When I started homeschooling my son, I assumed that helping him to learn math was going to be my hardest job. It was never my strongest subject, and I never loved it. Nevertheless, math has proven to be one of the easiest subjects for my son to learn, and one of the areas where he has excelled.

At age five, he can add up to five two-digit numbers in his head, without paper, accurately and, in some cases, faster than I can. With paper and pencil, he can add or subtract any series of whole numbers, and he can multiply and divide not only with ease, but also with a clear grasp of what the numbers he is working with represent. He knows the relationships between fractions and percentages and decimals. He can read and create graphs. He understands the rudiments of algebra; for instance, he can solve for x in problems such as 3x = 12. And he can apply these skills to real-world problems, organizing the data visually and solving them in a straightforward manner.

According to the Common Core standards (for what they’re worth), he is currently working through problems at a fourth-grade math level, and by any reasonable standard he is quite advanced for a five-year-old.

Although my wife and I have guided him in various ways, the effort that got him to this level has been his own. But a variety of math apps have been of inestimable help. And, of all the apps our family has tried, the following dozen have proven most useful.

## Intro to Math by Montessorium

One of the first math apps my son used was Montessorium’s Intro to Math, and we love it—in large part for what it does not do. It does not tell children they are good for getting an answer right—and thus does not imply that they are bad for getting an answer wrong. It also does not use silly illustrations, goofy voices, or visual explosions in a vain attempt to keep children interested.

What it does, instead, is to help children understand the numbers from zero to nine by, in traditional Montessori fashion, presenting objects in quantities that correspond to those numbers; letting children trace the numerals; showing them how even numbers are grouped differently than odd numbers; and letting them see, compare, and organize rods of differing lengths and differing segmentation.

For example, in one exercise the whole screen is white except for one red rod. The narrator says, “This is one.” Then, as a dot of light shows up on the rod, she counts it, “One.” Next, a rod segmented with two colors, red and blue, appears. The narrator says, “This is two.” And, as the dot shows up on the red part, she counts it, “one,” followed by “two,” as the dot transfers over to the blue part. This continues once more. A rod segmented with a red, blue, and red part is displayed. And the narrator counts each part of it, “one,” “two,” “three,” as the dot follows. Finally, all three rods are shown. The teacher says, “Tap on one.” After doing so, the teacher repeats, “One.” Then she says, “Tap on two.” Then she says, “Tap on three.” Each time she repeats the number after it is chosen. The program then progresses to the next section, which builds on what was learned in the previous sections.

My son finished the programs in this app with a strong, though obviously limited, grasp of numbers. And he enjoyed the learning process. I heartily recommend it.

## Maths 3–5 and Maths 4–6 by One Billion Apps

For young children still learning the basics of mathematics, One Billion’s math apps for children three to five and four to six are the gold standard.

The first of these helps a child learn how to count from one to ten and how to add or subtract within that same range. It shows different shapes, provides their names, and gives children practice identifying each. In similar manner, it also teaches basic concepts regarding size, number, weight, and length.

The second app builds upon the concepts introduced and knowledge gained in the first. It helps children learn to count up to one hundred, and again how to add or subtract within that range. It shows how to count by twos, fives, and tens. It presents additional shapes, including 3-D shapes, provides their names, and gives children practice identifying each. It similarly introduces children to concepts of direction, time, and position.

All of the concepts are conveyed visually and verbally, using exercises and games, in ways that are easily understandable, engaging, and enjoyable for children. And, at the end of each section, the app includes a quiz in which children have to get every question right to pass. The result is some dramatic fun, and, when the child passes, a certificate and a well-earned sense of pride.

My son finished these two apps a mentally different child. He had a greater understanding of mathematical concepts and a greater appreciation for the kind of focused thinking that mathematics requires and develops. He has also returned to these apps many times since, often commenting on how formerly hard parts are now much easier.

## Montessori Math: Add & Subtract Large Numbers, Montessori Math City, and Montessori Math Multiplication by Edoki Apps

After a child has gained a basic understanding of mathematical concepts and operations, he can deepen and solidify that knowledge using these three apps: *Montessori Math: Add & Subtract Large Numbers, Montessori Math City, and Montessori Math Multiplication*.

*Montessori Math: Add & Subtract Large Numbers* uses two staples of the Montessori method, the stamp game and the bead frame, to make crystal clear what is happening when we add or subtract numbers and what the numbers refer to in reality.

For example, the stamp game shows how to represent (i.e., specify), add, or subtract numbers with stamps. The “stamps” in this case are colored squares with 1,000, 100, 10, and 1 written on them. These are organized in four respective boxes at the bottom of the screen.

To play the game, a child first represents a number, say, “24,” with the stamps below—in this case by dragging four of the stamps with 1 written on them into the 1’s column above and then dragging two of the stamps with 10 written on them into the 10’s column. Then he represents the number below a plus sign, say, “41,” by repeating the same steps. With that done, the plus sign starts to shake, a big red arrow points to the other side of the screen, and the app says: “slide to add.” After the child follows this instruction, the stamps placed below the plus sign move up to join the stamps above it. The app then says: “add up the results and confirm.” At this point, the child counts the number of 1’s in that column, of which there are five, and the number of 10’s in that column, of which there are six. Finally, he finishes the equation 24 + 41 = __ by typing “65.”

The beauty of the stamp game, and of this app’s translation of it into an electronic medium, is that it breaks down a complex mathematical operation into single steps that children can grasp. Once a child has mastered this process, he can learn with relative ease how to add hundreds and thousands, how ten ones become one ten, how ten hundreds make one thousand, and so forth. And, importantly, the child feels a sense of accomplishment, perhaps even a thrill of excitement, with every step.

The other two apps in the series—*Montessori Math City* and *Montessori Math Multiplication*—are similarly excellent, and, in my son’s case, proved equally effective at teaching their respective mathematical concepts and processes. For example, the former uses progressive exercises based on the Golden Beads (designed by Montessori) to introduce or reinforce the concepts of units, tens, hundreds, and thousands. The latter uses the stamp game and the bead frame, as well as a multiplication board, to make the process of multiplication and of carrying numbers more concrete, graspable, and automatic.

These apps are not strictly in line with the Montessori method, in that they include a reward element that is absent (for good reason) in strictly Montessori classrooms. But they are nevertheless superb apps, and well worth trying.

## Khan Academy App

The Khan Academy app contains a host of videos on math from basic addition to advanced calculus, as well as problems and exercises for practicing the concepts and operations covered in the videos.

The videos are often narrated walks through a problem, showing different ways that Salman Khan, the founder of Khan Academy, might solve it. The videos are short, Khan’s enthusiasm for the subject (as well as his quirky sense of humor) makes them enjoyable, and the instruction is typically exceptional and always clear.

When introducing basic multiplication, for example, Khan shows three star patches and asks, “If I had one group of three star patches, how many star patches do I have?” Then he continues:

I literally have one group of three star patches [which he writes as 1 x 3]. Well, that means I have three star patches [saying, as he counts them]: 1, 2, 3. This is my one group of three. Now let’s make it a little bit more interesting. Let’s say I had two groups of three. So that’s one group, and then here’s a second group. [Both are now showing on the screen.] Here’s two groups of three. So how many total star patches do I have now? Well, I have two groups of three [which he writes as 2 x 3] or another way of thinking about it is that this is three plus three, which is equal to 6 [all of which he writes, making the equation 2 x 3 = 3 + 3 = 6].

So we see one times three—one group of three—is three. Two groups of three—which is literally two 3’s—is six. Let’s make it evenmoreinteresting. Let’s havethreegroups of three. Now what is this going to be equal to? Well, it’s three groups of three, so I could write this as three groups, three times three [written as 3 x 3]. And how many of these star patches do Inowhave? Well, this is going to be three plus three plus three [which he writes as 3 + 3 + 3]. . . . So this is three plus three plus three, which is equal to 9 [and which he writes, finishing the equation 3 x 3 = 3 + 3 + 3 = 9].You can count them: 1, 2, 3, 4, 5, 6, 7, 8, 9. Or you could just count by 3’s: 3, 6, 9. And I think you see where this is going . . .

^{1}

Khan is a master at building up conceptual understanding step-by-step like this, and doing so while coming across as a friendly guide. That is one reason the huge collection of videos and practice problems at Khan Academy have been a go-to resource for my son, but there are many, many more. Speaking of which, did I mention that the entire curriculum—including the app—is available for free?

## LearnZillion

LearnZillion is another free collection of short, four- to eight-minute math videos. This app was put together by a “dream team” of some of the best math teachers across America. These passionate teachers break down the entire math curriculum into specific, hierarchically structured steps, focusing on a single step at a time. They then prepare the best lesson imaginable for it in electronic format—at least that is what it seems like, because it is hard to imagine any better than these.

Most (if not all) of the lessons start with a review of relevant material previously taught, contain a real-world problem, show the method or methods to solve it, and dispel some common misconceptions that students likely have.

The result is a fantastic resource for anyone interested in learning mathematical concepts—or reviewing them, and thus learning math in a deeper, more connected way. I watch one or two with my son every day, and always with great satisfaction. My son always learns something new or deepens his understanding of something he already knows. Unconditionally recommended.

## Mystery Math Town and Mystery Math Museum by Art Gig Apps

I am no expert on the psychology of pedagogy, but I think one of the things that causes many children to hate math is the amount of repetition required to ensure that they have automatized the basic operations of addition, subtraction, multiplication, and division.

I for one never find myself reminiscing fondly on the process of filling out yet another worksheet in math class. But my son actually may reminisce in that way. And if he does, it will in part be because of Mystery Math Town and Mystery Math Museum, which are two of his favorite apps.

These apps have a set structure: You have to find objects in various houses or museums. As you travel through the buildings you collect numbers, and to get through each door you have to answer a math problem involving those numbers. The problems might be addition, subtraction, multiplication, or division depending on what settings you’ve selected.

My son *loves* these games. He started playing them when his multiplication was so shaky that I would have to draw—“just like Mr. Khan does”—exactly what 5 x 8 or the like meant, and then we would count out the circles, socks, boxes, or whatever, one by one. But, with these apps, he rapidly mastered basic multiplication, and I no longer draw socks.

We delete these apps from the iPad each time he has finished collecting all the objects involved. But, every few months, he will ask to play them again. So we load them back up, and he speeds through them, quickening his multiplying mind and reaping the rewards of intense focus.

Again, I don’t remember exactly what my response to math worksheets was, but I am pretty sure it wasn’t this.

## Front Row Education

Front Row is a free, beautifully designed and amazingly comprehensive math practice app. It contains a host of well-thought out questions and word problems. It has very short videos that children can watch if they are stuck on a problem or need a reminder of an approach to solve it. And it is adaptive—meaning that the app adapts to the answers it gets, changing the kind of problems it will present next based on whether previous answers were correct or incorrect.

One of the things I love most about this app are the detailed reports it provides on my son’s progress. I can see what he is working on, what grade level it’s for, whether he is blasting through the material or struggling with it, and even what outside resources might be useful to him given his level of understanding.

That information lets me see where he is in detail and allows me to step in and offer some pointers or, as is usually the case, point him to excellent help somewhere online. So if he is having problems with a certain kind of word problem where knowing how to use bar models would help, I know there’s a Khan Academy or a LearnZillion video for that and can introduce that to him straightaway.

This app is pure gold—especially for homeschoolers.

## Squeebles Times Tables 2 by KeyStage Fun

Squeebles are made-up characters with their own personalities, hobbies, favorite foods, and abilities. They are instantly likable. But there is one problem, a very big problem.

As the app explains to children at the start, “The Squeebles have been taken prisoner by the nasty Math Monster.” Worse still, this monster stole the ball-shaped flying machines and the flip-fish that the Squeebles use to play a game called bubble-ball. So the Squeebles are prisoners with no games to play!

This is where your child enters the picture. He can help rescue the Squeebles and retrieve the things needed to play bubble-ball. How can he do it? By means of times tables. If he achieves a certain score on a series of times table problems, he not only rescues the Squeebles and their game, he also gets to play bubble-ball with the Squeebles he rescued.

If that doesn’t sound like fun to you, you need to get in touch with your inner child. Seeing my son’s joy in playing this game puts me in touch with mine.

I recommend this app as wholeheartedly as those above.

### You might also like

*Perfectly Reasonable Deviations from the Beaten Track*, by Richard Feynman

[groups_can capability="access_html"]

## Endnotes

1. See video, “Intro to Multiplication,” Khan Academy, https://www.khanacademy.org/math/arithmetic/multiplication-division/mult-div-concept/v/multiplication-intro.

[/groups_can]