Math personified: a teaching story


My friend Mark R., his identical twin Expo, and I are sitting on the chalk rail. It’s not really a chalk rail but I think that’s what you call it even though we’re mounted to a white board. The three of us love exercise and we get a great workout in math support but sometimes we get abused. Sam will put me in choke hold and press my face against the white board. Ellie will forget to put the cap back on Mark R. so he’s all stuffed up. And poor Expo has lost his head completely. We now call him the Headless Horseman.

By this account you would think the students don’t like us. They actually do. Students can’t wait to get their hands on us because for the better part of the day they learn using an iPad.

Story I

This past week our caretaker, Mrs. Dooms, used us for a form of visual storytelling. Ellie, a seventh grader, struggles with mixed numbers to improper fractions. Her class is working on rational numbers, specifically adding and subtracting negative fractions. Mrs. D. picked me up to visually explain.

“Remember last week when I served you guys pizza and cut it up?”

“Yeah! That was good. Where was it from?” Ellie asked.

“It was a frozen pizza and I heated it up during my plan period. Let’s get back to fractions.”

Mrs. D. uncapped me and began to draw circles to represent 2 1/4.

Mrs. D. didn’t photograph our Expo marker work so we gave her permission to use the SAMR model of substitution using the iPad.

“This is where the short-cut comes from,” she said. “Two wholes cut into fourths plus the remaining one fourth equals 9/4. Ellie needs more examples. Mrs. D. writes two mixed numbers for Ellie to model. She hands me over to Ellie for more practice.

Story II

In sixth grade support, Sam picks me up with his usual choke hold. We’re multiplying decimals, 4.31 x. 2.2 and I brace for the decimal point. His math is fluent; the ink flows freely onto the whiteboard. Sam generally follows the algorithm but I hear Mrs. D. ask, “How do you know your answer is reasonable?”

“Because there are a total of 3 place values to the right of the decimal,” he said.

“But how do you know your answer is reasonable?” Mrs. D. asks.

“There are a total of 3 place values to the right of the decimal so I move the decimal three times.”

“If we were to estimate, to what whole numbers would we round 4.31 and 2.2?”

Sam’s eyebrows furrow, his hand grips me tightly. “We round 4.31 to 4.3 and we’d leave 2.2 alone?”

Mrs. D. picked up Expo from the chalk rail, saw his head was missing so she grabbed Mark R. instead. She lightly wrote next to Sam’s work ‘Estimate: 4 x 2’.

Sam reveals to Mrs. D. that he relies on memorizing a procedure and needs more support with the meaning of decimals and place value. My buddies and I are gearing up for next week’s lessons.


If Mrs. D. was writing this she would want you to know that most students enjoy math support. In fact some seventh graders drop in on their non-scheduled days.

As far as I’m concerned, students appreciate learning with us Expo Markers, even though I wish they would treat us with a bit more affection.


Favorite lessons create an intellectual need

As the math interventionist in our building I have the luxury of co-teaching with an incredible group of 6-8 math educators. When the 6th grade math teachers meet face-to-face much of our PLC time is spent examining student work and reflecting on what went well in today’s lesson and what can be improve upon.

Today we discussed with our counterparts the results of our attempt to give students a “headache” and create an intellectual need for vocabulary when learning about exponents. Our original plan was to go straight to the notes as a mini-lesson then introduce the vocab but, as my colleague and I were discussing over email, we decided to introduce exponents within the context of a real world situation.

Here’s what today’s lesson looks like from our lesson plans in Google Drive.


Here’s a summary of our discussion in our PLC tracking document.


Our improved instructional design of the lesson wasn’t perfect, but it was a much better learning experience than simply launching with notes followed by practice.

Here’s how we framed the learning. We began by posing to the class the problem in the image below and constructed meaning using a team teaching think-aloud. I read the problem, my co-teacher Cathy modeled confusion and read the problem again, and discovered a starting point. I interjected that there are many ways to represent problems and suggested we use a tree diagram to get us started.


As we created the tree diagram we eventually stopped to emphasize that the pattern continued. At that point we asked the class (and fell out of the think-aloud mode!) how to represent the continuing pattern for Thursday and Friday. Then we finished creating the tree diagram.

problem2 My co-teacher Kathy mentioned to the class that her eyes were beginning to look at the rows of “F’s and we counted the numbers in each row.


We then looked at the diagram, noted the values and introduced the vocabulary term “Standard Form” by labeling that column.


Next  we noticed two was a factor in 2, 4, 8, and 16. We introduced the vocabulary term “Expanded Form” and began with 2 x 1. Students quickly shared 2 x 2 = 4, but to write the expanded form of 8 most students wanted us to write 2 x 4 instead of 2 x 2 x 2. We had a discussion on only focusing on using 2.

After expanding each of the numbers, we worked from the bottom to complete the Exponential Form column noting 2 x 2 x 2 x 2 is equivalent to 2^4. Students quickly caught on as we worked our way up the column. Comprehending that 2^0 is equivalent to 1 was a bit troublesome for a few students. I don’t have the image but we then attached meaning to the vocabulary words base, power and exponent.

The goal for the original lesson was to acquire new vocabulary. We could have accomplished it using only notes followed by independent practice, but providing a context for learning and creating an intellectual need for the vocabulary made the lesson more meaningful.

I wonder, however,  if too much vocabulary was shared in the lesson. I hope not. Students were only asked to identify the base, power, and exponent in their practice. The next day they examined more closely the terms standard form, expanded form and exponential form.


Classroom Management or Classroom Environment?

As the school year begins I’ve been thinking a lot about classroom environment. Our district uses the Danielson Framework as part of our teacher evaluation and one of the four domains identified is Classroom Environment. As an aside, when The Framework was first published it was designed to be a reflective tool not an evaluative one.  If you are not familiar with The Framework for Teaching, Charlotte Danielson identifies four domains of teacher responsibility. Domain 2, Classroom Environment, includes five components that make up this domain.


For some reason whenever I hear “classroom management” I immediately think of procedures, student behavior, and physical space. Do the students know where the supplies are? Are they efficiently transitioning between learning activities? Do students know the procedure for when they return from an absence?  That’s just the business of learning.

Two components I want to focus on are the first two from Danielson’s list: 1) respect and rapport and 2) a culture for learning. I LOVE how the wording below captures the essence of both while merges the two ideas into one. It even addresses the expectations of both teacher and student as it relates to learning:

respect learning

I came across this language three or four years ago at a summer technology workshop hosted by our district. The words never escaped me. Kate Kieres traveled from Pennsylvania to lead the PD and referred to it several times.

When sharing this with students we obviously discuss examples of respect, but instead of posting a list of do’s or don’ts,  I prefer to capture it in question form as the basis for our classroom norms.

Creating a culture for learning requires students to be in an engaging, collaborative environment that empowers all learners. That’s my responsibility. When students ask…

What am I learning from others?

When students consciously ask themselves, What am I learning from others? attention is immediately drawn to the learning experience. It demands critical thinking and processing. It invites inquiry, debate, and feedback. An example could be a simple turn and talk where students share a problem solving strategy, but the caveat is the intentional self reflection while the conversation is taking place, “What am I learning from you?”

How am I contributing to the learning of others?

Similarly, students should be cognizant of how they contribute to the learning of others. Either in pairs, small groups, or whole class discussion it is my responsibility to create a learning environment where every student voice is valued.  That means every student needs to be into a position where they can participate and not feel they have nothing to offer. A group-worthy task with roles and responsibilities or an open middle problem are ways to give students the opportunity to contribute to others’ learning.

As classroom norms

My support classes are only 30 minutes. Time is precious. By referring to the norms What are you learning from others? and How are you contributing to the learning of others? I can quickly redirect behavior. It makes the statement that this is a learning environment and these are the expectations, on both the students’ part and mine.



Intervention Goals and Goal Setting

I’m so grateful to @jreulbach and her blog  for resurrecting #SundayFunday.   My position as a 6-8 math interventionist (a relatively new position) has prevented me from blogging regularly but I hope the weekly prompts will get me to write more often. My responsibilities continue to evolve, but my main projects are providing grades 6-8 homeroom math support, co-teaching four classes, and overseeing our Title I after school math program.

Our homeroom intervention program is a work in progress. It is NOT a formal class, but the identified students (below 30% percentile MAP and teacher recommendation) are scheduled to attend math support twice weekly. The two previous years, support time was dedicated to re-teaching and supporting the daily classroom instruction. This year I will also be targeting specific skills for intervention.

One goal I have for myself is to spend time identifying quality intervention programs that best suit our students. One resource I’ll be looking at more closely is this document from Hanover Research. I’m also quite fond of Marilyn Burns Math Reasoning Inventory .  I’m not sure if my district will be purchasing a program or if we will be creating our own. Either way the work is cut out for me. If you have a favorite or have struggled with a particular set of resources please add your thoughts in the comments.

Along those lines, one goal I have for my homeroom support students is for them to set their own goals and monitor their progress. I created this google spreadsheet for them to complete.

Below is a screenshot. The student would fill in the blue and green and the graph would self generate.

Goal setting

Students can use the continuous learner characteristics as the basis for the goal setting plan.

If the electronic version becomes too cumbersome I guess I’ll go back to paper.

I’m interested in learning how goal setting has worked in your classroom as well so please chime in!


Using derived facts to compensate for lack of fact fluency

Jacob (not his real name) is a 6th grader who struggles with his multiplication facts. Recently I sat one-on-one with him to capture his thinking as he completed Jo Boaler’s math cards multiplication matching activity. He had one prior experience with the activity when he worked with a small group during homeroom math support, but this was the first time he was completing it with me.

I shuffled the cards and  handed Jacob the deck. I did not mention any strategies to get him started. I wanted to see how he would get himself started. He began by keeping the cards in the deck–faced up. He removed the top card, looked at it, then placed it at bottom of the deck, as one would do if you were working with flash cards.  He went through three cards before stopping at 9×7, where he spent about 30 seconds mentally calculating. “Sixty-three,” he said.


“How did you figure that out?” I asked.

“I knew 5×7 was 35, and I kept adding 7s.” I noted he is using a derived fact, using a fact he knows to find out the answer to one he doesn’t. 

Jacob placed 9×7 on the counter top then immediately drew 7^2 then 8^2. “I don’t remember what those are,” he said placing the cards to the side.

The next card he drew was 9×4. It took him about 25 seconds to complete the mental math computation. “36,” he said, then created a new column for 9×4.

two cards.jpg


“How did you calculate that?” I asked.

“I knew 9×5 was 45 then took away nine.” It’s becoming obvious he is using landmarks or friendly numbers, then adds or subtracts accordingly.

The next card he drew was a 7 by 9 array. He knew to place it in the 9×7 column he just started, but when I asked him the product of 7 and 9 he had forgotten. He counted the squares, but he counted each square in an inward spiral motion, working his way from the outside in.

“Sixty-four,” he said.

three cards

I suggested he count again. He counted each square in spiral form again and said, “Sixty-two.”

I asked, “If there were 10 columns of 7, what would that be?”

“Seventy. (PAUSE) Oh, it’s 63.”


“Because I need to take away a seven.”

Before we moved on, I wanted Jacob to think about a strategy that would help him complete the activity efficiently while at the same time revealing his mathematical thinking.

“What do you think we can do to help us with this activity before we forget what 9×7 is? I’m pretty sure there will be another card that represents 63.”

At this point he spread out the unmatched cards, found 63, then placed it at the top of the column. Then he began looking for another card.

four cards

“What are you looking for?” I asked. He found the 4 by 9 array to place it in the 9×4 column.

five cards.jpg

He looked at the array for about 15 seconds then said, “9 times 4 is 36.”

“How do you know?”

“I made it 10 times 4, which is 40 then I took away 4.”

I didn’t ask him but I assume he was searching for the 4 by 9 array to give him a visual representation to support the same friendly number strategy he previously used.  He then proceeded to look for 36 and placed it at the top of the column.

We spent the remaining 15 minutes completing the activity.

At the end of the period I mentioned to Jacob that I was pleased to learn he was utilizing a strategy to compensate for an overall weakness in his multiplication facts.

To my pleasant surprise Jacob actually has strong number sense. He is decomposing numbers and using derived facts to think about multiplication. He is utilizing that strength to compensate for a lack of fluency with other math facts. In addition, he is using visual models to construct his math arguments and uses them to defend  his derived fact strategy.

I’ll be working with Jacob next year and it is our goal to improve this fact fluency. This observation gave me insight into Jacob’s mathematical thinking. He has more number sense than I previously gave him credit.

Special thanks to Capri Paluch (@CapriPaluchD95) for helping me process this observation.

Making Thinking Visible # 4: Recognizing the need for clear learning intentions

Chapter 4 is a great resource to tap into. The routines for “Introducing and Exploring Ideas” are varied and strategically target making observations, uncovering new ideas by connecting prior knowledge, and considering problems from alternate perspectives. Some of the routines are variations of what I’m already familiar with or am currently do in the classroom, so I was reading to learn what I can add or focus on to improve student thinking.

What it really boils down to is clear learning intentions. If learning intentions solely focus on skills, we’re missing the opportunity to develop students as thinkers.

A few of the routines are described below.

The See-Think-Wonder routine, similar to I Notice…I Wonder, invites students to inquire. To maximize the benefit of the routine students may need to experience it several times in order to break through the surface level to dig deeper. My co-teacher and I used this routine to introduce exponents to our 6th graders. We showed this Best Offer video twice.

lump sum
Most students noticed a stack of money. Few noticed the money was bundled or how the bundles were labeled.
Many students saw the amount of money growing, but were puzzled with what was going on.

Using the video, students were asked whether they would prefer a lump sum payment of $50,000 or be paid exponentially, starting with two dollars. As the students experienced the routine many of their wonderings were surface level. At that point it became clear that the routine needs to be reintroduced so students have opportunities to practice digging deeper.

Another routine I enjoyed reading about was Zoom-In as it will be useful when exploring rational numbers. I’ll likely create a powerpoint to overlay a variety of rational numbers, integers, and whole numbers–attempting to connect prior knowledge with classification. The idea behind Zoom-In is to show a small section of an image and slowly zoom out, or pan from left to right, revealing other aspects of the image.

Yet another routine, the Explanation Game, piqued my interested. The goal is to elicit thinking on the parts of something. It reminded me of David Wees (@davidwees) and his collaborative efforts on the Contemplate then Calculate instructional routine which emphasizes the mathematical practice make use of structure. I’m particularly familiar with this task…

…which asks students to explore structure to find short cuts when calculating. Connecting it to the Explanation Game, students describe the features of the problem, explain their short cut, give reasons for why it works and describe alternatives.

Chapter 4 is a great reminder of existing instructional routines. What is also quite useful is each routine’s description includes its purpose, how to select appropriate content, implement or vary it–all essential when making learning intentions explicit.

Making Thinking Visible #3

A few thoughts and connections while reading chapter 3: Introducing the Thinking Routines. Thinking  routines can be used as: 1) a tool to elicit thinking (i.e. a Think-Puzzle-Explore which is related to a KWL) , 2) a structure to support student thinking (i.e. Generate-Sort-Connect-Elaborate, related to a concept map) or 3) a pattern of behavior where thinking routines are embedded in the classroom culture. When embedded the routines “build an arc of learning” throughout a unit of study.

As I learn about each of these routines in greater depth I’ll be able to explain in greater depth, but already I’ve noticed the tool and structure are closely intertwined. I’m going to take a risk and predict one way these routines would play out when students are learning factors, greatest common factor, and least common multiple.

Present the Think-Puzzle-Explore routine to reveal the level of understanding of  factors. What do you think you know about factors? What’s puzzling about factors? Let’s explore and investigate what puzzles you. I’d likely give each student a document with the prompts to record their thoughts before discussing.

In the middle of the unit prior to GCF and LCM, the Generate-Sort-Connect-Elaborate thinking routine is introduced. Here students would brainstorm identify the differences between factors and multiples. The would be done individually at first then shared and organized  with a partner. A finished product may look something like this:

thinking map

At the end of the unit perhaps the perspective taking thinking routine called Step Inside would help students distinguish between when a problem requires finding the LCM and when it requires the GCF.

I think this scenario is pretty accurate in demonstrating how thinking routines can be used as a tool, a structure, and as a pattern of behavior.

More discussion is on Twitter #eduread.