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.

Making Thinking Visible #2

My reflection and critique of Chapter 2, Putting Thinking at the Center of the Educational Enterprise,  begins with this quote:

What kind of intellectual life are we presenting to our students in our individual classrooms and in our school as a whole? What are my students learning about learning? What messages am I sending through the opportunities I create for my students about what learning is and how learning happens? (p48)

What am I modeling in my classroom? What types of thinking occur in the classroom? As I read chapter 2 I kept relating it to the importance of having clear learning intentions. In my previous post I noted how our math classes do a fine job of identifying the specific skill to be learned, but the thinking objective is not clear–it’s left as an assumption. When it’s left as an assumption what happens in class is the teacher does all the thinking.

To make thinking visible, the first of three strategies, questioning, was introduced. One of the authors provided a scenario in which he modeled the questioning strategy to a group of teachers. Having a strategic list of questions–knowing how you are going to direct the learning–is crucial. Yet when the teachers implemented the strategy, they reported students were not responding with higher levels of thinking. The teachers neglected to listen deeply and ask follow up questions. Listening, strategy #2, was explained in depth using this scenario.

Be it anecdotal records, formative assessments, whiteboard work, photos, videos, etc.–any artifact can serve to document (strategy #3) student thinking and be used to advance learning.

I’m not sure it this will be elaborated upon on in later chapters, but I’m looking to read about balancing surface and deep level thinking–juggling when to take students into the deep end and when to stay in shallow waters. I’m certain that depends on the individual student. The reason for my raising the issue is thinking deeply is serious mental exercise. Students need to develop the mental stamina, so perhaps a  variety of thinking activities is in order.

I do believe students need to see their teachers as authentic thinkers and learners as well. Case in point, towards the end of the school year a colleague and I were wordsmithing a conference proposal. This was taking place in the colleague’s classroom, after school, with a student present. At the end of our lengthy discussion the student remarked it was the first time he had seen adults struggle as much as kids do.

Continue the conversation on Twitter at #eduread.

Also check out key quotes found on this post.


Making Thinking Visible Chapter 1 takeaways

After reading Ritchart, Church and Morrison’s first chapter of Making Thinking Visible, here are some takeaways:

Chapter 1: Unpacking Thinking

“What kinds of thinking do you value and want to promote in your classroom?” What kinds of thinking does that lesson force students to do?” These two thinking questions are at the heart of Making Thinking Visible.

Every time I annotated or highlighted a takeaway from Chapter 1 I critiqued my practice. For example, “…understanding is not a precursor to application, analysis, evaluating, and creating but a result of it.” I noted how this belief can lead to greater access and equity, After all you don’t have to be fluent with your math facts to solve complex problems. I also thought about how our math team sometimes schedules a group work task at the end of the unit—after the students have completed a string of procedural problems. Often times a task at the end limits the student’s opportunity to think.

Another quote from Chapter 1: “In most school settings, educators have focused more on the completion of work and assignments than on a true development of understanding.” I kept thinking about Michael Pershan’s relentless pursuit of analyzing student work and making sense of it. I also thought about this in terms of our PLCs. We can do a better job of collaborating to improve student learning by studying student work.

I also noted that our lesson objectives do a fine job of articulating what students will be able to do from a procedural standpoint, “I will be able to convert between fractions, decimals, and percents” but the thinking piece needs to be clearly defined. The book’s eight high leverage thinking moves, were previewed and I’m looking forward to studying them at length.

  1. Observe closely and describe what’s there.
  2. Build explanations and interpretations.
  3. Reason with evidence.
  4. Make connections.
  5. Consider different viewpoints and perspectives.
  6. Capture the heart and form conclusions.
  7. Wonder and ask questions.
  8. Uncover complexity – go below surface learning.

I’m also interested in discussing Bridget Dunbar’s comment about metacognition.


Pam Wilson’s post does a fine job of capturing the salient points of Chapter 1. Check it out and join the conversation on twitter using the #eduread and #makethinkvis hashtags.

US Congress: is it representative? A math lesson

As I read Dan Meyer’s post, “A Response to Danny Brown,” and the stream of comments that followed, I began to think about middle school and the lost interdisciplinary units, or the singular, math related, social justice lesson that “matters”-a topic that appears to be dear to Danny Brown. He writes,

…I really want to model something *important*, in the sense that it *matters*, perhaps politically, or socially, perhaps having a *local* relevance for the students in the classroom

I hope there is still a place in middle school for cross-curricular thematic learning. I haven’t seen many IDUs play out in my 15 years of teaching. Some middle schools are more middle school than others. I only know what I’ve experienced.

The point of my post is to share a lesson I slightly modified from the October, 2015 issue of Mathematics Teaching in the Middle School. I do know one thing about middle schoolers–they always seek fairness.

The proportional reasoning lesson is titled US Congress is it Representative?  Students studying the US Constitution use proportions or percentages to examine equity, fairness, and how the racial and ethnic make-up of our country compares to the ethnic backgrounds of the current Congress.

The lesson begins by providing background knowledge on the US Census, which counts every resident of the US then, from that count, determines the number of seats each state has in the U.S. House of Representatives. The NCTM article did not include this background so this is where I deviated from the original lesson.

The most recent apportionment is below.



If your state lost or gained seats, that would make for good discussion. On a side note, I’m surprised Illinois did not lose more seats in the House!

One major part of the lesson is examining the population statistics and discussing why the sum of the various ethnicities exceeds the total. (The way the Census is counted residents can and do identify with more than one ethnicity.)


Students are then asked, “Based on this data, and the knowledge that there are 435 members of the House of Representatives and 100 members of the Senate, what should a “fair” racial/ethnic make-up of the House and Senate be?

Once they’ve determined what they think is “fair” the ethnic background of Congress is revealed for comparison.


Students use claim, evidence, and reasoning to argue whether the House and Senate are representative of the US population.

I’m with Danny in the sense that I want students to learn about “important” matters. But I also understand that what’s important to a middle schooler may not be the same as an adult.

In my role as math interventionist, I’ll use this lesson to review percentages in our after school math program, but it may get more traction if the lesson is presented by the social studies teachers. So I shared it the social studies teachers instead of the math teachers with the hope they’ll consider it when teaching the Constitution.

Time and timing are the biggest constraints when implementing an interdisciplinary lesson.

Justifying moving the decimal

I need your help. I’ve been thinking of ways to better connect decimal multiplication with fraction multiplication, especially in terms of justifying moving the decimal point. Sixth graders may recall counting the number of decimal places, but do they really know what they are counting and why they are counting it?

The following is work in progress and I would appreciate your input. How can my attempt at creating curiosity  be improved upon? Are the examples sufficient or should they be revised? What am I missing?


I’m anticipating students will say, “Count the decimal places,” so my follow up is this question.


I don’t think students will connect fraction multiplication with decimal place value so my next move will be to ask them to compare the two expressions.


Some students may recognize the two expressions are equivalent, but will they make the place value connection?


I prepared a second example that “culminates” with this slide.


Since the above problem has numbers in the ones place, some students may recognize 2 x 2 = 4, so it would be reasonable to place the decimal between the 5 and 6. I would certainly accept that, but my point is to discuss powers of ten. Yet I need to be careful. My sixth graders haven’t been exposed to negative exponents. Would it be appropriate to describe it as powers of tenths? Is this they best way to address the mathematics? I’ll likely begin with establishing a pattern before I present these slides.

I’m also thinking of creating a similar lesson which connects dividing decimals with dividing fractions using the common denominator method.  The students are familiar with the common denominator method, but I wonder if it will be too messy?

Mindset, abundance

Even though I’ve switched middle schools a colleague in my former building and I continue to commiserate on the implementation of Carol Dweck’s growth mindset research. Nurturing a growth mindset is a daunting task. It requires creating and sustaining the conditions for it with multiple learning opportunities, feedback that’s timely (and implemented!), and frequent formative assessments.

What my colleague and I have both feared, and Dweck  too, is that growth mindset has been reduced to the grit mentality of telling the students to work harder.

A growth mindset isn’t just about effort. Perhaps the most common misconception is simply equating the growth mindset with effort. Certainly, effort is key for students’ achievement, but it’s not the only thing. Students need to try new strategies and seek input from others when they’re stuck. They need this repertoire of approaches—not just sheer effort—to learn and improve.

Could this be a reason why the growth mindset effect size is only 0.19? Hattie thinks teachers treat their students with a fixed mindset. Is focusing our attention on effort a fixed mindset?

Now I wonder, should we be asking a different set of questions?


I am so conflicted right now I’m putting mindset in the back of my mind. Instead my focus is on abundance.

The biggest gift of abundance I can give students is patience. I have to remember they are children. They don’t share the same enthusiasm for learning as I do. Or worse, they’ve quit on themselves because of a self-concept they’ve developed. That patience will also serve me well when students learn implementing feedback is a requirement, not an option.

I can also give an abundance of learning opportunities that weave the concrete, representational, and abstract. For example, when re-teaching estimating percent of a number several students got the concept when I introduced a bar model. In fact one said, “Can I do this on the test?” Of course.

One final gift of abundance is continuous self-reflection. We put students on a treadmill and never give them time to step off and check their heart rate. How can they take ownership of their learning when we don’t give them time to reflect on the learning process?

Intervention: reteaching, review, and problem solving

In my role as interventionist I’m writing the middle school math curriculum for our Title 1 after school program. Circumstances outside of my control have prevented us from rolling out the program sooner, but invitations will be going out next week and we anticipate a start date within the next two weeks.

I know students will need review and reteaching of concepts, but I also want them to develop a repertoire of problem solving strategies so I will be including James Tanton’s Curriculum Inspirations. which is supported by the Mathematical Association of America. It’s a fabulous resource that includes a series of 10 strategies with examples.

Since the first lesson will focus on multiples and factors students will be introduced to Tanton’s think aloud on the Divisible by 13 task using the problem solving strategy Engage in Successful Flailing. They’ll solve the problem using the attached.


Of course students will have opportunities to practice LCM. Then they will have a chance to explore multiples and 3 consecutive numbers problems from Don Steward’s site.

consecutive numbers

Several teachers will be implementing the after school program, so it makes sense for the curriculum to be delivered using Canvas. Each lesson begins with a content page that identifies the learning objectives and mathematical practices. The page also includes a resource link so teachers have access to answer keys stored in google drive.


As the title of this post suggests I want to balance review, reteaching, and problem solving.  Students need to be exposed to challenging problems but they need to have access to them as well.

Math Intervention and The Remainders Game

I’m in a new building with a different role this year, math interventionist. A big part of my job is working with the 6th-8th graders during their homeroom periods, pushing in to the sixth grade classrooms for reteaching, plus creating math curriculum for the after school Title 1 program.

Officially the intervention period is not a replacement class, but I do see specific students twice a week supporting the day’s learning objectives while shoring up foundational skills and problem solving strategies.  A twice weekly intervention period is not an ideal situation, but it’s the best we can offer students right now. The period also gives me a chance to pilot some of the lessons/activities I’m considering for the after school program.

One great example which fits perfectly with factors is NRICH’s The Remainders Game. The object is to deduce what number between 1-100 the computer is thinking in the fewest moves possible. The player chooses the divisor between 2-10 and the computer gives a clue–its remainder. We’ve played the game several times as a class and I’m at a point where I can now share a Remainders Game worked solution which the after school tutors can follow, plus a Remainders Game worksheet so students can play independently.

Discuss the first move. What number should we divide by to narrow the list of possibilities?
remainders game
What possible numbers between 1 and 100 can the computer be thinking?

In this scenario we’re working with the numbers 18, 28, 38, 48, 58, 68, 78, 88, and 98. We don’t know what we’re going to do with those numbers yet but we need to keep track of our work. I led students to roughing out a table by hand. I later created the worksheet.

Introducing factors as a strategy comes into play when you’ve determined the possibilities. For example when the computer is thinking of an even number, is dividing by 2 a wasted move? Can finding the factors of each of the numbers help us rule out some of the possibilities?

For time’s sake each student was assigned a number to factor and we shared out. In the future students will be able to do this independently now that they know the process. Next was a great discussion on which number to divide by next.


We chose 4, and the computer told us there was a remainder of 0. What numbers did we eliminate? What number should we divide by next?

This task builds stamina while studying the composition of numbers. I think a scaffolded approach is giving my students access to a challenging problem.