How to balance between teaching to the standardized test vs understanding?

Question

For people teaching high school standardized curriculums such as AP, IB, or A-Levels, how do you find the balance between preparing students for the standardized test compared to ensuring they understand the material?

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In an ideal world, the two are the same. And whether standardized testing is good or not is another question. I would like to highlight that in these three programs, the goal of the course is typically to prepare the students for the respective examinations. This is contrasted with ACT/SAT preparation taking only a small part of class time.

I want to know more about i) structuring the class to target a relatively wide range in students background and goals (including differences in motivation, ability, plans, and outcomes); ii) selecting questions that are good for learning vs that are common on the exam; iii) the role that past exam questions play in instruction and assessment.

EDIT: to allow responses from people who do not teach/research standardized curriculums, the alternate question is "In an environment where the teacher can predict but not change the scope of a math course, how to teach high-level problem solving skills? How big of a role should problem solving exercises play, comparison to theory or other exercises?" The context is important since I am not interested in the teacher changing the course coverage to allow more time spent on harder problems.

Answer 1

Upon the request of the OP I'll elaborate on my comment:

If the AP and other test makers were doing their job the test would be vastly different from year to year precisely to discourage teaching to the test.

I have a variety of different data which informs my disdain for the AP test as it is currently practiced as well as the "SOL's" (Standard Of Learning) tests which shape elementary math teaching in Virginia (where I used to live).

I'll begin with the SOL's. I had a neighbor who taught math to middle school kids, he described how he taught them "tricks" to get the problem right on the SOL. From what I heard, the focus of the learning was on the tricks to help pass the SOLs rather than on building a unified mathematical framework infused with both concept and calculation.

With the AP test, I think they're allowed a graphing calculator now ? Yep:

https://apstudents.collegeboard.org/exam-policies-guidelines/calculator-policies

So, that's not a good sign. I can't see a reason you would need a graphing calculator for a well-designed calculus test. Unless you're testing ability to use a graphing calculator (why on earth encourage that ? there are far better alternatives online and... again, really inappropriate to a calculus placement test). So, I don't have a lot of firsthand testimony from students who took AP calculus and described their course, but I know that as a department we saw reason to be less and less trusting of anything but a 4 or 5 on the exam. At that point, it's more an indication of intelligence and study habit than actual knowledge of calculus.

Why ? Because the test is largely unchanged, so if you study the standard problem types you'll be able to make a good score if you are intelligent. Your inability to add fractions etc. can even be hidden by the ever present calculator.

Or, if you're really clever and a cheat, you can program the answers of the test into your graphing calculator after a scout student takes the test and publishes the questions and answers on the appropriate "help" site. Here I conjecture, but anyone who has experience with the Chegg homework feedback loop knows my speculation is not an idle one.

Beyond the specific situations I list above, I think the general response we receive in teaching students speaks to the abuse of math in schools. I have had countless students (good students) come to me mid-semester the first time they have my course and say something along the lines of "your teaching is so different". Why ? Well, I think there are a couple things I do which are never going to be high on the priority list of a teacher who teaches to the test.

• emphasis is on definitions
• proofs of theorems which involve the methods of the course are given
• compare and contrast definitions
• ask inverse questions
• contrast graphical verses analytical methods
• warn about pathological cases
• discuss historical evolution of math concepts

In contrast, a person who teaches to the test has two main questions:

• which questions are allowed on the test ?
• how can we reliably do the questions as quick as possible ?

In the above mode of teaching, definitions may not even be stated, theorems are seldom proved and the bulk of time is spent solving problems. I am guilty of such teaching in my absolute lowest level teaching, but to teach calculus this way is to rob students of beauty.

This danger is not isolated to calculus. In fact, any suitably standardized test or curriculum has the danger of squelching creativity and the best teaching for the sake of the pragmatism of passing the test.

But, the existence of a standard test is not the automatic death of teaching. For example, my wife took advanced math in highschool in Hong Kong. They have to take a test which is quite serious just to get into the math track if I understand correctly. That test didn't discourage good teaching, rather it promoted it because the test was a bit unpredictable and their were many different kinds of problems you might face. That test was administered in such a way that the teachers who taught the students were not the teachers who proctored the test (a much more serious business than what we find at American institutions).

I'm torn on the merit of standardized testing. In principle, I see great merit in replacing course credits with passing certification tests because many students are trapped in universities whose standards are so low that professors are held hostage by the weakest students.( I can envision replacing a math degree with holding certifications in different subjects. That would free students from the burden of liberal arts dogma) In practice, we cannot teach the course as it ought to be because, well, the students. The existence of an external test which created without direct regard to the weakness of students at a particular institution gives serious students a way to prove their skill. That said, such a test once it becomes too familiar after years of creation and the perhaps inevitable recycling of questions, becomes gamified and discourages good teaching of the topic.

This is why I made my comment, in my view the creators of standardized tests must engage in a certain randomness to fight against the gamification.

Answer 2

You have zero responsibility to teach to the test. A kid who actually understands calculus should get a 5 on the AP exam. Problem solved.

Kids who want test prep can simply download sample exams and familiarize themselves with the types of questions on the exams.

I want to know more about i) structuring the class to target a relatively wide range in students background and goals;

If you have a student with too poor a background to do well in AP calculus, then realistically that student doesn't belong in AP calculus. If that student takes AP calculus anyway, achieves a poor understanding of calculus, and therefore doesn't pass the AP exam, then that's not some horrible tragedy. They can take calculus again in college, and they'll probably pass because of the previous exposure. That's a good outcome.

Answer 3

I acknowledge the subjective nature of the question and hereby provide an answer from my own experiences as a tutor, hoping to inspire better answers from more qualified teachers. I am in the midst of completing my teaching certificate so I'm somewhat familiar with general teaching principles and the different approaches. I don't intend to turn this into a discussion since I have no intentions of persuading anyone or responding to others' claims.

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The three programs for the most part give access or charge a small fee to past exam papers; this is in part to uphold a transparency and consistency in grading and expectations.

Generally in my tutoring sessions, I first emphasize a firm grasp of the background and central concepts, and strong algebraic and error-finding skills. This is more than enough to achieve an above-average grade in such programs. But to achieve the top marks, I rely upon the abundance of past papers. Some reoccuring problem solving skills and motifs are identified and assigned to tutees. When I was in high school, I was taught to just drill past papers questions which was not only time-consuming but possibly also inefficient. I am keen to see if others have the opposite issue of having too much theory and not enough practice.

For weak or average students I assign relatively common questions and for stronger one I give an even spread of difficulties, not emphasizing the easy nor the hard questions. This is mainly to give students a realistic expectation of the exam, and to not overwhelm them. This is easy to accomplish in a private tutoring environment, so I was curious as to how teachers manage this balance in the classroom. In a classroom there are more differences than just skills or math experience. In my experiences as a student and my observations, the most practical and common approach is to give every one the same material and instruction, so to prioritize timely and adequate coverage of the syllabus.

The overemphasis on past papers is more apparent in students, especially if their teacher is subpar and they are looking to "be able to do past papers". I say this despite my high school teacher giving us loads of past paper questions and dedicate about half of in class time on such problems. It is better for the teacher to overemphasize on past papers, because a student has a much tougher time navigating the resources if they were to buy past papers without guidance.

Therefore I believe the primary role of the teacher is to make sure the students understand the material, via using a variety of resources and technology, and to highlight sufficient past paper problems so the students understand specific expectations and reoccuring themes. This should pave a good starting place for if they want to do additional problems on their own.