School Math is Void of Common Sense

A research mathematician turned teacher has a word problem for you: “There are 125 sheep and 5 dogs in a flock. How old is the shepherd?”

Most adults would quickly come to the conclusion that there is not enough information to find an answer. Not most students.

Now consider that, according to researchers, three quarters of schoolchildren offer a numerical answer to the shepherd problem. In Kurt Reusser’s 1986 study, he describes the typical student response:

125 + 5 = 130 …this is too big, and 125–5 = 120 is still too big … while … 125/5 = 25 … that works … I think the shepherd is 25 years old.

Remarkable. In their itch to combine the numbers presented to them, students negotiate three solutions. They show some awareness of context in dismissing the first two candidates. But a 25-year-old farmer is plausible enough for students to offer it up as their answer. The calculations are correct, but they are also irrelevant. Common sense has deserted these students in their pursuit of a definitive answer.

He says those findings are the direct result of the type of problems we ask students to solve during their travels through school math.

Students believe that all math problems are well-defined, usually with a single right answer. They strongly associate mathematics with numbers, to the extent that they will instinctively derive numerical answers to problems regardless of the context. They are subservient to computational procedure and trust that accurate calculations will always lead them to relevant truths. They accept that confusion and ambiguity is a staple fixture of mathematics, willingly offering up solutions that are void of context, meaning or even common sense.

So, what’s the alternative to our current standard math curriculum, featuring repetitive pages of calculations and ambiguous word problems with one right answer? The writer has some excellent ideas. However, they would require completely reimagining the way we teach math.

At the core of those changes is an emphasis on understanding process rather than finding answers.

Mathematics is a journey; it is defined by process, not rigid outcomes. That process can not be reduced to a series of discrete computation steps. It is governed by a flow of reasoning that guides the thinker towards a solution. Problem-solving often an unstructured, messy affair that requires several iterations of developing and testing assumptions. Error and ill-judgement are the most natural components of problem solving; they should be embraced. All mathematicians need pause to reflect on their problem solving strategies; to step back and retain full view of the big picture. Students must be afforded the same opportunities; their development as mathematical thinkers depends on this sense-making.

Punching Holes in Your Comfort Zone

I disagree with the very negative opinion about the education system expressed by James Altucher, an economic writer who says he needs very little in life. However, in this post, he does make a couple of great points about something he finds essential: curiosity.

For one thing, he says it leads to happiness: “Dopamine is being released because I am in anticipation of the reward of curiosity getting satisfied. Higher dopamine equals greater happiness, better brain and heart health. Live longer.” I admit, I feel pretty good when I’m satisfying my curiosity.

I think he’s also correct that curiosity leads to greater creativity, maybe to better relationships and community. Not so sure about fighting Alzheimer’s but anything that exercises your brain can’t be bad.

But for me, this is the core of his thoughts on creativity.

Our comfort zone is where we are safe in the womb of life. Our real self is everything beyond that.

The Curiosity Zone is bigger than the Comfort Zone.

Every time you are curious, you punch another hole in that comfort zone.

I am certainly saving that idea to use sometime, somewhere.

Pushing Back at Anti-Intellectualism

One excellent selection from the commencement address President Obama delivered at Rutgers University last Sunday.

Which brings me to my third point:  Facts, evidence, reason, logic, an understanding of science — these are good things. These are qualities you want in people making policy.  These are qualities you want to continue to cultivate in yourselves as citizens. That might seem obvious. That’s why we honor Bill Moyers or Dr. Burnell.

We traditionally have valued those things. But if you were listening to today’s political debate, you might wonder where this strain of anti-intellectualism came from. So, Class of 2016, let me be as clear as I can be. In politics and in life, ignorance is not a virtue. It’s not cool to not know what you’re talking about. That’s not keeping it real, or telling it like it is. That’s not challenging political correctness. That’s just not knowing what you’re talking about. And yet, we’ve become confused about this.

Reading the text is good but the listening to the president speak is even better (for those not part of the Fox “news” fan club). If you have 45 minutes, watch the whole address:

Speaking of the current “strain of anti-intellectualism” in American society, I also recommend reading an opinion piece, “A Cult of Ignorance”, written more than 35 years ago by the great Isaac Asimov.

There is a cult of ignorance in the United States, and there always has been. The strain of anti-intellectualism has been a constant thread winding its way through our political and cultural life, nurtured by the false notion that democracy means that “my ignorance is just as good as your knowledge.”

I’m hopeful Asimov’s essay and the president’s remarks will not still be valid in another 35 years. But I’m not optimistic about that.

School Math is the Wrong Subject

Conrad Wolfram, a mathematician and “director of what’s arguably the world’s ‘math company’” (that would be Wolfram Research), believes “today’s educational math is the wrong subject”. Meaning that what we present to students as “mathematics” is not anything like what it is in the real world.

In the real world we use computers for calculating, almost universally; in education we use people for calculating, almost universally.

This growing chasm is a key reason why math is so despised in education and yet so powerful and important in real life. We have confused rigor at hand-calculating with rigor for the wider problem-solving subject of math. We’ve confused the once-necessary hand mechanics of the past with the enduring essence of math.

At its heart, math is the world’s most successful system of problem-solving. The point is to take real things we want to work out and apply, or invent, math to get the answer. The process involves four steps: define the question, translate it to mathematical formulation, calculate or compute the answer in math-speak and then translate it back to answer your original question, verifying that it really does so.

Teachers and textbooks give lip service to math as a tool for problem-solving but do little to help students understand the process Wolfram describes. As a result, the work kids do in “math” class is dry, boring and largely useless. For the most part, students learn to step through algorithms that the real world turned over to computers and calculators many decades ago.

I love how he describes “word problems” (now often euphamistically called “applications”) “as toy problems and largely outside any context most students can relate to”.

But it’s not just about turning the computation part over to computers. Wolfram says we need to completely replace the current mathematics curriculum taught in most schools.

Instead of rote learning long-division procedures, let’s get students applying the power of calculus, picking holes in statistics, designing a traffic system or cracking secret codes. Such challenges train both creativity and conceptual understanding and have practical results. But they need computers to do most of the calculating — just like we do in the real world.

All of us who have taught math in K12 have heard one common question from our students: “when are we ever going to use this?”. The fact that the honest answer is “probably never” should be a clue that something needs to change.

Digitally Faulty

From the Washington Post’s Grade Point blog we have a list of “five critical skills every new college graduate should have”. It begins with

Every graduate needs to be “digitally aware.” Students entering college and the workforce now often are referred to as “digital natives” because they were raised on technology from a very young age.

And stop right there.

Anyone still using the “digital native-digital immigrant” trope is, at the very least, being intellectually lazy. As with many other concepts about people, especially kids, that phrase is a binary, either-or shortcut that excuses the writer from the responsibility of explaining the complexity of the subject, and their readers from having to understand it.

Being “raised on technology from a very young age” does not convey the expertise implied by calling them “natives”. For those kids who have easy access to digital devices and networks growing up (which excludes large numbers of children, even in the US), most acquire a comfort level with the tools that connect them to their friends and personal interests. They are not computing geniuses – or “hackers” when a negative slant is needed.

For most students, their “native” digital skills don’t automatically translate into using the technology tools for learning skills needed to live in the broader world. They still need parents and teachers to guide them in those areas.

Continuing in the same brief section, the writer also leans on another, more recent, flawed assumption about the needs of graduates, from both college and high school.

It’s no longer good enough to know how to use a computer. Understanding the programming language behind the apps on your iPhone, or the basics of Artificial Intelligence are all now seen as basic foundational skills by many employers. Learning to program is much like learning a second language was in the 20th century: You might not become proficient enough to move overseas, but you could get by if you traveled to a particular country.

I’d love to see some statistics about the “many employers” who see programming as a “basic foundational skill”. Plenty of politicians, business-types, and other education experts, tell us that kids need to learn to code. The president is asking for $4 billion to provide computer programming classes for all students in K12, without a clear definition of why it’s that important.

And equating learning to program with learning to speak a second language is yet another lazy, not to mention very wrong, shortcut. Beyond both being classes offered in many high schools, the two require different skill sets and processes in the brain. But it’s probably not as bad as equating coding with being able to read and write in your native language.

Ok, so the writer goes on to present his four other “critical” skills for graduates, but that first one is bad enough. I really don’t want to waste time on figuring out how one becomes a “learning animal”, or explain why lacking the ability to “navigate through life without a syllabus” is a failure of their schools, not the graduate.