How Standardized Testing Works


50,000 page views

I’m taking a one-day break from my usual posts on mathematics and mathematics education to note a symbolic milestone: has had more than 50,000 total page views since its inception in June 2013. Many thanks to the followers of this blog, and I hope that you’ll continue to find this blog to be a useful resource to you.

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Some other (probably useless) statistics: this blog has been viewed by readers from 167 different countries. Top viewership: the United States, India, the Philippines, Canada, the United Kingdom, Australia, Brazil, the European Union, United Arab Emirates, Germany, Taiwan, and Pakistan.

Twelve most viewed posts or series (written by me):

  1. All I want to be is a high school math teacher. Why do I have to take Real Analysis?
  2. Analog clocks
  3. Arithmetic and geometric series
  4. Exponential growth and decay
  5. Finger trick for multiplying by 9
  6. Full lesson plan: Platonic solids
  7. Fun with dimensional analysis
  8. Geometric magic trick
  9. My “history” of solving cubic, quartic and quintic equations
  10. Square roots and logarithms without a calculator
  11. Student misconceptions about PEMDAS
  12. Was there a Pi Day on 3/14/1592?

Twelve most viewed posts (guest presenters):

  1. Engaging students: Classifying polygons
  2. Engaging students: Congruence
  3. Engaging students: Distinguishing between axioms, postulates, theorems, and corollaries
  4. Engaging students: Distinguishing between inductive and deductive reasoning
  5. Engaging students: Factoring quadratic polynomials
  6. Engaging students: Laws of Exponents
  7. Engaging students: Right-triangle trigonometry
  8. Engaging students: Solving linear systems of equations by either substitution or graphing
  9. Engaging students: Solving linear systems of equations with matrices
  10. Engaging students: Solving one-step and two-step inequalities
  11. Engaging students: Solving quadratic equations
  12. Engaging students: Square roots

Top twelve search engine terms that landed people on my blog:

  1. systems of equations project / system of equations project ideas / system of linear equations project / system of equations projects / solving systems of equations projects / real life system of linear equations / solving systems of equations project / linear systems project / etc.
  2. log table / how to find square root using log book / how to find square root in log book / how to find square root using log / logarithms with square roots / log tables / how to find square root of a number using log table / how to use log table for square roots / etc.
  3. geometry tricks
  4. law of exponents foldable / exponent rules foldable / foldable exponent rules
  5. mean green math / green math / meangreenmath (hey, it works!)
  6. cavalieri’s principle / cavalieri’s principle proof
  7. tables / square root table / mathematical table / math tables
  8. law of exponents fun activity / law of exponents activity
  9. 5e lesson plan math
  10. congruent segments in real life
  11. student art work with circles and parabolas
  12. finger trick for multiplying by 9

Twelve other search engine terms that caught my attention:

  1. deercrossing sign phone call
  2. engaging students in reading lesson
  3. examples of inductive and deductive reasoning in the declaration of independence
  4. grape and triangle pun
  5. math and 2048
  6. common core and math number line
  7. meetkunde trucjes
  8. dominoes falling over
  9. slide rule advertisement
  10. the stereotypes about math that hold americans back
  11. youtube/danceofchanceprobabilityfromanormalcurve
  12. honey i shrunk the kids dilation project

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If I’m still here at that time, I’ll make a summary post like this again when this blog has over 100,000 page views.

Collaborative Mathematics: Challenge 14

I’m a few months late with this, but my colleague Jason Ermer at Collaborative Mathematics has published Challenge 14 on his website:

See also

Second-Grade Teacher Uses NBA Games To Teach Basic Math To His Students

I love this:

Value Added Meets the Schools: The Effects of Using Test-Based Teacher Evaluation on the Work of Teachers and Leaders

The March 2015 issue of Educational Researcher was devoted to the perceived usefulness/uselessness (depending on the perceiver) of high-stakes testing. The issue contains multiple perspectives from teachers, principals, and education researchers. The abstract from the journal’s editors sets the tone for the issue:

Teacher accountability based on teacher value-added measures could have far-reaching effects on classroom instruction
and student learning, for good and for ill. To date, however, research has focused almost entirely on the statistical
properties of the measures. While a useful starting point, the validity and reliability of the measures tell us very little
about the effects on teaching and learning that come from embedding value added into policies like teacher evaluation,
tenure, and compensation. We pose dozens of unanswered questions, not only about the net effects of these policies on
measurable student outcomes, but about the numerous, often indirect ways in which these and less easily observed effects
might arise. Drawing in part on other articles in the special issue, we consider perspectives from labor economics, sociology
of organizations, and psychology. Some of the pathways of these policy effects directly influence teaching and learning
and in intentional ways, while other pathways are indirect and unintentional. While research is just beginning to answer the
key questions, a key initial theme of recent research is that both the opponents and advocates are partly correct about the
influence of these policies.

Engaging students: Graphing with polar coordinates

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Laura Lozano. Her topic, from Precalculus: graphing with polar coordinates.


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How could you as a teacher create an activity or project that involves your topic?

An activity that I believe will go really well with graphing polar coordinates or any type of graphing lesson will be to convert the classroom floor into a graph. Also, I will have a selection of random objects like, a rubber ducky, boat, toy, etc. The size of the graph will depend on the size of classroom of course. If the classroom is really small then I would have to take this activity outdoors or maybe even the gym or anywhere with enough room for the graph and my students. The graph doesn’t have to be super big but I would use a graph no smaller than 8 feet by 8 feet area. I could create the graph lines with tape on the floor or draw them on big paper and tape the paper on the floor. I would start the activity with first talking about points on a Cartesian graph. An example could be to first have a students plot a couple points like (5, 4), (3, 6), or (-4, 2) on the board. Then transition them from Cartesian to polar coordinates by using the floor graph and have them discover how they relate by using the x and y coordinates to find the radius and the angle. Then later, after they get the hang of it, I would have the class split up into groups of two and let them choose an object, like a rubber ducky, boat, or toy, to set on the graph and have them write and tell me the point of their object.


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We see radars in the news almost all the time. One category that it is usually used in is weather. The weather center uses their radars to detect for any water particles, debris, and basically anything that is in the air that could be approaching. The way that they tell if a storm or any other weather change is coming is by the radar’s omitting radio waves. The radar omits waves that then come back to the radar if the waves clash with anything in the air. The radar can detect how far an object is by the time it takes for the wave to come back. It works just like an echo! Also, recently with the search of the Malaysian airplane, we saw it used more. The news will show a clip of aircraft radar or ship radar searching for something in the air or in the ocean. Radars look almost exactly like a polar graph does. On the left is a regular polar graph. On the right is a ship’s radar. Both graphs have angles with circles.




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How can technology (YouTube, Khan Academy [], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

Graphing calculators can be used to discover polar coordinates and polar equations. I would first tell them to take out their calculators and just type in a random number from -10 to 10. I choose this interval because the graphing calculators have this window preset for graphing. I number that I randomly chose was the number 4. So I would go to the “Y=” button and type in 4. Then I would hit “GRAPH” and I should get a straight line horizontal line going through the y-axis at 4. I would then change the calculator mode and change from “FUNC” to “POL”. Then I would tell them to do the exact thing again with whatever number they chose. Once the hit “GRAPH” a circle should then come up. They then see how different polar graphs are from Cartesian graphs. Now, the graphs on a polar coordinate graph will all be circular instead of lines and curved lines like on the Cartesian graph.



Polar plot


Math anxiety

I recently read the following interesting articles on math anxiety from the perspective of cognitive scientist: and I don’t have a particular opinion on this, as my own expertise is on post-secondary students, but this definitely gave me some food for thought.

Your Humble Servant, Isaac Newton

From the YouTube description:

Almost fifty years ago, Cambridge University Press published the correspondence of Isaac Newton, a seven-volume, 3000-page collection of letters that provides insight into this great, if difficult, genius. William Dunham shares his favorite examples of Newton as correspondent. He ends with Newton’s most-quoted line about standing on the shoulders of giants and how his search for its place of origin led him, improbably, to a library in Philadelphia.

Another proof of the Law of Cosines