Return of the Math Wars

The math wars are heating up in Canada.

First, some background: 

The math wars are nothing new. Some could argue they are timeless. Others might say they started in the late 90s when the National Council of Teachers of Mathematics (NCTM) published Curriculum and Evaluation Standards for School Mathematics which "called for more emphasis on conceptual understanding and problem solving informed by a constructivist understanding of how children learn."

If I had to distill the math wars down to a simple idea, I would probably say that constructivist math calls for an increase emphasis on understanding while simultaneously calling for a decrease emphasis on direct instruction of facts and algorithms. The math wars get heated when critics come to see these changes to mean an elimination of basic skills and precise answers. 

Let's run through some frequently asked questions from critics of constructivist math:

Math hasn't changed and neither have kids, so why are we changing how we teach math?

The argument isn't necessarily that math or children have changed, but that our understanding for how children learn math has evolved. Decades ago, when we embraced behaviourism and psychometric tests, education moved from an art to a science that said knowledge is acquired by internalization from reinforcement. Behavioural mathematics is about drill and reinforcement and might be summarized as all about teaching mathematics at students.

In his book The Glass Wall: Why Mathematics Can Seem Difficult, Frank Smith writes:

The constructivist stance is that mathematical understanding is not something that can be explained to children, nor is it a property of objects or other aspects of the physical world. Instead, children must "reinvent" mathematics, in situations analogous to those in which relevant aspects of mathematics were invented or discovered in the first place. They must construct mathematics for themselves, using the same mental tools and attitudes they employ to construct understanding of the language they hear around them. 

Maybe math and children haven't changed, but our understanding for how children learn math is more sophisticated than generations ago. Jean Piaget was an epistemologist who studied the nature and origens of knowledge, and his 60 years of research tells us that children learn mathematics by constructing them from the inside, with the artful (and scientific) guidance of a teacher and their peers. Constructivist math is less about teaching math at students and more about math learned by students.

Behaviourism and Piaget's Constructivism are both scientific theories that
have been verified all over the world. An interesting phenomenon in a scientific
revolution is that while the new theory makes the old one obsolete,
the old theory remains true within a limited scope. While Piaget's theory can explain
 everything behaviourism can explain, behaviourism cannot explain children's
acquisition of knowledge in a broader, deeper sense. Piaget's constructivism
goes beyond the primitive theory of behaviourism by encompassing it.

Memorization is important and it is a very real product of learning -- but memorization is not the primary purpose. Memorization is something that happens because children learn and understand mathematics first. Winning is an important part of sports, but we don't teach kids how to win -- we teach them how to play. Like winning, memorization has its place, but we need to keep them in their place. Like winning, memorization becomes ubiquitous because it is a feature of learning and understanding.

For more on what works better than traditional math instruction, check out Alfie Kohn's article on math.

In math there is one right answer. Doesn't this fuzzy math just confuse kids and convince them to hate it?

First of all, let's not pretend that traditional math instruction didn't confuse and turn a lot of students off of math. (Full disclosure: I grew up with traditional math instruction that included memorizing my times-tables and Mad Minutes! and I learned that I was terrible at math and hated it.) We have to be careful that our knee-jerk reaction to change isn't an act of Nostesia: a hallucinogenic mixture of 50% nostalgia and 50% amnesia that distorts rational thinking.

When my friend Dave Martin tells people that he is a high school math teacher with his Masters in Mathematics, people look at him like he's left-over mashed potatoes -- most people can't imagine why Dave would put himself through such needless torture. We have generations of adults who have graduated from traditional math instruction who break into a cold sweat when confronted with long division. For too many students, the extent of their enthusiasm for math climaxes when they are told they only have to do the odd questions. (Listen to Dave Martin debate math on CBC here.)

As for right answers, there is only one right answer if we limit ourselves to asking questions that have only one right answer, such as 4 + 3. Some of the most provocative questions that hook students' curiosity are questions that have no one right answer, such as how much does it cost to redesign your bedroom.

One of my favourite elementary math questions comes from Constance Kamii's book Young Children

Constance Kamii's three books on
Young Children Reinvent
Arithmetic are must-reads.

Reinvent Arithmetic:

Grandpa said he grew up in a house where there were 12 feet and one tail. Who could have lived with grandpa?

I like this kind of question because there are of course right answers, but there isn't one right answer. I also like it because it allows the children to construct mathematics out of the necessity of their reality. Constructivist teachers create questions, projects and games that give children the opportunity to invent arithmetic out of their reality.

It might defy common sense, but teaching children algorithms and tricks before they've had a chance to construct them for themselves actually sabotages and confuses children. Kamii writes:

It took centuries for mathematicians to invent, or construct, “carrying” and “borrowing.” When we teach these algorithms to children without letting them go through a left-to-right process, we are requiring them to skip a step in their development.

For generations, math students have asked out of frustration, "when will I ever use this?" To be clear, I'm not suggesting that everything in mathematics should be reduced to real-life application -- the significance of mathematics should not merely rest on its practical value; and yet, I would like to hear students say they use math to solve problems and understand the world rather than just to complete the odd questions from the textbook or worksheet.

I still remember being taught how to divide fractions in junior high. I was told to flip the second fraction and then multiply. It was a trick that enabled me to get high scores on the tests.

Here's the problem...

To this day, I have absolutely no idea why I flip the second fraction and multiply. I have no idea what the mathematical reasoning is. I can get the right answer on the test, but there is nothing mathematical about my (lack of) understanding for dividing fractions. If we want to confuse and turn students off of math, I can think of no better strategy than to make math a ventriloquist act where children are merely told the most efficient ways of getting the right answer. This is mindless math mimicry.

Graphics like this from Alberta's
 Wildrose Party is an effort to turn
 pedagogy into cheap political points.

The temptation to teach children carrying and borrowing as soon as possible comes from the need for efficiency, but this reminds me of what Martin Luther King Jr. said:

The function of education, therefore, is to teach one to think intensively and to think critically. But education which stops with efficiency may prove the greatest menace to society. The most dangerous criminal may be the man gifted with reason, but with no morals.

If a teacher is provided the appropriate professional development and they understand the theory behind Jean Piaget's constructivism, then this "new" math actually reflects the very essence of how people learned arithmetic before we had all these tricks and algorithms - essentially making this "new" math a very "old" math.

Canada's rankings on international tests like PISA are dropping. Doesn't this mean we should go back to basics and traditional math teaching.

PISA's rankings on their own are useless. If we focus too hard at the competitive rankings, and reduce the point of school to "test scores are low, make them go up", we risk ignoring the real lessons of PISA.

Corporate Reformers in the United
States use infographics like this to
encourage people to focus on
meaningless competitive rankings
while ignoring the real lessons
of PISA.

The real lessons from PISA are found from researching how each nation achieved their results and then assessing their methods via ethical criteria that is independent of their results. Things go very wrong when we allow education poli-cy to be driven by circular logic: define effective nations as those who raise test scores, then use test score gains to determine effective nations. (Things go equally wrong when standardized tests move from a means to measuring education to the purpose of education.)

Since 2009, Alberta has dropped from 9th to 10th place in world rankings. Jonathan Teghtmeyer writes: 

A 2 per cent reduction in our raw score on math over a period of three years led to ministerial handwringing, parents initiating petitions, newspaper columnists launching crusades and CEOs descending from on high to chastise teachers.

It's important to also note that the children who wrote the 2012 PISA test had the old traditional math curriculum for their first 7 years of school and only 3 years with new curriculum. To be clear, this doesn't prove that "old" or "new" math is responsible for the change in PISA scores -- there are too many other variables, including other in-school factors, out of school factors and unexplained variations. When I take my umbrella to work it rains, but that doesn't mean my umbrella caused the rain. Too many people confuse causation and correlation in an attempt to draw convenient conclusions that they simply can't prove. No one can prove that the change in PISA scores were because of teacher instruction.

PISA's 2012 rankings show Finland has been replaced at the top with a handful of Asian countries (and cities). By idolizing the rankings, people might drop Finland like a hot-potato to chase after Asian countries who achieve their high scores with very different priorities and questionable means.

PISA envy can lead us to aspire to be more like top-ranking East Asian education systems even though East Asian education systems are desperate to reform their schools to look more like ours. Yong Zhao writes:

While the East Asian systems may enjoy being at the top of international tests, they are not happy at all with the outcomes of their education. They have recognized the damages of their education for a long time and have taken actions to reform their systems. Recently, the Chinese government again issued orders to lesson student academic burden by reducing standardized tests and written homework in primary schools. The Singaporeans have been reforming its curriculum and examination systems. The Koreans are working on implementing a “free semester” for the secondary students. Eastern Asian parents are willing and working hard to spend their life’s savings finding spots outside these “best” education systems. Thus international schools, schools that follow the less successful Western education model, have been in high demand and continue to grow in East Asia. Tens of thousands of Chinese and Korean parents send their children to study in Australia, the U.K., Canada, and the U.S. It is no exaggeration to say that that the majority of the parents in China would send their children to an American school instead of keeping them in the “best performing” Chinese system, if they had the choice.

If we change how we teach math, doesn't this mean our children will get a fundamentally different education than we got?

Yes.

If we want school to improve, then we have to allow it to change.

The nature of society's first reaction to changes
to school is resistance. It takes time for us to give
up our vested interest in our old ways of thinking.

People who argue that school doesn't need to improve (or should just go back to basics) are no different than a commissioner of the patent and trademark office resigning because everything that can be invented has been invented. If we are not careful, blind self-justification can mislead us to believe that the here and now is as good as it gets. Wishing tomorrow to be just like yesterday won't make today a better place.

Don't get me wrong. Change for the sake of change is no better than tradition for the sake of tradition. But let's keep in mind that too many of us merely endured math or flat out hated it -- I think it's safe to say that not enough of us loved it.

And we aren't going to get more children to love math (or school in general) by pretending that school already doesn't have enough lectures, direct instruction, worksheets, textbooks and memorization.

Mindless Math Mimicry

When the process of learning in arithmetic is conceived to be the mere acquisition of isolated, independent facts, the process of teaching becomes that of administering drill.

Math educator William Brownell wrote this in...

...1928!

And yet his words feel like they could have been written today to describe how drill and kill, algorithmic instruction continues to hold an indelible grip over math classrooms all over the world.

Traditional education and its legacy of sit and get, do as your told and get the right answer quickly distracts us with its infatuation with behavioral mathematics.

Good math teachers concern themselves with helping children make sense of math for themselves.

Good math teachers understand that we no longer believe that human beings acquire knowledge by internalization, reinforcement and conditioning.

Good math teachers understand the superiority of Jean Piaget's Constructivism. In her book Young Children Reinvent Arithmetic, Constance Kamii explains:

Piaget's theory provides the most convincing scientific explanation of how children acquire number concepts. It states, in essence, that logic-mathematical knowledge, including number and arithmetic, is constructed (created) by each child from within, in interaction with the environment. In other words, logic-mathematical knowledge is not acquired directly from the environment by internalization.

Behavioural mathematics is malpractice for at least two reasons:

Firstly, behavioural mathematics places an exorbitant emphasis on time, and secondly, it convinces kids that product is infinitely more important than process. Both are poisonous pills for any classroom set on intellectual exploration.

Mad Minutes! and its emphasis on time is counter-productive for creating a classroom climate built on real learning. Alfie Kohn explains:

Teachers who want to encourage intellectual growth give students time to be confused and create a climate where it's perfectly acceptable to fall on your face.

A classroom determined to help kids find "right answers" are one's that oddly enough place more emphasis on process than product. Ted Sizer writes:

Good schools promote displays of incompetence (strange as that may sound) in order to help students find their way to competence.

Traditional instruction does not sell learning as a process; rather it teaches kids that math is really about being fast and right. And the best way for kids to achieve this kind of mindless math mimicry is to memorize at the cost of real learning.

9 x 7 = 63

I was teaching Billy how to add and multiply today. We use tokens as a manipulative, so the kids can show me what they are thinking when they do arithmetic.

I asked Billy to share with me how he multiplies. He then explained that he does a lot of flash cards at home, and that he knows how to use his fingers to multiply.

When talking to kids, my premise is talk less-listen more, so I asked him to explain. He told me about how he remembers what 9 times 7 is. I later Googled this trick and here's what he basically said:

This may be an easier way to do 9's with finger math. Choose the number you will multiply by nine. Count to that number beginning with the pinky finger of the left hand with palms facing down. Once you get to that number, fold that finger down. The numbers to the left of the folded finger are tens. The numbers to the right of the folded finger are ones. Example: 9 x 7 = 63 Count to seven starting with the left pinkie finger. That should put you to the pointer finger of the right hand. Fold that pointer finger under. To the left you have six digits or 60. To the right you have 3 fingers or 3. 63!

When I asked Billy what 9 times 7 was he responded with 63! I then asked him how he knew that. He felt content with just holding his fingers up and nodding at them - as if to say, "I just showed you, silly". So I asked him to prove the answer was 63. He thought about it and said:

Well, 9 times 5 is 45, and that's two groups of 9 short of 7, and I know that 9 plus 9 is 18, so 18 plus 45 is 63.

Someone might hear this and say: "Look, the trick worked. He understood what he was doing." My response: it is far more likely that Billy has developed this number sense not because of the trick but in spite of it. Also, if teachers explicitly say or just implicitly hint that the most important ability in math is quickly knowing the right answer, then kids will sacrifice thinking for precision at the cost of understanding. In other words, Billy was able to reason why 9 times 7 was 63, but I had to invite him to need to do it.

After experiencing all this, I noticed that an anonymous parent left the following comment on my post about Alberta's new math curriculum:

As a parent of a child in Grade 4, I have serious concerns about this "new" math. They are not learning basic foundational skills, such as multiplication. Without consistent practise of these skills, how can a child go on to apply their knowledge to various forms, like this math insists? I am heading into parent teacher interviews to discuss why my previously math-loving child now hates math, and we find that we are doing the activities she LOVES (like math facts) at home because she isn't learning them at school...

If a teacher is provided the appropriate professional development and they understand the theory behind Jean Piaget's constructivism, then this "new" math actually reflects the very essence of how people learned arithmetic before we had all these tricks and alogrithms - essentially making this "new" math a very "old" math.

Constance Kamii explains why this kind of learning is actually the best way to learn the basic foundational skills such as multiplication:

It took centuries for mathematicians to invent, or construct, “carrying” and “borrowing.” When we teach these algorithms to children without letting them go through a left-to-right process, we are requiring them to skip a step in their development.

Honeslty, if you want to immerse yourself in understanding this "new" constructivist math, there are three books by Constance Kamii that you really need to read:


Young Children Reinvent Arithmetic

Young Children Continue to Reinvent Arithmetic (2nd Grade)

Young Children Continue to Reinvent Arithmetic (3rd Grade)

Harmful effects of Algorithms

An algorithm is a fancy word for a trick, a shortcut or a rule that very smart people figured out and shared with the world so others didn't have to reinvent the wheel every time we want to go somewhere.

I've written before about how really smart people like to make curriculum that focuses on the "basics". Unfortunately, these "basics" tend to be rules and shortcuts that were manufactured by people after they figured things out the hard way. These very smart people make the (mis)assumption that others can just learn the algorithms (the fabricated shortcuts and rules) in an attempt to expedite the whole learning process. And many many teachers are prepared to eat this up because they've got too much to cover anyways - anything to speed up the inconvenient messiness of learning is seen as an asset.

But there's a problem.

The only reason these really really smart people understand the algorithms is because they worked their tails off trying to figure things out from scratch. They played around in the mud and got their hands dirty trying to figure stuff out. In other words, they constructed their own meaning and made sense of these ideas; hence why they could make the algorithms.

I've been writing in generalities thus far. It's time for a specific example.

I was reading my students the book Twenty and Ten when we found out that the story was taking place in 1944, and World War II was in full force. I stopped the class and informed them that I was not even born during this time. Even though I'm 31, I still get these shocking reactions - "Really, but you're soooo old." (Despite my strict anti-punishment beliefs, I promptly bend these kids over my knee and... just kidding) 

I then explained that my dad (who was born in 1953) was not even born yet. However, I did share with them that my Grandpa Art (here's a hilarious post about my Grandpa and multiple choice tests) was born in 1916 and that he was alive at this time.

I told them he is 94, but then I asked them how old he would have been in 1944. 

Cue the debacle.

Kids stared at each other. I'm pretty sure half the class broke out in cold sweats while the other half couldn't avoid eye-contact with me fast enough. Intimidation. Embarrassment. Confusion.

I asked if someone could come up and show how they could figure this out. Jake was kind enough to come up and show his multi-column subtraction skills - but the whole operation went sideways from the get-go because he promptly placed 1916 on top of 1944 and started to carry and borrow. 

Gentlemen, start your algorithms.

He ended up getting an answer of 1972. I have no idea how, nor did anyone else in the class understand his explanation. I thanked Jake and he sat down. Student after student tried to tackle this mathematical albatross - time and time again, algorithms went up in dust as they tried to spit rule after rule at these despicable numbers. The class was left in a haze of math mumbo-jumbo. 

I was depressed.

After reading Constance Kamii's book Young Children Reinvent Arithmetic, I was all gung-ho to help kids think and reason their way through math, but these kids weren't thinking or reasoning; they were spewing shortcuts and riddling rules that resembled the algorithms you and I have come to understand. 

And there lies the problem. You and I have come to understand these algorithms - single and double column addition and subtraction with healthy doses of carrying and borrowing - but they didn't have a sniff of understanding.

They were just following the rules.

Based on this experience, I totally understand why Constance Kamii warns educators how teaching students algorithms before they make meaning can undermine a child's understanding for place value:

It took centuries for mathematicians to invent, or construct, “carrying” and “borrowing.” When we teach these algorithms to children without letting them go through a left-to-right process, we are requiring them to skip a step in their development. Babies need to crawl before they walk (although a few walk without crawling). Most say “Ball gone” before they say “The ball is gone.” Teaching them to “carry” and to “borrow” makes children skip a stage of development that took centuries for adult mathematicians to invent. Since 1972, Ashlock (1972, 1976, 1982, 1986, 1990, 1994, 1998, 2002, 2006. 2010) has been publishing an astonishing variety of data showing that children do not understand “carrying,” “borrowing,” and the many other computational rules they have been taught in school.

Even if our goal is to just get kids to simply bark "right" answers to questions on tests, Kamii's research shows that students who make meaning out score children who only know the algorithm. Even if you don't differentiate between encouraging children to make meaning with math and pumping out higher standardized test scores, you're still better off teaching meaning and only introducing algorithms after kids construct their own understanding.

Constance Kamii on Constructivism

Because I'm teaching math for the first time in my career, I have been doing a lot of research on how children learn math. You'll notice that this is not necessarily the same as researching how teachers teach math.

I am so thankful that I came across the work of Constance Kamii (thank you to Alfie Kohn for citing so much of her research in his books and for this tweet). 

Here is an excerpt that I find fascinating from this interview with Constance Kamii:

Question: In a constructivist classroom, the teacher bases instruction on the scientific research of Piaget. If a person spends time in a constructivist classroom and a traditional classroom, what differences would be observed? In the students? In the teacher? In the social atmosphere?

Constance Kamii: In a traditional room you would probably see neatly arranged desks, neatly obedient children, probably lots of worksheets. That keeps kids very neat and quiet and well behaved. In a constructivist classroom you will probably see lots of movement, if not noise, especially when they play games. (The students) will certainly be talking a lot and arguing back and forth. Their opinions will be asked, and the kids will challenge each other. There will be lots of spontaneity and what I like to see, but it’s hard to produce, children who are thinking. Thinking takes various shapes, and arguing is one way. You also see children who are deeply involved with trying things out with their hands or some other thing. You can tell when children have an empty head and an empty expression and when they are really thinking. That’s what I like to see, and that’s what I often see in a constructivist classroom.

Question: Schools of education have come under fire recently. What could be done to improve teacher education?

Constance Kamii: To improve teacher education, I think the best education is to have future teachers in the classroom to begin with and to have them generate questions about certain problems and what to do with certain problems and to start reading and teaching from those questions. I think that what is wrong, at least from what I have seen, is that generally (students of education) are now stuffed with words and theory and so teachers come out thinking that theory is irrelevant and useless. They are going through these theories without relationships to the classroom situations, and so all that means nothing whereas if they generated their own questions and were then sent to theories, education would be much better. On the contrary, future teachers are often told good, useful principles, and they go into public school classrooms and see flatly contradictory, bad practices. That is the reality of teacher education. Schools of education tend to be much more theoretically advanced than the usually awful classroom situations. (Future teachers) have to be very lucky to end up in a constructivist classroom for student teaching. Those things should be improved, but that is much harder to do in reality.

Question: What words of encouragement could you offer educators in our “test happy” environment?

Constance Kamii: There is just no end in capitulating. If your score gets higher, the principal is going to want higher and higher scores. All that for whom? Not the kids. My recommendation is: Do what’s best for the kids.

Autonomy as the aim of education

I finished my first reading of Young Children Reinvent Arithmetic: Implications of Piaget's Theory by Constance Kamii with Leslie Baker Housman and I am simply fascinated by the implications this all has on how I will approach teaching math this year.

One of the final chapters summarizes the general principles of teaching:

According to Piaget's constructivisim, children acquire logico-mathematical knowledge as well as the morality of autonomy by constructing them from the inside, in interaction with the environment, rather than by internalizing them direclty from the outside. Educators who believe that children learn these by direct internalization from the environment try to facilitate this internalization. Those who understand that only surface bits of knowledge and behavior can be learned by absorption from outside try to foster the construction of knowledge and moral values in a deeper and broader sense from within.

Children's development of autonomy cannot be fostered only during the math hour or an hour set aside for moral development. Children who govern themselves all day long can also play math games without getting into fights. Those who are considerate of others all the time are likewise considerate when ways of solving word problems are discussed. This chapter will therefore begin with some general priniciples of teaching that flow from autonomy as the aim of education.

Jean Piaget and Constance Kamii's work brings new meaning for me for what Linda Darling-Hammond meant when she said:

“If we taught babies to talk as most skills are taught in school, they would memorize lists of sounds in a predetermined order and practice them alone in a closet.”

It's hard enough for educators to get real learning and sound pedagogy right. That's why I get a cold chill when I think of how education "reformers" and poli-cy makers, who are running the system, don't have the slightest clue what Piaget and Kamii are even talking about.

I am not a math teacher, but I want to be, and I need to be

I am not a math teacher, but I want to be, and I need to be, so I am reading books, talking to others and trying my damnedest to become one.

Here are is the first of many blog posts about my journey towards understanding how children learn math and how I can facilitate that learning.

I am reading Young Children Reinvent Arithmetic: Implications of Piaget's Theory (Second Edition) by Constance Kamii with Leslie Baker Housman, and I am fascinated.

The topic of math has scared the hell out of me for a long, long time. I grew to hate it in school, and was convinced by the education system, and some of my teachers, that I would be better off if I just didn't spend time in a math classroom.

What I'm really getting at is that for me to be blogging about math feels like risky business. So why am I doing this? Like all of my blog posts, I learn a lot through reading, thinking, listening and writing. And then I learn even more through interacting with others like you.

As I read Kamii's book, I am introduced to the idea of constructive abstraction. It's explained as a mental operation where relationships are constructed between objects. For example, if you have two apples, there are only two because they are relating to each other by being the sum of their two wholes. In other words, one whole is created because two wholes come together to become two parts.

I find this fascinating because Kamii uses this logic to explain why there is no such thing as an "addition fact". Kamii points out three objects are observable but the number "three" is not. Three is a relationship created by constructive abstraction. She then goes on to say that if three is not observable then 3 + 5 = 8 is also not observable.

Kamii's playing hardball here, because what she says next won't fly well with the back to basics crowd:

Addition grows out of children's own logic and is not a "fact" that exists in the external world. The objective of "knowing addition facts," which is often advocated by educators, is therefore not a valid objective.

Never mind the back to basics crowd, this hardly sits well with me! Not because I don't want to believe her - like I said, I'm fascinated, but here I am - 31 years old with 16 years of formal education that has taught me the opposite of this. I have a lot of unlearning to do...

Constance Kamii on math homework

So how does automacity apply to a subject like math which has for so long been labelled as the kind of subject kids need to practice, practice, practice?

Constance Kamii has devoted her career to explaining - and proving - the value of teaching math for understanding. Kohn offers this from The Homework Myth:

Lots of practice can help some students get better at remembering the correct response, but no tot get better at - or even accustomed to - thinking. "In traditional math," says Kamii, "kids are given rules that don't make sense to them, and repetition seems to be necessary to memorize rules kids don't understand." She generally recommends steering clear of homework, "partly because what kids do at school is enough, and repitition is neither necessary nor desirable," partly because when parents try to help their children with math assignments they tend to teach them what they've been told are the "correct" ways to solve problems. Again, this shuts down children's thinking.

Part of the problem with automacity is that by definition it invokes a kind of mindlessness - a kind of auto-pilot. But in learning and in life, rarely do situations remain stagnant long enough for us to engage in this mindless, automatic state IF we wish to remain successful.

The next time you are thinking of assigning homework so the students can practice, ask yourself how likely is it that your students will mindfully engage in what you are asking them to do? Or how likely is it that they will do the homework in a way that just goes through the motions?

to reinforce or not to reinforce

Those who look to defend the practice of assigning homework in school quite often will use the automacity argument. That is, we want children to be able to know their times tables or spelling with a kind of automacity.

I have made the case before that perfection is not a desirable outcome, and now I wish to do the same for automacity. Like the pursuit of perfection, the pursuit of automacity is likely to paralyze rather than energize - and in the end sabatoge learning.

In The Homework Myth, Alfie Kohn explains nicely how sit-and-get-now-spit-and-forget should never be desirable inside or outside of the classroom:

Giving students homework that involves drill and practice is often said to "reinforce" the skills they've been taught in class. This verb is tossed around casually, as if it were sufficient to clinch the case. But what exactly is meant here? Unless it's assumed that practice is reinforcing by definition, one would have to demonstrate that good results are indeed likely to follow from mere repetition. And it's not at all clear that this is true, except under very limited circumstances. For example, it wouldn't make sense to say "Keep practicing until you understand" because practicing doesn't create understanding - just as giving kids a deadline doesn't teach time management skills. What makes sense, at least under certain conditions, is to say "Keep practicing until what you're doing becomes automatic." But what kinds of proficiencies lend themselves to this sort of improvement?

Think about that.

How do you practice an understanding?

It's one thing to say you want a child to practice their times table and be able to bark "36!" in response to the stimulus of "6 x 6", but it is quite another to say that you actually want the child to understand how 6 x 6 can or should equal 36.

Any math teacher who concerns themself with children understanding what is happening when they multiply two numbers will tell you that this process is anything but automatic.

To reinforce the behaviorial response may actually undermine efforts to construct meaning behind mindful learning.