I’ve been itching to get into some basic abstract algebra goodies. With the help of the Cuisenaire Rods, Simply Fun Sumology number tiles and the Discovery Toys Busy Bugs, I’m able to do that.

Start with wrap around addition.

This type of math is officially called “modular arithmetic.” We are only going to use the numbers 0, 1 and 2.

It begins as regular addition. And since we are only using those three numbers, all our answers have to be either 0, 1 or 2. So when we add 1+2, we wrap around.

If we were to count in our system, we’d say: “0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, …”

The addition table looks like this:

(Notice you could do this with numbers from 1- 12 and it would be clock addition!)

Now things get buggy.

Switch out all the number tiles with some pretty color Cuisenaire Rods. They don’t have to be the “right” rods. We’re only looking at the colors. Here’s the progression I did:

The end result is a very abstract chart!

You can “bug” two things together.

Like this:

(I know – a spider isn’t a bug. But run with me on this, okay?)

Notice that each of these are directly from the “spider table” above.

You can read this as, “Purple spider green equals green,” just like you would say, “Zero plus one equals one.”

This is one of my favorite (and first) math “tutoring” stories. It has a great ending, for me. The biggest fraction of the pizza came my way! Continue Reading

One Response to “Let’s NOT divide the pizza equally” – A Hard Lesson in Fractions

Growing up, my dad was always throwing math at me as a game. It was playful (as math should be!). We had a game where we’d try to “trick” the other person out of the most pocket change. The idea being to make your 7 pennies sound like a way better deal than 2 quarters. This is an over simplified example, but you get the idea.

I loved it & although now I know my dad could add up the change MUCH faster than I could, I always felt like I had won something.

This is the first in the Cuisenaire Rods series. Video showing how to engage kids in learning coordinate pairs by tapping into their creativity. Continue Reading

Check out this way to see how union and intersection work – using SnackTraps and blocks. Also, how the empty set works with them! Continue Reading

5 Responses to Union, Intersection and the Empty Set

In the paragraph “What happens with the empty set?”. I like the idea that two cups, one empty and the other with some objects in it, have nothing in common, hence the empty set as intersection!

What puzzled me a bit is what if we try to find the intersection of two cups with all different objects in them? The result is again an empty set. Moreover, if the question arises, what are the ALL subsets of a given set, it will be {T, S, R}, {T, S}, {T, R}, {R, S}, {T}, {R}, {S}, O (empty set). Here I used your example Square, Triangle, Rectangle as a shorthand. We can see that power of set (set of all subsets) is exactly 2^3 = 8 i.e. 2^(# of elements) and this includes empty set too!

In other words it can also be said that empty set is common element for empty set and any other set, which looks like a different statement that empty set has nothing in common with a set of objects. Actually, empty set is a subset of any set. If we empty the cup we will see that there is an empty set right there as well and that it is common with the other (empty or emptied) cup, i.e. other empty set, on the left side of intersection sign.

Thanks, Nash! I love the power set – just can’t get enough of it (maybe because I’m so jealous, I wanna be the Power Bon).

You write, “it can also be said that empty set is common element for empty set and any other set.” I think you might have mistyped. The empty set is a common subset of both the empty set and any other set, but not a common element. This was the point of the previous article on empty sets. There is not an empty set inside the empty cup – the empty cup IS the empty set.

I think you might mean that the empty set is a common element of the power sets of the empty set and any other set.

Power set of empty set = { {} } (the set containing the empty set)
Power set of {T, S, R} = { {T, S, R}, {T, S}, {T, R}, {R, S}, {T}, {R}, {S}, {} } (what you wrote out)

In which case the empty sets are elements of the bigger sets – and are the intersection of the two.

Fantastic examples and graphics! A person could literally scroll through and learn about unions, intersections, and the empty set just through the pictures and reading the captions. Super brilliant.

My suggestion? “Write” a math picture book just with labels, captions and titles. It will be like summarizing novels in 160 character tweets! (or is it 140?)

Sets are intuitive – you have a basket and you put stuff in it. But what about the empty basket? How do you write it? And why is that important how you write it? Here’s a hands on example of the empty sets in your kitchen. Continue Reading

4 Responses to Empty Sets – When to Use Curly Brackets

I think that the analogy might be misleading, because you can have many empty containers, but there is only one empty set. A set is like a container in some ways; but unlike a container, a set is determined by its elements. Two sets that contain the same elements are identical, but two distinct containers could have the same contents, or no contents.

I do agree that the analogy can be useful, but one should avoid phrases such as “two empty sets” which can cause misconceptions.

Which makes me really think about the possibility of having different empty sets. Since the empty set is an element of the power set of any set, might the empty set as an element of the power set of integers be technically different than the empty set as an element of the power set of the set of polynomials (for instance)? I guess, no. But something feels like it might be yes.

If we say, “There’s nothing in it,” it might be very different if we know what the possible things are that MIGHT be in it.

Hmm. The axiom of extensionality states that two sets are equal if and only if they have the same elements. This is a standard axiom of set theory, and it implies that the empty set is unique.

But it does seem to me that there are instances where we would want to allow different sets to have the same elements. For example, suppose that a high school has a math club and a chess club. It might happen that the two clubs have the same membership list, but we would still consider the clubs to be different. Maybe there is a non-standard set theory that would model this situation.

Don’t confuse the name of something with the thing itself. Two clubs can have the same set of members, just like you can define sets S=\emptyset and T=\emptyset. Four containers gives you four copies of the empty set, which we lazily call four empty sets. (Except that some students will correctly think that the containers are not empty, and have different sets of air molecules.)

Math concepts are everywhere. Identifying them when you see them can make teaching math a breeze. Here’s one place to find some math concepts: at the dinner table! Continue Reading

4 Responses to How to Teach Math Concepts at the Dinner Table

I try to tell my students that math is everywhere. It is great how you described how your daughter is learning math by experimenting with everyday objects and not being told anything yet about what she is teaching herself. I forget that the young children have not been exposed to a lot of things mathematically yet. They do have to start from somewhere.

I wonder how it would be to have a “You Can’t Find Math in That!” contest. Students would have to find something that totally looked like there was no math in it and then explain some math in it. Win points for being the most creative.

Love is noncommutative and nontransitive. As an old joke goes: “Men love women. Women love children. Children love hamsters. But hamsters don’t love anyone!”

Rock-paper-scissors is a nontransitive game.

Lots of nonsymmetric relations can be amusingly roleplayed with kids: a book reading a person, a dog giving the owner a walk (mine frequently does), a chair sitting on you!

We know that if you start counting at one object the number of thing is the same as if you start with something different – but kids don’t. Here are some thoughts on counting that can put you in a different mindset when dealing with children and counting. Continue Reading

One Response to Counting Isn’t an Inherent Concept

[…] Crowder encourages teachers to think deeply about preschool math in Counting Isn’t an Inherent Concept and What is […]

I like your idea of using of the balance – it’s beautifully visual, but I would hesitate to call what you’ve done subtraction. Of course, it’s formally equivalent to subtraction, but this to me looks more like the “missing number”, or complementary addition flavour of the operation, which is not necessarily understood as the same thing by a young learner. i.e. To me, you’ve demonstrated the formal equation 5 + ? = 12, rather than the concept of 12 – 5 = ?.

As you’ve pointed out, 5 + ? = 12 is the same thing as 12 – 5 = ?. If we tell a young learner that they can’t understand this, then they will believe us.

They can understand this. Actually, they are much better at understanding this than we are. They have not been socialized out of understanding what is inherent to them.

Ordinal numbers are “1st, 2nd, 3rd, …” but we need these to count items to arrived at the final cardinal numbers. All this fancy math talk – why not make it fun with some colorful bugs! Continue Reading

Growing up, my dad was always throwing math at me as a game. It was playful (as math should be!). We had a game where we’d try to “trick” the other person out of the most pocket change. The idea being to make your 7 pennies sound like a way better deal than 2 quarters. This is an over simplified example, but you get the idea.

I loved it & although now I know my dad could add up the change MUCH faster than I could, I always felt like I had won something.