Numbers, Counting and Mathematical High Weirdness
MANY people find mathematics incredibly difficult and have negative feelings towards it. These two things feed into each other! It's tragic, because mathematics is not only a Great Thing, but is also well within the mental powers of MOST PEOPLE. The reasons for this calamity are many...but a BIG ONE might be that people are being taught a fatally mutated version of mathematics, one that consists of a million or so disconnected MATH FACTS that came out of nowhere. This 'Maffs' treats the most unintuitive things as EASY, and then proceeds to turn the simple things into a maze of AWFUL COMPLEXITY. 'Basic' ideas like number and concept are examples of the former!
Counting, shorthand
While counting is an important skill in dealing with mathematics, it's not necessarily an EASY or NATURAL thing to be doing. Without instructions, humans can lean upon an included 'number sense' to 'see' quantities up to 4 as distinct from one another. To reliably quantify larger amounts of things, more sophisticated techniques are needed. Physical and mental technology for keeping track of numbers greater than four has been evolving for thousands of years. Some of the older technology is evident in children's introductions to mathematics; props are used to provide CONCRETE methods of dealing with numbers. However, these give way suddenly to modern methods, with all of their ABSTRACTIONS and SYMBOLS.
It's assumed that this transition from concrete to abstract is reasonably straightforward. I don't believe this is the case. There are many subtleties buried within even this seemingly basic mathematics. For instance, you might think COUNTING is as simple as it gets. However, the essence of this activity is the assignment of shorthand symbols to incrementally larger abstract amounts. ADDITION, another 'basic idea', is in fact a sophisticated method of rearranging these shorthand symbols to avoid the more laborious (and concrete) task of COUNTING. MULTIPLICATION is an advanced technique for compressing repeated additions. The abstraction begins to pile up rapidly; without a guide who is wise to the intricacies and connections between these things, a student is likely to become overwhelmed by them. Remember that even these 'elementary' ideas are human inventions, which took CENTURIES to develop. The foundations of mathematics are STILL under constant debate, partly because of their weird abstraction!
The symbols
One thing that might make it harder for students to grasp mathematics are the number symbols themselves. There are ten different glyphs that represent quantity (in base-10) - their order must be remembered, along with their names! Additionally, various ways of combining the symbols to produce other symbols must be correctly remembered...using addition or multiplication, for instance. Unfortunately, the glyphs don't VISUALLY represent the quantities they stand for...
Imagine if our ten symbols actually looked something like the amounts they represent. The concrete examples would then be 'built into' abstract number problems. Suddenly, the many ways of producing a given number through addition would be apparent within the number symbol itself! For instance, the fact that '7' is just shorthand for '1 + 1 + 1 + 1 + 1 + 1 + 1' would be OBVIOUS. Instead of having to memorise a whole bunch of symbols and their relationships, a student could easily build whatever shorthand number symbol they needed at the time. This might have very interesting consequences!
For starters, students might be able gain a more intuitive grasp of numbers, and gain confidence in their abilities, as they can always fall back on counting to confirm their results. Secondly, the play-dough characteristics of numbers (where you can express them in a multitude of ways, depending on what's interesting or convenient) would be READILY APPARENT. People might develop a very creative mindset towards mathematics! Finally, and importantly, it may reduce the anxiety associated with trying to succeed in what is actually a very unintuitive activity! If people aren't hit with an overwhelmingly negative experience when they meet mathematics, maybe they'll stick around and come to enjoy it!
Developing a sane system of 'direct quantity representation' would only be a minor challenge. In fact, I've made up some of my own...visible on the right hand side of this image. There's no reason why anybody can't make up their own system of number symbols, they probably just need to be legible when drawn quickly.
Practicality dictates that the 'classic' number symbols are introduced at some point. Hopefully, this can be done when the student has overcome the mental challenge of abstracting and working with quantities far greater than can naturally be perceived.