Always under construction.
These simple concepts can change the way a student feels about mathematics.
The Five Basic Concepts:
1. Mathematics is the study of numbers and all we can do with numbers is COUNT
2. The concept of SAME. The highest number we can count to (in base ten) is 9, the numbers tell you how many the places tell you what kind. Before we can count they must be SAME.
3. We form rectangles to facilitate counting.
No Fun Get Back To One. NFGBTO.
These basic concepts can be applied to any math problem. Math text books are always just variations on these five themes depending on subject matter. The trick (if there is one) is knowing how and when to apply them.
My students hear "hero zero, no fun get back to one" during problem solving and solving equations algebra constantly...and they realize very quickly just how easy it is when you apply these concepts. The reactions range from relief to joy to rage. Relief for some when they realize they aren't stupid after all, joy for others who go from "F" to "A" so instantly that the teachers accuse them of cheating, rage by others who get mad when they realize all that fear, frustration and anxiety was for nothing.
Presenting the five basic concepts.
When presenting the basic concepts always ask questions. Whatever the answers are give positive feedback, and lead them to the answer you want. Remove the NO from the lesson. This takes practice.
Eventually there will be a 5 or 6 vids here, about ten minutes for each concept. This should tell you there is a lot more to each of these concepts than just the few words used to outline them.
Three Period Lesson
Use a “three period lesson” to introduce whatever it is ALWAYS.
The three period lesson's origination has been credited to several people. I don't know who for sure can claim the right to say they invented it. I am; however, sure glad someone did. It is the fastest way to get everybody on the same page I know of for certain.
It's had to instruct if the students don't know what your talking about. Or if they aren't sure what the name of each piece is or what a term or symbol means.
It seems deceptively simple:
But it is powerful. Lets say you were presenting the manipulatives x-square, x, and unit for the first time. You put the three pieces out and show them them each with a short explanation for each. That's the "this is" portion.
Then you ask "what is this?" as you point to the x. They should respond x. If they don't show them what they did respond. If they said unit show them this is a unit this is x. Remove the no from the lesson. Go back and fourth a few times. Once they have identified all three correctly move to the next portion.
Say show me an "x". They point to or lift up the "x". Then the "x-square" then the "unit". Go back and fourth a few times.
Do this every time you introduce a new concept or symbol or manipulative. Works great with ABC's too. Get out the letters (refrigerator magnets, foam letters or what-have-you) or draw them on a page 4 or 5 at a time.
Do not be fooled into thinking this is for little kids or young students only the three period lesson is for all ages and students, and with a little practice can be incorporated into many and varied lessons, not just the mathematics. Learning the names of tools in carpentry or mechanics for example, dates in history, names of plants or herbs and their uses. It is super simple and very effective because it is so simple.
Another concept for teachers is "directed discovery". You get them to "discover" math concepts. It doesn't matter that we all know 3 three's make nine and that nine is square, the student doesn't know this yet. You direct them to discover these facts. You must become masterful at this. It takes a huge amount of patience. You have to stop yourself from telling them things they they are supposed to memorize and start getting them to discover math concepts and facts for themselves.
You aren't going to teach them anything new. The mathematics has developed and been written down over centuries. Concepts have been lost and rediscovered more than once. It has been anything but a linear progression. You have to allow your students to discover a²+b²=c² all by themselves...rather than just make them memorize formula.
Simple discoveries like there are 4 fours in 16 and 16 is square go right along side bigger ones. You do this by letting them PLAY. Don't just tell them these numbers are square: 1, 4, 9, 16 etc, let them build squares. Put out 16 and measure it with twos and fours and eights...and they'll discover eight 2s is the same as two 8's. You will hear "ahhhh" and "ah-ha!"....and other clues that let you know endorphins are going off in their brains because the infinite consciousness has made it so that learning is fun. It feels good.
There is an actual chemical reaction that takes place...very similar though not as powerful as orgasm. No kidding. I'll have links to the science behind brain chemistry on this site soon enough.
Meantime: direct their discovery don't do it for them. Show them. Give examples. Let them reach mastery...and demonstrate that they have done so.
I like to think of it like this. You have a group of students and just for example the subject is Pythagorean theorem, you send them down a road where they discover square numbers, square roots and then triangles and some geometry. And they get over there and come back with a²+b²=c² and then you send them out again and they come back with distance formula...you know where you want them to go they figure out how to get there. They may end up going down some interesting side roads and less tread paths on their way there and back.
Here is a talk regarding directed discovery in the extreme:
"To state a theorem and then to show examples of it is literally to teach backwards." ~E. Kim Nebeuts
Note: directed discovery is not letting the students do whatever they like. The lesson is on whatever it is YOU the teacher are teaching, you know where they are going...you just point them in the direction and let them figure out how to get there.
Remove the "NO" from the lesson.
This is a very difficult thing for some people. Stay positive. You don't want to be negative. This is a little joke because we also stay positive for the first few lessons before we focus on negative numbers or negative expressions in algebra.
But basically that means what it says: refrain from saying NO during a class or lesson or when you are directing their discovery. I mean it. Figure out how to express yourself in the positive. If you think they are getting it "wrong" keep in mind they are just getting more information. If they are building tens and they try the 4 with the 7, let them. Get out of the way. Watch them discover the right answer for themselves. The blocks are "self correcting" because the child can see that 7 + 4 is either longer than 10 if it's out of the trays or doesn't fit in the tray. If you ask them a direct question like show me the six and they show you an eight, say "that's an eight, show me a six." Tell them what they have not what they don't have.
Tell them when they get it right not when they get it "wrong". Keep your lesson positive. You will get into the habit of staying positive if you try. For some kids the hour that they came to see me was the only hour of the day when they didn't hear and adult tell them "no". This is very difficult to start but it gets easier.
At this late date in my math career I still catch my self, usually BEFORE I say it, but not always. Happily, it's a pretty darn rare occasion. Like everything else it just takes practice.
Also in extreme cases you may want to say "STOP" instead of "NO". For example when a child is taking 6 away from 13 and they have already been instructed several times to take the 6 out of the 10 instead of the three and the ten. You want small addition instead of subtraction. Stop them from getting into bad habits but refrain from using the word "no" to do so. "Be good" is so much more effective than "don't be bad".
Stages To Mastery.
Pre-Stage one: ignorance/incompetence. "So stupid they don't know their stupid', as grandma used to say. But seriously: student is unaware of the subject at hand. Example: student doesn't know how to multiply, doesn't care, doesn't know any facts.
Stage one: Unconscious incompetence, makes errors without realizing they are making them, or takes a test and is not sure how well they have done. Example student adds 7 + 8 and says 16, counts on fingers and is unaware they lost their place. Student using calculator gets answer that is absurd puts it down anyway because they got it with a calculator. Doctors can graduate with a "C" never moving past this stage.
Stage two: Conscious incompetence. Becomes aware of mistakes right after or while the mistake is being made. Example student adds 7 + 8 and says 16, but then realizes it's 15...or does multiplication, answers 6 x 4 says 25 but realizes there are no odd numbers on 6's or 4's...recognizes that answer may not look right or may not sound right.
Stage three: conscious competence. Mastery. Gets answers right consistantly but has to think about it, work at it. In this methodology mastery is exhibited not only by getting answers consistantly right but by being able to make problems correctly for others. For example I knew when a learning challenged student of mine had finally reached mastery when he was able to set up a game of what's under the cup for me. Another slightly autistic girl demonstrated mastery by explaining HOW to do subtraction and getting the self talk questions correct and in the right order.
True Mastery: Unconscious competence. Gets answers right without thinking much about it, knows what to do, how to do it, when to do it and why. Can teach others and can teach others to teach others. I am not completely there yet with computation but I am pretty much there with the methodology. I know what to do without thinking about it and can easily explain why when asked. Martial arts call this by various names Japanese sometimes use the term "moshin", without thought.
Place the student in a situation where they can not fail.
Place the student in a situation where they can not fail means just what it says: put the student in a position where they have no option but to get it right. The blocks are self correcting. If you are building addends, use the tray to build tens. If you put a nine block in there it is "visually obvious" that a one is what fits there. The student my try other blocks just for fun but they wont fit. They can not fail to make a ten...same with the other addends.
If you are building squares and doing square roots they can see if the shape they have is square. When factoring or doing division they can see if they have a rectangle or not...
It also means the first two or three times you give an example and then give a very easy problem for them to try by themselves. No challenge, you are finding out if they understood your examples and got the CONCEPT. Redundancy is good to start. The three period lesson is a good example of this.
Often text books will give an example or two and then give practice problems that are several degrees of difficulty above the examples. This is great for guys like me because students need help because they don't understand or are unable to do the work on their own, but it's not so great for the student who gets frustrated to the point where their parents have seek out help: tutoring isn't cheap. Worth it, but not cheap. It's like climbing a ladder where several of the rungs have been left out.
Which ladder would you rather climb?
In this method we always start super easy and STAY super easy, the answers are obvious (visually obvious) and pretty soon 7 year olds are showing high school students how to factor polynomials...build their confidence by putting them in a position where they can not fail.
Increase degree of difficulty gradually.
Once they have mastered a concept, if you ask them if they want a problem that is hard or easy often times they will say HARD. Once in a while they say "medium". Everybody loves a challenge when they are ready for it...and they feel confident. Building confidence and self esteem is a major part of this method.
In Australia there are a people we call aborigines. The Australian government was unable to teach these people math and even classified them as "a-mathematical". They had a number system that went like this: one, two, many. Can you see the problem? Jerry spent about 3 months down there and soon we have pictures of aborigines doing multiplication, aborigines doing fractions, aborigines showing white kids how to do algebra...I need to find the articles and post them here or in the pdf section...and if anybody has video of this let me know. I will edit it and put it up too. I have seen short clips but it's been years. This method works because it teaches math the same way we teach language and we have gotten pretty good at teaching language. In fact, we are starting to accelerate the process using mind maps and other techniques in conjunction with understanding HOW we learn.
I like to study foreign language programs and see how the techniques can be applied to the language I choose to teach: MATH. I am also fascinated with a process/technique called photo reading, which explores the role the subconscious mind plays in learning. Along these lines is the work of the many people who study the subconscious and changing the beliefs rooted there...which leads down some very interesting roads. Soon I will have a library section that reflects the real life library of Crewton Ramone's House of Math. I don't have enough space for all my books. I think I'll start with a "must read" section. With links to the books and reviews etc and then I'll get a nickle every time somebody buys one via my links..
Economy of Symbol.
Use the fewest symbols possible to represent the idea or expression we are describing. I often used to say mathematicians are lazy they don't like to write a lot of symbols. Now I usually say they are "efficient"...
It's easier and faster to use 2⁴ than to write:
2x2x2x2 or 2*2*2*2 or (2)(2)(2)(2).
It very similar to using the fewest pieces possible to "build" a problem with the blocks. Rather than grabbing a bunch of units it faster and easier to grab 4 four blocks for 16. And super easy to see 16 is square...with individual units arranging and even counting the blocks takes TOO long.
Eventually this leads to one of the points I make about mathematics: math is counting but it leads to critical thinking skills, and this involves the economy of thought expressed with symbols. Math IS NOT just computation.
The masters come up with formula that have the fewest symbols possible. They are often described as simple, beautiful and elegant, descriptions that sometimes baffle the layman who think about math in no such terms. The formulas mean nothing if you don't know what the symbols mean; however, and are useless.
CONTROL OF ERROR
Basically you allow them to fail their way to success. Putting them in a situation where they can not fail doesn't mean they get right the first time, it means you expect the student to fail, allow them to fail, allow them to correct their own errors without a "no" in the lesson. They didn't really fail just got more information.
Math Concepts Helps Ensure Success.
So sorry the vid that went along with this has been taken down by it's host Basically it detailed what I've been saying along with Jerry Mortensen and Maria Montessori: The earlier you start the better, as with any language, but he had a PHD so it gave it more credibility than us silly people who have been in the field for 25 years...anyhow here is this:
Here are my five concepts/principles for teaching with manipulatives.
I explain them in more detail on this page:
Math Rich Environment.
Economy of thought/method/motion.
Use manipulatives any kind...
Use Base ten manupulatives or "base ten blocks"
Return home from The Five Basic Concepts.
“Intellectuals solve problems, geniuses prevent them.” ~Albert Einstein
“Example isn't another way to teach, it is the only way to teach” ~Albert Einstein
"Almost all creativity involves purposeful play." ~Abraham Maslow American psychologist 1908-1970
"Whoever wants to understand much must play much." ~Gottfried Benn German physician 1886-1956
"Play gives children a chance to practice what they are learning." ~Fred Rogers American television personality 1928-2003
"People tend to forget that play is serious." ~David Hockney Contemporary British painter
"Do not…keep children to their studies by compulsion but by play." ~Plato Greek philosopher 427-347 BCE