Number Pyramid Puzzle

I just found this on Andrew Jeffery's Email and thought it was worth sharing.

A Number Pyramid

The Number in the box is the product of the two numbers below it.

Write in the missing numbers.

Taken from Y6 UK KS1 and KS2 Assessment materials

The Problem with Math Education-Lack of Curiosity??

From Math Mojo Chronicles 3^rd Feb 2009

Well, of course it’s not the problem, just one of many, but here it goes…

Somehow along the way, people got the feeling that math is supposed to always be right, and that math teachers are supposed to know all the answers.

Math has gotten the reputation of being an authoritarian science. I don’t think this is the fault of mathematicians, I think it is the fault of many math educators who have tried to turn mathematics from an art and science into a “subject.”

Math education is all too often about “standards” and “curricula” that students take “tests” about that they are “graded” on.

If Archimedes was to take a high-school math test today, he would be unfamiliar with the jargon, and would find little value in the trite little multiple-choice and partial credit nonsense that passes for assessment.

On the other hand, he could run rings around the math teachers with his knowledge of actual mathematics, and could twist their pedagogical dogma into moebius bands.

Trying to shove math into the education industry’s rubric is one of the worst educational crimes I can think of. School math seldom has anything to do with actual math, except for the very rare cases where an inspired person is doing the teaching. And when that happens, that person is invariably in trouble with the administration.

How do you know if your school is doing a really good job?

Easy - your school definitely is not doing a really good job. Even if the school representatives like to trot out some meaningless statistics saying how well their students score on this or that test.

The tests show nothing whatsoever about the essence of math. They are created by the same industry flunkies who sell the schools the textbooks and other curriculum material. It is the foxes guarding the henhouses.

Show me a school that can compete against the Russians and Chinese, and you might have a case. But you can’t .

The Germans have a great quote, which you will find yourself quoting frequently if you understand it. It’s - “Never trust any statistics that you haven’t falsified yourself.”

In other words, the education industry can manipulate their statistics to seem to “prove” anything it wants.

If you truly understand mathematics, you can understand why those statistics are worthless.

So what does matter in math education?

Here’s one thing that matters - Realize that education is not inculcation. It’s not about getting children to jump through hoops that somebody in the state capital thinks are important this week. It’s more about turning the mind on to enjoying and investigating a subject, and fostering a thirst for learning.

A Chinese acquaintance of mine said, “Curiosity is the best teacher.” I love that quote.

If you can get a child to be curious about a subject, all you then have to do is point him or her to the library where they can find out more about the subject, and they will learn more than all the textbooks in school closets have to offer.

They will also get more diverse input than from a “curriculum.” Curriculums, as far as I can see, are designed as much to limit knowledge as to teach. Maybe more.

School gives children the false sense that math is carved in stone, and all the answers are there, you just have to know the formulae. It teaches them how to answer questions that someone else has posed. It doesn’t teach them how to ask questions.

Math is about asking questions - asking how can you make and understand patterns of the things in your world. When was the last time you ever even heard about a teacher mentioning that in a classroom?

I’ll admit, it occasionally happens, but that is in spite of curricula, not because of them.

++++++

Curiosity ? ?                                     (Len Cooper)

It is suggested that there are two events that increase the activity of the

synapses in the brain.

One is:            Fear - Fight or flight 

The other:       Curiosity  
  

How much curiosity are we encouraging in our maths classrooms?
 

Questions to Nurture Student’s Curiosity

• Can you show me what you did?

• Where did you start?

• How did you decide that?

• Can you try that another way?

• What happens if you take part of it away?

• What would happen if. . .?

• Can you write about or tell me what you did first? second? third?

• What will you do next?

• Why do you think that happened?

• How would you describe what you saw (read)?

• Can you think of another way to do this?

Getting Your Math Skills Down Cold

From Math Mojo Chronicles 3rd Feb 2009

Getting your Math Skills Down Cold

There is a real problem in our society about what people think they know, and what they know.

What I mean is, so often I hear from students, and adults, that they, “know” the “math facts.”

Well, aside from the term “math facts” being counterproductive, most people can retrieve, say, basic multiplications from their memory, but they are not in their bones.

To illustrate, try this:

Relax, get in tune with yourself. Breathe easily. Start noticing your thinking.

Now, what is 7 x 8?

If you had to think, “Um, let’s see…”, or if you hesitated at all, you do know really “know” the answer. You can find it in your memory, but really knowing it is like knowing that “cat” spells “cat.” You don’t have to say, “um, let’s see, ‘c’ - ‘a’ -’t;’ that’s ‘cat!’” (At least I hope you don’t.)

If you don’t know your basic multiplications by second or third grade, someone has cheated you. Every seven-year old should be able to really master them. It’s just that our society makes wussie excuses, like, “Well, he has ADD, you know…”

Yeah, maybe he does. Glenn Cunningham had his legs burned so severely when he was eight years old, that the doctors suggested he have them amputated. His mother wouldn’t allow it.

Do you know who Glenn Cunningham was? He set a world record for the mile and indoor world records for the 1.500 meters and the mile. He was on the 1932 and 1936 Olympic teams .He was considered by many the greatest American miler of all time.

Did his mother use his burns as an excuse? Yes, she did. She used it as an excuse to help him excel against something much more horrific than ADD.

I am not trying to make it sound like ADD isn’t awful. I have it and it drives me nuts. But I’d really feel like a wuss if I said, “I can’t do this or that because I have ADD.”

I might say, “I can’t do this or that yet, because I haven’t worked hard enough to overcome my ADD to accomplish it, but I’m not giving up.”

That’s still pretty wussified, but at least it’s not blaming something outside myself.

There are some things that I have tried that have really helped, and recently I have started working on some of them again.

If you have ADD, you know that there are some things you don’t have symptoms about. I’ll bet you can surf the web for hours without interruption, or even food or drink.

You can get into that “trance” state when you play computer games, possibly. Many people with ADD can get into the “zone” when they do art, music, or sports. My wife wonders why I have a hard time getting my work at the desk done, yet can spend six hours in a row solving Sudoku puzzles. (I wonder, too. It’s not like my work isn’t interesting.)

So I’ve been trying some experiments to combat my short attention span. Lately I’ve been going to the gym and using the stationary bike and the elliptical machine - but with a twist. I bring something to learn or memorize on a small clip-board.

Let’s say I am memorizing a poem. It’s hard enough to do when you are in a quiet space and not distracted. But when your heart-rate is over 120 for more than a half-hour, and you’re sweating (and your a big, fat, out-of-shape middle-aged guy like me) it is really hard to concentrate.

It definitely takes me more time to memorize something when I am physically straining at something else. But you know what? I retain the stuff much better.

Later, when I am relaxed and trying to remember what I learned, it comes easier. I can’t explain why in scientific terms, but it seems like my body is saying, “If I can do it while panting for my breath, I can certainly do it now!”

It seems to work on the same principle as the Biathalon. Do you know what that is? It is one weird sport. It’s cross-country skiing combined with target shooting.

A guy (or girl) sprints long distances on cross-country skiis (that is a hell of a workout, let me tell you!) then has to shoot at a target with a very accurate rifle. Most people’s body would be shaking with exertion to the point of fainting, but these guys have to steady their hands and nerves, and shoot at a distant target. And they’re being timed! Now that’s pressure.

When I was a kid, I used to laugh at that sport. It seems so silly. Cross country ski and shoot? Why not sumo-wrestle and play ping-pong? But it really makes sense, and tests skills that other sports combinations cannot test as accurately.

So what am I getting at? You sit and home and look at multiplication tables, maybe look at flash-cards, or (God forbid) use some vacuous songs, rhymes or video games to memorize your “math facts.” Maybe you have the TV on at the same time, or are texting your friends.

What you are not doing is challenging yourself. It’s just so passive. You’re not engaged. You just have some passive distractions.

On the other hand, if you set a timer, and say, “I’m going to push myself for the next twenty minutes, ” and really exert some effort, you will be making an investment in yourself. And your subconscious will not let yourself be cheated out of your investment.

And if you can manage to do the same thing again the next day, but this time while taking a walk for 20 minutes, you’ll find that you can’t concentrate as well. But try it for another day, and another, until you can concentrate as well while walking as when you were in your room, then you have really made some progress.

The next time after that, when you try to study in your room, you will feel more empowered, and will not be as tempted to be distracted by the radio, your iPod, or that idiot that is texting you about some stupid YouTube vid.

Try it.

Questions

Anyone who has had their own children or taught young children will know that they go around asking questions.  The one most used of course is "Why?"

With my 3 year old grandchild, questions are a constant, often to the extent that her parents and grandparents get totally frustrated with them all.  BUT we know this is her way of learning and exploring the world she lives in.

Children NEVER lose the need to know why and question, but, often schooling does drum it out of them and the higher the level of education, we often see less chance of a learner asking questions.  

Questions are often replaced by what the Teachers says, or what will be in the exam/test.

If children learn through asking questions, why do we in Formal Education keep trying to prevent them?

Peter Duncan a retired New Zealand Educator wrote the following poem which has been published in a book of Poems (all written by Peter) "From The Space Between"

It really encapsulates the Education Systems and their views of Questions.

Questions

Fresh-faced
Smiling
Eager
Five-year old enters school
with a thousand questions

In year four
the child's questions become 
less valued diversions
from the standard's curriculum

By year nine
no time to ask
no time to answer
or you will never get through
what has to be covered

At university
two questions remain

What do I need to get an A pass?
What's the minimum attendance required in this class?

The nine hundred and ninety eight other questions are
no longer relevant.

(posted with permission)

Some of Justin'sDiscoveries through using Maths Games in Teaching

NOTE: Justin is encouraging teachers to purchase his games, that is for individuals to decide. I concur with his philosophy and findings. Len Cooper
When I first started to work on the use of math games in the classroom, I was amazed at what I began to see happening! Here are a few of my discoveries about games where children can learn and practice math:
Many of the games lead students to talk mathematics.
Games forced students to justify their reasoning.
Games put pressure on players to work mentally.
Games did not define the way in which a problem had to be solved or worked out.
Students began to explore and learn new strategies by working and talking with each other as they played.
A game could often be played at more than one level allowing the teacher to differentiate instruction.

Games and Assessment
Teachers who observe and interact with children while they are playing math games can diagnose a wide variety of their mathematical strengths and weaknesses. In assessing learning through math games, teachers' concerns are not just confined to the children's levels of factual knowledge. Rather, they may also note, record, and analyze the following:
reasoning and problem-solving skills,
the forms of children's responses,
the processes that children employ in solving problems and arriving at answers,
children's patterns of persistence and curiosity, and
their ability to work with peers, adults, and a variety of resources.

In addition, the recording sheets that children produce while playing games can be placed in assessment portfolios, where they can be of great value to children, teachers, and parents.

Finally, games provide children with a powerful way of assessing their own mathematical abilities. The immediate feedback children receive from their peers while playing games can help them evaluate their mathematical concepts and algorithms and revise inefficient, inadequate, or erroneous ones.

Good games evaluate children's progress. They provide feedback so that teachers, parents, and the child know what they have done well and what they need to practice.

Calculators
Calculators can be quite helpful for settling questions about answers, executing complex calculations, or keeping track of players' cumulative scores. Use your judgment as to whether calculators will speed up or defeat the purpose of the game.

Recording Sheets
Many of the games include recording sheets. Recording the problems solved while playing a math game can leave a mathematical trail that is of great value to children, teachers, and parents. Children can feel a sense of accomplishment as they look back at all of the math work they have done; teachers can use the records for assessment; and parents will appreciate this "evidence" that their children are actually doing mathematics and not just playing games.

Communication
Many people think that a quiet room is one in which learning is taking place. I strongly disagree with that tenet. When children are playing games, they need to be able to talk with each other. This talk can be very constructive if children take responsibility to make sure that all players in a game understand the algorithms, concepts, and facts being used within the game. Sharing strategies with each other helps everyone see different ways to play.

The bottom line is: Teach each other and learn from each other.

Competitive Versus Noncompetitive Games
Most of the 42 File Folder Math Games have been designed as competitive games where the high scorer wins. All can be transformed into games where the high scorer is not the winner or into noncompetitive games.

For example: Children can roll a die. If the number rolled is an even number, the player with the highest number or score wins the game. If an odd number is rolled, the player with the least number or score is the winner.

Many of the games can be played in such a way that players keep track of their own individual scores over a period of days and try to better their previous day's scores. Children can enjoy keeping graphs of this information themselves.


Math Should Be Fun!
Justin
MathFileFolderGames.com

Basic Facts Learning

A hardy annual is the lament from teachers and parents alike, that the kids do not know their tables. As a shopper I am often perplexed at how the shop assistant has to use a device to find the cost of 2 items at $1.99, instead of 2 @ $2 minus 2 cents.
As a consultant I encouraged teachers to help their students see connections between the various facts rather than  learning/memorising 100's of isolated facts: When learning 6 + 3, why are we not encouraging them see 3 + 6 as well as the reverses 9 - 3 and 9 - 6.  We usually call these a Family of Facts.  This should mean that learning 6 + 3 gives 3 other facts at the same time.
I also encouraged looking for patterns in the tables; doubles, fives, Tens, and then ideas such as +9 is like adding 10 and taking away 1.   Similar ideas for Multiplication Family of facts can also be applied.
Once explored chunk the learning down to small pieces! I have never come across a student who can not learn 6 x 9, 9 x 6, 54 ÷ 9, 54 ÷ 6 overnight, provided there is support from home.

Justin Halliday also suggests the use of maths games which require mental use of facts is also a great way to assist memorisation:
All too often I hear teachers lament that their students don't know their basic math facts. In class, students guess, freeze up, count with their fingers, or appeal to their friends or the teacher to help them out.

Why is it so important for children to memorize math facts in order to succeed academically? Quite simply, a lack of fluency in basic math fact recall significantly hinders a child's subsequent progress with problem-solving, algebra, and higher-order math concepts.

This can have a serious impact on a child's overall self-confidence and general academic performance.

The guidelines of the National Council of Teachers of Mathematics (NCTM) state that second graders should be able to quickly recall basic addition and subtraction facts, and fourth graders must have quick recall of multiplication and division facts.

Everyone agrees that students need to learn the basic facts, but there's far less agreement among educators about how this can best be accomplished. Many drill and practice programs have been developed to help kids memorize the basic combinations by rote. The theory is that if children hear or practice 9 plus 7 equals 16 repeatedly, they'll eventually just remember it. Doing this with worksheets or flash cards can be boring and student engagement is low.

Almost every elementary teacher struggles to find effective ways to encourage students to master these basic math facts. I have found that math games meet the varied needs of learners, offer opportunities to differentiate instruction, and are effective, motivational, and engaging.

Whether you're a new teacher, a teacher new to teaching math at a different grade level, or a veteran teacher looking for a fresh perspective, I would encourage you to give math games a try. Games engage children and enhance their math learning.

The following game is a great one for helping third, fourth, and fifth graders learn their basic multiplication facts.

Multiplication Fact Feud
What you need: deck of cards
2 players

Teacher decides the particular multiplication fact to practice (i.e. x7, x4, x8, etc.)

Once the constant factor is determined, that card is placed between the two players. Players then divide the remaining cards evenly between themselves.

Each player turns over one card and multiplies that card by the constant in the middle. Players must verbalize their math sentence. The player with the highest product collects both cards.

Example: 5 is the constant
Player #1 turns over a 4 and says "4 times 5 equals 20″.
Player #2 turns over a 7 and says "7 times 5 equals 35″.
Player #2 would collect both cards.

In the event of a tie (i.e. both players have the same product), each player turns over one more card and multiplies that by the constant factor. The player with the highest product wins all four cards.

When the cards are all used up, the player with the most cards wins the game.

Maths Games Develop Strategic Thinking

I like Justin's thinking about the teaching and learning of Maths (or Math, as he is Canadian) it would be great if we incorporated more games into our lessons.  Perhaps we should be teaching and assigning 'Games' as homework, instead of the usual repetitive uninteresting practice that is usually assigned!!!
My youngest, now 30, started as a 3 year old learning colours, numbers etc while playing the card game "UNO" He wanted to play with the rest of the family, so his cards were laid on the table and he was assisted to choose "the same colour" or a "matching number' etc.  Many times he could win the game because of the input he was being given.

Justin's latest email post.
Strategic thinking is one of the most important skills for children to develop. It requires the ability to observe, take in different pieces of information, analyze information, plan and analyze possible solutions, and choose the appropriate action.

Strategic thinking is a way to solve problems. Every day we have to solve the problems. Every day, we need solutions. Problem solving is an essential skill in our professional, family, and social lives.

Games like bridge, chess, and backgammon are ideal for teaching strategic thinking. But learning bridge is more than fun and games; students who play, practice math and reasoning skills and show improvements on standardized tests.

However, games such as bridge have complex rules that can take time to learn and master. Instead of using complicated games, there are many math games at every grade level that are much easier for children to learn and play. All of the math games are focused on providing engaging activities to entertain strategic mathematical thinking both inside and outside of the classroom.

If you are a teacher or parent, I encourage you to have a look at the assortment of math games. You will find many that will pique your interest and help you develop strategic thinking and problem solving abilities in your students/children while having fun!

Math Should Be Fun!
Justin
http://www.mathfilefoldergames.com/42-math-games

Yes Justin does create Apps for Smart Phones etc, but it is not always necessary to use the modern technology for games.  Consider:  Battleships, Mancala, all card games, Monopoly, Draughts...