How Do Kids Even Learn Math...?
Concept of number combined with concept of algebraic thinking = mathematical thinking
In reading, sounds/language are mastered before letters and written word. Something similar happens with mathematics, although most people do not think of it this way. Algebraic thinking and problem solving are mastered before understanding of our number system. This is because our number system, like our written language (phonics) was created. But long before students learn the words for numbers and operations, they are able to solve problems. There is even evidence that humans can solve complex mathematical problems without a number system at all. The youngest of humans develop cognitive skills that translate to speech/language and early problem solving.
It is in our best interest to build on students’ natural abilities. We do this by helping students connect their sounds/language to the letters and graphemes. We do all of this while simultaneously strengthening their ability in language skills like speaking, listening, vocabulary, etc. It would also be in our best interest to help students connect their thinking, problem solving abilities, and natural approximation of quantity to the numbers and operations. We do all of this while simultaneously increasing their ability in problem solving, cognitive thinking, and processing.
This is why it is confusing to me to separate number standards from algebraic standards. Students strengthen their understanding of number by using their algebraic skills and they can use their number skills to in turn strengthen their algebraic process skills. There should never be a day when students are not engaging in deep number understanding and algebraic thinking together.
It is also important to make a distinction here. Explicit does not equate to memorization. Explicit means to make clear sense. When the idea of explicit phonics came into our conscience, many people took that to mean that we should begin dedicating our time to students’ memorizing phonics. This is not true as it is not that simple. Explicit phonics means to make phonics clear to students. This means to expose students to our phonics system in such a way that they internalize it and begin to understand it (read it) and therefore use it (spell it, write it). Many classrooms had abandoned phonics systems and turned to sight words, poems, rhymes, tricks, and ...memorization (see the irony?). They were being quite “explicit” in the broad sense of the term, but little thought was given to creating connections or neural pathways. The phonics system is similar to our number system (although the number system is far more reliable) in that it is a system that makes sense when you approach it with a strong background (in language) and relies on patterns (though I will say again, the number system is much more reliable - phonics often fails - and ouch , haven’t we all suffered from it?)
Similarly, the number system makes sense when you approach it with a strong background (in algebraic thinking/problem solving) and relies on patterns.
However, over time we have allowed our math classrooms to become the counterpart of a strictly whole language classroom. Instead of spending time to bolster students’ deep thinking, connections, and strong sense of patterns, we have resorted to memorization, poems, rhymes, and tricks. We are very “explicit” in math, but we are not very good facilitators of connection. I would implore you to move the term “memorization” from your teaching vernacular and begin to use the phrase “internalizing”. This should and will cause another big shift in your teaching. To memorize, one must be told it. To internalize, one must use it. You can memorize many things without understanding them. You have internalized something when you have used it, questioned it, connected it, and understood it. This internalization can be done (tediously) with phonics patterns and (in a much simpler and more fun (ok, bias) way) number systems.
But, algebraic thinking cannot be memorized, It’s an impossible oxymoron to even say it. Algebraic thinking is thinking….how can you memorize thinking? And it may surprise you...but addition, subtraction, multiplication, division, comparing, and quantity all fall under algebraic thinking. Students cannot memorize addition or multiplication. They can possibly memorize facts but facts are not addition. And this is very hard to understand, which is why we all struggle with it. Addition (or the other operations) is not one concrete thing we can tell our students. Addition is a logico-mathematical knowledge. And addition is not counting, no matter how much we teach our kids to draw and “count all” (you’re now covering a counting skill, fyi). Addition is an idea that exists outside of a number or other number. It is a process. If we do not help our students practice the processes, we will almost definitely lose them to the masses that have deemed themselves “not math people”. The beautiful thing is….students come to use with major ability to do these processes. They just don’t know our names for them. They join, separate, split, and group things from a very early age. With little help over time, students can start doing things like pass napkins out by giving one to each kid or join blocks because they know 2 groups will make their tower larger. They are doing the processes - we just bring them to school and immediately get in their way. We bombard them with remembering what we call this and that and “how we do” this and that and we do little to start where they are and help connect (I get paid for using that word...I wish).
Learn more about early algebraic thinking in this blog post here :)
When I first began to grapple with these comparisons between developmental early reading research and early math, I was disheartened. We still have major disagreements in how reading should be taught to our youngest students and reading is at the front of everyone’s mind. If we can’t even rely on good research for helping students read, when will we ever begin to care about shifting math? And that’s not even to mention how math has joined the ranks of so many other things to turn (somehow, some way) very political. Most parents could not tell you a major difference between whole language and phonics, much less what kind of reading classroom their student is in, but somehow they know some distinction between “old math” and “new math” (which isn’t even a possible thing at all...but I digress) and will take you down on Facebook. It is my passion to advocate for all students to receive a deep and satisfying mathematical education that begins with their natural approximation of number and builds on their natural thinking skills. It is why I consistently read and research how students develop number and algebraic skills and will continue to share. I hope you will shed your preconceived notions of what math is and join me in a series called “How Do Kids Even Learn Math?”
In each post I will dive into a component of early math and share with you how we can help our students become competent and confident mathematicians.
This is post 1.
Happy teaching :)
