Adventures in Math

As a visiting teacher, I love to ask students what they think a “number talk” might be. It’s usually silent at first, and then a brave soul will say something like, “Is it when you talk…about numbers?” Their faces light up when that profound idea is confirmed. I have been in countless elementary and middle school classrooms in the last few years both observing and facilitating Number Talks (Sherry Parrish, 2010), and I am part of an amazing team that has trained over 1,000 teachers on their implementation since 2013. Number talks are changing how teachers and students interact with mathematics by opening up the discourse in the classroom to more than just a volley between teacher, student, teacher, student. They invite students to construct and consider multiple strategies for a single problem or string of related problems, and serve as formative assessment for teachers who want to know how their students are thinking about a problem. While these “open strategy sharing” conversations...

I'm knee deep in writing an article about an area lesson I taught to rising 4th graders in summer school a couple weeks ago and, as usual, something totally unexpected and profound has emerged as I take a closer look at student work. I wanted to do a quick post today as a shout-out to the importance of spatial structuring for students who are being asked to think about multiplication in terms of arrays  and  area . In the lesson, students were given two tasks related to area. The first task was to find different rectangles with an area of 12 square units, where students were given 12 square units to build their rectangles and then asked to draw pictorial representations on a square grid. The second task was to determine the area of a table given only enough square units for two side lengths and to draw a representation of their thinking on a blank sheet of paper (for an amazing read on the power of a blank sheet of paper, check out Tracy Zager's blog ). There was a star...

Let's play a game. It's called "What math does this assess, and is that the math the standard calls for?" Wordy, I know, but it's a game worth playing as we consider how to make assessments actually assess what we need them to. The item below comes from a 1st grade benchmark exam, and is tagged as an assessment of standard 1.G.3, with no other items assessing this standard. Before I give you the text of the standard, we're going to play the first part of the game: what does this assess? English translation: Color a quarter of the square. Color half of the square. Color half of the rectangle. Color a quarter of the circle.  Some teachers' answers to "What math does this assess?": Understanding of the phrases "half of" and "quarter of" Ability to color "one part" of a shape Low level assessment of identifying a part of a shape (not necessarily understanding of equal parts) And now ...

The Background It's confession time: I was a compulsive fraction simplifier. Not in lowest terms? Simplify it. Improper fraction? No question. Change it to a mixed number. No rhyme or reason, except that the simplest form or mixed number was the only correct answer most of the time... and an improper fraction just felt wrong. As a student I learned the tricks for converting improper fractions to mixed numbers and vice versa as isolated procedures, and as a 5th and 6th teacher B.C.C. (Before Common Core) I taught it that way. There was very little concept or context... as Phil Daro would say, it was answer-getting, not math. Something like this: If I was lucky, students put all the digits from their division problem into the correct spot in the mixed number. Spoiler alert: I wasn't always that lucky. Why do we convert improper fractions to mixed numbers anyway? (wait time) No, really, why? Turns out, there's no mathematical reason why we MUST convert an im...