Sunday, September 2, 2018

Praxis in Pedagogy: Theory, Action, Doing, Refection Praxis in Education?

Aristotle's Praxis "Pedagogy" Process in Education: Theoria (THINKING ), Poiesis (MAKING), and Praxis (ACTION/DOING)! 

Using the Praxis Process requires involving all stakeholders in the ongoing communication of ideas, theories, tasks, structures, and institutions to be transformed. Praxis requires cogent communication that is entwined with reflection, cyclical review, and reciprocity.  The praxis process seeks to find the best practices that benefit and liberate the neglected, marginalized, and or "left behind."

Time to Get Off the
Dead Horse.

American education is trapped in a cycle of accountability theater, EdTech grift, and top-down control — and we keep piling on instead of getting off. A MECE analysis through Aristotle's Praxis lens.

By Sage Sean TaylorLong Read · ~14 minPedagogy · Systems Thinking · Policy

There is a Dakota Sioux proverb that ought to be posted above the door of every Department of Education in America: When you discover that you are riding a dead horse, the best strategy is to dismount. But in this country, we have a different strategy. We form a committee. We hire a consultant. We rename the dead horse. We buy expensive saddle technology from a company that offers no refund and no guarantee. We blame the rider. We grade the rider. We fire the rider. And then we wonder — genuinely, earnestly wonder — why the horse will not move.

This essay is not a polemic. It is a diagnosis. Using Aristotle's three-part framework of Theoria (critical thinking), Poiesis (making and creating), and Praxis (reflective action) — alongside a rigorous MECE analysis — we will examine precisely what is broken, why it stays broken, and what two small nations with no natural wealth chose to do differently. And we will dare to ask what it would look like if American education finally dismounted.


The MECE AnalysisMapping the Broken System — Mutually Exclusive, Collectively Exhaustive

A MECE analysis demands that every dimension of a problem be identified without overlap and without gap. When we apply this framework to the American education crisis, four non-overlapping root cause categories emerge. Together, they explain the whole failure. Separately, each one is a wound that festers on its own.

MECE Root Cause Map — American Education Crisis

I. Governance & Incentive Failures

Politicians and school boards optimize for optics, not outcomes. Administrators protect institutional hierarchy. The incentive structures reward compliance and test performance over actual learning. No stakeholder with power has a meaningful accountability stake in whether children flourish decades later.

II. Curriculum & Pedagogy Failures

Top-down mandates like Common Core impose uniform frameworks that flatten the diversity of learners. Teaching to the test crowds out critical thinking, the arts, physical education, and the kind of exploratory, hands-on learning that builds actual human capability. Poiesis — the act of making and creating — is nearly absent.

III. Resource & Market Failures

EdTech vendors extract public funds without performance guarantees. Publishers lock districts into expensive, dated curricula. Per-pupil spending of $17,700 (2025) produces no measurable improvement in outcomes when the underlying pedagogy is broken. Money flows toward the broken system, not around it.

IV. Human Capital & Equity Failures

Teacher attrition is at record levels. Shortages persist in STEM and special education. Poverty remains the single strongest predictor of underperformance, yet social investment in children's earliest years remains politically contentious. The children who need the most receive the least, and the system is designed — whether intentionally or not — to keep it that way.

These four quadrants are not separate problems. They form a reinforcing system — a doom loop. Weak governance enables market capture by EdTech and publishers. Perverse incentives corrupt curriculum. Poor pedagogy drives teachers out. Teacher shortages deepen inequity. Inequity produces the test-score gaps that politicians use to justify more testing. The horse is not merely dead. The dead horse is hitched to a wheel that spins and spins.

30%of 12th graders lack basic reading proficiency (2024 NAEP)
80%of educators report frequent classroom disruptions
$17,700average per-pupil spending in 2025 — with no measurable outcome gains

Theoria — The Art of Looking DeeperWe Are Auditing the Wrong Things

Aristotle's concept of Theoria is not mere thinking. It is cyclical, Socratic, probing. It is the discipline of asking why before asking how. It demands that we audit our premises before we audit our teachers.

What does America actually audit? Test scores. Graduation rates. Compliance with federally mandated reporting structures. These are outputs of a system, not indicators of its health. We are reading the thermometer and ignoring the fever.

Our obsession with testing doesn't mean we are doing the right thing. We are measuring the measurable and calling it education.

Paraphrased from research on Finland's PISA success — Krueger et al.

A genuine Theoria audit of American education would ask different questions. Not what score did this school produce? but what conditions allow children to develop into full human beings? Not did this teacher's class hit the proficiency target? but what does this teacher need to do the most important work in civilization?

The Stanford d.school's Design Thinking framework — built on human-centered empathy, iterative prototyping, and genuine problem definition — offers exactly this kind of auditing capacity. It has been applied to healthcare, technology, and urban planning. It has been applied to redesigning hospital gowns and airport experiences. It has rarely, in any systematic way, been applied to the American K–12 system. The irony is staggering. We have a world-class tool for reimagining broken systems sitting forty-five minutes from Silicon Valley, and the Silicon Valley titans who claim to be disrupting education have never seriously picked it up.

Theoria demands that we perform a genuine cost-benefit analysis — not of individual programs, but of the entire paradigm. When Aristotle's audit process is applied honestly, the result is damning: the costs of our current system are incalculable in human potential; the benefits accrue almost entirely to administrators, publishers, test companies, and politicians. The children are the input. They are not the priority.


Poiesis — The Art of MakingWe Have Stopped Letting Children Create

Aristotle's Poiesis is the act of bringing something into being that did not exist before. It is the craftsman's wisdom, the maker's instinct, the artist's compulsion. Martin Heidegger described it as a "bringing-forth" — the blooming of a blossom, the butterfly emerging from its cocoon, the waterfall released when the snow melts. A moment of threshold: something moves from what it was into what it becomes.

What we have done to education is the precise opposite of Poiesis. We have taken the threshold moment — the astonishing window of childhood when the human mind is most plastic, most hungry, most capable of transformation — and we have filled it with worksheets, standardized bubbles, and test prep. We have replaced the craftsman's workshop with the assembly line. We have stopped trusting children to make things, and we have started trusting Pearson Publishing to tell us what children should know.

Dreyfus and Dorrance Kelly argued that the task of the craftsman is not to generate meaning but to cultivate the faculty for discerning meaning that is already there in the world. This is precisely what great teaching does. And it is precisely what accountability-driven, standardized, top-down pedagogy cannot do — because it is not interested in the individual child's encounter with meaning. It is interested in the child's compliance with a rubric.

What Poiesis Looks Like in a Classroom

Project-based learning where students build, design, and solve real problems. Reggio Emilia ateliers where children direct their own inquiry. Maker spaces. Cross-disciplinary challenges that have no single correct answer. A student who builds a working water filter to understand chemistry has engaged in Poiesis. A student who circles the correct answer about water filtration on a multiple-choice test has not. Both can score the same on a standardized exam. Only one has become a craftsman.

The tragedy is that we know this. The research literature on project-based learning, experiential education, and student-directed inquiry is substantial and consistent. What is missing is not knowledge. What is missing is the institutional will to act on it — because acting on it would require dismantling a testing-industrial complex worth billions of dollars to the companies that profit from its perpetuation.


The EvidenceTwo Nations That Decided the Child Was the Point

It is instructive to look at two nations that had nothing — no oil, no mineral wealth, no legacy of imperial advantage — and decided that their only real resource was the human mind. What they did with that decision is among the most remarkable stories in the history of modern governance.

🇫🇮 Finland

  • Teachers hold master's degrees; treated as elite professionals
  • No standardized testing until the end of high school
  • Play-based, child-centered learning in early years
  • Emphasis on well-being, creativity, and critical thinking
  • Free public education, free meals, integrated special needs support
  • Education policies consistent and stable for over 40 years
  • Rooted in the Reggio Emilia and whole-child philosophies

🇸🇬 Singapore

  • Largest share of GDP invested in education in Asia
  • Rigorous teacher selection, compensation, and ongoing development
  • Curriculum regularly overhauled — "Teach Less, Learn More"
  • Holds top PISA scores globally across three consecutive assessments
  • Strong family and cultural investment in learning from age 3
  • Focus on STEM with deep integration of real-world application
  • Stable political commitment to education as national priority

These two countries arrived at their excellence through radically different philosophies. Finland chose the whole child. Singapore chose academic rigor with relentless systemic refinement. They share two things: the decision to invest in teachers as the most important variable, and the refusal to allow testing culture to substitute for genuine learning culture.

Finland's education success has the rest of the world looking — because its educational policies were consistent for more than forty years. The US changes course every four years depending on who wins an election.

Adapted from PISA comparative research

America's response to these examples has largely been to either ignore them as "culturally incomparable" or to cherry-pick one element — charter schools, a coding curriculum, a new app — without engaging with the systemic logic underneath. You cannot bolt a Finnish outcome onto an American incentive structure. The incentive structure is the problem.


Praxis — Reflective ActionThe Cycle We Refuse to Enter

Aristotle's Praxis is not simply doing. It is doing with ongoing reflection, revision, and accountability to those the action is meant to serve. It is the antithesis of top-down policy mandated from a position of no accountability and implemented with no feedback loop. Praxis is inherently democratic. It demands that those closest to the problem — teachers, students, parents, communities — be genuine partners in the inquiry.

Action researchers who engage in praxis-oriented work involve the community under study in the research process itself. The knowledge is not handed down. It is built together. The interventions are tested, reflected upon, revised, and tested again. This is neither utopian nor impractical. It is simply rigorous. It is what good doctors do. It is what good engineers do. It is what American education has never, structurally, been willing to do.

The Praxis cycle, applied honestly to school reform, looks like this:

  1. Take the actionImplement a pedagogy change, structural reform, or new resource — at the classroom or school level, not the district-mandate level. Small, specific, with teacher agency.
  2. Consider the impactsObserve what actually happens — not through a standardized test administered months later, but through real-time teacher observation, student response, and community feedback.
  3. Analyze through reflectionAsk: did this serve the child? Did it serve the teacher? What changed? What didn't? Who benefited and who did not? Whose voice is missing from this analysis?
  4. Alter and revise the conceptionRedesign — not by committee mandate, but by the people with genuine knowledge: the practitioners, the families, the students themselves. Revise the theory based on the reality.
  5. Implement and continue the cyclePraxis is not a one-time reform. It is an institutional posture — a permanent commitment to looking at what we are actually doing and asking whether it is serving human flourishing. The cycle never ends. That is the point.

This process is unglamorous. It does not produce a press release or a TED Talk or a billion-dollar contract. It does not give politicians a talking point or give billionaires a product to scale. It is slow, iterative, deeply human work. This is precisely why it has never been seriously attempted at the systemic level in American public education.


The Political EconomyWhen Billionaires Play Curriculum Designer

We cannot leave this analysis without addressing the particular damage done by the entrance of tech wealth into education reform. Bill Gates's Common Core initiative — one of the most expensive and consequential education experiments in American history — was built on the same logic as enterprise software: standardize, scale, measure. It treated the classroom as a deployment environment and children as end users.

The results speak for themselves. The 2024 NAEP Nation's Report Card confirmed a decades-long crisis. Major initiatives including No Child Left Behind, Common Core, and Every Student Succeeds Act were all designed to improve achievement. They did not. The most significant losses occurred among the students who were already most disadvantaged — the very students every reform claimed to be serving.

This is not an accident. It is the predictable outcome of designing reform without asking the people it affects, without building in reflective feedback, and without genuine accountability for failure. Gates did not refund the districts. The publishers did not offer refunds. The politicians moved on to the next initiative. The children remained.

The problem with education reform in America is not a lack of ideas. It is a lack of Theoria — a refusal to truly look at what we are doing, to whom, and in whose interest.

Sage Sean Taylor

Meanwhile, Stanford's d.school has spent decades developing human-centered design frameworks explicitly suited to complex systemic challenges. IDEO has produced methodologies that begin with genuine empathy for the human at the center of a problem — in this case, the child — and work backward from there. These tools exist. They are not obscure. They are simply not profitable for those who control the policy levers, and so they are not used.


The Path ForwardWhat Dismounting Actually Looks Like

To dismount the dead horse is not to abandon accountability. It is to build a different kind of accountability — one that flows upward from teachers and children, not downward from politicians and publishers. It means genuinely investing in the people who do the work, as Finland and Singapore did, rather than surveilling them with instruments designed by people who have never spent a day in a classroom.

It means acknowledging that poverty is not a teacher problem. That chronic absenteeism is not a curriculum problem. That a child who is hungry, unsafe, or unhoused cannot perform on a standardized test — and that no amount of testing will change that reality. It means accepting that early childhood investment is not a liberal or conservative value — it is an economic one. Every dollar invested in early childhood education returns between seven and twelve dollars in reduced remediation, incarceration, and social costs over a lifetime. This is not ideology. This is arithmetic.

It means treating teachers the way Singapore and Finland treat them: as the most important professionals in the country, whose judgment should be trusted, whose working conditions should be respected, and whose expertise should be the foundation of curriculum, not an afterthought to it.

And it means applying the Praxis process — genuinely, systematically, at scale — to the question of what education is for. Not what it should produce for the economy. Not what it should look like on an international ranking. But what it owes to each specific child, in each specific community, right now.

The best education system for any individual depends on what they value most: academic rigor, student well-being, equity of opportunity, creative development, or career outcomes. The research is consistent on one thing: excellent teachers, adequate resources, high expectations with strong support, and cultures that value learning.

The Tutor Bridge, Global Education Rankings, 2026

The horse is dead. It has been dead for a long time. Parents know it — which is why charter school enrollment and homeschooling have surged to historic levels. Teachers know it — which is why attrition is at record highs. Children, who are not permitted to say so out loud, know it in the particular way that children know when the adults around them have stopped paying attention to what actually matters.

The question is not whether the horse is dead. The question is whether we have the institutional courage — and the political will — to dismount.


Food for ThoughtQuestions Worth Sitting With

This is not a conclusion. It is an opening. The Praxis process demands that you bring your own reflection to what you have read. So before you close this page, consider:

If you are a parent: What are you being asked to accept as "normal" in your child's school day? What would you fight to change if you believed change was possible?

If you are a teacher: When did you last feel trusted? When did you last feel that your professional judgment mattered more than a test score? What would you teach if you were free to teach it?

If you are a policymaker or administrator: Who sits in the room when reform decisions are made? Who does not? What would happen if the people most affected by your decisions were genuine co-authors of them?

If you are a citizen: Education is not a product. It is a public good — the most foundational investment a society makes in its own future. Who benefits from the current system? Follow the money. Then follow the children.

The Praxis cycle does not end here. It begins here. The thinking, the making, the doing — and then the reflection, the revision, the return. This is the work. It has always been the work.

Get off the horse.

Sources & Further Reading: NAEP Nation's Report Card 2024; Pew Research Center Survey of Educators 2024; Learning Policy Institute Teacher Shortages Scan 2025; PISA Global Rankings (OECD 2022–2024); Comparative Analysis of Singapore and Finland Education Systems, International Journal of Education and Humanities (2024); Stanford d.school Design Thinking Framework; Heidegger, M., The Question Concerning Technology; Dreyfus, H. & Kelly, S.D., All Things Shining; Aristotle, Nicomachean Ethics.

Sage Sean Taylor

Education · Pedagogy · Systems Thinking · Praxis

© Sage Sean Taylor · All Rights Reserved · For Podcast Compani



Like action researchers, those who engage in praxis-oriented research involve the community or group under study in the research process. ... Praxis-based researchis a long process that involves establishing mutually beneficial relationships between the researcher and members of the community of study.  methods.sagepub.com

Theoria (Thinking): The word theoria is a cyclical form of critical thinking, it is the foundation for Socratic inquiry, to look deeper, or to seek or see a truth, virtue, or knowledge. Adroit knowing is a mind's eye way of seeing, a sort of intellectual seeing.

Inquiry, Questioning. and Thinking: audit, inspect, examine, survey, go through, scrutinize, check, probe, vet, investigate, inquire into, assess, verify, appraise, evaluate, review, analyze, study; WHY, WHAT, and HOW!


Audit Process

The following is a list of steps that comprise a cost-benefit analysis of ideas or theories.

  • Define clear goals, objectives, competencies measures, and or desired outcomes of critical learning procedures, activities, pedagogical reform theories, or new educational systems being considered for adoption. 
  • List alternative ideas, projects, systems, theories, programs.
  • List all stakeholders.
  • Select clear benefit measures,  and measure all cost/benefit elements.
  • Predict all possible outcomes negative and positive, examine the cost, the amount of time, real wages, buy-in, and the measurable benefits over the relevant time period.
  • Convert all "possible costs" and "possible benefits" into a common language and value.
  • Apply a value for the negative reciprocity over time.
  • Calculate net present value of project options.
  • Perform sensitivity analysis. +1 to -1 correlation 
  • Adopt the recommended choice based on real value and reforms that have a strong statistical correlation.

Poiesis (Making): Poiesis  is "the activity in which a person brings something into being that did not exist before, a kind of making/creating, developing new knowledge, skills, virtues, or a poiesis."

Demiurge (Makers/Creators): With teacher-students and students-teachers developing and co-creating learning activities with real-world applications that are hands-on. These real-world tasks encourage active learning and active into (skilled educated worker) a craftsman, artisan, producer, and or a creator of ideas, knowledge, and skills.
Martin Heidegger refers to poiesis as a 'bringing-forth' (physis as emergence), using this term in its widest sense. He explained poiesis as the blooming of the blossom, the coming-out of a butterfly from a cocoon, the plummeting of a waterfall when the snow begins to melt. The last two analogies underline Heidegger's example of a threshold occasion: a moment of ecstasis when something moves away from its standing as one thing to become another. (These examples may also be understood as the unfolding of a thing out of itself, as being discloses or gathers from nothing [thus nothing is thought also as being]). Additional example: The night gathers at the close of day. 


Dreyfus and Dorrance Kelly urge each person to become a sort of “craftsman” whose responsibility it is to refine their faculty for poiesis in order to achieve existential meaning in their lives and to reconcile their bodies with whatever transcendence there is to be had in life itself: “The task of the craftsman is not to generate the meaning, but rather to cultivate in himself the skill for discerning the meanings that are already there.”
Praxis in education starts with critical thinking about problems within pedagogical institutions and or educational systems. New actions, structures, systems, ideas and or theories that may solve the problems are always being analyzed. Comprises, review, further reflections, action,  and reexamination of the efficacy of actions are key parts to praxis. Praxis can be viewed as an ongoing progression of cognitive and physical actions:

  1. Taking the action
  2. Considering the impacts of the action
  3. Analysing the results of the action by reflecting upon it
  4. Altering and revising conceptions and planning following reflection
  5. Implementing these plans in further actions
This creates a cycle which can be viewed in terms of educational settings, learners and educational facilitators.

13 Math Heuristics

What are the 13 Problem-solving Math Heuristics!

Top-Down Approach to Teaching Problem Solving Heuristics in Mathematics

A heuristic technique, or heuristics, is an approach to problem-solving, learning, and/or discovery method that employs a mental shortcut/tactic, it is not always optimal, perfect, logical, or rational, but instead efficient for reaching an immediate solution or answer. Where finding an optimal solution is not apparent, impossible, and/or impractical, heuristic methods can be used to speed up the process of finding a satisfactory solution. Heuristics can be mental shortcuts that ease the cognitive load of making a decision. Examples that employ heuristics include using a rule of thumb, an educated guess, an intuitive judgment, a guesstimate, stereotyping, profiling, or common sense. Math Heuristics, Act it out, Draw a diagram or use a model, Use guess-and-check, Make a systematic list, Look for patterns, Work backwards, Use before-after concept, Make suppositions, Restate the problem in another way, Simplify the problem, Solve part of the problem, Think of a related problem, and finally use equations.




(PDF) Math Heuristics - ResearchGate

[PDF]Top-Down Approach to Teaching Problem Solving Heuristics in ...
identified thirteen heuristics that are applicable to mathematical problem solving. ... 13. Use equations. (Heuristics 12 and 13 are not in the primary syllabus.) ..... January 7 th. , 2005, from http://www1.moe.edu.sg/syllabuses/doc/Maths_Pri.pdf.

[PDF]A Metacognitive Approach to Support Heuristic Solution - NIE Digital ...
Together they describe the process of mathematical problem solving. ... Ministry of Education Singapore (MOE), have identified 13 heuristics that are applicable to mathematical ..... from http://www1.moe.edu.sg/syllabuses/doc/Maths_Pri.pdf.

[PDF]CIRCLES - Maths Heuristics
Heuristics is a specialisedmathematical problem-solving concept. Mastering it .... Page 13 ...

[PDF]Loren C. Larson - Problem-Solving Through Problems - (Problems ...
The book is both an anthology of problems and a manual of instruction. .... will be concerned with theheuristics of solving mathematical problems. ... Page 13 ...


[PDF]LEARNING PROBLEM SOLVING HEURISTICS FROM ... - CiteSeerX
Much human learning of mathematics takes place from worked examples, yet this is a ... the heuristicswere learned through attempts at problem solving, and not through ..... Proceedings of the 13thInternational Joint Conference on Artificial ...

[PDF]Learning Problem-Solving Strategies by Using Math Games 1.2312016
Computer Programming as a Game. Chapter 9 of this book briefly discusses ...... In attempting to understand a math problem, a sometimes-useful heuristic is to draw a picture.


[PDF]Heuristic Strategies and Deductive Reasoning in Problem Solving
Use of Heuristics in Mathematical Problem Solving. 4. 2.1 Introduction to ... 4.2.1 Deduction as a Formal Syntactic Process based on Rules 13.

[PDF]Mathematical problem solving in primary school - Utrecht University ...
Early attempts to teach problem solving focused on heuristic training (Kantowski, 1977; .... 13. More specifically, the non-routine problems in this study contain ...

[PDF]Problem Solving in Mathematics Education - DiVA
Problem Solving in Mathematics Education. Proceedings from the 13th ProMath conference. September 2 " 4, 2011, in Ume , Sweden. Editor: Tomas Bergqvist.
Missing: heuristics ‎| ‎Must include: ‎heuristics

[PDF]problem solving in school mathematics based on heuristic strategies
“Problem Solving in School Mathematics Based on Heuristic .... 28 pupils aged 13) and two upper secondary grammar ..... pisaproducts/48852548.pdf.

[PDF]heuristics in problem solving for the teaching and ... - Estudo Geral
heuristic mathematical problem solving and simultaneously learn with your personal ... purpose to create a problem solving Manual, to be further tested. ...... established compulsory education for children between 7 and 13 years old, reinforced ...

[PDF]Learning to prove: The idea of heuristic examples - Mathematical ...
proving in designing the single sub-steps of a heuristic example. We illustrate our ... items in the 12th grade (respectively 13th grade) TIMSS test that could be ...

[PDF]Nonmanut Pongsakdi – Bridging Mathematics With Word ... - UTUPub
. 13 solve word problems that resemble maths-problem situations encountered in their everyday life (Verschaffel ... implemented one year of heuristic problem-solving instruction with middle school students.

[PDF]Untitled
Mathematical problem solving can be seen as a process in which cognitive and affective .... otherheuristic is available to solve a specific problem or that the allotted .....

[PDF]An investigation into the use of problem-solving heuristics to improve
Heuristics, problem-solving processes and mathematical modelling. 12. 2.1.1Heuristics ..... These four phases have 13 grades in total and in theory, the ages of ...... http://www.math.kent.edu/~edd/HeidPaper.pdf. Arnon, I.

Wednesday, August 29, 2018

Place Value Number Talks, Games, & Activities

Free Place Value Number Talks, Games, & Activities PDF Printable


[PDF]Number Talks Grades 3-5 Resources
Kindergarten through fifth grade teachers will facilitate Number Talks with all students .... Encourage students to break each number into its place value using numbers that do not have an ..... Mia saved $182. She bought a video game for $53.

[PDF]
place value games - BOOST Conference
Games support the instruction of place value concepts with base- ... Use number lines: 0 - 9, 0-100, or tape ten together for a 0-1000 line.

[PDF]Place Value Yahtzee Score Cards - Shopify
At the end of the game, compare the number you wrote in each category with the number ... Place ValueYahtzee Score Card.

[PDF]Place Value Place Value Math Games Math Games Place Value ...
Math Games. Place Value Comparisons (Grades 2 +). Skills: comparing numbers, odd and even. Players: 2 or more. Materials: cards (Ace=1)-9, die. Procedure:.

[PDF]Copy of Math Board Games - The Mathematics Shed
Star Struck - Division – Practice division. 9. Odd and Even Tag – Revises odd and even numbers. 10. Build A Pyramid Red Game – Practice place value to 100.


[PDF]Fun Math Game Printables - The Mathematics Shed
Math Board Games at www.math-board-games.com ... Pencil and Paper Math Games .. 29 ..... This is a great game for exploring number order and place value.


[PDF]Station: Base 10 game - PDST
Number. Strand unit: Place Value. Objective: Explore, identify and record place value 0-999. Read, write ... When you have no ones/longs left the game is over.

[PDF]The Big “Z”
Players total their scores on paper and at the end of the game, player with the highest ... Building Fluency: review place value - compare multi-digit numbers.
[PDF]Silent Signals - Wisconsin Mathematics Council
Using Place Value and Negative Numbers. 5. ... Number Talks: Helping Children Build Mental Math and Computation Strategies. ..... The activities given above are illustrations of games and tasks in which ... structionalShifts_Mathematics.pdf.


[PDF]Math: It's Elementary! - Georgia Association for Gifted Children
by G FiddymentStrategies and tools used to develop number sense and place value, including: ... Activities for number sense: estimation prior to solving, determine reasonableness of ... http://elemath.hallco.org/web/wp-content/uploads/2014/05/Number-Talks-Quick-Start-Guide.pdf (from. Number Talks .... Game Night Packet, Henry County.

[PDF]Math Talk
Explain that this game is about forming and reforming groups as quickly as .... Lessons for ExtendingPlace Value, Grade 3 (Math Solutions Publications, 2005).

[PDF]conference - Kentucky Center for Mathematics
classroom activities will further develop the classroom teacher's approach to teaching place value as well as implementing math talks throughout daily math lessons. ... The soft skills games will include hands-on cooperative learning,.

Wednesday, July 11, 2018

Number Talks: Building Number-Sense and Mathematical Reasoning

Number Talks are a great anticipatory set for building number-sense, introducing new heuristic math strategies, and developing deeper mathematical reasoning. The development of mathematical heuristics, or mathematical rules of thumb, greatly improves a student's ability to solve more complex math equations and multistep word problems. Daily Math Talks can be used to front-load or review math concepts that reinforce or stretch a student's math problem-solving abilities. With teacher-students and student-student collaboration and guidance, the Zone of Proximal Development for learning is greatly expanded. Math Talks are a Socratic math dialectic that is designed to build critical thinking in mathematics and a safe harbor for students to try and test new mathematical strategies and problem-solving. 

Gersten and Chard say number sense "refers to a child's fluidity and flexibility with numbers, the sense of what numbers mean and an ability to perform mental mathematics and to look at the world and make comparisons."
13 Heuristic Math Strategies: Singapore Math Method


[PDF]Number Talks
Developing Number Sense and Math. Vocabulary through. Number Talks. Jane Osborne jane.osborne@hoodriver.k12.or.us.

[PDF]Making Number Talks Matter—Resources by Chapter
 Number Talks Build Numerical Reasoning ..... talks on first grade students' number sense.

[PDF]What is a Number Talk? - Rhodes University
Number Talks Build Numerical Reasoning” (Sherry Parrish, 2011). How can educators make shifts in their instructional practices that foster sense making ...

[PDF]Silent Signals - Wisconsin Mathematics Council
Number Talks: Helping Children Build Mental Math and Computation Strategies. ... http://www.svmimac.org/images/Cristo_Rey_-_Middle_Level_Bank.pdf .... Number sense is the foundation for all higher-level mathematics (Feikes ...

[PDF]Number Talks - Liberty County School System
Building Mental Math and Computation Strategies while ... How can I use Number Talks to build student mathematical fluency and number sense while.

[PDF]What is a Number Talk.pdf
ability to reason about quantitative information, possess number sense, ... ing efficient, flexible, and accurate computation strategies that build upon the key ...

[PDF]What Are Number Talks?

[PDF]Middle Level Routines
Sense of the relative size and value of very large and very small numbers .... Routines are an excellent opportunity to build on-going rich practice into the ... Number Strings. Number. Talks. Concepts of. Equality. Thinking. Relationally. Math.

[PDF]Improving Number Sense Using Number Talks - Digital Collections ...
classroom on number sense/critical thinking in mathematics. ... the teaching of math through number talks, in addition to their regular math instruction. ..... their students with ample opportunities to engage in number sense building activities.

[PDF]Participant Guide for Number Talks - CESA #4
Building Number Sense First. Understanding the Progression of Skills. Number Talksand Mental Math. Sharing Best Practices from ...

[PDF]KINDERGARTEN Number and Number Sense - Virginia Department ...
Number and Number Sense Module – Kindergarten ... and use linking cubes to build a tower to represent their numbers. After the towers are built, the ..... Come back together as a class and talkabout how students might know that they found.

[PDF]Developing Number Sense and Conceptual Understanding through ...
Developing Number Sense and Conceptual Understanding through Number Talks and Manipulatives is a new endeavor of the Columbus Regional ...

[PDF]Number Talks Grades 3-5 Resources - BPS.solutions
with two or multi-digit numbers. In addition to developing efficient computation strategies, Number Talks encourages students to make sense of mathematics, ...

Wednesday, July 4, 2018

Dynamic Assessment Continuous Formative Assessment

Dynamic assessment is an interactive formative assessment used in education. Dynamic assessment is a product of the research conducted by developmental psychologist Lev Vygotsky. As an alternative assessment modality, it seeks to identify the skills that an individual child possesses as well as their full learning potential (Zone of Proximal Development). The dynamic assessment procedure emphasizes the hands-on learning curiosity-driven process and accounts for the amount and nature of investment between teacher-students and student-teachers. It is ongoing, formative, highly interactive and hands-on process-oriented. It has become popular among educators, psychologists and speech and language pathologists. 


(PDF) Dynamic Assessment: A History of Fundamental Ideas
The development of dynamic assessment rests on the prior and successive appearance of a small number of fundamental ideas that are ...

(PDF) Dynamic Assessment in Practice: Clinical and Educational ...
Dynamic assessment embeds interaction within the framework of a test-intervene-retest approach to psychoeducational assessment. This book offers an ...

(PDF) Dynamic Assessment Training Manual - ResearchGate
 This manual provides information to support a variety of formats of training in Dynamic Assessment for professionals working (or being ...

[PDF]dynamic assessment - Westminster Public Schools
Dynamic assessment is based on the work of Vygotsky and looks at cognitive development within the context of social interactions with others who are more ...

[PDF]Dynamic assessment in the language classroom - Semantic Scholar
The focus of this paper is on the implementation of Dynamic Assessment. (henceforth, DA) in the L2 classroom setting. DA is an approach to assessment and ...

[PDF]Dynamic Assessment - International Journal of Pediatrics
Dynamic Assessment (DA) and Evaluation of Problem-solving ... The term dynamic assessment(DA) refers to an assessment, by an active teaching process, of a ...

[PDF]Dynamic Assessment of L2 Development - CALPER, Penn State ...
The present paper outlines a theoretical framework for a research program on Dynamic. Assessment(henceforth, DA) within in the fields of L2 research, ...

Tuesday, June 26, 2018

How to Disarm Hostile Parents?

How do you Disarm Angry Combative Parents?

  • IT STARTS WITH ASKING CALIBRATED QUESTIONS THAT EXPOSE THE PERCEIVED PROBLEMS, THEY START WITH HOW OR WHAT QUESTIONS.
  • START WITH AN "AUDIT OF PROBLEMS AND ACCUSATIONS", STAY DETACHED FROM THE ACCUSATIONS.
  • NEVER TRY USING LOGIC WHEN EMOTIONS ARE HOT!
  • ACKNOWLEDGE THE PARENTS EMOTIONS AND USE "EMPATHY" TO BUILD A RAPPORT WITH HOW & WHAT QUESTIONS. HOW CAN I HELP YOU & YOUR CHILD?E BE SUCCESSFUL IN SCHOOL? WHAT ARE WE GOING TO CHANGE MOVING FORWARD TO INSURE SUCCESS?      
Some schools are becoming more and more toxic as principals and school administrations opt-out of managing, communicating, and dealing with "problematic students and parents." Teachers are leaving the profession after years of little or no support, working in unhealthy hostile work environments. What is the solution when parents are angry, toxic, and or abusive? Some parents have every right to be upset in our over-tested, confusing, and ever-changing educational goals. 

“He who has learned to disagree without being disagreeable has discovered the most valuable secret of negotiation.”... “Negotiate in their world. Persuasion is not about how bright or smooth or forceful you are. It’s about the other party convincing themselves that the solution you want is their own idea. So don’t beat them with logic or brute force. Ask them questions that open paths to your goals. It’s not about you.” - Chris Voss Retired FBI hostage Negotiator 
First, don't panic and keep calm. Stay focused on the positive and set up a working relationship at the very beginning of the year. What are the issues being discussed throughout the year and what is the agreed agenda when parents have questions or a grievance? Everyone knows good communication and feedback is needed, listening is key for both parties. Parents may have zero interest in listening to anything you say if they are angry. Asking open-ended questions can get the conversation started and help cool things down. The FBI has a 95 percent success rate in terms of resolving a hostage crisis without fatalities using open-ended questions. But what happens when parents go negative and start attacking your ideas, teaching methods, curriculum, classroom management, and or grading, your first instinct is to go on the defensive. If you cannot get the conversation back on track quickly with redirecting questions and the Golden rule, you need to stop the conversation politely and set up a new time. If your principal is "BUSY" find a colleague to attend the new meeting and have a prepared agenda.  Teachers should never continue a conversation when there is verbal abuse or personal attacks. Do not take verbal abuse, remove yourself from the raw emotions and seek a colleague to talk with and decompress. Teachers don't always act like their time is valuable but it is very valuable. 

  • Start with open-ended questions, keep questions slanted to the positive and directed to solutions
  • Listen to understand their perspectives, even more so when you disagree with their point of view
  • Plan out an agenda whenever possible with clear guidelines
  • Document and record critical ideas, focus on decisions being made not on the decision makers! 

Improve Your Meetings With Parents Using an Effective Agenda!
Questions to ask your parents before a meeting: 
This only works with reasonable parents! 
  1. What are your questions, concerns, ...? Create goals and objectives for your agenda based on those concerns.
  2. What are your priorities...? Break down the key priorities into key points and who is making decisions based on those priorities. 
  3. What has worked in the past...? Use the agenda to explore solutions and strategies.
Disarm your parents with The Quick 4!
  1. Interesting, please tell me more..., how can I help...
  2. Interesting, please explain why you said that...
  3. Interesting, please help me to understand..., explain your concerns further...
  4. Interesting, I agree we need to fix this, what are your ideas...
Respond to your parents and administrators the "Mr. Spock Way!" Socratic inquiry, the art of disarming and redirecting negative and unproductive conversations. Using open-ended questions to create a positive dialogue. 

Fascinating, what is your desired outcome for this conversation.... 
1. That is fascinating/interesting, please explain your thinking further...
2. That is fascinating/interesting, please explain why you would say that to me...
3. That is fascinating/interesting, please explain why you do it that way...
4. That is fascinating/interesting, please explain you "ideas, strategies, goals, experience, praxis" further...
5. Fascinating/interesting, why would you say that...
6. That is fascinating/interesting, why do you believe that please explain you reasoning...
7. That is fascinating/interesting, please help me see your point of view...
8. Fascinating, what do you want me to change...and why do you believe it is...
9. Interesting, what do you think is getting in the way of our success...
10. Interesting, please suggest a strategy that you think will work... and why....
11. Fascinating, how do we move forward and create success out of...
12. Interesting, I want to know more about your solutions, strategies, theories, plans....
13. Interesting, please give me some insights, goals, objectives, recommendations, that will work for...all of us/both of us.

Monday, June 25, 2018

Praxis Process in Education?

One of the oldest Pedagogical Theories in use today is the Praxis Process! The Praxis Process is a Socratic or revisionist-reflective process by which a theory, idea, hypothesis, lesson, are turned into an actionable skill by developing heuristics strategies (logical rules of thumb). Going from theoria (thinking), poiesis (making), and praxis (doing) or eupraxia ("doing good works," "creating good fortune"). Adopting a new pedagogical theory is easy and "common", yet analyzing it, reflecting on it, revising it, and discarding it when needed, that is the part most schools and teachers fail at. 
"There is no such thing as a neutral education process. Education either functions as an instrument which is used to facilitate the integration of generations into the logic of the present system and bring about conformity to it, or it becomes the 'practice of freedom', the means by which men and women deal critically with reality and discover how to participate in the transformation of their world."
— Richard Shaull, drawing on Paulo Freire
Superlative Eupraxia starts with a transformational theory, (being oppressed, enslaved, and ignorant, to becoming free and enlightened requires and erudite education). Eupraxia is enacted, embodied, and or realized in positive transformational action that must be reflected on and revised. "Eupraxia" often refers to the actions, practices, exercises, and applications a teacher and student collaborate on that are transformational. John Dewey believed that students learn best through a 'hands-on' approach. We as educators sometimes skip or underestimate the time needed in the poiesis or the making of knowledge. We also are not skeptical enough when new pedagogical theories are thrust upon us by people like Bill Gates. The Common Core Mess that never went through the Praxis Proces. Today we are plagued with dyspraxia ("bad praxis, pedagogical misfortune")

Praxis (from Ancient Greek: πρᾶξις, translit. praxis) is the process by which a theory, lesson, or skill is enacted, embodied, or realized. "Praxis" may also refer to the act of engaging, applying, exercising, realizing, or practicing ideas. This has been a recurrent topic in the field of philosophy, discussed in the writings of Plato Aristotle.
Praxis may be described as a form of critical thinking and comprises the combination of reflection and action. Praxis can be viewed as a progression of cognitive and physical actions:
  • Taking the action
  • Considering the impacts of the action
  • Analysing the results of the action by reflecting upon it
  • Altering and revising conceptions and planning following reflection
  • Implementing these plans in further actions
This creates a cycle which can be viewed in terms of educational settings, learners and educational facilitators.

Thursday, June 21, 2018

Setting Daily Micro Goals with Your Students

Setting daily Micro Goals with your students jump starts their intrinsic and extrinsic motivation and self-confidence! Setting daily Micro Goals with students builds Self-Esteem, Perseverance, Grit, Resilience, and a renewed love of learning in students that may have lost interest in school. Setting daily Micro Goals shows students their accomplishments and achievements, students will take real ownership of their learning. Students see that they are truly capable and start setting bigger and more demanding Macro Goals. Building on real accomplishments and measurable results give students a stratagem and path to further academic success. 
The best way to create sustained academic success, is to experience real academic success. Students that develop the habit of setting and meeting daily Micro Goals thrive in the positive feedback loop that is created.  That feedback loop positively impacts a students desire, commitment, perseverance, resilience, and mindsets about school and academics. 
Micro Goals are about doing things in the NOW! 
Academic decisions, goals, and actions or inaction's are based on academic attitudes and positive or negative academic habits. Micro goal setting circumvents unproductive intuitive responses, bias assessments of one's own ability, poor self-concept, flawed heuristics, and stinky thinking. Setting Micro Goals with your students is a Praxis (process)! 


Why set Micro Goals with students?
  • Macro goals can only be met with micro goals
  • Micro goals when met show a measure of achievement 
  • Meeting micro goals "small success" leads to big wins and bigger goals 
  • Micro goals build a bridge between commitment and follow through 
  • Micro goals help keep you focused 
  • Micro goals that are met give you a sense of accomplishments
Micro-goals usually take 1-20 minutes to complete. Everyone can focus on a task for one-twenty minutes.

One-Five Minute Micro Goals Samples:  

  • I will praise my peers using a character trait I admire for one-five minutes
  • I will exercise, dance, or do calisthenics for one-five minutes
  • I will practice a difficult spelling word for one-five minutes
  • I will study a close reading passage for one-five minutes
  • I will deeply study a new math concept for one-five minutes
  • I will organize my backpack, desk, cubby, or desk 
  • I will share 5 things I am passionate about with my family, peers, teachers, or friends 

ü Goals, Tasks, Actions, and Missions  
Yes, I Can! Yes, I Will:
Yes, I Did!
Yes, I Will Do it Again
ü  Show Gratitude, Empathy, & Esteem to ALL People


ü  Seek & Do [… ] Benevolent & Virtuous Activities


ü  Ask Questions & Seek Clarity on NEW Concepts I’ve Learned: My Daily Queries Goal is [… ] Requests


ü  Teach ONE New ELA/Math Concept to a Friend or Family Member TODAY!


ü  Study, Review, & Reflect on [… ] new Math Concepts and [… ] new ELA Concepts


ü  Study [… ] Phrases in Spanish, Latin, French or 漢字


ü  Read [… ] Chapters of a GREAT BOOK


ü  Write, Blog, or Journal about my Daily Learning (200-1000 words daily)


ü  Make a New Friend or Renew an Old Friendship: Be Virtuous, Be Benevolent, Be Kind, Be Supportive   


ü  Find [… ] Things to Laugh & Smile About



[PDF]Workbook for Goal-setting and Evidence-based Strategies for Success 
Best Possible Future Self Sets the Stage for Goal-setting ......................................... ..... Not only has she proven that fleeting micromoments of happiness ..... that new meditators who persisted with a short meditation practice every day experienced ...... The following examples are from photos I've taken from cars on the highway.

[PDF]A powerful yet simple daily planner to help you ... - Schreiben wirktWe'll share additional content including videos and PDF guides on how ... between goal setting and goalachieving by giving you a system of execution that will ...

[PDF]Setting Goals for Yourself, and Motivating Yourself to Succeed (PDF)Setting Goals for Yourself, and Motivating Yourself to Succeed. Page 2 ..... cards and placed where you can see them daily and be reminded of positive aspects.

[PDF]Your 100 Personal Goals Worksheet - HubSpotTypically, the process for setting goals is part of the ... micromanaging. You are a professional .... and enthusiasm if you create an “every day” environment which.

[PDF]A powerful yet simple daily planner to help you ... - Schreiben wirktWe'll share additional content including videos and PDF guides on how ... between goal setting and goalachieving by giving you a system of execution that will ...

[PDF]Setting Goals for Yourself, and Motivating Yourself to Succeed (PDF)Setting Goals for Yourself, and Motivating Yourself to Succeed. Page 2 ..... cards and placed where you can see them daily and be reminded of positive aspects.


There was a study done at Harvard between 1979 and 1989. Graduates of the MBA program were asked “Have you set clear written goals for your future and made plans to accomplish them?”
The results of that question were:
• Only 3% had written goals and plans
• 13% had goals but not in writing
• 84% had no specific goals at all Ten years later
Harvard interviewed the members of that class again and found:
1. The 13% who had goals but not in writing were earning on average twice as much as the 84% of those who had no goals at all
2. The 3% who had clear, written goals were earning on average 10 times as much as the other 97% of graduates all together. The only difference between the groups is the clarity of the goals they had for themselves.  http://www.peer.ca/Singles/MM255.pdf

Monday, June 11, 2018

13 Heuristic Math Strategies: Singapore Math Method

Heuristic mathematics models, process, and methods, heuristic math techniques and strategies, or simply heuristics, are cognitive problem-solving strategies that become intuitive with repeated exposure. The problem-solving strategies "intuitive shortcuts" are formed in the students' mind after repeated rehearsal, practice and exposure to multi-step or complex math problems. Heuristics develop into the rules of thumb, today they incorporate 13 problem-solving models designed to help students make sense of difficult math problems. The 13 heuristics are grouped into 4 categories: “representation heuristics”, “simplification heuristics”, “pathway heuristics”, and “generic heuristics”. Together they form a model of problem-solving strategies in mathematics.

21st Century Competencies & Singapore Mathematics Curriculum





The 13 heuristic math problem solving strategies:

1. Act it out 2. Use a diagram/model 3. Use guess-and-check 4. Make a systematic list 5. Look for patterns 6. Work backwards 7. Use before-after concept 8. Make suppositions 9. Restate the problem in another way 10. Simplify the problem 11. Solve part of the problem 12. Think of a related problem 13. Use equations

External links for lessons and further research:

[PDF]21st Century Competencies & Singapore Mathematics Curriculum
Curriculum Framework established by the Singapore Ministry of .... cognitive work and routine manual labor – the types of tasks that are ... Inventing new problem-solving heuristics when standard protocols have failed.


[PDF]Singapore Math Research and Efficacy - Michigan's Integrated ...
Singapore math emphasizes problem solving and positive attitudes toward ... Ministry of Education uses a pentagon with problem solving in the ...


[PDF]Top-Down Approach to Teaching Problem Solving Heuristics in ...
Singapore mathematics syllabuses identified eleven heuristics which are applicable to problem solving at ... (Heuristics 12 and 13 are not in the primary syllabus.) Though ... methods and strategies that can be helpful in problem solving. In sum ..... 2005, from

[PDF]A Metacognitive Approach to Support Heuristic Solution of ...
can be used in mathematical problem solving, they can be classified into four categories: ... Ministry of Education Singapore (MOE), have identified 13 heuristics that are ... problems, For Bruner (1960) they are methods and strategies that can be helpful in ..... from

[PDF]Heuristic Strategies and Deductive Reasoning in Problem Solving
Use of Heuristics in Mathematical Problem Solving. 4. 2.1 Introduction to ... 4.2.1 Deduction as a Formal Syntactic Process based on Rules 13.

[PDF]problem solving in school mathematics based on heuristic strategies
forms the basis for successful mathematics education; solving of carefully selected problems ... It deals with solving of problems using the following heuristic strategies: Analogy, .... 28 pupils aged 13) and two upper secondary grammar schools

[PDF]Problem-Solving Through Problems - ASU
important problem-solving techniques typically encountered in undergradu- ... The book is both an anthology of problems and a manual of instruction. .... will be concerned with the heuristics of solving mathematical problems. ... Page 13 ...

[PDF]LEARNING PROBLEM SOLVING HEURISTICS ... - Semantic Scholar
Much human learning of mathematics takes place from worked examples, yet this is a ... the heuristicswere learned through attempts at problem solving, and not through ..... There has been a reluctance to concentrate purely on goal-based techniques that ... Proceedings of the 13th International Joint Conference on Artificial ...

[PDF]Learning Problem-Solving Strategies by Using Math Games 1.2312016
Learning some general strategies for problem solving is a unifying theme in this book ..... 13. Computer Programming as a Game. Chapter 9 of this book briefly ...  A heuristic is a step-by-step set of instructions designed to solve a specific type of problem.

[PDF]Problem Solving In Singapore Math - Houghton Mifflin Harcourt
 - strategies for problem solving, what Polya called heuristics. Many of ....

[PDF]Sixth graders and non-routine problems: Which strategies are decisive ...
 of non-routine problems and strategies in mathematics textbooks and syllabi ... are 12-13years old as it is in most countries. Sixth grade level ....

[PDF]Mathematical problem solving in primary school - Utrecht University ...
Key words: mathematical problem solving / early algebra / textbooks / ICT / online game / .... including domain-specific knowledge, heuristic strategies, metacognitive skills and .... 13. More specifically, the non-routine problems in this study contain interrelated .... from http://www.oecd.org/dataoecd/25/12/34009000.pdf.

[PDF]Heuristic Method for Decision-Making in Common Scheduling ... - MDPI
 elements: preliminary analysis of the problem, techniques for the choice of ... scheduling [11], stochastic scheduling problems [12,13], ... There are many different approaches used to solve scheduling problems, mathematical.

[PDF]Proceedings - Le-Math
13 have all undergone a rigorous reviewing process by members of the ...
Summarizes problem solving strategies, with the use of algorithms ... for solving problems outside the domains of mathematics and science.

[PDF]an investigation into the use of problem-solving heuristics to improve ...
Heuristics, problem-solving processes and mathematical modelling. 12. 2.1.1Heuristics ..... These four phases have 13 grades in total and in theory, the ages of .... algebra. It focuses on non-lecture instructional strategies where the researcher- ...... http://www.math.kent.edu/~edd/HeidPaper.pdf. Arnon, I.

[PDF]Problem Solving in Mathematics Education - DiVA portal
from the 13th ProMath conference, September 2011 (pp. ... conduct of an action strategy, the design of meaningful mathematical tasks and the situatedness.
Missing: heuristic

[PDF]The Effects of Problem Posing on Student Mathematical Learning - Eric
- Keywords: mathematical problem posing, teaching strategies, meta-analysis. 1. ... 13; 2014. 228 teachers incorporated a problem-posing intervention, classrooms .... Also, a manual search was .... heuristic (pose a related.

[PDF]Pentagon Curriculum Framework - Mathematics and Mathematics ...
Maths & Maths Education ... 1986, 4+ years, Institute of Education (Singapore) ... factors to help students become better problem solvers. • This session on a new ...

[PDF]Philosophy and Pedagogical Approach of Singapore Math
a problem-solving approach for the teaching and learning of mathematics. Thoughtful application of ... The fact that problem solving is the central idea in Singapore math can be seen in the pentagon from Singapore's. Mathematics Framework ...

[PDF]The Singaporean Mathematics Curriculum: Connections ... - CiteSeerX
Singaporean mathematics curriculum document is probably no different from .... of the model isproblem solving and the five sides forming the pentagon are: ...

[PDF]Math in Focus: Singapore Math by Marshall Cavendish, for Grades K-8 ...
to U.S. classrooms. Math in Focus is based on the curriculum and pedagogy in Singapore. Singapore math emphasizes problem solving and positive attitudes toward mathematics, while ... the pentagon; which is their curriculum framework.