Play is one of the most important parts of mathematics; allowing students to learn through play allows them to brush through other concepts they haven't seen before, ask questions, and come up with their own conclusions. Also, play helps them physically work with abstract concepts, and links both halves of their brains to allow them to learn more. Below are some of the tools I recommend for play.
Teach place value, counting, addition, subtraction, multiplication, division, and order of operations through simple takes on the game "War". Instead of flipping over one card and seeing who has the highset value, flip over multiple, do a predefined operation, and see who wins! You can even use the red cards as negative numbers and teach operations with integers. See what else you can come up with!
Marcy Cook Tiles are puzzles that let students explore new concepts and work together to solve challenging tasks, allowing them to make new connections in their head and causing them to jumpstart their journey of problem solving.
A geoboard is just a grid of pegs and a set of rubberbands, but the possibilities for learning are limitless. You can learn counting, area, perimeter, shapes, properties of shapes, geometrical invariants, trigonometry, and graphing all through this simple tool. For example, you can ask students, "How many squares can you make?" The answer is trickier than it seems - take a close look at the picture above!
A very useful tools for students learning volume, these solids allow for students to intuitively "see" that a cone has 1/3 of a volume of a cylinder with the same dimensions. Hey, did you know that the cone and the sphere fill the cylinder perfectly?
A tried and true method, the balance allows students to see the relations between numbers. You can use this to teach addition, subtraction, multiplication, division, base-10 counting, and more! You can even use this for introductory decimals, and this can help students understand the "distance" between numbers; for example, that taking 2 away from both sides doesn't change anything.
A step up from the previous manipulative, the four-pan balance has four pans, allowing students to solve single variable equations visually! These equations are often a big hold up for students who are having trouble with algebra, but "seeing" the Zero Property up close lets them really understand what's going on.
Shameless self-promotion ;)
Overall, the key takeaway here is to have something that students can interact with themselves. Touching something can bring away the abstractness of math, and students will have a deeper connection to what they are being taught.
Have any other ideas? Let me know at @gamifyingedu or on the Contact Me page!