The debate surrounding the memorization of the times table has been around for decades. Do a quick Google search and you will find arguments on both sides. However, most of the arguments against the memorization of the times table are weak and often by people who are not in STEM careers or who had trouble through school with math. After two decades in education as a math teacher and tutor, here are my first-hand observations.
To memorize the times table requires a student to memorize their zeroes through 12. That is 13x13, or 169 math facts. This is quite a feat for a 3rd grader. However, children that young are capable of learning and memorizing a lot more. Some studies show that students around that age has the capacity to learn 7 languages simultaneously. If you're not convinced, ask any child who plays video games all the details about everything within the entire game.
Suppose a student does not want to memorize the times table. They'll end up having to memorize the facts by counting in particular increments or drawing grids. Imagine having to do this every time the student needs to calculate something. While these methods shows the why behind multiplication, they should not be the primary method for doing calculations every time. Now imagine doing this on a timed exam. Having students count in increments have actually led to the memorization of the sequences of numbers. In fact, ask a student to multiply 4 and 17, they can't count past the twelth number in the sequence of 4s!
What about modern technology, such as calculators? For harder problems, yes. But for simple calculations, no. Imagine having to pull out your smartphone every time you read something. Not memorizing basic calculations is like not memorizing basic words. Oh wait...if you don't know the meaning to basic words, how would you understand the definitions to those basic words when you look them up? I don't believe students should have to memorize the quadratic formula or the Pythagorean Theorem for life, but they should master and be fluent with basic operations, especially with single-digit numbers.
Memory is the basis of all intelligence. To learn and not remember is no better than not having to go through the process at all. Every test is a test of memory. And memory can make and break personal and professional relationships. Therefore, training your ability to remember things is crucial to life. We often think that the ability to remember things is fixed (the fixed mindset). The ability to remember is a skill like any other--it can be improved upon (the growth mindset). We can all hone that skill and become better with memory. From a biological standpoint, when we build memory, we are actually building physical connections within our brain. When we repetitively study something, we biologically reinforce those connections with proteins, similar to using "duct tape". Our brain physically changes when we study and practice repeatedly.
Repetition is most of the time not fun. Ask any Olympic gold medalist how many times have they practiced the exact same thing over and over again and their response probably will be, "I lost count." To do well means to hone a skill through practice. Rote memorization (or drilling) has gotten a bad reputation in school because students (and parents) think that it is boring, dull, and lead students to not enjoy school. If memory is built on the basis that a person must repeatedly do something, then the question is "How can we make the repetitive process more enjoyable?" This is the question that most anti-memorizing-times-table people should be asking. There are games (card games and video games), activities, and even songs to make the learning more enjoyable.
Students and parents who refuse to memorize the times table will end up with years of frustration. Let's take the hardest multiplication fact that most students miss: 8 x 7 = 56. The following is a list of just some topics they will struggle with if they don't know this fact. Remember that math builds on itself; the struggle gets worse and worse along the way if fluency with basic operations is not taken care of.
If you're not convinced that memorizing the times table should be required of every student, then there's not much else I can say. Math is something that every student will take at every step of the way to 12th grade. Parents who don't reinforce this idea will find themselves paying a lot of money for tutors and other resources, when the main culprit comes back to poor foundations in basic operations. In fact, in years of tutoring, once a student has memorized the times table, they all started to do better in math. As a teacher of over 20 years with working with students, I am convinced that memorizing the times table is the first step towards success in math. Mastering the foundational math concepts such as adding, subtracting, multiplying, and dividing should be required of all people.
If individual numbers are like individual notes to a musician, then mastering basic math (add, subtract, multiply, and divide) is like mastering chords. A musician who does not memorize chords can only play songs one note a time, limiting their ability. However, if they memorize the hand positions of different chords, they open up an entire world of possibilities.