Tag Archives: elementary math

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.

Different Paths, Same Endpoint

Way back in first grade, we started with MEP Math, which I adored, but which turned out to be all wrong for both my kids at that point. When both the kids were frustrated by MEP’s tricky problems, I pulled out Math Mammoth and tried that.

BalletBoy took to the Math Mammoth immediately. He liked that it was so straightforward. It made him practice a good bit, but he didn’t mind that, especially when I didn’t make him do all the problems. In fact, BalletBoy kept doing Math Mammoth all the way through fourth grade math. At that point, the Math Mammoth shine seemed to wear off. The order of topics got a little confusing for him. So we jumped ship. We tried a number of different things, including almost a full year’s worth of the Singapore program, Math in Focus, which was great, but also didn’t quite work for him.

In the end, we went back to MEP Math. BalletBoy finished out his elementary math doing MEP. This time, the tricky problems worked for him. Knowing that he was good at math that emphasized following strong examples and just getting in the practice, I got a vintage copy of Dolciani’s Pre-Algebra: An Accelerated Course to use with him. It was perfect and he did the whole book.

On the other hand, Mushroom found Math Mammoth just as stressful and confusing as he had found MEP Math. For several months, I didn’t make him do anything formal for math. He read living math books and played math games. At some point, I let him try Miquon Math and finally we had found something that clicked. Mushroom did so incredibly well with Miquon that I came to adore the program. Unlike BalletBoy’s Math Mammoth, this was a program that inspired me as a teacher. The number relationships and the huge flexibility of the Cuisenaire Rods as a learning tool was perfect.

When Mushroom ran out of Miquon books, he turned to Beast Academy, which unfortunately only had a few volumes out at the time. However, he did them all. He started talking about how much he loved math. While he never became a fast worker, he was sometimes an incredible problem solver with math. He could think creatively about it. I really credited that to Miquon. He thought about math in a much simpler, straightforward way than his twin.

When we ran out of Beast Academy books, he continued his eclectic math path. He did a lot of the Key to Math books, as well as some problem solving books, like the Ed Zaccaro books. I started him on Jousting Armadillos, which is a pre-algebra program. Unfortunately, the amount of writing focus in that program was all wrong for him. He finished it, but barely. We took a math break, then he started in on Jacobs’s Mathematics: A Human Endeavor, which started to re-invigorate his love of math, though he never quite regained it. Mushroom has a lot of anxiety about academics in general, even though he keeps making good progress.

Having liked Jacobs’s other books, I chose Jacobs for Mushroom’s algebra program. Since BalletBoy did so well with Dolciani’s pre-algebra, I assumed he would continue with Dolciani’s algebra program. However, partway through the year, BalletBoy hit a major snag with algebra and I hit a major snag in teaching it. Since I was loving Jacobs, we made the switch.

That means that, for the first time since the very beginning of first grade, my twins are heading into high school finishing the exact same math program, at more or less the same pace.

It’s fascinating to me how different their paths have been. BalletBoy continues to be a “get it done” math student for the most part. He sometimes gets very stuck in his thinking and I have to tell him to stop and try again the next day. He argues with me about math, only to realize he’s completely wrong when he tries to do the problem. Mostly he likes to do his work and he tends to score well, especially if the problem sets are repetitive. If he gets to one he doesn’t understand, he’s liable to skip it and happily go on to the next problem. Overall, he’s in very good shape for finishing algebra.

Mushroom meandered through so many different math concepts. He continues to be a slow worker. While he doesn’t like to admit it, he does better when he can get engrossed in a few very challenging problems instead of a lot of repetitive practice. He second guesses himself and refuses to move on until he understands, which can be good, but can also bring down his scores on tests.

Despite all these differences, Jacobs’s Elementary Algebra has been great for both of my students. It’s not a perfect program, but it has enough challenges and enough practice. It has engaging introductions and enough example problems. It’s really a thorough and great program. I’m also just thrilled to be back teaching the same math again!

I also think there’s something to be said here about letting kids take their own paths through math. It’s okay to take different ways through the material. In the end, you’re going to emerge in more or less the same place.