Use of QRs in Math Class to add a bit of Fun

My schedule has been very hectic as of late; busy taking MOOCs and taking care of school work.

The current MOOC that I am taking through coursera is "Emerging Trends & Technologies in the Virtual K-12 Classroom". The topic of using QRs came up and I recalled that two years ago I prepared a little game a la scavenger hunt to help my students review for the Precalculus test. 

Following is a brief description of the said activity:

1) I used classtools.com (I strongly recommend this cool site; Coolism!) to generate the scavenger hunt for this activity. The site guides you through the whole process.

2) I entered my five questions, generated the corresponding QR codes (try them out by pointing your own mobil device that has a QR reader), printed them, cut them up, and then posted them around the building in conspicuous places that the students will have to find. 

3) The students were split into groups and the adventure started.

4) The five questions were appropriate for the 50-minute class. The group that completes all solution processes first and correctly got sweets.

5) The session was fun as the students attested and they got a bit of a review.

QR Code of Question 1

QR Code of Question 2

QR Code of Question 3

QR Code of Question 4

QR Code of Question 5

6) These are just a few examples of how such a technology may be used. Adding games and fun to any learning activity is a big plus.

Please, reply to this post and make sure you add your own game and fun-filled activities in the comments area. Thank you 

MTBoS Mission #1 Blog Post: Clickerisms & Conceptual Understanding of Mathematical Ideas

This blog is my contribution to the Explore MTBoS Mission #1 [The Power of The Blog] initiative by our good fellow math bloggers Tina Cardone (@crstn85), Julie Reulbach (@jreulbach), Justin Lanier (@j_lanier), and Sam Shah (@samjshah).

Specifically, this blog addresses "What is one thing that happens in your classroom that makes it distinctly yours? It can be something you do that is unique in your school… It can be something more amorphous… However you want to interpret the question! Whatever!" of Mission #1 as stated by Sam.

Clickerisms! Using a set of clickers (class response systems as some folks would call them and they are shown below), I have always challenged my students with conceptual questions that would enable them and me gauge their deep understanding of mathematical concepts. The process goes as follows.

* Clickerisms as Unit Starters: My students are required to read a given section/chapter prior to start of coverage and a set of of true/false questions are presented to gauge students' understanding of what they read. Following are examples of such questions that precede a unit on number sets for more questions (check #precalcchat posts where I put sample questions from trig.)

*** Start of 1st Clickerism Set ***

1.  Answer True or False. The sum of two irrational numbers is always irrational.

A)  True

B)  False

2.  Answer True or False. The sum of any irrational number with any rational number is always irrational.

A)  True

B)  False

3.  Answer True or False. The division of an irrational number and a non-zero rational number is irrational.

A)  True

B)  False

4.  Answer True or False. The square of an irrational number is always rational.

A)  True

B)  False

*** End of 1st Clickerism Set ***

Students work on these problems individually at first, commit their answers to the output area of their respective interactive notebooks (INB), then a rapid fire sequence of clicking of answer choices would follow to obtain a set of bar graphs that would show the distributions of students' answers to each of the questions. Students are then asked to discuss the questions, that had no consensus class-wide, with someone who answered differently than they did. An updating of students answers would ensue. If the new distributions reflect a consensus, then the said questions would be skipped for now and the rest of the questions would become a springboard for delving into the topics at hand in greater depth by me.

** Clickerisms as Lesson Pulse Checkers: After the above process is completed, new activities would start to guide the students' learning process of the related concepts. But, before the lessons are considered done, a new set of Clickerisms is presented to check how well the concepts that were investigated earlier are well understood. These Clickerisms are more like mini quizzes that are generally extracted from whichever textbook I use for the given class or I generate them from encounters with Mathematical Blasphemies (MB) in previous years. Since, I use Axler's Precalculus textbook, here is an example of such a Clickerism as it appears in one of my PowerPoint slides that relates to the same topic mentioned above. Notice the three follow up questions that include a final question that sometimes I have students click anonymously to help me gauge their feeling about their level of confidence in understanding the underlying concepts of the questions at hand, the processes that went in completing them, and the final outcomes. 

*** Start of Clickerism ***

*** End of Clickerism  ***

The Clickerisms are a consistent set of formative assessment tools that I use to ensure students' deeper grasp of mathematical ideas they have read about and then have worked on in class. 

Thank you for taking the time to read this blog entry and I hope it would provide you with a possible way of increasing your students' confidence in what they learned and your confidence in their deeper conceptual grounding in mathematical understanding. Please, comment and add suggestions that would improve my teaching and my students' learning.  

  

When Trig Meets Series and Sequences

Combining the ingenuity of one of the problems posted by Maths Challenge site, "Corner Circle" and the versatility of Desmos this problem led to further neat sub problems that end up connecting trigonometry to topics of series and sequences. Below is the link the investigation "Mr. Le Nadj! Desmoses "Corner Circle" Problem." 

Please, give this activity a whirl in your class and report back as to how it goes for your students. Thank you

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

Unit Circle Desmos Investigations

I have just posted two Unit Circle investigations at desmos.com, Unit Circle Investigation 1 & Unit Circle Investigation 2.

Please, take them for a spin, assess their educational value, and then post some feedback in the comments area. If you use these with your students report back regarding its success or lack thereof. Thank you

1st #precalcchat Script

Thanks to my partner @untilnextstop, the first Precalculus Chat materialized tonight and this is the link to the chat tweets for anyone who missed them. Tweets about "#precalcchat"

Thank you to all who attended and contributed and we look forward chatting with you and other fellow Precalculus teachers in the following weeks at #precalcchat.

Invitation to a Twitter Chat #precalcchat

All Pre-Calculus teachers, instructors, and professors are cordially invited to the launch of a series of twitter chats that would address everything Pre-Calculus. The first chat is Thursday, August 22nd starting at 8:30 PM. 

Event Title: "1st Week Matters," & it will discuss topics such as syllabi, pre-tests, technology, TIs, opening activities, etc.

Event Time: Thursday, August 22nd starting at 8:30 PM

Event Hosts: @MrLeNadj & @untilnextstop

Event Sponsor: Global Math Department with Cool Initiative of @jreulbach

Event Twitter Location: #precalcchat

We look forward having you and happy new Pre-Calculus school year!

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