Engage in Math: NCTM 2017: Day 1

Group of Algebra I teachers met over the course of two years where they individually identified which math practice(s) were encouraged by different IM tasks, and then talked about where they disagreed
Over time their agreement increased—this process a potentially powerful tool for coming to common understanding of the math practices
Are some math practices more likely to occur together in a single task? Here was the correlations for this group

Structure across the table: each row goes from iterating --> partitioning --> associative property --> distributive property
“Students must be taught structure explicitly”—planning and being aware of it yourself isn’t enough, but annotations can be used to help students look for and make use of structure
Because students are already in the mode of partitioning/iterating, 2 1/3 x 15 is easier for students than 2/3 x 15 (and 16÷3 is easier to evaluate than 2÷3)

The number line is already too abstract for some students. A beaded number line is more concrete and really helps the students see and understand what value mean. Can go from beaded number line (concrete) --> number line (pictorial) --> computations (abstract)
“Structure: When I look at 7 is there a doubles fact hiding in there? What do I notice about all of these numbers?”
“Repeated Reasoning: Does this always work? And why or why not? Are there patterns to the way we say and write our numbers?”

Computation Strategy: purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another vs. Computation Algorithm: a set of predefined steps applicable to a class of problems that gives the correct result in every case when the steps are carried out correctly
Activities to do with kids: counting center (estimate the # of something then put it into ten frames), 10-frame build it (flash a quick image and then have students recreate on 10 frame, use some sort of interesting progression ex: all have a value of 6), shake and spill (5 red/yellow counters in a cup, shake and spill, how many red and how many yellow?), macaroni squeeze (10 noodles in a bag, line in the middle, what are all the possible combinations)
Progressions Document for single digit addition and subtraction

65-36 alternative algorithms: subtracting in parts, counting on by 10s, and compensation
Disrupting people’s conceptions about standard algorithms: when dividing a fraction by another fraction, it does work to divide the numerators to get the numerator of the quotient and divide the denominators to get the denominator of the quotient (ex: 6/21 ÷ 2/3 = 3/7)

ShadowCon

In regards to only a subset of students participating, went from feeling indifferent --> insulted --> inspired (maybe not a linear progression)
How can you access the students who hide?

Students care what their peers think and they compare what they think to their peers

K-12 50/50 math achievement M/F, but post-secondary 75/25 or worse
Maybe this is because we are missing an explicit effort to humanize math, so try:

Pose a pursue questions that catalyze emotional resonance (not just about the past and present, but future)
Empower our students to make and explore conjectures

Play is characterized by diminished consciousness of self, improvisational potential, continuation desire
Goals of play: ownership and identity in math

Growth mindset tells us that effort leads to ability, but it’s hard for students to see that when they are always being compared to a changing standard. Students need an opportunity to zoom out and see how they have grown over time
Students can see this through the following iterative process: assign complex tasks that produce complex work, written reflection, conversation