Open House night; cue anxiety and sweaty palms! Hope my students’ parents don’t mind.

I just began my seventh year of teaching middle school mathematics. Middle school is a limbo land filled with prepubescent pre-teens, drama, and students trying to find their individual voice without drawing too much attention to themselves (sigh). There are sixth grade boys and girls in my class who are taller than me, 5’9”. Some of the boys have mustaches while others still look like they’re in third grade. It’s a difficult year for the students. This is their last year before moving onto the even weirder, and much more confusing junior high. Students are anxious about this being the last year of elementary school, and so are the parents; maybe even more anxious than their little boys and girls becoming young men and women. I think it is my job to help ease this transition, and to get them excited about what is to come.

There is no shortage of available math resources for teachers to use in their classrooms. The difficult and time-consuming job for teachers is weeding through all of the tools to decide which best supports students in learning mathematics. It is a difficult job because it first involves thinking about how students learn mathematics and then, after choosing a resource, ensure it is being used to best support students’ learning. Our team at Illustrative Mathematics works closely with partners to align their resources with the IM 6–8 Math curriculum so teachers can feel confident using them in their classrooms to support student learning. In aligning these resources, we keep the focus of how students learn mathematics at the forefront, while considering the type of support the additional resource is providing.    Continue reading “Planning for Meaningful Practice”

By William McCallum

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 the number of pieces were 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.

To understand this, recall that in the standards fractions are built up out of unit fractions. In grades 1–2 students learn that numbers can be built up out of ones, tens, hundreds, etc. They learn to see 43 as 4 tens and 3 ones. They see ten as a single unit made by bundling together 10 ones, and they learn to unbundle that unit when necessary for addition or subtraction.

In grade 3, students start to see the number one itself as a bundle of smaller units. Their rulers become subdivided into halves or fourths of an inch. On their number line diagrams they see the length from 0 to 1 subdivided into smaller equal lengths, say thirds or fifths. They learn to  see $\frac13$ not as “one out of three” but as a piece of a certain size, three of which make a whole. And then they build other fractions out of pieces; $\frac53$ is literally five thirds, the number you get put putting 5 pieces called $\frac13$ together on the number line.

The work in geometry in grades 1–2 has prepared them for this thinking. To quote from the latest version of the fractions progression:

“. . . first graders might put two congruent isosceles triangles together with the explicit purpose of making a rhombus. In this way, they learn to perceive a composite shape as a unit—a single new shape, e.g., recognizing that two isosceles triangles can be combined to make a rhombus, and simultaneously seeing the rhombus and the two triangles. . . . [they] use fraction language to describe partitions of simple shapes into equal shares—halves, fourths, and quarters in grade 1, extending into thirds in grade 2.”

Where does the “out of” language come from? It is fairly natural language to use when you are describing fractions less than 1. You divide a rectangle into 4 equal parts and shade 3 out of the 4 to represent the fraction $\frac34$. It is also natural when using fractions to describe sets: 3 out of the 4 roses were red, so $\frac34$ of the roses were red. The danger here is that students might not see beyond the formal process: count the red roses, count all the roses, then put the two numbers together. It is not a coincidence that the standards do not use set representations to introduce fractions, but rather use measurement representations (length or area) where the whole is more easily seen as one thing.

So perhaps one way to understand fractions better is to simply listen to ourselves when we name them. When we say “five sixteenths” we are not only naming a particular fraction, we are giving a constructive definition for it: you can figure out how big $\frac{5}{16}$ is by taking a unit called a sixteenth, 16 of which make a whole, and putting 5 of them together. It doesn’t always happen that our names for things describe their mathematical meaning, but this is one case where it does.

Here’s the record of the skip counting from the beginning of this post.

What patterns do you notice? How would you explain why these patterns occur? Do the names for these numbers help reveal the mathematical underpinnings?

### Next Step

Can you think of an example where the words we use for a number get in the way of its mathematical meaning?

When I first started teaching, at the end of each day, I would open my teacher’s guide, grab my pen, and thumb through the stack of completed worksheets. My eyes would dart quickly from the red answers in the teacher’s guide to the corresponding answers on each student’s page. I would dole out my x’s and checks with finality and authority. When I got to the end of a page, I would tally a percentage score and enter it into my electronic grade book. I approached every piece of student work as if it were a summative assessment.

I asked my 15-year-old what she learned today at school. She paused for a moment and then answered,  “What did you learn at school today?”

It took me a while to think about what I had learned (which will make me more patient when I ask her again tomorrow), and then I remembered and shared with her:We are working with some teachers who are using the Illustrative Mathematics 6–8 Math curriculum. The 7th grade teachers are in Unit 1, Scale Drawings. They are working with scale drawings and maps. Today I learned to look more closely at the scale given for a map.

Growing up we usually think we are either a math person or not a math person. But, in preparing for this year I saw a picture that said ‘How to be a math person: Step 1: Do math Step 2: Be a person’ and I really started to look at math differently.

NCTM’s Principles to Actions names several productive beliefs about assessments that will promote mathematical success for all. At the top of the list is that the “primary purpose of assessment is to inform and improve the teaching and learning of mathematics (82). Continue reading “Planning to Use Pre-Unit Assessments”

There are always so many things to do in preparation for a new school year.  At this point of the summer, to-do lists start getting made, materials get purchased, rooms are organized, and math class planning begins. Whether you are using the IM 6–8 Math curriculum for the first time or entering your second or third year with the program, there are always new things to learn. While the Illustrative Mathematics blog is packed with great information from curriculum authors, teachers, and coaches, it can often be a job in and of itself to narrow down what to read. Continue reading “IM Preparing for the School Year”

By Kristin Gray, Jenna Laib, Sarah Caban

Open House. Back-to-School Night. Family Welcome. Math Night. No matter what the name of the event that launches the school year, family members will arrive at your school with the same burning questions: What do I need to know to set up my child up for success in math this year? and How can I continue to support them throughout the school year? Continue reading “Building a Supportive Home/School Partnership”