The Value of Discovery through Investigations [Part 1: Inverse trig Functions]

Introduction:

After a review of inverse functions, their domains & ranges, and how their graphs are related, Precalculus students were broken onto three groups to investigate inverse trig functions of the basic three trig function (sine, cosine, & tangent) themselves prior to me teaching about them. This process was pursued to achieve the following educational goals:
(a) To offer students the chance to review past material, trig functions, their graphs, and domains and ranges of functions and their inverses.
(b) To help the instructor to gauge how well prior material was grasped by students that other means of assessment may not have captured.
(c) To assist the instructor in figuring out how much transfer of concepts students can do on their own without much intervention from them.
(d) To enable students to use technology as means of verifying and rectifying some misconceptions they may still have held prior to the coverage of inverse trig concepts.

The Prompt & Group Findings:

The following figure shows the smart board with the prompt written on it.

Activity Prompt

The groups then began drafting their own papers in which they addressed the tasks of the prompt. Following are copies of the raw work the groups did without being allowed to use any references of any kind nor any calculators.

Sine Group work shows what they understood and what they misunderstood about sine functions.

Cosine Group work shows that the concept of inverse of a function was understood in terms of reversal of roles of inputs and outputs. But graphically speaking, the idea did not sink in yet in terms of the use of the y=x line and restricted domain.

Tangent Group work shows that the concept of inverse of a function was understood in terms of reversal of roles of inputs and outputs. But graphically speaking, the idea did not sink in yet in terms of the use of the y=x line and restricted domain.

Once the groups were done with the prediction phase, they were instructed to use technology to verify the outcomes of their work and then update their findings regarding inverse trig functions and reflect upon them. Below are a few images that illustrate the checking process followed by a sample revised group results with reflections.

Desmos and a TI graphing calculator were used during the verification phase of the activity.

Desmos and a TI graphing calculator were used during the verification phase of the activity.

Desmos and a TI graphing calculator were used during the verification phase of the activity.

Desmos and a TI graphing calculator were used during the verification phase of the activity.

After using Desmos and a TI graphing calculator, the Tangent Group seems to have hit the jackpot of discovery.

To ensure that no group members are left twiddling their thumbs when peers are still finishing their activity, another activity was added. The activity consists of looking at a flier that features the school's organ and then connect what they observe to the subject matter at hand. The organ picture is shown below and the Tangent Group's reflection is shown in the picture above this text.

Cool Math Connection with Organ Pipes

Conclusions:

• The formal coverage of inverse trig functions was preceded by this activity to achieve the goals stated above. The results reported after the technology portion of the activity showed that three quarters of the students made great strides toward understanding inverse trig functions, their graphs, and their domains and ranges. 
• The task of teaching the subject matter went smoother now because it was a mere strengthening of ideas for students who grasped the concepts while it was an eye opener for the ones who missed to understand the concepts initially on their own.
• This activity also made it clear to me that there were still gaps in students' understanding of the basic trig functions (especially their graphs) and the concept of inverse functions in general. So, a nice review of the unit circle and its connection to the basic trig graphs followed. The method of review that was used consisted of a series of class-wide UC-Bowl games that will be featured in another blog post. Inverse functions (especially how to draw their graphs) was also reviewed but now through the official coverage of inverse trig functions.
• The spiraling in tackling concepts was definitely a pleasant outcome of this activity and for this alone I believe the whole activity is worth replicating with other mathematical topics.

Note:

I hope the ideas shared in this post would encourage you to post in the comments area your own wonderful approaches to how you assist your students in internalizing challenging topics such as inverse trig functions. Thank you 

Math Activities in Images

I apologize to my readers for the long time since I last posted; a busy schedule has not been kind at all!

I would like to let a few images speak for some aspects of this year's math.

Image 1

Image 2

Image 3

Image 4

Image 5

Image 6

Of course, these are not the only pictures that give away how much coolism did occur in math this year but at least they convey some ideas.

What do you think these images conjure up in your mind curriculum-wise, learning-wise, and math-wise?

Thank you for taking the time to read the post and please leave comments that reflect your thoughts.

Musings of a Nadjling!

As we were driving back home from his percussion lesson, Omar (the 12-year old Nadjling) said, "Baba, future does not exist." 

"How so?" I replied. 

He said, "Well, future is not here, it is unknown, and when it occurs it is present. So, there is only past and present." 

This took me by complete surprise and I retorted, "This is an interesting thought and I would have to think about it."

Then, the math connection came when he continued, "The future is undefined and when it is experienced it is not a future; it is the present then." The Physicist in me could not help but think of contrast between the deterministic interpretation of the world by classical Physics versus the quantum mechanical probabilistic approach to physical phenomena. In addition, I thought of the wave function collapse from a "nebulous/combined state" to a definite state once one performs an experiment with elementary particles.

For a kid to hit on an idea such as this is fascinating. So, I encouraged him to jot down such things when he thinks of them in the future.

Afterwards I gave Omar the example of how knowing the current position, velocity, and acceleration of a probe we are sending to Pluto for instance, we can predict its future position, velocity, and acceleration and this is how we are able to get such probes that far and with incredible precision. I was trying to convey the idea of the deterministic nature of classical Physics. 

But, before I went far with the idea, he interjected by mentioning the take-home test that he mentioned earlier in our trip and he was going to take when we get back home and said, "I am going to cheat." Alarmed by unethical behavior I was going to start giving him a big lecture but he interrupted by explaining, "I have to do a decay experiment and instead of taking out sixes of dice, I will just take out evens." I said, "Why would you do that?" He replied, "This way the decay rate will be a half." To this I commented, "This is not cheating; this is clever! But, why take out the evens and not the odds?" To which he answered, "Evens have only one prime number." Ok!

Fast forward and Omar is at home. He thought he had 100 dice (the required starting number given in the test directions) around the house but he found only 49. So, he shared with me the following thought, "Would it be ok to just use 49 and whatever I get I divide by 49 and multiply by 100 to make the experiment match a 100-dice roll?" This being a test, I told him that would have to be his decision. Deep inside, the father in me could not help but feel proud of the kid's divergent thinking and resourcefulness. In the end he decided to keep the experiment simple, straightforward, and he used 100 coins.


Omar is lucky that his father knows about math and is willing to give-and-take with him in such musings and I just hope that the many Omars out there would be equally fortunate and benefit from such exchanges with adults (teachers, parents, older siblings, etc.) around them. So, if we (the math/science teachers) are the only adults that kids may get this mental gymnastics opportunity from we must seize it with earnest passion and great patience for it would help them grow as thinkers and flourish as math/science lovers.

Happy Valentines Day & please share musings of your students or your own kids in the comments section.

Thank you and take great care! :-)

Updated Unit Circle Investigation in Desmos

Once my students worked on Unit Circle Investigation 2, new cool things emerged that I decided to add to the first two investigations and committed them to Unit Circle Investigation 3 within Desmos. Please, check out this new investigation and provide feedback and how did it work for your students? Thank you

Mr. Le Nadj! Assists with Math Apps

I thought starting the academic semester/year with a bunch of math Apps and strategies for acquiring them might be the best gift I give my fellow math teachers and instructors the world over.

So, within this PDF file I listed all the Apps I currently own, which I hope you can acquire as many of them as possible to make your educational experience with your students a richer experience at least as far as mobile devices technology is concerned.

Please, keep in mind that most of the apps were free when I first got them but some either were removed from the App store or a price was added to them.

Mr. Le Nadj! suggestions regarding app acquirement:

1) Check the app store every two weeks or so and search for new apps dealing with specific topics you are interested in.

2) Sift through the junk by setting the search criteria to be Free & the display to be based on Ratings so that you get the free apps that got the best ratings.

3) Download the chosen app(s) even f you may not think you would need it right away because they may either disappear or become pricy. You still can delete them if need be.

Happy Educational App Year! :-)

Students' Reflections & Citations in Math Assessment (07/09/13 Global Math Department Presentation)

I had the privilege of contributing to the July 9th conference at the Global Math Department site. 

Thank you to Megan Hayes-Golding for facilitating this web meeting and for guiding us (presenters) through the presentation process so professionally and in a very user friendly manner.

My presentation focused on the use of reflections and citations in math assessment and how it should (a) nurture depth in expressing one's thoughts regarding the solution process to a math problem through reflections and (b) foster the scholarly approach to mathematical writing and problem solving amongst our students through citations from the course's textbook, instructor's notes, or peers' presentations.

The conference's recording shows samples of such references and citations. But, after the conference was over, fellow attendees expressed interest in seeing samples of peer citations. So, here are a few examples of such citations.

Note: The student solver name was whited out to respect anonymity & the numbers next to the referenced students (inside oval) denote the problems they solved that their current presenting classmate is referencing. 

Note: The student solver name were blocked out to respect anonymity & notice how this student referenced her peer by stating the date of the referenced student's presentation date. 

Note: The student solver name was grayed out to respect anonymity & the number next to the referenced student (inside oval) denote the problem he solved that his classmate is referencing. 

Notice how the students' methods of referencing vary but the context still helps other students in class recognize the specific aspect of the current problem solution that drew from other sources (their peers in this case.)

A neat outcome of this peer citation and referencing system in general was the start of a healthy and scholarly competition amongst students and students with me (their instructor) as to who ends up being the cited champion as the year progresses (a la Erdos citation number idea.) 

Please, provide your own take on the use of Reflections or Citations in your respective curricula or comment on how you would implement such idea in the future. Thank you

The Curse of Rote Teaching [PEMDAS & its Siblings]

An excellent question was raised in the Global Math Department bulletin board that I decided to elaborate upon in here.

Rote teaching of certain mathematical rules and the lack of understanding of their underlying concepts lead to difficulties that students will continue to struggle with even at higher level courses such Precalculus, College Algebra, and even Calculus. These difficulties manifest themselves in what I have called MBs in an earlier post. 

In this post, I will address an MB related to PEMDAS. So, what to make of a student's treatment of let's say 4-6(3x+1) the same way as -2(3x+1)? And, generally how do we disrupt the misconception inertia that PEMDAS and its siblings of rote-taught mathematical rules engender within our students?

1) Geometry, Geometry, & Geometry: I am a firm believer that Geometry should be used as extensively as possible to enable our students grasp algebraic concepts better and at a deeper level for geometric, graphical, and pictorial representations offer visual clues and cues that facilitate students' understanding of abstract concepts of Algebra and other mathematical fields. So, have the students predict the nature of the graphs associated with the above two expressions and then have them execute the actual graphing processes either by hand or with a graphing technology tool (to add potential "What if?" investigations afterwards.) 

Once the above tasks are completed, one must ask the students to interpret the outcomes, reflect upon them, and tie them to other math concepts (such as translation, scaling, and meaning of slopes to mention a few) to complete the learning cycle. What other geometric tools have you used to aid your students' understanding of mathematical concepts?

2) Carefully crafted assessment items can also aid our students in confronting their misconceptions and try to sort out the difficulties therein until they get a strong handle on the underlying concepts. Following is such an assessment item.


*** Start of Assessment Item ***

PEMDAS refers to the concept of order of operations. Four students argued about how PEMDAS should be carried out when evaluating mathematical expressions. To find the answer to "1 – 10 ÷ 5 ∙ 2 + 44 = x, what is x?" the students used the four different steps shown below. Which of the four solutions steps do you agree with and why? Make sure you state how the student of your choice must have interpreted PEMDAS based on his/her solution steps.

Student 1 Solution: 1 – 10 ÷ 5 ∙ 2 + 44 = 45 – 10 ÷ 5 ∙ 2 = 35 ÷ 5 ∙ 2 = 7 ∙ 2 = 14

Student 2 Solution: 1 – 10 ÷ 5 ∙ 2 + 44 = -9 ÷ 10 + 44 = -0.9 + 44 = 43.1

Student 3 Solution: 1 – 10 ÷ 5 ∙ 2 + 44 = 1 – 2 ∙ 2 + 44 = 1 – 4 + 44 = 41

Student 4 Solution: 1 – 10 ÷ 5 ∙ 2 + 44 = 1 – 10 ÷ 10 + 44 = 1 – 1 + 44 = 44


*** End of Assessment Item ***

With such assessment items both the student as well as the instructor would have a better idea as to what to address conceptually speaking based on the problem-solver's answer choice. In the comments area, please, provide your own assessment examples of how you address specific students' misconceptions.

3) One day one of my children came home ecstatic that he learned the four geographical directions (NESW) through a chant. Unfortunately, the whole chant hinged on knowing where the first direction is supposed to be to proceed with the rest. So, when I asked him about a specific direction, he was at loss for he was not sure what direction he should point for the first direction in the chant. So, it was time for us to go through a brief on the field (both during day light and night time) conceptual encounter with the Astronomy of orienting oneself in space and toss the cute chant in the bin of seeming handy but useless methods of sponging in information.

Share your own anecdotes of such missed opportunities in learning because of "cute" rote teaching approaches. Thank you

What is math?

"What is math?" is an essential question a la Grant Wiggins for it is an open ended question that does not have a single answer, it definitely is ageless, it elicits different answers from different people, and it encourages deep thinking, and it may even arouse passions especially in our country where math and Physix tend to charge the air during conversations.

It is imperative that our students address this question to establish the kind of frame of mind that we ask of them and assist them to approach math more as we and math experts/practitioners approach the subject.

So, I start each semester (year) with a Critical Thinking Activity (CTA) that asks this very question and guides the students through a sequence of math problems that I adopted from RonRitchhart's PZC*2009: Views on Understanding (pg. 79). The key parts of the activity are listed below.  

*** Start of Sample Items from the CTA  ***

1) What is math? What does it mean to you?

2) List the 7 mathematical words/phrases that come to your mind when doing math.

...

13) Now that you have completed this activity, complete these last two tasks:

a) What do you think math is? 

b) List 7 mathematical words/phrases that the above activity suggests to your mind.

*** End of Sample Items from the CTA ***

The new background that I adopted for this blog is a picture of my old blackboard (click here to view the full image) with my students' lists of mathematical phrases pre/activity (item 2 above) and post/activity (item 13b above).

The list on the right side of the board clearly demonstrates that students' perception of what math is all about has evolved and all one hopes is that they would carry this new frame of mind for the remainder of the semester (year).

Share what you do in the first few days of classes to guide your students toward a renewed look at what math is. Thank you

*PZC = Project Zero Classroom