I really question the merit of a system of teaching that damages a child's self esteem.
This student is a high school girl whose confidence is being shaken by her inability to understand algebra they way it is presented in her text book and by her teacher. I am getting through to her because she easily sees the concepts and is starting to see it's not hard once you get the basic idea.
I hear her go "ohh" and "ahh" and "I get it!" and I know I'm am succeeding at making her understand. The two main concepts we are going to cover is no fun get back to one; knowing what one is; having an understanding of one and hero zero followed closely by the concept of the rectangle. More basic math concepts here.
This screencast and post is a little incomplete because I am a little pressed for time when I work with her and I don't have time to stop and take pictures because currently it detracts from the lesson. The first session I failed to snap even one picture and here I barely got any either...if you have read many posts or spent any time on my website the themes should be familiar by now.
In this case I am working on preventing problems by exposing her to factoring and completing the square now so it makes much more sense when it's presented to her later, and by later I mean in her next chapter. Here you see her contemplating the concept, she understand the factors and is seeing what I mean by dividing by two and multiplying the result to complete the square. She can SEE it.
The symbols themselves cause minor panics at the moment. x2 + 8x + ____
x2 + 8x + 16
still seems a little scary but now she's finding it easy.
She also SEEs that the factors can be written
(x + 4)2
and it makes sense, and so does this:
x2 + 8x + 16 = (x + 4)2
more so because we just did a lesson on exponents and economy of symbol.
“Teaching is the greatest act of optimism.” ~Colleen Wilcox
I like this screencast for several reasons, one it has a lovely shot of powerlines at the beginning but even more than that it shows how you can go from working on simple subtraction to third power algebra all in the same lesson, and even more than that, once the student understands the concepts he can make up his own problems and start making discoveries about algebra and the distributive theory of multiplication and what factors end up where, and more. Don't be fooled this student DEFINITELY hasn't recovered his joy of learning when it comes to mathematics but at leaste he doesn't hate it (as much) anymore. Sometimes he has fun in spite of himself.
Now with regard to rules an process when it comes to subtraction, and making change...it is important to make sure students of all ages but especially very young children get the concept of subtraction, that we are "taking away" or "minus-ing" in kid speak. The algorithm make nines and a ten make subtraction easy but it's not magic or a trick and they can see that and get a hold of that concept IF you start in the concrete with the blocks. They can see that the "difference" and the "subtrahend" are the same as the "minuend", in other words if you add the difference to the subtrahend you get the minuend. The minuend is the number you are subtracting from and the subtrahend is the number you are subtracting and the difference is what's left AND my students have never even heard these math terms because they are not important for conceptual understanding.
That's why I refrain from calling this magic traingles because it's not magic IT'S MATH. So I call them math triangles. You can find more on my subtraction page and even download a 10 page pdf with practice problems for making change. A one page math worksheet is FREE access to a lot of pdfs including the 10 pager and fractions and algebra vids using negatives costs a whole buck. Get a password here. Where it says multi-page password.
This is a good video illustrating the power of manipulatives to help conceptualize numbers and relative amounts. It uses pennies.
Base ten blocks and other math manipulates also help with this. Here you see base ten manipulatives called basic operations pieces, complete with a ten thousand square. You can change the scale several times and work your way up to trillions. They are very useful for teaching exponents getting a handle on 102, 103, 104, as well as higher power algebra.
In the picture you can see we have changed the scale and made what used to be a unit 100 and now the 10,000 square is a million!
Bringing the method into focus. Making it all fit together. This post attempts do do more of that for you.
More math enrichment with a 6th grader. He has come 12 times now, I should point out 4 of those sessions were devoted mostly to some homework he had that his parents couldn't help him with and not on using this method per se...the point being it didn't take all 12 lessons to get him off his fingers and get his multiplication down. Also he spent some time playing games like Timez Attack and practicing multiplication on his own because 12 lessons wouldn't be enough to get all 144 facts (12x12) and soon all 400 facts (20x20) into his instant recall memory. Note we didn't spend much time using formal multiplication tables or math flash cards or even math worksheets.
Again this screencast covers degree of difficulty and how to use the algebra to cement the addends and multiplication facts into their experience.
Here is a lesson with my Autistic student, she is coming along quite nicely. This screencast is abrivated and does not contain the pictures for the negative expresssions. If you want to see that screen cast it is available here, Crewton Ramone Playing With Negative Factors...you need a password. You can buy one at the house of math here. That same password also gets you into the PDF's page where you can download printable math worksheets.
If you just see the symbols it looks like the child is a genius, and people are even more amazed when I tell them the child is Autistic. Once you SEE what you are doing you can see how very young shildren can do these with ease after just a few sessions. This math is great for learning factoring, addends, multiplication, and intergers...
Here is a short screencast covering a lesson with two young students of 7 and 8, we managed to spend the whole time just playing addition and multiplication, everything we did was addition and multiplication...even though we did fractions and division...it was still just addition and multiplication.
We practiced concepts we already knew and practiced writing...this is the only way to attain mastery: practice. So this class was pretty easy for them no stretches nothing new really although each time we do multiplication some of the facts SEEM new event though they've seen them before.
Here they have threes on their fingers, and had to work together to show me eight threes...we get practice with threes AND we get practice with addends because they have to decide who will will show what, three and five or four and four or whatever...
Here we took our threes and laid them out and put them in groups of 9, talked about 3x + 3 and x being 9 in this case, also talked about 9 times 3 plus 3...and of course 10 threes all being 30.
Writing out what they see in symbol form and beginning to get what equivalent fractions mean, we also talk about common multiples.
Here we build rectangles and count the sides, we count them several ways the sides only for example 4x5 and 5x4 and then what if I have a rectangle where one side is 4 and the whole thing is twenty? What's the other side. What if te whole thing is 35 and one side is 7? What's the other side...? Etc...
This sheet shows the fractions and a pause to write out sixes in a matrix where they CLEARLY see the pattern with the symbols...
Once in a great while I give them a worksheet to take home, I am making webpages at Crewton Ramone's House of Math that have free math worksheets, starting with some first grade math worksheets and fraction worksheets...I also have a kindergarten math worksheet for them to practice writing their numbers and learning their addends, of course it can be used with older students too, it takes lots of practice to write neatly. Right now there's only a few worksheets but I remember when I only had a few videos too, last count I was over 70...
One day there will be that many and more printable math worksheets available there. Not all of them are free but you can get a password for a buck and tat lets you get into a page that has all of them PLUS you can get into the advanced algebra page and my password protected screencast channel that shows how to factor negative expressions...a buck goes a long way. Here is the open channel, I made this as an alternative to my YouTube channel so teachers and students could watch my vids at school.
Anyhow here is the screencast covering the 90 minute session.
Practicing addends builds fine motor skills, besides teaching the combinations for ten, he knocked it over several times and I helped him once or twice so he wouldn't get frustrated. The more time he builds it the more impressions are made on both his subconscious and conscious mind. This how too many people who have these powerful tools use them. To teach counting, addition, maybe some multiplication, and place value.
Finishing a task is satisfying. At my house of math, and here on this blog I show you how to use these base ten blocks to teach math concepts like "higher" algebra and Pythagorean Theorem. I'm not saying they shouldn't be allowed to play and have fun doing these simple tasks what I am saying is do this AND MORE. Playing and having fun also leads to a positive association with the manipulatives and math in general.
After each task he would smile and say "COMPLETE." MU the cow helps teach math when she is on my hand, it's just fun when the boys play with her. They Call her Mooey. They get the joke about MU since they both know the Greek Alphabet...Lambda the sheep has been lost for quite a while now...there's a joke in there too someplace...
If you have trouble viewing try this link:https://screencast-o-matic.com/watch/cXVqlhloe
Here is the same set of pictures with the boys helping. It was made about a week later, so it's a great way to to refresh their memories and make more impressions for the info's journey to the long term memory where it can be made available for instant recall.
Why English is hard to learn (you've seen this email):
The bandage was wound around the wound.
The farm was used to produce produce.
The dump was so full that it had to refuse more refuse.
We must polish the Polish furniture.
He could lead if he would get the lead out.
The soldier decided to desert his dessert in the desert..
A bass was painted on the head of the bass drum.
When shot at, the dove dove into the bushes.
I did not object to the object.
The insurance was invalid for the invalid.
Since there is no time like the present, he thought it was time to present the present.
There was a row among the oarsmen about how to row.
They were too close to the door to close it.
The buck does funny things when the does are present.
A seamstress and a sewer fell down into a sewer line.
To help with planting, the farmer taught his sow to sow.
The wind was too strong to wind the sail.
After a number of injections my jaw got number.
Upon seeing the tear in the painting I shed a tear.
I had to subject the subject to a series of tests.
How can I intimate this to my most intimate friend?
You can listen to the sentences
or download the files in .mp3 or .wma (Windows Media) format.
Let's face it - English is a crazy language.
* There is no egg in eggplant nor ham in hamburger; neither apple nor pine in pineapple.
* English muffins weren't invented in England or French fries in France.
* Sweetmeats are candies while sweetbreads, which aren't sweet, are meat.
We take English for granted. But if we explore its paradoxes, we find that
*
quicksand can work slowly,
*
boxing rings are square and
*
a guinea pig is neither from Guinea nor is it a pig.
*
And why is it that writers write but fingers don't fing,
*
grocers don't groce
*
and hammers don't ham?
*
If the plural of tooth is teeth, why isn't the plural of booth beeth?
*
One goose, 2 geese. So one moose, 2 meese? One index, 2 indices?
*
Doesn't it seem crazy that you can make amends but not one amend, that you comb through annals of history but not a single annal?
*
If you have a bunch of odds and ends and get rid of all but one of them, what do you call it?
*
If teachers taught, why didn't preachers praught?
*
If a vegetarian eats vegetables, what does a humanitarian eat?
Sometimes I think all the English speakers should be committed to an asylum for the verbally insane.
*
In what language do people recite at a play and play at a recital?
*
Ship by truck and send cargo by ship?
*
Have noses that run and feet that smell?
*
How can a slim chance and a fat chance be the same, while a wise man and a wise guy are opposites?
*
How can overlook and oversee be opposites, while quite a lot and quite a few are alike?
*
How can the weather be hot as hell one day and cold as hell another?
*
Have you noticed that we talk about certain things only when they are absent?
*
Have you ever seen a horseful carriage or a strapful gown? Met a sung hero or experienced requited love?
*
Have you ever run into someone who was combobulated, gruntled, ruly or peccable?
*
And where are all those people who ARE spring chickens or who would ACTUALLY hurt a fly?
You have to marvel at the unique lunacy of a language in which
* your house can burn up as it burns down,
* in which you fill in a form by filling it out and
* in which an alarm goes off by going on.
English was invented by people, not computers, and it reflects
the creativity of the human race
(which, of course, isn't a race at all).
*
That is why, when the stars are out, they are visible, but when the lights are out, they are invisible.
*
And why, when I wind up my watch, I start it, but when I wind up this essay, I end it.
The was not written by Crewton Ramone it was cut and pasted from an email I have gotten more than once. I'm pretty sure (not beautifully sure) it was one of those emails that got longer the more it was sent around if it was written by one person that person never got credit because his or her name was left off the email...by the way sorry and apologize NOT same if you think they are go to a funeral and instead of saying sorry say I apologize...
Dearest creature in creation,
Study English pronunciation.
I will teach you in my verse
Sounds like corpse, corps, horse, and worse.
I will keep you, Suzy, busy,
Make your head with heat grow dizzy.
Tear in eye, your dress will tear.
So shall I! Oh hear my prayer.
Just compare heart, beard, and heard,
Dies and diet, lord and word,
Sword and sward, retain and Britain.
(Mind the latter, how it’s written.)
Now I surely will not plague you
With such words as plaque and ague.
But be careful how you speak:
Say break and steak, but bleak and streak;
Cloven, oven, how and low,
Script, receipt, show, poem, and toe.
Hear me say, devoid of trickery,
Daughter, laughter, and Terpsichore,
Typhoid, measles, topsails, aisles,
Exiles, similes, and reviles;
Scholar, vicar, and cigar,
Solar, mica, war and far;
One, anemone, Balmoral,
Kitchen, lichen, laundry, laurel;
Gertrude, German, wind and mind,
Scene, Melpomene, mankind.
Billet does not rhyme with ballet,
Bouquet, wallet, mallet, chalet.
Blood and flood are not like food, Nor is mould like should and would.
Viscous, viscount, load and broad,
Toward, to forward, to reward.
And your pronunciation’s OK
When you correctly say croquet,
Rounded, wounded, grieve and sieve,
Friend and fiend, alive and live.
Ivy, privy, famous; clamour
And enamour rhyme with hammer.
River, rival, tomb, bomb, comb,
Doll and roll and some and home.
Stranger does not rhyme with anger,
Neither does devour with clangour.
Souls but foul, haunt but aunt,
Font, front, wont, want, grand, and grant,
Shoes, goes, does. Now first say finger,
And then singer, ginger, linger,
Real, zeal, mauve, gauze, gouge and gauge,
Marriage, foliage, mirage, and age.
Query does not rhyme with very,
Nor does fury sound like bury.
Dost, lost, post and doth, cloth, loth.
Job, nob, bosom, transom, oath.
Though the differences seem little,
We say actual but victual.
Refer does not rhyme with deafer.
Foeffer does, and zephyr, heifer.
Mint, pint, senate and sedate;
Dull, bull, and George ate late.
Scenic, Arabic, Pacific,
Science, conscience, scientific.
Liberty, library, heave and heaven,
Rachel, ache, moustache, eleven.
We say hallowed, but allowed,
People, leopard, towed, but vowed.
Mark the differences, moreover,
Between mover, cover, clover;
Leeches, breeches, wise, precise,
Chalice, but police and lice;
Camel, constable, unstable,
Principle, disciple, label.
Petal, panel, and canal,
Wait, surprise, plait, promise, pal.
Worm and storm, chaise, chaos, chair,
Senator, spectator, mayor.
Tour, but our and succour, four.
Gas, alas, and Arkansas.
Sea, idea, Korea, area,
Psalm, Maria, but malaria.
Youth, south, southern, cleanse and clean.
Doctrine, turpentine, marine.
Compare alien with Italian,
Dandelion and battalion.
Sally with ally, yea, ye,
Eye, I, ay, aye, whey, and key.
Say aver, but ever, fever,
Neither, leisure, skein, deceiver.
Heron, granary, canary.
Crevice and device and aerie.
Face, but preface, not efface.
Phlegm, phlegmatic, ass, glass, bass.
Large, but target, gin, give, verging,
Ought, out, joust and scour, scourging.
Ear, but earn and wear and tear
Do not rhyme with here but ere.
Seven is right, but so is even,
Hyphen, roughen, nephew Stephen,
Monkey, donkey, Turk and jerk,
Ask, grasp, wasp, and cork and work.
Pronunciation (think of Psyche!)
Is a paling stout and spikey?
Won’t it make you lose your wits,
Writing groats and saying grits?
It’s a dark abyss or tunnel:
Strewn with stones, stowed, solace, gunwale,
Islington and Isle of Wight,
Housewife, verdict and indict.
Finally, which rhymes with enough,
Though, through, plough, or dough, or cough?
Hiccough has the sound of cup.
My advice is to give up!!!
The mathematics begins with counting. Because as you can see from the basic concepts math is the study of numbers and all we do with numbers is count. How do we teach counting? Start with patterning. Just say the numbers from one to five, then one to ten then ten to twenty. Repeatedly. Start with your fingers, at bed time is a good time but any time is fine.
With most beginning students I make them write out the five basic concepts. With older students counting is redundant, they can all count, and you can make a joke of it. After they write down "Math is the study of numbers and all we do with numbers is count", or some variation thereof, ask them to count to twenty. Then ask them to count backwards from twenty.
“See, I can teach you math, we are one fifth of the way there...you can even count backwards.” Then ask them to recite the ABC's backwards. So far I have only had a very few kids that could.
“Hmm, looks like you're better at math.”
Younger students (and many older students) need to be taught the patterns whether it's counting or addition with addends, multiplication or what have you. When it's counting start with the simple concept: the highest number we count to is nine. The numbers tell us how many the places tell us what kind. After that it's just vocabulary, we have English names for all the numbers, even really big numbers. One, two...seven, eight, nine and then ONE of the next kind, one ten. One ten and one unit is called eleven. One and one is just two, you may have to explain this several times to a three year old even if you are using manipulatives. 11 looks like two to them and this looks like three: 111. Several explanations may be required to
Then just pattern: Ten, twenty, thirty...seventy, eighty, ninety ONE hundred.
One hundred, two hundred...eight hundred, nine hundred, ONE thousand...it never gets past nine.
Two tens and three units has a name, twenty three...what's important is understanding the concepts place value is easy and almost visually obvious and becomes clear after just a few explanations.
We count the big ones first. One hundred, one ten and one unit are one hundred eleven. 111. With manipulatives this IS visually obvious but when writing 111, little kids often think THREE...can you see how much easier it is to teach place value?
Teaching a child to count properly takes MONTHS, you can teach them to pattern and memorize by counting from one to 20 over and over again, which is fine; however this can lead to some confusion. Ever hear, "my kid can count to 10 but gets a little confused in the teens"? I have. A lot. Make sure they see all the teens are, are just ten and some more, or specifically ten and one through nine units more, and that the pattern repeats with two tens and one through nine units, and three tens and so on. This way they get the concept and the pattern and link the vocabulary to the numbers.
I can't tell you how many times I got a student that was failing algebra that had to use their fingers to add numbers. Simple numbers like nine and three. Asked to multiply they didn't have their tables memorized either without thinking hard about it. OF COURSE THEY WERE FAILING ALGEBRA: they had no foundation upon which to stand. Back to basics. Addends. No magic. Then again, when the basics are mastered correctly the magic begins. First counting, then addition with addends, then multiplication: seems obvious.
“Obvious” is the most dangerous word in mathematics.” ~Anon
Addition and multiplication are just way of counting very quickly. In order to get there you start off crawling then you walk then you can run, then you can hit the hyperspace button...or at leaste turn on the after burners, the easy way, visit Crewton Ramone's House of Math for help with teaching counting and much more.
Like language, it's important that the students hear the patterns as well as see any patterns before they write down the symbols. Multiplication is a great example of this idea. Long before students see a multiplication table or worksheet young students should be singing songs, using blocks and hearing the patterns that are associated with the various multiplication tables. Then when they are exposed to the symbols, they understand what they mean. They understand that three times three is written 3 x 3 and that it really is the same as nine. The equals sign and all the symbols are understood. If your children are playing with blocks they can see that nine is a square number and what that means. Nine really is square even though the symbol for nine has curves and a circle. 9. Look at it, it doesn't look very square, little kids and even some older kids can get caught up in the symbols instead of understanding the concepts and what the symbols represent.
Parents and teachers can use worksheets to reinforce math learning but they should be used sparingly to introduce new ideas, if at all. With manipulative based teaching we always start in the concrete with base ten blocks, then move to sketching then at last to the symbols. Since most worksheets are symbol based it's only natural that they should be used last not first. Concept based teaching techniques emphasize understanding the concepts long before students see the symbols. In fact, a lot of little kids can start getting complex math concepts well before they can write complex symbols.
Using math worksheets to introduce math concepts is literally teaching backwards. First Grade Math Worksheets should be introduced after quite a bit of playing has been done. That way the worksheet is just practice that allows you to see how well they understand the math concepts you are teaching. They can be used for drills but using drills at an early age has unintended consequences, you don't want to turn them off at an early age and excessive drills will do just that eventually. The worksheets should be easy to begin with and then become more challenging as the student's confidence builds. They should be thought of as practice instead of as tests or drills or something to be fretted over. Some students develop math anxiety at an young age and worse test anxiety.
If the child firmly grasps the concepts any math worksheet should be simple or at worst “challenging.” If children don't have the writing skills required you can actually do the writing for them. Be sure THEY tell you what to write, you are not doing it for them just writing the symbols. This is more for parents or home schoolers because teachers in school will find this impossible unless the number of children is very small and we all know they rarely are.
Often times it's a good idea to play with the concepts for several days and then give the worksheet on Thursday. If the students don't complete it they can take it home and you can finish it off on Friday. If you are homeschooling or just giving your child a head start, be sure you play for a couple of days at least before you get out a worksheet. With first graders you can often use the same or very similar worksheets every few months and it will be new again. This is normal it takes quite a few impressions to get information into the long term memory. Many teachers lament that after Summer Vacation or even Christmas or Spring Break their students don't remember most of what they have been taught. It's still “in there” and this is a great time to bring it out with a practice sheet they are already familiar with.
Worksheets are great for reinforcement, and great teaching tools when used properly.
Using the Making Change Worksheets.
FREE WORKSHEETS HERE at Crewton Ramone's House of Math. More free worksheets coming soon.
A few pictures and a screencast from a two and a half hour session I had with some students. For the entire first hour or so I forgot to take a single picture.
We basically played getting to know you, and then went over the five basic concepts, played with the pieces to gain familiarity with the blocks, built some tens, did some multiplication and then moved on to algebra.
Before we started playing on the floor we did some drawing and of course a three period lesson to get comfortable with the pieces we were about to use on the floor. They got a chance to draw whatever they wanted and then we played a simple game where they had to tell me the factors and I had to draw a picture to see if our pictures matched. Similar to this post where the student got to draw anything he wanted. When I say "anything", I mean any picture of a third or fourth power expression.
Had a few games where we took the factors and built the rectangles...just drawing. But then for even more fun we played on the floor.
Note in the initial picture we have a drawing and some notation that does not fit the picture because the picture is taken after we did a drawing where one factor was x + 2, and then I added to the picture making the factor 2x + 2...and we hadn't changed the notation yet...they really get an intrinsic feel for the distributive theory when you present math this way. As time goes y and they see it a few times when given a formal presentation on the distributive theory they "get it."
Right now for the first introduction it's much more about counting, adding and multiplication. The algebra and factoring come along for the ride as it were, we tie it together with the "higher algebra."
"Main thing: we have fun", and even a ten year stayed "tuned in" for over two hours. Afterwards he told his mom, "it was fun."
Mastering the 45 addends is an important step on the way to making computation easy. Addition is simple, if the concepts are understood. 5 + 7 is the same as 7 + 5 and when 7 and 5 get together it's always going to end in 2...so 17 + 5 and 15 + 7 are easy and students can also see that 37 + 5 is basically the same problem as the single digit problems with tens "just along for the ride." You would be amazed at the number of students who don't get that simple concept. They'll come up with 21 or 23 instead of 22 when adding 15 + 7. They can also use the simple "want to be a ten" algorithm to make it easy: 7 takes 3 from 5 making one ten and two, OR 5 takes 5 from 7 making one ten and two. Either way it's 12, and the best way to do it is the way the student likes best.
This method allows the student to get off their fingers by making "a ten and some more" when adding two numbers. As it turns out there are only 45 combinations...once students understand this simple "want to be a ten" algorithm addition becomes a lot easier and they can tackle bigger problems on their own. Then it just comes down to practice and repetition. Use a wide variety of problems to practice this skill and teach other concept the same time in order to keep the practice from becoming mind numbing drill work which will also turn students off to math.
Using their fingers is a step on the way to mastery of addition facts, unfortunately many students remain stuck at this step all the way into adulthood. For kinesthetic learners using fingers and hands IS IMPORTANT: that's HOW they learn, and you need to help them move past this: manipulatives are a great way to move them into "doing it their heads." For young students using fingers and hands is just natural...you can also spot the kinesthetic learners because they will rely more on their fingers and be slower to move on from them. This does not mean they are "slow" or any less able than visual or auditory learners, they grasp concepts just as fast or faster than those with other learning styles. We also find when it comes to sports and other activities requiring hand eye coordination (like arts and crafts) they often excel. Using your fingers is great! AND you need to get past that stage if you are going to be fast at addition and attain mastery. Being fast at addition leads to easy mastery of multiplication as an added bonus. They may even like math, why wouldn't they if it's fun and easy?
Many speed reading courses incorporate the use of the finger to guide the eye along the page, some use this to start, and then drop it for other courses this is the main stay of the course. Adding more sensory input increases learning, and in the case of reading the hand and the eye are integrally connected. The point is you want to encourage students to move through this step when it comes to the mathematics NOT discourage or skip the step all together. Some students will naturally NOT use their fingers when doing mental calculations...for those that do use their fingers later it will become a handy-cap. Counting quickly makes math easier, because all math is is counting; however, don't confuse computation with the mathematics. The mathematics is the use of computation and critical thinking skills to solve problems and express reality numerically.
Addition and subtraction as well as multiplication are just counting quickly. They are among the first steps to understanding math, and they should be mastered to ensure success. Using fingers can lead to a loss of accuracy too, often children (and adults) are off by one sometimes even two.
Practice with the addends verbally, build walls and towers, play games like what's under the cup, simple story problems and work sheets with pictures give the student the experience they need to make the transition from fingers to symbols to being able to do it "in their heads." Drawing rectangles and other math concepts as well as making drawings of the manipulatives they use, help the student make sense of the symbols and see what they are doing. It also adds variety, and helps students (and teachers) see that you use the same skill sets all through the mathematics, which is why you often see me use third and fourth power algebra to teach addition and multiplication facts.
Indeed if you carry the concept far enough they can also get off the symbols as it were and do it ALL in their heads if need be, no paper or pencil. This was illustrated perfectly by a five year old who is able to factor trinomials in his head because he can see the pictures when he hears expressions like x^2 + 3x +2, he can see it and tell you the sides. Or if you tell him the sides (x+3)(x+2) he can tell you the whole rectangle not because he is seeing symbols but because he is seeing PICTURES. Further he is "cementing" his addends and multiplication facts into his memory. How much easier is it to see 6 taking a 4 out of a 7 to make 13 when presented with a problem like x = 6 + 7 than to do algebra? It's also quite easy to see 6 + x = 13 or x + 7 = 13, especially if you give them a simple algorithm to solve these based concept of "want to be a ten." He also gets a ton of positive reinforcement because people think he is a little genius which motivates children to do more. Never underestimate the power of simple praise.
Once they learn some basic concepts and understand what the symbols mean math becomes easy and even fun. Being able to visualize what you are doing makes all the difference, it also makes it MUCH easier to commit to memory because the mind works in pictures not symbols, so memorizing the 45 addends and multiplication tables is easier because the mind can store pictures much more readily than symbols. Then when it is time to be recalled, a picture or the symbols or just words can easily be retrieved from that place we call the long term memory.
Have you ever known anybody that remembers phone numbers by picturing the keypad in their head? They may even point to the numbers and move their pointer finger on an imaginary keypad in the air as they are recalling the number. This is a visual kinesthetic way of storing long numbers. The brain works with pictures and this makes it easier to get the information out. How much simpler is it to add two numbers together than recite seven to ten digits? Especially if you have a method for visualizing them if you somehow forget?
A simple exercise: ask a student to picture a cow. Then ask if they saw C O W or a picture of a cow? Ask what color was it? This lets you know they weren't seeing symbols. The problem is with math most students have nothing to picture whether it's algebra or simple addition. The "trick" if there is one is to get the information into the long term memory so it easily recalled and it's pretty well proven that symbols, that is letters and numbers, are a difficult way to get information there.
Manipulatives are the perfect bridge to get information there. After all, it's never storage that's the problem it's retrieval.
Crewton Ramone is the alter ego of a fed up math tutor. More can be found by visiting these websites:
The point of this post isn't so much the mathematics that the students learned; there are plenty of "how to" posts on Equivalent Fractions, as well as other fractions concepts, Card Games and Algebra on this blog and on my website.
The point of this point is to draw your attention to the student's notation as a big clue to how well you are doing at getting whatever concept you are teaching across. Here it starts out "all hamajang" as we say in Hawaii and quickly turns into clear, neat notation. You can actually see the confidence build in the notation.
You can also see it here in the drawing and notation if you know what you are looking for, as he becomes more comfortable with the concepts and just drawing and counting the sides his pictures and notation get "neater" or "nicer", as I often admonish my students "Write neatly. Neatness counts." And we laugh at how punny that is.
Here is a quick screencast covering the lesson and the concept that the notation is an important clue for you as the teacher. I use these because the example is dramatic and you can't miss it you may find it more subtle with your own children or students.
Well, sort of. When most people think of self directed they think of free schools where the kids do what they want and end up learning very little because for the most part they just goof off.
This is Directed Discovery (one of the concepts I talk about on the Basic Concepts page ), where they get to do what they want within certain guidelines. In this case it was just draw rectangles and count the sides.
There will be a few more links added to this page as time goes by. For now Here are the usual links and the two I talk about in the screencast.
You can find Timez Attack on my multiplication page and this is the blog post where you see him playing it. Pretty cool when his parents have to make him stop playing a game that teaches multiplication...and now it can be used as a reward for good behavior or doing chores or whatever. Learning is fun. He was markedly better at multiplication but still has quite a ways to go as they say. He is also almost completely off his fingers when doing addition, the skills sets he is acquiring will last his entire life. It's also helping in the classroom because math is easier when you can count fast.
There is also the element of increased self confidence and self esteem...because he knows that doing algebra is something "none of my friends can do." Apparently he was showing some of his friends what he does after school on Fridays...
And here is a link to the rules of the card game multiplication facts war I talk about in the video. The woman who wrote this page wrote me an email to complain that I had infringed her copyright by cutting and pasting the rules from her page. There will be a blog post about that later. So I went back and changed my page and reworded the rules and renamed some of the games. I am interested to see if she is claiming she made up these games, I've been playing them for many years...
More work with my Autistic student. She was happy to report her school work with fractions was "too easy."
I treat her pretty much like any other kid and use the same basic lessons to provide powerful understanding of math concepts. After the concepts sink in math itself is pretty easy.
It was late when I made this screencast and my verbalization of the story problem in the middle is quite screwed up, lol!! One day will learn how to edit. I left it instead of re-cutting it because even with the verbal errors I think the concept comes through at last, and you can see even if you make a mistake you can still do a lesson. So many parents and teachers never get started for fear of "doing it wrong", some of you may have even gotten it better due to my errors because you thought to yourself "what he's trying to say is..." Anyhow here it is correctly:
If each student got one bottle of water we could give 120 students one bottle each.
If each student got 2 bottles each we could give 60 students 2 bottles each.
If each student got 3 bottles only 40 students would get water, if we gave each student 4 then we could only give 30 students water...when I was doing it with her I was saying correctly and she understood what I meant. In the picture we see 120 broken into 4 groups of 30 each...we could also form a rectangle and count the sides.
Here is a good reason to use rectangles instead of circles or pies when presenting fraction concepts. being able to draw it demonstrates comprehension and moves the student from the concrete to pictures to symbols naturally. For more on what that picture represents and how we develop the concept of equivalent fractions go here.
If you donate a buck you get a password...the password is worth more than a buck...unlocks "advanced algebra" page and soon a screencast channel that will have all manner of instructive scereen casts on topics you can't get anywhere else...the videos on the password pages contain less errata.
“The authority of those who teach is often an obstacle to those who want to learn.” ~
Marcus Tullius Cicero
A session with a 7 year old boy and an 8 year old girl where we play math and have fun.
Surprisingly, there were a few tears on the part of the boy when he got an answer wrong while we did 3rd power algebra but as per usual with kids this age he was laughing and having fun a few minutes later as he easily saw what to do to get the right answer...I cheered him up with 3rd power algebra! Try that in high schools across America. Big difference when you can see what you are doing.
Here is a quick sketch of a "big" problem.
(2x3 + 9x2 + 16x + 12) ÷ (2x2 + 5x + 6)
Tears over math is a silly thing especially when you can see the answers. Mostly we played games drew pictures...and learned a lot of math concepts along the way.
Here is another enrichment session with a 6th grader That flew by.
"It felt like 5 minutes..."
That's when you know you are doing it right. Learning is fun and time flies when you are having fun.
First we looked over the left over lesson from the 4 and 5 year old and talked about equivalent fractions.
Then we wrote out a few equivalent fractions all the way to 12...1/2, 1/3, 1/4, 3/4, 5/7's...later will do other combinations.
Then we built all 45 addends. He tried to stand them up and goofed around a bit while he made them. I encourage this. Then we did "drills" verbally what's 7 + 5, what's 37 + 5, what's 25 + 7, what's 97 + 5? etc...did lots of them for various addends the idea is to get him off his fingers. Using your fingers is great! But you need to get past that stage if you are going to be fast and attain mastery.
Using fingers is a step on the way to mastery, unfortunately many students remain stuck at this step all the way into adulthood. For kinesthetic learners using fingers and hands IS IMPORTANT, that's HOW they learn, and you need to help them move past this, manipulatives are a great way to move them into "doing it their heads." For young students using fingers and hands is just natural...you can also spot the kinesthetic learners because they will rely more on their fingers and be slower to move on. This does not mean they are slow or any less able then visual or auditory learners, they grasp concepts just as fast or faster than those with other learning styles. We also find when it comes to sports and other activities requiring hand eye coordination (like arts and crafts) they often excel.
Many speed reading courses incorporate the use of the finger to guide the eye along the page, some use this to start, and then drop it for other courses this is the main stay of the course. Adding more sensory input increases learning, and in the case of reading the hand and the eye are integrally connected. The point is you want to encourage students to move through this step when it comes to the mathematics NOT discourage or skip the step all together. Some students will naturally NOT use their fingers when doing mental calculations...
After addends we played cards. Not poker, but war. We played multiplication facts war where we each put down two cards and multiplied them together; as an added bonus we couldn't start another round until he told me the difference between our scores. It was fun although I beat him.
Then because he had behaved will wrote neatly and did all the work I asked him without a single complaint I let him play Timez Attack for about 10 minutes...he really had fun with it and when it was time for class to be over he wasn't ready to leave...
"Always leave them wanting more..." as some famous circus guy once said...
“If I am walking with two other men, each of them will serve as my teacher. I will pick out the good points of the one and imitate them, and the bad points of the other and correct them in myself.” ~Kung Fu-tzu Confucius
Equivalent Fractions teaches multiplication. It all goes together so while they learn fractions concepts they learn multiplication or in this case they get to practice it...but when we get to the higher numbers they'll be learning the higher multiplication tables.
It's easy to see that 1/3rd and 6/18ths are the same. They can see it.
People often ask the ridiculous question: "Do they need to have the blocks with them to do math?"
The other night without blocks or even symbols just verbally I asked him what the factors of x2 + 3x + 2 where and he looked up for a moment and said X plus one and X plus 2...then before naps today I asked him, "if the sides are x plus 2 and x plus 3 what's the whole thing?"
He thought out loud and said, "one x square, three and two...five x's and four...no I mean six...six would fit in there cuz it's three and two."
No blocks or symbols. It's the same when you teach any language to babies: they hear it FIRST before they ever see an ABC...just because they can't write symbols doesn't mean they can't learn math.
Further the average student can't multiply 15 x 17 in their heads...because they need to have the symbols and a pencil and something to write on. My students can see the answer is 255 or at worst can add 100, 120 and 35...
Here in this video they have seen this a couple of times before without symbols and for the first time they see the symbols for fractions today, and the symbols make sense and are easy to comprehend because they have heard them before.
You can see their progress with multiplication if you have been following this blog...they still can't tell you 4 x 5 off the top of their heads but give them a little time and they can figure it out...and the more we learn fractions the better they'll get.
“The point is to develop the childlike inclination for play and the childlike desire for recognition and to guide the child over to important fields for society. Such a school demands from the teacher that he be a kind of artist in his province.”
~Albert Einstein
