Think Language or Modality Rather Than Stages
& Build Time for Change Into Your "Game Plan"



A "Game Plan" for Growth

    The only constant is change. This writing urges you to THINK LANGUAGES or MODALITIES (used simultaneously or individually and chosen to reflect the most appropriate level of sophistication) rather than a sequence of STAGES (which implies proceeding from one to a next to a next stage) and includes suggestions on how to develop your skills in each language.

    Below, the table [1] embraces both WRITING TO LEARN (WTL) and Concrete, Pictorial, Abstract (CPA) and is a unified approach to choosing the format of a class or an assignment. It backs up the agendas and topics of each of the AMTNJ Spring 2025 Newsletter [2] authors Julie Norflus-Good, Meagan Carr, and Madelene Taub-Chan and Jackie Molnari.

    This is not a very radical request.  Math educators have used WTL and CPA for the last 30 years. The radical material comes after some suggestions on how to use the languages. After that, the text recalls a time and traditions of 50+ years ago.



The Languages of the Math Classroom
© '98, '08, '09, A2, Agnes Azzolino
MOTHER TONGUE & OTHER TONGUE(S)
Most Sophisticated and also the Most Basic
 
MOSTLY MATH TONGUES
Most Sophisticated, Most Basic
  VERBAL / Auditory       WRITTEN / Symbolic     PICTORIAL / Visual     CONCRETE / Kinesthetic    
     formal spoken mathematics      written word   DIGITAL MANIPULATIVE     object
     informal spoken mathematics          written symbol   moving picture   model
     spoken symbol      semisymbolic   static picture   DIGITAL MANIPULATIVE
     symbol speak      calculator symbol   numeral
     calculatoreze/computereze   graph
     web speak   nonverbal body language

 


General Suggestions [3]
  • Choose a modality first.
    Mother Tongue(s)
    VERBAL / Auditory
    PICTORIAL / Visual     		WRITTEN / Symbolic
    CONCRETE / Kinesthetic
  • Strive for comfort in all modalities, not just your favorite.
  • Usually, introduce in the most concrete.
  • Introduce in the Mother Tongue(s) because it is both the most concrete as well as the most abstract.
  • Introduce/use spoken vocabulary two years before using written vocabulary or symbols.
  • Usually, summarize in the most abstract.
  • Sometimes use multiple modalities at the same time.
    Repetition improves retention, especially in different modalities.
  • Regularly use "Discuss this with your partner" to debrief so students can discuss what they have learned, but, also as a warm-up so students can figure-out the agenda or strategy or lesson BEFORE the teacher introduces it.
  • Don’t force a student to use a less sophisticated language.
    If a student can "do it mentally," don't force use of your chosen language, they already have the concept.
  • Need a review before new material?
    Don't review with a COMPUTATION OF SYMBOLS,
    review with a PICTURE OF THE COMPUTATION.


More Topic-Specific Suggestions
  • To improve your mother tongue,
    tell your students to “discuss this with your partner“
    then listen in on their conversation.
     
  • If you’re writing assignments are not working, make the assignment shorter, or,
    use a less sophisticated assignment or a task lower on Bloom's taxonomy [4]
    See: "A Journey with Self-Assessment as a Compass" [5]
     
  • To increase your work with manipulatives, plan ahead.
    Pack/organize, on Sunday night while writing lesson plans, the manipulatives you wish to use on Thursday. If the manipulatives are not ready, they can't be used.
     
  • To decrease anxiety with your work with manipulatives, include a "search of the floor" as part of the debriefing or "discuss this with your partner" activity. If the manipulatives are incomplete, they can't be used.
     
  • If you have "no money" for manipulatives but "there is paper in the copier today," make your own.
        See the cheat sheet page [6] for masters of: fraction bars, nomographs, Napier's bones, function manipulatives, Term Tiles.
        See the stuff page [7] for a video on digital manipulatives and Geometer's Sketchpads of middle school, algebra, precalc, and calc topics.
     
  • Use "symbol speak" and "mathematics" concurrently. For example:
    Given the phrase "x + 3"
    say "three more than a number"
    also say "eks plus three" as you
    write in symbols " x + 3"
     
  • Introduce vocabulary 2 years before you need it. For example: In algebra I,
        to reinforce the algebra I and ready the student for precalc, when graphing a function, do not just use a line like "x + 3", use a reciprocal function like 1/(x + 3)" where the slope is not constant but varies and where the domain is not "all real numbers" but "all numbers except -3."
     


The Languages of Math "Classroom(s)"
© '98, '08, '09, '25 A2, Agnes Azzolino
MOTHER TONGUE & OTHER TONGUE(S)
Most Sophisticated and also the Most Basic
 
MOSTLY MATH TONGUES
Most Sophisticated, Most Basic
  VERBAL / Auditory       WRITTEN / Symbolic     PICTORIAL / Visual     ANIMATION / Video     CONCRETE / Kinesthetic    
     formal spoken mathematics      written word   DIGITAL MANIPULATIVE     viewer can not "touch"     object
     informal spoken mathematics          written symbol   moving picture   perhaps a traditional video   model
     spoken symbol      semisymbolic   static picture   learner replay possible   DIGITAL MANIPULATIVE
     symbol speak      calculator symbol   numeral   may have a DIGITAL or concrete version
     calculatoreze/computereze   graph   independent use possible  
     web speak   nonverbal body language   it may be written for young viewers

 


ANIMATION / Video - A New Modality

    It is now the second quarter of the 21st century. Many things changed in the first quarter of the century.  Important to the topic of language, modality, and change are the availability of material on the internet, including free & not free courses, free downloadable software, ZOOM classes, entire free textbooks in pdf form, and really useful videos.

    Videos using animation -- things that move -- are, I believe, so powerful and different from last century, that, for me, they are a new language.

    Below are images of two samples of this new language. These videos/animations are not the "typical" talking heads symbol-speaking while writing symbols while completeing a written computation.  In the first instance, blocks of numbers -- Number Blocks [8] -- which may be purchased in concrete form as manipulatives, but, which exist in math movies as movie characters.  These videos were on Netflicks but are still very present and used/viewed by millions on Youtube because they are SO GOOD at presenting the math and entertaining. Think the impact of Sesame Street in the last century moved to this century. I see them as between the pictorial language (static) and the manipulative (concrete and active).

    They present math in a way a teacher can not unless moving things digitally with Excel or Sketchpad or a Java-enabled web page. The power of this language is not just the content. The possibility of repetition is also important. Number Blocks exist in concrete form, in an interactive digital form, and in the animated video form. The animated version makes it possible to just skip the concrete form which would usually require a human to narrate or introduce its initial use. There are Number Blocks videos valuable for two-year-olds.

    The second example of this new language is a Taylor series video [9] . It "speaks" pictorial/graphing and symbolic and mathematics and mother tongue simulataneously, and beautifully.

    PLEASE listen to math presented this way and consider suggesting it to the parents of young learners and also use on occasion in you classroom.





    Number Blocks[8]     Taylor Series [9]


Advise from Max Sobel

    Dr. Max Sobel [10] was AMTNJ's 45th President and in 1990 the first recipient of AMTNJ's "Award for Outstanding Service and Leadership in Mathematics Education," which is now named in his honor.

    He was also a Chair of New Jersey's MAA Section, a world famous author, the NCTM President who lead NCTM through the publication of "The Agenda for Action." He was a math and math ed prof, a teacher at Montclair State's College High and a really wonderful man.

    Max Sobel taught not only mathematics, but, that a math teacher must also be a showman in the classroom and a leader of teachers outside the classroom.

    He advised his students, future math teachers and mathematics education leaders to:

  • Self-evaluate your class/teaching at the end of each class.
  • Build change into your work. Plan to grow. He suggested a 5% change -- develope 1 class a week (given 5 days a week, 4 classes each day ) to grow and keep current.

    Embracing the math class languages, building your skills in each language, and embracing new teaching strategies is what I think Max would urge you to do.

    Stay safe, a2, Agnes Azzolino, PP#89, 2017MAX



[1] "19+ Languages of the Math Classroom", 11/15/2001, A2, Agnes Azzolino
https://www.mathnstuff.com/papers/nov1601.htm
[2] AMTNJ Spring 2025 Newsletter,
https://amtnj.org/wp-content/uploads/2025/05/Copy-of-AMTNJ-Spring-Newsletter-2025.pdf
[3] Arc Functions Through Auditory, Symbolic, Visual, and Kinesthetic Modalities,
10/27/2023, A2, Agnes Azzolino
https://www.mathnstuff.com/papers/MATYCNJ23.htm
[4] Bloom's taxonomy in wikipedia
https://en.wikipedia.org/wiki/Bloom%27s_taxonomy
[5] A Journey with Self-Assessment as a Compass, © 1995, ..., 2016. A2, Agnes Azzolino
https://www.mathnstuff.com/papers/ai/invent.htm
[6] cheat sheet, Sketchpads, videos, manipulatives, A2, Agnes Azzolino
https://www.mathnstuff.com/cheat.htm
[7] Stuff at mathnstuff.com , © 11/20/2023, A2, Agnes Azzolino
https://www.mathnstuff.com/stuff.htm
[8] Number Blocks
https://www.youtube.com/channel/UCPlwvN0w4qFSP1FllALB92w
[9] "Taylor Series" video, Grant Sanderson, https://www.3blue1brown.com
https://www.youtube.com/watch?v=3d6DsjIBzJ4&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr&index=11
https://www.3blue1brown.com
[10] Dr. Max Sobel, © 11/2020, Dr. Joseph Michael Nuspl, PhD, PP#66, 2016MAX and
© 1/27/2017, Dr. Janice-Lynn Shuhan, 1/27/2017
https://amtnj.org/wp-content/uploads/2020/11/tt26.pdf