Teaching mathematics is a continuous cycle of identifying where each student is in their learning trajectory and determining meaningful ways in which to build on their current understandings. While we often have little control over students’ mathematical experiences before they walk into our classrooms, we do have complete control of our own learning.

In one of our professional development workshops, there is an activity in which the facilitator asks teachers to skip count by $\frac34$. The facilitator records the count, $\frac34$, $\frac64$, $\frac94$, . . . and then asks for patterns they notice in the recording. In a recent workshop, a group of grade 5 teachers noticed that the numerator increased by 3 each time but that the denominator remained unchanged. When the facilitator asked why, they could easily explain why the number of pieces was increasing in the numerator, but couldn’t really give an explanation for the denominator other than “it is just always out of 4.” The funny thing is, they weren’t saying “3 out of 4, 6 out of 4, 9 out of 4” when they skip counted. They were saying “3 fourths, 6 fourths, 9 fourths” and writing it in fraction notation. The key to understanding why the denominator stayed the same was hidden in plain sight, in the very language they were using to name the fractions.Continue reading “Say What You Mean and Mean What You Say”→

“I’m afraid I can’t explain myself, sir.
Because I am not myself, you see?” Alice in Wonderland.

The idea of equivalence in mathematics is tricky for learners, because when we talk about two things being equivalent, for example the fractions $\frac35$ and $\frac6{10}$, we are emphasizing two contradictory things: Continue reading “Fractions: Units and Equivalence”→

There are some standards I think we do such a great job developing in early elementary, but never revisit explicitly when students learn about different numbers such as fractions and decimals. I blogged about this in reference to even and odd numbers last year, but this past week I have found another: Continue reading “5th Grade: Decimal Place Value”→

When I first started the Math Methods course at University of California, Irvine, all of my ideas on how to learn math took a complete 180.

During the first two months, a million questions swirled in my head as I worked through problems with my classmates: We don’t just teach the algorithm anymore? What do you mean “use representations to build conceptual understanding”? What is an area diagram? What are all of the multiple strategies to solve a problem? How am I supposed to anticipate misconceptions when I have never taught the curriculum?, just to name a few. Continue reading “The IM 6–8 Math Curriculum Changed My Math Methods Experience”→

In everyday language, $\frac{a}{b}$, $a\div b$, and $a : b$ are all different manifestations of a single fused notion. Here, for example are the mathematical definitions of fraction, quotient, and ratio from Merriam-Webster online:Continue reading “Untangling fractions, ratios, and quotients”→

We know instructional materials play a key role in student learning experiences but how do we ensure our students are learning from coherent high-quality instructional materials that engage them in critical thinking and provide opportunities to “do math?”

Let’s think about this from the lens of a 4th grade standard, 4.OA.A3: Solve multi-step word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing in for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.Continue reading “Instructional Materials Matter: Interpreting Remainders in Division”→

You know it is time for a change when half of the students in class are lost by the third lesson of a new unit.

I teach third grade in a charter school in Philadelphia. We use Go Math! and each year I have followed Chapter 8: Understand Fractions, exactly as written. In the first lesson, students name equal parts in pictures such as halves, thirds, and fourths, and then move into finding equal shares. By the third day, when we discuss unit fractions, I feel like I have already lost about half of my students. Despite this, I usually trudge along and move into the fourth lesson where students are asked to identify the shaded fraction of different shapes. By the end of the lesson, my students typically have learned the rote skill of counting the number of shaded and total pieces in order to write the fraction. This becomes incredibly evident when we move to putting fractions on a number line and problematic when problem solving with fractions.Continue reading “Adapting Curriculum For Students to Know, Use and Enjoy Fractions”→

As a teacher, curiosity around students’ mathematical thinking was the driving force behind the teaching and learning in my classroom. To better understand what they were thinking, I needed to not only have great, accessible problems but also create opportunities for students to openly share their ideas with others. It only makes sense that when I learned about routines that encouraged students to share the many ways they were thinking about math such as Number Talks, Notice and Wonder, and Which One Doesn’t Belong?, I was quick to go back to the classroom and try them with my students. It didn’t matter which unit we were in or lesson I had planned for that day, I plopped them in whenever and wherever I could because I was so curious to hear what students would say. Continue reading “Warm-up Routines With a Purpose”→