Foundations First, Always
Hey Reader,
I cannot believe it’s already July! We are settling into summer over here, only to realize school starts back NEXT MONTH, and of course that means I have been doing what I always do this time of year: thinking about what's ahead.
Life
There is something about the slower pace of July that gives me just enough mental margin to zoom out and think about the bigger picture: not just what we are teaching, but how we are teaching it, and whether it is actually working. If you are in that same reflective space right now, I am glad you are here, because that is exactly what we are going to talk about for the next several weeks.
Homeschool + Math
If I had to name the single concept that causes the most downstream damage in middle and high school math, it wouldn't be algebra, integers, order of operations, or even negative exponents.
It would be fractions.
Fractions are where mathematical confidence goes to die for a lot of kids, and for a lot of homeschool moms, if we're being honest! The reason this happens isn't because fractions are inherently complicated. It’s because most of us were taught fractions as a set of procedures to memorize rather than as an idea to understand.
Add fractions? Find a common denominator. Multiply fractions? Multiply straight across. Divide fractions? Flip and multiply. These rules work, but if a child doesn't know why they work, then they can't remember which rule applies when, they can't recover when they make a mistake, and they have no way to check whether or not their answer makes sense.
Here's the shift that changes everything: fractions are numbers. They are not just operations (although fractions ARE division!). They are also not just procedures. They are numbers that live on the number line, just like whole numbers do, and they follow the same logical rules that all numbers follow. When a child understands that 3/4 means "three pieces, each of which is one-fourth the size of a whole," or “there four pieces in the whole, and we are counting three of them,” then they have something to hold onto that no amount of forgetting can take away.
That understanding is the foundation of truly mastering fractions. And if that foundation isn't there, no amount of fraction practice will build it. Instead, it will only bury the confusion under more procedures.
If you have a student who struggles with fractions, the question worth asking isn't "how do I get them to remember the rules better?" It's "do they actually understand what a fraction is?" Those are very different questions, and they lead to very different interventions.
Next week we are going to talk about integers and negative numbers, which is the other concept that tends to unravel silently in the middle school years. But this week, if you have a moment, sit with your student and ask them to explain to you in their own words what a fraction means. Not how to add them. What they are. Their answer will tell you a lot.
For classroom and co-op teachers: this same principle applies regardless of your setting. If your students are struggling with rational number operations, it is almost always worth going back to the concept of what a fraction represents before adding more practice problems.
Grace
There is a particular kind of discouragement that comes from working hard at something and still not being able to make it hold. If your child has spent hours on fractions and still can't seem to get there, I want you to hear this: the problem usually isn’t effort. It’s almost always a crack in the foundation. And foundations can be rebuilt, no matter how long the confusion has been sitting there.
I have been thinking about that in my own life lately. The places where I have been striving and still feeling stuck have been slowly surmounted by the quiet invitation to stop pushing forward and go back to what is underneath. God does that with us, I think. He is more interested in the condition of the ground than in the height of what we are trying to build on top of it. We need a firm foundation - one only found in Christ, built on God’s word.
"Unless the Lord builds the house, those who build it labor in vain."
Psalm 127:1 (ESV)
"For no one can lay a foundation other than that which is laid, which is Jesus Christ."
1 Corinthians 3:11 (ESV)
Our foundation matters. In math, in our jobs, and in everything else. The good news is that when we build on the right foundation, when we go back and lay it properly, the work we do on top of that foundation actually holds, builds, and grows. That is true of fractions, and it is true of everything we are building our lives on. Start there. The rest will follow.
See you soon!
- Mrs. Holman
