Today was laundry day - the first in quite some time - which meant there were about a thousand loads to wash. As the las few items were rolling through the dryer, John was folding load number 999 while watching me finish a math lesson on subtraction with borrowing with Henley. When Henley and I got through the last equation, I turned to John and expressed a combination of relief and amazement that Henley was able to pick up this concept in less than three days. John was still busy sorting through shirts and folding a mountain of shorts and underwear while I was letting go of the relief and feeling more and more proud, when he turned around and asked, “But does she really know it?” It wasn’t meant to be a challenge, not personally anyway, but the question caught me off guard. I felt it was obvious that she had learned how to borrow with subtraction, and I definitely defended my position immediately. He was right there listening and watching us so it couldn’t be that hard to convince him. He listened, and then went on to explain his question and defend his position. He was arguing that while she proved she can define borrowing, and she can answer my questions in the middle of the lesson within context (What doesn’t work and why? How can we fix this?), she never actually did any of the equations on her own. Okay, yeah, that’s true, but I am pretty sure that she walked me through, step by step, how to solve “82 - 29 = ?” until she had the correct answer just sitting there for me like dessert on a silver platter. While it wasn’t a fight, John and I came to a small divide, or gap if you will, that was defined by the question “what is learning?”

What is it? Is learning the ability to memorize something? Is it being able to explain what you’re doing? Is it being able to sit in a room by yourself for 30 minutes and take a test? A tough question to answer for sure. At times it seems like one or the other of these things, and at other times it seems like all or none of them. In short, all of these are learning, it just depends on the context. (I have to take a moment to properly address “it depends”. This phrase - “it depends” - just brought back grad school nightmares that lasted a full three years. Going into grad school to get your a Doctor of Physical Therapy, you would think you’d be learning the answers to every possible situation that PT would apply. Actually sitting in the classes and doing all of the clinicals, you realize quickly that the answer to every question you ever had is answered by our professors and mentors with “it depends”. Cue the full body shudder like a ghost just passed through me. Okay, back to learning!) There really are an infinite number of variables involved in any form of learning and also teaching, but not having a concrete answer to this question drives me crazy. Hopefully our explanations help you to define it for yourself better and allow you to believe that your child is capable of learning just about anything you teach them.

Let’s begin with the previous example from our math topics - Henley learning how to subtract with borrowing. First we are going to take an inventory of what we can confirm about Henley and her math knowledge. Here is what she doesn’t know: “mental math” or how to complete simpler calculations in her head quickly. Here is what she does know: how to use blocks of set values to add, subtract, multiply, and divide into the thousands. She also understands different place values and where they are, well, placed. She knows units-tens-hundreds-thousands (which we will talk about in a later blog for you, and why this is one of the most important math lessons teach, simultaneously if not before addition). In general, we have been working on math that John and I have deemed important to learn for foundational purposes, and we change the base topic each month. It can vary from decimals in April to negative numbers coming up in May and shapes later in June. In addition (haha, haha - get it?), we have been working our way through a third grade math book to make sure that we aren’t letting any other components of math education fall through the cracks for Henley. You may be asking yourself, Why third grade? Is that a magic year in math?” Third grade is a little bit arbitrary, but it seemed to fit the concepts she knows, so third grade book it is. This information along with the quick tally of “doesn’t know/know” give us a good baseline to see what is being learned or not.

Which brings us to concepts. We’ve used this word a lot and this is where John and I ran into our math and subtraction borrowing debate. Henley knows a lot of concepts. I spend ninety percent of my time teaching Henley concepts. I think it is important that she knows as wide a range of conceptual math as possible so that we can delve into the details of any one of these concepts at any time in the future. I feel this makes teaching math or any subject easier for me. For example, if Henley knows the basic make-up of a human cell, I can talk to her about heart cells versus bone cells versus brain cells, and we don’t have to cover the concept of what a cell is each time. She knows it’s something small that has different parts within it and that each different part does a different job. This concept can be applied across the board to all cells and she now has foundational knowledge that is supported multiple times over. Check out what we did the other day:

For math, this idea of teaching concepts makes sense because we see each topic so many times throughout even a single math book. It’s not like we learn to multiply one time in the sixth week of the second semester of third grade and never come back to multiplication ever again. Quite the contrary, we multiply throughout our entire math careers - algebra, geometry, trigonometry, calculus, matrices, etc. So, instead of teaching Henley the step by step process of how to multiply a million different variations of one and two digit numbers, like 17 times 6 for example, I figure it’s better if she can say “I have 17 friends and each friend has 6 tigers. How many tigers are there in total amongst all my friends?” If she an understand the goal in this situation, then Henley knows how to multiply. Just to be clear, Henley can do this kind of math with blocks and beads, but it takes a looooonnnnnggg while. Still she can get the job done, and there is also a certain value in being able to learn to persevere, even if it’s just through a math problem. We just wanted to make sure you all understood that it’s definitely not something she’s a whiz at. . . yet :)

So, yeah, borrowing. This is exactly what Henley was doing. She was explaining it perfectly. She told me that 82 as a number can be represented as 8 blocks of ten and 2 single unit blocks. She also told me that she could also take 1 of those blocks of ten and trade it in for 10 more single unit blocks. Now she had 82 represented with 7 blocks of ten and 12 single unit blocks. BOOM - BORROWING!!! I felt vindicated a bit, but that feeling was short lived. John persisted a bit that Henley didn’t actually know how to subtract with borrowing on her own, in a way that she could replicate it autonomously. And guess what. He was right. A little.

Henley had grasped the conceptual knowledge of borrowing, that much we had established. Something like this takes most elementary school students months or more to learn. In that way, what happened for Henley in just a couple of days was extraordinary. But she didn’t know what to do next, once she borrowed. She was missing the steps that come with the concepts, the steps that complete the problem when it’s been laid out in front of her. She was lacking the procedural knowledge!

Say what now? Procedural knowledge? Like another kind of knowledge? Yup. Procedural. It is exactly what it sounds like - the knowledge to actually complete the procedure or steps to a problem. Think of any situation where you wanted to learn something from a friend or co-worker, and the lesson essentially amounted to, “just do what I do”. You didn’t really get an explanation of what you were trying to do, why you were trying to do certain steps, or how to navigate completion of the steps. You just got an in-person instructional video that was meant to be imitated. Often times, we end up learning procedural knowledge before conceptual knowledge which can make assimilation of the concepts more difficult; however learning the concepts without the procedural knowledge also has its negatives. Being able to explain a concept but not be able to apply it isn’t practical at all, so we need to have this happy coexistence of conceptual and procedural knowledge.

Luckily, John helped me figure this out, and in a quick 30 minutes, Henley was using her combined conceptual and procedural knowledge to subtract and borrow! One technique that helped quite a bit was to just have her write her work out on paper rather than use blocks. The blocks were great and still are great for learning the concepts - they are visual, tangible, and real. But when it comes to getting the work processes learned well, pencil and paper work amazingly because that is exactly how she will complete these problems in the future. Cross out the 8 in the tens place, write a 7 instead, and place a 1 in front of the 2 in the ones place. Having the strong conceptual knowledge made it possible for the procedural learning to be supported, trusted, and executed extremely quickly!

Whether your child is memorizing letters or the sounds animals make, they’re learning. If they can count to 10, that’s great. If they can count 10 toy cars, even better, but regardless, they’re learning. One is conceptual, one is procedural, but both are considered success and together they are considered awesome. Don’t sell yourself short, or your child either - one letter or 26, one word or a whole book, or better yet, reading a book or comprehending a book - no matter what, your child is learning something new in each of these cases and if they haven’t already, they will soon learn and be able to apply what they learned.

Also, if you’re interested in any of the materials we use for teaching specific lessons, we would be happy to share what we use and where we purchased them.