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.  

  

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

Nadji's Personalized Conceptual Problems (PCP) [How to make assessment personalized, inventive, & concept-driven?]

Generally, there seems to be a tendency in our mathematical curricula to encourage the inertial, linear, and algorithmic-heavy approach to learning and assessment. It is imperative that, as educators, we break the cycle of these mind-numbing approaches and disturb them with various pedagogical tools that offer students the chance to let their resistive, non-linear, creative selves shine through. 
One such method (aka PCP, read on) would be to insist that students every now and then invent their own mathematical functions, expressions, equations, or real-life application (word) problems that address specific concepts on one hand and allow the students to demonstrate their true conceptual grasp of those concepts on the other hand.
I conceived and started using the Personalized Conceptual Problem (PCP) idea to address some of the shortcomings of routine-laden problems that students typically encounter in most textbooks. These PCPs are personalized and tailored to each student's individual conceptualization of the concepts at hand, open-ended, and still do not sacrifice any procedural or algorithmic processes that are equally important in our students' progress in their mastery of mathematics. Following are two examples of such PCPs that deal with algebra topics.

PCP 1: Invent your own absolute-value equation that uses one of your initials as its variable (unknown) and would have no solutions at all. Show the detailed process of solving your equation for the given variable, graph the solution on a number line, and share how you conceived your equation in the first place. (EC: How about inventing your own inequality that has no solutions at all?)

PCP 2: Write your own math application problem (word problem) whose solution process must eventually involve the use of the equation 7x + 13 = 41, solving it, showing the detailed solution process, and providing the real life meaning of each of the equation's constants.

In the comments area, share your thoughts about the PCP assessment approach, your variation on it, and sample problems of your own. Thank you