Mathematics IS literacy. I don't remember if I've said this before, but Mathematics isn't just working with numbers. Mathematics is a language.
It's even got its own symbols (what other languages call an "alphabet") and syntax, which gets just as confusing and confused as any other spoken language. There are specific pronunciations for the symbols, and different branches have different notations (think of them as dialects). Literacy instruction is critical for any student of mathematics to get anywhere in understanding and communicating.
As a teacher, I fully intend to use representation and writing instruction with my students. Mathematics is generally know as or regarded as a process of proof. If students can't prove anything, then I've failed them as a mathematician. I'm going to encourage, and even sometimes require, "English language" transliterations of their mathematics work, which can be considered a simple form of proof. The intent will also help them with their verbal expression of mathematics.
Critical literacy can often be dismissed as unnecessary or extraneous in Mathematics, but it's often a huge area that can be analyzed and exploited for exercises in statistics. "How much does socioeconomic status affect student performance?" is a wonderful question for the budding statistician. Lots of data has been collected, tons of studies have been done, and the question relates to the students' world. When they see and analyze the data, they can speak from the perspective other than themselves, and also from their own perspective, and can speak and discuss the question in terms of themselves and others.
In math, everyone is a language learner. Its important to use visuals, to make obvious some of the less obvious connections, and to use translations into the students main language. There's no reason not to, it's one of the best ways to teach anything, be it history or math or even physics. Without a connection to what they know, there's no learning. Learning IS making connections.
Text selection is also a vital part of mathematics instruction. If the students can't understand what they read in their OWN language, how will they be able to work in another. Catering to every level is important. Make interesting choices available.
In short, literacy is just as important to Mathematics as it is to English or Language arts classes. It's Central, and makes everything else work. Teaching without literacy instruction is bound and destined to fail in an epic fashion.
Sunday, December 9, 2012
Wednesday, October 31, 2012
Wait... These Little Squiggles Have Meaning?
When I was a little kid (the year before kindergarten), my parents put me in preschool. I learned the letters in the English alphabet and some basic addition and counting. Nothing monumental, but it was definitely learning.
Then I hit kindergarten. I was already reading, pronouncing lots of "advanced" words, spelling, reading, all of that. I was immediately targeted as an "advanced" student. I didn't realize it at the time, (actually, I just kind of put this together five minutes ago) (that time being 11:10 pm) but I was probably, at that time, setting myself up to be an educator.
ANYWAY... Reading. I was always reading. In first grade, I was part of an advanced reading program in the classroom. When the other students were reading out of the classroom readers, I was reading other selections and testing my comprehension of the short stories or situations presented. (Not Accelerated Reader. That comes later.) I also got put in Gifted and Talented, a type of special out-of-classroom special education for the "advanced" students.
When I got to second grade, and all the way through to 7th grade, I was in the Accelerated Reader program. We read books on "the list" and took short comprehension quizzes on the books, and earned AR points based on how we scored on the test (usually out of 10 or 20 questions) I bought into it until about halfway through 4th grade, at which point, I'd basically read all of the books that were "in my reading level" (which was like 6th grade to high school senior) (It was probably actually higher, but the tests wouldn't score me that high). I just got sick of reading for the program, and read for me. I devoured books in hours that were actually pretty deep and long. Reading for a grade bothered me. I'm not sure why, but it just did. There wasn't anything there that I thought should or could be graded.
I very rarely read (past tense) nonfiction, unless I had to find some source or other for some research project or something. There just were so many amazing, imaginary places I could go to that weren't my life or my world. (Not that my life was bad. My life was actually pretty awesome. Just not full of magic and lasers and cool spaceships.)
Today, now, I still try to avoid nonfiction; it's mostly depressing. Math, on the other hand, (and yes, I do make that distinction) isn't depressing at all. It's challenging, sure. It makes you consider and wonder, absolutely. But it's really hard to depress someone with mathematics. (Yeah, you can show them figures and trends of society and such, but that's more statistics... which is another story altogether.)
In math specifically, I was always kind of questioning what was shown me. I never just accepted something unless I'd run through several possible failed counter-examples. That, right there, is how you see what goes on behind the general statements teachers make, and the specifics of the problem you're currently working.
I'm definitely planning on encouraging my students to look at the mathematics behind the world... to read it, you might say. There aren't a whole ton of trade-books about or explaining mathematics, but then, that's just old ground that people have crossed hundreds of times. Personal experimentation with numbers, with graphs, with all of the things you can consider in mathematics (read: EVERYTHING): That's how you learn mathematics. All the exercises are just a way to artificially inject that into students. But until you find something that students say to themselves, "I wanna learn more about THAT." "What makes THAT work that way?" "How come?" and then they look at it and decide for themselves that they can see what makes it work the way it does, to kind of prove it to themselves, THAT's what we're looking for.
Not everyone likes math. But everyone can do it, can have ideas about it. And I plan to tell my students that, every day if I have to. Just because you do or like math, that doesn't make you nerdy or lame or uncool. It makes you informed, and gives you a way to think about the world. And thinking about the world, while occasionally or frequently depressing, is how you make sense of what happens to you.
Thursday, October 18, 2012
The Scribbles We Call "Writing"
Writing. It's something that we all do, whether we want to or not. Some of us have wonderful writing; in fact we pay them for it. Others, well... Not so much. Some enjoy it. Others hate it with a passion. (Fire, lots of fire. Fire will fix the evil assignments, right? ... Yes, that was probably very nearly a direct quote at one point in an English/Writing class.)
Writing is both one of my loves and one of my banes. If you give me constraints on my writing and want me to pick my own topic, kiss any kind of happiness goodbye. Ask any of my friends, and they'll likely tell you stories of me complaining to them about my writing assignments and how I hate writing.
On the other hand, I regularly create quite a bit of literary content. Most of the content I create for enjoyment is just never really written down. Being a math major, I create a lot of technical writing in the process of proving mathematical conjectures. The rest of it is just a lot of snarky, ironic statements about the world and its state of affairs. Or puns. Puns are great. (Sadly, I can't think of any at the moment.)
When I was a junior in high school, one of the writing projects that were assigned was a persuasive research paper. The topic was one of my own choosing, and I chose the topic of Dihydrogen monoxide. I can't recall whether I was arguing in favor of banning this chemical, or opposing the ban, but I loved researching the topic, finding new facts to support my position. The reason:
Dihydrogen monoxide is Water. Yep. H2O.
The instructor didn't catch on until the very end of the assignment as he was awarding final grades for the research paper. The look on his face as he realized what had happened was PRICELESS.
If I can find a topic, I'll be happy as a clam. I can find the research, I can build the sentences, and work for a long time on making the writing of good quality. But topic generation is a pain for me. I just don't do it very well. Hence, the complaining about writing assignments to friends and roommates.
That's the thing though.. If you can't come up with things to talk about, how do you say anything meaningful? And yeah, I know there are always the topics that are rehashed so many times that teachers and society are sick to death of the the entire topic in general. (This happened my senior year of high school with respect to abortion.) But are those rehashings actually meaningful? Sure, they mean something on the report card, but at the end of the day, has anything been contributed to society, to the world at large? Has a distinguishable mark been made by anyone?
I say no.
Assignments that simply maintain the status quo and say the same thing as a million other assignments by a million other students can be good for the student, but usually are not so great for the world. How does mathematics change and grow? By original thinking and by doing something that has never been done before. I'm not saying that every student has to create a new, fantastic, game-changing proof of epic proportions, but there are a lot of different ways to look at concepts in mathematics. A new mnemonic here, a new metaphor there, here a proof, there a spoof, everywhere a ... wait a second...
ANYWAY. It all adds up. Original thinking drives the world. By the time I get access to my students, they will probably have had all of the capacity for original thinking beaten out of them by their experience in grade school. This will make my life hard and possibly quite boring. I don't like boring. (See my explanation of "Normal" here) Nurturing it back to life is a hard job, but a necessary one. I want it to happen, and assignments that will further that goal are a must in every classroom. So, to make life interesting again, and NOT boring, interesting assignments that foster creativity are going to be present.
(Only issue is... I have to be creative myself to make that happen. Crap.)
Writing is both one of my loves and one of my banes. If you give me constraints on my writing and want me to pick my own topic, kiss any kind of happiness goodbye. Ask any of my friends, and they'll likely tell you stories of me complaining to them about my writing assignments and how I hate writing.
On the other hand, I regularly create quite a bit of literary content. Most of the content I create for enjoyment is just never really written down. Being a math major, I create a lot of technical writing in the process of proving mathematical conjectures. The rest of it is just a lot of snarky, ironic statements about the world and its state of affairs. Or puns. Puns are great. (Sadly, I can't think of any at the moment.)
When I was a junior in high school, one of the writing projects that were assigned was a persuasive research paper. The topic was one of my own choosing, and I chose the topic of Dihydrogen monoxide. I can't recall whether I was arguing in favor of banning this chemical, or opposing the ban, but I loved researching the topic, finding new facts to support my position. The reason:
Dihydrogen monoxide is Water. Yep. H2O.
The instructor didn't catch on until the very end of the assignment as he was awarding final grades for the research paper. The look on his face as he realized what had happened was PRICELESS.
If I can find a topic, I'll be happy as a clam. I can find the research, I can build the sentences, and work for a long time on making the writing of good quality. But topic generation is a pain for me. I just don't do it very well. Hence, the complaining about writing assignments to friends and roommates.
That's the thing though.. If you can't come up with things to talk about, how do you say anything meaningful? And yeah, I know there are always the topics that are rehashed so many times that teachers and society are sick to death of the the entire topic in general. (This happened my senior year of high school with respect to abortion.) But are those rehashings actually meaningful? Sure, they mean something on the report card, but at the end of the day, has anything been contributed to society, to the world at large? Has a distinguishable mark been made by anyone?
I say no.
Assignments that simply maintain the status quo and say the same thing as a million other assignments by a million other students can be good for the student, but usually are not so great for the world. How does mathematics change and grow? By original thinking and by doing something that has never been done before. I'm not saying that every student has to create a new, fantastic, game-changing proof of epic proportions, but there are a lot of different ways to look at concepts in mathematics. A new mnemonic here, a new metaphor there, here a proof, there a spoof, everywhere a ... wait a second...
ANYWAY. It all adds up. Original thinking drives the world. By the time I get access to my students, they will probably have had all of the capacity for original thinking beaten out of them by their experience in grade school. This will make my life hard and possibly quite boring. I don't like boring. (See my explanation of "Normal" here) Nurturing it back to life is a hard job, but a necessary one. I want it to happen, and assignments that will further that goal are a must in every classroom. So, to make life interesting again, and NOT boring, interesting assignments that foster creativity are going to be present.
(Only issue is... I have to be creative myself to make that happen. Crap.)
Thursday, October 4, 2012
The Point, and the Beginning
So, as a kid (like first grade), I was always one of the "bright kids"--
Point of order: NEVER label your students. It's a cancer that will do horrible things for their self-esteem and/or make them into self-righteous know it alls. Yes, this did both to me. No, it's not a good idea.
-- and I was placed into a gifted-and-talented program. This allowed me to look at math and science, as well as reading and research topics, that were pretty advanced work for someone of my age. I mean, we were doing algebra and actually calling it algebra in first and second grade.
(For those of you that haven't thought about it, you were doing algebra in kindergarten... but no one told you about it till you were in at least seventh or eighth grade. If they ever did. Example: 3 + ? = 7. ... Replace the "?" with "x" and what do you have? 3 + x = 7. Solve for x. MAGIC ALGEBRA GO.)
All my life, I dealt with fractions, counting, and heck, even multiplication at a simple level. Everyone does. Anyone who's used a recipe for basically anything has used all of the above. ("Above" here doesn't include algebra. Usually. I've heard of recipes written in algebraic notation, I just can't find any at the moment.) Anyway, you've got a recipe that calls for 3/8 tsp. of vanilla, but no 3/8 measuring-spoon. So what do you do? You use three 1/8 tsp. measuring-spoonfuls, because 3·1/8 = 3/8. You use algebra and advanced (ish) math techniques, and it's was fairly intuitive.
There were other times in early classes where teachers would outright lie to students about what was "possible" or "allowed". For instance, in my first grade class, we covered subtraction of integers, and we weren't "allowed" to subtract a "big" number from a "small" number. It just wasn't done. But I knew better, and had been told about these shady, "criminal" things called negative numbers.
OOOOOHHH!!
"You're just not allowed to." That was the response. ... .... NO! It's obviously allowed, or the concept wouldn't be named and defined! (Sorry... It just bugged me and still does when teachers outright lie.)
Anyway, the culmination of all of this was in my sixth grade math class. Over the years, I stayed in the gifted-and-talented program and did more advanced math than my peers, and I'd occasionally get frustrated and angry and hate math. (Yes. You read that right. Once upon a long time ago, I actually thought I hated math.) Then Mr. Dye came to me one day in class.
Dye: Colin, why do you hate math?
Me: I dunno. I just do. It's dumb.
Dye: But... You're so good at it. You really are!
Me: It's dumb though!
Dye: Why don't you try not hating it, and maybe even consider liking it? You don't really seem to have a reason to hate it, so maybe if you just stop hating it, you'll enjoy it more.
Me: I... Okay. I guess, I can try that...
And that was the beginning. From then on, I didn't hate math. It wasn't a pain. It wasn't dumb. (It never was any of those things, I was just confused.) I actually started to like it. I got it. Deep down inside of me, it always just made sense and worked the way the teachers said it did. Occasionally (and usually far less obviously than the negative numbers fiasco of first grade) teachers would lie to me, but that would get fixed further down the road. I was just always able to--sooner or later--figure out what was going on, and get things done.
I guess my point is this: Just cause it's not on the list for this grade level, doesn't mean you can't teach it. It might be a little advanced, or a little over some peoples' heads, but lying to students and saying you can't do something that's TOTALLY and COMPLETELY allowed, and possible? Not kosher. Just don't do it.
There's this myth out there that math is hard. It's perpetuated by some teachers, and some of the students. Particularly the students that struggle and get frustrated easily with mathematics. The other kids, the kids who excel there, will hear them, and believe them. The best thing you can do for any kid, is to challenge them to do better. (Believe it or not, reverse-psychology can actually work in some cases.) If a kid is good at--but also "hates"-- a subject, run through the logic of their hatred with them. And if you can, just change the original assumption.
(Here I devolve into the process of proof, don't mind me...) When you're proving a conjecture, you can assume anything, basically. These assumptions lead to other conclusions, some of them contradictory and/or unwanted. But if you pick the right assumption, the proof basically writes itself. Same with kids, as far as I can tell. If you change the "I hate math because I hate math" assumption to something more like "I don't necessarily hate math", then you've done a great service to the child. And a mathematician might just pop out at the end of the logical implications.
Also? Forest fires.
Point of order: NEVER label your students. It's a cancer that will do horrible things for their self-esteem and/or make them into self-righteous know it alls. Yes, this did both to me. No, it's not a good idea.
-- and I was placed into a gifted-and-talented program. This allowed me to look at math and science, as well as reading and research topics, that were pretty advanced work for someone of my age. I mean, we were doing algebra and actually calling it algebra in first and second grade.
(For those of you that haven't thought about it, you were doing algebra in kindergarten... but no one told you about it till you were in at least seventh or eighth grade. If they ever did. Example: 3 + ? = 7. ... Replace the "?" with "x" and what do you have? 3 + x = 7. Solve for x. MAGIC ALGEBRA GO.)
All my life, I dealt with fractions, counting, and heck, even multiplication at a simple level. Everyone does. Anyone who's used a recipe for basically anything has used all of the above. ("Above" here doesn't include algebra. Usually. I've heard of recipes written in algebraic notation, I just can't find any at the moment.) Anyway, you've got a recipe that calls for 3/8 tsp. of vanilla, but no 3/8 measuring-spoon. So what do you do? You use three 1/8 tsp. measuring-spoonfuls, because 3·1/8 = 3/8. You use algebra and advanced (ish) math techniques, and it's was fairly intuitive.
There were other times in early classes where teachers would outright lie to students about what was "possible" or "allowed". For instance, in my first grade class, we covered subtraction of integers, and we weren't "allowed" to subtract a "big" number from a "small" number. It just wasn't done. But I knew better, and had been told about these shady, "criminal" things called negative numbers.
OOOOOHHH!!
"You're just not allowed to." That was the response. ... .... NO! It's obviously allowed, or the concept wouldn't be named and defined! (Sorry... It just bugged me and still does when teachers outright lie.)
Dye: Colin, why do you hate math?
Me: I dunno. I just do. It's dumb.
Dye: But... You're so good at it. You really are!
Me: It's dumb though!
Dye: Why don't you try not hating it, and maybe even consider liking it? You don't really seem to have a reason to hate it, so maybe if you just stop hating it, you'll enjoy it more.
Me: I... Okay. I guess, I can try that...
And that was the beginning. From then on, I didn't hate math. It wasn't a pain. It wasn't dumb. (It never was any of those things, I was just confused.) I actually started to like it. I got it. Deep down inside of me, it always just made sense and worked the way the teachers said it did. Occasionally (and usually far less obviously than the negative numbers fiasco of first grade) teachers would lie to me, but that would get fixed further down the road. I was just always able to--sooner or later--figure out what was going on, and get things done.
I guess my point is this: Just cause it's not on the list for this grade level, doesn't mean you can't teach it. It might be a little advanced, or a little over some peoples' heads, but lying to students and saying you can't do something that's TOTALLY and COMPLETELY allowed, and possible? Not kosher. Just don't do it.
There's this myth out there that math is hard. It's perpetuated by some teachers, and some of the students. Particularly the students that struggle and get frustrated easily with mathematics. The other kids, the kids who excel there, will hear them, and believe them. The best thing you can do for any kid, is to challenge them to do better. (Believe it or not, reverse-psychology can actually work in some cases.) If a kid is good at--but also "hates"-- a subject, run through the logic of their hatred with them. And if you can, just change the original assumption.
(Here I devolve into the process of proof, don't mind me...) When you're proving a conjecture, you can assume anything, basically. These assumptions lead to other conclusions, some of them contradictory and/or unwanted. But if you pick the right assumption, the proof basically writes itself. Same with kids, as far as I can tell. If you change the "I hate math because I hate math" assumption to something more like "I don't necessarily hate math", then you've done a great service to the child. And a mathematician might just pop out at the end of the logical implications.
Also? Forest fires.
Monday, September 3, 2012
Hello, World!
(*sigh*.... Here we go...)
Mathematics is probably the most important thing anyone will learn, and (at least in my experience) one of the most neglected skills in the life of your average bear. (Or person, whatever. I'm sure bears neglect math, too. Bears are jerks like that.) Math is complicated, but carries no baggage of its own. Sure, people bring baggage to math all the time, like poor teachers or poor previous explanations, but in math, only what is defined or explicitly involved by the writer is actually there. Every problem or proof starts with a blank slate.
Everything in the world deals with math: Take grocery shopping, for instance. First, you have to know "how much" money you have (this is a simple counting problem, but it's still math). This total then dictates that you can afford less than or equal to that value in groceries (using an inequality) Then, once you've added an item to your cart, you have to multiply the cost of that item by the number of items of its type (estimation plays a big role in this), and then sum the values of groups of items (polynomials can be used for this). Then, you have to return to the original inequality, and compare the total value of your cart to the amount of money you started with. If it violates the inequality, you have to decide what to put back, until the inequality is satisfied again. To some, this is simple. To others, not so much. But it's still math, and it's still useful.
I should introduce myself and the point of all this, so here goes:
Presently, I'm a pre-service secondary-level teacher going to school full time at Utah State University, and (if you couldn't already tell) I plan on teaching mathematics and statistics. I grew up in Idaho, in the Snake River Valley. I am a nerd, and I'll probably take offense if you call me normal. "Normal" is one or more of three things to me: 1) A setting on the dryer, 2) A reference to the Gaussian distribution, or, for you laypeople out there, the "bell curve", 3) Boring. (I'm sorry if you identify as "normal". Normal is boring.)
Hobbies... I probably like video games too much. I appreciate them as art, and as something that is beautiful. (...and they contribute to my procrastination habit.) I play tabletop role-playing games with my friends, and I'm always up for a good book. I'll always join a conversation about Star Wars or the Force. (If you like math and star wars, check this out.)
There's a couple reasons that I chose to pursue mathematics and statistics education as a career. First, I understand it and enjoy showing others how it can be understood. That was the biggest reason by far. Second, I had good math teachers in school that inspired me, and made me want to share my talents.
So I've talked about math a lot, and you may have looked at the title of this blog, and noticed it involves literacy. "What does math have to do with literacy?" you ask. A couple of years ago, I would have asked the same question, probably with some disdain for English as a subject. I know better now.
Literacy is more than just being able to read and write. (Definitions are kind of sacred to me as a mathematician, because they have to state everything you're going to use and work with from the beginning. I may cheat and change that over the course of time, but for now, they're sacred.) My definition for literacy is: Literacy is the ability to understand, demonstrate, apply and/or use information gained from a text. (Wait, what's a text? A text is defined (here, at least) as any resource that relays information the same way every time it is used.)
(If I change those definitions later, you can burn me as a heretic.)
Does literacy apply to Mathematics? ABSOLUTELY. First and foremost, you have to understand the axioms and basic rules that comprise the basis of mathematics. Definitions are here, and this is why they're so sacred. All of math is built on them. Then there are symbols that denote certain processes and definitions. There's hundreds, if not thousands or millions, of theorems, postulates and corollaries that allow for more and more complex uses of our basic rules and axioms. And that's all just understanding how math relates to itself. Using mathematics in the real world is a topic that's just as big, if not bigger, than just understanding the mathematics itself. (Here, "big" is used to refer to the amount of information in a topic, not the importance. I'd say they have equal importance.) So yes, in short: Literacy is important to math.
Now I need to find a way to go slap past-me for being so shortsighted...
Greetings, Blogosphere! First off, here's something you should see and understand if you're gonna be sticking around for this:
 |
| This really is how it is. Well... How it SHOULD be. Most people forget about us way over here, in return. (source) |
Mathematics is probably the most important thing anyone will learn, and (at least in my experience) one of the most neglected skills in the life of your average bear. (Or person, whatever. I'm sure bears neglect math, too. Bears are jerks like that.) Math is complicated, but carries no baggage of its own. Sure, people bring baggage to math all the time, like poor teachers or poor previous explanations, but in math, only what is defined or explicitly involved by the writer is actually there. Every problem or proof starts with a blank slate.
Everything in the world deals with math: Take grocery shopping, for instance. First, you have to know "how much" money you have (this is a simple counting problem, but it's still math). This total then dictates that you can afford less than or equal to that value in groceries (using an inequality) Then, once you've added an item to your cart, you have to multiply the cost of that item by the number of items of its type (estimation plays a big role in this), and then sum the values of groups of items (polynomials can be used for this). Then, you have to return to the original inequality, and compare the total value of your cart to the amount of money you started with. If it violates the inequality, you have to decide what to put back, until the inequality is satisfied again. To some, this is simple. To others, not so much. But it's still math, and it's still useful.
I should introduce myself and the point of all this, so here goes:
Presently, I'm a pre-service secondary-level teacher going to school full time at Utah State University, and (if you couldn't already tell) I plan on teaching mathematics and statistics. I grew up in Idaho, in the Snake River Valley. I am a nerd, and I'll probably take offense if you call me normal. "Normal" is one or more of three things to me: 1) A setting on the dryer, 2) A reference to the Gaussian distribution, or, for you laypeople out there, the "bell curve", 3) Boring. (I'm sorry if you identify as "normal". Normal is boring.)
Hobbies... I probably like video games too much. I appreciate them as art, and as something that is beautiful. (...and they contribute to my procrastination habit.) I play tabletop role-playing games with my friends, and I'm always up for a good book. I'll always join a conversation about Star Wars or the Force. (If you like math and star wars, check this out.)
There's a couple reasons that I chose to pursue mathematics and statistics education as a career. First, I understand it and enjoy showing others how it can be understood. That was the biggest reason by far. Second, I had good math teachers in school that inspired me, and made me want to share my talents.
So I've talked about math a lot, and you may have looked at the title of this blog, and noticed it involves literacy. "What does math have to do with literacy?" you ask. A couple of years ago, I would have asked the same question, probably with some disdain for English as a subject. I know better now.
Literacy is more than just being able to read and write. (Definitions are kind of sacred to me as a mathematician, because they have to state everything you're going to use and work with from the beginning. I may cheat and change that over the course of time, but for now, they're sacred.) My definition for literacy is: Literacy is the ability to understand, demonstrate, apply and/or use information gained from a text. (Wait, what's a text? A text is defined (here, at least) as any resource that relays information the same way every time it is used.)
(If I change those definitions later, you can burn me as a heretic.)
Does literacy apply to Mathematics? ABSOLUTELY. First and foremost, you have to understand the axioms and basic rules that comprise the basis of mathematics. Definitions are here, and this is why they're so sacred. All of math is built on them. Then there are symbols that denote certain processes and definitions. There's hundreds, if not thousands or millions, of theorems, postulates and corollaries that allow for more and more complex uses of our basic rules and axioms. And that's all just understanding how math relates to itself. Using mathematics in the real world is a topic that's just as big, if not bigger, than just understanding the mathematics itself. (Here, "big" is used to refer to the amount of information in a topic, not the importance. I'd say they have equal importance.) So yes, in short: Literacy is important to math.
Now I need to find a way to go slap past-me for being so shortsighted...
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