# Stanford Professor: We Need to Take the Fear Out of Learning Math

Young math learners are done a major disservice by speed trials and drills, says education expert Jo Boaler. We need to redesign education so that students work on problems they enjoy.

Jo Boaler is a professor at Stanford Graduate School of Education and a major proponent of education reform. She's a big fan of equitable classrooms and intuitive learning strategies, less so of rote memorization and other processes that make learning math a joyless endeavor. Some of her recent work is highlighted in a piece at *Stanford News* in which she explains to reporter Clifton B. Parker that the key to improving America's dismal levels of mathematics prowess is in extricating fear and pressure from the learning process:

"'There is a common and damaging misconception in mathematics—the idea that strong math students are fast math students,' said Jo Boaler, a Stanford professor of mathematics education and the lead author on a new working paper. Boaler's co-authors are Cathy Williams, co-founder of Stanford's YouCubed, and Amanda Confer, a Stanford graduate student in education...

But according to Boaler, when students are stressed—such as when they are solving math questions under time pressure—the working memory becomes blocked and the students cannot as easily recall the math facts they had previously studied. This particularly occurs among higher-achieving students and female students, she said.

Some estimates suggest that at least a third of students experience extreme stress or 'math anxiety' when they take a timed test, no matter their level of achievement. 'When we put students through this anxiety-provoking experience, we lose students from mathematics,' she said."

What's important to grasp here is that the ways we've evaluated math students in the past has been all wrong. We've rewarded students who can get away with regurgitating simple formulas under pressure. The problem is that mathematics is supposed to teach us new, evaluative ways of thinking. It's not useful to just cram numbers for the near future. Mathematicians don't let time confine their thoughts—why do we let time dictate our evaluations of children?

Read more at *Stanford News* and let us know what you think.

Photo credit: Victor Naumik / Shutterstock

## Why the number 137 is one of the greatest mysteries in physics

Famous physicists like Richard Feynman think 137 holds the answers to the Universe.

- The
**fine structure constant**has mystified scientists since the 1800s. - The number
**1/137**might hold the clues to the Grand Unified Theory. - Relativity, electromagnetism and quantum mechanics are unified by the number.

## Americans under 40 want major reforms, expanded Supreme Court

Younger Americans support expanding the Supreme Court and serious political reforms, says new poll.

- Americans under 40 largely favor major political reforms, finds a new survey.
- The poll revealed that most would want to expand the Supreme Court, impose terms limits, and make it easier to vote.
- Millennials are more liberal and reform-centered than Generation Z.

## Can you solve what an MIT professor once called 'the hardest logic puzzle ever'?

Logic puzzles can teach reasoning in a fun way that doesn't feel like work.

- Logician Raymond Smullyan devised tons of logic puzzles, but one was declared by another philosopher to be the hardest of all time.
- The problem, also known as the Three Gods Problem, is solvable, even if it doesn't seem to be.
- It depends on using complex questions to assure that any answer given is useful.

### The Three Gods Problem

<iframe width="730" height="430" src="https://www.youtube.com/embed/UyOGZk7WbIk" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe><p> One of the more popular wordings of the problem is:<br> <br> "Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for <em>yes</em> and <em>no</em> are <em>da</em> and <em>ja</em>, in some order. You do not know which word means which."<br> <br> Boolos adds that you are allowed to ask a particular god more than one question and that Random switches between answering as if they are a truth-teller or a liar, not merely between answering "da" and "ja." <br> <br> Give yourself a minute to ponder this; we'll look at a few answers below. Ready? Okay. <strong><br> <br> </strong>George Boolos' <a href="https://www.pdcnet.org/8525737F00588A37/file/31B21D0580E8B125852577CA0060ABC9/$FILE/harvardreview_1996_0006_0001_0060_0063.pdf" target="_blank" rel="noopener noreferrer">solution</a> focuses on finding either True or False through complex questions. </p><p> In logic, there is a commonly used function often written as "iff," which means "if, and only if." It would be used to say something like "The sky is blue if and only if Des Moines is in Iowa." It is a powerful tool, as it gives a true statement only when both of its components are true or both are false. If one is true and the other is false, you have a false statement. </p><p> So, if you make a statement such as "the moon is made of Gorgonzola if, and only if, Rome is in Russia," then you have made a true statement, as both parts of it are false. The statement "The moon has no air if, and only if, Rome is in Italy," is also true, as both parts of it are true. However, "The moon is made of Gorgonzola if, and only if, Albany is the capitol of New York," is false, because one of the parts of that statement is true, and the other part is not (The fact that these items don't rely on each other is immaterial for now).</p><p> In this puzzle, iff can be used here to control for the unknown value of "da" and "ja." As the answers we get can be compared with what we know they would be if the parts of our question are all true, all false, or if they differ. </p><p> Boolos would have us begin by asking god A, "Does "da" mean yes if and only if you are True if and only if B is Random?" No matter what A says, the answer you get is extremely useful. As he explains: <br> </p><p> "If A is True or False and you get the answer da, then as we have seen, B is Random, and therefore C is either True or False; but if A is True or False and you get the answer ja, then B is not Random, therefore B is either True or False… if A is Random and you get the answer da, C is not Random (neither is B, but that's irrelevant), and therefore C is either True or False; and if A is Random...and you get the answer ja, B is not random (neither is C, irrelevantly), and therefore B is either True or False."<br> <br> No matter which god A is, an answer of "da" assures that C isn't Random, and a response of "ja" means the same for B. </p><p> From here, it is a simple matter of asking whichever one you know isn't Random questions to determine if they are telling the truth, and then one on who the last god is. Boolos suggests starting with "Does da mean yes if, and only if, Rome is in Italy?" Since one part of this is accurate, we know that True will say "da," and False will say "ja," if faced with this question. </p><p> After that, you can ask the same god something like, "Does da mean yes if, and only if, A is Random?" and know exactly who is who by how they answer and the process of elimination. </p><p> If you're confused about how this works, try going over it again slowly. Remember that the essential parts are knowing what the answer will be if two positives or two negatives always come out as a positive and that two of the gods can be relied on to act consistently. </p><p> Smullyan wrote several books with other logic puzzles in them. If you liked this one and would like to learn more about the philosophical issues they investigate, or perhaps if you'd like to try a few that are a little easier to solve, you should consider reading them. A few of his puzzles can be found with explanations in this <a href="https://www.nytimes.com/interactive/2017/02/11/obituaries/smullyan-logic-puzzles.html" target="_blank" rel="noopener noreferrer">interactive</a>. </p>## New tardigrade species withstands lethal UV radiation thanks to fluorescent 'shield'

Another amazing tardigrade survival skill is discovered.