how many ways can I order this product, etc.). Our tax system is more complex. There were very few percentage problems in there, but I feel like that's one of the math applications I use the most often in daily life - percentage discount, percentage taxed, etc. And there are more statistics bandied about in our daily lives and we have to be able to evaluate. And a lot more jobs involve applied statistics - in the sciences and medicine. We are used to looking at bigger numbers, I think - bigger budgets, more people in the world, giant numbers of data. Anyway, I'm curious what people think. Do you think the math we need has actually changed in the last century? Are there other differences you can think of? 10 Quote Link to comment Share on other sites More sharing options...
Tsuga Posted July 31, 2015 Share Posted July 31, 2015 This is a great post. It could actually develop into an article. (Is that what you're doing with this thread?) You should put it on your blog. I need to think about it... 1 Quote Link to comment Share on other sites More sharing options...
Farrar Posted July 31, 2015 Author Share Posted July 31, 2015 Ha. I'm really curious. If people give me some more examples, maybe I'll put them in the numberless problems and put that on my blog. I replaced a few of the acres and bushels and rods ones to become things like how can you figure out the tax home pay after this and that and how can you figure out which toilet paper is cheapest with various discounts. 1 Quote Link to comment Share on other sites More sharing options...
redsquirrel Posted July 31, 2015 Share Posted July 31, 2015 Well, I for one am always absolutely amazed at how many people don't know how to compute a simple tip in their head and it is something lots of us have to do all the freaking time. I can see how at one time that was something that was done infrequently, but now it should be automatic. That is just another example of percentages and how they are important to calculate. I also think being familiar with bigger numbers and knowing how they relate to each other is important. How much bigger is a billion than a million? What does it mean when a debt is trillions of dollars? DH and I talked about it and we both think that lots of those math skills are still important to have. It is good to have the math you would need to build a house even if you will never build one. That seems sort of basic to me. It's like when I read an opinion piece in a major national newspaper last year that took the position that we should stop requiring Algebra in schools. It pointed out that most people study algebra and then never use it in their every day lives, so it is mostly a waste of time. I disagree. First of all, I think we use algebraic thinking much more often than we think we do, we just don't identify it as such. Secondly, I think it is good to teach the brain to do algebra, just like I think it is good to teach the brain to read and understand Shakespeare or TS Elliot. But I doubt you dispute that, so back to the topic at hand... But, yes, percentages, rates, ratios and statistics are somethings that come to mind as being pretty common in the home and workplace. 4 Quote Link to comment Share on other sites More sharing options...
Tsuga Posted July 31, 2015 Share Posted July 31, 2015 Hey, if your sales tax is 10%, you just have to double it! Easy peasy. :D Quote Link to comment Share on other sites More sharing options...
Farrar Posted July 31, 2015 Author Share Posted July 31, 2015 Oh, and tipping. That's another one. Redsquirrel, I do totally agree that measurement is still super important. I was just struck by how differently it was used and how there were more measurements. Like, yards per rod and so forth. And bushels, which is just totally a weird measurement system - it's volume but for non liquids? How weird is that. But maybe measurement is worse now. I think a lot of us need to do a lot of informal switching between metric and customary and there's no easy way to do that. And maybe we do more math with rounding and estimation now. Large numbers and more choices mean more averages and estimations for businesses. Quote Link to comment Share on other sites More sharing options...
Guest Posted July 31, 2015 Share Posted July 31, 2015 I've been thinking about this a lot lately as I've picked up a few maths books from the 50s and 60s as well as one from the 30s. Compared to your example, mine are infinitely more practical. The book from 1951 "Mathematics At Work" combines algebra, geometry and trig to teach how to balance a budget, figuring interest, reading a gas/electric meter, stocks/bonds, the cost of buying on credit, construction problems, etc. The book from 1962 - Mathematics to Use - is slightly less advanced in that it only uses arithmetic, algebra, and geometry for the upkeep of cars, statistics, mortgages, etc. In both books, the geo/trig problems are a little more abstract like which perfume bottle holds more and just general lessons in lines and shapes. I think a lot of the problems in my books, at least, aren't useful anymore because my mortgage papers give me an amortization chart, my credit card statements tell me automatically how much in interest I'm paying without me having to figure it out, the perfume bottles tell how many ounces/grams are in each, etc. For me personally, I'm finding that I'm using a lot more geometry than I ever thought I would when building my chicken coops and remodeling my house. It's ranking right up there with percentages....which sucks because I'm so much better at percentages :lol: However, I did like the fact that your book makes the student come up with a formula instead of saying in this type of problem, plug in the numbers in this way. Its more of a thinking book, rather than a drill book. Thanks for posting it!! Quote Link to comment Share on other sites More sharing options...
fdrinca Posted July 31, 2015 Share Posted July 31, 2015 Not strictly mathematics, but I'd say an understanding of data analysis is another huge component of modern mathematics that wouldn't have been possible 100 years ago. 1 Quote Link to comment Share on other sites More sharing options...
Sahamamama Posted July 31, 2015 Share Posted July 31, 2015 What blips me out when I read some of those "old-fashioned" books -- think, Anne of Green Gables or Little Town on the Prairie (or one in that series) -- is that so much of this type of math was meant to be done mentally. And, they seem to have been able to do those complicated calculations in their heads. But some of this has to do with practice, I think. The other day at the grocery store, my husband showed the twins a huge wheel of cheese. He said, "Hey, this cheese wheel weighs 30 pounds." And two seconds later, Boom said, "Oh, so since this smaller piece of the same cheese says it's $12.50 per pound, if that wheel sold at the same price per pound, it would cost $375." :blink: My husband, naturally, pulled out his phone, used the calculator, and -- lo and behold -- the kid was right. She does stuff like this all the time, so I do think that, for her, it is a matter of practicing how to manipulate the numbers in her head. I have another kid who... operates differently. She's the oldest, the perfectionist, and instead of thinking about the problem, she jumps straight in to calculate. For example, if you told her, "These three baskets sell for $3.99 each, how much for all three?" you can see her mentally trying to add up $3.99 in a column... nine plus nine plus nine is, um, twenty-seven, put down the seven, carry the two, um......... It's painful. We have to say, "Stop adding, and think." And then she can say, "Oh, I get it. Four dollars times three is twelve dollars, minus three cents. So, $11.97." I'd like to work in more mental math this upcoming year. I think we're going to have to split it out, though, do it individually. Or, I might just read from, say, a Zacarro book (or something similar to your link), and ONLY allow one student to solve it. But that probably won't work, because I have children who, when they know the answer, like to act as though they will BURST to not say it. "Oooh, oooh, ooooh." Just be quiet, and let your sister think. One of the most frustrating things about mental math for the more deliberating student(s) is being "beaten out" by the faster student(s). Oldest student here will not BLURT out random guesses, whether right or wrong. No, she is trying to solve the problem, but then a younger sister BLURTS. That can be frustrating, so we'll have to work on our consideration skills. :glare: I do think that if we grapple with these kinds of problems more often, we get better at figuring out how to solve them. I hope you write this up, Farrar. I look forward to reading your blog post! ;) 3 Quote Link to comment Share on other sites More sharing options...
Sahamamama Posted July 31, 2015 Share Posted July 31, 2015 Another thought: I have noticed how most cashiers do not know how to make change, or count your change back to a customer. When I was younger, a cashier would hand you the change while saying how much your total was from what you handed over, then count back up to that amount. For example, if your total was $23.87 and you handed the cashier two $20 bills, she would say, "Twenty-three eighty-seven from forty, that's [counting out three pennies] eighty-eight, eighty-nine, ninety, [counting out a dime] twenty-four dollars, [handing you a $1 bill] twenty-five, [handing you a $5 bill] thirty, [handing you a $10] forty, forty dollars." That was always the way a cashier counted back change, and even I was trained to do it this way. But it has been years since anyone has even known what change to make. Now, the cash register tells the cashier what change to give. And, since most people seem to pay with a plastic card, when I pay with cash and need actual change, this seems to confuse most cashiers, even though the machine is telling them how many pennies, nickels, dimes, quarters, and bills to give me. I honestly think that not all cashiers know which coin is which. I have seen cash registers with pictures of the coins and how many to give back. A far cry from the country general store where the seller added it all up and made change in his or her head! One thing we've come across in CLE Math that I never learned is digit sums. That was new to me, but it's somewhat useful. I realize that there is a push for STEM this and STEM that, but I think most students need a better foundation in practical money management -- that is, financial literacy, personal accounting, bookkeeping, call it what you will. We go all through to Algebra 2, but don't know how to manage money, which certainly is an increasingly complex area of study. CLE has some Consumer Math electives that look like they might be worth working through during middle school/early high school, along with the traditional math sequence (maybe 8th & 9th?). I'm still thinking about it, we'll see. And, I think there is a greater need now for an understanding of statistics, how researchers create them (LOL), and how they can be manipulated and/or tell us what we need to know. Also, I'm sure that in 1909 no one was concerned with applications of math to computers and similar technology. I dislike intensely the author's comment on page 4 of the Preface: "A few 'catch problems' are put in to entrap the unwary." Dislike this intensely! And I would rephrase the rest of his stupid statement as, "To stumble occasionally into a pitfall makes a pupil more distrustful of those who would entrap him and intensifies his distaste for math. To purport to teach math by setting traps for the unwary is in itself an absurdity." Imagine taking your kids on a nature walk, along a trail where you have "set traps," just to see if they notice the traps, instead of the pileated woodpecker or the nasturtiums. If they fall in a hole, how long before they once again trust you? Nice pedagogy. Blech. [Farrar, I know you know I don't mean you, personally, I am just venting in a generic "you" sort of way. That kind of condescending teacher talk seems to have been quite common during that era, and it rankles me greatly]. :rant: 4 Quote Link to comment Share on other sites More sharing options...
rose Posted July 31, 2015 Share Posted July 31, 2015 Interesting topic. As a Canadian I am constantly having to switch back and forth between metric and imperial. My US friends seem to have to do much less of this. I think that we probably have to deal with travel issues more often. For example, google maps says that a certain trip will take 17 hours of driving and is x km/miles away. We have to figure out if google is realistic for our family (it never is) and then where to stay for the night based on how far we estimate that we will have travelled by nightfall. We need to be able to catch a scam more. For example, The Brick might be selling in couches for "low monthly payments of $49". We need to be able to figure out what that means when you add up the interest accrued over 3 year. How much more will we be paying by buying on credit then by just saving up the money and buying the couch outright. People would definitely be better equipped if they had more experience dealing with statistics. People throw statistics around all the time to persuade people of their opinion. Simply knowing the different strengths and weakness of mode, median and mean averages would make a load of difference. There's a great little book that we have called, "How to Lie with Statistics." I plan to have my children understand that book thoroughly before they leave home. Our society is also more science based then ever and so math related to technology, like binary or hexadecimal, has more relevance. Understanding logarithmic scales is relevant to understanding news about earthquakes. I'm getting a little esoteric here. I doubt my mother or father would understand these examples but nonetheless I think that it's best if we can roughly understand the science around us. 5 Quote Link to comment Share on other sites More sharing options...
Farrar Posted July 31, 2015 Author Share Posted July 31, 2015 I noticed that phraseology about traps and also disliked it. But I have to say that a huge number of them are tricky and I sort of like it. Like, take this one: A farmer sold a grocer a cake of beeswax at a certain price per pound. It was afterward found that in the center of the cake was a large cavity. If you know the volume of the cavity in cubic inches, and the weight of a cubic inch of beeswax, how would you adjust the transaction fairly, between the farmer and the grocer? Um, you wouldn't adjust it. It was sold by the pound and it's a substance that's easily manipulated into new shapes so it's not like you need it in a big cake so the transaction was fair. I think thinking like that is really important to math understanding. I wouldn't present it the way the author does though. I generally warn my kids when something is likely to have tricky problems. 3 Quote Link to comment Share on other sites More sharing options...
Farrar Posted July 31, 2015 Author Share Posted July 31, 2015 We need to be able to catch a scam more. For example, The Brick might be selling in couches for "low monthly payments of $49". We need to be able to figure out what that means when you add up the interest accrued over 3 year. How much more will we be paying by buying on credit then by just saving up the money and buying the couch outright. Good examples all, but this one is especially true. There are so many deals that you have to look out for. Quote Link to comment Share on other sites More sharing options...
Loesje22000 Posted July 31, 2015 Share Posted July 31, 2015 Most math books I've seen currently only learn the 'round' percents (25%,75%, 10%) We learned to calculate any percentage, even the odd ones from odd numbers. Normally we only learn the metric system, not the British or the American way. But before the Euro we learned how to convert money between countries. The vocational track has more real life world problems now. I often use them to make clear why one needs certain math skills. I only learned mostly problems like: A travels at that pace, B at such, the distance is that long, where will they meet. Dd will probably chose a collegebound track with less math then the math track. She will learn more practical math like statistics and financial algebra (can I finance my house/car, can I pay this mortgage) 2 Quote Link to comment Share on other sites More sharing options...
Kerry Blue Posted July 31, 2015 Share Posted July 31, 2015 The book from 1951 "Mathematics At Work" combines algebra, geometry and trig to teach how to balance a budget, figuring interest, reading a gas/electric meter, stocks/bonds, the cost of buying on credit, construction problems, etc. The book from 1962 - Mathematics to Use - is slightly less advanced in that it only uses arithmetic, algebra, and geometry for the upkeep of cars, statistics, mortgages, etc. Could you share the authors of those books? Quote Link to comment Share on other sites More sharing options...
SporkUK Posted July 31, 2015 Share Posted July 31, 2015 Interesting thoughts. I agree more maths related to percentages and taxes than I was raised with would be useful. We've been wrestling with this lately as A-8's way of learning has moved us from MEP maths which UK based to US Mammoth Maths and as we went through reviews we ended up flummoxed in units in ways I hadn't foreseen. Even though the customary/imperial units are related, they aren't the same. A UK gallon isn't the same as a US gallon, they have a different amounts of fluid oz, UK doesn't use cups at all which I somehow hadn't realized in 12 years living here and was surprised when my British partner mentioned he only used them in adulthood to translate US recipes. Coming from the US I knew them all but ended up down a rabbit trail of my own trying to explain the differences to my kids (apparently it has to do with changes in the definition of a gallon that happened a few times and the weight of wine and water). Discussing how things became standardized and moving between them (a lot of Brits I know don't have an understanding of how metric compares to UK imperial and ask for things in 'old money'), would be a useful modern maths. 1 Quote Link to comment Share on other sites More sharing options...
letsplaymath Posted August 1, 2015 Share Posted August 1, 2015 Another change is that we need to be able to summarize data and extract meaning, and express that meaning in a way other people can understand. And we need to be able to look at a situation and make decisions about which things are most relevant to the problem, so we can ignore the parts that are less relevant. Keith Devlin has been talking about these changes for years, especially the need to switch from a focus on executing calculations to a focus on thinking about how things relate and interact within a problem. That's not a new type of thinking, of course, since it's what the Problems Without Figures book was trying to develop. Both types of math have always been important, but schools used to emphasize the former, and now we need to emphasize the latter. Here are some of my favorite Devlin posts, from earliest to most recent: Telling stories with numbers What is mathematical thinking? Most Math Problems Do Not Have a Unique Right Answer Your Father’s Mathematics Teaching No Longer Works Hard fun – video games creep into the math classroom 3 Quote Link to comment Share on other sites More sharing options...
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