Myrtle
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Posts posted by Myrtle
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I think you are on the right track by easing up on speed drills. But rather than looking at this as a curriculum issue, I'm thinking that it's a temperament issue. I've had two kids that were uptight about making mistakes in math and with one of them it was because he had a very low tolerance level for frustration and the other thought there was something wrong if she didn't get a problem correct.
My goal at that point was to not worry about making progress in a particular topic...getting all those addition facts down before October or something, but simply to convey to her that it's okay to stop and think as looong as you need to about any given problem. "And you know what? There was a boy who was ten years old that went to the library and saw a neat problem. Do you know how long it took for him to solve it? It took him 30 years! (Andrew Wiles) Math is about sticking with the problem until you can get an answer so if it you spend a long time working on a problem, do you know what? That makes you a really hard worker and good at math."
I would break down problem sets into more manageable and less intimidating chunks. I'd give hints, but not without asking first if she wanted one, and make a mental note of what I needed to reinforce. I'd go through a melodramatic routine (and this wtih both kids) when they'd make perfect scores saying, "But if you got everything right that means you didn't learn anything" If they get something wrong, I get excited and talk about, "Well, let's go over this so that you can learn something in math today." And at the end happily ask, "Tell me what you learned today!" (expecting discussion of the mistakes, not the lesson)
Here was another one of my favorites when they show up all happy they did a worksheet in record time, I'd say "How is this a problem if you solved it so quickly? This wasn't a problem for you. Problems are things that you have to think about to solve." I am not sure how to convey my tone of voice when I say that, it's more playful than serious.
I've taken the mistakes she made from one sheet, rewritten only those mistakes and then had her solve those and then pointed out, "Yesterday you had to think really hard for these problems, but not today!" So that she could see that she makes progress. I've taken out old math from a year ago so that they can see how easy it looks now and said, "And what you are doing now is going to look just that easy. Do you believe that?"
I have been very motivated by Michael Atiyah, a Field's Medalist who said, and I can only paraphrase, he said something like, "There are two ways you can be good at math, one way is to just be smart. The other way is to be dumb, but very, very persistent."
Now, I didn't cure the tears overnight by saying those kinds of things, it took months, and with my older son I still have to give him pep talks, but I think they do "get" the fact that math is not easy and that real math isn't about being trained to react like a human calculator but by sitting down and puzzling through problems, which takes patience.
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Is she a perfectionist?
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How about demonstration of half-life, exponential decay.
Holly
LOL
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instruction seems different. For the life of me I can't remember how to do this, but I do clearly remember being taught how to estimate square roots in algebra, and also how to use a slide rule.
For all three of my girls we have not had to rely too heavily on manipulatives. I used flashcards for a while; for our family, this was the most productive manipulative around:
We used them for addition, subtraction, multiplication, and division. I can't keep them around the house anymore---I don't need the temptation!
If anyone remembers how to estimate square roots, I'd love to know!
Estimate them or extract them by hand?
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Yoo hoo, genie!!
Genie is up-to-speed on the rep issues, so maybe she'll pop in and answer.
She's told me it's 30.
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I'm going to redo your proof the way we'd do it:
6a - 3 + 3 = 3a + 1 + 3 Justification: Equals added to equals are equal (equality axioms)
6a + (-3+3) = 3a + 1 + 3 Justification: Associative property. There's also a sneaky thing going on in there with the signs that I'm leaving out for the sake of simplicity.
6a + 0 = 3a + 1 + 3 Definition of additive inverse.
6a = 3a + 4 ...
In our algebra program the addition or subtraction of natural numbers is simply called "an addition fact" They can actually be proven using a set of axioms not used in algebra called Peano's axioms, but that is outside the scope of algebra. In general, in algrebra we ASSUME the basic properties of natural numbers and PROVE that these properties still hold for the integers, rationals, and real numbers as we go along through the school year. We wouldn't use a phrase such as "combine like terms" to explain why a different expression such as 6a - 3a is equivalent to 3a. "Combine like terms" isn't a justification or explanation of equivalency, it's a command to do something. Instead, we'd use the distributive law 6a - 3a = a(6 -3) ...and here, to be rigorous, we should stop to prove that the distributive law can in fact apply to expressions with subtraction, rather than assuming that it can, since the distributive law as officially stated only works for addition with two terms. You have to prove that it distributes over more than two terms, you have to prove that it distributes over subtraction. (I have one personal question about assuming that the properties of real numbers can be applied to "x"...JANE! Help! Is it because when you adjoin x to R that by definition the properties still work?)
This takes up days, if not weeks, of algebra instruction.
6a - 3a = 3a + 4 - 3a "Equals added to equals are equal" (Having already shown/proved that adding a negative is equivalent to subtracting a postive, the student can invoke "equals added to equals"
3a = 4 Again, we'd use the distributive law, not "like terms"
3a x 1/3 = 4 x 1/3 Equals multiplied by equals are equal
a x 3 x 1/3 = 4 x 1/3 Commutative property
a x 1 = 4 x 1/3 definition of multiplicative inverse
a = 4 x 1/3 definition of multiplicative identity
a = 4/3
This last step can be justified after you've proven that for any a,b
a x 1/b = a/b (and that alone requires you to begin by proving that the multiplicative inverse is unique.) Here is how that is done. Although there is some notational ambiguity that is clouding up the proof in this example.
The idea behind the proofs it to teach the kid what constitutes justification, and what constitutes assumption. An algebra program that simply throws out "combine like terms" as justification without actually giving mathematical justification is training the student to think fallaciously. Now it's not that a good math book never has the student assume anything, you have to assume some things in order to make any progress, but the difference between a good math book and a bad one, is that the bad ones don't tell you what and when you are assuming. I can't tell you how many times that Serge Lang in the middle of teaching math says something like, "We shall assume the basic properties of distance without proof" (and this is in a high school book) thus helping the student to distinguish between justification and assumption.
If you want to know more about proofs in high school algebra get Mary Doliciani's Introduction to Analysis from the 1960's or Serge Lang's Basic Mathematics. The latter is compressed high school math (4 years in one book) and very appropriate for an adult reviewing it for themselves. If you want to focus solely on learning what consititutes justification and what does not and why then Patrick Suppes First Course on Mathematical Logic is a good start. It's written at the high school level and we're using it as a course this year in seventh grade along with finishing up algebra.
So far, I have taught proofs by having my son memorize the general form of the proof and then use those steps in the following exercises by solving equations two-column style. But you are right, at some point, you skip steps because you know them and use the theorem that you now know in applications.
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Why is math instruction so complex? I went to elementary school in the 70's. Do you guys remember math in elementary school? I don't completely remember it, but there are certain things I know that we did not do. I don't remember any of the mental math strategies from Singapore. I don't remember using manipulatives like Math U See and RS. I don't remember seeing division in 3rd grade like BJU or long division like Abeka. And did I truly see Horizons introducing integers in 3rd grade? What the....:001_huh: I must have read that wrong!
Does anyone remember math in elementary school?
Holly
I make an analogy of math ed to computing power.
30 years ago we knew a lot less than we do now when it comes to computers and as a result we can do things harder, better, faster, stronger.
Math ed has a much longer half life, about every 50 years it gets a new makeover.
In the 1800 only geometry was really taught at the high school level and then it was taught to only a few students. Arithmetic wasn't even required to enter Harvard until the early 1800s. Very basic algebra wasn't required to get in until the mid 1800s ad now of course people talk about needing AP Calculus.
Calculus was taught in the junior or senior year of college, and wasn't taught to freshman until, I tink 1920. It was the Americans that transformed calculus into a more or less theoretical discussion about math into a course about practical applications in physics and engineering.
In the 1960s as the result of New Math Calculus was pushed down to the high school level for the first time.
Other countries have found ways of trimming the fat and making slick connections between arithmetic concepts, they are very carefully scaffolded in some cases, and we are seeing that in those countries algebra is taught in 7th and 8th grade.
Sound great?
It does, in a way. But consider this in regard to the use of manipulatives:
Math, by definition, is abtract. The real concepts that explain why math works the way it does are not to be found in the physical world in the form of manipulatives. Axiomatic systems and formal defintions are what makes math work.
So we can teach kids that "greater than" means that the alligator mouth eats the big number, it isn't possible to teach it based on ordering axioms, something that you could do if you waited until the high school level. The curriculum publishers, for the most part, have no interest in revisiting concepts such as these and teaching them correctly because that sort of thing doesn't sell. What sells with consumers is saying that a kid is ready for calculus earlier and earlier. Who knows? Maybe we'll see delta episilon manipulatives soon. I've already seen companie selling hoola hoops as "set theory" manipulatives and doing it with a straight face.
Math Manipulatives I'll never own:
640 plastic farm animals, dinosaurs, cars, fruits, learning links, pattern blocks and multicolored chips for about $60.
I find the idea of grabbing handfuls of toys off the floor of my house and selling them at this price appealing. I have a sorting activity for you: Go clean up your room.
Would you like to supersize those manipulatives?
It's a Happy Math toy served with a smile. $16
I thought I could get away with printing out my fractions using pie charts in Excel. I missed the complete McMath experience with FOOD fractions. $20.
Manipulatives for teenagers..I can't imagine why they aren't prepared to think abstractly... The price tag? A mere $222.
An interesting discussion among high school teachers and mathematicians about the use of algebra tiles. (scroll down to see the responses) If you wind your way through that thread don't miss the snark about cutting tiles to irrational lengths.
PS In general we try to avoid manipulatives as much as possible (after 3rd/4th grade) and I spend a lot of time thinking about ways of making abstract ideas accessible at a younger age. As a result of the "multiplication ain't no repeated addition" debate,for example, I sat around and wondered what else couldn't be done to teach the properties of multiplication at in the 3rd and 4th grade in an abstract way. But at any rate, it is never the case that you simply can't teach a math concept properly without expensive plastic. There is always a cheap alternative already in your house.
Manipulatives are crutches and as such are hinderances as much as they are help, if only because they fasten the imagination on a particular example in the physical word when what follows is wonderfully general. "Keep that crutch if you must; discard it when you can; limp if need be."
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What does Devlin mean by wrong? In what way is it wrong? Are there any studies to back him up?
I'm going to follow your link. This is interesting.
Holly
Holly, Devlin is using some sneaky rhetoric and making both a mathematical claim and a pedogical one all in the same breath.
Pedogogy can be backed up with studies, like you say. But what's really roiled up the math people is that his math is wrong. He's making a claim for something that a junior year math major would know. And that is just astounding considering his math background. A math claim isn't backed up by a study, but by a proof. He hasn't given that either and Mark goes on and on about Devlin's rhetoric. (In math you don't assert over and over again your claim, you actually need to demonstrate what you are saying is correct, so this is why you see an explosion of symbolic vomit on the part of some of the math people in these discussions)
I had one person give me the name of a book which supposedly developed the real numbers, i.e. would give me a demonstration, with some quirky set of axioms but from what I've seen so far it's just very faulty since he uses informal assumptions without proof.
The debates so far have consisted of teachers who don't know the higher math trying to figure out if it might make sense to try to do something new and grad students and math people who are not familiar with what's invovled in teaching the concept to kids, they just look at the math claims, not the teaching claims. Mark is looking at both.
My favorite response so far was in reference to Field's Medalist Tim Gowers saying that it should be taught axiomatically! Technically, that is how it works, but you can't do that with an 8 year old!:lol:
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In a thread below I had posted about a very long internet debate that was taking place as a result of Keith Devlin (NPR's math guy and Stanford logician/mathematician) telling teachers that teaching kids that multiplication is repeated addition is wrong.
The debate spread across four or five different forums, privately through email, and finally Mark Chu from Good Math/Bad Math picked this up. The Good Math/Bad Math blog is the top of the math blog food chain as far as quality and traffic goes and he does a very good job, as usual, of making complex mindless formalism understandable for us chickens.
This whole debate was fascinating to me because 1) it's math ed poltics in real time. Ground zero, baby.
and 2) higher math..as in higher than calculus...gets invoked really quick to come up with a solution and 3) It illustrates what H Wu means when he says that teachers need to have better math educations in order to make better decisions even at very low levels of teaching math.
I'm motivated to keep larnin' and I just wanted to share this with anyone who had been following this on the links I had given earlier.
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The natural numbers are used often in series or iterative processes
Best,
Jane
Unless you are Keith Devlin.
HIJACK: Have you seen his latest? "Multiplication Ain't no repeated addition"
And if you could read it and shed some light on what he could possible mean I'd love to know. We're scratching our heads at our house and I just did a blog entry on it. It turned into an entertaining and amusing flame war on Denis's "Let's Play Math" blog (she closed the comments but it's all still there)
It would be perhaps elucidating for me to see this discussed by cooler heads. I asked my FIL who dismissed it as crackpottery and in fact, I'm having problems getting anyone with a degree in math to take what he's saying seriously and explain it to me.
The only person who is taking Devlin's side on this recommended an analysis text which supposedly develops the real numbers from four axioms. It's by a Polish mathematican called Mikusinksi and I went ahead and ordered the book (it was only $12)
And the person that originally recommended this book also kindly emailed me the first 10 or so pages. After reviewing them it would seem that perhaps Mikusinski is able to use only four axioms because he assumes the rest informally. In fact, in the first paragraph of the book he dismisses the need to define inequality or introduce ordering axioms because "any school child understands this"
I will suspend judgment until I see the rest of the book though.
Plaid Dad:
I think set theory was invented by Cantor and it was Richard Dedekind who then showed how the different sets are related. Prior to Dedekind everyone was using Euclid to explain rational numbers and Dedekind was the first to say, "It's not a good idea (proof wise it didn't work)to use geometry to explain arithmetic, arithmetic and algebra need their own set of axioms" He was disatisfied with the gaps in logic that the explanations involved and tightened up the mathematical arguments used. In other words, there were mathematical problems that came up and Euclidean axioms and ideas couldn't be used to solve these by 19th c mathematicians. These problems were solved with Cantor's set theory, Dedekind, and Peano.
Cantor and Dedekind are very important in the history of ideas, not just math, but because they contributed a lot to how we view "infinity" and the origin of these musings of infinity was with the Greeks. If one is watching how the infinity concept baton is handed off in Western Civ, Cantor and Dedekind can be seen running with it.
I guess I should add that infinity was explored in the idea of "infinite" sets, and infinite numbers (continuity) between any two given numbers. At the time in math they were some glaring mistakes made by Euler, (for one,) on how to handle this. (It comes up as a division by zero problem, for example, back in the day people didn't say that this was undefined) and also dealing with the "infintesimals" ...an idea used in 18th century calculus. The ancient Greeks, preferring to avoid assumptions, avoided the idea of infinity as much as they could in their math and there is some neat stuff in Euclid where you can see him contorting his proofs to avoid bringing it up directly.
I'm having a bit of notational issues with set theory myself. The Germans seemed to have used some interesting German letters for this and I can't make out what's up with it.
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It is sold through Geography Matters but their website says it is "temporarily unavailable." Is it sold out or are they bringing out a new version?
I was just looking at this for DD11. It says 6th grade through high school. What levels have folks successfully used this program? How helpful is the extra activity book?
Thanks,
Pegasus
I ordered it and got mine last week but a note said that the student book was on back order.
I have no idea what's in the student book.
The rest of the program looks great. This may be one of the best homeschooling finds ever for me.
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We started out fine on the first few pages. Then we hit the number bonds and it is way.over.her.head.
I have never used the early bird math. Should I start with that or wait a little with 1A? Opinions?
All lessons in Singapore at this level are supposed to be preceded by doing the activity with manipulatives.
concrete--->pictoral--->abstract
Start with groups of 5 objects and show her how they can be partioned physically into 1,4 etc.
In fact, I made a large triangle number bond and placed the five objects in the big circle and then split it up by moving the small groups to the little circles.
It took a lot of practice with this...months.
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My youngest dd was diagnosed as being on the Autism Spectrum when she was two. The diagnosis was reconfirmed when she was four due to my request.
After the first diagnosis, I sank us into every type of therapy recommended for her: physical therapy, occupational therapy, play therapy and speech therapy. At the age of four the only issues she was continuing to have were speech and a slight delay (6 months to 1 yr) in her thinking skills. I think the most frustrating thing for us in regards to this journey we are on is one of set backs. Before any therapy Zoe had fine motor skills superior to her older sister (older by 2 years), but after therapy they were gone. Apparently the ''retraining'' that they had to do in therapy erased her fine motor skills..
So, that is where we are. We now only do home therapy because she is so improved! I am thrilled.
~~~~~~~~~~Question~~~~~~~~~~~~~
Zoe has always had an issue with memory recall, so the counting has been one of our biggest problems. :) It took awhile to get her on track to count to ten and then a LONG time to count to twenty, which we just accomplished!!! How on earth am I going to help us reach thirty???? I almost lost it with it taking forever to learn to twenty.
We rote count, count objects, count out a number of candies and she can have as many pieces as she can count. LOL
Any other ideas???
Thanks!
I had a child with serious language/verbal issues, but not autism. I taught him to count to ten and it did take forever. When I got to eleven I just taught him to say "ten and one", "ten and two" , "ten and three" etc.
If I recall he was able to add quite well mentally before he finally could remember the names of the numbers...he could indicate the correct number by writing it or by using this alternative method.
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PM me. I in the mood for being the "too bawdy or not" arbiter tonight. :lurk5:
What does a capon say?
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:iagree:
The one I ate this evening was called a capon. :D
I have a joke about capons but I'm trying to figure out if it's too bawdy for this board.
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Is it normal for roosters to bite or peck at someone? I hope this isn't abnormally aggressive behavior or he may be destined for the ol' stewing pot. :001_smile:
Are neutered roosters less aggressive?
Those are called "capons" aren't they?
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I'm confused, I was placing my order for Singapore Math from their site and they only sell the Standards Edition for KG which has been condensed into 2 books (from the previous 4) however Rainbow Resources only sells the 4 book series? Which one to get? Rainbow is actually cheaper with the whole order (I'm also buying US Edit 2A-2B for my older ds)??
I think the 2 book series is the US edition.
At least that's the way it works with the elementary grades.
The only difference, I think, is with metric measurement. I'd go with the cheapest.
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from the Teaching Company latest sale catalog about the course Argumentation: The Study of Effective Reasoning by Prof. David Zarefsky, Northwestern University.
In the description it states,
" The tools of formal logic, while essential and even definitive for mathematics and programming computers, are inadequate to decide most controversial issues. This course shows more useful approaches."
"'Argumentation' is a course in classical rhetoric. This is not rhetoric the way the word is used commonly today, but a course in how to debate, how to influence others by sound reasoning, how to gain support for your position in a disagreement. ..."
So then why study formal logic?
I ordered the course from inter-loan library. Has anyone viewed this course?
I have't seen teh course but no one has answered your post and it sounds interesting.
I wonder what they mean by "formal." Is the author of this program counting on the fact that most people conflate 'symbolic' with 'formal'?
Formal logic would include the study of the nature and general features of deductive arguments, argument validity and soundness, symbolization, truth-functional logical (Boolean) connectives, quantifiers (all, some, and none and how to negate them), checking argument validity, giving counterexamples, and constructing formal deductive proofs. It would also include learning how to “translate†English sentences into sentences in the formal language, and vice versa. Whether these concepts are expressed with symbols or with natural language they are still 'formal' logic since it is the concepts in abstract that are being studied rather than the symbol on the paper.
Learning how to translate ideas into English sentences and English sentences into symbols is crucial in math and physics. It's what enabled Godel to handle:
This statement is false.
I can't even begin to imagine how one would reason through an argument of any kind without being being fluent in the above topics.
I certainly won't say that being able to puzzle through moral dilemmas isn't central to education, but on the other hand there are some fantasticly interesting paradoxes in philosophy, math, logic, and physics, which were resolved by someone able to see the underlying structure of what was going on, symbolize it, and resolve it.
I have read that philosphy majors have the highest verbal on the GRE and I can't help but wonder if part of that isn't due to the fact that all philosophy majors are required to take courses in logic.
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Oooh, could I ask you a favor? If you catch any mistakes, would you please let me know? I haven't had a chance (taken the time) to proof it, yet.
Normally, DS1 proofs the schedules for me as he goes through them, but I did that schedule after he finished the course. (Neither of us liked the original schedule I had.)
Sure.
I don't know that we will stick to it religiously. What I needed was for my son to have a rough idea of what he needs to accomplish in one sitting.
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fried pork rinds
chicken skin (lots of people peel it off before they eat it)
raw fish marinated in lemon/lime juice
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This neuro Dr. says he doesn't know specifically, but is referring it to the pulminologist.
Experience with this? Stories?
Thanks - Stacey in MA
I had a doc refer me to a specialist for a positive lupus panel and the rheumatologist then said that it wasn't. Being referred to a specialist is no fun. I've got an unfun appointment with yet another one that I'm not looking forward to.
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My 16 dd is going to a camp with her poms squad. They are going to be staying in dorms at a university a few hours away. I was fine with this because I just assumed that their coach would be going with them. Yesterday I found out that their coach is staying about 20 minutes away. Apparently each squad will have an adult who will be in charge of them. I have no idea who this adult is and how old they are and how many kids they will be in charge of. Apparently the organizers of this camp have strict guidelines curfews, etc. but I'm still a little uneasy about this. On the other hand since dd will be going to college in 2 years maybe I'm being unreasonable.
I just got off the phone with another mom whose dd is in this squad. She is actually a teacher at this school and is very upset about the whole situation.
I stayed in a university dorm completely unsupervised when I was 14.
We acted responsibly because we knew we had to get up on time and make it to class and we were motivated to earn good grades.
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Sorry Jane, slight hijack:
Mrytle, thanks! You are a wealth of information. I had not discovered penmanship books at google yet. Off to check that out. :auto:
I used HWT with my oldest because he seemed to have issues with handwriting and I am all set to teach it this year to my 2nd grader and 3rd grader.
I hope I didn't make it sound like there is anything "wrong" with HWT. Frilly script is very beautiful and I wish I could have made that happen but it wasn't worth the tears.
If one of my other two seem like they would be interested in something frillier and would enjoy that I would consider switching.
A search on google books for penmanship in full view mode turned up a bunch of neat stuff. This one makes me want to stock up on quill pens: http://books.google.com/books?id=WZwBAAAAYAAJ&printsec=frontcover&dq=penmanship&as_brr=1#PPA17,M1
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Christine from Core Foundations has put together a schedule.
That's looks fabulous. I'm printing it out right now.
Thank you sooo much.
I am shocked when I see someone with over a 1000 posts..
in General Education Discussion Board
Posted
But then...
book discussions were organized, Batman was oh so carfully watched, sailing trips planned, dyslexia support given, cans of silly string theory sprayed, Logic PMed everywhere....