This whole "lesson" lasted about 45 minutes, which is long "lesson" for kids this age, but as long as their attention was engaged and it was easy and fun we kept going. So, really it was just play time although the older student knew he was "learning stuff". Obviously, a little more went on here than I was able to capture in pictures. With children this age the camera is a HUGE distraction so I only managed to snap 6...and then took a couple of the book so you could see that too. We spent much more time with the blocks than the books BTW.
Here we are b uilding 10's and 9's...the older boy is 5 the younger 3 almost 4. The older boy can answer without looking, "what does two need to be a ten?"
2 + x = 10,
In fact he knows his tens addends so well he doesn't have to think about it. Mastery.
We are doing this for the benefit of the younger student. Incorporate algebra immediately. The concept of "x" is easy if they've been playing with it since they were little.
Doing nines is also easy but takes a little thinking. They are almost mastered. Why people don't teach little kids algebra from the get go is beyond me. Here one child is counting units the younger child is counting x. (In the mathematics "x" IS the plural.)
The younger student is "building stairs" with x's, basically he is playing with them and using his imagination. X is simple concept. We continue on with the basic concept that the numbers have TWO parts, the "how many" part and the "what kind" part. When the tens are on the smooth side we call them x. (At this age he still calls them "exes.") He is counting out x.
x + x = 2x.
This is no big deal.
2x + x = 3x...I have spent time in high school and college classes explaining where the "two" came from in x + x, and how we got 3 when adding 2x + x because the one is invisible and x + x being 2x doesn't make any sense. Text books say "the one is understood" ...except it isn't. These children will never have this problem. Further we talk about x2 + x2 making 2x2, didn't get a picture of it but it's them playing with the big red squares. I have seen numerous college aged students add x2 + x2 and get x4!!!!! I'm the only one right? The system in place for mathematics instruction in the USA is broken. Anyhow, before I go off on a diatribe, these little kids can see that two red squares have nothing to do with 4...if the red squares are called x2 (x-squared) then two of them are simply 2x2. Painfully obvious. In the Montessori method they say "visually obvious". Little kids doing algebra? YES, THEY CAN.
When we build eights he has to get out blocks to make sure... Just because he is familiar with the addends for 10 and 9 does NOT mean he is automatically able to do 8's...many teachers fail to recognize this. They need to learn addends just like they have to practice ABC's...there are only 25 addends from 1 to 10 and 45 single digit addends in all ending with 9+9.
Note I told him to build 8's I didn't tell him to start in any certain order, like 1+7, 2+6, etc he's just building...note we aren't using any symbols or drawing. Just because the child can write doesn't mean they can't learn "complex" math concepts. Later we can go back and use symbols...and know what they MEAN.
Just fooling around, they got out a ruler because I said, "measure it to make sure"...I meant measure with the blocks and they knew it, but since we are just having fun we played with the ruler too. Measurement is an important area of study in math.
Then of course we measured with a pen. Have fun with it. Math time doesn't have to be serious at all...
The 3 year old can't make numbers but he can point to the right answer. This page is fun because it has a lot to count...6 "problems". Again, the numbers have the "how many" part and the "what kind" part. Here we are learning units or ones, tens and hundreds. Just matching the picture and the symbols...900 is square and we can count it by three's...
The 3 year old is given pause when counting tens: "Oh man this is gonna take forever" he starts counting the tens one unit at a time then I show him we can just count 10, 20 30...this is not the first time he has seen this book and it will be new every time for about another 18 times...repetition is the mother of skill. At this age repeating it over and over again is natural and fun. Which is why this is a great time to teach multiplication songs...it will be fun and easy and avoid tedium later when the brain has moved onto a stage of development where repetition is not as large a part of the child's learning modality.
This page which came after the other page shown, (note the page numbers) is not as much fun because there isn't as much to count...or so says the 3 year old. I'd say he's the fun expert.
Here is more on counting and combining like terms.
Much more at Crewton Ramone's House Of Math.
"Almost all creativity involves purposeful play." ~Abraham Maslow American psychologist 1908-1970
"Whoever wants to understand much must play much." ~Gottfried Benn German physician 1886-1956
"Play gives children a chance to practice what they are learning." ~Fred Rogers American television personality 1928-2003
"People tend to forget that play is serious." ~David Hockney Contemporary British painter
"Do not…keep children to their studies by compulsion but by play." ~Plato Greek philosopher 427-347 BCE
"Necessity may be the mother of invention, but play is certainly the father." ~Roger von Oech Contemporary American creativity guru
More math quotes and more education quotes at my site.