I gave a presentation at a gathering of a local group who invited me to speak about math education. You can view it here. And if you are interested in what I talk about, you can buy the book as well for the low price of $7.83 (excl shipping).
In my last “smart and thoughtful post” (the parlance that edu-pundits use when they refer to each other’s writings), I talked about “understanding vs procedure”. The quote at the end from a teacher in New Zealand seemed to ruffle the feathers of some who took to Twitter to state that they believed otherwise.
In all these discussions of “understanding”, those who believe it is not taught and that students are doing math without knowing math rarely if ever explain what they mean by understanding in terms of how it translates or transfers to problem variations or new areas in math. For example, a student who has learned the invert and multiply rule for fractional division may not be able to explain why the rule works, but may have an understanding of what fractional division represents. The student then uses the latter to solve problems requiring fractional division.
Anna Stokke, a math professor at University of Winnipeg has also addressed the issue of student understanding in math and echoes what the teacher in New Zealand said. She has kindly given me permission to quote her:
When we teach, most of us generally do teach students why things are true but I sure don’t want my students going through the understanding piece every time they solve a problem. What a waste of time! The point, I think, to get across to students is that there is a reason why everything in math works the way it does and you could figure this out if you need to (because you WILL almost certainly forget).
With the consultants I’ve met, who always push this stuff and insist that kids aren’t fluent unless they can explain everything to you, it seems that they themselves just figured out that there are reasons behind procedures in math as adults. Then they’re angry that their teachers (supposedly) didn’t explain all these things to them. They’re certain that they would have liked math more and done better if only their teachers would have focused on understanding. So, their mission is to make sure that all kids are forced to explain their thinking at every step. Pure torture, really.
Funny thing is, that the understanding piece is a lot more difficult for students. They generally don’t like it and it’s something that really comes with much experience and mathematical maturity. It won’t make students like math more if we spend more time on understanding…it will just confuse and frustrate them more. In my experience, I’ve found that students like step-by-step procedures and algorithms more than anything else.
What people in the “understanding uber alles” crowd likely mean when they talk about understanding probably has to do with words. They would probably be happy with words that didn’t ensure that the kids could actually DO the problems: a “rote understanding”.
What with Robert Pondiscio’s welcome and well-written article extolling the benefits of Direct Instruction (Zig Engelmann’s method for instruction) and thereby praising direct instruction in general, there are indications that others may be following suit. I just read a blog piece by a math teacher who has reached the eye-opening conclusion that conceptual understanding doesn’t always have to precede procedural fluency. In fact, procedures may not be the bogeyman that math reformers have been saying they are for the past hundred years or so.
And it isn’t as if math teachers have routinely refused to teach the conceptual understanding. It’s just that if you’ve spent any time at all in a classroom, you will have noticed that your students glom on to the procedures. And unless the conceptual understanding piece was part and parcel of the procedure (as is the case with adding and subtracting with regrouping) few if any remember the underlying concepts. This has led to math texts now “drilling understanding” by making students do the conceptual understanding piece as if it were the algorithm itself; i.e., 3 by 5 rectangles and shading the appropriate parts to represent 2/3 x 4/5 as a means to “understand” what fractional multiplication is.
The belief still persists that in order for students to understand, math must be made relevant. It just can’t be that if students know the procedures and can do them, and use them to solve problems, they really do not care if the problems are relevant or not. And so we have statements like this which appeared in a recent Education Week testimonial/polemic that passes as evidence-based, research-based, relevance-based, brain-based and any other kind of “base” you can think of:
Math lessons, on the other hand, have historically focused less on real-life connections. Like many students, I excelled in math by memorizing rules and tricks. In college, I trained to teach social studies, but became a math teacher by accident because I had earned enough math credits to qualify for a math teaching certification.
Never mind that the author of the article may have benefitted from what she calls “rules and tricks”.
In any event, it appears that there may be more teachers who had insisted we are at war with Eastasia now coming out from the woodwork to say that we’ve always been at war with Eurasia, though it comes out more like: “Hey, procedures aren’t that bad, and most kids don’t really get the understanding til later.”
I will leave you with the words of a math teacher I know from New Zealand who puts it this way:
A few years back I started explicitly telling my students “I don’t care if you understand it, provided you can do it” when they complained that they “didn’t understand”. I tell them that when their exam papers are marked there are no marks for “understanding”. I follow that up with saying that understanding will inevitably follow in time, provided that they could do the skills, but that it would not follow if they couldn’t do the skills.Now that isn’t to say that I don’t teach the reasons for things — I teach invert and multiply explicitly, but I also explain why it works. What I don’t do is fret about whether they understood my explanation, and I don’t let them not do something because they “don’t understand”. I most certainly do not try to teach understanding of a procedure to a student who can do it accurately.Some students find that truly liberating — they can get on with learning the Maths without any pressure to have to understand the whole picture first. Most just do what they always have done, which is do what the teacher asks them to do and not worry about understanding because they never have. To the fury of many reformers, most kids really don’t want to understand very much.
Another Happy Land story about how schools are seeing the light and not teaching math in the way it used to be taught. It is a given in education and apparently also for reporters, not to ever challenge the premise that the way it was taught never worked. This story is no different.
We start with the classic notion that math shouldn’t be rote memorization (as if that is what traditionally taught math is about) but about critical thinking.
At Fair Oaks Elementary in Brooklyn Park, teacher Michelle Kennedy pushes her math students to give her more than just the right answer.
Her tactic was on display during a recent lesson when she asked her class: What is three plus three?
“It’s a six!” a kindergartner blurted out.
“Why?” Kennedy asked, unsatisfied.
“I saw it in my head,” the timid student explained.
“How did you see it in your head?” Kennedy persisted.
“I see a three and a three,” the student answered.
Really, folks, kids really do get what addition and subtraction are about without having them explain it each and every time. But memorization is a no-no unless students show that they “understand” what is going on. And the pay off is evident; I see high school students counting on their fingers to get the answer to 7 + 8; they are definitely showing understanding of what addition is about by combining the numbers, rather than just pulling it from memory.
The new approach: less memorizing formulas and more focus on understanding math concepts and building up kids’ confidence to do math. School leaders say the changes are necessary to shift the emphasis from boosting test scores to better preparing students to excel in college and in the workforce. “Our instruction needs to change to meet the needs of today’s workplace,” said Kim Pavlovich, director of secondary curriculum, instruction and assessment for the Anoka-Hennepin School District.
Most of the school districts mentioned in the article use some form of discovery-based programs, such as CPM. They also use other chestnuts such as Everyday Math, whose spiral approach–in which students partially learn a concept and then bounce to an unrelated topic the next day and eventually spiral back to the first concept which by now they have completely forgotten–has gone unquestioned by the powers that be. The publisher simply tells the teachers to “Trust the spiral” and those words are repeated to parents.
There is one district using “Math in Focus” which is based on the programs used in Singapore. And though such series has been Americanized to include the aspects of discovery and explaining things that defy explanation, it is at least a step in a better direction.
But for the most part, people are happy in Happy Land with programs like CPM:
On a recent Thursday, eighth-graders at Roosevelt Middle School in Blaine tackled math problems together using the CPM program that focuses on teamwork. They sat in small circles in a classroom, decorated with motivational words such as “I’m going to train my brain to do math” and “Mistakes help me improve.” They measured the rebound ratio for a small ball in groups of four, carefully explaining their answers to each other. They demonstrated to the teacher who was walking around the classroom that they knew how to at least solve a decrease in a quantity using graphs and measuring sticks.
Math teacher Carrie Paske peppered each group with such questions as “Tell me why you think that?” to gauge their critical thinking skills.
In her class, students use objects like algebra tiles to help them visualize algebra. Homework also has become less of a burden because students take home no more than five problems.
Yep. Everybody’s happy all right! They even have the Jo Boaler-inspired quotes to keep them going. The question that’s never asked, however, is how many students who make it into AP calculus in high school and major in STEM fields have had help at home or from tutors or learning centers. And how many students how have not had such help are still counting on their fingers and doing poorly in math in high school?
In this article, we learn that the number of North Dakota kids who are homeschooled more than doubled in less than a decade.
The State Superintendent of schools put her spin on the trend:
“There’s an increasing desire from parents across the United States to really make sure that their child has an individualized, personalized learning system,” Baesler said. “Public schools are moving in that direction.”
Well, if “individualized, personalized learning system” means teaching kids facts using direct instruction, with math and grammar practice thrown in the mix, I would agree. I tend to think public schools are probably not moving in that direction though am open to evidence that proves otherwise.