I wrote a few practice tasks for my online ICS class for using loops and arrays, as well as a challenge task for anyone who’s interested. You’re welcome to use them in your classes if you like.
I teach high school math. Students bring scientific calculators to class, or they sometimes have to borrow one from me. I have two types available: immediate execution calculators and formula calculators. I’ve been wondering lately whether one type of calculator is better for learning algebra than the other.
Here’s how they work (see Wikipedia for a longer explanation: https://en.wikipedia.org/wiki/Calculator_input_methods).
These calculators work by performing calculations along the way as you type in values and operations. For example, you can evaluate the expression
by typing 3, multiply, 45, then the sine key. As you press operations and operands the calculator will evaluate what it can according to the rules of order of operations, or BEDMAS. For binary operands (those taking two values to produce a result, like multiplication), you put the values in order. For unary operations (those taking just one value, like squaring or taking a sine), the value must be present on the calculator screen when you press the operator key. These calculators usually have a bracketing feature to allow the user to work through complex expressions without using memory storage.
These calculators work by waiting until the user has typed in a complete expression to evaluate, then evaluating the entire expression. The order of button-pushing is pretty much as the symbols are written in the expression, making them easier to use for a lot of folks. Once a value is calculated, it’s stored in an “answer” variable in case it’s needed for the next evaluation.
Algebraic Expressions and BEDMAS
When we write out algebraic expressions, we have a number of conventions to follow. The most important convention is order of operations, which people usually learn to remember with the mnemonic BEDMAS or PEDMAS:
- Brackets (Parentheses)
- Division and Multiplication
- Addition and Subtraction
When evaluating (simplifying) an expression, you first simplify the smaller expressions inside brackets. Then you evaluate exponents, then division and multiplication in the order they appear, and finally addition and subtraction in the order they appear. It’s useful to think of brackets as isolating sub-expressions, which then follow the same rules. It’s also useful to think of this order as the “strength” of the operation: multiplication is a stronger operation than addition, so it holds its operands more tightly together, and it gets evaluated first.
When a student is learning order of operations, it often feels like a set of arcane rules. There is no reason, from the student perspective, that it has to be this way. In fact, it didn’t really need to be this way, but the convention was established and now it’s important to abide by it (if you want to be understood, that is).
How a calculator helps (and hinders) learning arithmetic
People often lament that today’s youth can’t perform basic arithmetic in their head. It’s unfortunately true; I often see students reach for their calculator to evaluate or even . These are facts which prior generations had drilled relentlessly and now have available as “instant” knowledge. Younger people typically haven’t spent enough time practising these computations to develop facility with them. This is partly because the calculator is so readily available.
(Aside for parents: If you have kids, please do make them practise their age-appropriate facts. It’ll help them in the same way practising reading makes things easier)
This will draw a lot of heat, I’m sure, but I think calculators do have a strong place in even K-6 learning. They let students explore quickly without the burden of computation getting in the way of non-computational learning. It’s the same effect that web-based, dynamic geometry software can have on learning relationships between figures, lines, etc. (if you’re looking for awesome dynamic geometry software, try GeoGebra – free and wonderful).
But calculators are a hindrance when students are learning to compute fluently. They allow a student to bypass some of the thinking part of the exercise. Don’t let students (or your kids) use a calculator when they don’t have to. Only use them when students need the speed for the task they’re completing.
How a calculator helps (and hinders) learning algebra (?)
Here’s the part I don’t know about, but I’m speculating about.
I think immediate execution calculators require students to understand the algebraic expressions we write, where formula calculators bypass the thinking part of evaluating expressions.
As with arithmetic, if practising evaluating expressions is not part of the learning, and might be getting in the way of the goals for learning, then either type of calculator is fine.
But as students are developing their understanding of algebra and the order of operations, the immediate execution calculator displays the results of operations as they are evaluated, while the formula calculator obscures the evaluations in favour of a single result.
When a student types 5, add, 6, square, equals into an immediate execution calculator, they see the value 36 as soon as they press the square button. There is a reminder that the square operation is immediate. Similarly when a student wants to evaluate they must type 30, add, 45, equals*, then sine, emphasizing that the bracketed portion has to be evaluated first (i.e. before the sine function is applied).
*A student can use brackets, which is equivalent to pressing equals before sine. Also, I hope anyone using the sine function knows that 30+45 is 75 and doesn’t need a calculator’s help for the addition.
Is there research?
I perused the InterTubes to find research into this question, but either it’s not out there or I’m not skilled enough to find it.
I want to know whether one calculator is better than the other for a student who is learning to evaluate expressions.
Has no one looked into this? Help?
I spent about 13 hours working on school stuff today. I planned lessons, collaborated with colleagues, taught math concepts, worked one-on-one with students, developed practice materials, communicated with online students, searched out supplementary resources, marked tests,….
It’s 12:28am. I finished the school work that my brain can safely manage a little while ago, and I’m getting ready for bed. I’ll be up again at 6 o’clock to restart the cycle, and I’ll still be tired.
I like my job, I love my students, and I desperately want them to succeed. But I’m also exhausted already and feeling guilty for taking ten minutes to tap this into my phone. The work is so important, but letting it take precedence over everything else in my life is unhealthy and is unfair to everyone, including me. It always takes up more than its fair share of my brain’s CPU cycles, and it’s taking up too much of my personal time as well.
If anything eludes me this year it’s balance. I need some real down time, real soon.
I’ve been listening to Commanderin’, a podcast about the Commander/EDH format of Magic: The Gathering. It’s a great show with excellent production quality.
Commander is an eternal format, meaning cards don’t rotate out (expire) over time. It also requires the use of Legendary creatures as commanders, which are typically rare and expensive. The rest of a Commander deck is comprised of 99 other unique cards and basic lands (not unique).
Sounds expensive, eh? It is.
So I wandered onto the Interwebs and found this site, with rules for Pauper Commander:
Pauper and Peasant Commander
Sadly, the site and its successor are not being maintained, but the rules are there. Basically, it’s the Commander format with only common cards except for a possibly-uncommon Commander. The commander doesn’t have to be Legendary either, which vastly increased the number of possible choices.
I dig into my miscellaneous, unsorted multicolour cards and found these six which seem to have fun effects:
I’m sure there are more powerful commanders in many sets, but I’m playing with what I have (i.e. I’m not buying more cards for this). Any thoughts about the viability of any of these cards for Pauper EDH?
I play Magic: The Gathering (MTG or Magic) with my wife, my kids, a few friends, some students at school (hey guys!) and online.
New Magic sets are released a couple of times a year, and they are legal for Standard play for a while. Eventually they “rotate out” as they are replaced by newer sets, and the cards are only legal for play in other formats (don’t worry, there are many formats to choose from).
I have a bunch of paper cards which are “Standard” at the moment (they are from recent sets). I bought playsets (4 each) of common cards from Theros, Born of the Gods, Journey into Nix, M15 Core, Khans of Tarkir, Fate Reforged, and Dragons of Tarkir. This lets me play Standard Pauper, a format in which you can only use common cards (i.e. no uncommon, rare, or mythic rare cards).
Until tomorrow, that is… then Theros through M15 Core rotate out and aren’t legal for Standard play anymore.
For me, that’s not really a big deal. I’m not playing in tournaments. They’re perfectly good cards for a perfectly good game, it just won’t be Standard Pauper (unless I stick with just Fate Reforged and Dragons of Tarkir). I don’t need to buy more paper cards.
The Online story is different.
I started playing MTGO (username: bgrasley) a week or so ago. I now have all of the common cards from Khans through Origins (the one right after Dragons). I didn’t bother with anything before Khans because of the upcoming rotation. Battle for Zendikar comes out tomorrow (I think it’s the same time online), so then I’ll wait a few days until the common price is in line with the rest of the sets. I wonder if I might even be able to pick up Theros through M15 cheaply (right now, the price is 0.002 tickets per card, which is slightly less than a dollar for an entire set… “cheaply” is relative).
I’m most interested in playing Standard Pauper so I’ll definitely need to pick up BFZ’s commons soon so I have more cards to work with. It should be interesting.
The two card I’ll really miss
I haven’t played with the Theros block cards yet (they’re still wrapped up in the box), so I won’t miss anything from there yet. I understand people who play other colours have some painful losses to rotation as well.
I’m going to read some set reviews, especially those focused on Standard Pauper, and see what I might brew up for BFZ.
What are you going to miss?
Round two… I modified my BW deck from yesterday a bit (took out Divine Favor and Unmake the Graves, replacing them with Selfless Cathar, Necrobite, and Crippling Blight) then spent just a few minutes putting together a Mono Red Pauper deck again using only M15 Core commons. Here’s the (as yet unplayed) deck:
4 Borderland Marauder
4 Foundry Street Denizen
4 Generator Servant
4 Thundering Giant
4 Torch Fiend
4 Wall of Fire
4 Crowd’s Favor
4 Lightning Strike
4 Inferno Fist
I don’t know if I should have that many Thundering Giants, but I figure I can trade up my Generator Servants for them in round 3 or so. There are a bunch of cards in here I’ve never used, so we’ll see what happens. Comments and concerns welcome :)
I mentioned a while ago that I bought playsets of all common cards from Theros up to Dragons of Tarkir. I decided this weekend to make a Black-White Pauper deck using only cards from the M15 Core Set to try out against a couple of Fate Reforged Intro decks I have and some kids at school. So far it’s performing fairly well, if a little slowly. First the card list, and then I’ll give my thoughts on it so far.
2 Carrion Crow
2 Child of Night
2 Heliod’s Pilgrim
2 Oreskos Swiftclaw
2 Sungrace Pegasus
4 Typhoid Rats
2 Covenant of Blood
2 Divine Favor
4 Eternal Thirst
2 Oppressive Rays
4 Raise the Alarm
4 Sign in Blood
4 Triplicate Spirits
2 Unmake the Graves
4 Evolving Wilds
Total cost was peanuts (this is Pauper, after all). If you paid $0.10 per card it would be $6, which is about what I paid for all the M15 Core commons together.
I originally had 2 Festergloom in there, but I realized (belatedly) that they would kill off my own white tokens. Maybe it would fit into a Mono-Black deck. I swapped them out for Unmake the Graves. I have yet to draw that card, but I think it might not be helpful given the number of tokens I’m using versus creature spells.
Oppressive Rays works well to slow the pace and make sure my 1/1 wimps can chip away at the other player.
I’ll tweak it some more. Suggestions always welcome!