All Easy Math Problems Are Alike…

Even if you’ve never read Anna Karenina, you’ve likely heard Leo Tolstoy’s famous assertion, “All happy families are alike; each unhappy family is unhappy in its own way.” My much less controversial take has nothing to do with complex social dynamics but rather with a far simpler subject: math.

All easy math problems are alike; each difficult math problem is difficult in its own way.

If we consider math problems in the context of learners with the requisite instruction and experience to even have a chance to successfully solve, the easy math problems are the ones most of them solve correctly. All problems in this category share obvious traits:

• Easy math problems test easily understood and remembered content.
• Easy math problems have obvious solution methods.
• Easy math problems may require few steps to solve completely.
• Easy math problems may require little time to solve completely.
• Easy math problems primarily require recall and reproduction, the lowest level of depth of knowledge.
• Easy math problems are presented in a straightforward fashion with few or no traps.
• Easy math problems strongly resemble previously seen problems.

Easy math problems have a place both in the classroom and on even the most challenging standardized exams. However, easy alone doesn’t build muscle, character, skill, or success. Easy problems prepare test takers for more easy problems but do little to foster the ability to tackle the toughest problems. We harness the testing effect by engaging in deliberate practice—pushing the boundaries of performance by taking on greater challenges until what was once difficult becomes easy.

Of course, that means that students seeking top scores on the math sections of tests like the SAT & ACT will have to correctly answer more than just the easy and average difficulty questions. In the testing journey, everyone eventually learns that difficult math problems may be common but they are hardly alike. In fact, novelty is one of many features that make a problem difficult:

• Difficult math problems may test esoteric or confusing content.
• Difficult math problems may test easy content in esoteric or confusing ways.
• Difficult math problems may have no obvious solution methods.
• Difficult math problems may require a lot of time (maybe too much) to solve completely.
• Difficult math problems may require multiple steps to solve completely.
• Difficult math problems may require higher level DOK skills and a greater cognitive load.
• Difficult math problems may be presented in an inscrutable or easily misunderstood fashion with many tempting traps.
• Difficult math problems may seem entirely unique, with little resemblance to previous problems.

Of course, returning to Tolstoy, the easiest and most difficult math problems are still related in a family of sorts. The solution to any math problem requires a specific set of facts, skills, and strategies that are usually applicable to a host of other problems. Learners should climb the ladder of mastery by tackling problems at or slightly above their level of proficiency until what was difficult becomes easy, which is the sign that they are ready for the next level of challenge. (This is why we recommend Mathchops for so many of our students.)

But the hardest math questions possess traits that set them apart from the rest–the black sheep of the math problem family, if you will. On standardized tests, these items are tautologically the ones most test takers get wrong. Give yourself an edge by recognizing that each of these problems are as different from each other as they are from easy and average problems:

1. Even though every difficult problem may be different, they all become easier when you bring the same effective problem solving approach to them. Have a proven strategy for math problem solving and stick to it every time.
2. Give yourself plenty of time to unpack a difficult math problem. I’ve seen really strong students require five or more minutes to solve some of the toughest SAT math problems. Sometimes, just figuring out what the question requires is the time waster, while, in other cases, a complex sequence of algebraic operations cannot be rushed.
3. Think about other difficult math problems you’ve solved. Experience unlocks a host of heuristics–solution strategies and mental shortcuts that expedite problem solving.
4. Difficulty is relative. Just because the problem in front of you feels impossible doesn’t mean the next one will be even harder. Don’t hesitate to move off of a really challenging problem; take a guess and come back if time permits. Sometimes a little perspective makes a world of difference.

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