Teaching understanding or algorithms?

I recently read Schoenfeld’s “When Good Teaching Leads to Bad Results: The Disasters of “Well Taught” Mathematic Courses”. 

This paper reports the results of a year-long intensive study of the teaching and learning that took place in a tenth-grade geometry class, which will be called the target class. The class took place during the 1983-84 academic year in a highly regarded suburban school district in upstate New York. The study included periodic observations of the target class and of eleven other mathematics classes, interviews with students and teachers, and questionnaire analyses of students' perspectives regarding the nature of mathematics. The target class was observed at least once a week, and was videotaped periodically for subsequent detailed analysis. Two weeks of instruction near the end of the course, dealing with locus and construction problems in geometry, were videotaped in their entirety. The analyses focused both on the mathematics that was learned, and on what the students learned about the mathematics -- including how and when they would use, or fail to use, the mathematics that they had studied.

This paper explored the difference between performing mathematical algorithms and truly understanding the underlying fundamental ideas of mathematics.  Due to the structure of the study, they were able to quantify certain classroom behaviors; time spent in questioning, active learning time, amount of praise, and amount of feedback.  The study also included the type of grouping, the size of grouping and so on.  Originally, they defined “learning” as how well the students performed on achievement tests.  I say originally, because through the article it illustrates how these tests fail in significant ways to measure subject matter understanding.

The article also suggests that providing students with repetitive routine exercises that can be solved out of context and no significance provided, actually causes the subject matter to seem frivolous to students.  This monotonous work actually deprives the students of the opportunity to apply their learning in a context that is meaningful to them.

Brown and Burton (1978) developed a diagnostic test that could predict, about 50% of the time, the incorrect answers that a particular student would obtain to a subtraction problem -- before the student worked the problem!  Teachers are still currently doing this, when we provide students with “distractors” on a multiple choice exam.  If we already know what mistakes students are going to make, before they make them, we need to start changing our instructional models.

The predominant model of current instruction is based on what Romberg and Carpenter (1985) calls the “absorption theory of learning”: “The traditional classroom focuses on competition, management, and group aptitudes; the mathematics taught is assumed to be a fixed body of knowledge, and it is taught under the assumption that learners absorb what has been covered” (p. 26) This view is essentially implying that the “good” math teacher has multiple methods of covering the same outcome.  Through these multiple methods the students will eventually “get it”.  Unfortunately, what most math teachers fail to realize, is that through these multiple methods we are forcing the students to “get” something entirely different; resistance to change.

Math teachers need to stop testing students on algorithms and more on the understanding of math concepts.  Below is an example of two problems Werthemier (1959) gave to various elementary school students.

Many of the students, who were deemed as high achievers, added the terms in the numerator and then performed the indicated division.  They followed the conventional order of operations of BEDMAS, or to some PEDMAS.  An idea which should not be taught to students out of context, which I have wrote about here.  Even though these students calculated the correct answer, I would argue that these students do not demonstrate any depth of understanding of mathematics.  To truly understand the underlying substance students should recognize that repeated addition is equivalent to multiplication and division is the inverse of multiplication. 

This example truly illustrates that being able to perform the appropriate algorithmic procedures, does not necessarily indicate any depth of understanding.  Also, the sad truth; virtually all standardized exams for arithmetic competency focuses primarily on algorithmic mastery, and not deep understanding of the math concepts. 

This is the first part of the article summarized, more to come….

Student revolts about Standardized Exam

Kathryn Coffey is passionate about public education, and she is especially passionate about teaching reading and writing. She also interested in Cognitive Coaching, education leadership and education policy.
Here blog can be found here.

Here is her story:

 

In the spring of 2005, when my youngest son A.J. took our state’s high-stakes standardized test, the MEAP, (now the MME) he had an ax to grind. Unfortunately, no one really knew how angry he was until after he took the exam. I received a call from the principal requesting a meeting. It seems that my son had chosen to draw a very large and detailed fist with an extended middle finger where his science graph should have been. And, he chose to bubble in his multiple-choice answer sheet with “AC/DC” and “ABBA”, as well. The principal was quite distressed since my son’s unconventional test responses would most definitely effect the school’s test results and would reflect poorly on the district. So, why would he do this? It turned out that A.J. was upset about one of his teacher’s policies that he had tried unsuccessfully to address with the teacher and administration. Feeling he had not been heard, he took his revenge. I had assumed A.J.’s stunt was an isolated incident—until yesterday.

Fast-forward to March 9, 2011. A colleague who supervises pre-service teachers for a nearby university shared a disturbing conversation he overheard while he was visiting a high school classroom that morning.

At the beginning of the class, he was sitting near a group of four young ladies. As Juniors, they had just taken the Michigan Merit Exam (MME) and ACT last week. One of the other adults in the room asked these girls how the testing had gone. One said, “they’re so stupid, I paid attention for the first couple of pages, then I just started bubbling in randomly.” A second girl said, “Yeah, I couldn’t take it seriously, I don’t want (our school) to do well.” One of the other girls responded, “Yeah, if this was for us and they were still giving out the scholarships, I would have taken it seriously.”

What’s particularly interesting to me is that while A.J. decided to take his frustration out on the science exam, it was a foreign language teacher’s policy he was protesting. In addition, the comment above “I don’t want (our school) to do well,” didn’t target a particular teacher or subject area, but apparently the school in general.  

The standardized test movement is based on the assumption that students are actually going to take the test seriously, that the test will measure achievement and will measure teacher effectiveness. Neither A.J.’s performance on the science MEAP, nor that of the girls mentioned above meets those assumptions. This has me wondering, and it raises questions that I believe need to be addressed.

·      Is there any research about whether or not students actually take high-stakes standardized tests seriously?

·      How many teens out there have an ax to grind with the adults in their lives, particularly with their teachers and their schools?

·      How many angry teens would it take for testing results to be corrupted for a given school district?

·      How often does this happen?

·      Do students understand their teachers and school are being held accountable for how well they do on these exams?

Perhaps some understand it only too well. It’s entirely within a teen’s nature to subvert the adults and authorities in their lives. Why would anyone think it’s a good idea to put teacher’s careers and the viability of a school district in their hands?

The art of wait time

I have blogged about the importance of questioning here:

Just as asking a certain type of question is important, we must also allow for sufficient wait time to occur to give a student a chance to think and answer the question.

When I first started asking higher level questions in my class, my students appeared as if I had I just delivered them a dose of shock therapy.  I never realized that it was going to take some time for my students to adjust from answer YES/NO to giving me more in depth solutions.  Once the class had adjusted (which took weeks, not days), I still had to wait on my “Wait-time I” and “wait-time II”.

Wait-time I: the time that teachers wait after having asked a question to receive an answer… three seconds here, feels like an hour!

Wait time II: the time that a teacher waits after a student has answered a question.

When you increase both of these times, with wait-time I being at least 3 seconds, the research (Rowe, 1974a, 1974b, Rowe, 1986) states you will witness the following outcomes:

·         The length of student responses increases by 700%

·         The number of unsolicited, but appropriate, student response increases;

·         Failures of students to respond decreases;

·         Students’ confidence, as reflected in decrease of inflected responses, increases;

·         The incidence of speculative student responses increases;

·         More students inferences are supported by evidence and logical argument;

·         The incidence of student-student comparisons of data increases;

·         The number of student questions and proposed experiments increases; and

·         The incidence of responses from students rated by teachers as relatively slow increases.

It is troublesome for some teachers to wait for 3 seconds, but I encourage you to try it!

If you have ever waited 3 seconds after asking a question, before receiving an answer, you will understand how long this feels.  However, the price we pay for waiting 3 seconds is more valuable to learning than you answering your own questions for an hour.

The art of questioning.

Have you taken the “Questions Only” Challenge?

Here is more on the Art of Questioning:

Our students want to be engaged, challenged and intrigued!  If you disagree, try lecturing to your students for an entire class, and then ask them about how much information they have retained or even bothered listening to.

I believe true engagement and challenge comes from the way we pose our questions towards students.  Teachers need to realize that not only do the tasks or assignments we provide to students determine the outcome of the lesson, but how a teacher addresses the challenges the students may have with these tasks also contributes to the quality of learning the student receives.  Looking back at my previous years, I have given higher level thinking tasks to students, but when a student required assistance my questioning techniques destroyed any critical thinking that had occurred

In 2003, Weiss et al. wrote “effective questioning is relatively rare in mathematics and science classes”, and Redfield and Rousseau in 1981 wrote “asking higher level questions has been shown to facilitate learning”. 

Educators need to stop asking questions where the answer can be a simple Yes/No, and start asking questions that requires responses with deeper thought.  Easy analogy: The deeper of understanding required to answer a question means the deeper level of learning that has occurred.

This does not imply that we should only be asking higher level thinking questions, but instead we could start asking initiating questions of a reasonable difficulty and then scaffold to more challenging contexts.  Most teachers are aware of the taxonomy introduced by Bloom, et al. in 1956, but another strategy was created in 1996 by Penick et al, which can be used in math and science classes.  Penick suggests asking questions that build on the students’ history, relationships, application, speculation and explanation.  Here is how Penick describes questions of these categories:

History – questions that relate to students’ experience:

·         What did you do….?

·         What happened when you….?

·         What happened next….?

Relationships – questions that engage students in comparing ideas, activities, data, etc:

·         How does this compare….?

·         What else does this relate to…..?

·         What do all these procedures have in common?

Application – questions that require students to use knowledge in new contexts:

·         How could this idea be used in design….?

·         What recognized safety issues could this solution solve?

·         What evidence do we have that supports…..?

Speculation – questions that require thinking beyond given information:

·         What would happen if you changed…..?

·         What might the next appropriate step be?

·         What potential problems may result from….?

Explanation – questions that get at underlying reasons, processes, and mechanisms:

·         How does that work?

·         How can we account for…”

·         What justification could be provided for….?

Questions should promote students learning outcomes using their own thought processes.  As teachers should not take the pencil/pen from the student and complete their work, they should also not be taking their learning and thinking away from them either.  To illustrate, how effective questioning appears in a math class I remember when a student was squaring negative numbers and keeping the product negative.

Student: If I square negative 3 the answer is negative 9.

ME: Why is the answer 9?

Student: Because in my calc, -3 squared is – 9.

ME: Forget your calculator, what does it mean to square a number?

Student: (Pause) To multiply by itself

Me: What happens if you multiply a negative number by another negative number?

Student: I am not dumb, I know a negative by a negative is a positive, but when you square…. Oh wait…. -3 times -3 is ….. stupid calculator!

From the outside some would say I taught the student how to square a negative number, but to the student he taught himself.  This is a powerful idea to the individual student.  Our goal, through questioning, is teach our students how to critically solve problems without asking any questions at all.

the true goal of a teacher is to put themselves out of a job, by having students solve problems without their help.

I fear that a common questioning technique by teachers, to students, is to ask "What do you think?"  Here is a take on that by this site:

Everyone asks questions (especially my four-year old granddaughter). It’s how we learn, understand issues, solve problems, and even socialize. But of all the questions you can ask, there is one that invariably leads to confusion, especially between men and women: What do you think?

It’s innocuous enough, and everyone uses it, but it can cause big trouble. Here are just a few examples:

A female employee just finished a project she had spent several weeks perfecting and brought it to her male boss expecting to receive lavish praise. She asked, “What do you think?” and he found a small criticism. She was devastated.

A female employee brought to her male boss an issue about which she wanted to gain some more insight, and asked “What do you think?” He told her what to do, and she felt that she now had to do the wrong thing.

A male employee asked his female boss for advice on a problem and asked, “What do you think?” She started a discussion about the topic when all he was looking for was confirmation that his approach made sense. He left frustrated and disillusioned.

A female marketer had her sale “wrapped up”, everyone loved her product, and they were ready to sign on the bottom line. She asked, “What do you think?” One person raised a minor issue that ended up taking a long time to resolve and almost cost her the sale.

My friend went to buy a dishwasher and had a specific brand in mind. She wanted to “involve” her husband so she insisted he come shopping with her. She asked, “What do you think?” Her husband answered, “I think we should get the GE”. Unfortunately she didn’t want that brand and now had to convince her husband otherwise.

A woman tried on a new outfit and asked her significant other, “What do you think?” He said, “It’s OK”, and she was deflated because she loved how it looked.

What do you think? is a question that really doesn’t ask anything so it can be interpreted in a myriad of ways. There are much better ways to get opinions and information.

Does testing hold teachers accountable?

If you think testing is about holding teachers accountable, here is a take on that idea:

An excerpt from the daily papert:

It is this freedom of the teacher to decide and, indeed, the freedom of the children to decide, that is most horrifying to the bureaucrats who stand at the head of current education systems. They are worried about how to verify that the teachers are really doing their job properly, how to enforce accountability and maintain quality control.

They prefer the kind of curriculum that will lay down, from day to day, from hour to hour, what the teacher should be doing, so that they can keep tabs on it. Of course, every teacher knows this is an illusion. It’s not an effective method of insuring quality. It is only a way to cover ass.

Everybody can say, “I did my bit, I did my lesson plan today, I wrote it down in the book.” Nobody can be accused of not doing the job.

But this really doesn’t work.

What the bureaucrat can verify and measure for quality has nothing to do with getting educational results–those teachers who do good work, who get good results, do it by exercising judgment and doing things in a personal way, often undercover, sometimes even without acknowledging to themselves that they are violating the rules of the system.

Of course one must grant that some people employed as teachers do not do a good job. But forcing everyone to teach by the rules does not improve the “bad teachers”– it only hobbles the good ones.

The real lesson on finance

As I look through our mandated outcomes for finance and the monetary system, we have outcomes such as:

"Use the compound interest formula......"

"Calculate the simple interest.........."

I was searching for some application and stumbled over a blog by Jason or FrugalDad.  After reading his blog I decided to share his ideas about "10 things students should really learn about money".

1. No one owes you a thing. Too many people go through their entire lives with the expectation they are owed something. This is not the case, or at least it shouldn’t be. All you should ever expect is to be judged, compensated and respected based on your work ethic and your ability to create, inspire and hustle.

2. Debt is a cancer. Debt is a cancer on our society, on households, and on us as individuals. It saps creativity. It creates pessimism. It robs your future dollars. It limits your freedom. Avoid debt like the plague. Remember the old adage:

“He who understands interest – earns it. He who doesn’t understand interest – pays it.”

3. Save for emergencies…big emergencies. When you are young and many years from considering retirement (and not earning much), it’s tough to save money. But I have discovered no softer pillow than having money in the bank for emergencies. Aim to save about a year of your basic living expenses in a simple savings account (no risky investments here). With a one-year cushion, you’ll be able to weather storms many others will not.

4. Live simply. In 2011, life seems pretty complicated. By the time you are adults, I imagine it will be even more so. There will be new gadgets and toys and cool services and “got to haves.” The problem is, all these things compete for your earnings. I’m not advocating living like a pauper, but limit yourself to only a few of life’s luxuries.

5. Sleep on big financial decisions. When it comes time to buy a car, or a house, or book your first major vacation as a family, sleep on the plans for a couple nights. People selling you these things want you to act immediately to lock in their commission, as I would expect them to, but remember that you are the one who has to pay the bill. Some of my biggest financial regrets came because of a knee-jerk reaction. Be slow. Be methodical. Listen to your gut.

6. Protect your credit. Not because you hope to borrow money, but because you may find people extending a service to you may do so for less cost if they think you aren’t a big risk. And if those people don’t know you well, your credit score may be their only determining factor. It’s not necessarily fair, but it’s a part of life. Credit blemishes can hang around for a decade, so it’s best to avoid them in the first place.

7. Learn to do things yourself, but don’t be afraid to call in the experts. You may remember the time your dad rescued a toy from the toilet trap, saving us an expensive plumbing repair bill. Or the time I climbed up in the attic to unclog the air conditioner drain. But your dad knows his limitations, and calls in the experts when necessary. That’s what emergency savings are for.

8. Shallow people judge your things, real friends judge your character. Some of the saddest, loneliest people I’ve ever known have been surrounded by the nicest things money can buy. They often acquired these things to impress people they thought mattered, and in many cases it did – temporarily. Meaningful relationships are based on things money cannot buy: trust, respect, integrity, compassion, love.

9. Don’t trade the things you care about for a big salary. Remember what mattered to you most when you were a kid: Family, fun, dreams. These things should remain important to you as a grown-up, but often adults sacrifice these things to earn a big salary. Now, everyone has to sacrifice some to earn a living, but by learning to be content, you may be able to earn a comfortable living while still enjoying other things.

10. Start saving early. Remember those money games we used to play when you were a kid? One of them was an attempt to get you to understand one of the great financial wonders of the world: compound interest. You see, when you save money you earn interest on it. The next month you earn interest on the money you first put in, plus the interest you earned the month before. That’s right; you earn interest on interest. Now carry out that example for many years, even decades, and you can understand how some people are able to accumulate wealth. The trick is, you have to start early.

Parents, consider opening a kid’s savings account to get them started early. Our kids deposit a portion of allowance earnings every couple weeks and it has taught them a lot about the mechanics of banking – completing a deposit slip, balancing their savings register, etc.

Finally, keep in mind something your great grandfather taught your dad about finding balance. Be frugal, but remember to occasionally stop and smell the roses. Life is short, and it is meant to be enjoyed. Take an expensive vacation every now and then. Buy something of your “heart’s desire,” even if it doesn’t make sense financially. Be frugal in other areas of your life to make room for things you truly enjoy.


Click here to see what a trillion dollars look like.