Pyramid Playing Produces a Preponderance of Productivity.

Playing with blocks makes math easy and fun for little kids...and given the proper tools and games and activities "just playing" with blocks can teach quite a bit of math. In fact this is how the Mortensen Method was born. One of the plethora of fun games and activities in your arsenal should be building pyramids of blocks. Walls and towers are fun too...entire math towns can be built. These towns can teach addends or multiplication or combinations of these topics and MORE.

Playing and having fun with math is foreign to so many parents they are sure it can't be this easy. Actually it is. You have to direct a lot of the learning but over all you will see a lot of it can be self directed.

Here are a few of vids that focus on just this one activity: building pyramids. Starting with this short one where we made nines:



Here is a slightly longer vid where we made 12's:



And here is a longer sample lesson with a child that has a few cognitive challenges, but you can hear him catching on after a very short time. You won't find this vid because it is UNLISTED. You will also find it on the sample lessons page.

In the past I have said I don't want to hit you over the head with it but due to some comments made by well meaning parents and teachers of late I am going to hit you over the head with it:

"Play is our brain's favorite way of learning."~Diane Ackerman
Contemporary American author

"Almost all creativity involves purposeful play."~Abraham Maslow
American psychologist 1908–1970

This is not a new concept:

"Do not…keep children to their studies by compulsion but by play."~Plato
Greek philosopher, 427–347 BC

"Man is most nearly himself when he achieves the seriousness of a child at play." ~Heraclitus, Greek philosopher 535–475 BC

"Whoever wants to understand much must play much." ~Gottfried Benn
German physician 1886–1956

"Play is the only way the highest intelligence of humankind can unfold." ~Joseph Chilton Pearce, Contemporary American scholar

"Play gives children a chance to practice what they are learning." ~Fred Rogers
American television personality, 1928–2003

"Children need the freedom and time to play. Play is not a luxury. Play is a necessity." ~Kay Redfield Jamison, Contemporary American professor of psychiatry

"Deep meaning lies often in childish play." ~Johann Friedrich von Schiller
German poet 1759–1805

"Children learn as they play. Most importantly, in play children learn how to learn."
~O. Fred Donaldson, Contemporary American martial arts master

Now if you so are so foolish as to think that you are wiser and smarter than that partial list of people who understand how important play is, I am sorry for you. Play is not just for recess and math can indeed be taught through play. Further, children who learn this way are not confused when it comes time to take tests, because they actually understand the math the questions on these tests are easy and they usually score not just well but in the top percentile of their peers.

With the manipulatives they can actually SEE what they are doing. It's not like this is another way to do math it is a decoding of the symbols in such fashion that both the conscious and sub-conscious mind can understand and better yet it is presented through play.

Algebra Is Child's Play

Here we see a six year old knocking out factoring problems with ease. He doesn't know factoring polynomials is supposed to be hard and that I've had to work with kids 10 years his senior on these exact same problems...only they were frustrated and confused.

We just did a few problems for fun first thing in the morning. His brother chose not to play, opting for cars instead which I am told is equally fun not really MORE fun. Algebra doesn't have to be painful. The basic concepts can be mastered at an early age thus freeing up the teen age years for other things, more math if they want or whatever topic catches their interest. This child will have ALL the math currently taught in public schools through sophomore year MASTERED by the time his is about 9 or 10. Notice I use the word mastered. This is completely different from "memorized to be forgotten", which is what is currently the norm in classrooms through out the country, including our universities...

Here is a very short vid showing him developing his problem solving skills, along with a host of other math skills used for factoring polynomials.



Here you can see his progress. Algebra gets easier and easier as he gets better at counting and making symbols.


Learn more at The House Of Math.

The Difference of Two Squares

They call it that because it is.

This is a good place to start with the toddlers because the blocks are bigger and easier to get a hold of...also conceptually students can see the problem with more clarity.


The sample lessons are here. Look for the one called "Difference of 2 Squares." The vid is 34 minutes long. Scroll to the bottom for this one. As you may or may not know your password opens up a bunch of pages not just that one. Several people have told me that one page is worth a lot more than 3 bucks but so far only my friends in Namibia have been kind enough to actually put it writing.

Here we get into some very simple analysis of the blocks:

These very same questions have gone on in teacher trainings and it has at times taken me a lot longer to get a room full of adults to have the same light bulb go on as it did with the 6 year old after just a minute...literally. Watch. Sadly the camera was not pointed at his face when he had his ah-ha moment. YOU have to explain and show them and ask questions that they answer themselves NOT just tell them.

"Here: memorize this. Here is the formula. Now go do it." DOES NOT WORK. 20 years later and not much has changed except things have gotten worse...

Here is an opportunity to change. Start by going to the house of math and watching the vids or if you are more advanced break out three bucks and get a password...

x² - 25 = (x + 5)(x - 5)

Is obvious to these kids: they can SEE it. No big deal. The concept of zero is being introduced and later when they see zero identities it won't be scary or confusing just really, really obvious.

x² - 1 = (x + 1)(x - 1) and x² - y² = (x + y)(x - y) are also easy once the concept is understood. Several more impressions and it will START to be internalized. One of the main problems with high school and college courses is the fact that they don't go over concepts enough times (if at all) so the formula is stuffed into the short term memory recalled for a test and then forgotten. You also get to pay thousands per credit for the privilege.

Also added to their quiver will be adding zero to help with problem solving. Math concepts are never beyond a child if presented at the child's level. The next thing is negative expression and very shortly I was able and will be able to let them build and count problems that high school teachers will consider advanced. And we got there in baby steps...by the ripe old age of 4 in one case and 6 in the other...I have done this with HUNDREDS of students over the years. I know it works...we are past the testing stage. Soon there will be some pdf's with more evidence if somehow you can't believe your eyes here. Newspaper articles from 20 years ago and such.

Time To Get Negative.

2x² + 5x + 3, sure, but what about negative coefficients?

I am constantly asked how to show and factor or work with expressions like

x² - 25

or

x² - x - 6

and some people start with a presumption, "That's great but I bet you can't show negatives very well." Well enough to get this autistic kid to get it. Or "...that's great but what about trig?" What you have here is a uniform methodology for the visualization of mathematics...not just the positive part not just the rectangles. Eventually there will be the derivation of πr² up on the sample lessons page, that 1/2 hour alone will be worth the price of admission. There are quite a few vids that I am told are worth the three bcks all by themselves. I am also told by marketers that the "problem" is the price is so low that people think there's no values there. I mean really, what can you get for 3 bucks?

Hours and hours of instruction that's what. Plus PDFs. Did I mention the PDFs?

I raised it from $1, apparently $19.95 is the going rate. MONTHLY. When I start getting some editing and some other paint and polish I will be raising the rates, meantime take advantage of the ridiculously low prices, NOW. It won't be long, this blog is already 3 years old, if you look at the first posts there were barely even pictures...lol...seems like yesterday.

Anyhow, here is Philip getting ready to go negative. Note how he is less than enthusiastic, because he has had bad experiences with integers due to that thing called the number line. I have lots of great stories about children who were taught about integers with this methodology compared to the way they did in school. One where the younger brother started solving problems faster than his older sister who "had learned all about negative numbers in school" but the method she had been taught was confusing and laborious...after seeing what her brother had been taught by me she said what most people say upon learning math this way: "I wish I'd had the blocks when I was learning this..."

For example:

Seems it would make sense to use blocks to teach about squares and the difference of two squares, or square roots or factoring...but they also come in handy for trig, pre-calc, and some basic concepts in calculus which I may never get around to here because there's hardly demand for it...well not nearly as much as algebra. But once you SEE an example of a derivative many light bulbs will start going off. Anyhow, once we are comfortable with factoring positive expressions THEN it's time to get NEGATIVE:



I see so many people who want to do too much too fast. "Okay, since they got that lets do this..." Remember degree of difficulty. Also the other thing to remember is you don't want to use the blocks for every single problem. The point is exactly the opposite...where you don't need the blocks once you understand the symbols anymore than you need pictures in your novels...but they were REQUIRED when you were first learning to read.

When you get to the Sample Lessons Page just look for the tittle "Show me the negatives." There are plenty more example on the password protected screencasts page too...going all the way up to 3rd and 4th power algebra.

BTW: I'd appreciate if a couple of people who do have passwords would comment on their experience with the password protected pages. Is it worth the 3 bucks or 12 bucks or what? If not here than on the facebook page and I'll screenshot it an paste it here.



Via email:

Hi there

Just read your blog where you ask for feedback about the password protected pages.

I visited once or twice with the password below, and the pages are great! The videos/screencasts are especially helpful. My only problem is that here in Africa (Namibia), our bandwidth is not enough to watch the screencasts without constant buffering etc. Far better for me is to be able to download the videos to watch later (and I might add, watch again and again!)

I have downloaded all your You Tube videos which are so helpful. Is there not some way that you can make it possible for us to download those screencasts?

OR, if you had some kind of full blown E-book/video course with an entire curriculum (k-12) in it to download, I’d definitely buy it. (?)

Anyway, thanks again for all the stuff that you’ve posted in various places already. I really is great!

Regards
Carol

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4 Year Old Factors Polynomials No Problem

Here is part of a longer video that you can find on the Sample Lessons Page. In it we see two little boys on a hot Hawaiian day playing algebra. You can see all the learning that takes place in this clip...they learn counting, addition, multiplication, factoring and algebra...and you can see it's fun and satisfying when they get it right.

God made it that way. Learning is fun. You can see when the endorphins go off by the expressions on their faces...you can hear the light bulbs going off in other videos with older students...ohhh, and ahhh, and "I get it now" are your verbal clues big smiles and clasped hands are physical clues.



Some day I would like to have a directors cut type session where I can pause the vids and talk over what's happening. The Mortensen Math methodology makes it easy for little kids to play math, it really does make math child's play...it's simple, once you understand the basic concepts. It's so simple some think it's magic, it's not magic: it's mathematics...and all we are doing is counting.

Too cheap to spend 3 bucks on a password? Go here for a FREE one hour overview on how to teach math using manipulatives, specifically using Mortensen Math.

Addends Past 10

Here is a simple part of a lesson where we worked on addends with a 11 year old...he does math that is much more "advanced" by some standards but we know all math is, is counting so addends and algebra aren't much different on the "hardness" scale. Same goes for multiplication...the idea is to attain mastery in basic counting...addition, subtraction, multiplication and division.

The concept is simple: numbers are made up of other numbers. Here we practice the single digit combinations past ten...what you will find with any student of the mathematics is their knowledge has some gaps or holes in it. So occasionally it's good to go through and fill in some of the holes and gaps with fun practice.

Note that this is quite different from just memorizing facts or formula...they learn to figure out the formulas themselves, and if they forget the facts they have algorithms or tools to figure out the answers quickly for themselves if it is not part of their instant recall memory...and the more you practice the better the chance of it getting into the instant recall memory.

Adding some symbols to go along with the blocks certainly doesn't hurt anything. Notice it's just for keeping track and that he doesn't write out all the combinations, the blocks give a quick visual for that, especially with the colors for added instant recognition...we also did subtraction which is basically going "backwards and forwards."

If I have 15 and take away 8 I am left with 7 which is just the addend. You can also each them to just add 2 which is the 10 compliment for 8 to the 5 for a total of 7, or you could take five out of the 8 which leaves 3 and know that the 10 compliment for 3 is 7 or (leaste preferred) you could count backwards from 15, 8 places...students HATE the number line for a reason. It is ineffectual and can be confusing even when presented well. Knowing all 45 addends with emphasis on 9's and 10's will solve many problems later, in fact you won't just solve them you will avoid them altogether. I have had countless students who were failing algebra come to me unable to do simple addition and subtraction in their heads without aid of fingers or toes, not to mention multiplication...these addends lead to multiplication which makes all the math easy because it allows you to count very, very quickly....

Parents always ask me why don't you just cover the blocks why does it have to be a math bag (and why go to all the trouble to decorate the bag)? BECAUSE IT'S FUN. WHAT'S MORE FUN? COVERING THE BLOCKS WITH A PIECE OF PAPER OR GETTING INSIDE YOUR VERY OWN MATH BAG? Seriously: some adults are a sad shadow of the bright, fun loving, energetic, curious, infinitely inquisitive children they used to be.

Here as you can see is a youtube vid, with apologies to all my public school teacher friends who have to go home to view it. It's just a short (less than 3 min) vid where the unknown mathematician shows he knows some of the combinations without looking, without blocks or symbols (or fingers) just his imagination.



Find me on Facebook or Youtube. I'm told there are benefits to subscribing to youtube because sometimes I put videos up and then go back and un-list them as I put them on password protected pages. BTW lots more video has been added to the password protected pages...

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More On Division.

How is it possible we can divorce multiplication from division or addition from subtraction...? Or visa-versa?

Here is a little more on division concepts, look forward to more screencasts on this in the future we will just get bigger...and bigger...

 Here we start nice and small and add a little twist by talking about parameter and sides.  This could be a nice beginner lesson for little kids and also a good lesson for older students.

Here we can see that division and multiplication go together and that they are indeed inverse functions, play with it a little and area parameter (and volume) can be easily understood because they can be easily seen.

Long division is despised among 5th and 6th graders...it should be easy as pie...but not the way we currently teach it. Which by the way is part of a methodology that is an UTTER FAILURE. Yet year after year very little changes. If you start small understanding simple problems like how many times is 2 contained in 8, bigger problems like how many 12's in 132 is also easy and then ones that have remainders like how many 6's in 341 or how many 23's in 687 are no problem.  You also realize that you don't want to model every problem with blocks! Once you understand the concepts and have gone from blocks to pictures to symbols ANY problem is easy (or easier) with pencil and paper and "multiply and subtract" makes sense...you might also find students able to do the easy ones that have no remainders in their HEADS.

They are going to have to get very conversant with multiplication though...

In the conceptual method using base ten blocks little kids are exposed to division concepts at the same time they are learning to add and subtract because math is just counting and multiplication and division are just adding and subtracting very quickly...all of it is just counting. 35 divided by 7 is just counting sevens and even easier than that if we use blocks to make a rectangle it's just counting one side when we know the whole rectangle and the other side.

Then there are improper fractions...which is just division, and fractions concepts inherently involve division (and multiplication) concepts. Then of course there are ratios which are also steeped in division concepts...

Go to my division page which is under construction (along with the rest of the site) for more.  You'll find this vid as well as a couple of others there.



As you read and watch this know that worksheets are being created for both multiplication and division and they will soon appear on the pass word protected pdfs page...passwords are currently a buck but the password is about to change at the end of the month and the price is going UP.

Currently a password gets you into the pdf's page where you can down load lots of different worksheets, onto a password protected screencast site that has algebra and percentages on it, as well as the advanced algebra page which at the moment isn't that exciting it just links you to the screencast channel and has some vids on factoring all in one place...but I am told it's MORE than worth the dollar, lol. people also tell me that the price is too cheap so people think there's no value there since it's only $1...and a buck doesn't get you anything anymore.

Divinely Dandy Non Difficult Division

 Get Divinely Dandy Non Difficult Division for just $19.99.  This book will show you everything you learned here and MORE laid out step by step with links to videos and pages that give simple concise explanations for how to use the rectangle to organize thought,  how to introduce division concepts at a very young age, and how to make fun while you are doing it.  I guarantee that video alone will expand your thinking when it comes to division and math.

Watch the video on the Preview and Purchase page that gives you a page by page over view of the PDF so you can "try before you buy", see exactly what you are getting and be confident it will be money well spent. 

Passwords $1

Math Bunny towing the line for Crewton Ramone's House Of Math.


Caution screencast contains blatant sales pitch by a cute rabbit and contains subliminal messages.
VIEW AT YOUR OWN RISK.
You may end up spending a dollar.



Passwords are available here and here and here.

More links to FREE stuff appear on the FB page.

Crewton Ramone Cross Teaching Concepts.

Here is a longer video showing multiplication, algebra and division. The lesson is simplified by keeping in mind the five basic concepts, and having blocks the child can get her hands on. In my travels I found that the saying math is not a spectator sport was QUITE TRUE. Just watching is not enough, you need to get your hands on the blocks. It was became apparent the difference between everybody in the room "getting it" and "only" getting 80 or 90% of the room was to make sure everybody had a set of blocks or were sharing a set of blocks, but everybody had to be able to have their hands on the blocks.

If you don't "get it" you may be one of those people that needs to get your hands on the blocks.

This child is supposed to be special ed or SPED, I doubt many of her SPED-mates are doing math like this. Children doing math like this at 6 or 7 years of age would be considered gifted...but actually when they can see what they are doing ANY little kid (or big kid as the case may be) can do math. I often say if you can speak English, count to 9, identify a rectangle and tell whether things are same or different, you can do math...




You may want to watch this one first. The video above is a continuation of this lesson:



Crewton Ramone's House of Math

You also might want to subscribe to my youtube channel.

New Video on Subtraction With Addends

Here is a quick Video explaining the concept of "Small Addition" when doing subtraction.

Go to my website for more.