Definitions and examples for relation composition and the two types of relation reduction that commonly arise can be found in the following articles:
Peirce’s idea of reducibility and irreducibility is the more fundamental concept, having to do with the question of whether relations can be derived from others by relational composition, and this type of operation is invoked in every variety of formal construction. Consequently, projective reducibility does nothing to defeat Peirce’s thesis about the primal nature of triadic relations.
But people sometimes confuse the two ideas of reducibility, compositional and projective, so it’s good to clarify the differences between them. Projective reducibility, when you can get it, is more of a “consolation prize” for dyadic reductionists, who tend to ignore the fact that you can’t do anything constructive without triadic relations being involved the mix. Still, it’s a useful property and good to recognize it when it occurs.
]]>But, here we go again!
Yes, it’s the “META” prefix.
It’s sort of like the “i” in iphone and ipod and ipad and iwatch and i-everything else!
Except that it’s Meta!
Now our PM, Tony Abbott, got himself into a bit of a hole while discussing “Metadata” not so long back. What is it? Who knows? But it seems to have become a part of the lexicon already. Maybe we don’t really NEED to know what it means!
There doesn’t seem to be much info around to help we poor, unsuspecting teachers to understand the “Meta” world.
The VCE English syllabus talks lots about Meta-language. Yes, I said “Meta-language”. Closer examination of this amazing term reveals that we older, more experienced teachers would probably substitute the terms “functional grammar” for the aforementioned “Meta” term.
If “Meta” implies some grand, fantastically large knowledge base for every category of learning, then perhaps it wouldn’t be too much to ask for a clear directive from the educational authorities, eloquently but practically expressed, to give teachers a fighting chance of making a fist of it in their teaching.
Until such clear direction is forthcoming, we teachers here in Australia need to do our very best to continue to impart genuine mathematical and linguistic skills to our students in the best, most efficient possible way, given the resources available to us.
Our EdShop materials, outlined at http://www.EdShop.net.au are but one source of heading in the “Meta” direction, teaching and encouraging positive attitudes towards maths AND reading comprehension in one fascinating, fun package freshly-released each week.
There are many others! Unfortunately, not all are adapted to the Australian curriculum and lingo. A pity, that!
Let’s hope that considerably more “Meta” detail is forthcoming in the near future. It SHOULD benefit all! But will it? Experience tends to suggest that the REAL danger is that a whole new layer of expectation will be dumped upon teachers yet again, with no realistic opportunity to have access to resources and the professional development opportunities required to allow them to develop the necessary skillsets for successful implementation.
A pity, that!
How are YOU responding to this “META-CHALLENGE”?
]]>Why can’t we as humans and other assorted creatures divide by zero?
Short answer: we can, but it’s illegal.
Numbers have power. Division is the act of splitting quantities of anything into groups of a specific number. Depending on the number, those groups will have particular properties shared by the number. For example, in groups of ten, the object being grouped will be rounder than usual, while groups of four will be very square.
Groups of zero are powerful. Very powerful. As we all know, magic is real. But most magic is limited, reliable, indistinguishable from sufficiently-advanced science. When grouped in groups of zero, it is not so. The discovery of the limitless power that becomes available when dividing things by zero soon led to horrible abuse, the nadir of which were the dreaded Zero Wars.
The Zero Wars were bloody and destructive on an exponential level. Families were torn apart, livelihoods destroyed; entire cities were decimated, the survivors left with nothing. Eighty percent of the world’s produce was locked into groups of zero, and it seemed, for a time, that matters would never be made right.
Fortunately, that prediction was–narrowly–proven wrong. The Plant King–for this was just at his ascent to power–came onto the scene, setting right what he could of what had been made wrong, bringing order into the chaos that reigned, and helping people to put their lives back together. In order to protect our future, our world, he instated the law, enforced across all four kingdoms of living things–his own plants, along with animals, rocks, and mold–that no one might ever divide anything by zero again.
____________________
Disclaimer: Over 99% of this blog post is false. The writer recommends against dividing by zero.
]]>In the activity Comparing Two Numerical Distributions, students compare two populations by looking at the similarities and/or differences between them. The populations in this activity are drawn from familiar real world scenarios to which students can relate. This way, the comparisons and interpretations make more sense to students.
Students will extend the skills they have already learned for finding measures of center and variability for one population to finding these values for two populations, and using the values to compare the populations. With visual representations, such as dot plots and boxplots, students can also draw visual comparisons between the populations to make interpretations.
Yes, knot theory is a real area of mathematics, with books and current on-going research and everything! It is a little different to what you’re thinking as knots, but not as different as you might expect. There’s a lot of common terminology here that has a much stricter meaning in mathematics, so look out for those.
A knot in mathematics is almost the same as the type of knot you tie with your shoelace. The main difference is that in maths, you have to glue the two ends together after tying.
Let’s think in three dimensional space first (the kind of space we’re most used to), where we’ll have a one-dimensional string that we tie our knot with. Here, a knot is a circle placed, or embedded, in this space. The circle that looks like an “o” is still a knot, but it is the most simple type of knot and we call it the unknot.
The next most simple knots are the trefoil knot, then the figure-eight knot. You can see they’re still a circle, but they have some twists and crossings in that mean you can’t untie them without cutting the string. This is because they are different embeddings of the same circle, in three dimensional space.
Embedding here is pretty cool, and really quite powerful. We know our knot is basically a circle, but something that’s happened to it. It was a circle that was taken out of three dimensional space, messed up some way, then placed back into three dimensional space in a way that means we can’t get it back into the unknot anymore. It’s knotted.
Of course, a knot could appear to be knotted when in fact it is actually the unknot. Anyone who’s ever left a pair of headphones or a necklace alone for too long will know this!
In everyday language, the word “circle” can mean either a filled-in circle, or the non-filled-in kind. In maths, confusing these two is almost as bad as dividing by 0. The filled-in circle is called a disk, and empty one is a circle. In fact, the circle is what we call the one-dimensional sphere, or for shorthand. The disk is the two-dimensional ball, or .
Unsurprisingly, we have the two-dimensional sphere (a hollow football), , and the three-dimensional ball (a solid football), . These are often just referred to as the sphere and the ball respectively.
Excitingly, we can knot the sphere as well as the circle!! Get your head around that one!
(Note: I am not a Canadian. I have tried my best based on the public data, but I may be missing things. Corrections are appreciated.)
For PCAP 2010, close to 32,000 Grade 8 students from 1,600 schools across the country were tested. Math was the major focus of the assessment. Math performance levels were developed in consultation with independent experts in education and assessment, and align broadly with internationally accepted practice. Science and reading were also assessed.
The PCAP assessment is not tied to the curriculum of a particular province or territory but is instead a fair measurement of students’ abilities to use their learning skills to solve real-life situations. It measures learning outcomes; it does not attempt to assess approaches to learning.
Despite the purpose to “solve real-life situations” the samples read to me more like a calculation based test (like the TIMSS) rather than a problem solving test (like the PISA) although it is arguably somewhere in the middle. (More about this difference in one of my previous posts.)
Despite the quote that “it does not attempt to assess approaches to learning”, the data analysis includes this graph:
Classrooms that used direct instruction achieved higher scores than those who did not.
One catch of note (although this is more of a general rule of thumb than an absolute):
Teachers at public schools with less poverty are more likely to use a direct instruction curriculum than those who teach at high-poverty schools, even if they are given some kind of mandate.
This happened in my own district, where we recently adopted a discovery-based textbook. There was major backlash at the school with the least poverty. This seemed to happen (based on my conversations with the department head there) because the parents are very involved and conservative about instruction, and there’s understandably less desire amongst the teachers to mess with something that appears to work just fine. Whereas with schools having more students of poverty, teachers who crave improvement are more willing to experiment.
While the PCAP data does not itemize data by individual school, there are a two proxies that are usable to assess level of poverty:
Lots of books in the home is positively correlated to high achievement on the PCAP (and in fact is the largest positive factor related to demographics) but also positively correlated to the use of direct instruction.
Language learners are negatively correlated to achievement on the PCAP (moreso than any other factor in the entire study) but also negatively correlated to an extreme degree in the use of direct instruction.
It thus looks like there’s at least some influence of a “more poverty means less achievement” gap creating the positive correlation with direct instruction.
Now, the report still claims the instruction type is independently correlated with gains or losses (so that while the data above is a real effect, it doesn’t account for everything). However, there’s one other highly fishy thing about the chart above that makes me wonder if the data was accurately gathered at all: the first line.
It’s cryptic, but essentially: males were given direct instruction to a much higher degree than females.
Unless there’s a lot more gender segregation in Canada than I suspected, this is deeply weird data. I originally thought the use of direct instruction must have been assessed via the teacher survey:
But it appears the data instead used (or least included) how much direct instruction the students self-reported:
The correlation of 10.67 really ought to be close to 0; this indicates a wide error in data gathering. Hence, I’m wary of making any conclusion at all of the relative strength of different teaching styles on the basis of this report.
…
Robert also mentioned Project Follow Through, which is a much larger study and is going to take me a while to get through; if anyone happens to have studies (pro or con) they’d like to link to in the comments it’d be appreciated. I honestly have no disposition for the data to go one way or the other; I do believe it quite possible a rigid “teaching to the test” direct instruction assault (which is what two of the groups in the study seemed to go for) will always beat another approach with a less monolithic focus.
]]>On occasion a friend or relative who’s got schoolkids asks me how horrified I am by some bit of Common Core mathematics. This is a good chance for me to disappoint the friend or relative. Usually I’m just sincerely not horrified. Much of what raises horror is students being asked to estimate and approximate answers. This is instead of calculating the answer directly. But I like estimation and approximation. If I want an exact answer I’ll do better to use a calculator. What I need is assurance the thing I’m calculating can sensibly be the thing I want to know. Nearly all my feats of mental arithmetic amount to making an estimate. If I must I improve it until someone’s impressed.
The other horror-raising examples I get amount to “look at how many steps it takes to do this simple problem!” The ones that cross my desk are usually subtraction problems. Someone’s offended the student is told to work out 107 minus 18 (say) by counting by ones from 18 up to 20, then by tens from 20 up to 100, and then by ones again up to 107. And this when they could just write one number above another and do some borrowing and get 89 right away, no steps needed. Assuring my acquaintance that the other method is really just the way you might count change, and that I do subtraction that way much of the time, doesn’t change minds. (More often I do that to double-check my answer. This raises the question of why I don’t do it that way the first time.) Though it does make the acquaintance conclude I’m some crazy person with no idea how to teach kids.
That’s probably fair. I’ve never taught elementary school students, and haven’t any training for it. I’ve only taught college students. For that my entire training consisted of a single one-credit course my first semester as a Teaching Assistant, plus whatever I happened to pick up while TAing for professors who wanted me to sit in on lecture. From the first I learned there is absolutely no point to saying anything while I face the chalkboard because it will be unheard except by the board, which has already been through this class forty times. From the second I learned to toss hard candies as reward to anyone who would say anything, anything, in class. Both are timeless pedagogical truths.
But the worry about the number of steps it takes to do some arithmetic calculation stays with me. After all, what is a step? How much work is it? How hard is a step?
I don’t think there is a concrete measure of hardness. I’m not sure there could be. If I needed to, I’d work out 107 minus 18 by noticing it’s just about 110 minus 20, so it’s got to be about 90, and a 7 minus 8 has to end in a 9 so the answer must be 89. How many steps was that? I guess there are maybe three thoughts involved there. But I don’t do that, at least not deliberately, when I look at the problem. 89 just appears, and if I stay interested in the question, the reasons why that’s right follow in short order. So how many steps did I take? Three? One?
On the other hand, I know that in elementary school I would have had to work it out by looking at 7 minus 8. And then I’d need to borrow from the tens column. And oh dear there’s a 0 to the left of the 7 so I have to borrow from the hundreds and … That’s the procedure as it was taught back then. Now, I liked that. I understood it. And I was taught with appeals to breaking dollars into dimes and pennies, which worked for my imagination. But it’s obviously a bunch of steps. How many? I’m not sure; probably around ten or so. And, if we’re being honest, borrowing from a zero in the tens column is a deeply weird thing to do. I can understand people freezing up rather than do that.
Similarly, I know that if I needed to differentiate the logarithm of the cosine of x, I would have the answer in a flash. It’d be at most one step. If I were still in high school, in my calculus class, I’d need longer. I’d struggle through the chain rule and some simplifications after that. Call it maybe four or five steps. If I were in elementary school I’d need infinitely many steps. I couldn’t even understand the problem except in the most vague, metaphoric way.
This leads me to my suggestion for what a “step” is, at least for problems you work out by hand. (Numerical computing has a more rigorous definition of a step; that’s when you do one of the numerical processing operations.) A step is “the most work you can do in your head without a significant chance of making a mistake”. I think that’s a definition that clarifies the problem of counting steps. It will be different for different people. It will be different for the same person, depending on how experienced she is. The steps a newcomer has to a subject are smaller than the ones an expert has. And it’s not just that newcomer takes more steps to get to the same conclusion than the expert does. The expert might imagine the problem breaks down into different steps from the ones a newcomer can do. Possibly the most important skill a teacher has is being able to work out what the steps the newcomer can take are. These will not always be what the expert thinks the smaller steps would be.
But what to do with problem-solving approaches that require lots of steps? And here I recommend one of the wisest pieces of advice I’ve ever run across. It’s from the 1954 Printer 1 & C United States Navy Training Course manual, NavPers 10458. I apologize if I’m citing it wrong, but I hope people can follow that to the exact document. I have it because I’m interested in Linotype operation is why. From page 308, the section “Don’t Overlook Instructions” in Chapter 7:
When starting on a new piece of copy, or “take” is it is called, be sure to read all instructions, such as the style and size of type, the measure to be set, whether it is to be leaded, indented, and so on.
Then go slowly. Try to develop even, rhythmic strokes, rather than quick, sporadic motions. Strive for accuracy rather than speed. Speed will come with practice.
As with Linotype operations, so it is with arithmetic. Be certain you are doing what you mean to do, and strive to do it accurately. I don’t know how many steps you need, but you probably won’t get a wrong answer if you take more than the minimum number of steps. If you take fewer steps than you need the results will be wretched. Speed will come with practice.
]]>Why New York State has been changing exams so frequently over the last 15 years (disruptive innovation) should be the subject of another post.
But my bread and butter exam has been Course I. Then Math A. Then Math A (adjusted). Then Integrated Algebra. Now Common Core Algebra.
What you do with these exams depends on who you are, and where you are. At Columbus I taught a course that took kids who had already failed Course I multiple times. Me and Bill Gerold taught it. And we figured out ways to get a kid who tried hard to break that 65. It was kind of amazing. And given our success, and it was success keeping the school off the SINI list or whatever it was called then for a year, the administration refused to offer the course again. At my school today I teach a fairly old-fashioned algebra course, with fairly heavy emphasis on mathematical understanding, challenging problems, lots of fractions, rich discussion of process, but not much emphasis on real-world connections. And then I build in the supplements for the exam. Obviously this means different supplements every time they change the exam.
It works fine. Kids get scores in the 80s or 90s. Once I had a kid with 100, but that was not my fault. The kids, the school, the parents, they all care about the scores. More than they should. Since my kids already know some math (they can all add fractions) when they arrive, and they all can take a standardized test (they get into the school by passing a test), the pressure is not on passing, but on getting scores that look good. And on getting the average for all of their regents exams to be at least 90, which qualifies them for an “honors designation” on their diplomas. It’s a sticker, and I give out nicer stickers, some more colorful, some scratch and sniff, some glittery, but the kids want this sticker in particular.
So the new Algebra regents (common core) hits last June, 2014, and I was on sabbatical. But I heard from around the state, from the AMTNYS listserve, and from talking to people, and from my school, that scores for strong kids were down 5 – 10 points.
So we set about scrambling to see why the scores were down, and what we could do to bring them back up. I went to the AMTNYS conference in Syracuse in October. I talked to people. Teachers, professors. Consensus was that those who used the “modules” ran out of time, those who taught the old curriculum watched the grades fall a full 10 points, or 10+ even, and those who used a reduced sampling from the “modules” did best. The modules are NY State supplied materials that probably would require a 300 day school year to teach completely (we have 180. There are 260 weekdays in a year).
There was my answer – adopt portions of modules. Of course I did nothing of the sort. I took an already rich function unit, and expanded it. I added a couple of stats topics, taught differently than I had in the past, emphasizing equally what the stuff means, and how to calculate it. I ended the year with a week and a half of intensive test prep. And my students did well. But the scores were 5 – 10 points lower than I would have expected. Mid 70s through mid 80s. Something was wrong.
I’ll tell you what went wrong in the next post.
Teaser: NY State lowered top scores intentionally.
]]>Imagine the following setting: we have a sample of independent normally distributed variables, all with mean and variance . Suppose is known and we want to test whether some value is a feasible guess for .
H0:
H1:
The way we would typically go about this is by calculating the sample mean and saying that under H0 it follows a distribution. This means that under H0 the test statistic
follows a standard normal distribution and we can proceed with calculating the p-value of this test. Easy!
What happens if we don’t know ? It is tempting to use the (unbiased) sample variance
to estimate and substitute in the statistic above.
Assuming we can compute an approximate p-value that should be fine… right?
Let’s forget about the algebra of it for a second. In the first case, we had some uncertainty about the mean and no uncertainty whatsoever about the variance. What the hypothesis test does is it checks whether the uncertainty that we have is enough to explain the difference between and . Not knowing the variance gives us some extra uncertainty on top of what we had before. The more uncertain we are of what we know, the more tolerant we should be to reality not matching our expectations.
Intuitively at least, the approximate p-value above is going to be somewhat conservative, since we’re not accounting for the variability of the sample variance. However, we still expect to behave more or less like a standard normal — especially if our sample size is large, in which case we’re very confident about our estimate for . We can make several guesses about the density of based on this idea alone:
it looks more or less like a standard normal (i.e. like a bell curve)
it has wider tails
it’s not fixed, but depends on : the more data we have, the better our normal approximation
I do love informal approaches, but the academic community doesn’t necessarily share my enthusiasm. In any case we can all agree that knowing the exact p-value is preferable to making an approximation. This is where the t-distribution kicks in.
Definition. Let and be two independent random variables. The t-distribution with k degrees of freedom is defined to be the distribution of .
While you can write the density function explicitly, it’s the form above that is the useful one. I won’t go through the algebra of it, but using it you can check that . This means we can compute exact p-values for the hypothesis test (or rather we can let R compute them for us). This is what Student’s t-test amounts to!
Now for the really cool part. We can plot the standard normal and several t-distributions with varying degrees of freedom. Here’s what happens:
Our intuition was pretty spot on!
]]>The Inequality:
a/(b+c) + b/(a+c) + c/(a+b) ≥ 3/2
Proof: To solve this inequality we have to see a few facts first
(a+b+c){1/(b+c) + 1/(a+c) + 1/(a+b)} ≥ 3/2 +3 = 9/2
m+n+p = 2(a+b+c)
(m+n+p)(1/m + 1/n + 1/p) ≥ 9
(m+n+p)/3 ≥ 3/(1/m + 1/n + 1/p)
So the last result which we get in terms of m, n, p is basically what we would get if we applied the A.M ≥ H.M inequality on the three numbers m, n and p.
That basically completes the proof of the A.M ≥ H.M Inequality.
In my next blog I would be discussing a Putnam competition problem based on the A.M ≥ G.M ≥ H.M Inequality. More problems based on A.M-G.M-H.M Inequality will be posted by me in the next couple of weeks. So don’t forget to keep an eye.
Well as promised by me in my earlier blog you can post inequalities as comments , and I would try to give the solution of most of the inequalities which you post.
]]>This image of three rhombicosidodecahedra “orbiting” a common center was made with Stella 4d, a program you may try for free at this website.
]]>A year or two after I’d finished at art school I would still occasionally pop into the pub frequented by the art students. One night I bumped into an acquaintance who was just starting the final year of her fine art degree at the college. She mentioned that she was a bit worried because she didn’t have any ideas for her artwork – both her real and her metaphorical canvases were blank – she was suffering from artist’s block…
Having consumed half a pint of Guinness, and feeling rather confident I said, ‘Hey, tell you what, I’ll set you a project!…’
Happy to clutch at any straw that came along she said, ‘Oh, okay…’
I thought back to a few ideas that I’d had back at college, but had never got around to making. This is the one I suggested.
I’ve always had a glancing interest in mathematics and numbers (but without much actual skill in the field) and one day I began wondering what a million things would look like, and if it would be possible to clearly see exactly one million things, and (bearing in mind that those were the days of ‘conceptual art’) if an artwork could be made showing this. My idea was to buy some sheets of graph paper printed in tenth of an inch squares and mount them on all a large board edge to edge. The board would be 100 inches square – one million small squares. A rather boring looking piece of art, but possibly of some ‘conceptual’ interest to someone.
Of course for my friend this wouldn’t do at all, as she could have it all finished in a couple of days. I suggested that she should instead do series of works based on large numbers. Perhaps get a large sheet of paper or a canvas, and inform everyone, including the tutors, that she would be counting out and painting on it a specific large number of dots – say 5,000, or 10,000, it doesn’t really matter so long as it is impressively large. She should do this daily in a place where the maximum number of people could see her doing it; they could chat and ask her questions, and be supportive in her great laborious task…
This project had three things going for it: It was novel, and eye-catching; it was painting, sort of (even the painters on the staff might be impressed); and people could see her putting the hours in – this is always impressive at art college.
Anyway, she was busy and visible doing quite a few of these ‘number paintings’ over the following academic year, and apparently at the end of it she was awarded a first class degree.
I wonder if she’s out there somewhere, and still painting dots?
Note: Bring Geometry NB (It is Like Science NB) tomorrow.
Revision Task:
Do the following questions in the NB1.
1) Find the least number which when divided by 6,15 and 18 leave remainder 5 in each case.
2)Find the least number which divided by 12, 16, 24 and 36 leave remainder 7 in each case.
3) Find the smallest 4 digit number which is divisibly by 18,24 and 32.
4) For each of the following pair of numbers , verify that
LCM x HCF = Product of the numbers.
Time : 30 min MI: Logical RBT : Application
(Numbers are given, Find LCM and HCF and then multiply them.Multiply the numbers also.Now check the above statement )
(i) 21, 28
(ii) 36, 90
А каким можно доверить все свои деньги? Мнение эксперта. From Ancient Greek times, music has been seen as a mathematical art, and the relationship between mathematics and music has fascinated generations. The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics (Sterling Milestones) [Clifford A. Pickover] on. Before 520 AD, on one of his visits to Egypt or Greece, Boyer, C. B. (1968) mentioned that Pythagoras might have met the ca. 54 years older Thales of Miletus, it is. Amazon.Com and Bn.Com. The Scientific American book club sometimes offers The Math Book for $1.99. Fancy edition: Bound in genuine leather and accented with 22kt. 1 Introduction; 2 History; 3 Characteristics; 4 Brownian motion; 5 Common techniques for generating fractals; 6 Simulated fractals; 7 Natural phenomena with fractal. Washington, D.C. – Richard Hunt, President and CEO of the Consumer Bankers Association (CBA), issued the following statement after MeasureOne released “The. An Introduction to Contemporary Mathematics John Hutchinson (suggestions and comments to: John.Hutchinson@anu.edu.au) March 21, 2010. 12.06.2012 Pythagoras had a problem with beans and irrationality. What really happened? I don’t know! The square root of two is irrational, and beans are delicious. I About the Author David Wright is professor of mathematics at Washington University in St. Louis, where he currently serves as Chair of the Mathematics Department. Your mathematics seem to me very like a bottle of mixed pickles the more you fish for what you want the less chance you have of getting it.
]]>I constructed the “Ann and Bob” examples of sign relations back when I was enrolled in a Systems Engineering program and had to explain how triadic sign relations would naturally come up in building intelligent systems possessed of a capacity for inquiry. My adviser asked me for a simple, concrete, but not too trivial example of a sign relation and after cudgeling my wits for a while this is what fell out. Up till then I had never much considered finite examples before as the cases that arise in logic almost always have formal languages with infinite numbers of elements as their syntactic domains if not also infinite numbers of elements in their object domains.
The illustration at hand involves two sign relations:
Each of the sign relations, and contains eight triples of the form where is an object in the object domain is a sign in the sign domain and is an interpretant sign in the interpretant domain These triples are called elementary or individual sign relations, as distinguished from the general sign relations that generally contain many sign relational triples.
If this much is clear we can move on next time to discuss the two types of reducibility and irreducibility that arise in semiotics.
To be continued …
]]>I am Kurtis Evans. In elementary school, I was assigned a project to profile a famous contributor to society. My teacher generated a list of names to choose from. Most of the names were unfamiliar to me, so I picked randomly – Stephen Hawking. It felt as if I was inside a black hole as I learned about astrophysics for the first time in my life. That discovery awoke in me a fondness for the physical world and space through the lens of science and numbers. Today, I am enrolled in Seattle Pacific University’s (SPU) Master in Teaching Math and Science (MTMS) program as a prospective mathematics and physics teacher. I intend to continue my personal education in these fields, and to share my appreciation with students. I hope that my students will develop a basis of knowledge that will allow them to push forward as future engineers, doctors and scientists
Math and science teachers are responsible for providing instruction and knowledge in their particular subject area in as much that students who wish to continue in that discipline are prepared for the next level of education. Standards developed at the state and national level detail what students should know, and provide sequential learning progressions. Four of these standards are: Next Generation Science Standards, Washington Science Standards, Common Core State Standards (CCSS) in Mathematics, and Washington State Mathematics Standards (Balliram 2015). These provide a basis for science and math teachers to know what should be taught. A better education, however, may include historical references and tie in non-related disciplines to humanize the learning experience. My intentions are to demystify historical figures and their discoveries in mathematics and science, and to encourage my students to visualize how they too can become part of the historical narrative.
A qualified math or science teacher must have a deep understanding of the subject matter. When teaching mathematics, for example, an expert teacher will be able to identify when students struggle with particular content, and identify the source of confusion. Math teachers who understand the foundations of their subject can tailor instruction to meet their students’ needs (Watts-Taffe et al., p.304). Gritter (2013) indicates that teachers must have critical encounters with adolescents who struggle in content areas and provide literacy tools to close the knowledge gap. In order for teachers to engage these adolescents, they must have a fluid understanding of the subject material.
I plan to incorporate writing, reading and critical thinking activities in my math and physics classrooms. Massey explains that teachers “need to be specific about what [they] do with the text [they] use in the classroom and explicit about the way [they] think” (Skinner 2013). Literacy in mathematics and science education is often overlooked and many students stagger accordingly. Checkley claims that “people who don’t understand algebra today are like those people who couldn’t read or write in the industrial age” (2002, p.6). It is imperative that I incorporate new literacy strategies in my classroom to give students a solid foundation in their math and science education. Typically, this would give students strategies for breaking down textbooks. However, I would also like to incorporate different types of texts, such as related articles or historical biographies with the intent of teaching strategy; visualization, what’s this in math and connecting new information to a personal level (Skinner). I would also like to assign hands-on learning projects, such as origami, and have students explain the geometry behind their folds.
As a secondary math and science educator, I see myself dealing primarily with hard and applied knowledge. Abstract and theoretical applications come with advanced studies in these fields, but my students will need to develop fundamental knowledge before arriving at that level. My students will be able to learn specific formulas and definitions, and apply them accordingly. They will require this type of instruction in order to meet standards associated with the CCSS. I also plan to treat knowledge of mathematics and physics as celebrating experience. With hard applicable knowledge in these fields, I want my students to understand that specific, valued careers are available with these skills. This knowledge is a means of personal advancement. Although I do intend to encourage conversation and debates regarding historical figures and outcomes, the intent will be to develop mathematical and scientific curiosity in my students. This would in turn promote hard knowledge and its application.
The issue of improving literacy in the content areas of math and science is crucial to closing achievement gaps and providing quality education for all students. Gritter (2010, p.164-166) identifies four steps for all content area teachers to improve literacy: develop a deeper respect of existing student knowledge, engage struggling adolescents in content areas and provide literacy tools to close knowledge gaps, increase cross-discipline collaboration, and review what is considered text in content areas. These suggestions indicate that exceptional teachers will attempt to develop relationships with their students and respect them as learners. In addition, I recommend more research into how the developed mathematical mind functions. Massey’s (Skinner) experiment with a professional math teacher identified his personal process in solving math problems. These finding indicate that successful problem solving begins with visualization. Math teachers should incorporate visualization strategies into their classroom, including hands on learning projects, as a function of math literacy. Additional interviews with accomplished mathematicians will provide a basis for understanding successful problem-solving strategies and for establishing a context for teaching mathematical literacy.
I have found the content of this course fascinating. I had not considered the literacy of math and science previously. What I think would be particularly useful are more resources specifically addressing mathematics and literacy. I would like to see more examples to build my understanding for how to be an expert literacy teacher in mathematics.
References
Balliram, N. (2015). EDU 6132 – Module 2. Retrieved July 1, 2015, from https://spu.techsmithrelay.com/rTsN
Gritter, K. (2010). Insert Student Here: Why Content Area Constructions of Literacy Matter for Pre-service Teachers. Seattle Pacific University, Seattle, WA. 147-167.
Massey, D. & Riley, L. (Speakers). (2013, March 27). International Reading Association [Audio podcast]. Retrieved from http://www.reading.org/downloads/podcasts/jaal-56-7-DixieMassey.mp3
Watts-Taffe, S., Laster, B.F., Broach, L., Marinak, B., Connor, C.M., & Walker-Dalhouse, D. (2012). Differentiated Instruction: Making Informed Teacher Decisions. The ReadingTeacher, 66 (4).
]]>Okay so we say that we have a cuboid with dimensions where we conventionally say that it has volume meaning it has the same volume as a cube with side . Let us use dissection to motivate this; that is describe a procedure of cutting up the cuboid and reassembling it into the prescribed cube.
We will essentially perform regular rectangle to square dissections face-wise splitting the cuboid into prism.
Let us start with the face and dissect it into a rectangle having one side and the other (by the area formula) being .Applying this to the whole cube the cuts through the face are extended through the solid along planes perpendicular to the face forming prism which are rearranged to form a new cuboid with as one of it’s side.
Note the depth of the cuboid is unaffected by this procedure.
(You might need more pieces than the image indicates)
Next we turn out attention to the face with sides and and perform the same procedure where this face is transformed into a rectangle with one side being by construction and the other side
And so this face is square! All the sides of this cuboid are therefore equal and we’ve ended up with a cube.
*This procedure is not original but my interpretation hasn’t been checked so some caution should be taken
Sidenote: Just as triangles and polygons can be dissected to rectangles a prism can therefore be disected into a cube as well and paying attention to the details of such a procedure you recover the conventional formula for general prism. For the cylinder we don’t have any elementary means of dissecting into a cube but just as you may approximate a cirvle by triangles you can approximate the cylinder by triangular prisms and obtain the standard formula.
]]>Practical experience and consultation with peers has shown students struggle with the literacy involved in FM5. As a result, FM5 has been allocated with 10 lessons, not the 6 lessons indicated in the Level 1 S&S. This is still aligned with the Level 1 S&S, in lieu of the revision lessons planned.
Once FM5 has been taught, a teacher would ideally give students revision in the form of a holiday homework assignment covering all 4 topics previously taught (AM3, MM4, FM4, FM5). This would also be an appropriate time to give students a preliminary yearly exam to revise pre-requisite knowledge for later topics.
]]>
Once, in the previous week, we started to discuss an education topic with my coworker during the launch. And, when we came to our working places, he shared with me a link to a work of Paul Lockhart. This is Paul Lockhart. A Mathematician’s Lament, which was translated in Russian language, and you can find russian translation here, and the original version you can find here.
I liked this work, I read with great interest. A similar obstacle is raised in a “Dead Poets Society” film. And Gauss said about it: “What we need are notions, not notations.”