Mathematics in Civilization (2/n): Numbers in different bases

I reached the end of the first chapter in Mathematics in Civilization only to be greeted by a practice set at the end. I know practice sets are used inappropriately in many contexts, often only seeking level 1 information from kids, but in this case this practice set served as a formative assessment of my comprehension.

This is interesting for me as a teacher, because as adults, we don’t often encounted what we refer to as “instructional level text.” I might even be willing to wager that, given the idea that instructional texts are supposed to be read with 95% fluency, that this book is a frustrational book for me.

So what am I doing for myself so I can have a deeper understanding of this text? Well, in class we often encourage kids to draw from a bank of resources (in ELA, this often includes a teacher-or-team-vetted stack of books at a variety of levels on the same topic).

Unfortunately, part of the struggle I had in teaching math is that for many years I didn’t feel like I had access to resources that could help **me** in my elementary math **instruction,** so this text is really instructional or frustrational level for me. Deeply understanding this book and its concepts are partly hampered by my lack of experience in reading dense // technical texts.

I know it’s difficult to picture my work without seeing the book, but here are my notes for the first section of the first chapter and the practice set (which is really enough for me to chew on for basically the past week). For context, this is considered pre-algebra and is explored in 5th grade Eureka math curriculum.


I got stuck on base 2 and base 8 expression, and so I went to Khan Academy. I was fairly sure that I’d find a video that would fit my specific needs. I discovered that the Khan Academy was great for specific skills, and this one featured a history lesson before jumping straight into the binary lesson system.

Some relevant quotes that I think show good direct instructional practice:

“And to solve this, (finding a way to move on from primitive counting methods such as using tally marks, etc) human beings have invented number systems. And it’s something that we take for granted. You might say, ‘Oh, isn’t that the way you’ve always counted?’, but hopefully over the course of this video you’ll start to appreciate the beauty of a number system and to realize that our number system isn’t the only number system that’s around.”

There, the instructor shares his learning outcomes so students will know the teacher’s goal for this lesson, which will provide students with historical context to broaden understanding of number sense, and will hopefully help them manage numbers in base 10.

The thing that’s interesting to me is that the learning goal ISN’T “learn how to write in base 2.” The learning goal is broader, and it reaches out to work in other disciplines civilization, economics, language standards here), but it’s still grounded in one specific aspect of mathematics: being able to “see” numbers in both base 10 and base 2.

This video was enough for me to better understand what I needed to do that I didn’t need too look at other number systems videos (there were a great number at first glance), and these additional videos could provide both extension support as well as RTI.

It took me perseverance to make it through the whole video, because honestly, a part of me thought it wasn’t going to deliver. So here’s an example of the next video I had lined up in case this video didn’t meet my needs.

This video is more of a practice set. I had it going in the background while I returned to my practice set and finished it. I appreciated that the faceless Khan Academy voice spoke to me using an informal tone to invite me into the learning, saying things like, “I’m assuming you have at least tried, and we can work on it together.”

Upon reflection of my own work, I realized the biggest stumbling block in my way of learning mathematics is that often times I can’t “see” how numbers fit together. You know how in your brain you can just KNOW that 1+4 = 5 AND that 2+3 also equals five? I don’t yet have that kind of automaticity in this kind of number sense practice.

I’ve had enough “we do it” practice, so it’s time for me to try again on my own.

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