Tuesday, December 13, 2016
Why Kids Should Keep Using Their Fingers to do Math
Nearly all kids learn how to count using their fingers. But as kids grow older and math problems become more advanced, the act of counting on fingers is often discouraged or seen as a less intelligent way to think.
Read more via Pocket
Sunday, November 6, 2016
A school's "no homework" policy led to kids getting better grades and it's a super inspiring story
In grade school, homework was the bane of our existence. It wasn’t uncommon to glance around to see a classmate working on assignments for other classes during lectures, or to see students hard at work in the library or at the lunch table.
Read more via Pocket
Sunday, October 9, 2016
Stop the Homework Insanity and Let Kids Be Kids
I have so many fond memories of my childhood. Growing up in a relatively rural area of Northwestern New Jersey sure had its benefits. As we returned home from school each day, my brothers and I would jump off the bus and diligently make our way about a halfmile back to our house.
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Wednesday, October 5, 2016
#EurekaMath: How to pace the lesson and meet the needs of EVERYONE
As much as I love the Eureka Math curriculum, teachers seem to have a real tough time figuring out how to pace the lesson while also meeting the needs of all their students. I am finding teachers struggle in this regard with Eureka Math far more than they did with the preCommon Core textbook. I don't see this as a knock against Eureka Math. If anything, it merely speaks to the cruddy quality of the old textbooks and that the old textbooks were nothing but snake oil, convincing us we were doing a good job, when in reality we weren't.
Recently, I received an email that is very typical of teachers new to Eureka Math. Here is the email and my response. I hope it helps someone else out there in Internetland.
The original email
Hi Duane,
I am an Instructional Coach in the XXXXX School District in Washington state and I hope you don't mind if I ask you a question about Eureka Math. This is our first year of implementation and teachers are really struggling with the differentiation issue how to meet the needs of the low and high students. I'm wondering if you have found ways to address this issue?
Additionally teachers are feeling that there is so much to cover in every lesson and the program moves too quickly and they want to teach the program 4 days a week and then on the fifth day spend time reviewing concepts taught or missed. They are also struggling with what to cut out... Any words of advice?
Hi there!
Your email is a common one that I get weekly from teachers inside and out of my district! Essentially, you are asking two things:
1. How to meet the needs of ALL students?
2. How to fit all the components of a lesson into a single 45 minutes math time?
I'll start with #2 because it also somewhat addresses #1...
Recently, I received an email that is very typical of teachers new to Eureka Math. Here is the email and my response. I hope it helps someone else out there in Internetland.
The original email
Hi Duane,
I am an Instructional Coach in the XXXXX School District in Washington state and I hope you don't mind if I ask you a question about Eureka Math. This is our first year of implementation and teachers are really struggling with the differentiation issue how to meet the needs of the low and high students. I'm wondering if you have found ways to address this issue?
Additionally teachers are feeling that there is so much to cover in every lesson and the program moves too quickly and they want to teach the program 4 days a week and then on the fifth day spend time reviewing concepts taught or missed. They are also struggling with what to cut out... Any words of advice?
My response
Your email is a common one that I get weekly from teachers inside and out of my district! Essentially, you are asking two things:
1. How to meet the needs of ALL students?
2. How to fit all the components of a lesson into a single 45 minutes math time?
I'll start with #2 because it also somewhat addresses #1...
In general, I see teachers move through the components of a lesson too slowly. This is because teachers LOVE their students and do not want to move too fast for the struggling students in the class. Unfortunately, this means a fluency activity that should take only 3 minutes ends up taking 10 minutes. Or an Application problem that should take 8 minutes ends up taking 20 minutes because the teacher turned what should merely be a short formative assessment opportunity into a fullon teaching moment. Finally, the teachers often trudge through too much of the Concept Development, doing too many examples at too slow a pace. Or worse, the teacher skips the Concept Development altogether and just works with the whole class in completing the Problem Set questions.
What is the result?
What is the result?
The students who understand the math and are ready to move on become bored because the teacher is going too slow. I call this "over teaching". The teacher is teaching too much (trying to help the strugglers) and not letting the top students move to independent work time. Moreover, the low students, for whom the teacher is "over teaching", still aren't understanding the math because what they need is something altogether different  perhaps a small group intervention, reteaching with manipulatives, or backfilling with content from an earlier grade.
Here is a blog post I wrote a little while ago that goes more into specifics about how to pace an individual lesson...
http://primetimemathpusd.blogspot.com/2016/08/pacingwitheurekamathengageny.html
With proper pacing, much of #1 is addressed automatically. Counterintuitively, the proper pacing is probably faster than what teachers typically do.
Now to address #1 a bit more...Let's assume the teacher has kept a zippy pace with the fluency activity and the application problem. Now she is ready to do the Concept Development. This is where I suggest teachers resist the urge to "over teach". The teacher should choose the minimum number of example problems from the Concept Development vignette in the teacher edition and then release most of the students to independently work on the Problem Set, while she immediately works with a few students in a small group setting.
Of course, this means the teacher is now doing double duty: 1.) continuing to teach the strugglers in the small group and 2.) monitor the rest of the class to ensure they are working productively. This is hard, but can definitely be done.
Keep in mind that the Problem Set is a timebased activity rather than a productbased activity. This means students are working on the Problem Set for a fixed amount of time (about 10 or 15 minutes) regardless of whether they finish all the problems. Indeed, the teacher should identify the problems in the Problem Set as "Must Do", "Could Do", or "Extension". During the 15 minutes, students should first do the Must Do questions, then the Extension Questions. Time permitting, the Could Do questions are last.
As students complete the Problem Set, I encourage teachers incorporate some sort of problemsolving opportunity for students to work on. Some awesome online resources:
These allow students to work on something OTHER than drillandkill math. Of course, the teacher can certainly use other problemsolving activities she may already have.
Once the teacher finishes working with the strugglers in the small group, often there is no time for those students to work on the Problem Set. This is okay. Just move on and have the students skip the Problem Set that day.
The important thing is for the teacher to resist the urge to reteach the entire lesson the next day simply because a few students do not understand today's lesson. The curriculum is specifically written to spiral and review, which makes reteaching rarely necessary.
So...let's boil this whole thing into a simple game plan for teaching:
Here is a blog post I wrote a little while ago that goes more into specifics about how to pace an individual lesson...
http://primetimemathpusd.blogspot.com/2016/08/pacingwitheurekamathengageny.html
With proper pacing, much of #1 is addressed automatically. Counterintuitively, the proper pacing is probably faster than what teachers typically do.
Now to address #1 a bit more...Let's assume the teacher has kept a zippy pace with the fluency activity and the application problem. Now she is ready to do the Concept Development. This is where I suggest teachers resist the urge to "over teach". The teacher should choose the minimum number of example problems from the Concept Development vignette in the teacher edition and then release most of the students to independently work on the Problem Set, while she immediately works with a few students in a small group setting.
Of course, this means the teacher is now doing double duty: 1.) continuing to teach the strugglers in the small group and 2.) monitor the rest of the class to ensure they are working productively. This is hard, but can definitely be done.
Keep in mind that the Problem Set is a timebased activity rather than a productbased activity. This means students are working on the Problem Set for a fixed amount of time (about 10 or 15 minutes) regardless of whether they finish all the problems. Indeed, the teacher should identify the problems in the Problem Set as "Must Do", "Could Do", or "Extension". During the 15 minutes, students should first do the Must Do questions, then the Extension Questions. Time permitting, the Could Do questions are last.
As students complete the Problem Set, I encourage teachers incorporate some sort of problemsolving opportunity for students to work on. Some awesome online resources:
These allow students to work on something OTHER than drillandkill math. Of course, the teacher can certainly use other problemsolving activities she may already have.
Once the teacher finishes working with the strugglers in the small group, often there is no time for those students to work on the Problem Set. This is okay. Just move on and have the students skip the Problem Set that day.
The important thing is for the teacher to resist the urge to reteach the entire lesson the next day simply because a few students do not understand today's lesson. The curriculum is specifically written to spiral and review, which makes reteaching rarely necessary.
So...let's boil this whole thing into a simple game plan for teaching:
 Properly pace the fluency activity and Application problem. Keep it zippy. Assess student understanding, but don't turn it into a teaching moment.
 Keep the Concept Development to a minimum. Do as few examples as possible in order to release most students as quickly as possible. I call this "UNDER teach" the math...meaning the teacher should send students to work independently a little earlier than she might otherwise have done.
 The teacher now does double duty: reteach a small group while also monitoring the rest of the class.
 Have additional problemsolving activities (online or paperandpencil) for early finishers of the Problem Set.
 Limit the Problem Set time to 10 or 15 minutes only. Then do a short Debrief conversation with the whole class.
Sheesh...there is more we could talk about. Specifically, Universal Design for Learning (UDL). This is a framework for how a teacher can design/modify a lesson in order to meet the needs of more students within the lesson, thereby reducing the need to differentiate for the high and lowstudents after the lesson. In UDL, teachers proactively plan strategies for removing barriers to student learning in three ways:
https://goalbookapp.com/toolkit/strategies
I hope this gets you started on your journey. Please feel free to email again. We all need to support one another as we help our teachers implement Eureka Math!
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 Strategies for engagement
 Strategies for representation
 Strategies for expression.
https://goalbookapp.com/toolkit/strategies
I hope this gets you started on your journey. Please feel free to email again. We all need to support one another as we help our teachers implement Eureka Math!
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Wednesday, September 21, 2016
How to catch up in Eureka Math
This weekend I was asked a question via my YouTube channel. The question is a common one that I have heard numerous times. It isn't unique to Eureka Math either...I've heard it my entire 26 years of teaching.
Here is a screenshot of the question and my quick response at the time.
As I reflect on my response, here are some additional thoughts...
 The strategy of "catching up" a student by going backwards two years just doesn't make sense to me. Essentially, the strategy is flawed from the start because it takes a student who is already struggling with mathematics and merely piles more math on top of the student. In order for this strategy to work, not only does the struggling student have to learn the current year's standards, but also the standards from the previous year or two. For a student already struggling with math, this is just plain silly.
 Eureka Math (EngageNY), being OER, affords us the ability to print "just in time" content for a student in need. Rather than subjecting the struggling student to the ENTIRE previous two years' worth of math, let's just print an occasional worksheet from a previous year. The strategy here is to have laserlike purpose for what "old" math to teach the student.
 Most importantly, before we try to "fix" the student, let's first reflect on what went wrong in the first place that caused the student to need fixing? This is UDL!!! Moving forward, how can we select strategies for engagement, representation, and expression that remove the barrier to learning in the first place?
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Just a thought.
Monday, September 12, 2016
Research Finds Effects Of Homework On Elementary Students
After over 25 years of studying and analyzing homework, Harris Coopers’ research demonstrates a clear conclusion: homework wrecks elementary school students.
Read more via Pocket
Saturday, September 10, 2016
Are Timed Math Tests Harmful to Students? — MashUp Math
Question: Which of these statements best describes an exceptional math student? If you chose one of the first three statements, then your beliefs about the essence of math understanding may be rooted in misconceptions.
Read more via Pocket
Thursday, September 8, 2016
Why Parents Should Not Make Kids Do Homework
President Obama’s pick for Education Secretary, John King, Jr., is headed for confirmation Mar. 9. King’s track record shows he loves standardized testing and quantifying learning.
Read more via Pocket
Here’s why I said no to homework for my elementaryaged kids
Here’s a shocking thought: What if not doing homework was better for your kids? Would you still make them do it? We want what’s best for our children, and when kids hit school age we follow the teacher’s lead. If the teacher assigns homework, we enforce it at home. Even in elementary school.
Read more via Pocket
Wednesday, September 7, 2016
Looking for the magic bullet
Today I was asked for my opinion on ST Math. I have heard this type of question many times before. It goes something like this...
"I have heard that ST Math improves student test scores. What is your opinion of XXXXX Elementary School asking the PTA to fund the annual subscription?"
Here is my response today...

When investigating ST Math, schools are attempting to outsource the instruction to a computer, while completely ignoring that the teacher's instructional strategies are far more important than ANY software. Unfortunately, all the money is spent purchasing the site license allowing no money left over to improve the quality of classroom instruction.
"I have heard that ST Math improves student test scores. What is your opinion of XXXXX Elementary School asking the PTA to fund the annual subscription?"
Here is my response today...

Hey there...
I know ST Math (Ji Ji Math) pretty well. I've observed it in action and have been asked numerous times to give my opinion about it. In short, it costs a lot of money for minimal (if any) gains. Nearly all of the studies showing a benefit to students involved lowincome students only. It is never made clear whether the students benefitted from ST Math specifically, or if students merely improved because ST Math provided "extra math time" and ANY extra math time would have created the same benefits.
It is pretty easy to find evidence on both sides of the ST Math question...
Has no effect: https://www.researchgate.net/publication/260638069_A_Randomized_Trial_of_an_Elementary_School_Mathematics_Software_Intervention_SpatialTemporal_ST_Math
Has an effect:https://www.edsurge.com/news/arizonastudentscatchupfastwithstmath
What worries me about schools that are considering ST Math is that there is a belief that ST Math will improve student results WITHOUT teachers having to change their instructional strategies. I've observed teachers in computer lab time using ST Math as a glorified babysitter.
The NUMBER 1 thing that improves mathematics achievement is improving the quality of the math instruction in the classroom. ST Math is low on the list.
My suggestion: Use the money to pay for PD that will improve the teaching that goes on in the classroom. Alternatively, the annual cost of ST Math could purchase 50  100 Chromebooks each year.
Sorry dude. ST Math is not the magic bullet. What we need is just good teaching.

Please...let's stop looking for a magic bullet. Instead, let's focus on using our mathematics instructional coaches effectively to improve classroom instruction.
It is hard work, but much better than expending our time and energy looking for a magic bullet.
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Wednesday, August 24, 2016
Pacing with Eureka Math (#EngageNY)
The question
Our teachers are one week in with implementing Eureka!!! Some are already very anxious around pacing of the lesson. Any advice around what to tell them when they are wanting to slow way down & take 2+ days to do one lesson? Teachers are worried about moving onto the next lesson without feeling like their students are really understanding the concepts. What something encouraging to tell them? Doing a new lesson every day is making teachers very nervous!
My response
Pacing is a common firstyear concern. Here are some quick tidbits...
1. Don't do ALL the fluency activities listed in the lesson. Only choose one, perhaps two of them. Perhaps, consider doing NONE of the fluency activities for that day.
2. Do not turn the Application problem into a teachable moment. It is a time for the students to practice using their brain. It is a time for the teacher to collect formative data about the progress of her students. It should NOT turn into a 20 minute minilesson on how to solve the problem correctly.
3. Be efficient with the Concept Development. Aim for 20 minutes maximum. To be efficient, the teacher needs to have decided the night before exactly the sequence of example problems to do for the lesson. Don't do ALL the examples in class.
4. Limit the time students do the "independent" practice (the Problem Sets) to ONLY 10 or 15 minutes! This means the Problem Set is a timebased event rather than a productbased event. Students are not expected to do ALL problems during the 1015 minutes.
5. The night before, the teacher should decide which problems in the Problem Set are "must do's", "could do's", and "extensions". While students are working on the Problem Set in class for 1015 minutes, they should do the "Must Do" problems first. Then the "Could Do" problems.
6. Save at least 5 or 10 minutes for the Student Debrief time. The teacher should pick one or two key debrief questions to ask the class. The teacher edition lists some questions the teacher might ask. Or the teacher can simply ask, "Would someone please explain their thinking for Question 4?"
All of the above can easily be fit into a 45 minute time period.
Now the question is "What do I do with my students who are struggling and need an extra day with this concept?"
The answer to this is often "Move to the next lesson anyway!"
Often in Eureka Math, the lessons move very incrementally from one lesson to the next. If a student doesn't understand Lesson 4, move on to Lesson 5 because it is likely the student will suddenly have the Ahha moment in that lesson rather than in Lesson 4. This is a very different mentality than what teachers are familiar with.
To help the struggling students even though the teacher has moved on to the next lesson, the teacher ought to consider how she can make the concept accessible to the students. Some ideas...
 hook up the student with a student partner
 allow the students to use manipulatives to solve the problems rather than drawing the pictures
 consider teaching the student a different method altogether (perhaps a method taught in a future module) even while the teacher also continues attempting to teach the student the original method
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Pacing with Eureka Math (#EngageNY)
The question
Our teachers are one week in with implementing Eureka!!! Some are already very anxious around pacing of the lesson. Any advice around what to tell them when they are wanting to slow way down & take 2+ days to do one lesson? Teachers are worried about moving onto the next lesson without feeling like their students are really understanding the concepts. What something encouraging to tell them? Doing a new lesson every day is making teachers very nervous!
My response
Pacing is a common firstyear concern. Here are some quick tidbits...
1. Don't do ALL the fluency activities listed in the lesson. Only choose one, perhaps two of them. Perhaps, consider doing NONE of the fluency activities for that day.
2. Do not turn the Application problem into a teachable moment. It is a time for the students to practice using their brain. It is a time for the teacher to collect formative data about the progress of her students. It should NOT turn into a 20 minute minilesson on how to solve the problem correctly.
3. Be efficient with the Concept Development. Aim for 20 minutes maximum. To be efficient, the teacher needs to have decided the night before exactly the sequence of example problems to do for the lesson. Don't do ALL the examples in class.
4. Limit the time students do the "independent" practice (the Problem Sets) to ONLY 10 or 15 minutes! This means the Problem Set is a timebased event rather than a productbased event. Students are not expected to do ALL problems during the 1015 minutes.
5. The night before, the teacher should decide which problems in the Problem Set are "must do's", "could do's", and "extensions". While students are working on the Problem Set in class for 1015 minutes, they should do the "Must Do" problems first. Then the "Could Do" problems.
6. Save at least 5 or 10 minutes for the Student Debrief time. The teacher should pick one or two key debrief questions to ask the class. The teacher edition lists some questions the teacher might ask. Or the teacher can simply ask, "Would someone please explain their thinking for Question 4?"
All of the above can easily be fit into a 45 minute time period.
Now the question is "What do I do with my students who are struggling and need an extra day with this concept?"
The answer to this is often "Move to the next lesson anyway!"
Often in Eureka Math, the lessons move very incrementally from one lesson to the next. If a student doesn't understand Lesson 4, move on to Lesson 5 because it is likely the student will suddenly have the Ahha moment in that lesson rather than in Lesson 4. This is a very different mentality than what teachers are familiar with.
To help the struggling students even though the teacher has moved on to the next lesson, the teacher ought to consider how she can make the concept accessible to the students. Some ideas...
 hook up the student with a student partner
 allow the students to use manipulatives to solve the problems rather than drawing the pictures
 consider teaching the student a different method altogether (perhaps a method taught in a future module) even while the teacher also continues attempting to teach the student the original method
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Tuesday, August 23, 2016
Three stages of counting
Today I had the pleasure of coteaching a class of 1st graders. Being only the second week of the school year, I was amazed at how deftly the teacher peppered her math lesson with minilessons on the various routines and protocols of the classroom. This old formermathteacherturnedelementarycoach learned tons about how to run a 1st grade class. Humbling, truly humbling.
I was able to return the favor by sharing some math thoughts. Here is how our time progressed and my resulting mathematical thoughts...
The teacher began by posting two fish bowls on the board and used chips to represent goldfish. She put seven “fish” in one fishbowl and two in the other.
Teacher: “How many fish are in the left fishbowl?”
Class: “7!!”
T: “How many in the bowl on the right?”
C: “2!!”
T: “How can we use those two numbers to begin filling in this number bond?” (She posts a laminated number bond on the board.)
C: “Write a 7 in the top circle and a 2 in the bottom circle!” (The teacher does so.)
T: “What number should I put here?” (The teacher points at the big empty circle.)
Here is where the cool thing happens…
Some students began pointing at the chips onebyone, clearly counting. Other students simply raised their hands.
T: “At the snap of my finger, say the answer.” (She snaps.)
C: “Niiiiiine!”
T: “How many fish are there in all?”
C: “Nine”
T: “7 plus 2 equals….” (She writes ‘7 + 2 =’ on the board.)
C: “Nine”
So what was the cool thing?
All the students got the right answer, and yet it was obvious that the students were in a variety of developmental stages of counting.
There are three stages of counting:
.....Stage 1: Count all
.....Stage 2: Count on
.....Stage 3: Make an easier problem (Use a strategy)
Stage 1: Count all
When given a group of 7 chips and a group of 2 chips and asked “How many are there?”, students in this stage count all 9 chips onebyone. Students in this stage recognize the need for onetoone correspondence as they count the chips. This is typical for students in Kindergarten.
Stage 2: Count on
At this stage, students are able to see one group as an entity (recognizing the cardinality of the group) and count on from there, often touching each chip of the second group as they count. In 7+2, a student might say “Seeeeven, eight, nine” as he touches each of the two chips in the second bowl. Stage 2 is typically introduced in Grade 1 (although some Kinders may begin Stage 2) with the hope that all 1st graders will have this stage under their belt by the end of the year.
Stage 3: Make an easier problem (Use a strategy)
This stage is introduced in Grade 1 with the hopes that students will internalize this strategy later in Grade 1 or in Grade 2. This stage is easier to describe with a problem such as 8 + 5. A student in Stage 3 might take two from the five and give it to the eight, making 10. Then add 10 and the remaining 3 to get 13.
For a problem an addition problem within 10, students in Stage 3 might explain knowing 7+2=9 by saying something like “I just knew it in my head”.
Why do we need to know the three stages of counting?
It is not enough to see that a student has written “7 + 2 = 9” beneath the fish bowl on her paper. We teachers need to dig a bit deeper and determine with which stage did the student use to get that answer? A student who gets 100% on her paper is not considered fluent with basic facts if she uses “Stage 1 Count All” on every problem.
Basic fact fluency requires the presence of flexibility, appropriate strategy use, efficiency, and accuracy. It is not enough to verify whether a student can correctly solve the problems in a timely fashion (ala “timed tests”...but that is a different blog post...ugh). Somehow, the teacher needs to also assess the flexibility and strategy use of each student. This is where number talks, small groups, and informal formative assessment comes in. Somehow for each student the teacher must identify the student’s current developmental stage (count all, count on, or make an easier problem) and then nudge that child to the next level up.
It was fascinating to watch the three stages in action during this single 1st grade lesson. It is humbling to me when trying to advise the teacher how to determine the stage of each child. I take comfort in the fact that if I was the teacher I wouldn’t worry about trying to assess the stage of EVERY student in a single day. Perhaps I’d use an anecdotal list to record the stages of the various students I happen to come across. Then specifically target the remaining students during centers time.
My challenge to us all
When we wander around the room, looking over the shoulders of our kiddos at their answers, let’s try to go one step beyond merely checking if the answers are correct. Take a moment with one or two students per day to focus not only on WHAT is the student’s answer, but also HOW did the student arrive at that answer?
Oh yeah...while ensuring proper classroom control with the other 24 students. But THAT is for another blog post.
Wednesday, August 10, 2016
Hattie's Interactive Visualization
We only have so much time in our day. How can we get the biggest bang for our buck with our limited time?
Take a look at John Hattie's interactive visualization of influences on student achievement and their effect sizes. What is "effect size"? In layman's terms it is a unit of measure that allows us to measure the expected increase in achievement for a particular influence. The bigger the "effect size" the better.
Here it is...
Of particular note...try to find homework. You will see that it is way, way down the list. Seems to me, all the energy we put into homework is for very little gain. Perhaps, we should refocus our energy on things that are demonstrably more beneficial to our students.
Just sayin'.
"SelfEfficacy and Homework" or "The 'ifs' of Homework"
Today I was reading this summary of studies about the roles of homework and selfefficacy in closing the mathematics achievement gap. (http://www.ernweb.com/educationalresearcharticles/howmuchhomeworkshouldyougiveyourstudentsmathachievement/)
It seems to suggest two things...
1. Selfefficacy is essential in closing the achievement gap
2. Math homework is the way for students to develop selfefficacy
Number 1 seems very reasonable. Indeed, I have come across many other studies regarding selfefficacy and its role in math anxiety, achievement gap, and gender differences.
Number 2, however, has me scratching my head. It is unclear to me why the authors of this study single out homework as the means for developing selfefficacy. Especially because it is entirely dependent upon the student having access to all necessary support resources at home in order to complete the math homework thereby developing the selfefficacy.
It seems that many who support math homework do so with several "ifs" attached. For example, this blog post ends with a series of such "ifs".
Unless and until we work our the "ifs", issues such as math anxiety, ethnic/racial achievement gaps, and gender achievement gaps are likely to continue vexing our profession.
While the adults are working out the "ifs" it seems we need to ensure we are doing no harm to the students. Reading for K5 homework? No question...yes! No "ifs" there. Math for K5 homework? Hold your horses...let's carefully work out the "ifs". In the meantime, definitely do this!
It seems to suggest two things...
1. Selfefficacy is essential in closing the achievement gap
2. Math homework is the way for students to develop selfefficacy
Number 1 seems very reasonable. Indeed, I have come across many other studies regarding selfefficacy and its role in math anxiety, achievement gap, and gender differences.
Number 2, however, has me scratching my head. It is unclear to me why the authors of this study single out homework as the means for developing selfefficacy. Especially because it is entirely dependent upon the student having access to all necessary support resources at home in order to complete the math homework thereby developing the selfefficacy.
It seems that many who support math homework do so with several "ifs" attached. For example, this blog post ends with a series of such "ifs".
Unless and until we work our the "ifs", issues such as math anxiety, ethnic/racial achievement gaps, and gender achievement gaps are likely to continue vexing our profession.
While the adults are working out the "ifs" it seems we need to ensure we are doing no harm to the students. Reading for K5 homework? No question...yes! No "ifs" there. Math for K5 homework? Hold your horses...let's carefully work out the "ifs". In the meantime, definitely do this!
Wednesday, May 25, 2016
Introducing EMBARC.Online
I’d like to introduce you to EMBARC.Online.
EMBARC stands for Eureka Math Bay Area Regional Consortium.
Our vision is:
 To build a collaborative community of Eureka Math users
 To provide a common website to support all users of the Eureka Math curriculum
During our time dabbling in using this wonderful curriculum, it became very clear to me  as the TK5 Mathematics Instructional Coach  that our teachers were going to need a lot of support in order to fully understand and implement it. Thus began a steady stream of emails from me to my teacher community, sharing the “latest and greatest” web URLs, links, and resources.
This is where EMBARC comes in.
Rather than sending teachers out to a variety of thirdparty locations (Illustrative Mathematics, Inside Mathematics, Zearn, ThreeAct lessons, to name a few), we will curate the best of the web and organize it on EMBARC.online. Great supplemental resources will be placed right at the topic or lesson where they will be most useful.
Are you a 5th grade teacher doing Module 5 Lesson 18 tomorrow? Navigate there and you will find great resources to choose from to supplement, augment, or totally rewrite the lesson. The teacher can access the exact support she needs at the exact moment she needs it.
Curating outside content into the curriculum is the first way that EMBARC distinguishes itself from other EngageNY/Eureka Math website. Nearly all EngageNY/Eureka Math websites have the core materials as PDFs ready for download and that’s about it. For additional resources, teachers are usually sent to external links to fend for themselves. EMBARC curates to make things easier on the teacher. We bring the resources in for the teacher, rather than sending the teacher outside to other sites.
A collaborative community
A second, more profound distinction is the collaborative community on EMBARC. Every module has a “Faculty Lounge”, in which our users can share ideas: new resources, shared Pinterest link, ideas for pacing.
Our EMBARC editors will find the best ideas shared in the Faculty Lounge and integrate them through the rest of the module. Teachers learn so much around the lunch table! We recreate that experience in the Faculty Lounges on EMBARC.
Now what?Join the community! Take part in a community of Eureka Math users who support each other, share ideas, and collaborate.
EMBARC is entirely free to use. No account is necessary. However, you will need to create a free account in order to contribute to the discussion in the faculty lounges.
A collaborative community
A second, more profound distinction is the collaborative community on EMBARC. Every module has a “Faculty Lounge”, in which our users can share ideas: new resources, shared Pinterest link, ideas for pacing.
Our EMBARC editors will find the best ideas shared in the Faculty Lounge and integrate them through the rest of the module. Teachers learn so much around the lunch table! We recreate that experience in the Faculty Lounges on EMBARC.
Now what?Join the community! Take part in a community of Eureka Math users who support each other, share ideas, and collaborate.
EMBARC is entirely free to use. No account is necessary. However, you will need to create a free account in order to contribute to the discussion in the faculty lounges.
The more people who join in on the conversation, the more we support each other, the better we will be for our students.
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Tuesday, May 10, 2016
Is a parallelogram also a trapezoid?
A famous bard once said, “A trapezoid by any other name would still be a trapezium”.
Okay...totally not true. But it brings to mind the question I am often asked, “What is the definition of a trapezoid?” In fact, I was asked this question today. So here is my answer…
For mathematics, being a subject that is supposedly the “universal language”, this question opens a huge can of worms and has a surprisingly involved answer.
There are three – yes three – different ways one can define a trapezoid. Let’s get started.
If a person walks up to you and says, “Let’s discuss trapezoids”, the first things you should do is listen his accent. Is it American? Is it Canadian? Or some other Englishspeaking accent? This matters.
For the words trapezoid and trapezium, America and Canada defines them one way, but in other English speaking countries these same two words have their meanings switched.
In America and Canada...
In other English speaking countries...
In America a trapezium is a quadrilateral that has no parallel sides. Sometimes this is called an irregular quadrilateral.
So now that we have define a trapezoid to be the figure that is not a trapezium. There is still the matter of the two definitions of trapezoid in America.
Let’s start with this figure…
Most people would immediately recognize it as a trapezoid. There are two ways we can classify as trapezoid: the inclusive definition and the exclusive definition.
T(I): a figure with at least one pair of parallel sides
T(E): a figure with exactly one pair of parallel sides
Both definitions are legitimate, but they each lead to other differences in classifications.
For example, a parallelogram is just a parallelogram in T(E), but a parallelogram is also a trapezoid in T(I).
In T(I), even a square is considered a trapezoid!
Why can’t we all just get along?
We don’t need to argue over which definition is correct. They both are. So, this means when we speak about trapezoids, we must preface it with an agreement of which definition we are using...at least for that instance.
T(E) seems to have its origins in the 1500’s prior to the advent of calculus. When calculus came along, we began using trapezoids to estimate the area under curves.
Public Domain, https://commons.wikimedia.org/w/index.php?curid=647823
CC BYSA 3.0, https://en.wikipedia.org/w/index.php?curid=44244531
As the animation shows, some of the trapezoid slices begin to look suspiciously like rectangles. Aha!!! This is when T(I) got invented.
Common Core Standards
In 5th grade the standards are noncommittal on the trapezoid issue.
The 3rd grade standards also sidestep the issue.
But the Progressions Documents for the Common Core Math Standards make it clear that mathematicians prefer we use T(I).
Since a key component of the Common Core Standards is that students will be college and career ready, it seems the best trapezoid definition to use is the one that leads to calculus...T(I).
These are all trapezoids…
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Okay...totally not true. But it brings to mind the question I am often asked, “What is the definition of a trapezoid?” In fact, I was asked this question today. So here is my answer…
For mathematics, being a subject that is supposedly the “universal language”, this question opens a huge can of worms and has a surprisingly involved answer.
There are three – yes three – different ways one can define a trapezoid. Let’s get started.
If a person walks up to you and says, “Let’s discuss trapezoids”, the first things you should do is listen his accent. Is it American? Is it Canadian? Or some other Englishspeaking accent? This matters.
For the words trapezoid and trapezium, America and Canada defines them one way, but in other English speaking countries these same two words have their meanings switched.
In America and Canada...
trapezium  trapezoid 

trapezium  trapezoid 

In America a trapezium is a quadrilateral that has no parallel sides. Sometimes this is called an irregular quadrilateral.
So now that we have define a trapezoid to be the figure that is not a trapezium. There is still the matter of the two definitions of trapezoid in America.
Let’s start with this figure…
Most people would immediately recognize it as a trapezoid. There are two ways we can classify as trapezoid: the inclusive definition and the exclusive definition.
T(I): a figure with at least one pair of parallel sides
T(E): a figure with exactly one pair of parallel sides
Both definitions are legitimate, but they each lead to other differences in classifications.
For example, a parallelogram is just a parallelogram in T(E), but a parallelogram is also a trapezoid in T(I).
In T(I), even a square is considered a trapezoid!
Why can’t we all just get along?
We don’t need to argue over which definition is correct. They both are. So, this means when we speak about trapezoids, we must preface it with an agreement of which definition we are using...at least for that instance.
T(E) seems to have its origins in the 1500’s prior to the advent of calculus. When calculus came along, we began using trapezoids to estimate the area under curves.
Public Domain, https://commons.wikimedia.org/w/index.php?curid=647823
CC BYSA 3.0, https://en.wikipedia.org/w/index.php?curid=44244531
As the animation shows, some of the trapezoid slices begin to look suspiciously like rectangles. Aha!!! This is when T(I) got invented.
Common Core Standards
In 5th grade the standards are noncommittal on the trapezoid issue.
The 3rd grade standards also sidestep the issue.
But the Progressions Documents for the Common Core Math Standards make it clear that mathematicians prefer we use T(I).
Since a key component of the Common Core Standards is that students will be college and career ready, it seems the best trapezoid definition to use is the one that leads to calculus...T(I).
These are all trapezoids…
A trapezoid is a quadrilateral with at least one pair of parallel sides.
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Monday, May 2, 2016
What research says about the value of homework: Research review
Retrieved from Flickr with permission

Homework in elementary school has zero effect.
To read more about the confusing state of what research says about homework, go HERE.
Tuesday, April 5, 2016
Wednesday, March 30, 2016
Eureka Math Grade 2 Module 7 #eurekamath #engageny
Here are the tutorial videos for Grade 2 Module 7. For the rest of our Eureka Math resources, visit http://bit.ly/eurekapusd
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Tuesday, March 29, 2016
Finding your page in a HUGE pdf
Ugh...I have a 415page PDF and I need to find where Lesson 20 begins. How can I do this quickly?
The sloooooow way is to simply begin scrolling page after page until I finally get there more than 200 pages into the document. Avoid the horrible scroll of death.
The FAST way is to use the search field!
The sloooooow way is to simply begin scrolling page after page until I finally get there more than 200 pages into the document. Avoid the horrible scroll of death.
The FAST way is to use the search field!
Enter "Lesson 20" into the Search field. Make sure you include the quotation marks!
Then hit RETURN until you finally get to your desired page.
In this case, Lesson 20 finally began on page 274!!!
I got there in just a couple of clicks.
This will become an increasingly important skill as school districts gradually move to OER and other electronic and pdfbased curricula.
Wednesday, March 23, 2016
Remember the multiplication facts NOT memorize them
My son, a 4th grader, is still struggling with memorizing his multiplication facts. Being his math teacher dad, this kills me. Sorta.
While it is true that he is struggling to memorize his multiplication facts, my real concern is that I'm not entirely sure he understands what multiplication even means. When he comes across a fact that he doesn't know, I'm not entirely sure whether he even knows of a strategy for finding the answer. It appears that his only strategy is based on skip counting. For example when confronted with...
...his primary strategy is a laborious strategy based on skip counting. With his right hand he taps his fingers and counts to six. On his left had he then holds up a finger, indicating he has counted to six once. On his right hand he counts to 12, raising a second finger on his left hand. He continues counting to the next multiple of six on his right hand, while tallying on his left hand until has has reached 7 tallies on his left hand. Essentially, this is a physical version of skip counting...
I'm not sure if there will be any value in keeping the completed templates and reviewing them in subsequent days. Research shows that simply rereading the same information is an ineffective way to learn. I suspect the best idea is to keep the completed template for a day or two and then throw it away. It is likely that my son will have to redo the same fact another day, but that is okay.
While it is true that he is struggling to memorize his multiplication facts, my real concern is that I'm not entirely sure he understands what multiplication even means. When he comes across a fact that he doesn't know, I'm not entirely sure whether he even knows of a strategy for finding the answer. It appears that his only strategy is based on skip counting. For example when confronted with...
Clearly, however, this technique is fraught with error, since each hand only has five fingers and he is attempting to repeatedly count six on his right hand, while tallying to seven on his left hand.
I believe the problem is that his teachers have tried to move my son too quickly into memorization without first ensuring he has developed the number sense necessary to achieve fluency.
Basic fact fluency requires the presence of flexibility, appropriate strategy use, efficiency, and accuracy. (I copied and pasted this from somewhere, but for the life of me I cannot find where.) Clearly, my son's single coping strategy indicates he is neither flexible nor is able to select from a variety of strategies. For example, if one does not immediately know
then perhaps skipcounting will help...
or try repeated addition using "doubles" to increase efficiency...
or try using an area model with a little bit of the distributive property thrown in...
or connect the fact to a related known division fact...
Research shows that the way to develop fluency is to increase the student's number sense.
Memorization does not increase number sense. In fact, the lowest achieving students are more likely to focus on memorization, while the highest achievers focus on strategies based on number sense.
Using this as my premise, I created a system that will give my son more strategies to use for deriving the answer to the multiplication fact if he doesn't already remember it.
The plan:
 Show a flash card and silently count to five seconds. However, give as much time as he needs to answer the question. (Speed is NOT a goal at this point.)
 Place the card in one of two piles: "Correct" or "Not Correct/Not in Time"
 Continue until 3 flash cards are in the "Not Correct/Not in Time" pile
 Using those 3 facts fill in this template
Essentially, the template take those strategies I mentioned above and puts them in a row allowing one to compare and contrast the different strategies. Marzano has long shown that "compare and contrast" is one of the most effective teaching tools.
Of course, the commutative property allows us to think of the same fact in terms of sevens:
I'm not sure if there will be any value in keeping the completed templates and reviewing them in subsequent days. Research shows that simply rereading the same information is an ineffective way to learn. I suspect the best idea is to keep the completed template for a day or two and then throw it away. It is likely that my son will have to redo the same fact another day, but that is okay.
The goals:
 Focus on strategies and number sense NOT on speed
 Focus on REMEMBERING not on memorizing
Am I just being a helicopter parent?
Am I needlessly worrying about this?
Will my plan cause more harm than good?
Tell me what you think on the comment section below.
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