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Webinar Follow-Up: The Big Picture in Math: Four Concepts the Books Need to Teach

This discussion will focus on questions, key concepts, and resources from the Webinar, "The Big Picture in Math:  Four Concepts the Books Need to Teach." 


Please feel free to share any questions that you have for Dorothea Steinke from her webinar, 8/6/18, she will be responding to your comments from August 6 to 8, 2018.

Thank you for your participation!

Brooke Istas
Math and Numeracy COP Moderator

Tags: #numeracy


Dorothea Steinke's picture

There are 3 documents at this URL - the handouts for a couple exercises (like How many ways to make 8?), the PowerPoint slides, and the Resource Pages (easier to print in a Word doc than from the slide show).

Keep the questions coming. Let me know how well any of this worked in your classroom or tutoring situation.

Dorothea Steinke

Lafayette, CO



S Jones's picture
One hundred

There is a nifty book called "Routines for Reasoning"   that has activities that I think apply well to the concepts you talk about.   One of them is "Connecting Representations" where students figure out different ways the same idea can be represented. A post about it in elementary school is here   ... I like this one because it can be used in *any* level math.   

Kat Davis's picture

Thank you for your webinar today.  I wear several hats and I'm looking forward to applying those manners of thinking to my teaching classes....

* K-2 class for an afterschool program of refugees

* ESL 4 class to adults

* ESL/Civics class to adults

I'm looking forward to getting a copy of all the slides to see more of other avenues that you had discussed (while I was signing in for the webinar).


I do have another question regarding long division.  A 4th grader did a long division with boxes on the side.  Is anyone familiar with this form of long division???




Kathryn Davis


Dorothea Steinke's picture

Kat - I don't know what you mean by "boxes on the side." Can you draw what this looks like, put it in a PDF and email it to me?

That being said, I have seen students from other countries use different methods of doing long division. One young woman from Africa used the standard "division box" but put the answer inside the box. The Whole (what is being divided) went on the  top of the box in her method (where American long division puts the answer).


Jethra Kapp's picture

The long division method you are describing may be a method called "partial quotients." I used to teach 4th grade. It emphasizes place value. So, for example, a student could set up 3,081 divided by 13 like long division, but then he/she could do the division in parts. It might go something like this:

  • The student reasons that there are at least one hundred 13s in 3,081. Keep track of 100 on the right side (partial quotient) and subtract 1,300 from 3,081.
  • 1,781 is left, so there must be another one hundred 13s. Keep track of 100 on the right side (partial quotient) and subtract another 1,300 from 1,781.
  • Now there's only 481 left, so a student might start by realizing that there can't be another one hundred 13s but there are at least ten 13s because that's 130. Keep track of the 10 on the right side (partial quotient) and subtract 130 from 481.
  • Since there's 351 left maybe he/she tries twenty 13s. Keep track of the 20 on the side (partial quotient) and subtract 260 from 351.
  • Now there's only 91 left. Maybe the student tries five 13s because that's 65. Keep track of the 5 on the side (partial quotient) and subtract 65 from 91.
  • Since there's only 26 left, the student might realize that's exactly two 13s. Write the 2 on the side (partial quotient) and subtract 26 from 26.
  • Since there is now no dividend left, add up the partial quotients on the right side. 100+100+10+20+5+2 = 237 so 3,081 divided by 13 is 237.

The advantage of the partial quotient method is that it emphasizes place value and does not require the student to figure out each digit of the quotient correctly/exactly on the first try.

Edward Latham's picture
One hundred

As I read Jethra's example, I thought that Another benefit of the partial quotient approach is that learners begin to build their skills in estimation. Teachers often share that building estimation skills can be a challenge for them and this division process seems helpful in that regard. Learners may start off with 100s as Jethra shared, but after a few of these problems, some students start to jump to, "I wonder if 200 of those would fit in first cutting out one step for me..." Humans like to find short cuts and given enough time and practice with the partial quotient, I think learners will derive "short cuts" that approach many of our other algorithms. 

ecappleton's picture
One hundred

Hi all,

Here's an example of what Jethra is describing (in case it's helpful to see it) with a different division problem (425 / 8):

I learned yesterday that the partial quotients method is also called the forgiving method.


SharonShoe's picture

Thanks for sharing, Jethra and Eric.

This will be helpful to our students who struggle with division.  I teach this, but your representation clearly expresses the concept of estimating in division.

Abby R.'s picture

HI Kat -  I thought you might be interested in this blog post and lesson plan from Graham Fletcher about long division.

The lesson plan uses the method you asked about, I think. Glad you asked the question, I learned a lot from the answers to your post. 


Dorothea Steinke's picture

One of the comments that came in yesterday during the webinar was:

"We're quiet because we're thinking of how to integrate this. I'm looking to make these 4 concepts a way of covering numeracy with multiple levels in the room. Thanks!"

There will always be multiple levels in a room in adult education or community college basic math and pre-algebra.

One way to help establish "equal distance" (the Stage 1 to Stage 2 concept) in a whole-class activity is using toilet paper rolls: When you start at the edge, how many pieces do you have? None. You have to count the space/distance between the perforations on the roll. (This was explained in an Adult Numeracy Network newsletter in 2015.)

For Stage 2 students to advance to Stage 3, pairing up and describing to each other the parts and whole of large objects in the classroom (I mentioned clock and chair) starts the process.

For Stage 1 students "parts and whole at the same time" may not make sense because they have not established the "equal distance" concept. A start for whole-class participation in "equal distance" is to do "brain breaks" using cross-body movements. One example is: blink one eye at the same time as you snap your fingers on the opposite-side hand (left eye, right hand); then reverse. Start slowly, even verrrry slowly, at first. Work to keep the tempo even (steady rhythmic beat).  You can find other examples on the internet. The activity becomes a contest for some students - how fast can they go? BUT - the steady beat (the "equal distance" in time) is the goal.

Stage 1 students who are capable of gaining conceptual understandings in math can practice "equal distance" if they are willing to work with you one-on-one. That requires that they trust you first. It also requires a place where no other students can see them. "Walking" a large number line (think 1 foot floor tiles, or oversized post-it notes spaced equally on the floor) gets them feeling the spaces. (Again - walk at a steady beat.) There is a deeper explanation in the  2016 LINCS presentation: Number Sense: A simple tool that uncovers it.

Scroll down to DAY 5 - Learning Equal Distance for an in depth explanation and links to video samples.


Dorothea Steinke




Dorothea Steinke's picture

One of the questions that was asked during the Webinar last Monday was:

"Is there any advanced concepts that someone would have that would show they get this? In other words, if someone can do proportion problems or percentage problems, can we assume that they are at least stage 3?  (Just thinking about differentiating instruction for those)"

And that's the problem. Students can "do" proportion or percentage problems with learned crutches, like the percent triangle (or circle). Except that device only works for percentages less than 100%. So - No, being able to "work" these problems is not a guarantee that students understand the number relationships.

Students who use the percent triangle may arrive at the correct answer. They will likely have little understanding of why the answer is smaller than the starting number when multiplying by a percent less than 100%.  Example: $20 x 80% = $16

These students need work on multiplying and dividing by numbers less than 1. They may be stuck at the elementary level "multiplication makes it bigger" and "division makes it smaller" thinking. They are not thinking in terms of the number relationships. They can be taught to look for the "whole" - yes, a fraction or a decimal less than 1 can be the whole (the entire amount I have). They then need to ask: "Will I have more of the item (pencils, whatever) or fewer of the item when I'm done?" Again, looking at the relationship of the numbers, they can decide whether to multiply by the fraction/decimal (smaller answer) or divide by the fraction/decimal (larger answer).

To test for Part-Whole thinking, I again refer you to the Sept. 2016 LINCS discussion on Number Sense: A Simple Tool that Uncovers it.