The Backward-Forward Method

Unfortunately, little or no time is spent in most math classrooms discussing "heuristics," the art of problem solving. 

This often leaves students grasping at straws, struggling even to know where to begin when staring down an unfriendly, unfamiliar math question.

George Pólya's How to Solve It is a classic on this subject, required reading for all serious math students.

In another classic, How to Read and Do Proofs, author Daniel Solow advances a powerful problem solving approach he calls the “Forward-Backward Method.” 

I’ve found it helpful in my own teaching and mathematical work to reverse the method, first thinking backward from the ultimate goal to various subgoals which, if achieved, would enable direct progress to the original objective.

Whether writing complex proofs or tackling simple algebra problems, this “Backward-Forward” process provides students with a simple yet powerful structure for solving problems.

Just as an archer would prefer to move the target closer, so can a math student make a problem easier by finding a nearer target to shoot at. The next step would be to think further backward, to find another even closer target tied directly to the first, and so on. These subgoals are set by repeatedly asking the same question: “What would I need to know to find that?” And then “What would I need to know to find that?” Subgoals should be written down, to keep the trail clear.

After looking backward as far as possible, it’s time to reason forward from each given fact, with the last subgoal in mind. A different question governs the forward process: “What can I imply from that fact?” And then “What could I imply from that?”

Eventually, forward progress enables us to hit the target. All that’s left is to follow the string of subgoals up the ladder to the desired result.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Bring a Mouse to the SAT

Most people find use of a mouse to be a time saver, especially if you’re accustomed to employing one.

On the digital SAT, every second counts, and use of a computer mouse can save time on this important assessment. 

The College Board allows use of a "USB A" mouse (wired or wireless) with mouse pad on the digital SAT. Practice taking digital SATs at home using your favorite mouse. Then bring the same mouse with you on test day, plug it in, and go. Don't forget to bring a small mouse pad, as well.

This seems like a minor detail, but there’s almost no such thing when it comes to high-stakes testing.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

How to Handle Super-Hard Questions

For high scorers, only the hardest problems near the end really matter. That’s where top math students get lost, waste time, feel stress, and lose points.

How to avoid freaking out over crazy-hard questions?

Three concrete suggestions follow.  

1. Handle intimidation

It’s normal to feel intimidated by a long, wordy, or complex problem. Most of the time, however, you’ll be able to answer the question correctly in a reasonable amount of time, if you give yourself a chance. Always approach such a problem with the mentality that you can solve it. Boil down the question. Reread and make sure you understand each sentence, part by part. You probably can, in fact, do this. If you slow down a bit, read carefully, take notes, and focus intently, you’ll be successful most of the time.

2. Dodge nightmare questions

Instead of getting hung up on an unsolvable problem, it’s better to surrender, as quickly as possible, and live to fight another day. Prevent the wasted time, loss of energy, frustration, and stress caused by battling impossibly difficult “nightmare questions.” The fight isn’t worth it. Avoid the bottomless pit. Earn points somewhere else. Once you realize a question is over your head, just guess and move on. Do not come back later, even if you have time.

3. Employ Skip-Guessing

Naturally, top students want top scores, and often assume they’ll have to work hard to crack every single question. But this is not true. Students can generally afford to skip-guess lots questions without worry (to confirm this, study the scoring tool for any official practice test). When faced with an uncomfortably difficult question, don't get stuck, whatever you do. Quickly make your best guess, skip the question for now, flag it for later review, and move merrily along. 

What matters is making the utmost of what you’re able to do on test day. To optimize your score, spend your limited time where it can be maximally effective, not on crazy-difficult problems you’ll just get wrong, anyway.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

The Official Digital SAT Study Guide

The Official Digital SAT Study Guide by The College Board has been the sine qua non of SAT work for decades. 

But with the SAT's switch to a new digital format, just how useful is this latest incarnation of the venerable test prep tome? Why buy a book printed on paper when the test is now taken on-screen?

These are good questions.

The new edition contains four non-adaptive paper versions of the digital SAT – the very same "linear" tests made available as free downloads to students everywhere – which are roughly 70% identical to the four official adaptive on-screen tests contained in the College Board's BlueBook app (the only official tests available in on-screen adaptive form).

So that's even less reason to buy the book, right?

Right. Except for one thing.

Currently, there's a severe scarcity of official SAT practice materials (this happens each time the College Board decides to overhaul the test). Every new official SAT question made available to the public for practice gives valuable clues as to what to expect on the test, and is therefore worth gold.

As it happens, the new College Board Official Guide to the SAT contains 192 printed practice questions different from those provided anywhere else. That's nearly the equivalent of two additional full-length digital SAT tests.

So yes, at this point, you should buy the new edition – if only to gain additional practice with those 192 precious questions.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Well Begun is Half Done

Aristotle’s admonition to make a good start on any journey counts doubly on difficult SAT/ACT math problems. 

Beginning is often half the battle, and almost anything you can do to get yourself going will probably be helpful.

 
This is sometimes easier said than done, but there are several things you can generally try. It's good to keep in mind a few tips to help grease the wheels when stuck at the beginning of a tough question.

Primary among these is to reread the question slowly and carefully, at half-speed. Many times, you’ll find you simply missed something, and can now solve the problem. Easy as that.

To get a feeling for what’s going on, experiment with simple, realistic numbers in place of unknown quantities. Let the cost of the sweatshirt be $20, for instance. Use that number in the problem, and see what happens. Based on what you learn, the solution may reveal itself.

You can try making up a “simpler similar problem.” Solve that simpler problem, and apply the same approach to the more complex one you’re tackling.

For multi-part questions, pick the easiest part, and work that out first. With such a “jump start,” you may find you’re able to make progress and find your way to the answer.

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Contact Chris

borlandeducational.com

Copyright © 2006-present: Christopher R. Borland. All rights reserved.