Archive for September, 2011

In recent years the testing of students has become increasingly important and a major component of American education. Examinations are being used to ensure accountability to school constituents as well as to assess the academic achievement of students. As a consequence, it has become essential that appropriate interpretation of test results takes place. The following provides answers to frequently asked questions (FAQs) regarding norm-referenced achievement tests.

What are norm-referenced achievement tests?

  • Norm-referenced tests compare a person’s score to the scores of a group of people who have already taken the same exam called the “norming”group.

What is the purpose of norm-referenced achievement tests?

  • The purpose is to assess student achievement by comparing one’s scores to others who have taken the same exam. The test does not provide information about what a student does or does not know but rather places students in “rank order”.

What does “norm-referenced” mean?

  • A sample (subset) of the target student population (e.g. 5th graders) who has previously taken the test is used for comparison purposes. Testmakers create exams whose distribution of scores is graphically represented by a bell-shaped (“normal”) curve. The test design is such that most (5th grade) students will score near the middle, and only a few will score low (the left side of the curve) or high (the right side of the curve).

What other types of tests are there in addition to achievement tests?

  • Other types of commonly used exams include aptitude tests (a measure of a student’s ability) and criterion-referenced tests (an assessment used to see whether students have mastered a certain body of knowledge).

What is a “percentile rank”?

  • A percentile is a score that indicates the rank of a student compared to others using a hypothetical group of 100 students. For example, a percentile rank of 50% means that the student’s test performance equals or exceeds 50 out of 100 students taking the same exam; a percentile rank of 88% indicates that the student’s test performance equals or exceeds 88 out of 100 students taking the same exam. Note that percentile rank is not equivalent to percent. Typically in American schools a student scoring a 50% on a classroom exam indicates that the student got ½ of the exam questions correct earning him/her a failing grade.

What are the age/grade equivalent scores?

  • Such scores indicate that a student has attained the same score (not skills) as an average student of that age/grade. For example, if a student obtains a grade-equivalent score of 6.8 on a mathematics test, this means that she obtained the same score as the typical student in the 8th month of the 6th grade. Note that this does not mean that the child is ready for the 6th grade! It simply means that an average 6th grader would have scored as well on the same test and that the child mastered the material very well.

How accurate are norm-referenced achievement test scores?

  • All tests have “measurement error” and are not perfectly reliable. Sometimes results are reported in “bands” which show a range within which the student’s “true score” is likely to have fallen. For instance, a score of 73 may in actuality fall between 66 and 80 but could even be further off.
  • There are many factors that can cause measurement error including an ill student, unfamiliarity with exam design, distractions during the exam, or a student simply having a “bad day”.
  • The items on the exam are only a very small sample of the whole subject area. There are a very large number of questions that could be asked but generally there will only be a few dozen. With so few questions being asked a student getting even one more right or wrong can cause a big change in the student’s score.
  • If one is disappointed in the results of a particular exam, retaking the exam can help give the student a clearer picture of what their “true score” is.

Most achievement exams are focused heavily on memorization and routine procedures. This means that they often do not provide a thorough assessment of problem-solving, decision-making, judgment, or understanding.

When asked what the average of a set of data means what answer do you think most people (including mathematics teachers) give? Our experience from leading many professional development workshops and in classroom instruction is to hear a response such as “you add up all of the numbers and divide by how many numbers there are”. Well, that is true if the question had been “How do you calculate the average of a data set?” but is not the answer to what the average means. The response typically given describes the process of how to find the average NOT what the numerical value represents in a real world context.

If we use as an example the average G.P.A. of the students in our classroom, most of our students know that the process for finding the average G.P.A. is to total up all of the G.P.A. points of the students in the classroom and then divide that number by the total number of students in the class.  Once we get a result like 3.12 the question becomes what does that number mean? The meaning is closely linked to the calculation process but is not the same thing. When finding the average we find the total number of G.P.A. points (envision all of the points accumulated on a desk in the classroom) and then divide them up equally (think about handing one G.P.A. point to one student, then the next student, etc. until every point has been physically handed out). We can now clearly see what the average G.P.A. of 3.12 means –  if every student in the class had the same number of G.P.A. points they would posess 3.12.

Likewise, the average shoe size of students in class would be the size of shoe every student in the class would have if every student in class had the same size shoe. Note that the average does not have the same meaning as the mode (the most frequent shoe size) or the median (the shoe size directly in the middle).

How does having a thorough understanding of average help one more clearly make meaning of the average rate of change? The average rate of change truly is an average but it is the average of a change!  We calculate the average rate of change by determining the total change in the ouput value and dividing by the total change in the input value. This results in a value that describes how the output values would change for every change of the input value if the output values changed by the same amount for each change in the input

Let’s consider calculating the average rate of change in the population of Austin, Texas. In 2009 the population was 1,705,075 and is anticipated to be 1,730,267 in the year 2015. To calculate the average rate of change we find the total change in population 1,730,267 – 1,705,075 = 25,192 and the total change in time 2015 – 2009 = 6 years. We then divide the total change in population by the total change in time which is 25,192 people/6 years =  4199 people/year. This means that the population of Austin will increase by 4199 people each year IF the population changes by the amount each year.

Understanding the concept of average is fundamental to the proper understanding of average rate of change in Algebra and Precalculus (and subsequently the notion of derivative in the Calculus series).

In August 2011 the median salaries for 2011 bachelor’s degree graduates were released. The ranking of the majors by midcareer salaries is found in the following table:

Top 10 Majors

Starting Salary

Midcareer Salary

Petroleum engineering



Chemical engineering



Electrical engineering



Materials science and engineering



Aerospace engineering



Computer engineering






Applied mathematics



Computer science



Nuclear engineering



Notice anything special? It is not hard to see that every one of these majors require a significant amount of mathematics…but what kind of mathematics?   Most of the jobs that come from the Top 10 majors require unique skills and thinking abilities.  Successful workers in these fields must be able to actively and skillfully conceptualize, apply, analyze, hypothesize, exhibit creativity, synthesize, and/or evaluate information gathered from, or generated by, observation, experience, reflection, reasoning, or communication, as a guide to belief and action.

Consequently, if we are to serve the students needs that enroll in our mathematics courses we must help students learn to think critically, solve challenging problems, and are required to communicate clearly and effectively with other students.

Many educators seem to believe that it is their “job” as a teacher to make learning easy. Teachers expend much time and effort to come up with creative ways to help students remember facts and and develop cute tricks to help them perform procedures.

It is my belief that a math teacher’s primary job is to help students learn by thinking through and solving challenging problems.  If a “problem” or exercise is truly a problem, there will not be a simple solution. To arrive at a solution, students will be required  to persevere, struggle, experience wrong turns, and take extended periods of time for contemplation. Major League Baseball player, Edwin Jackson, was quoted recently about the necessity to struggle in his development as a pitcher: “As long as you learn something and take positives from it, it’s alright to struggle. You’re supposed to struggle.”

In the September 10, 2011 article, The Trouble with Homework, Annie Murphy Paul notes that a common misconception about how we learn holds that if something is easy to learn then we have learned it well.  According to brain researchers, this couldn’t be farther from the truth! In fact, when we have to work hard to understand information, we recall it better and the extra effort places added emphasis on that information within our brain pathways. Psychologists have found the phenomenon of facing challenging obstacles in the learning process (known as “desirable difficulties”) that they have begun to purposefully introduce hurdles  into the learning process with amazing results. An example in the area of homework is something called “interleaving”. An interleaved assignment would include mathematics problems that have mixed up situations and problems, instead of grouping them by type. This forces students’ brains to work harder to come up with a solution resulting in students learning material more deeply.

Of course, we as teachers should not let our students flounder. Support is given by the  teacher so students are encouraged to try, fail, try again, and experience successes. We create an environment in which students can feel comfortable trying alternative approaches and have their classmates and teacher challenge their thinking.

Students are best served by this type of mathematics classroom experience. Our goal should be to achieve creating such a place.