There is another pause.
Contributions begin slowly, but gain momentum and volume as the exercise continues, several people talking at once.
“More slimy than juicy.”
“Sort of tastes like a peach.”
“But, with an orange sort of flavor.”
“Kind of stringy.”
“So, Maria, now do you understand about mangoes?” the speaker asks.
“Sort of, not really,” she replies.
The group is invited to try again.
“Smells a bit like socks.”
Maria is obviously trying hard to understand but doesn’t look convinced.
What is Maria’s problem?
There is a general consensus from the group that Maria has to actually experience a mango to really understand it; she needs to smell it, peel it and taste it before she can develop a real “sense” of mangoes.
“What about skating?” asks the speaker.After a brief exchange about the sub-set of skills and ideas that make up an understanding of skating, the speaker continues.
“Could you learn to skate if I explain it to you? What if I show you how?”
“Not really, but I could probably repeat your instructions,”replies one of the teachers.
“So then, what are you telling me about learning to skate?”
“You have to put the skates on and feel what it is like to skate.”
“You have to experience skating?”
The speaker continues,“How do you learn to paint or draw?
What about learning a new language or playing a musical instrument? Building a house?
For each scenario, the group agrees that experiencing the idea is critical to “knowing” it.
Consider a time when you have personally experienced a breakthrough with something you were trying to learn, a time when the lights suddenly came on and you felt the rush of clarity and associated excitement that comes with discovery.
What made that learning experience different?
Chances are, you were interested and engaged in an activity that allowed you to have direct personal contact with the idea; you were experiencing it.
This exquisitely simple principle holds true in just about every type of learning situation — even with numbers — especially with numbers.The straightest path to intuitive understanding in math is through engaging investigations that allow any learner the opportunity to come into direct personal contact with the idea, discovering for them selves the mechanics that make it behave the way it does.
In other words, you can’t get wet from the word “water.” Intuitive understanding comes with direct personal contact.
The more complex the idea, the more important this becomes.
What does personal contact with math look like?
Doubt. It all starts there — wondering why, wondering how, imagining what if. In fact, it is absolutely the only place where real learning can happen. Without doubt in the learner’s mind, nothing happens other than the rote storage of sounds or symbols in the brain, to be retrieved at a later date and applied to a similar situation. But what happens when the situation changes, even though the same idea applies? Typically, not much happens.
Suppose that same learner was given the opportunity to wonder about (doubt) something, then investigate to discover the idea him/herself? Now you have a “knowing” that is full of meaning and capable of adjusting to new situations.
A rich math investigation will create doubt in the learner’s mind, uncover a variety of possibilities and create the need to record thinking, thereby breathing life into the symbols we teach.
What might this look like in your classroom?
Let’s imagine that a group of students have been challenged to investigate and explain the pattern in the following numbers.
4x3 = 2x6
6x5 = 3x10
8x3 = 4x6
7x6 = 14x3
5x3 = ?
From previous investigations, they already know that multiplication can be seen as area — rows of square tiles.
There is a swell of activity as groups use their tiles to build the first model.
Now the question becomes,“How can we transform this 4 x 3 rectangle into a 2 x 6 rectangle?”
Several methods for rearranging the rectangle are generated.Although all are correct, one stands out as most efficient.
“Just take half of the rows and add them on here.”
“That gives us half as many rows,
with twice as many in each row, 2 x 6!”
“Does half/double work with all of them? Let’s find out!”
The next hour is spent testing each of the equations presented here as well as others generated by the students themselves. By the end of this investigation, students have generated a few ‘rules’ they’ve found to hold true:
- Half/double works if the first number is even (half as many rows with twice as many in each row).
- When the second number is even, the opposite works... double/half (twice as many rows with half as many in each row).
Well, that’s nice, but how does it really help?
3. If you are using these tiles, one of the factors has to be even or it doesn’t work.
Good question! Anywhere you find understanding (as opposed to rote memorization), you will also find the ability to transfer those ideas to new situations... and that, my friends, is where the real power resides.
When learners from our investigative classroom encounter situations with more complex numbers they are far more likely to transfer what they already know to the new situation, often solving the problem mentally and always with greater confidence.
8x150 or 4x300 Which makes your brain sweat less?
And the moral of the story is…let them eat mangoes!
Exercise: Without using a pencil or calculator, what can you do with 160 x 300? What else?