This year at NCTM in San Antonio I realized that I have reached my 25th anniversary of NCTM membership. For twenty-five years I've attended inspirational talks, read well-informed ideas, and engaged in important conversations. And each year I think to myself... "maybe I could contribute something."

Well, this is it. I've officially submitted my

__first conference proposal__for NCTM 2018 in Washington DC. The process of writing the proposal was grueling but undeniably useful. A couple things I've learned? It is**SO**difficult to:- fully develop an idea now for a potential workshop one year away... so that it feels fresh and current both now and then.
- write 'explicit' and 'specific' descriptions that are engaging and thorough, and
__within the character limits defined by NCTM.__ - appreciate just how important it is to focus on struggling and under-represented learners.

**"From Number Lines to Logarithms: How Forgotten Instincts Can Spark Deeper Understanding"**

Research suggests that logarithmic thinking is innate. We
are born with a number line in our heads, but the numbers are spaced in equal
RATIOS instead of equal distances. We will explore how reigniting this instinct
can pave an enlightened pathway from number properties and fractions, through
ratios and means, to rational exponents and logarithms.

**Write the participant learning outcomes of your presentation, including an explicit description of what participants will learn. Please also provide an overview describing how time will be allocated during this presentation.**

In four parts, this session seeks to answer these guiding
questions:

**Introduction**- Given a number line from 1 to one billion, where is 1 million? Where would your students place it? What research-based implications can we make? How can we turn our confusion into flexibility and sense making around additive and multiplicative mindsets?**Foundations**- How does this flexibility illustrate basic number properties? How does it change our understanding of fractions? How does your vision of later connections impact the way you teach foundations?**Connections**- How do ratios compare to fractions? How does our new flexibility change the way we think about "the middle" and "equal sharing"? How do factors and terms help us recognize what kind of thinking is required?**Breaking Barriers**- Can we use our new flexibility to break down the confusion surrounding exponential models, logarithms, rational exponents, inverse functions, geometric vs arithmetic sequences and means... and more?

**What is the key mathematics content that is a focus of this presentation?**

This presentation is about how exponential models/patterns
have many connections and similarities to linear models/patterns. Noticing and
celebrating these similarities helps make sense of:

- Additive and Multiplicative Number Properties
- Fractions and Ratios
- Linear vs Exponential Models
- Logarithms as Inverses and Related Facts
- Integer and Rational Exponents
- Arithmetic and Geometric Sequences
- Arithmetic and Geometric Means

**How does your presentation align with NCTM’s dedication to equity and access?**

I attribute part of my success in mathematics to an occasional willingness to accept that “it will make sense later.” This kind of persistence through adversity in mathematics is admirable, but it also pushes the limits of student engagement and promotes boredom and withdrawal. My presentation is rooted in the idea that logarithmic thinking is innate. Embracing this instinct makes sense-making and flexibility with multiplicative and additive mindsets accessible to all... now, not later.

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