I have two pre-school boys, 5 and 4. I noticed my five year old developing a lot of enthusiasm for numbers and I recognized his "number sense" was pretty good. I wanted to take a queue from his enthusiasm and help him develop that better so I got this book on number sense. Even before it arrived, I was reading the preview and was impressed with the Early Number Sense Learning Trajectory. Right away I recognized some things that could help both my boys. After I received the book, I've been reading through it and it's full of ideas that I can use for years to come. The book is helping me understand the learning process my boys are going through, and how to plan and implement routines that will help them develop an advanced sense of number.
The book has useful ideas for pre-school through most of the elementary grades. The cover indicates K-3 but there's some 4th grade examples inside. More importantly, there's routines that will help people develop their number sense from wherever it's at now, no matter their age.
My boys are examples of the early stages. My little one had some sense of magnitude, one-to-one correspondence in counting, and cardinality, but if you showed him four or five things he was still counting them. I started to work with him in subitizing with dot cards. In just a few days, he's gone from subitizing three up to six. We're making this a routine, for him with a single die (from a pair of six-sided dice), dot-pattern cards, and the dominoes whose two halves total to maybe 8. I'm using a ten wand and introducing ways to make a number. We also do choral counting as a family and we're starting to include him on counting around the circle.
My older boy was excited about the dot patterns when I showed the kids the cards I had made. At first they just went up to six, but I told him I had some harder ones with more dots. Right away he tells me, "I know if it has five and five that's ten!" He was developing his sense of unitization.
I had worked with him, introducing number lines, both in abstract, and as a thermometer, ruler, tape measure, weight scale, and the clock. I also showed him number circles on the circular clock, scale, thermometer dial, and a circular day of the week and month calendar I made. He could read any analog clock to the minute with or without numbers printed on it and had a pretty good grasp of modular arithmetic.
For him, I brought out the dominoes that go up to 18 (two 9-dot patterns). I'm also using two dot-pattern cards or both dice at the same time. He's telling me things like, "I know this is 17 because nine and nine are eighteen but this one is missing one dot." He has good mastery of ten, ways to make ten, and factors of ten (he tells me that he knows 80 and 80 is 160 because he knows that 8 and 8 is sixteen). He also counts into the thousands, races through backward counting by one from 100, and skip counts by two and ten. He's developing a better sense of compensation, and I think he will also get better at mental math with numbers other than factors of 2, 5 and 10. I really look forward to having more in-depth discussion with him about things like ways to make a number, and having him explain his thinking in addition to the answer.
In recollecting my own education in math, I distinctly remember a decisive turning point during the 4th grade. In 3rd grade I was loving it and excelling at the level of adding and subtracting fractions. By 4th, math became algorithmic, tedious, and I began to struggle. The only time I really enjoyed math after that was when I discovered Euclidean geometry. What I was missing was the kind of number sense the routines in this book help to develop when they're practiced daily. Because of that, the only strategies I had for problem solving were inefficient, tedious, and algorithmic. If you think about it, a long-division problem is whole bunch of simple subtraction and multiplication problems bundled into an algorithm. If you have strong metal math skills and the ability to pick efficient strategies to solve the component problems, everything will go well. If not, a whole sheet of long division problems is tedium beyond what one can bear. It's the same thing with quadractic equations and polynomials. Unfortunately my own educators focused on explaining the algorithms without recognizing the gaps in my skills and number sense. I believe this happened because they simply didn't know what to do about that anyway. This book has the answers.