Math is notorious for being agonizing to learn for some, while appearing to come naturally for others. Many have provided excellent and compelling theories on how & why this happens (Carol Dweck [1] and John Mighton [2] certainly come to mind). However, we believe there's a crucial mechanism at play in why certain learners struggle ➔ one that Grokkoli focuses quite intently on, but that we haven't seen elaborated on very much by experts in math education. For this reason, we are presenting our case for this mechanism here.

Viewing math as a ladder

Math is quite unusual. More than any other subject, it is made up of concepts that are tightly linked, where each new concept builds on the last. Subtraction is backwards addition, exponents are shorthand for compound-multiplication, which is itself shorthand for compound-addition, and so-on. Math just keeps building upon itself, unlocking ever more complex and challenging stuff to learn.

This isn't the case for a subject like history, where one can study early mesopotamian civilization without having a detailed understanding of the French Revolution. But try teaching exponents to kids before they've understood multiplication, and you will see how hard it is.

Enter the naive application of Mastery Learning in software

Learning for Mastery (or LFM), first elaborated on in the 60s has enjoyed a bit of a revival in learning software more recently. As is often the case with ideas becoming popularized over time, what was originally a complex set of ideas & principles has gradually been boiled down to a simple proposal: that every student should achieve mastery (usually defined as being able to get 90% or more of the questions correct) before being allowed to move on to the next lesson. [1]

And while this may break the classroom model (where kids are supposed to advance through learning content in lock-step), it's an obviously good idea. Particularly for a medium like software, where learning experiences can be personalized more easily. So it's not surprising that much of the math-learning software built in the last 20 years follows this core proposal (Khan Academy, for instance, has this as its central learning-design-principle, while IXL's scoring system is designed around achieving mastery rather than mere proficiency.)

Unfortunately, despite the rise and mass-adoption of these kinds of solutions, the learning outcomes of kids coming out of our school system have largely failed to improve when examined at scale. [4] We believe the problem is that these naive interpretation of mastery learning form an incomplete understanding of where things go wrong for kids. They view math as a ladder to be climbed sequentially, one rung at a time. And there's an important reason this isn't enough to work on its own.

Viewing math as a cathedral

It's not just that understanding compound multiplication helps us become ready to learn exponents, it's that raising a number to an exponent is compound multiplication ➔ it's a shorter way of writing down the exact same thing. If you can do one, you already know how to do the other.

In math, everything is connected:

  • 4 + 4 = 8 in the exact same way that 400 + 400 = 800
  • 6 + 3 = 9 and 9 - 3 = 6 may be different mathematical statements, but they are both based on the same underlying relationship between the quantities of 3, 6, and 9
  • 7 blocks - 2 blocks = 5 blocks in the exact same way that 7×100 - 2×100 = 5×100
  • 0.42 is made up if 4 tenths and 2 hundredths in the same way that 609 is made up of 6 hundreds and 9 ones
  • 2×7 = 2×3 + 2×4 is obviously true in the same way that (2+2+2+2+2+2+2) = (2+2+2) + (2+2+2+2) is obviously true

So we believe it's crucial that kids not only understand how to solve a certain kind of math-question, but how that type of question relates to just about every other type of question they've encountered so far. Every new skill learnt should extend what's already understood, so that one's understanding of math remains a single, unified and internally-consistent structure.

Imagine a well-assembled house, versus the pile of bricks, wooden beams, and other materials it is constructed from. The difference between the two is everything. If the materials that make up a house are improperly assembled, or not even connected at all, you dont have a house ➔ what you have is a heaping mess of materials. This is what happens to a kid's understanding of math when they aren't able to connect the concepts they're learning. Just as a house is much more than the long line of materials it is built from, math is a lot more than a long line of question-types to be individually mastered with scores of 90% or higher. In our view, math is not a ladder. It's a cathedral.

What happens when we lack a unified understanding of math?

The difference this makes in learning math couldn't be more night and day.

With a well-integrated understanding of math, everything fits together and makes sense. There's almost nothing that needs to be memorized, new concepts make sense pretty much instantly, and every new lesson reinforces the whole. It's what allows certain kids to breeze through their math curriculum and learn nearly effortlessly. It's what makes math fun.

But the kids that lack this foundation are forced onto a painful path. In lieu of a single integrated & internally-consistent understanding of math, they must instead memorize a long list of disparate rules and procedures, often with only a fuzzy sense of how & why they work. Things will seem fine initially, as these kids will still be able to answer questions correctly for a while. However, this list of disparate rules and procedures just keeps building with every new lesson. Kids may be able to manage it for a few years, but eventually the cognitive load of maintaining all of this becomes too much to bear, and they will eventually resign in bitter frustration.

This is also why kids that were initially doing just fine in math suddenly find themselves really struggling - it's often because of something that happened 2-3 years ago that dis-integrated their understanding of math, the consequences of which are only now coming to bear. (Typically triggered at specific points in the curriculum, where big jumps in complexity occur.)

And perhaps most alarming of all, it doesn't take much to put kids onto this path ➔ just 1-2 concepts that couldn't be integrated can create enough initial fragmentation to bar most (if not all) future concepts from getting properly integrated, all with little-to-no warning signs that it has happened.

Grounding all of this in a personal example

I was fortunate to have been obsessed with Lego as a kid. At age 3 I was already building spaceships taking up the entire table I was working on. If I couldn't find two pieces that were 6-long, I knew that I could use three pieces that were 4-long instead. All of this play demonstrated to me how quantities and distances could be combined, overlaid, grouped or otherwise spatially arranged relative to one another. A physical basis not just for doing basic arithmetic, but for how addition, subtraction, multiplication and division all relate to one another, had formed in my mind long before I started school.

Despite this, school wasn't easy for me, at least not at the outset. I struggled with reading & writing, as well as the linguistic aspects of math (i.e. how we represent expressions with symbols and operators). I despised doing tedious operations like long-division, and made lots of careless errors due to disinterest in the questions I was made to answer.

But in highschool, as things were getting more complex, my grades started improving - the questions were getting tougher, which in turn held my attention, and I started making fewer careless mistakes. Everything new we were learning instantly made sense to me because it fit in perfectly with everything I already understood. I didn't need to study or practice, and still got 100% on most math-tests.

During all of this, I saw the exact opposite happen to students that had only recently been top of the class. They began to struggle, only slightly at first, but cascading to the point that within a single year they were barely even passing their math-tests. These students had excelled at the procedural aspects of math, but had never been encouraged to connect or think deeply about the concepts they were learning along the way.

None of this, not my sudden ability to excel in math in highschool, nor my classmates' precipitous decline, was the result of innate ability.

What schools and parents can do about it

The first thing is to be aware of how dangerous a dis-integrated understanding of math can be. It's very difficult to detect when it starts and often goes on for years before causing problems. And once firmly lodged, it often takes a skilled instructor months of employing a wide variety of diagnostic approaches to pin-point underlying areas of confusion and truly get to the bottom of it. Unfortunately, teachers rarely have the time needed to focus on a single learner in this way, and most families aren't in a position to pay for the caliber of private tutoring that is needed.

(This is of course the main reason we created Grokkoli.)

The second is to look for curriculums that emphasize how everything connects in math. It's definitely encouraging to see new programs being developed that center on group-discussions, giving kids the chance to really consider what the math is saying to them, what the underlying logic is, and how it relates back to the real world. Stepping away from answering math-questions every once in a while to look at the bigger picture is a great way to help kids integrate what they're learning. And there are other learning theories growing in popularity among educators (such as Variation Theory [5] and the Spiral Approach [6]) that show a high appreciation for the importance of integrating kids' learning.

Nevertheless, the problem is far from solved. In a classroom where even the most well-meaning of teachers still can't be everywhere all at once, and until intelligent tutoring systems that continually running diagnostic checks are ubiquitous, the risk of kids falling through the cracks will always be there.