Math Mastery - How good is good enough for a math kid
For math kids, how much mastery should be necessary to move onto a new topic
How good is good enough?
We had a year-long disagreement with one of our kid’s teachers a few years ago.
Over several teacher-parent conferences, we pressed for more advanced work for our student and were given the same answer - they haven’t shown they are capable of the current work, so why should we advance them.
The teacher contented that because our student wasn’t getting 100% right on every math assignment it meant that the student “didn’t get everything and hadn’t achieve mastery” so they couldn’t be presented with more advanced work.
We fought back with, well, they’re only 8 (and 9) years old; sometimes, because they are a kid, they will make mistakes, but that doesn’t take away from the fact that the mistakes are not on the content; they are on simpler skills that they seemed to have blanked on at that particular moment.
We lost the argument, and our student was not given extra work or allowed to advance.
There are several directions we can go from here
One direction would be talking about what it really means to master a discipline. We could also talk about what it means to get the content right but make errors on less advanced stuff (they can do algebra, but they messed up a multiplication step, so the answer is wrong). We could discuss what it takes to go from 90% to 100% in a given subject area. We could discuss how in some assignments, the 100% doesn’t show how much the student knows (they could have gotten 150% if the problems continued getting harder). We could discuss how the difference between 98% and 100% is philosophically bigger than 2%. We could discuss parent-teacher conferences and aligning what the parents/guardians think is good for their child and what the teacher/administration/school/pedagogy says is good for the child. We could discuss self-advocacy for children and how they can push back on teachers and instructional methods. We could discuss the breakdown between a parent’s wishes, a teacher’s wishes, and the wishes of an administrator. We could discuss how students can self-entertain when bored in a classroom.
We’ll talk about a child’s dreams
And treading softly.
Aedh Wishes for the Cloths of Heaven
W. B. Yeats, 1865 –1939
Had I the heavens' embroidered cloths,
Enwrought with golden and silver light,
The blue and the dim and the dark cloths
Of night and light and the half light,
I would spread the cloths under your feet:
But I, being poor, have only my dreams;
I have spread my dreams under your feet;
Tread softly because you tread on my dreams.
That Math kids are kids who want to do more math is tautological. As they say, “It does exactly what it says on the tin.”
More math.
More math games.
More math books.
More exploring math.
More playing with math.
More time spent doing math.
Schools do not support math dreams
School gets in the way of math dreams. The way it gets in the way is that the whole classroom moves lock-step through the math of the year/semester/week/day, whether the student wants to explore more or has already explored the day's topic.
Does it work with…
What if….
Can you…
What about…
I’ve already seen it. What comes next...
I have spread my dreams under your feet;
Most teachers will be dealing with students on the lower end and/or determined to act out to draw attention to themselves.
Only an excellent teacher willing to take 1-on-1 time with the student, who has explored the math area themselves, will be able to dream with the student.
Schools do support math dreams
Math is hard, and to learn more math, you need to know, understand, be able to apply, and internalize all the subjects that have come before. That is, you need to have mastered the previous topics.
Schools have spent decades teaching kids math and have seen what generally works for most students most of the time.
They must cover a set curriculum each year/semester/month/week/day, and it will generally lead to the next step seamlessly.
They know that if a student gets 100% on a homework or test, they don’t need to worry about going back over the basics, whereas if a kid gets 20% on a homework or test, they do need to worry about returning to the basics.
For math kids, how much mastery do they need to show?
This is the important question for math kids.
The goal is to have them get as close to 100% on everything and explore more math.
But if they are only getting 99% or 95% or 80% or 60% or… there is probably a cut-off where it makes sense to review the fundamentals and ensure they understand the content.
On the one hand, we want to make sure the student knows their “basics” well enough to grasp what comes next.
On the other hand, we want to ensure that we don’t bore the student by only working on things they “already know” so they can move from 95% to 100%.
Mastery Learning and their cut-off for a “master”
Mastery Learning is a thing1.
Per Wikipedia,
Mastery learning (or, as it was initially called, "learning for mastery"; also known as "mastery-based learning") is an instructional strategy and educational philosophy, first formally proposed by Benjamin Bloom in 1968. Mastery learning maintains that students must achieve a level of mastery (e.g., 90% on a knowledge test) in prerequisite knowledge before moving forward to learn subsequent information. If a student does not achieve mastery on the test, they are given additional support in learning and reviewing the information and then tested again. This cycle continues until the learner accomplishes mastery, and they may then move on to the next stage.
From that paragraph, you can see that the cut-off is 90%. That said, there are other sources, Otus.com (“Otus provides technology that empowers educators to make meaningful decisions by acting on student growth data”), who have the cut-off at 80% for the student to be considered a “master” of that level2.
Split the difference, and we get a hand-wavy 85%.
Tread softly because you tread on my dreams.
Math Mastery - How good does good enough need to be for a math kid?
So, we get to an answer of somewhere between 80-90% on homework/tests, which should be sufficient for us to move on to the next topic.
That feels about right - there are still areas to understand fully, but there is enough to put it to use comfortably.
But what about the missing 10-20%, you ask?
They are a math kid, so remembering our tautology, they’ll do more math and be excited to do it.
Which means they’ll want to tackle the next subject that comes up.
This means they’ll need to apply what they got 80-90% on to new types of problems that will force them to practice over and over again what they missed.
You don’t really get a subject until you see the next subject
When I was in college, a Ph.D. student told me their advisor’s policy was that if the Ph.D. student struggled in any of their grad classes, the advisor would force them to be a teaching assistant in a class that taught that subject the next semester.
While perhaps apocryphal, I never asked their advisor; it rhymes with my experience that for many subjects, I didn’t thoroughly understand the subject until I saw how it was used by a higher level of problems in the class that followed.
Similarly, you’ll often hear that people really only learn Algebra I and II when they take Calculus and are forced to use it ad nauseam. Likewise, most people struggle with Calculus because they do not have strong enough Algebra skills.
Go forth and dream of math with your math kid
We eventually settled on the 85% cut-off at home and stopped trying to battle the school. We explained to our student that while school will expect 100%, when we are working ahead of the school and doing math explorations, the joy is in the math and not the 100%. Someday, when college tests come, we’ll focus on getting to 100% on everything, but that day is not today.
Until next time,
Sebastian
Anderson, L. W., & Block, J. H. Mastery learning. In D. Treffinger, J. Davis, and R. Ripple (Eds.), Handbook on educational psychology: Instructional practice and research. New York: Academic Press, 1976.
https://otus.com/guides/mastery-learning/