Does Drawing Pictures Elucidate Math Concepts?

Barry Garelick says no, not always, and I agree:

A Wall Street Journal opinion article drew considerable attention to Common Core (CC) math standards—particularly the sixth-grade standard for fractional division— in early August. In it, math professor (emerita) Marina Ratner criticized approaches she saw in her grandchild’s classroom, where students were expected to represent fractional problems with pictures. She says:

The teacher required that students draw pictures of everything: of 6÷ 8, of 4 ÷ 2/7, of 0.8 × 0.4, and so forth. In doing so, the teacher followed the instructions: ‘Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for 2/3 ÷ 3/4 and use a visual fraction model to show the quotient . . .’

Ratner then asks: “Who would draw a picture to divide 2/3 by 3/4?”

I tend to agree. Requiring students to draw a picture to prove they “understand” the meaning of 2/3 ÷ 3/4 is likely to confuse more than it enlightens. Many others agree, including Barbara Oakley, a professor of engineering at Oakland University in Rochester, Michigan. She expressed her ideas in another Wall Street Journal opinion article called “How We Should be Teaching Math” published just this week. (Here is a link to a non-paywalled version.) In it, she states, “True mastery doesn’t mean you use crutches like laying out 25 beans in 5-by-5 rows to demonstrate that 5 × 5 = 25. It means that when you see 5 × 5, in a flash, you know it’s 25—it’s a single neural chunk that’s as easy to pull up as a ribbon. Having students stop to continually check and prove their understanding can actually impede their understanding, in the same way that continually focusing on every aspect of a golf swing can impede the development of the swing”…

The standard I have discussed here illustrates the theme of “understanding” and “explanation” that pervades many of the CC math standards. Opinion is divided in the education community about how to teach understanding. It is certainly worthwhile to explain to students why the fractional division algorithm works. Even more important, however, is recognizing that a student who knows what problems fractional division can solve and can perform the procedure possesses some understanding. Marking students down who have enough understanding to solve problems but cannot do the things judged to indicate understanding is placing the cart before the horse. It is tantamount to saying, “If we can just get them to do things that look like what we imagine a mathematician does, then they will be real mathematicians.”  (emphasis mine–Darren)

Yes, what he said.  Let’s just teach what works, shall we?

Competency degrees help working adults

Working adults are turning to online competency-based programs to cut the cost and time of earning a degree.

Carnival of Homeschooling

Freedom is the theme of this week’s Carnival of Homeschooling.  Homeschooled Mom writes:  “I don’t really see a point in having the legal right to homeschool, only to check with everyone and their mother before deciding on a curriculum, event, or even a political stance.  Homeschooling is an individual endeavor, and I believe that while input and opinions are nice, they are just that.  We have to remember that as parents, WE ARE IN CHARGE.”

Sex and the Campus

MARGARET WENTE: The New Campus Sex Puritans.

Sixty years ago, sexual behaviour among the young caused deep alarm among the puritanical religious right. Today, it causes deep alarm among the puritanical progressive left. Like their forebears, they are doing their best to restrict and regulate it.

This weekend, California Governor Jerry Brown signed a bill that makes universities redefine consensual sex. From now on, students must effectively obtain the “affirmative consent” of their partners, which must be “ongoing” every step of the way. Those accused of violating the consent rule will be judged on the preponderance of the evidence. Perpetrators face suspension or expulsion, and universities face heavy penalties for failure to enforce.

The new measure is designed to stem a tidal wave of rape on campus that, in fact, does not exist. (Violent crime, including sexual assault, has been in decline for 20 years.) Even so, universities across North America have set up vast new administrative apparatuses to deal with the crisis. Many of them have also expanded the meaning of “sexual violence” to include anything that makes you feel bad.

You don’t have a right not to feel bad, after sex or at any other time.

That’s from Instapundit.   I’m curious, though, why we’re regulating sex only amongst college students.  Are they raping more than others?  Not at UC Davis, they’re not:

Domestic violence, dating violence and stalking – three categories added to the annual Jeanne Clery Act crime disclosure list – show that UC Davis, like all campuses, is not immune to any of the crimes.

Starting this year, colleges and universities were required to compile reports on the three categories. The report indicates there were 11 cases of domestic violence, five instances of dating violence and 17 cases of stalking on property associated with UC Davis.

The report further indicates there were 24 cases of sex offenses in 2013, compared to 18 in 2012. There were seven cases of aggravated assault in 2013, compared with nine the year before.

UC Davis has over 34,000 students enrolled.  Given the numbers above, it sounds like a relatively safe place.  Doesn’t it make you wonder where all this talk of “rape culture” comes from?  A cynical person might conclude that such talk is designed to rile people up and get them to support a certain political party.

Good thing I’m not that cynical, right?

Learning and working at the factory

Unable to find skilled workers, Kentucky manufacturers are training their own technicians with help from a community college. Advanced Manufacturing Technician (AMT) trainees work part-time on the factory floor, earn an associate degree and qualify for jobs that start at close to $65,000 a year. Many applicants don’t have the math skills to qualify.

Not A Common Core Fan

I don’t agree with everything this author says, but let’s look at her credentials and see if she merits at least a listen:

My degree is in engineering.  I spent five and a half years refurbishing nuclear submarines, and then I quit work to bear, rear, and eventually homeschool our three children.

As a homeschool mom, I participated in co-ops, taking turns teaching groups of homeschooled children subjects such as nature study and geography. As our children entered their teen years, I began teach to teach algebra, trig, and calculus to small classes of homeschoolers at my kitchen table.  And as our children left home for their four-year universities, two to major in engineering and one in art, I began teaching in small private schools known as classical academies.

This last year, I have also been tutoring public-school students in Common Core math, and this summer I taught a full year of Common Core Algebra 2 compressed into six weeks at an expensive, ambitious private school…

OK, she has some qualifications and some practical knowledge.  Let’s see what she has to say:

Fifty years ago, transformations were not taught, although math-bright students would figure them out for themselves in analytic geometry (second-semester pre-calculus).  Today, they are taught systematically beginning in elementary school.

The treatment of transformations reminds me of the New Math debacle of the 1960s.  The reform mathematicians of the day decided that they were going to improve mathematical education by teaching all students what the math-bright children figured out for themselves.

In exactly the same way, the current crop of reform math educators has decided that transformations are an essential underlying principle, and are teaching them: laboriously, painfully, and unnecessarily.  They are tormenting and confusing the average student, and depriving the math-bright student of the delight of discovering underlying principles for herself.

One aspect of Common Core that did not surprise me was a heavy reliance on calculators.

Huh?  What’s that?  Reliance on calculators?  She definitely has my attention there.  I’ve written plenty on that topic (here and here, among many others); the links are there if anyone wants to go read them, there’s no need to rehash the arguments here.  Let’s get back to the Common Core piece:

Common Core advocates claim that they are avoiding that boring, rote drill in favor of higher-order thinking skills.  Nowhere is this more demonstrably false than in their treatment of formulas.  An old-style text would have the student memorize a few formulas and be able to derive the rest.  Common Core loads the student down with more formulas than can possibly be memorized.  There is no instruction on derivation; the formulas are handed down as though an archangel brought them down from heaven.  Since it is impossible to memorize all the various formulas, students are permitted – nay, encouraged – to develop cheat sheets to use on the tests…

The oft-repeated goal of Common Core is that every child will be “college or career ready.”  Couple that slogan with the oft-expressed admiration for the European system of education – in European countries, students are slotted for university or a dead-end job at age fourteen, based ostensibly on their performance on high-stakes tests, but that performance almost inevitably matches the student’s socioeconomic class.  Do we really want to destroy upward mobility and implement a rigid class structure in the United States of America?

That doesn’t sound like an admirable goal, but the author is correct.  She follows that explosive comment up with her denouement:

I predict that if we continue implementing Common Core, average students will drop out of math as early as they are allowed.  Even math-bright students will hate math.  Tutoring companies will proliferate to serve wealthy families.  The educational gap between rich and poor will widen.  If we want to destroy math and science education in this country, keep Common Core.

How many kids will have been harmed before we admit that this was a political mistake?

Student Free Speech

What are appropriate boundaries for curtailing K-12 student speech in public schools?  I discussed the idea over 8 years ago, and in the time since then I cannot see how much has changed.

If you like free speech issues in general, a large collection of related posts can be found here.

Onion: Teacher fired for learning more from students than vice versa

From the Onion: A teacher is fired for “gross incompetence” after declaring, “I just love being around the students—I honestly think I get more out of these classes than the kids do.” She adds, “I learn something new from them each and every day. They teach me so much—far more than I could ever teach them.”

This brings to mind a (real) quote from Michael John Demiashkevich’s Introduction to the Philosophy of Education (1935):

An old schoolmaster dedicated his book to all his old pupils, at whose expense, he said, he had learned everything he knew about education. This is either a case of exaggerated modesty or it is a belated confession of incompetence. It is necessary to distinguish strictly between broadmindedness and ignorance.

I suspect, though, that the Onion teacher was really fired for her use of fluffy phrases like “so much,” “honestly think,” and “each and every day.” If she had said, simply, “I enjoy learning from the students as well as teaching them,” she might still have her imaginary job, and she could still learn “something,” or even “a lot.”

Professors on food stamps

Some adjunct professors make less than minimum wage — with no benefits or job security. These days, the majority of college instructors are part-timers.

Repetition and Memorizing

After Joanne’s introduction and Diana’s open cheerfulness at being a co-guest blogger here, you might think I’d offer up more intellectually stimulating fare than funny pictures of Starbucks references.  For those of you who were disappointed, I’ll endeavor to make it up to you in this post.

In a recent Wall Street Journal piece, Barbara Oakley posits that the Common Core standards, and the pedagogy that is often pushed with those standards, prioritizes “conceptual understanding” at the expense of slighting repeated and varied practice that leads to computational mastery:

Conceptual understanding has become the mother lode of today’s [Common Core] approach to education in science, technology, engineering and mathematics—known as the STEM disciplines. However, an “understanding-centric approach” by educators can create problems….

True experts have a profound conceptual understanding of their field. But the expertise built the profound conceptual understanding, not the other way around. There’s a big difference between the “ah-ha” light bulb, as understanding begins to glimmer, and real mastery.

As research by Alessandro Guida, Fernand Gobet, K. Anders Ericsson and others has also shown, the development of true expertise involves extensive practice so that the fundamental neural architectures that underpin true expertise have time to grow and deepen. This involves plenty of repetition in a flexible variety of circumstances. In the hands of poor teachers, this repetition becomes rote—droning reiteration of easy material. With gifted teachers, however, this subtly shifting and expanding repetition mixed with new material becomes a form of deliberate practice and mastery learning….

True mastery doesn’t mean you use crutches like laying out 25 beans in 5-by-5 rows to demonstrate that 5 × 5 = 25. It means that when you see 5 × 5, in a flash, you know it’s 25—it’s a single neural chunk that’s as easy to pull up as a ribbon. Having students stop to continually check and prove their understanding can actually impede their understanding, in the same way that continually focusing on every aspect of a golf swing can impede the development of the swing.

Understanding is key. But not superficial, light-bulb moment of understanding. In STEM, true and deep understanding comes with the mastery gained through practice.

I recently wrote on a similar theme, quoting from a newspaper article about a Stanford paper that demonstrated, among other things, that children should memorize multiplication tables and addition tables.  Quoting:

Next, Menon’s team put 20 adolescents and 20 adults into the MRI machines and gave them the same simple addition problems. It turns out that adults don’t use their memory-crunching hippocampus in the same way. Instead of using a lot of effort, retrieving six plus four equals 10 from long-term storage was almost automatic, Menon said.

In other words, over time the brain became increasingly efficient at retrieving facts. Think of it like a bumpy, grassy field, NIH’s Mann Koepke explained.

Walk over the same spot enough and a smooth, grass-free path forms, making it easier to get from start to end.

If your brain doesn’t have to work as hard on simple maths, it has more working memory free to process the teacher’s brand-new lesson on more complex math.

‘The study provides new evidence that this experience with math actually changes the hippocampal patterns, or the connections. They become more stable with skill development,’ she said.

‘So learning your addition and multiplication tables and having them in rote memory helps.’

As a math teacher I explain it this way:  to truly understand algebra a student must already have mastered operations with fractions, decimals, and negative numbers.  The simple calculation of calculating the slope of a line between two given points could include all three of those, and if one expends all his/her brain power on that simple calculation, there won’t be as much brain power “left over” to understand what the answer, the slope, actually means or represents.

Some things must be memorized–not for their own sakes, but because they are useful tools, they are means to an end.

In professional development sessions I’m often told, as if it’s an obvious fact that cannot possibly be doubted, that if you cannot explain how something works, then you truly don’t have a “deep” enough understanding of it.  You have rote memorization, nothing more, and rote memorization is useless.  Sometimes I’m even told this by math teachers, who will at lease concede that memorizing the multiplication tables is a valuable exercise.  I put up a division problem, usually something simple like 515/3, and ask “Who can perform this calculation?”  Everyone can and all hands are raised.  Then I ask, “Who can explain why the standard algorithm (which everyone our age knows and uses) works, and why?”  Even most math teachers cannot, but everyone recognizes why that standard algorithm is important, useful, and efficient–everyone, that is, except for those who think that some Indian lattice method leads to “deeper understanding”.  Beyond knowing that division is akin to finding out how many “groupings” of a certain size can be made from a certain number, how “deep” does one need to understand division?  It’s useful only as a tool to get to bigger and better things, IMNSHO.

So repeat and repeat and repeat until the repetition begets memorization.  That’s what Mrs. Barton did until every one of her students knew the multiplication tables.  Don’t allow a pet pedagogical theory to harm students’ ability to calculate.  Teach them what works.  Give them the most efficient tools out there.

Teach.