This was an interesting read. I’m a retired math teacher, 15 years in the public education system. I came into teaching as a mid life career change from engineering. I was tossed into middle school math when the industrial electronics program I was teaching was killed off during a summer break and they had to offer me something by contract.
The first thing i realized was that many, if not most of the students wanted the easy way out. The second was the looping style of instruction with the same material being offered at a slightly higher level the following year was boring the daylights out of the students. Yep, the system worked so well that the average student was 2.5 years behind grade level.
The approach with Common Core came just after I retired, and I will vouch for my own experience that teaching number sense was a hard sell. What did work for me was to step out of the bounds of the canned curriculum and introduce new concepts to pique their interest. One of the state test points was place value which would seem fairly straightforward but their brains would turn to mush when it was brought up as they “had already seen it for the last three years”. It still didn’t mean they really understood it.
The solution fell in place when I realized the four light switches at the door each controlled a long bank of lights running up and down the classroom. Ah-ha! I’ll teach them to count in binary up to four place values ;-) Amazingly enough once the concept was explained in simple terms such as a bucket of numbers filling up and resetting to zero whenever it overflowed and sending another count to the next place holder. To help things along I pretended I was a space alien with two pseudopods and no fingers. The look on their faces was priceless when they realized you could only have two counts and not ten. This was a new concept for them, and therefore interesting. I had special ed students participating and counting correctly with random numbers as I would turn the classroom lights on and off in random patterns. Keep in mind this is with most of my 8th grade students functioning at the end of year 5 and early year 6 grade level. As an aside I also had 8th grade algebra students and we covered some of the same material except I had them adding and subtracting in Base 2, 8, and hexadecimal.
To up the ante after they were doing well at this, I wrote a few short stories where someone traveled to Mickus Mousus Land where all the Toon characters had four fingers on each hand. Look at the comics sometimes and notice how many digits they have on their hands. So we moved into counting in base 8. There was no doubt in their minds how place value worked after that, and jumping into exponents and scientific notation clinched the deal to reinforce everything. I considered it a major success that my ESL, Special Ed, and my challenged students were picking this up, and having fun learning something that was genuinely new to them. The memories of those kids shouting out answers quickly to be the first is one on my fondest memories, right along with the looks of amazement from some other teachers in the room.
Unfortunately, my next principal had been a math teacher herself and squashed those efforts to follow a very rigid canned curriculum that was dry as tomb dust and just about as exciting. I retired early about three years later.
While I came into education by a roundabout path and certainly non-traditional, I did earn a Master’s and achieve full certification as a math teacher for grades 5–12. What I did recognize quickly was a lack of understanding of fundamentals by many students, mental shutdown about grade 7, and student turn-off by a boring, repetitive, and shallow looping of the same material year after year. Don’t even get me started on applied problem solving as any problem that involved words would make their heads hurt, and that was in their own words. Needless to say, my Master’s project was examining story problems and how analytical thinking to break down problems was not being taught.