Math With No Numbers

Can you do math without numbers? The answer is obviously yes.

Several years ago, I read about someone asking for more math problems without numbers and thought to myself, huh? What’s that even mean? What would it look like? From there, I discovered the vintage book Problems Without Figures by S.Y. Gillian.

Reading on, I discovered exactly what a math problem with no numbers looked like.

If you know the width of one stripe on a United States flag, how can you find the total width of the red stripes?

See what I mean? In order to answer the question, students need to know how many stripes are on the flag and how many are red. They need to understand that to find the total width, they’ll need to choose an operation. In this case, they need multiplication. And in order to answer the question, they’ll have to explain it, because the problem doesn’t tell you the width. The only way to answer is by explaining your process.

See how sneaky a numberless problem is? Sometimes numbers worm their way in there, and several of the problems in the original book did include a number or two. However, most of them were like the problem above. They made students really think about the process of solving the problem.

When students face a word problem, they often revert to pulling all the numbers out and “doing something” to them. They want to add, subtract, multiply, or divide them, sometimes without really considering which operation is the right one to perform or why. When you don’t have numbers, it sidesteps that problem. For students who freeze up when they see the numbers, this can be a really good way to get them to think about their process with math.

That’s been an increasing focus in the wake of Common Core to get kids to be able to show that they understand the math they do. This is a very old fashioned approach that does exactly that.

However, when I first read Problems Without Figures, I saw that Denise Gaskins, the author of the excellent Let’s Play Math, pointed out that it could really use a rewrite. Excited to give it a try before using it with my own kids, I did just that for the first few dozen problems and went on to use them off and on with my kids over the last few years.

Recently, I pulled out the book again and decided to give it a full facelift and publish it. Some of the problems just have updated language. However, for many others, updating didn’t seem to make a ton of sense. Take this gem:

I know the length of a field in rods and the width in feet, how can I find how many acres it contains?

Kids are barely familiar with acres today and rods are entirely bygone as a system of measurement. Some problems like this got rewritten. I added problems with meters, for example. However, some of the problems just needed a totally new take. I tried to add a lot more problems about figuring out how to navigate all the choices we have nowadays.

If you plan to leave approximately a 20% tip on your restaurant bill, what’s a quick way to calculate that amount?

Overall, this was a really fun project. I hope other people find it useful! You can find it on Amazon.

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Numberless Problems for Today – Free Download

In my last post, I linked to a really cool vintage resource called Problems Without Figures. As I pointed out, I decided to update it so I could use it with my kids more easily. I didn’t update the whole work because it was very long. I did the first 150 problems, which is a little less than half, but I thought I’d share my update here with you. It’s free to download. For more about the types of problems, the reasons to use a book like this, and how to use the book, have a look at my previous post. Below, I’m going to talk a little more about the changes I made to the book and why. I can’t promise that every problem in here is perfect. I’m pretty sure I thought through everything, but I didn’t have a professional editor. Feel free to send me any corrections and do tweak any problems to make them make more sense for your kids if needed. Note that the nature of these problems is to include superfluous information, have answers that are impossible to get without more information, have multiple potential solutions or paths to a solution, and to have “trick” problems where the answer is much simpler than the problem implies. None of those things are necessarily errors. Again, see my previous post for more about these problems.

I changed the problems in a couple of ways. A large number I left the same or simply updated a word or two to be less old fashioned (such as “shall”), including some of the old fashioned problems about farms and orchards and the like, simply because it’s nice to have different perspectives occasionally. I updated the language in a few problems. For example,  instead of buying marbles and penny candy, kids are buying Legos and other contemporary items. For others, I changed the situation or the measuring units for a problem to be more up to date but tried to keep the spirit of the question. However, for several problems, I couldn’t come up with a modern equivalent so I completely rewrote the problem to be more contemporary.

Doing this was an interesting process for me. It let me think about how math has changed over time. Much of the original book was out of date because of the situations it portrayed. It assumed the readers were familiar with agriculture and farm animals. However, much of what has changed is the math. There were several measurement units that are simply not used anymore, such as the rod as a measurement of length. Other measurement units are in use today and students may have a general sense of what a bushel or an acre is, but they are less a part of everyday life. In general, while measurement math is still a key component of everyday math and we have the complexity in the US of having to go between customary and metric measurements, this book drove home for me how it was an even bigger share of the everyday math people used a century ago. For one thing, we tend to deal with standardized sizes of things much more often. Doors, windows, pieces of paper, tablecloths, picture frames, etc. are all standardized. If we want to know how much land we own, we look it up instead of measuring. If we build something, we tend to follow preset directions instead of making them up ourselves with all the accompanying measurement challenges.

On the other hand, there are types of math that were never or almost never covered in the original book. For example, there were only a couple of problems involving averages. There were no problems involving estimation. There were no problems involving statistics or ratios. Very few problems involved fractions or percents, which surprised me. There were no problems involving permutations or combinations.

Life today is more complex with more choices. We tend to need to know more about combinations and permutations just to order off a menu or decide how to buy something or what to wear. We have to evaluate more complex data and statistics to understand a news article or a scientific claim on the internet. We’re used to bigger numbers. None of the numbers in the original book were very big. These days we’re used to hearing about numbers in the millions and billions. Our tax code and economy are much more complex, meaning that percentages come into play a lot more often. When the original book was written, people lived with a cash economy and prices were more straightforward. It’s different now.

When I tossed out those few problems from the original, I tried to replace them with ones that asked kids about making the sort of decisions that we often have to make using math these days. So there are problems about how to make basketball matchups, how to choose which toilet paper to buy, how to know if a statistic is reasonable, and other more modern conundrums than how many acres in your fields or how many fence posts you need.

Numberless Math

BalletBoy has always gravitated toward just getting his math done. Ideally, he likes to have a page of all the same sort of problem, let me remind him how to do them, and then just do them all. However, as he’s gotten older, this has meant a struggle for him to some extent. As the math gets more complex, with more to remember, not having a strong foundation in the whys of math has led to more and more difficulties for him. If you’ve been reading this blog for awhile, you may note that he’s jumped around programs in the last year as a result. I’ve been trying to honor the fact that he’s pretty good at getting the algorithms memorized while still helping him understand the whys. Finding the right approach hasn’t been easy. Most things have been too easy or too hard.

Recently, the author of the excellent Let’s Play Math blog pointed me to the vintage book Problems Without Figures, which you can find as a pdf here. I immediately fell in love a little bit. Many of them have numbers, but there’s usually missing information so they can’t be “solved.” Instead, the question is focused on the process. If given this and this, can you find that? How would you do it? What other information might you need? Many of the problems can’t be solved unless you know more information. Others are easy, but they require a lot of steps. A ton of them require that you move between different measuring units. Many of them are filled with superfluous information.

There are also several trick problems. My favorite, by far, is the one that asks how you can find how old a coin dated 56 B.C. is. Obviously, a coin couldn’t be dated that (think about it…). The author of the book suggests that these should be sprung on students. I’ve already given the kids a few of these but warned them to look out for them. They were delighted to discover them and felt very clever doing so. I think it’s really good for kids to realize that the answer isn’t always straightforward, that it might be easier or harder than they anticipate, or that they might need different information than what you’re given or expect to need.

In general, I like the focus of having kids doing math that’s not about getting “the answer.” It changes the focus of math and makes it feel more approachable for BalletBoy. It helps kids with their writing as well. Mushroom has been using the Arbor School algebra series, which requires a lot of writing. I wish we had been doing these for a little while in preparation because they really focus a student on writing out a clear, step by step set of instructions for solving a problem. However, the small nature of the problems makes it feel like a doable task. Mushroom has struggled with the writing in the Arbor School series because it asks that he summarize everything he learned and give his own examples. This is such a small chunk and so specific that it builds good logical writing and thinking skills. Basically, it’s a great thing for kids to do for math, writing, logic, and thinking skills. Explaining how to solve a problem is just good across the board.

You may notice that many of the problems in the book are outdated. In order to easily use it with my boys, I updated the language for us. My next post is about that process and I’ll link to the updated version if you’d like to use it.