Hey Reader,
It is the middle of July already, and this month is moving faster than I expected, which seems to be a theme every summer no matter how much I tell myself I will slow down. Every year I think I’m going to have a calm summer, but it so easily gets busy and flies by!
Life
This year I have been trying to be intentional about noticing the good things as they happen rather than looking back at them later and wishing I had paid more attention. It’s definitely easier said than done, but I think it’s worth the effort. If you are in the thick of summer right now and feeling the pull to either rush forward or look back, I hope you can find a moment to just be here. ❤️
Homeschool + Math
Last week we talked about fractions: specifically, about the importance of understanding what a fraction is before drilling how to operate with one. This week I want to talk about a concept that causes a surprisingly similar problem: integers and negative numbers.
On the surface, negative numbers seem simple. You go below zero. You owe money. The temperature gets really really cold. Most kids can recite the concept without difficulty, but the problems start when they have to operate with negative numbers, especially in combination with each other. The conceptual understanding falls apart when students are asked to add, subtract, multiply, and divide integers.
Why does a negative times a negative equal a positive? Most students (and many adults!) cannot explain it. They know the rule, but they don’t know why the rule is true, which means they tend to apply it inconsistently, and since many adults cannot explain it, the students don’t get the explanation of WHY it is what it is. Spoiler: it has to do with opposites!
The conceptual gap that many curricula skip over that causes this issue is this: negative numbers are not just “less than zero.” They are the opposite of their positive counterparts, and the operations we perform on them follow from that relationship in a logical, consistent way. When a student understands that -3 means “the opposite of 3,” then the idea that -(-3) equals 3 makes intuitive sense: you’re just taking the opposite of the opposite. If the opposite of 3 is -3, than the opposite of the opposite of -3 is 3! So, when we are explaining why a negative times a negative is a positive, you can think of it like this: -4 times -3 is like saying the opposite of 4 groups of -3. 4 groups of -3 is -12, so the opposite of that is positive 12! Thus, -4 x -3 = 12. Pretty cool, right?
The number line is one of the most powerful tools for building this understanding. We shouldn’t use it as a place to count spaces, but rather as a visual representation of the relationship between numbers and their opposites. Students who can see that 4 and -4 are the same distance from zero in opposite directions have a mental model they can actually use when things get complicated.
When trying to explain adding and subtracting integers, utilize physical counters in two different colors, say blue for positives and red for negatives. When students can SEE that -4 + (-3) is just four negatives plus three more negatives, making 7 negatives (or -7!), they not only can answer accurately, but they can see the concept and thus build a truly solid understanding of integers.
Before your student moves into 6th grade, 7th grade, or pre-algebra (or if they are already there and struggling), it is worth checking whether they can explain, in their own words, not just what a negative number is, but also why operating with negative numbers works the way it does. That explanation will tell you a great deal about whether the foundation is there.
For classroom and co-op teachers: integer operations are often taught quickly, with the expectation that students will solidify their understanding through practice. But practice on a wobbly conceptual foundation produces inconsistent results. A single class period dedicated to the “why” behind the rules that includes number lines, real-world examples, and physical manipulatives can prevent months of confusion later.
Grace
We spent this week’s math conversation on a concept that looks simple until you actually have to explain it, and grief works the same way with faith. It is remarkably easy to believe our understanding of God is settled and complete right up until loss shines a harsh reality on it, and only then do we discover how much of what we believed was more assumption than conviction.
I think of the moms (like me!) in this community who have walked through the kind of loss that does not make sense, the kind that no single verse fully explains and no timeline resolves on schedule. Grief has a way of exposing exactly what negative numbers expose in math. The underlying truth was never wrong, but our grasp of it was often shallower and more procedural than we realized. We knew the rules. We could recite what we believed about God’s goodness, His nearness, and His sovereignty. But it is one thing to know a rule and an entirely different thing to understand why it holds when everything around you says it shouldn’t.
Questioning and doubting not a failure of faith. Testing and challenges take the beliefs we held on the surface and asks them to bear weight, and in doing so, it shows us which parts of our understanding were genuinely rooted in who God actually is and which parts were just borrowed language we had never had to lean on before. The good news is that the foundation itself was never in question. Only our grasp of it was. And all that questioning and doubting? God can handle it!
“That the tested genuineness of your faith, more precious than gold that perishes though it is tested by fire, may be found to result in praise and glory and honor at the revelation of Jesus Christ.”
1 Peter 1:6–7 (ESV, condensed)
“The Lord is near to the brokenhearted and saves the crushed in spirit.”
Psalm 34:18 (ESV)
“I had heard of you by the hearing of the ear, but now my eye sees you.”
Job 42:5 (ESV)
After walking through the loss of my daughter, Myla, that last verse struck me as the truest picture of what testing produces. Job did not get an explanation for his suffering. What he got instead was a far deeper knowledge of who God actually is. He got the kind of knowledge that can only be reached by going through your own loss. Whatever season of testing you are in, whatever it has revealed about the misconceptions in your understanding, it is not evidence that your faith was never real. It is evidence that it is finally becoming yours.
Keep walking toward the why instead of settling for the rule. God is patient and He desires for you to truly know Him, and the understanding you gain through the trials will encourage you in a way the surface version of your faith never could.
See you soon!
- Mrs. Holman
