Math is a dangerous subject to teach in today's high-stakes testing accountability environment. It's dangerous because it is the kind of subject that can be used to create right and wrong, black and white kinds of assessment tools. I'm talking specifically about multiple choice exams.

The best math teachers are those who resist the pressures to reduce mathematics to such a narrow dichotomous thinking. The art of mathematics is not found in being 'right' or 'wrong' but in the thinking and problem solving. Because of this, it can be said with confidence that these kinds of high-stakes multiple choice tests actually end up measuring what matters least.

It is important to never again brag about high scores or to become saddened by low scores, and that we must resist the temptation to allow these poor tests to encourage poor teaching.

Here's what I mean:

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 test out correctly on the multiple choice tests. Heck, I could even get the right answer on the short answer 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. So I can get the right answer on the test, but there is nothing mathematical about my (lack of) understanding for dividing fractions.Another problem with testing mathematics is that questions may look elaborate or challenging, but they may be superficial.

In his book The Case Against Standardized Testing, Alfie Kohn illustrates how a seemingly difficult question can be testing rather shallow skills. He cites from Al Cuoco and Faye Ruopp's

*Math Exam Rationale Doesn't Add u*p from the Boston Globe (1998):1 2 3 4 5 6tn 3 5The first two terms of a sequence, t1 and t2, are shown above as 3 and 5. Using the rule:tn = tn-1 + t n-2, where n is greater than or equal to 3, complete the table.

Kohn writes, "This is actually just asking the test taker to add 3 and 5 to get 8, then add 5 and 8 to get 13, then add 8 to 13 to get 21, and so on."

Cuoco and Ruopp conclude:

The problem simply requires the ability to follow a rule; there is no mathematics in it at all. And many tenth-grade students will get it wrong, not because they lack the mathematical thinking necessary to fill in the table, but simply because they haven't had experience with the notation. Next year, however, teachers will prep students on how to use formulas like tn =tn-1 + tn-2, more students will get it right, and state education officials will tell us that we are increasing mathematical literacy.

These examples go to show that too often math tests lack validity - meaning the results of these tests don't tell us what we would want them to tell us. It is far too easy to make tests that don't provide us with valid data on students' mathematical skills.

All this means that we must be very careful placing too much, if any, emphasis or importance on these tests.

Joe,

ReplyDeleteI echo your sentiments about fractions. The other question that would interest me in finding out is how many math teachers have no idea why we "flip" the fraction? Or how many social studies teachers don't know the difference between communism and socialism? Or how many English teachers don't know how to use a semi-colon?

The next question I would have is: do they need to? (I think so, but some would debate this.)

I believe in curiosity being one of the driving forces in learning as I would imagine you do as well. I also believe in self-directed learning (to an extent). So if I believe in these concepts then shouldn't it mean that the students will develop the curiosity to find the answer out for themselves for it to have any legitimate meaning anyway?

Your contention is spot on. The creativity and thinking that exists in mathematics (as well as all other subjects) is being killed on a daily basis by a number of factors including multiple-choice tests.

I would go a step further and place some of the blame on teachers who are not curious about their own subject. It'd be a fun little "game" to play.

Cool post.

Joe, there is validity in Aaron's question. When I work with teachers who teach advanced mathematics, they often do not know the "whys" of their content or even where their content serves a purpose. This is critical knowledge if we are to make learning meaningful. However, math teachers are not alone. The other disciplines have their share of unaware teachers. This presents a major obstacle to bringing relevance to instruction. We've got to know more—and often dig for it—than the textbook provides.

ReplyDeleteWhile I understand (and agree with) the main point of your post, I think you picked a bad example.

ReplyDeleteI also don't know why you flip the fractions, but I would argue that this is an example of something you don't really need to understand, it's sufficient to know how to do it. Of course this doesn't apply to everything - knowledge of function can be applied to other areas in many things, but not in all. Some basic skills are just tools that you combine to achieve more complex tasks.

From my own field, I would never consider explaining to a learner confronted with the present continuous form for the first time why it's formed by using the present simple of to be and adding the present participle of the verb. It doesn't help the learner to know why - they simple have to know how.

I agree with you about maths focussing on right and wrong answers, and marking maths exercises is a tricky exercise in itself. How much weight should be given to a correct answer? A student who has solved a problem using the correct workings, but has made a simple error at the start ends up often with an answer that is nowhere near the required answer. Setting the balance between the working and the right answer is far from a trivial exercise for the teacher. Ultimately, Someone who gets the working right but comes up with the wrong answer every time is no use to anyone who requires accurate answers. Sometimes, being right is more important than doing it right.

I agree wholeheartedly with Aaron's criticism of teachers who are not interested in their subject. For me, that would be grounds for removal from a teaching post.

Thanks for a thought-provoking post.

Olaf

Joe,

ReplyDeleteThis is my first year NOT to teach math. I LOVE math, its the way my brain thinks (shocking for someone who still doesn't have multiplication tables memorized). I quit teaching math because I was so over the stress of testing.

This past summer I attended a 2 wk workshop for math called "Patterns I." I went in there very curious, especially knowing manipulatives would be used. I almost had a bad attitude about it. I had the mindset of you must teach formulas, processes for students to be successful.

The workshop taught me how wrong I was. For a majority of the time we were given "menus" where we were to "Find the Formula." Here are examples of some: http://twitpic.com/1fbxo1 and http://twitpic.com/1fbzrc (I don't have digital copies - might be part of my bad attitude lol!)

I did not walk out of the course knowing anything new about the quadratic or slope intercept formulas. They were still the same. But I did have a better understanding of why the x was x. All of those chart of X and Y (input/output) made so much more sense.

One of my coworkers uses this concept in his classroom. He teaches the general level classes. My other coworker uses traditional math methods of memorizing, she teaches the Pre-AP course. When the standardized tests came back there was not a huge gap between the classes. Actually they were about equal. But I believe there will be a difference in future. The students whom he teaches seem to have a bigger grasp on WHY they have to learn concepts and WHY it works. His kids are on my "team" so in discussions I see this thought process happening.

Kathy Richardson has wonderful ideas and information on this. Even how it can be assessed. It is very impressive research.

Like Aaron said many teachers aren't THAT interested in learning something new for their subject. It is one of those "if it ain't broke" attitudes. I HAD the attitude when it came to math, I no longer do, but we aren't that lucky when it comes to others!

If kids don't need to know why they flip the fraction, then who will? How will we know if the trick is no longer a good trick if we don't know why we do it.

ReplyDeleteWon't these kids grow up to become our future math teachers? Won't they then teach the way they were taught by perpetuate tricks and shortcuts while having little to no understanding for what they are doing?

Don't we also need to ask why we are teaching this kind of notation and table filling out? Or why we need to teach how to divide fractions?

ReplyDeleteThere are a very few math majors in college who need this as a foundation. There needs to be some critical analysis of what we teach, not just what we test.

Most of my career was as a hs math teacher. I never believed in memorizing formulas and algorithms. You will memorize the things you need to through their frequent use. Other things can be looked up.

ReplyDeleteI tried to teach kids ways they could reason out how to do stuff because you're going to forget rules and procedures.

For instance, as I became more experienced, I stopped teaching the classic "rules" for adding and subtracting integers and stressed concrete reasoning with sketches and manipulatives.

My students didn't memorize the distance formula in geometry. We talked about how it worked in practical terms and why you did each part of it.

I strongly advocated sketches, diagrams, graphic organization of problem information, being able to explain why you did what you did, and finding alternative ways of doing things.

Math is exercise that should be developing logical reasoning skills as lifting weights is an exercise to develop muscular strength. The math classroom is the mind's weight room.

Math is not nice, pleasant, and gentle, it's hard and rugged, and dirty if done right.

National math test scores continue to be disappointing. This poor trend persists in spite of new texts, standardized tests with attached implied threats, or laptops in the class. At some point, maybe we should admit that math, as it is taught currently and in the recent past, seems irrelevant to a large percentage of grade school kids.

ReplyDeleteWhy blame a sixth grade student or teacher trapped by meaningless lessons? Teachers are frustrated. Students check out.

The missing element is reality. Instead of insisting that students learn another sixteen formulae, we need to involve them in tangible life projects. And the task must be interesting.

A Trip To The Number Yard is a math book focusing on the building of a bungalow. Odd numbered chapters cover the phases of the project: lot layout, foundation, framing, all the way through until the trim out. The even numbered chapters introduce the math needed for the next stage of building and/or reviews the previous lessons.

This type of project-oriented math engages kids. It is fun. They have a reason to learn the math they may have ignored in the standard lecture format of a class room.

If we really want kids to learn math and to have the lessons be valuable, we need to change the mode of teaching. Our kids can master the math that most adults need. We can’t continue to have class rooms full of math drudges. Instead, we need to change our tactics and teach math via real life projects.

Alan Cook

info@thenumberyard.com

www.thenumberyard.com

Like Thomas, I have been a HS math teacher and ALWAYS taught via concepts, not memorization, because I believed in teaching understanding. Now that I am in MS, I am shocked at how much some of my colleagues struggle with the why's & how's because it was never taught them. This year I was moved to science for various reasons and I'm sticking to my guns about learning concepts and not memorizing facts. I think in the long run my students will be far better off.

ReplyDeleteBTW, you flip the second fraction because division of any two types of of number is simply multiplying by the inverse of the number. So to divide by a fraction, you multiply by its reciprocal (inverse) which is the "upside down" version of the number. :-)

I think that the majority of math teachers explain why dividing by a fraction is the same as multiplying by the reciprocal. The problem is that we don't assess students on whether they can explain the rule, so they don't have an incentive to learn the explanation.

ReplyDeleteI strongly disagree with the criticism of the sequence question. The problem presents the student with a complex mathematical formula. To solve the problem, the student must interpret the formula as "add each number to the previous number to get the next number". This interpretation is an important skill, and it is not a trivial one.

Kohn ignores this step (which is the real mathematical content of the problem), and falsely concludes that question merely asks students to add numbers together. Perhaps he has the misapprehension that mathematics is just about calculating, so he doesn't even see the interpretation step as mathematical in nature.

Good post, Joe! I too knew the "multiply by the reciprocal" rule without a solid understanding, even after years of teaching math. I'd been giving it some thought recently, and thanks to your post I finally put my explanation in print:

ReplyDeletehttp://blog.mathed.net/2010/04/dividing-by-fractions.html

So now's the real trick, and gets at the crux of your post. How do I pass on my deepened understanding, developed over years of teaching math, to a student learning the concept for the first time? And how do I ensure they have an equally deep understanding?

Challenging and dangerous, indeed!

I have taught grade 8 Math for 10 years and have always endeavored to understand and have my students understand the "why" of math. Probably becoming a Math teacher with no university math courses and a horrible math experience in High School has something to do with that. I never understood why I had to flip the second fraction and would flip at will when trying any problem because I thought it was some sort of mathematical magic trick when I didn't know what else to do. When I first started teaching this, I admit I taught the "multiply and flip" method because I didn't know how else to teach division. Years of trying to answer student queries of "why do we do that?", attending PD and reading on the topic have taught me some new methods which are far more clear and make more sense.

ReplyDeleteChocolate, I mean fraction bars are extremely easy to use when explaining division. Common denominators work very well too.

My students always write unit exams which are problem based where they can show me how to find the answer in their work so even if their final answer is incorrect due to miscalculation, I still know that they understand the concept and how to get the answer. They can draw pictures or calculate it out with numbers but to me, it's the work and the steps they take to get the answer that matter. Multiple choice exams do not allow students to receive credit for knowing the how and the why of math because their work is not shown.

Dear Joe,

ReplyDeleteI agreed 1000% with you....I wonder if I've made a mistake in calculating that???? But I couldn't have said it better myself.

If you stick to rehearsing and rehearsing for only the questions on the darn state tests, the kids will loose their minds. For my 2 cents there isn't anything like the wide openess of problem solving...because no matter how good you are at math, there are more layers and yet another approach to be discovered.

It takes a HUGE amount of number sense to be willing to do problem solving problems though. And if students come from places where test prep reigned, they will needs loads of time to talk and experiment and learning to take risks before they can do this kind of math. And even longer before they see any real purpose in it. And even longer still before they enjoy it.

We should do it even if it takes until we all retire....because it's the right thing to do and the kids will be so much better off for us having gone the extra mile for them.

For those that do not know. When you divide by a fraction, you flip it and multiply. The reason is... you can maintain the value of the fraction as long as you multiply both num and denom by the same number. By multiplying both num and denom of the fraction by the reciprocal of the denominator, the result is a 1 in the denominator which removes the complex fraction.

ReplyDelete