Month: November 2013

I’ve been a high school math teacher for 11 years now, and I’ve also been tutoring students privately for even longer than that. Consequently I’ve seen the whole spectrum of math students. Everything from the freakishly gifted to the astonishingly weak, For the most part I think this is fine. Some people are wired for certain things and some are not. I am not wired to be a sprinter. I could train my butt off for years and still not qualify for a track team. I’ve made peace with that.

What I don’t think is fine however, is the growing number of math-phobic students I am seeing. Students whose deep fear of math is so intense that it is almost impossible to determine where their strengths and weaknesses in the subject are. To understand what I mean think of a person who suffers from stage fright so severely that every time they sing in front of even a small group of people their throat closes up and all they can manage is a pathetic croak. Anyone listening would conclude this person is a terrible singer. Yet it may not be true. The neurosis camouflages the talent and it’s impossible to know what the person’s singing ability really is. What makes it worse is that it is extremely difficult to evaluate the cause of poor performance. After all, some people just can’t sing.

In math these students whose fear interferes with their performance very often conclude that they have no ability in the subject, which further feeds the phobia. A seriously vicious cycle that is difficult to break, even after it’s been recognized. So my question as of late has been, what’s causing the increase in math-phobic students?

I don’t have research to back my conclusions. It’s all purely anecdotal. However these observations have been made from the trenches. I see these students every day, in a classroom setting and one-on-one, for over 11 years. Here are my thoughts.

Poor Evaluation Criteria
More and more I am becoming convinced that this may be one of the single biggest causes of arithmophobia. I am talking about the alarming tendency for students’ grades to not reflect their ability, due to poor evaluation criteria. I’ll give you an example.

and here is another student’s work, also graded:

The first student received a mark of 2/3, which rounds to 67%. What are we to take from this? Imagine the student coming home with a report card that says 67% in math. What would the parents conclude? What would an independent observer (like a university) conclude about a grade of 67% in math? The easiest and most likely answer is that this is a student who grasps roughly 67% of the concepts covered in math. With respect to this question and the topic it tests, it means the student grasps only 67% of the concept of solving linear equations. Now based on their work, do you believe that is a true assessment? What would we have this student believe? It’s disturbing to say the least.

But significantly more disturbing is the grade of 0/3 assigned to the second student. This student answered the question correctly, however the traditional approach is to assign one mark per step in the question, and since the student did not show any of the expected work, he lost all marks. Now he has 0%. What would that say to parents and universities? Most disturbingly, what does it say to the child?

Stop and ask yourself what it means to solve an equation. The above equation, translated to English, states that

“There is a number which is multiplied by 4 and then the product is reduced by 3, for a result of 29.”

The instruction “Solve for x” means

“Tell me what the number is.”

Student 2 has successfully done just that. Period. End of discussion. Not only has he correctly answered the question, but in doing so has demonstrated that he understands the question and has the higher level thinking skills to answer it without employing any traditional algorithms. And we work in an educational system which has evolved to tell this student that he is so bad at math he gets a zero. Shame on us. Shame on us all.

So what happens to this student? Well from my experience he either dismisses the subject as “a stupid bunch of rules” (and who can blame him? When the answer is so obvious what value is there in writing down a bunch of steps that do nothing more than add tedium?), or he “learns” that to be good at math you have to suppress your instincts and replace them with the all-important STEPS. And let me tell you something. By the time you get to senior math in high school, there are a lot of steps! There’s no way most of us – myself very much included – could memorize all those steps, know precisely when to apply them, and do so with complete accuracy and precision every time.

Enter fear.

Imagine for a second you are a dog. A puppy. You mean no harm to anyone and in fact are a bouncing bundle of happiness and joy. Unfortunately you have an owner who has anger issues. You’ve discovered that your owner hates it when there is pee on the carpet in the house. The reason you know this is because every time he discovers any he loses his temper and yells. So in order to help, you begin peeing in hiding places around the house. To a dog this makes a lot of sense and is very considerate. Unfortunately all this does is make your owner even more angry, to the point where he smacks you every time he discovers the hidden pee. Result? You are now afraid of the owner, and afraid of peeing. Nothing productive comes of this because despite your best efforts, and despite the fact that you are doing what you think is right, you are still getting in trouble. That is a recipe for fear. And that is what happens to students who do what they think/know is right, but get rewarded with marks like 0/3 for their efforts. How can a person continue with a positive attitude under those kinds of circumstances?

What also happens to a large number of students is that over the years, as they fail more and more to memorize the right “rules”, they become more and more disillusioned with themselves. The mathematics becomes totally obscured by the algorithms, to the point where students believe that the algorithms are the mathematics, and can hardly be convinced otherwise.

I tutor a student named Randy and she is in grade 7. Here is a question from a test she wrote recently.

Sam has answered the question “7 – 3 ½” with “4 ½”. Sam says this is because seven minus three is four, and then there’s an extra half to make four and a half. Is Sam correct? Explain.

Here is what Randy wrote:

Sam is not correct. To answer the question you have to convert 7 and 3 ½ to improper fractions, then subtract the numerators, then convert your answer back to a mixed number. This is what Sam should have done:
7 – 3 ½ = 14/2 – 7/2
= 7/2
= 3 ½
So the correct answer is 3 ½

For this answer Randy received a “2+” which is a mark out of 4, with these comments from the teacher: “What was wrong with Sam’s thinking? How could he modify his strategy so that it would work? Expand on your answer.”

Hmmmmmmm. My thoughts as a teacher were immediately “Those comments would have made good questions for students to answer on the test instead of criticisms of Randy’s answer”. In any case let’s have a look at how this result impacted Randy.

So marks out of 4 like this one can be roughly converted to percentages, which they ultimately will be for reporting purposes. A mark of “2+” converts to around 65%-70%. I implore you, dear reader, to tell me just exactly how Randy has shown her capabilities in subtracting mixed numbers from whole numbers to be 30% less than perfect. The message to Randy?

Because you were unable to extrapolate from the word “explain” that I, your teacher, was expecting you to delve into the mind of a person who, unlike yourself, can not subtract mixed numbers from whole numbers, I conclude that you, Randy are a mediocre math student, at best. Despite the fact that the question was in two parts (“Is Sam correct?” and “Explain”), and that you addressed both correctly, you should have known that what I was really looking for was for you to help Sam understand why his thinking was wrong, despite the fact that it did not say this anywhere in the question and despite the fact that Sam is a fictitious person. Please work harder from now on so that you may become a better math student.

Randy was in tears over her results. She said she was sure she understood the material going into the test but she’s just bad at math and she hates it and she is never going to be good at it. It took quite an effort on my part to show Randy that she completely and perfectly understands subtraction of mixed numbers from whole numbers and that the real flaw is the question. I’m not sure she is totally convinced and her grade in math will certainly not reflect what I know to be true so it will be a difficult pill for her to swallow. Randy is developing a serious case of arithmophobia based on experiences like this. She is not wired to “know what the teacher means”. She reads instructions and takes them literally, and then answers them as best she can, usually correctly. But since there is more wrapped up in the evaluation criteria than is revealed in the question itself, Randy is rewarded for her efforts with marks like “2+”. To her this makes math incomprehensible, and who can blame her? To her math is now a mysterious subject with weird expectations that you have to “just know”, and what hope does she have of being able to do that?

So what can we do? The answer is as simple to state as it is difficult to implement in today’s education environment:

Let’s start teaching MATH again. And when we grade a student’s work let’s stop comparing what they did to some sort of “template of perfection” and instead evaluate what the work we see says about the student’s fundamental understanding of the mathematical concepts. Solving an equation means finding the values of the variable that make the equation true. The fact that we have algorithms for solving equations is wonderful and essential for very difficult equations, but let’s not punish students who are able to understand and solve without the algorithm! Let’s celebrate those students because they are the ones who really get it. The algorithms can be introduced and reinforced later when the equations get harder, but it serves no purpose to tell a student like that they are bad at math, for they are truly not. And for students like Randy? Let’s throw away the rubrics and fancy words and assess what their work tells us about their abilities. If we want Randy to extend her knowledge to be able to help Sam modify his strategy so that it will work let’s help her with that, but there is very little value in tying her grade in math to that ability, unless that ability is very specifically what we are trying to teach and assess, in which case we need to ask ultimately how much is that worth and how should it be reflected in the grade that she will use to determine her performance?

Arithmophobia is real and it is getting worse each year. We must change what we are doing if we want to reverse the trend.

Thanks for reading,

Rich