Math is Everywhere
The goal is to get children to puzzle over mathematical problems as much as possible -- both at home and at school. They will create many of their own challenges if you encourage it. Read the picture book The Math Curse, which talks about how a person who gets involved in math sees it everywhere in her life and can't get it out of her head. Children who do well in math typically have the "math curse", but unlike the book's protagonist, they enjoy having it!Real Math Problems at School
Find the math in your classroom and school, and have children work on the real mathematical problems that arise. Example 1: If the class is raising money, ask: How much have we raised? How much do we have to go? How much have we raised relative to other classrooms, on a "per student" basis? Example 2: If the class has a limited amount of money to spend on supplies, and knows how much things cost, ask: Will we have enough to get through the month? the year? How do you know?Real Math Problems at Home
Have children work on math problems from their lives outside of school. Example 1: We're saving Tropicana box tops at home; each is worth 5 points. The catalog shows many items and how many points you need for each. We drink Tropicana at a certain rate. How many do we need -- and how long do we have to save -- to get a t-shirt? a radio? a bike? Example 2: Children doing sports have lots of mathematics to work on -- everything from win-loss records to hitting rates, to total and average scores. Ask: What do you need to make your goals?Construction Problems
For many number problems where an answer is known, students can come up with a range of different numbers that construct the answer. Example 1: There are four people in a certain family, and their ages add up to 101. What could their ages be? We know that the mother is 37. Now what could the other ages be? Example 2: Children at our school read 5439 hours in October (altogether). How do you think this was split up among the 20 classes in our school? We know our class read 475 hours. How much could the other classes have read?Conjecture and Proof
Whenever anyone notices a pattern, or speculates about a pattern, ask them to provide evidence that the conjecture holds by finding enough instances that work so that they and others are convinced. Or they can disprove the conjecture by finding an instance that doesn't work. Ask other children to convince themselves of these conjectures, or disprove them. For example, a child notices that two even numbers add up to even numbers, and speculates that two odds also add up to an even. Is this true? How do you know? Can you find any examples to disprove it? What happens when you add an odd and an even?
Be careful about using the words "prove" or "proof" with students who are trying to convince themselves that a conjecture is true. Mathematical proof means showing a conjecture holds for all possible cases. Showing thousands of examples that two odds add up to an even may be convincing, but it's not a proof. A proof would explain why two odds must add up to an even for any two odd numbers. Disproving a conjecture is usually an easier process. Finding just one counterexample is enough to disprove a conjecture, because this shows that it does not work for all cases.
Whenever possible, try to get students involved in proving conjectures, not just providing evidence. Proof is what real math is all about.
Natural Extensions
A first grade class is working on the problem "How many eyes in our classroom?" Some children realize that the number is going to be the same as the number of feet in the classroom, the number of hands in the classroom, etc. Someone points out that you'll get a different number if you figure out how many fingers in our classroom. What about how many fingers and toes? You can really extend these problems a long way -- how many fingers and toes in both first grade classes? In the whole school?Comparing Strategies
Whenever two children have different ways of arriving at the same answer, the question comes up, "So, how come both ways work out to the same answer?" Showing the underlying similarities of two seemingly different approaches to a problem helps children develop flexibility.An example: A fifth grader uses a long division algorithm to solve 198/ 17. A second child does 17 X 10 = 170, 17 X 1 = 17, 170 + 17 = 187 so the answer is 11 remainder 11. How are these two strategies similar? (They have a lot in common.)
Explaining "Tricks"
A rule of thumb for all Investigations classes is that you can use any strategy that works as long as you can explain how it works. Some of the special "tricks" used in number operations work for interesting reasons. For example: When you divide one fraction by another, why can you invert and multiply? (This problem can take days or weeks to work out, even for adults.)Inventing Variations and Extensions for Games
Whenever you use mathematical games in your class, encourage children to think of variations in the strategy or scoring or rules -- or even in the materials. For example, how does it change "Close to 100" if you add more wild cards? If you add more of certain digits? Are your chances better or worse of getting close to 100? Why did the authors ask you to use 6 cards to create two-digit numbers? Do your odds get better if you use 7 or 8 cards? What happens if you change the rules so you can never use zeros (or another digit?).
Many children will love to create their own number games. This leads into a lot of thinking about what are the characteristics of a good mathematical game. How do you make it have the right combination of skill and luck?
Math and the Bigger World
If you get any news publications on a regular basis (Scholastic, Boston Globe or other newspaper), or if your class subscribes to a magazine like Zillions, there are hundreds of mathematical problems lurking in these texts. Data problems abound in any consumer publication. For example if the "average" rating given by 5 raters was a 9, what scores could each of the raters have given? (What does average mean in this case?) If the cost of plane tickets is going up 5% and the price of a plane ticket is now $350, how much will the new price be?Some other advice
- Some students who are especially talented mathematically have trouble explaining their strategies orally or in writing. Encourage these students in their struggle to explain their thinking, and don't let them get away with saying, "But I just know it." Mathematicians must be able to communicate clearly, as well as have elegant proofs.
- Turn questions back to children. If one child asks a lot of mathematical questions, use that kid as a source of problems for others. "Does Anthony's conjecture make sense? Can you prove it?"
- Provide appropriate audiences for these children: Sometimes, this means explaining their strategies to other adults (like the principal) or writing about them for the school newsletter, or sharing them with children on the Internet. Get children interested in recording and publishing their work for peers and posterity.
Jan Mokros, TERC
This information was reprinted with permission of CESAME, Northeastern Univ., and the Educational Alliance, Brown University.
