Common Failure Modes in Early Math Education

by Justin Skycak (@justinskycak) on

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We’re looking forward to extending our system down to earlier levels of math so more kids can join earlier.

In the meantime, I’ll go through some general failure modes to watch out for.

(Haven’t looked closely enough to weigh in on resources for very early math, but hopefully these failure modes will assist in avoiding ineffective ones.)

Failure Mode 1

Staying in “block/finger-counting” land so long that the the manipulatives become a crutch. The kid gets so comfortable and attached to the manipulatives that they resist moving on.

(Alex has a nice write-up here.)

Failure Mode 2

Covering a bunch of special-case arithmetic strategies instead of getting the kid really solid on the standard algorithms.

When the kid is first learning this stuff

  • they can get overwhelmed and paralyzed with "what strategy do I use,"
  • they can latch onto their favorite strategies and try to use them for everything even though they only work efficiently in special cases, and
  • they might not even realize that the standard algorithm is general-purpose (they may just think "whatever, that's another strategy, I already have one I like").

Failure Mode 3

Avoiding memorization. Once a kid is able to compute arithmetic facts, they need to work on developing automaticity via instant recall practice.

If they don’t develop automaticity on their math facts, then it creates a lot of friction later on. Things that should be quick & easy, like computing exponents and enumerating factor pairs, become really hard & time-consuming & error-prone when a student doesn’t have their times tables memorized.

(I wrote more about this here: )

Failure Mode 4

Serving up too many easy facts during times tables practice.

It’s a problem that plagues many, many times tables practice systems: when your practice is randomized across the entire times table, or even if you limit it to a longitudinal or lateral subset (e.g., “just facts involving 7 and 8”), you end up serving up way too many easy facts.

In a 12x12 times table including 0, over a third of the facts involve 0, 1, or 2. Over 70% of the facts involve a small number (0, 1, 2, 3, 4, 5).

So if you say that the student needs to answer some number of facts correctly in some amount of time, maybe even subject to a loose accuracy threshold, and you choose these facts randomly across the table, or even if you limit to a longitudinal or lateral subset later in the table… you’re still going to be serving up a crap-ton of easy facts and the student could quite easily “pass” your success criterion by nailing the easy facts while struggling on facts that are even just moderately challenging.

The way around this is to be very careful about what subsets of the table you’re practicing on, and very careful about how you select questions to test the student on. In particular:

  • After the student learns the easy facts, they need to quizzed on subsets of the table containing harder facts and leaving out easy facts.
  • Each individual question must be timed. If a student doesn't respond quickly enough, they miss the question. (Of course, this timed retrieval practice should come AFTER the student is able to compute results untimed.)
  • And if they ever miss a question, it needs to come up again (spaced out with some other questions in between) and they have to get it right.

Basically, you need to go Terminator-mode on the times tables, hunt down what facts the student doesn’t know cold, serve them to the student, and force the student to learn them. You can’t let the student “get by” on an aggregate metric despite not knowing some facts.

It’s not good enough for a student to know the easiest 80% of facts in the table. They need to know 100% of the facts and their practice always needs to be targeted to the facts on which they are the shakiest (while, of course, scaffolding up to harder facts by first covering any easier “prerequisite” facts that the student hasn’t yet learned).

Anyway, those are the 4 biggest failure modes that come to mind, though I’m sure there are more that I’m not thinking about at the moment.


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