Often, when people mention difficult math, they are referring to advanced math. I would agree that there is some correlation between those two things, but they are not the same. It is possible to find examples of relatively easy math problems that rely on relatively difficult concepts, and vice versa. For example, the current New York State Integrated Algebra Regents Exam is easier than a list of the topics that it covers would suggest, because of the way that many of the multiple choice questions are structured (more on this topic in a future blog posting). On the other hand, New York Academics’ Very Challenging Math Worksheets (available at shop.tutornewyorkcity.com) are much harder than you would probably expect if you only knew that they are a collection of problems involving fractions, decimals, proportion, simple number theory and similar topics.
Doing math is more than simply learning and applying a set of algorithms. It can also involve creative thinking, estimating, organization, and persistence. A fairly simple example is this word problem: A dog has six puppies. Each dog (young or old) has 15 fleas. Each flea has six legs. How many legs are there? (*The answer is below). In theory, everyone who can multiply and add whole numbers should be able to easily solve this problem. Nevertheless, many students who are competent in the required arithmetic still get this problem wrong because they lose track of where they are in the middle, or they forget to include the mother dog, both of which are examples of how organizational skills are important in math. Sometimes less confident students panic when they see a problem like this one and resist making any attempt to solve it (an illustration of how important persistence is).
The problems included in the Very Challenging Math Worksheets are more difficult (some of them are much more difficult) than the above example. Nevertheless, the principle remains the same. For example, the next problem is a moderately difficult problem that is found on one of the Very Challenging Math Worksheets. (**The answer is below.)
The Acme Doodad Factory employs 500 people. 80% of them are factory workers who make $55,000 per year. 12% are office support staff who make $45,000 per year. % are management who make $95,000 per year. Everyone else makes $250,000 per year. What is the mean salary at Acme Doodad Factory?
Finding the solution to this problem requires an understanding of percents and mean (specifically, weighted mean). It also requires that students have the ability to stay organized and calm through a problem that has many steps and some unattractive numbers (i.e., a fractional percent). While all of these topics are theoretically covered in standard middle school math curriculums, only a relatively small number of middle school students are able to correctly solve this problem the first time they see it. (To be fair to middle schoolers, many high school students and adults aren’t much better off.)
In my experience, the main things most kids need to solve challenging, multi-step problems like this one is more guided practice. Mathematically gifted students are more likely than other children to be given problems like this, and I certainly approve of these kids being challenged, but average students also get very real benefits from exposure to challenging, non-routine math problems, too (as long as they get enough support and encouragement along the way!) Doing this type of math problem helps develop the ability to think in a logical but flexible way and organize information. It also helps to build the persistence necessary to work through situations where the path to a solution might not be immediately obvious. As a bonus, the ability to do these kinds of problems can help children get into gifted and talented programs (such as Hunter College High School, the Anderson Program, Specialized Public High Schools, and the Center for Talented Youth at Johns Hopkins) and can help adults have a wide range of career choices open to them. In my experience, the main things most kids need to solve challenging, multi-step problems like this one is more guided practice. Mathematically gifted students are more likely than other children to be given problems like this, and I certainly approve of these kids being challenged, but average students also get very real benefits from exposure to challenging, non-routine math problems, too (as long as they get enough support and encouragement along the way!) Doing this type of math problem helps develop the ability to think in a logical but flexible way and organize information. It also helps to build the persistence necessary to work through situations where the path to a solution might not be immediately obvious. As a bonus, the ability to do these kinds of problems can help children get into gifted and talented programs (such as Hunter College High School, the Anderson Program, Specialized Public High Schools, and the Center for Talented Youth at Johns Hopkins) and can help adults have a wide range of career choices open to them.
Doing math is more than simply learning and applying a set of algorithms. It can also involve creative thinking, estimating, organization, and persistence. A fairly simple example is this word problem: A dog has six puppies. Each dog (young or old) has 15 fleas. Each flea has six legs. How many legs are there? (*The answer is below). In theory, everyone who can multiply and add whole numbers should be able to easily solve this problem. Nevertheless, many students who are competent in the required arithmetic still get this problem wrong because they lose track of where they are in the middle, or they forget to include the mother dog, both of which are examples of how organizational skills are important in math. Sometimes less confident students panic when they see a problem like this one and resist making any attempt to solve it (an illustration of how important persistence is).
The problems included in the Very Challenging Math Worksheets are more difficult (some of them are much more difficult) than the above example. Nevertheless, the principle remains the same. For example, the next problem is a moderately difficult problem that is found on one of the Very Challenging Math Worksheets. (**The answer is below.)
The Acme Doodad Factory employs 500 people. 80% of them are factory workers who make $55,000 per year. 12% are office support staff who make $45,000 per year. % are management who make $95,000 per year. Everyone else makes $250,000 per year. What is the mean salary at Acme Doodad Factory?
Finding the solution to this problem requires an understanding of percents and mean (specifically, weighted mean). It also requires that students have the ability to stay organized and calm through a problem that has many steps and some unattractive numbers (i.e., a fractional percent). While all of these topics are theoretically covered in standard middle school math curriculums, only a relatively small number of middle school students are able to correctly solve this problem the first time they see it. (To be fair to middle schoolers, many high school students and adults aren’t much better off.)
In my experience, the main things most kids need to solve challenging, multi-step problems like this one is more guided practice. Mathematically gifted students are more likely than other children to be given problems like this, and I certainly approve of these kids being challenged, but average students also get very real benefits from exposure to challenging, non-routine math problems, too (as long as they get enough support and encouragement along the way!) Doing this type of math problem helps develop the ability to think in a logical but flexible way and organize information. It also helps to build the persistence necessary to work through situations where the path to a solution might not be immediately obvious. As a bonus, the ability to do these kinds of problems can help children get into gifted and talented programs (such as Hunter College High School, the Anderson Program, Specialized Public High Schools, and the Center for Talented Youth at Johns Hopkins) and can help adults have a wide range of career choices open to them. In my experience, the main things most kids need to solve challenging, multi-step problems like this one is more guided practice. Mathematically gifted students are more likely than other children to be given problems like this, and I certainly approve of these kids being challenged, but average students also get very real benefits from exposure to challenging, non-routine math problems, too (as long as they get enough support and encouragement along the way!) Doing this type of math problem helps develop the ability to think in a logical but flexible way and organize information. It also helps to build the persistence necessary to work through situations where the path to a solution might not be immediately obvious. As a bonus, the ability to do these kinds of problems can help children get into gifted and talented programs (such as Hunter College High School, the Anderson Program, Specialized Public High Schools, and the Center for Talented Youth at Johns Hopkins) and can help adults have a wide range of career choices open to them.
