Summer, 2017

- Ph.D 1994 in Mathematics - Ohio State
- Professor at UNC-Asheville since 1997
- Third year working with FCR-STEM

- Fractals and chaos
- Computational tools to
- Aide the research process
- Illustrate mathematics at all levels

- Teaching

- To teach you some math!

- As a participant in a professional development program, you're here to improve your teaching of mathematics.
- I'm told that many of the CGI techniques you're learning have been in practice for some time.
- Research shows they work.

- Research on CGI is ongoing.
- One new question: how does teacher content knowledge impact student learning?
- Much of the math we see will be applicable in your classroom fairly directly.
- Direct applicability is not necessarily the main objective, though.

I happen to be the happy father of a proud 8 year old.

- Practical arithmetic
- with Algebra!

$4 \times 23=?$

\begin{align}
4 \times 23 &= 4 \times (20+3) \\
&= (4\times20)+(4\times3) \\
&= 80+12=92
\end{align}

Looks like we used the

$$a\times(b+c) = a\times b + a\times c$$

$$3\times(2+4) = (3\times2)+(3\times4)$$

$$4 \times (20+3) = (4\times20)+(4\times3)$$

We also used

If $a=b$, then $P(a)=P(b)$

$23=20+3$, so $4 \times 23 = 4 \times (20+3)$.

$3\times(2+4)$ $=$ $3\times2+3\times4$

Distribute

How about:

\begin{align} 4 \times 23 &= 4 \times (25-2) \\ &= (4\times25)-(4\times2) \\ &= 100-8=92 \end{align}$4\times(2x+3)=?$

Note how similar this is to the last question.

In fact, if $x=10$, this is the last question.

- Substitution
- Associativity
- Commutativity
- Distributivity
- Arithmetic using these

- How the rules arise from direct modeling
- Using the rules to do arithmetic
- Standard algorithms

- Uses of the equals sign
- Algebra

- Geometric/symbolic feedback
- Place value
- Something fun