It would be trivial to do so, though it might not change very much. Proof Theory is the subdiscipline of mathematics devoted to exploring how one can prove statements using entirely syntactic manipulations such as "Add two to both sides of the equation." In Proof Theory, an expression is typically presented in a Formal Language as a finite string of "symbols" taken from a finite set of available symbols known as the "alphabet" for that language.
You can trivially change the depiction of the symbols without changing their meaning. If you want a bunch of insects multiplying exponentially as a symbol, you can do that. The set of inference rules simply has to change to match the new symbols. We can use Category Theory to demonstrate that there is a morphism that preserves the meaning buried in the syntax.
As for approaches which are less trivial, it will be difficult to do so if you phrase the question "How can mathematical equations be expressed without equations." All of mathematics is a very large domain, and we could literally spend the next 500 years building up such a system and proving that it covers all of mathematics. Meanwhile, mathematics would be evolving and growing, so we would probably never catch up! You might need to reduce the complexity of the task by picking a subset of mathematics that you need. For example, much of early mathematics was proven not with symbols, but with geometry. The Greeks developed a set of proofs known as Compass and Straightedge Proofs which made proofs via purely geometric means. They even identified classes of proofs that could not be made with a compass and straight edge, but which appeared to be true. Many of these are now captured in the study of Oragami, which permits manipulation of the paper in ways that were illegal in compass and straight edge proofs.
In all, you can build the concept of mathematical equations up many ways. Ours happens to derive heavily from the concept of "implication" in First Order Logic, which is a fascinating operator. It is the only operator in First Order Logic that is not defined by First Order Logic but rather left up to the reader of the proof to decide what it's exact definition is. This operator could be thought of as the prototype for all syntactic manipulation approaches in all of proof theory which tie us to symbology.
Beyond that, you really will have to use your imagination, on a case by case basis. For instance, there are some who believe that all will be revealed through some deity. For those people, symbolic proofs may be meaningless, but some vision that we cannot understand may be proof of a mathematical statement for them! Philosophy leaves the door for such thinking wide open.
And, if I may note, I proudly did not use a single mathematical symbol in that entire answer, though I did use a few words which refer to mathematical symbols.