Math is the science of patterns, and it involves observing, representing, and investigating patterns and relationship. It beholds countless applications in past and present real-world such as architecture, stocks, money, roller coasters, telling time and so forth. This subject is so significant that many professions require that college students have an adequate mathematics background. As the world progresses, math is evolving and technology is rapidly growing. Thus, accommodations must be made towards how math is taught in the typical American classroom.

Technology can help bring materials in mathematics come alive which can help motivate and excite learners to learn new concepts once thought to be difficult and boring.

Content of Mathematics and TPACK

In Principles and Standards for School Mathematics published by the National Council of Teachers of Mathematics (NCTM) in 2000, a critical strand discussed is content standards. These standards address the important aspects of mathematical content that should be learned and of which include numbers and operations, algebra, geometry, measurement, and data analysis/probability. Furthermore, they act as blueprint for instructors to refer to whilst teaching for they acts as objective goals for students to meet. It is the responsibility of the instructor to accommodate to these standards by providing the most efficient classroom environment.

Technology has had a considerable impact on the content of math which involves observing, representing, and investigating patterns and relationships. Take for example, a fractal, which is a geometric pattern that is repeated at every scale and so cannot be represented by classical geometry. Striving for accuracy in defining or measuring various things such as geographic features, plant growth, blood vessels, lightning, and, predicting flooding of rivers, fractal mathematics and computer technology can be very useful today. Using fractal algorithms that make allowances for such different scales, computers can provide a much more accurate and consistent measure of the above mentioned examples. Fractals are just one example of the many topics in math that have evolved due to technology.

With the evolution of technology has come a new program that addresses how it should be addressed in the classroom, of which is dubbed TPACK. TPACK stands for Technological Pedagogical Content Knowledge. These four variables all collaborate with another in order to establish a productive and effective technology-incorporated classroom. There are a huge variety of pedagogical strategies and technology available for instructors to use and, of course, there is an immense amount of possible content objectives that an be addressed.

Content Knowledge (CK)

Pedagogy of Mathematics

There are a variety of ways in in which teachers can teach math with technology. A math teacher needs to imagine technology connections, determine the related benefits, and put it all together. Students with different learning styles may become steadily linked to effective technology use, whether it is through calculators, electronic geoboardsexternal image geoboard.jpg, fractalexternal image sierpinski.gifor tessellation imagesexternal image image002.jpg, various websites, instructional videosexternal image tutor_pc.jpg, etc. These methods truly enhance the learning process because they allow students to interact with one another and collaborate in their thinking.

Pedagogy: Teaching Mathematics via (positional words) within a Large Group-

Describe: To achieve learning of content, the teacher encourages discussion by asking open ended questions, and using picture cards to engage students in forming understanding of the introduced content.
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Students can then connect new ideas and experiences to existing knowledge and experiences to form new or enhanced understanding of content.

Pedagogical Content Knowledge (PCK)

Teachers are no longer the only ones leading the class in which students are sitting and receiving the information. Students need to be more involved by interacting with the teacher in order to complete the lessons. The best way to do this by completing several activities related to the topic many of which can be enhanced by using technology. There are many technology resources a teacher can use to involve the students including; developing a wiki; completing a WebQuest; creating a podcast; searching various websites; etc.
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Pedagogical Content Knowledge (PCK)-Group Discussion

The discussion begins sitting in large group where the teacher will explain positional words using sign language demonstrating the movements of the positional words. The Associate Teacher will then begin a video from Maple Leaf Learning Songs at https://www.youtube.com/watch?v=xERTESWbqhU&list=PL2wUZYx3aARUro1A-C3wk3cItbfAyxgd. This online video gives students a concrete concept of positional words in front of, behind, and in between, by visually describing the concepts through sign language movements with music.
Click the picture
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An infographic that may be helpful when thinking about pedagogy and how it relates to various subjects:

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Technology of Mathematics

Handbook of Technological Pedagogical Content Knowledge (TPCK) for Educators points out the position statement made by NCTM saying, “Using the tools of technology to work within interesting problem contexts can facilitate a student’s achievement of a variety of higher-order learning outcomes, such as reflections, reasoning, problem posing, problem solving and decision making” (p.153).

Handbook of TPCK for Educators (2008) points out, in recent work of Heid (2005), several areas in mathematics education were identified that seemed to be the most promising in technology use for the mathematics classroom (p.154). These areas include: dynamic computation tools, or software that facilitates interactive computation; microworlds, which allowed focused experimentation of various mathematical relationships; intelligent tutors, which facilitate symbolic manipulation of algebraic symbols; handheld devices, which permit convenient technology access; web-based instruction, which allows systematic, cost-effective instruction; and interactive learning communities, that can facilitate educational collaboration in classrooms across many different states and even nations.

Other areas in math that have been affected by technology include: statistics, graphing, coordinate geometry, matrices, probability, and combinatorics. Just as technology is in a state of flux, these areas of mathematics are also in a state of change and evolution due to technology.
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Technology of Mathematics

According to the Michigan Educational Technology Standards (METS) CCSS.MATH.CONTENT.K.G.A.1 Early Childhood students will know how to identify and describe objects in the environment describing the relative positions of these objects using terms such as above, below, beside, in front of, behind, and between.

According to the APA (American Psychological Association) article entitled, “The Benefits of Playing Video Games” by authors Isabela Granic, Adam Lobel, and Rutger C. M. E. Engels surveyed the landscape of video games. They identified four types of positive
impact that video games have on the kids who play them: cognitive, motivational, emotional, and

Teacher Sites to Integrate Technology

Mathalicious - You have to purchase an account, but you can always check out the free trial first! Not all of the lessons integrate technology, but most do.

Examples of Math TPACK Lessons Pre-K -3rd Grade

Geoboards for 1st Grade
Content (C) - Partitioning rectangles into equal shares (CCSS.MATH.CONTENT.1.G.A.3 - Partition circles and rectangles into two and four equal shares, describe the shares using words)
Pedagogy (P) - Experiential Learning
Technology (T) - iPads/Tablets and the GeoBoard App

The Content (C) is based on CCSS.MATH.CONTENT.K.G.A.1 Common Core
(Positional Words) Standards stating students will be able to describe objects in the environment, and
describe the relative positions of these objects using terms such as above, below, beside, in front of,
behind, and between.
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The Pedagogy (P) is Group Discussion.
The Technology (T) is an Online Positional game at http://pbskids.org/clifford/games/whichclifford-game.html
Check out this cool song in front of, behind, and between
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Examples of Math TPACK Lessons

This lesson uses multimedia (specifically Edpuzzle) (T) to learn about simple trigonometric addition and subtraction identities in precalculus (C) through flipped learning (P)

This lesson has students exploring a Desmos Lab (T) to investigate linear equations, and domain restrictions (C) through inquiry based learning (P).

This lesson plan has students use a graphing calculator (T) to learn about measures of central tendency (C) through collaborative learning (P).

This lesson plan has students use GeoGebra (T) to learn about circles (C) through inquiry based learning (P).

This lesson plan has students use Mathematical Modeling Online Simulation (T) to learn about Transformations of Quadratic Equations(C) through Inquiry-Based Learning (P).

This lesson plan has students use Online Tutorials and a SmartBoard (T) to learn about Transformations of Quadratic Equations(C) through Active Learning (P).

This lesson plan has students use Graphing Calculators (T) to learn about Transformations of Quadratic Equations(C) through Self-Assessment (P).

Scale Factor Lesson This lesson plan has students demonstrate their understanding of scale factor (C) using Geometer Sketchpad (T) while incorporating cooperative learning (P).

Fractions Lesson Plan This lesson plan has the teacher utilize the Promethean Board (T) to teach fractions (C) using visual learning (P).

Understanding Multiplication

Solving equations

This lesson plan has students use GeoGebra (T) to learn about Shapes (C) through Cooperative learning (P).

This lesson has students use Desmos (T) to learn about the area ratio of similar polygons (C) through Problem based learning (P)

This lesson has students use Educreations (T) to review the foundation of Geometry (C) through Project based learning (P)

This lesson has students use ipads (T) to engage students in an interactive game based learning (P) to teach addition to Early preschool students.
Teaching addition for early preschool students using the TPACK model.

A TPACK Lesson Example

The following lesson demonstrates the lesson plan that a teacher went through in order to implement TPACK into her classroom. The overall objective of the lesson was for students to gain an adequate understanding and skill necessary to divide from standard to mixed number and know that the two numbers have the same equivalency. From a TPACK perspective, the Content aspect is clearly fractions. Whilst the instructor was to develop the latter aspects, pedagogy and technology. In recap, technology is the actual learning device used to enhance the learning process whereas pedagogy is the actual mechanism in which the students undermine the lesson. In this case, the teacher chose the pedagogy of "visual learning" and she made a rather wise decision for a thorough understanding of manipulating fractions does require examination and analysis. It would be easier for the students to comprehend the content through such a pedagogy than through mere textbook or chalk-board instruction. Of course, an effective pedagogy must be accompanied with an appropriate technology. The instructor chose a Promethean Board. There is an excellent correlation between the pedagogy(visual learning) and the technology(Promethean Board) for the technology promotes the pedagogy.

Correlation Between Pedagogy and Technology

It should be noted that it is particularly significant for the pedagogy and technology chose for specific lessons to "match". For instance, group work would not really work with the Promethean. Examples of effective pedagogies are given.
  • cooperative learning & IPads
  • visual learning & Promethean Board
  • group work & laptops
  • Collaborative Math and discussion-based English to help promote
    deeper learning, critical thinking, and community.
  • Cooperative, problem-based learning has been shown to improve student engagement and retention of classroom material (Prince, 2004).

Qualities of a Mathematics Teacher with TPACK

The following list shares characteristics of an effective mathematics teacher that incorporates TPACK:

  1. Would have a relative openness to experimentation with the ever-evolving technological tools available to them in the classroom. Such teachers will try new technology-based lessons with their students, confident that students can learn something value each time they try something new.

  2. Would strive to be consistently “on-task” for the mathematical topic. That means teachers with strong TPCK backgrounds are effective at focusing on the mathematics concepts while taking advantage of the instructional opportunities offered by technology.

  3. Would strive to know where their students are conceptually, what they need to do to achieve the next step in an instructional process, and how they generally want their students to proceed through careful sequences of classroom interactions and tasks.

  4. Would consistently offer explanations to their students on what they are doing with the technology, why a specific tool is appropriate for a particular mathematical situation, and how a selected technology fundamentally works.

  5. Would use student assessment data, such as criterion-referenced tests. Such assessment data can help a teacher identify gaps in student understanding which might form the rationale for switching instructional strategies or taking a different pedagogical approach with some or all of the students.

  6. Would also do their best to be caring teachers who are comfortable and optimistic for change. Teachers with strong TPCK backgrounds should be caring instructional leaders that are welcoming to all students as they enter this changing and evolving world of mathematics.

According to the AMTE (Association of Mathematics Teacher Educators), mathematics educators serve their students by considering the potential impact of forms of 21st Century digital technologies and plan accordingly. The article addresses specific guidelines for enhancing mathematical learning experiences via technology.

As teachers, we need to remember our own experiences and frustrations as learners, as well as link prior knowledge with newly constructed knowledge . Teachers should be aware that students learn at different paste and try to use various ways in the class with the students when explaining a new lesson.

Math Teacher Education: Learn More, Do More, The TPACK Way

The mathematics teacher with TPACK (Technology Pedagogy Content Knowledge) is a chess player. The constantly evolving trends during a chess game change the roles of the interdependent pieces; involving a persistent need to rework proficiency with the different pieces just as the mathematics teacher has to continually reason logically and progressively with current tools. If “TPACK is a way of thinking strategically while involving in planning, organizing, critiquing, and abstracting” (Niess. 2008:224) to fulfill teaching and learning needs with today’s technologies, the teacher makes preparations to face challenges, systematizes approaches, consistently does analysis or appraisals of situations, and condenses information by updates. Hence, for the mathematics teacher, to learn more is to do more. With TPACK, it is learning more and doing more with technology in a meaningful context.

Taking the cue from this standpoint, pre-service and in-service mathematics teachers have to be educated with the TPACK model regularly. Considering the importance of experiencing the progression of mathematical inquiry in assimilating technology with pedagogy and content, these programs are designed generally to include discussions of the latest issues in mathematics and technology, Internet browsing for navigational and research purposes, simple programming, exploring mathematical softwares, problem solving with technology, and even showcasing video vignettes. These set up are intended to accomplish investigative learning to foster ease and insight on the use of technology in mathematics; to exemplify the proper uses of reputable and new applications of technology within the mathematics context; to guide teachers in making practical but sensible determination as to the suitability of the technology in the teaching and learning of mathematics; and to give teachers the chance to acquire and expand their technological skills to improve the understanding of mathematics.

As mathematics dynamically transforms with the developments in technology, the mathematics teacher with TPACK incessantly strives to keep up with these changes. Strengthening these core values of TPACK, the National Technology Leadership Coalition (NTLC), which umbrellas educational technology associations and teacher educator organizations in specific content areas; endeavors to concentrate technology knowledge (TK) with the pedagogy content knowledge (PCK) of the teacher educator organizations in a subject area by setting standards, endorsing TPACK practice, and advancing research on the judicious use of TPACK. Its mathematics wing is known as the Association of Mathematics Teacher Educators (AMTE).

With teacher education programs emphasizing on the TPACK approach, the mathematics teacher becomes well-rounded, fully informed, adaptable, and effective. It just proves to say that to learn more is to do more with the TPACK way.

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Relevance of TPACK

Many instructors are rather hesitant towards incorporating TPACK in the classroom, because they are more accustomed to the "ordinary" way of teaching. They deem it unnecessary for students to become acquainted with technology. However, with the rise of technology in the past years, there must be interaction between the variables of knowledge and technology because, ultimately, the learning process is enhanced. The following link contains rather positive statistics in regards to instructor and student perspective of technology in the classroom. http://www.teachthought.com/technology/the-current-state-of-technology-in-the-classroom-infographic/

Relevance of TPACK for Early Elementary Students
Online learning is an effective technology to facilitate group discussions, because students are able to reflect, gain self- awareness in their abilities, and enhance individualized learning of the content, and then share with peers. Without online learning, according to the U. S. Gov. on Education, students would build on their prior understandings which may or may not be consistent with the normative disciplinary content they are asked to learn in school, and actively drive their own learning from teachers or from textbooks. However, with the integration of technology, from online resources like PBS Kids, students are able to interpret new material, and gain an in-depth understanding, increasing their development in vocabulary, and spatial relationships. https://tech.ed.gov/wp-content/uploads/2014/11/Learning-Technology-Effectiveness-Brief.pdf

Inclusion of TPACK
Essentially, TPACK is a matter of incorporation and the matter of how this process is facilitated is a crucial topic that must be addressed to educators of America. This process is not a one-day reaction, but rather requires extensive effort out of the education system. Whether through seminars, workshops and simple typical school meeting, TPACK inclusion is critical and must be taken into account for the benefits of this program are immense.The following video discusses various methods in which TPACK can be included in the classroom. http://www.learnnc.org/lp/editions/every-learner/6776

According to Neiss, "integrating technology is not about technology – it is primarily about content and effective instructional practices. Technology involves the tools with which we deliver content and implement practices in better ways. Its focus must be on curriculum and learning. Integration is defined not by the amount or type of technology used, but by how and why it is used."

It is imperative for teachers to develop an in depth understanding of TPACK and utilize it in their classrooms. Mishra and Koehler worked to develop a framework for TPACK. According to AMTE, "it is intended to serve as a guide for mathematics educators and researchers to plan, examine, improve, and evaluate mathematics instruction at all levels.Below are the main areas of concentration of the framework, however here the link to receive additional information: https://amte.net/sites/all/themes/amte/resources/MathTPACKFramework.pdf
  1. Design and develop technology ­enhanced mathematics learning environments and experiences.
  2. Facilitate mathematics instruction with technology as an integrated tool.
  3. Assess and evaluate technology ­enriched mathematics teaching and learning.
  4. Engage in ongoing professional development to enhance technological pedagogical content knowledge.

Informal assessments- Are not data driven but rather content and performance driven.
Formal assessments or standardized measures - should be used to assess overall achievement, to compare a student's performance with others at their age or grade, or to identify comparable strengths and weaknesses with peers.

Formal vs. Informal Assessments | Scholastic

Formal assessments- Have data which support the conclusions made from the test. We usually refer to these types of tests as standardized measures. These tests have been tried before on students and have statistics which support the conclusion such as the student is reading below average for his age.

Formal vs. Informal Assessments | Scholastic


Informal Assessment for learning positional words:
To gather a broader and meaningful picture of development of student’s knowledge or growth in learning positional words, the teacher will observe students by taking anecdotal notes, and use photographs of their work in (LG). By doing an authentic assessment, teachers are able to meet children where they are developmentally, and identify students who show knowledge of concepts, or those who may need further guidance.

Formal Assessment for learning positional words:
When scoring student’s knowledge of positional words, teachers can utilize Teaching Strategies Gold which is a one-dimensional measure of children's ability, showing knowledge of concepts learned. This online assessment tool uses numerical goals, showing baseline knowledge, proficiency, and advanced student growth. Each domain uses a scoring system of “1-5”. If a student receives a “1” in the content area, then the student is showing some knowledge of concepts. If a higher score is given, a student was able to identify the relative location of an object to another object when provided with a directional word, and possibly comprehends relative location is different from absolute location. If students score under proficiency, teachers can gear additional learning activities online through online resources like PBS Kids, to assist children in obtaining achievable learning goals. Individual student work, and anecdotes, can be recorded via application, and uploaded directly into the tool for quick and easy access

TPACK Instructional Design and Analysis #2

This is an introductory lesson on solving systems of equations with substitutions in Pre-Algebra classes. Pre-Algebra is a high school level course that intends to prepare students for both Algebra 1 and Algebra 2. This is to ensure that they have all the preparations to succeed in both classes.

Possible Math Technologies:

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Elliott, R., Kazemi, E., Lesseig, K., Mumme, J., Carroll, C., & Kelley-Petersen, M. (2009). Conceptualizing the work of leading mathematical task in professional development. Journal of Teacher Education, 60(4), 364-379.

Figure 2f from: Irimia R, Gottschling M (2016) Taxonomic revision of Rochefortia Sw. (Ehretiaceae, Boraginales). Biodiversity Data Journal 4: E7720. https://doi.org/10.3897/BDJ.4.e7720. (n.d.). doi:10.3897/bdj.4.e7720.figure2f

Fromboluti, C.S., & Rink, N. (1999). Early childhood where learning begins mathematics. US Department of Education, http://www.ed.gov/pubs/EarlyMath/title.html

Grandgenett, N. (2008). Perhaps a matter of imagination: TPCK in mathematics education. Handbook of Technological Content Pedagogical Knowledge (TPCK) for Educators, New York: Routledge, 145-165.

Michigan Department of Education. (n.d). MDE k-12 curriculum. Retrieved September 23, 2009 from http://www.michigan.gov/mde/0,1607,7-140-28753---,00.html

Niess, M. (2008).Guiding preservice teachers in developing TPCK. Handbook of Technological Content Pedagogical Knowledge (TPCK) for Educators, New York: Routledge, 223-250.

Niess, M. L., Ronau, R. N., Shafer, K. G., Driskell, S. O., Harper S. R., Johnston, C., Browning, C., Özgün-Koca, S. A., & Kersaint, G. (2009). Mathematics teacher TPACK standards and development model. Contemporary Issues in Technology and Teacher Education, 9(1), 4-24

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Keppel, F., & Dabbagh, N. (1952). Contemporary Issues in the Education of Teachers. Journal of Teacher Education, 3(4), 249-255. doi:10.1177/002248715200300403

College of Education and Human Development George Mason University

Dabbagh, N. (2007). The online learner: Characteristics and pedagogical implications. Contemporary Issues in Technology and Teacher Education, 7(3), 217-226.

Dodge, B. (n.d.). What is a WebQuest? Retrieved July 27, 2007, from the WebQuest home page: http://webquest.org

Johnson, E. (n.d.). Collaborative Learning via In-class Group Projects. PsycEXTRA Dataset. doi:10.1037/e413782005-889

T. (1995). Collaborative Learning Enhances Critical Thinking. TE v7n1 - Collaborative Learning Enhances Critical Thinking, 7(1). Retrieved from https://scholar.lib.vt.edu/ejournals/JTE/v7n1/gokhale.jte-v7n1.html.

[[http://scholar.lib.vt.edu/ejournals/JTE/v7n1/gokhale.jte-v7n1.html?ref=Sawos.Org#Totten, et al.|Totten, Sills, Digby, & Russ, 1991 ]]T. (n.d.). Learning is a collaborative responsibility. Retrieved from https://dpi.wi.gov/sites/default/files/imce/cal/pdf/guiding-principles4.pdf

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