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.
On the face of it, a math class for teachers sounds like a very much needed and good idea, which is what I thought when I started reading this article:
“You need mathematical content expertise to teach mathematics effectively,” said PMI Director Andrew Baxter, a Penn State math lecturer, noting that the workshops focus not on the mechanics of elementary math but the logic behind concepts. “It’s not just sometimes you do this and sometimes you do that, but there is a cohesive understructure to how everything connects.”
I agree that math teachers need to know the math they’re teaching. I started to get suspicious when I saw the words “cohesive understructure to how everything connects”. The words had a sense of foreshadowing, like movies in which the shot of an otherwise affable friendly character is held on camera for just a little bit at the end of a scene–a scene in which that character does something nice for someone. The character’s eyes shift just slightly–and maybe there’s even an ominous oboe solo on the music score–and you know this friendly character isn’t what he or she seems.
“One of the things to the problem-first approach is we’re a student-centric model,” Baxter said. “How do we know what to say next? Well, it depends on what the student says. It’s not just because of what the script says. It’s because the students are saying this; they seem to be at this point of their understanding. How do we get them to the next point?”
And so now the music in the movie has a foreboding minor key understructure. “Student-centric model”? Yes, I agree, it’s good when a student has an observation that gets at a deeper level of mathematics and teachers have been taking advantage of such moments for years. They’ve also been on the alert for when a student is going off track, and you need to get the lesson back or you’ll run out of time. Such scenarios are downplayed and cast in a negative light in schools of education as “teacher-centric”. The teachers role is to facilitate and if you run out of time, so what? If it was a “deep discussion”, then “deep learning” has taken place. Yes, well some of us do want to cover the material which I know is old fashioned and out of step. But then again, how many of these programs take a look at how successful students in math have been taught, and what these students have done to gain mastery. In fact, procedural mastery is something that rarely comes up in such discussions. It’s all about “understanding”. Or “deep understanding.” Sometimes “deeper understanding.”
Now, she said, she allocates more time for students to share their thinking about strategies, to justify their answers, and to compare ideas with those of classmates.
I do that with warm-ups at the beginning of class. Another thing left out of this discussion is that many students just want to know how to do it. Sure, we teach for understanding. But what teaching for understanding means in the context of PMI and other similar programs is delaying the teaching of the standard algorithm, and using pictures or other techniques that supposedly provide the “understructure”. By the time they get to teaching the standard algorithm, it may seem like an afterthought to some students who are grappling with “what method do I use?” from the various approaches they have been shown. Such approaches, seen through the lens of adults who have been educated in the manner held in disdain, make sense because they know the standard procedure. They think they have seen the light with these “deep learning” approaches and often say “I wish I had been taught this way.” No, you really don’t.
“You can present one problem, but it has to be a rich problem where there’s a lot of thinking and adjusting as you’re going along so that the students are engaged,” she said. Back in the classroom, she employs the PMI approach with problems such as figuring out permutations for an 18-foot chicken coop perimeter. She continues to draw on eight mathematical principles that PMI emphasizes, including making sense of problems and persisting in solving them, reasoning abstractly and quantitatively, and using appropriate tools strategically.
“That’s about thinking like a mathematician, and PMI stresses that a lot,” Romig said. “And I stress that with my students.”
And there you have it. Instead of presenting the standard old fashioned problems of “Given a perimeter of 20 feet, with one side of the rectangle being 5 ft, what is the area of the rectangle?” which requires students do understand what perimeter is and how to work with it, we have “The perimeter of a chicken coop is 18 feet; what are it’s dimensions?” This is considered rich, and in keeping with the eight “Standards of Mathematical Practice” contained in the Common Core Math Standards. It’s all about thinking like mathematicians. But in the words of one mathematician I happen to know, his view of the SMP and “thinking like a mathematician” is as follows:
Mathematical practices grow by the practice of mathematics; not by replacing critical content with lessons targeting platitudes.
This is something that has yet to become the next “shiny new thing” but I look forward to the day when it does.
In a recent Ed Week Teacher article, the author posits that because students are so diverse, we must cater to all their particular weaknesses, strengths and (dare I say it?) learning styles. And what better way than through ed technology. I got as far as these two paragraphs. (Mainly because the rest of the article was behind a paywall for which I’m extremely grateful):
“Most teachers will agree that student brains are as diverse as their fingerprints. Each student is compelled by different interests, aided by different strengths, and hindered by different struggles.
“Universal design for learning, or UDL, provides a framework for embracing the neurodiversity that exists in all our classrooms. It asks teachers to create flexible learning environments and practices to support a broad range of learners. It might seem like you have to be an expert to start employing UDL principles in the classroom—but in fact, you can start laying the groundwork as early as tomorrow.”
The UDL acronym reminds me when Understanding by Design (UbD) was the hot ticket item. But now UDL is the latest shiny new thing to feel guilty about if you’re not using it, and doing tried and true traditional teaching.
Where I teach, they separate the more capable math students in 7th grade for the accelerated math course. The others who have had poor results in their math classes and on standardized tests are placed in the regular math class. This does make it easier because I have chosen a curriculum/program that is designed for struggling students and breaks things down. I daresay that although the program provides the conceptual understanding behind various procedures and algorithms, and I teach those things, the students glom on to the procedures. And it may be offensive to many, but I’m just fine with that.
In my book “Confessions of a 21st Century Math Teacher”, I described my experiences as a long-term sub at a middle school in California. I filled in for a math teacher who was put on a special assignment for reasons never fully explained. (Or at least explained why her special assignment occurred with little notice; i.e, why the principal had two days to find a sub who had a math teaching credential.)
The principal at the time was in his first year at this school so things were in a constant state of transition. It was also during a time when California was about to implement Common Core the following year, and schools were also transitioning to the new standards.
He scurried around the school with a wide grin on his face and was always outwardly friendly and optimistic. He presided over staff meetings with messages of the future; i.e., what it would look like next year when Common Core went into effect. “No more teacher at the front of the room telling students to open their books to page X and work on the problems.” The Superintendent of that particular school district had a distinct philosophy of education that he outlined in a paper he published on his web site that was strictly constructivist. Students construct their own knowledge. Since we now live in an age where facts/information can be Googled, facts were no longer as important as they once were. Facts in the Superintendent’s vision were “low level” and his aim was for “higher order thinking skills”. I.e. teach students how to learn, and they’ll get the facts on their own.
The new principal was and has been an adherent to such philosophy. Although I didn’t give him a name or nickname in my book, he has stayed in my mind and acquired the name of Principal Gladhand. Gladhand writes a weekly newsletter to the parents, available at the school’s website. I read it every week to see what the current level of thinking is. The most recent one summarizes his last “principal’s coffee” in which he chats with interested parents (i.e., those who don’t have a 9 to 5 job and can take the time for such activity) about their concerns and his vision(s) for the school. I reproduce his latest here and offer an occasional translation, but am interested, of course, in your reactions.
We had a good turnout at our principal’s coffee on Thursday. For those of you who couldn’t make it, [i.e, those who have to work during the day] our casual conversation covered a broad range of topics. Up first was high school- many of our 8th grade parents are looking ahead, and rightly so- we’re only a few months from our students transitioning to their next step. We discussed student concerns and the fact that our students go to the high school well prepared.
We had a lengthy discussion about the topic of Powerschool, students turning in work on time, and how we communicate student progress to parents. This has been a challenging topic for our staff of late because our focus and our curriculum have shifted significantly from when parents were in school. One fundamental shift in curriculum is that students are not only expected to know facts, but are expected to be able to use or connect them in new and novel ways.
A bit of confusion here as to the difference between a novice and expert. Sweller has written about this, but in short, expert’s are able to make connections in new ways through experience and much practice which novice’s are still in the process of acquiring. At the novice stage, such connections are usually guided by the teacher, but in the ed reform world, if a student can’t do that it’s because teacher’s have failed to teach students how to learn and acquire the cognitive skills which apparently exist independent of any prior knowledge.
To truly master a concept takes time and all students learn at a different rate.
See above; difference between novice and expert
In the classroom, this has impacts on how we teach. More teachers are allowing students extra time on assignments, retakes on tests and re-dos on assignments.
Ah, so that difference is accounted for by allowing for extra time, and retakes on tests, etc. Then how is this preparing them for high school; he did say students left the school well-prepared. Could it be extra help via tutoring and learning centers? They are doing a burgeoning business where this school is located.
This shift does not mean students aren’t expected to do the work. In fact in most of our classrooms, lessons are structured so students do more of the thinking work than the teacher (as it should be). In class, this looks like students talking, questioning, challenging and defending answers, and looking for novel approaches to problems rather than simply answering comprehension questions or worksheets.
The standard dictum of ed reform: Worksheets-bad; discussion and talking-good
What it does mean is there is less focus on deadlines (though they are still important) and “one and done tests” where students don’t have an opportunity to ever go back and show they learned the material, and more focus on students actually understanding what they are supposed to. As you can probably guess, then, our old, easier “do this and get the points, then that’s your grade” methods don’t show well enough where our students are on their path to mastery in the way we want.
Does he think that maybe if teachers taught in a more explicit and direct style with important topics emphasized and procedural skills mastered that there wouldn’t have to be so many test re-takes?
Many of our teachers have a foot in both worlds- trying to make our old ways of reporting fit a learning philosophy that embraces deep understanding and allowing for more time and attempts.
This is likely true; some teachers are trying to teach in a traditional manner because it has been shown to work; at the same time, they need to show they are toeing the party line. A difficult situation for any teacher.
Parents need good information to help their kids achieve. Teachers want parents to partner with them. I will be bringing this issue back to staff to work on ways where we can meet the informational needs of parents while providing a clear understanding of progress towards standards. I will be bringing this topic back to our staff to work on ways we can more clearly communicate student progress to parents.
In other words, parents do what we say, and don’t question it. Though we will pretend we’re interested in what they have to say.
Your reactions on this are welcome as always.
Chicago Tribune carries this story about a controversial approach to education being tried at various high schools in Illinois :
They’re abandoning most aspects of traditional classroom instruction and reshaping the way kids learn.
The approach, called competency-based learning, puts the onus on students to study and master skills at their own pace, making their own choices along the way and turning to peers and online searches for answers before they lean on teachers for help. Students may show proficiency on a topic not simply through traditional testing but by using projects, presentations or even activities outside school.
Competency-based; another “base” in the long line of bases; “inquiry-based”, “research-based”, “evidence-based”, “brain-based”. The list goes on.
The pilots are in various stages of planning and implementation, but all focus on instruction changes in high schools, likely a tough sell for parents who learned the traditional way: Teachers lecturing to a group; quizzes, tests and homework; letter grades and GPAs.
Right; everyone knows that doesn’t work. The educationists have been saying it for years, and the press eats it up and apparently many people believe it.
In contrast, students in competency-based programs take the time they need to master skills and make their choices in their academic journey.
Right, this sounds like a winner, especially if they’re trying to get into college and they need certain courses completed and mastered. But “take your time”, we’ll just make believe there’s no time element anymore and everyone’s happy in happy-land.
At Ridgewood High School, math teacher Tristan Kumor started off geometry class on a recent day by asking his students, “How do you want to learn today?”
His 9th and 10th graders sat in groups, with one student using Popsicle sticks to build a bridge. The lesson related to triangles. Kumor traversed the room guiding students individually on their progress, which is part of the way competency-based education plays out.
Yeah, right, that’s the ticket. But I think this has been tried before. PBL or something like that. Or is there some twist to this approach that I’m just not getting?
Computer testing and other work show how kids are progressing, and teachers provide individual feedback to students, acting as facilitators or coaches who monitor student growth and ensure kids are self-directed enough to assist and even teach their peers. The idea is that if a student can teach a peer, it’s clear they know the material.
At Proviso East, signs in classrooms direct students to do several things before going to the teacher: They need to reread the question, check their notes, ask other students for help and search Google for information. If their question is not resolved after all that, they may go to the teacher.
Yes, it all sounds happy and fruitful. Teachers as facilitators or coaches. Students teaching other students. A “three-before-me” approach; God forbid they should ask the teacher anything. We don’t want teachers
teaching handing it to the student, now do we?
There’s no such thing as an F in the competency-based learning world, because failing is considered an attempt at learning that helps lead to mastery.
Right, the Jo Boaler approach which holds that mistakes grow your brain. And if mistakes are that powerful, then failing is even better!
Depending on the school, students might not receive letter grades on report cards. Some Illinois schools already use a numerical approach, such as 1 to 4, rather than letter grades to show academic progress.
And let me guess; no one gets a 4. At least that’s how it’s played out in other schools that have tried similar things.
Educators say the grades and transcripts have been a source of concern, with high schools reluctant to change because that data is used in college admissions.
Well yeah, there’s that, but let’s not let that stop such a promising program.
And even the strongest supporters of competency-based learning acknowledge the challenges ahead as educators, parents and kids process how the program will work.
Right, but let’s just keep doing it until the next shiny new thing comes along and this approach will be tossed on the dust bin with the usual lamentations: “Well we tried competency based learning but that didn’t work because –list of reasons follows — but this approach fixes those things.” Just as long as it’s not traditional ed. Because God knows, that certainly hasn’t proved out, now has it?
This particular rant/polemic is by someone who rests her authority on being good at math, and most likely because she learned it in the way she now says is ineffective. Like many of the defenses of Common Core math, hers rests on a denigration of traditionally taught math.
I’ve written much on Common Core math and I maintain that it contains watchwords (or dog whistles per Tom Loveless of Brookings) of reform math that cause it to be interpreted along those lines. That is, emphasis on alternatives to the standard algorithms prior to teaching the standard algorithms in the belief that teaching the standard algorithms first eclipses the “understanding” of the algorithm. And the “understanding” of the algorithm is believed to be essential for students to solve problems.
I’ve also maintained that Common Core can be taught using traditional methods, and the standard algorithms can be taught first, with alternative approaches later (as it used to be taught). Jason Zimba has agreed with me on this in articles.
To wit and for example:
The criticism that I referred to earlier comes from math educator Barry Garelick, who has written a series of blog posts that aims to sketch a picture of good, traditional pedagogy consistent with the Common Core. The concrete proposals in his series are a welcome addition to the conversation math educators are having about implementing the standards. Reading these posts led me to consider the following question:
If the only computation algorithm we teach is the standard algorithm, then can we still say we are following the standards?
Provided the standards as a whole are being met, I would say that the answer to this question is yes.
With that, let me turn to the blog piece I had linked to originally. The argument starts out with the usual:
The traditional way involves rote memorization and algorithms performed on paper. They require little to no understanding of why the algorithm works. It simply works.
This statement is so far off, it’s not even wrong. In fact, older textbooks do explain why the algorithms work as they do, and later, after students master the algorithm, alternatives to the standard algorithm are introduced. (See for example, this article.) In many cases, students discover these methods by themselves. And it isn’t as if teachers do not teach the understanding; most do. But as many teachers will tell you, many students do not glom on to the reasons, and instead rely on the procedure. Understanding is a process that works in tandem with procedural fluency. Many students will understand in time. And some will never fully understand. There are varying levels of understanding. But being able to solve problems using a procedure or algorithm does not necessarily make the student a “math zombie”.
She goes on:
We forget. All the time. Especially when we don’t understand why the algorithm works or if it has been a long time since we last used it. I can guarantee if I took a survey at Starbucks right now, and asked people how to perform long division, convert a mixed number to a fraction, recite the quadratic formula, factor a binomial, and complete the square — most would fail. Not because math is too hard or people are bad at it, but because memorization and algorithms are not the best ways to retain information. We remember through context, understanding and application.
We also remember through continued application. For example, if I haven’t worked with percent calculations for a while, I have to brush up on it. Same with finding derivatives of certain functions. The survey of people at Starbucks might be different if the majority of customers were practicing engineers. That people forget how how to do something if they haven’t worked with it for years is not evidence that the traditional method of math teaching is ineffective. And like many authors of similar rants and polemics, she also does not provide evidence that her methods are superior to those that she feels do not work.
She then goes into a demonstration of how mental math provides a superior and quicker way of adding two three digit numbers than the standard algorithm and states:
Number sense allows us to have an arsenal of ways to problem solve, including but not limited to the traditional algorithm.
Quite true. And many of us got to that point by discovering these shortcuts ourselves after having been taught the standard algorithm, not to mention that the shortcuts she mentions (and gives a demonstration of) were also included in the old textbooks in the era of traditional math that she and others find so destructive.
She leaves us with:
The Common Core math standards are an attempt to expose your child to this flexible way of thinking. It may not be perfect, but it is in the right direction.”
Yes, and the standards can also be met quite effectively perhaps more so, using traditional methods. The “understanding uber alles” approach to math is likely to result more in “rote understanding” and an inability to solve many simple problems. Interestingly, students in the 80’s and 90’s entering algebra classes in high school knew how to do basic computations with fractions, decimals and percents, whereas now many teachers will attest that this is not the case. Traditional math is the usual culprit, but such finger pointing fails to acknowledge that the reform methods have been around for almost three decades.
Oh right. It’s because reform math hasn’t been taught correctly. I keep forgetting that.