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How would you teach math differently to young kids if budget was not a concern?
It seems to me we teach kids math in a way that prioritizes mass teaching and resource management over the actual learning of mathematical concepts.
We rely on paper and pencil, and maybe some limited manipulatives like unit blocks, and there’s 1 teacher for every 15-30 kids or so.
What are some methods that might work better to establish a strong understanding of math if we were able to approach it differently?
Or what are some methods that have been proven to work in other settings and why are they able to be successful?
I assume the ideal class size for most things is around 4 or 5 students per teacher. Less than that and the social dynamics change significantly. In addition there should be a lot of projects where the student works 1:1 with a professional in that field. Being around someone who's been doing that kind of work for 20 years is the best learning environment. An apprenticeship is so much more valuable than school because almost all questions the student can come up with can be answered with hard-earned wisdom. In school the teacher is mostly a professional book translator.
I've gotten the chance to work in some excellent environments since graduating college. When I have someone far more experienced around I learn so much more and get more done. Struggling through something occasionally is a good learning experience because sometimes you need to do something no one else can help with. But most of what people do has been done many times before. Why do we make kids struggle through learning so often?
Generally I think most education needs to be applied. You should teach in these stages:
Generally, most things should be tied back to something around the student to keep it grounded. This requires creativity and a relaxed environment for the teacher themselves to thrive in. Done right you'll teach everything from first principles. You'll be teaching angles from the ground up. "We want to know the difference between a camera angled way up or angled very low. Let's call pointed at the sky 100 and pointed horizontal 0. Oh, now angled 1/3 of the way to the sky is an awkward number - 33.3. What if we picked a number of units that is easily divided by a lot of different numbers? 2 * 2 * 2 * 3 * 3 * 5 (360) allows us to divide by 3, 9, 15, 2, 4, 8, 18, etc. Now we're using 360 for a full rotation or 90 for the difference between ground and sky. Did you know degree doesn't mean temperature it means a unit or piece of something? These are 360 degrees."
I would generally try to learn this way in public school. That meant if I forgot something during a test I'd using my first principles foundation to re-derive it and saved me from needing to study for math-focused classes.
From my experience with scouting... 6 is pretty close to ideal group size.
Lets kids pair off in 3 pairs or 2 groups of three.
Oh yeah that sounds pretty good.
I'm a teacher and I have a class of 4. It's a nightmare. I think 10 would be an ideal size. Then again, I'm a high school science teacher so having little kids in smaller groups may be beneficial
I mean at that point it isn't really teaching in the traditional sense, it's tutoring a group right? I assume that's why you dislike it so vehemently?
When I was younger I used to to tutor for cash on the side, and assumed it would translate into enjoying teaching as well. When I actually got to teach a few classes in graduate school... I learned how comically wrong I was. I hate teaching. I love tutoring. I can explain the why properly to small groups interactively. I had to go at a snail's pace to teach things the same way to a whole class.
That's a good way to look at it actually. So yes, I don't like tutoring but I love teaching. It's just all the other BS that goes along with teaching that I don't like
For me, the lack of practical application was a huge point for having difficulty learning math in middle/high school and college. The basics were easy because application was part of daily life, but the further from the everyday math becomes the more difficult it is to grasp.
The reason is twofold. First, seeing the way an equation interplays with a system or the real world makes something “click” mentally and second, it helps the student see why they might want to learn these things. It’s hard to want to learn detached abstract concepts “just because”.
And a good intuition for why the equations are the way they are allows you to hypothesize an equation you’ve never seen before. Is something radiating from a source? Multiply the source amplitude by an inverse square of the radius. Is a linear signal derived from rotation? Assume the equation is roughly sinusoidal.
I felt much the same way in college - I found the modules on mechanical maths which mapped clearly to something in the real world very easy, while at the same time I found the more abstract math modules I couldn't relate to anything so frustrating that I ended up dropping the entire course. A year or two after that I remember a friend explaining some of the concepts that frustrated me so much by explaining how you'd actually use them for a problem and I got it immediately.
Exactly! Anchoring in the real world lets students connect this new and flexible idea into more concrete things they already know. And if they're teaching it then the math isn't so fringe that there's no real application yet. Everything they're teaching has some real-world application.
One of the "aha" moments that stuck with me from my early formal education was when a teacher had us calculate and plot about a dozen points along
y = -(x^2), eliciting lots of sighs and eye-rolling as we did it.Once all the points were plotted, he said "watch this", and threw the chalk he'd used to plot the curve on the blackboard... and the chalk followed the curve.
That can make things interesting and it's definitely something I was curious about. But I also vaguely recall reading an argument from a professional mathematician that if the kids are asking "when are we ever going to use this" then you've already failed. Math can also be be a source of fun puzzles, if you approach it right. That's closer to how mathematicians treat it.
(On the other hand, they have a weird idea of fun, which involves really struggling to solve very hard problems, often making no progress and having no idea what to do next. Part of it is learning how to live with being stuck.)
Personally I agree that math is fun in isolation. I think it's my enjoyment of math that helps me look back on my education and understand where they were trying to take me and where I went instead.
The idea of grounding math in the real world isn't to stop kids from asking "When are we ever going to use this?". It's to make the educational process more intuitive. If you can connect some abstract concept to the real world you're able to rest on the student's foundational understanding of the world around them. Free floating ideas are hard to understand because they have a bunch of dangling questions hanging off of them. An idea attached to your broader knowledge graph has at least some of those questions tied into solid facts. That means you can infer the answers to some of the other questions through the Socratic method. Being able to go one more step down the Socratic method ladder is, to me, indicative that you actually understand something.
Yeah, I agree. Mathematical abstractions are often simple but difficult, and I've always found examples helpful for understanding them.
I'd distinguish between "anchoring in the real world" and practical applications, though. This anchoring doesn't necessarily need to be of practical use. It could just be "how this game you know works."
Yeah, I found math much more intuitive and interesting when it was framed like a "story problem". Even when those problems were artificial in some way, it felt like a puzzle when framed that way -- and I like puzzles! But in general I don't think most of my math teachers were proficient enough to come up with good application-type problems. Even my one teacher who was a working engineer literally never tied our work into any practical engineering task.
I also think that a huge point of difficulty for me when it came to high school math was a lack of things building on each other. I enjoyed my calc 1 in college way more than precalc in high school not because it was easier, but because it felt like each thing we learned built upon the last thing we'd learned. That combined with some of the practical-adjacent framing in calc ("how do we find the area under this curve" is at least more practical than not having something like that as a practical framing) made it much more understandable than precalc, when everything was just a series of disconnected bases for other math classes most of us would never take -- we literally took a few weeks to learn matrix multiplication despite never discussing any other concept in linear algebra whatsoever. It was just "here's this new thing we have to learn for a couple of weeks and will never speak of again after the test on them!" And honestly not including practical examples for linear algebra is just negligent -- that shit pops up everywhere!
Regarding things building on one another, that is also one of the primary reasons why it takes more work for students to catch up in math than in other subjects. If you somehow changed school districts and missed formally learning about the US Civil War (you moved out of the district that introduced it in 5th grade and into one that taught it in 4th, for example), that has close to zero impact on your ability to understand the lecture series on Renaissance Florence. However, if you never mastered fractions with variables in them, the rest of math after that (and all of science) will forever remain an enigma.
I'm not much of a fan of curricula that are a potpourri of disparate topics that seemingly exist solely so the child can "have been exposed to the material" by the time they revisit it in greater depth.
I found that at that young stage of math, even though the building blocks were being taught, they didn't feel like they themselves built on anything. So while it was very important for me to know how, say, Cartesian coordinates worked for later, it didn't start from any foundation of existing knowledge. Calculus started with limits, which I had some intuitive basis for -- I'd insisted when first learning division that obviously the answer to x/0 should be infinity and my teachers had never given me a satisfying answer, so this finally addressed that. By contrast, even though cartesian coordinates are hugely fundamental, I didn't have any sense of why they mattered or what problem they were solving as a kid, so I found it tedious and boring.
I now think that anything abstract needs to be tied to something concrete, preferably a project that integrates knowledge piece by piece through demonstration of each piece's use to the whole.
It's like that for programming. Lots of things seem pointless until one encounters the problems that necessitated their inventions.
Trying to teach an entire subject purely in the abstract for years and years is massive a waste of time. Unanchored knowledge falls right out of the head.
Indeed. Continuing with the example of programming, despite struggling with math I’ve now been working professionally as a programmer for almost a decade and been doing it in a tinkering/hobby capacity for about twice that.
Programming even includes mathematical concepts, just expressed differently, and yet I don’t have nearly as difficult of a time with them there. Why? Because practical examples abound and one learns to program by building things.
Of course some peoples’ minds work well with abstract theory, which is why mathematicians exist and that’s fine, but I think it’s an error to expect that way of teaching/thinking to work universally.
I had great success as a kid at an accredited Montessori (but public) school. Typically, we were taught new lessons in groups of 2-4 and were also always taught how to self-check our answers. As an adult looking at the curriculum, it appears pretty brilliant - because you're working in the physical realm, big problems are still easy to solve, which motivates small children and a lot of concepts such as fractions, binomial equation and cubes and cube roots are introduced at about kindergarten to first grade the first time in passing.
I've had a very solid grasp of mathematics and the importance of units and operators that I saw in nearly all of my fellow classmates and didn't see in most of the "normal" kids when I re-joined the normal public school system. It also had the added benefit of lettings kids work as far ahead and as quickly or slowly as needed. Also, we sometimes "regressed" and practiced something familiar or taught the younger kids, which helped with solidifying the concepts and connecting information.
My favorite was long division and the stamp game :) and I ended up being about 4 grades ahead by 6th grade.
This is a nice video showing some of the early materials:
https://www.youtube.com/watch?v=mIFQjONnn1g&t=381
https://hollismontessori.org/blog/2018/3/19/montessori-basics-how-math-progresses-through-the-levels
This was going to be my answer, too.
I was a Montessori kid (a long time ago). My daughter is 8 and has been in Montessori since she was 3. So I have gotten a closeup view of her progression.
Lot of good things to say about the curriculum. They go from concrete to abstract, starting with bead chain counting in pre-k and building to concepts like addition, subtraction, etc. There's a lot of coordination (e.g. color coding) among the materials.
The three year cycle (three years in the same classroom, like 3yo, 4yo, kindergarten or 1st, 2nd, and 3rd grades) means they have a much longer timeline letting concepts build on each other. This also means that younger kids see the older kids doing the more advanced work long before they are ready for it themselves, and (as @luks mentioned) the older kids cement their own knowledge by teaching the younger kids.
Another major difference is the fact that each child is following their own interest in the classroom. They reference the sensitive periods for different things, which come at different times for different kids. So instead of everyone having to do subtraction because it's "time for subtraction", kids that are still trying to master addition are working on that, some kids are doing subtraction, and some kids are on to multiplication. During parent nights, I always hear a lot of doubt from parents, "what if they don't ever figure out X" or "it seems like my kid is behind in Y", but my observation is that the teachers are great at managing the kids nd bringing them along, and the kids themselves working together does a lot to build interest in the materials.
I could go on and on about it. It's been great for us.
What's your experience with learning languages in that setting? I've only ever met Montesori (or Waldorf, Steiner, ect.) alumni from the two extremes: either they caught on early, and practically mastered the language very young, or they reach adulthood severely lacking foreign language skills.
And in the German speaking world, this is a problem. Both the education system and actual professional life require something close to fluency in English, and if you leave school without even the ability to follow podcasts/TV shows/novels, catching up will be hundreds of hours of very unpleasant work.
We are in the US, so unfortunately the standards for foreign language are abysmally low in general. The biggest barrier is the American-English-centric cultural attitude toward foreign language in general, but another thing that fights foreign language adoption is that there's no "obvious" second language to study. French and Spanish are most common. Sometimes, German. In my childhood, Chinese would have been completely out, but I am seeing it some places. So the education curiccula are fragmented. It's also a matter of luck whether you'd get a teacher who was either a native speaker or fluent in the language they are teaching. Imagine learning French and ending up (as I did) with Texan accented French. Finally, there's no immersive environment to practice. Things like DuoLingo help with this now, but they weren't available to me back in the day.
As for Montessori, my memory of foreign language (through third grade) is nonexistent, but it's likely there was some basic Spanish vocabulary. For my daughter, she currently has Spanish as a "special" twice a week (other specials include gym, music, and art). She is definitely getting a better foundation than I did. She's learning verb conjugations and a much wider and more practical vocabulary. But I would not say that it's up to the standards you described.
My assessment is that the deficiency seems to have more to do with the structural and cultural limitations I described, rather than a limitation of the Montessori method itself. I can easily see the language materials being extended to bilingual education and imagine "Spanish days" for classroom immersion, but my understanding is that it's hard enough finding Montessori trained teachers. Finding ones with language skills good enough for immersion (and all in the same language) would probably mean a much higher cost.
Ironically, schools being too expensive is it's own problem. We actually took my daughter out of the most expensive/elite Montessori school in our area because it was too much of an affluence bubble. The school she is in now has larger child-to-teacher ratios (20 kids in a class vs 10 at the other and the classroom pretty crowded), and while still expensive ($10k/y), it has scholarships to achieve a more diverse representation. To be sure, it's not perfect, but we're happy with the balance of education quality and cultural norms, given the local options.
Thanks for writing that all down!
I'm sure that's part of it. But in general, learning a second language just is a very frustrating experience in the beginning. After the first excitement has worn off, but before you're functional enough for children's books/movies, there's a few hundred hours of grind that almost nobody enjoys.
And unfortunately, children following the self-guided "following their own interest in the classroom" approach seem to pretty consistently chose to not recap the newest batch of vocabulary. It would be interesting if a hybrid approach could work here. Letting kids choose, but reserve 5 hours a week for old-school forced learning. Just for those who fall behind to far on something critical.
Anecdotally, that's what a few of the Montesori alumni I know ended up doing. Just not 5 hours in a week, but a couple of months in a year: they shipped their teens off to the UK/US, and solved the English language problem by forced immersion (and by throwing money at it).
You're welcome. You unlocked the intersection of my enjoyment of writing and love of Montessori.
I think you are right that hybrid methods have their place. The school she was in through kindergarten topped at kindergarten. Since they knew many kids would go on to traditional schools, they actually broke the kindergartners out for half a day of sit in-a-desk, raise-your-hand-for-the-bathroom traditional education environment. From what I have heard, it is quite effective for the transition.
Right now, (third grade), my daughter has weekly spelling words that everyone in the third grade does. The freedom comes in choosing the activities that they use to master them (e.g. rainbow copying, crossword puzzles, and some others I can't remember) culminating in a "spelling test" that is corrected, but not graded. But they are definitely required to engage with the spelling words each week. I think these kinds of methods could be used for pushing vocabulary in foreign language as well if the students didn't take to it on their own. My daughter revels in knowing Spanish words for things.
I think my spelling example is a good example of how the method is not "the child can do anything they want and won't be forced to do anything they don't like" vs "the children know that there are expectations, but there is (some) freedom in how they choose to meet them". However, hearing "follow their interest" as the former is a common misunderstanding (which I was alluding to in when I was talking about new parent questions in my topmost comment).
In pre-k, she had a lot more freedom to choose her work, but also, many more of the works were focused on building fine motor skills and hand strength in preparation for writing, so which ones they choose are not as important. One of my favorite examples of this are the metal insets. The tiny handles mean the children have to grab them in a way that strengthens their fingers for the pincer grip needed to hold pencil. Learning about the materials is a whole joy on its own, and one of the reasons I always attend the curriculum nights even though we've been in the program for so long.
There are also definite academic standards and learning objectives at every level. But how (and when) they are met may vary from child to child. Starting in first grade, the teachers have done a twice-yearly assessment against the state standards for grade level and had a conference to discuss progress with us parents. It's still not framed as a grade. The scale is something like, "Introduced, Practiced, Mastered".
My daughter has always been at an age-appropriate level, so I've never discussed remediation in these meetings, but I know there are strategies they employ. For example, there are several reading groups in her class. She is in the most advanced group (which is all third graders), but not all the third graders are in the advanced group. Students in the less advanced groups have more 1-1 work with the teachers and more focus on the reading itself. In her group, they are using their reading skills to do biographical reports and dioramas about historical figures. I understand this as a "carrot" approach that encourages the other students to want to advance vs the "stick" approach of bad grades.
Also keep in mind that first and second graders are seeing the work the third graders are doing and aspiring to it well before they are ready for it. The respect and a little awe that she had as a second grader and that she now enjoys as a third grader is a definite motivator, but it is also tempered with an expectation for leadership. A good example of this was that there was a day in the fall when her teacher was late due to a traffic accident. They were unsupervised in their classroom. I asked her if she went and got another teacher, and she said, "No, we know what to do. We just helped the first and second years get started with their materials, and Ms. ____ was only fifteen minutes late anyway." Amazing.
I think one important factor to this conversation is that there's a lot of variation between what appear on first glance to be similar styles of education in the whole "Montessori-esque" space. So it requires parents to really be discerning when looking at these types of schools, because the aesthetics can often be similar despite quite different approaches. I think there may well even be multiple "types" of Montessori specifically, though there do seem to be accrediting bodies for it at least.
Unfortunately as the other person said, this is the case for almost all American schools, so there's little point to compare them. My high school (a private religious but not remotely Montessori-adjacent school) replaced our only foreign language teacher with Rosetta Stone in my second-to-last year. If you're wondering how well Rosetta Stone teaches the subjunctive, it's not well at all!
But of course it's a very different situation in the US than in Germany in that respect, due to English's status as a lingua franca. Someone whose English was as bad as my German is would really struggle living in the US in a way I definitely don't here in Germany.
Wow this sounds pretty idyllic!
Smaller class sizes. 20-25 max.
Some sort of class for those who don't want to be there. We have a serious issue with not holding students accountable and it can take 2 or 3 who just don't care to ruin the whole lesson for everyone.
Individualize learning. When i was taught math in elementary school, we all had the same class work to start, but then we'd have tests. If you finished a test and passed it, you got a harder test next time. If you passed those they just got harder. There was no cap. While some kids were still struggling with multiplication tables others were doing fractions. Your classwork would change to some extent depending on your testing level, and the teacher could focus on helping students when they needed help. Those who were struggling to pass, or who had just gotten to a new tier and struggling with a new concept.
People learn differently and we should use that, but forcing people to wait for others to learn is awful. I had to teach a class where we taught 4 or 5 different ways to do a process (i want to say division? It was years ago). So if you're the kid who was smart enough to get it the first time, now we're going to force you to sit there for 3 more days while we go over it again and again, forcing you to do division or whatever in other methods that probably aren't as easy for you, and then punish you for doing worse. I literally had students who got 100% on the first day and 50% or worse on the second because they were both bored to tears and didn't understand the new method or why they needed it when they already had one that worked.
Blocks and physical props. Conceptualizing math works a lot better when you're actually looking at something and can see the interaction/interact with it yourself. Pictures work for some but as before different strokes for different folks.
Focusing on what kind of math they want. This is for later on once students have learned the fundamentals, but I really think there's a distinct lack of "useful" math once you're in high school and have decided you're not going into a mathy field. Algebra 2 and onwards was, at least in my area, required but really not something that people will find applications for. It's obviously also about the logic and problem solving, but I think low level economics or logistics style math would do a lot to help keep the attention of those who are
Use games. There are so many useful learning tools that also happen to be entertaining. Kerbal Space Program probably helped me understand more about space orbits and the like than years of trying to learn on my own, and it's hardly the only thing in that niche that lets you teach and use it. Schools are so fundamentally against entertainment, but honestly I think we're in an age where it's easier than ever to actually meld the two and not just have something disguising a test. I have plenty of friends who didn't enjoy classes but sure as shit upped their math game when WoW came out. I mean...hell, in high school if a kid knew metric they were either way into science or a person you could get your drugs from. Practical uses are extremely good teachers and there's a ton of value in games to do that.
Who is going to want to teach a room full of bad kids? I remember making the mistake of telling my guidance counselor I had a empty period on my schedule by accident, and they forced me into the only open class left. It was for taxes and all the bad kids were in there. Not much got done in that class.
There are people better suited for it especially when that’s what you know you’re going into and have more leeway with the lesson.
Trying to handle everyone from the kid who doesn’t care to the one who excels just don’t work. It’s a lot easier if you know you’re dealing with like minded individuals because you can customize the plan.
You basically lower expectations and try to focus on behavior first
I teach early childhood education, and have done so for the last 14 years. I can speak to the basics of mathematical concepts and how I feel they should be taught based on the evidence I've seen.
There are three areas we need to look at when teaching math:
Group size and dynamics
Unfortunately I have no empirical data for this but it is my firm belief that the ideal class size (at ages 4-8, for which I have first hand knowledge) is between 12-16 students in a class. This allows teachers to help individual students, to form small groups effectively, and to manage whole group instruction without constant redirection.
Any smaller and the social dynamics between students become, for lack of a better word, flanderized. Rivalries increase and it's harder for children to find or create a social niche for themselves. Any larger and classrooms become too difficult to manage effectively without an extremely strong teacher, and even then it's a struggle.
Student ability and interest
I also prefer having a mix of student abilities. Students who can perform above level can consistently help those performing at or below level to improve, and often help drive a classrooms abilities forward without any teacher involvement at all. The biggest difficulty is when you have disengaged students who just don't care, because their peers that they look up to don't care.
How we present the material
How do we get children to care? My method has always focused on games. I utilize games of all types (board, card, video, tablet, carpet, etc) as a vehicle for learning. Children at this age (and beyond) learn best through play - meaning, the practical application of concepts they are learning towards a purposeful goal that they can see real results from. In my classroom at the Prek level I always begin teaching math with simple board games that teach basic concepts such as counting, one-to-one correspondence, patterns, and numeral identification. From there we move on to games which involve combining and decomposing numbers, identifying number neighbors, and other such concepts.
Depending on the child, we can extend this indefinitely; last year I had a student who, at 4 years of age, I was able to successfully teach multiplication and division to through a mix of games and manipualtives.
Many teachers will talk about needing real items to demonstrate concepts; this is what I mean by manipulatives. Board and card games utilize these in the form of dice, counters, or cards. We can use other long-forgotten tools as well - I love abacuses!
In short, we need to game-ify early math concepts.
I can provide far, far more detail but I have to head to work soon. If there's interest I can expand on these ideas.
I’m sorry I’m going to just link a pdf but it’s a good one http://worrydream.com/refs/Lockhart-MathematiciansLament.pdf
The author is well spoken on the topic and has written a number of thoughts outside of the linked article.
I had a number of great math experiences growing up that I wish had made it to core curriculum. They were enabled by limiting class size, and streaming away the kids that didn't want to be there. There's a ton of ethical questions about equity there, but setting those aside:
IDK. I once asked my highschool math teacher if she could give me some historical context for the things she taugh. It might help me engage with the class and maybe even get a little excited about math.
She seemed defensive, as if I was making fun of her. Or maybe she couldn't care less about history, otherwise she'd be teaching humanities instead?
She never taught us about math history. I don't blame her, school teachers in my country have a huge amount of mandatory content and don't have the luxury to engage in personal projects.
It's a shame cause I think it would've help to have some context on matrices or whatever.
So yeah, teach a bit of math history. Give it context. What was it created for? How was it used? What was the importance of these concepts, and how did they influence science, business, engineering, and the world as a whole?
Ultimately, I'm a humanities dude. So context means a lot to me.
In a perfect world, I'd like age and the calendar year untangled from education.
I know the academic calendar is extremely important to administration and standardized outcomes, but my biggest school associations is stress nightmares about imminent exams and the shame of watching my class graduate without me.
But with an entire extra year, I could learn the work at my own speed, spend time teaching and explaining to classmates and practice some of the hardest work without a looming deadline. In retrospective, it was the best thing that could happened to me and it got me into a good college program.
Thinking of all the courses and training I've done since, I do think we have the technology tools to let kids learn on a flexible time line, within reason. But I can't imagine a classroom structure that would accommodate this framework.
If it did work, I think it'd go a long way in breaking down the stigma of failing, or just having difficulty in a single area. At the same time, it could help the issue of gifted kids feeling unfulfilled and developing unhealthy learning habits by keeping them engaged throughout.
Algebra is incredibly important for learning to solve problems outside of just math though. It's so important they've started teaching basics in Kindergarden.( 1 + _ = 2 )
Statistics and Finance instead of pre-calculus in high school makes a lot of sense.
It's all about the internalized problem solving method. I'm not saying that we sit down and literally do algebra every day.
1 + 1 = _is a straightforward thing: You have two knowns, and you solve for the answer. You're seeking a conclusion._ + 1 = 2is a twist, because you're working with a conclusion and a known, and you are seeking an unknown that fits.While that's trivial enough you could just say I'm making simple addition/subtraction harder....that's kind of the point. You've internalized algebra in such a way you just know to subtract 1 from both size to solve.
I put in 2 scoops of coffee grounds. I want 5. How many do I have to add?
That is a rudimentary applied algebra problem.
Yes. but they're doctorates and not kids whom have never been exposed to these concepts before. I'm definitely just showing examples of pre-algebra. But I think those applications get internalized....I'm better at math than my wife, and while she's much smarter than me as a whole...but this means one thing I'm better at is performing tetris to pack in the most amount of stuff inside a car. Am I measuring and doing the math? No...but I am taking the concepts I learned from math and applying it in an approximate and internalized way.
Even if you don't buy that core argument, I'd say algebra is foundational enough to working with other things that should have better literacy, like Statistics, that it should definitely be a core part of a math curriculum.
Maybe the state of science reporting wouldn't be so bad if everyone knew what a Standard Deviation was.