Here you will see students as young as 4 and 5 years old doing algebra and "advanced" math, without ever knowing it's supposed to be hard. You are invited to learn how to use this method...
Basically what you see here is a math rich environment in action.
Playin is learning. Play with the stuff they like to play with... In this case playdoh and manipulatives, or base 10 bucks.
This screen cast got over 150 hits in a day, I'm pretty sure people were amazed by their ability to multiply more than by the playdoh but you never know with the Internet.
If you like it, share it.
Basically we played and had fun first and learned math concepts second. this play session lasted about two hours you'll the complete lack of symbols interplay we just talked about the concepts, the symbols will come later.
Here you see us playing with addends and subtraction. A lot is going on here... You see Hero Zero wearing his five disguise at the top, first it was a 10 and we made addends for 10 below it and the four-year-old showed me ten take away one in no uncertain terms.
Here we see an addend for seven and a couple cutouts.
Here you can see how the boys lined up and press them in.
And here we are getting ready for addends bigger than 10.
Talk about making impressions, here we see 6+4 really is the same as 7+3, and they're both the same as 10.
So what if they don't line up perfectly? They still "get it" and that this age the emphasis is on fun not on being didactic with clay. You can't play "wrong."
Cutouts for the one I don't talk about talk much about the video. And you don't see it much here either but we did all kinds of map of the cutouts, mostly adding and subtracting and smooshing together.
Then we started playing with fraction, under my direction... It's play but it's directed play. "Why don't we..?" "What if we..?" "What would happen if..?""Can you make a square and cut it up?"
Looks like an American cheese single. And right there I got about 100 ideas for lessons with a pack of cheese singles... Eat your math lessons, that will get them subtracting--assuming you like eating processed American mucus forming, chemically enhanced for optimum flavored cheese-like substances. I wonder if I could get Kraft as a sponsor. Anyway I can see lots of lessons that could be done with cheese singles.
Here we talked about one whole, then because it was yellow we cut it into a 4 x 4 square using the four block as a guide. First would cut it in half then we cut each half into it quarters and the make up the other way and we talked about equivalent fractions 1/4 equals 2/8, and they could see it. And then we got those in the house and you get what we see here:
This gives a good chance to practice coming by fours and to see 4/16 is the same as one quarter. You can also see a relationship of 1 to 3, 1:3, but we did not talk about it at all I mention it here because it's so blatantly obvious we could show it equivalent fractions the other way keep it simple and fun and don't try to stuff too much in there at a time. Older kids and adults might be ready to make the switches between the two but with the toddlers I made sure it was a coherent lesson with no jumps and perception. It's one square with a strip colored in as opposed into rectangles, one over the other to show ratio and equivalent fractions of one third that way. This one square teaches a lot of math to toddlers as it is. his could just as easily been a lesson about right angles and squares and square root.
The older boy wants in on the action since you did such a good job accounting by fours and sixes.
But the younger boy would like to know that he did all the "work."
Bigger is funner. Here we put together a 6 x 6 square and talked about fractions as we did it can you see that fractions and multiplication go hand-in-hand and that you can't teach one without the other than that you can teach multiplication with fractions. Why not teach fractions while you teach multiplication or why not teach multiplication with fractions?
Since it was all cut up already or use a six to show one out of six or 1/6 and that was the same as 6/36 or six out of 36. Again they didn't see any symbols and we didn't do any writing during the lesson. But when they do see them, the symbols will make sense.
Just for fun, here's one of 36. 1/36.
And since they had some purple Playdoh, they made six out of 36 themselves. Again I see you ratio listing here 5 to 1 or 1:5, but we didn't do it. You could if your students are a little older. Or as a separate lesson.
Them we made it smaller and talk about fives and fifths each blue-black is one of five or 1/5th. Of course conking my fives is easy and fun.
Which is the same as 2/10ths, and here you see them using their imagination to break it into tenths.
It was fun rolling out the skinny lines and making stuff with them. I missed taking pictures of them playing with the yellow flat and showing other fractions and fraction concepts. But you get the idea.
And after we played long enough with these concepts we just crunched it all together and started over again.
"What should we do next?" the four-year-old asked.
"I dunno." I said, "let's just play and see what happens."
Here is a video for the boys help me tell the story. Caution: they may blow your mind with their ability to count.
At the time I made this post using YouTube and Screencast-o-Matic had outages that were commonplace. I put screencasts here for all my teacher friends who have YouTube blocked at skool but then the screencasts stopped working so now I'm going back and finding screen casts that don't work and am replacing them with YouTube vids or links.
This screen cast is titled "Crewton Ramone play W/Clay Learn Math" on both YouTube and Screencast-o-Matic. People ask me if they can use my material on blogs and web posts, lancer is by all means just a link back to wherever you got it.
Here are couple of old posts where they play and learn math:
Math is fun when you use base ten blocks. Here is a young girl who isn't so very keen on math, she'd rather be playing with her bows and across or her sword but playing with nines and building walls can be fun too. Measuring out how many nines in 27 turn out to be pretty easy.
Building walls of 24 can also be fun...here she sees quite clearly four sixes and six fours are the same also three eights and eight threes....and of course we counted by 3's, 4's, 6's and 8's...and we see counting by eights is the fastest...no need for worksheets. No muss no fuss. We talk about it she hears it she sees it.
She gets a math experience using base ten blocks that evolves more than just paper and pencil and memorization. She can get her hands on it. She is in a math rich environment. Everything we do is math during this lesson and counting. Here we are doing multiplication and division and skip counting. It's visually obvious that the bigger the number is the fewer we need to count to 24...we see a relationship between numbers. Six 4's is the same four 6's...which is pretty cool when you are six. But lets not get stuck doing the same thing over and over again...let's do something else.
Remove the no from the lesson.
So lets make some puzzles. Build me x² + 7x + 12...and then count the sides...basically here the algebra is just along for the ride mostly we are cementing in the addends and doing some multiplication facts. Then we also see x² + 8x + 16 is (x+4)² which is just a shorter way to write (x+4)(x+4)...this also gives us a better understanding of just plain ol' x²...
10 to 15 Minutes per Topic.
Factoring is fun and easy. And it builds confidence because every kids knows algebra is supposed to be hard. So if they can do it with ease they must be just as smart as grandma says they are.
That moment when you see the answer.
Keeping it small and easy makes it fun but there is plenty of math going on here none the less. Dividing by two, squaring a number, addends, multiplication she gets exposed to all of it in one simple problem. If you put the child in a situation where they can not fail and feel like they are making their own discoveries rather than you just telling them math facts and practicing with drills and memorization you will find lower stress and more fun which leads to more learning and happy memories and associations. Or you can bust out your work sheets.., and see how much fun that isn't.
It is a very simple concept, the making of positive associations with the mathematics instead of negatives ones. It should be pretty clear from the pictures that she is having a good time. If you have doubts listen to her excitement in the video.
Once you get the thing factored counting is the easy part. While she is building it fine motor skills are being developed. Also recognition of spatial relationships...
Algebra is fun and easy.
Factoring Polynomials is Child's play.
Here is a picture that's fun because we are seeing quite clearly that 3 sixes and six 3's is the same thing. We made a purple sponge cake with pink frosting and as we frosted the cake we counted by threes. We put them on top of each other and counted the sides...
Multiplication doesn't have to be drudgery, if you do it right the child see's that actually it's a time saver and learning to multiply makes life easier not harder because it allows us to count quickly which allows us to do more math with less pain. Discovering constancy of numbers is fun when you are six.
It's these little repeated exposures using base ten blocks that over time add up to a deeper understanding of how math fits together.
So here is a sample for you, showing us "jumping around" from topic to topic. The topic is math. so we are just playing math...we start off counting 11's...and then fool around with multiplication concepts...then we solved for x playing what's under the cup, told some stories...did some thinking...then we factored some polynomials and we were done...moved along in lively fashion didn't get bored doing the same thing over and over again even though we were doing the same thing over and over again...which is math...
Not just Theory. Theory in Practice. I have HUNDREDS of FREE videos that show you how.
I don't want to tutor your kids I want you to tutor your kids. Play math and have fun. Spend quality time with your kids and learn together. This method will transform your math time into something your kids look forward to instead of dread. But don't take my word for it, here are testimonials, you can see many others have found that this is a better way to go. All you have to do is dive in. Christmas is coming and it will be a great time to get some blocks and GET STARTED. Get a set of blocks and a password and you are good to go.
Playing math often means learning math without really knowing you are learning math....when you are four or five. In this case we learned a lot of math, addends specifically, without worksheets without pencil, pens or paper. We just built a house for a duck and as an added bonus got a few more repetitions for these addends added to the long term memory. It takes many repetitions starting at an early age to get math facts (or any facts) into "the long term memory" and available for instant recall, but as you will see it's easier when they are younger and here is WHY.
Where most of you err particularly with multiplication is you try to cram the number of repetitions into a short time frame usually well after the brain is out of the "download mode". This can be effective but it has been my experience that this can also cause a host of problems particularly when it comes to math anxiety and attitude toward math in general. Instead of a fun and exciting language that underlies ALL of science and explains the universe around them via the physics and chemistry and biology it becomes drudgery and something that requires a lot of mental effort to memorize a host of math facts that don't seem to go together.
Using base Ten Blocks to Make A House For A Duck, You can also see a math bag in the background because we played what's under the cup too. We had fun while learning basic addends less than 10.
What you want to do is make positive associations with mathematics and avoid having to pound math facts into your children. This is what happens when you wait until they are past the age of 7 or 8 to teach them math. The very basics, 45 addends and 400 multiplication facts are learned easily before the age of 7; it is much harder after this age because the brain state has changed.
If you want to be an effective teacher you need to understand a little about how the brain works and how to put information in it. I will have more on this later but the basic is this:
When the child is naturally in "theta", they are basically in download mode. Learning anything is easier for ANYone while in theta, there is an entire industry dedicated to selling audio technology that puts the brain into these states for "super learning"...being happy and unstressed adds to this ability to learn easily. I cover this in Parent/Teacher training.
Removing "the NO" from the lesson is important because "NO" is a charged word that has all manner of negative associations attached to it even by the age of 5. Just hearing "no" is enough to make some children mad or sad or angry which activates the reptilian brain (cerebellum) which shuts down the prefrontal cortex which activates the fight or flight response where blood is pushed into the limbs NOT into the part of the brain that facilitates learning or higher order thinking.
By age six or seven "downloading" is no longer the default setting as it were and they begin spending more time in alpha and then by age 12 beta (and with training Gamma, a state not shown here and a state associated with "miracles" and the "paranormal" and other psychic abilities not easily explainable by modern science but which can be achieved with "practice."). Logically the best time to teach a large amount of factual information is in the early years when it is easy down loaded or assimilated. The names of states and their capitols, Presidents, the ABC's and of course math facts like 7x8 and 17 squared are easily learned and stored in the subconscious for easy recall at earlier ages. And if you make a game of it and make it fun, the learning is easier still. Play math. Not worksheets.
At Crewton Ramone's House of Math as with Mortensen and Montessori, we emphasize whole brain activities...using both hands for example is VERY important, but can you also see that involving the other sensory cortexes would make learning easier? This is because we use more of the brain. This has been borne out by research at major universities.
This seems obvious to those of us that use manipulatives to teach math but quite a few others are only just now discovering that the more senses you involve the easier learning is.
WHEN you teach is also an important factor...as the child gets older more repetition is required to put information into the sub-conscious mind to be made available for instant recall. In other words the older the child is the harder it will be to memorize multiplication tables...learn a new language or most any subject. This does not; however, mean it can't be done. Obviously, it just means it will be harder, take longer, and often involves a lot of tears and frustration.
Any student properly motivated and who is relaxed and happy can learn math at any age. I have proven this repeatedly for over 25 years with students of all ages including adults. But it is FAR easier if you get them young because by the time they are teens many factors are working against you, brainstate being primary among them but as you know, attitude and motivation may be lacking by the time they reach their teen age years, plus there is the added conditioning of past failures and the thought that math is hard instead of something easy and fun.
Here is theory in action. He sees the math facts. He hears the math facts. He experiences the math facts. He has fun. I should have put the camera on him when he told his parents he was done because you could see the satisfaction and pride in the smile on his face. Such a simple thing building a house for a duck, but terribly important for you to understand that through play we learn math, sans tears or fears and without "robbing them of their childhood", but instead making life far easier for him and others like him down the road.
Once (while) addends are mastered, multiplication which is just repetitive addition also becomes easy and we lessen the chance of losing them to multiplication tables and the stupidity that is the way we currently try to force feed students math at a late age.
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Playing what's under the cup is a hugely powerful game. In the beginning it teaches simple addition...but as we progress it teaches much more. It is an extremely powerful game although it it deceptively simple. Great for pattern identification and basic algebra concepts that will be important throughout their math careers. The basic concepts are not beyond the grasp of any child so long as you introduce them at the child's level.
Here is a short vid showing the introduction of Hero Zero.
We could have gone either way...to NFGBT1 or to Hero Zero. I usually go to Hero Zero first.
Here is a longer video which is just one of many you will find behind the SAMPLE LESSONS password protected page. You get to watch it here for FREE. If you want to watch a lot more videos like this just get a password. The Trig page alone make the password worth it, and six bucks a month is a no brainer. Be reminded you are watching most of a lesson for which the people in paid $50.00.
This video shows sample lesson with a 6 year old girl doing problem solving it is also on the sample lessons page; problem solving like x + 5 = 13 is EASY.
But also x - 7 = 6 which takes a little more thinking.
And story problems like 20% of my little ponies are blue, I have 70 my little ponies how many are blue?
Introduces 21 is x% of 70 and more...for more on Percentages click on the percentages tab (%) at the house of math or click the link (t)here.
In the picture she shows us a problem she made and we got it right that a 5 was in the cup...my son earned 1 buck. I made 49.00. If you would like me to pimp my son out to you so your kid can play math and stay engaged email me...she usually will play for 20 minutes before she starts asking if we can quit. With my son I had to cut the lesson off after an hour...46 minutes of it is on video unedited..so there are long pauses as they figure out the answers...perfect for your kids to play along in real time...
The points of this post are manifold.
One, for those who are thinking they might want to start a tutoring business, DO IT. NO ONE is out there teaching kids math like this...well, I can't really say that anymore because actually quite a few people are now using these techniques to teach math...more than six years ago when I started this nonsense anyway, but you know what I mean. The number of people doing it this way is very, very SMALL. You should have no trouble finding a few parents who understand that giving their child, particularly if the child happens to be female, a real head start and are willing to pay well you for it. Don't undercut yourself.
You can charge decent money because of this, especially if you have been to several Friday trainings and/or have gotten passwords and watched all the videos (and played along with your blocks) on the Parent Teacher Training Page (there are quite a few more Hours about to added there BTW). Ask yourself where else are they going to get this? From me? I can only do so many students at a time and my time for teaching students is just about come to an end...I am interested in training teachers and ultimately training trainers and master trainers to replace myself.
Two, when teaching older students who have massive holes like this child in their math knowledge and basic math facts it is important to preserve the child's self esteem. With younger remedial students playing like this is cool because in kid culture "everybody knows algebra is HARD" so although we are factoring polynomials not so much to learn algebra and distributive theory but more to learn addends and multiplication, the older students don't feel "dumb" because after all we are still doing algebra, and the younger students feel smart because after all we are still doing algebra. Get it? I think you will find this is a recurring theme here on this blog. I use third and fourth power algebra to accomplish this too.
Three, it should become abundantly clear after watching this video that having a calculator in hand wouldn't help him factor an expression like x² + 9x + 18...you have to know your addends, which will make repetitive addition (which some call multiplication) easy and fast and will bring home the concept that numbers are made up of other numbers and that there are many ways to express any number...as the sum of two (or more) numbers as the difference of two numbers as the product of two (or more) numbers or as the quotient of two numbers...
I also have another video with a 7 year old where you can see having played with square numbers and being familiar with addends and ratios and fractions makes trig EASY. But if you can't even recognize a square number when you see one...well...an "F" is much more likely than an "A".
Get a password, get a couple of my books and get going. Trainings on Fridays has become "a thing", cost is 20 bucks for 1 to 4 hours, lately they have been going for 4...if you come for an hour it's $20 if you come for all for hours it's $20, starts at 8am HST...you need gmail vid chat and you need to have the video plug in ALREADY INSTALLED. Email me for more info. I know you think 4 hours is a long time but trust me: time flies when you are having math...ask anybody that has come it won't feel like four hours.
Nice, now change the game and make it so he can only use one three...then the other ones have to be a two and a one or three ones, you might mention right now they are all three plus zero and it's more fun to make combinations...same with the four, only use one four and the others have to be addends for four and so on also nothing says you only have to use 4 on the corners, you can use 5 or 6 or 7 or more pillars...then you can play "jenga" an see if you can take pillars out without knocking the whole thing down...then when does fall down just build it again and see who can take the most pillars out before it falls...Also, make a video of it and maybe you could win something.
Many math concepts can be learned with just this one simple exercise. You can slide a 2 into the level that he built with three's and the concept that 2 is "smaller than" three is easily seen...in math we could say less than...and we could show the symbols on a white board if we wanted to but at this age playing blocks and verbal reinforcement is enough. You are creating a math rich environment, where several of the senses are used to learn math and basic math concepts like more and less and addition and of course subtraction.
The mind absorbs all this then later (days, weeks, months or even years) this experience can be drawn upon to make sense of these symbols: 2 < 3...
You could put on the skip count CD or multiplication rock vids on youtube in the background and sing along as you skip count the threes he has used. PLAY, make teaching addition fun; little kids love to sing and they don't much care if you can carry a tune or not as long as you are making a joyful noise.
Anyhow make a vid of your kids playing math and you might win some cool stuff. UPDATE: this part is over but you should still check out primal math for some cool apps...Zombie Fish Bits is fun....
If you'd like a FREE manual on showing how to teach even very young children a super easy way to subtract click this link to my Supremely Simple Subtraction manual. A new Games and Activities Manual is in the works but it will be a while so meantime use the old one. (Takes a while to load so give it a minute...)
My response:
Excellent...! Now you can talk to him about the 9 in the corner...you could flip the blocks over and count 1 hundred, 6 tens and 9 units...you could talk about square roots and square numbers playfully with the 9 and the 169...and of course the polynomial...x² + 6x + 9...the square root is one side which is x + 3....and of course (x+3)² = (x+3)(x+3) = x² + 6x + 9 and you talk about economy of symbol...that we don't need to write it twice...you could talk about 13² = (13)(13) = 169...you could have him draw a simple picture of it...and talk about these things while he's making his drawing...note I say could you don't have to...you could write some symbols and show him how those symbols describe his creation the same way words can describe a brown cow...and you do it casually, not drilling him not formal instruction just, "hey...look at these green ones there--they make a square and we can count the sides or one side...hey look at that! You made all those blocks into a square can you count the sides? What's the blue one called?"
If you are on the base ten side he should say, "10...and three units." Or something like that. And you could say, "what's the name for one ten and 3 units...?"
If they are on the smooth side and you are in base x, the blue one's name is x and three units...
A ton of math concepts can be taught off of this one rectangle...you don't need to cover all of them at once...just a few at a time. Check this post out, it talks about all the things you can use this simple rectangle (square) for, when it comes to teaching math concepts.
Now build one that's 14 or x+4 squared...and then 15...or maybe he wants to make a different shape with his 6x...and make it (x+2)(x+3) or some other combination for 6, 5+1, 4+2, 3+3...still 6x but now a different number fits in the corner...just play math, bigger is funner.
Here are a few recent vids. The point of them and this post is to illustrate and bring home basic concept number three: we form rectangles to facilitate counting. Rectangles make it easy to count and we can reduce many seemingly complex math problems to basic counting with the help of our friend the rectangle. In the first vid they make percentages EASY by using a rectangle.
What basic concepts do you see in these three vids?
Rudimentary Editing Skills Level 101: ACHIEVED. Mastery w/ percentages is being attained. THEN we can move on to ones that end in 5. Baby steps. Note the block serves as a "post" or memory anchor on what to do. They understand the CONCEPT so now I'm just changing the numbers...and we also get practice with multiplication and division.
Here is their little sister playing and learning addends...she had asked very nicely if I would play what's under the cup with her since her brothers were playing outside with two neighbor boys...and she wasn't being included. So I brought DBoyz in and shot that short vid and then sent them back out and played with her...we played what's under the cup and built tens and then she made a house for little ponies to stay in and then we cleaned up and it was time to go. She made the comment, "Time always goes so fast when I come to your house." Because time flies when your having math.
This next video is also edited. The full version is on Dboyz page at the house of math. All three of these videos can be used for educational and training purposes...lets use them to focus on basic concepts.
You can stop here. The rest of this post is long and aimed at those who are interested in learning more from the perspective of a trainer or an aspiring master trainer. A link to this post will appear on the teacher training page too. Here is the whole converstion on Crewton Ramone's FaceBook page.
I highly recommend you click that link and read the thread, what follows are some (slightly edited) excerpts lifted from that thread starting with this: First things first, I don't write or make videos for Unicorns. Celeste is a unicorn. Female engineer. Statistically speaking they don't even exist. There are few Unicorns who have become interested in the house of math...Staci was another one: black female electrical engineer...Irene is another one I'm confident she knows more math and certainly more chemistry than I do...these are the exception not the rule...and it must turn tables and become the rule not the exception. I have been paid large sums by men who fully understood the mathematics well enough to apply it to physics or engineering or economics but could not explain why you invert and multiply to a seven year old's satisfaction...much less why the x's don't change when you add them...what has to happen is people who actually understand math have to be given the tools to be able to explain it to others.
Most parents statistically speaking failed algebra. Many teachers got a teaching degree because it "didn't involve taking a lot of math..." Teachers currently still think there's a trick to it:
Question: if you understand math and math concepts will you still need special techniques? Answer: once you understand the concepts all they can do is change the numbers.
I'm starting to break lessons into small pieces for a couple of reasons, First: because we know the attention span of the average viewer is extremely limited...two minutes is optimal video length...what a laff: people want to learn math in two minutes....if not screw it...girls can become strippers and guys can always cut grass--wealthy people rarely do their own yard work. Second: because too many parents and teachers can't wrap their head around switching from topic to topic because they can't see how the lessons relate...we are learning about Pascal and summation but all we are doing basically speaking is addends and multiplication....the percentages practice was practice on multiplication and division. Once we get the CONCEPT down all we can do is change the numbers and then it just becomes practice with doing whichever of the basic operations are required because AGAIN computation is how we DO math but it's not THE mathematics.
You can do all the math with just addition and subtraction but it takes longer...further if you cannot DO computations quickly and easily mathematics is HARD because you have to contend with two things layered on top of each other: trying to remember the rules and process for the problem at hand and trying to remember the rules and process for the computation required itself. This overloads the system and results in failure, anxiety and incorrect answers which leads to negative feed back, which leads to more fear and anxiety, which leads to an aversion for the mathematics which leads to more of the preceding and a feed back loop is created that includes actual pain, mental anguish and damaged self esteem which leads to hating math, which leads to the dark side.
The concepts are very basic here. We form rectangles to facilitate counting. There are more as an exercise for those who are thinking about tutoring or maybe one day training others...what other basic concepts are covered in these three short videos?...you might also go over them in your mind even if you are "just a mom" teaching your own kids. You can put comments below if you like...but there are several more at work here. I mention some of them in that FB thread.
Now, the reason that this method is so effective is because if all math is counting and we can reduce any problem to simple COUNTING, then anybody can do math that can count. The trick, if there is one, is knowing how to reduce a problem to simple counting, which as it turns out is the genius of it all and which is why we say if you can count to nine form a rectangle and tell if something is the same or different or not we can teach you math...I add that they must already be able to speak a language generally it's English and if you can speak that language math should be EASY. Anyhow what basic concepts do you see in these vids?
Mathematics is the study of relationships and we describe these relationships with symbols. pie is not just 3.14....it is the relationship between the circumference and the diameter, it's not just sin30° = .5, sin is the relationship between the side opposite 30° and the hypotenuse...addition and subtraction are inverse functions but basically the same thing in order to do one you need to do the other basically you are combining, the same with multiplication and division, things look scattered and random (even though the tab bar on my website basically goes in the order you will see it presented in the public schools) because you don't get how the language goes together or the gestalt of it...you can't study one part without studying some other part...using this method. You can certainly compartmentalize the mathematics and they do for an example think back to or visit your nearest public education institution.
"Just show me how to do the problems" is the problem. My son is currently suffering under this line of thinking where he gets to do timed worksheets for multiplication facts. He doesn't do as well as other kids on these because he doesn't write quickly, therefore he must not be very good at math. They are majoring in minors.
When asked to be able to use and apply this knowledge to something like 28 is what percent of 70 or how many 4's in 72 they either can't do it at all or take forever and certainly can't do it in their heads...but they can whip out a 50 problem worksheet of single digit multiplication facts in under 3 minutes...but then ask them to figure out how much change they get if they buy 7 pounds of rice at 2.78 per pound and pay with a $20...no pencil, no paper...and they can't do it, because they can't organize and keep track of their thoughts.
Concept #3: Rectangles facilitate counting.
A = LW This gets totally LOST if you just memorize (n² + n)/2 which is what most of my students have done if they even know about this at all....they use summations on SATs and other standardized test to "separate the men from the boys..."
In the example for 14 we see various addends. a couple for 14 and also for 4, we also see the number 105 is 70 + 35 again numbers are made up of other numbers and that part will come in handy when doing division but also trainers should see the distributive theory of multiplication as we distribute the 7 across the 10 + 5 so the 70 = (7)(10) and the 35 = (7)(5)...students some how are mystified by this when they get to algebra even though they have been doing it all along, where the most common grade is "F" and when asked to multiply 7(x + 5) often have a very hard time even though it's easier than doing it in base ten because all we get is 7x + 35...whereas in base ten we have to combine like terms (the tens) which is a basic Concept #2: we only count things that are the same. If you were going to be a trainer and were asked to delineate the concepts on these vids and you came back with counting, forming rectangles and "The blocks help you see the patterns." I would FAIL you...because you failed to see the the basic concepts...
If you were being tested under Jerry or another Master Trainer named Brian you would fail and more than likely receive a serious dressing down to go along with it...I remember one woman in particular in tears after she got done being schooled...although Brian would try to be a little nicer about it. I'm going to (try and) develop some Master Trainers...if I get 7 in the next 9 years I will have done phenomenally. When you are done you should be able to take any textbook and then take just about any problem and either show how to do it with the blocks or step back and show the basic concepts needed to do the problem because by now it should be clear that you don't want to model every problem with the blocks nor should you need to...and you don't need to hit the student over the head with them or take away the aha moments by just telling them....let them explore and discover for themselves. You nudge them in the right direction or point them down the path that will lead to the discovery you want them to make....but they walk down there alone. You are not far off, just in case and to catch them if the fall---control of error is important but failing your way to success is also important...not wrong just getting more information...that attitude leads to discovery and opens up the wonder. Control of error does not mean don't let them make mistakes ever or avoid making mistakes: it means learn from the mistakes, take that information and then use it to get it right.
Fear of failure or frustration of getting it wrong or worse being put in a position where they can not succeed shuts down learning decreases wonder and joy to zero or less--usually less, and makes the math what it is today. I see math memes like this:
And like this:
If you understood what you were doing how could you NOT know? Yet this is a completely common response from many of my students who are not used to getting A's. Try it some time ask your child or students how they did on their math test...
I would encourage to click this link and read the thread. The point is not to "talk down" or to deride other people who don't quite get it...entire companies have been built that purport to use base ten blocks to help you see the math...but some methodologies with base ten blocks are better than others. I can swing a hammer and work a saw...and my trig is strong, but I know quite a few carpenters who using the exact same tools, would get whatever it was built faster, better and prettier than I would. My aim is to make you, if you so choose into a person who can teach math using base ten blocks and other math manipulatives, faster better and prettier than the majority of your peers. Unfortunately, I can't do it in an hour or even in 50 five minute videos...but I do have some parent teacher training more than 7 hours worth at VERY reasonable rates which many have found to be very, very useful.
If you would like to enter into a dialogue regarding concept number three, or make a comment about the teacher training (many have taken it now) please do so in one or the other of the comment boxes or ask questions about concept number three not 4 not 5 but 3 that would be concept number three the one about rectangles and how they facilitate counting.
I do not recommend my Tumblr because there's sex, and "naughty" pics, and swearing and all manner of content for grown ups, along with controversial topics...but the other ones are fairly tame, although still not tame enough for the Christian Homeschool crowd.
Part of what makes this method so effective is the idea of compound lessons. Why learn "one thing at a time"? One thing at a time is great when introducing concepts...but once concepts are understood it's more fun to practice more than one thing at a time, which is why I emphasize compound lessons. And as you can see using base ten blocks to do it makes math fun.
A six year old factors x² + 11x + 30 all by himself and celebrates with his magical math microphone.
People ask me what they should get. Get a combo kit. A set of Multi-Tens and a password or both passwords...and you are good to go. Seriously. That's it.
In this hour we practiced mostly addition and multiplication but we did activities that were fun to practice them rather than doing worksheets over and over again. Using worksheets to memorize multiplication facts becomes counter productive at some point.
Just building addends over and over again would make math anything but fun. So even with this activity which by itself many kids find fun, we spice it up by racing your brother or mom or who ever happens to be around. Then we build some pyramids. All kids can benefit from building pyramids, the benefits are manifold. You get fine motor skills, addends which naturally include addition and subtraction skills, and the all important repetition which puts the facts in the memory for instant recall later...
Here the older boy is feeding his mother and brother combinations for their pyramids. and supervises the construction. They are building 11's, later we could build and find patterns with 11's.
Pretty soon he gets in on the base ten block building action. And his little brother hands over some combinations...sometimes we build all 45 addends at one sitting but today we just did all the ones from 8 to 10 and then 11's and 12's. How many math facts is that?
Children are always quite pleased when they complete one of these...or several of these. Sometimes we build towns or cities or temple complexes or moon bases (or whatever is in their imagination), made entirely out of pyramids built using base ten blocks, but today we played algebra which will give us addends and multiplication too. Started out with problems like x² + 7x +12 and worked our way up. We started there because he already has some experience factoring polynomials and I wanted to start easy...then we went on to x² + 8x + 12, and x² + 8x + 15, x² + 8x +16...Went thru several with 9x then on to 10x, those polynomials have lots of combinations.
x² + 9x + 8 you get addends for 9, 1 and 8, and 1 x 8 = 8.
x² + 9x + 14 you get addends for 9, 2 and 7, and 2 x 7 = 14.
x² + 9x + 18 you get addends for 9, 3 and 6, and 3 x 6 = 18.
x² + 9x + 20 you get addends for 9, 4 and 5, and 4 x 5 = 20.
You don't have to do every single one in order...mix it up and have fun. I did a couple with 10x like x² + 10x + 16 and then x² + 10x + 24...didn't do x² + 10x + 25...which is a good place to start if you want to go down the square numbers and square roots path which are made plain and EASY with base ten blocks. I had a mom tell me they completed the square with 24 once. I said go back and take a look at that...24 will complete a rectangle but certainly doesn't complete a square. 25 would complete a square...and completing the square is quite a useful skill to have later on when you are graphing polynomials and want to easily convert to vertex form.
Playing with squares and radicals are also fun math activities where you can compound the lessons and learn algebra, addends, multiplication, square roots and more. You decide the emphasis depending on the student. For some older students who are supposedly past addition and multiplication because of their age you can use these techniques for remedial math without them even knowing it.
Instead I gave him x² + 10x + 24 which he got pretty easily but then I gave him x² + 11x + 24, which is a challenge at this age when they are still learning their times tables. You can see he figured out he needed 8 and 3 to make it work.
Then I gave him a challenge. x² + 11x + 30...the dilemma after he figured out it was going to be (x+5)(x+6) was finding enough fives or sixes....they were being used in the pyramid...and he found that he could count by 5's but not by sixes as easily...remember this boy is 6 years old, he hasn't mastered his 6 times tables yet but he will....and there won't be tears or frustration. It will be part of fun memories spent playing with his mom instead of being angry at his teacher because of yet another worksheet.
After a little fooling around he decided to make 30 with 6's...which meant he was going to have to bust up his pyramid.
Completing the corner was important to him, later you will find students especially older students who know their times tables skip putting the units in because they already know 5x6...and that 5x6 is the same as 6x5...when they are younger this is still a fun discovery.
Challenge completed! It's time for some celebration which is where we came in. Also some high fives with mom never hurt nothin'.
This lesson took place way back in the beginning of October. Just got around to finishing it here. In this hour we basically practiced math facts in a very disorderly fashion...we didn't just do a bunch of addition or subtraction for an hour. We did both AND we did multiplication and some division AND we factored polynomials but the algebra was just along for the ride...as you can see it wasn't scary or hard and I used it to teach concepts and facts...most teen-agers I know FAIL algebra right at this point because they don't understand the distributive theory and never saw a polynomial until they were 12 or 13 years old...
Positive memories and positive associations with math are so important i don't have words. Wouldn't you rather have your child's or student's early childhood math experience be like this instead of rife with tears and frustration? They won't all be diamonds but more often than not you will find YOU the parent or teacher can make math fun and teach a whole lot of math all at once in compound lessons like this.
Want to get the materials you see in this blog post? GO HERE.
