Hello colleagues, The Math and Numeracy, College and Career Standards and Assessment Communities of Practice are pleased to welcome Dorothea Steinke to lead this week's special discussion on Number Sense and how to assess learners' understanding of this foundational numeracy skill.

Dorothea asks: Do your learners struggle with number sense? Do they have difficulty thinking of numbers in relationship to each other even in the most basic math operations? Do they struggle with mental math? How can practitioners help learners develop a deeper understanding?

In this week's discussion, participants will learn how to evaluate adult learners’ understanding of number sense through a simple tool, i.e., the Number Line Assessment for Number Sense. The way adults place five non-consecutive whole numbers (between 0 and 20) on an empty number line reveals their understanding of whole number relationships. Participants will learn about the two concepts of number sense that many adults lack. The presenter will discuss how to use and interpret the Number Line Assessment, a research-based tool that reveals whether students have these essential concepts. The assessment results indicate where to begin math instruction with a student. This assessment can be administered to a whole class in under 10 minutes! The discussion will also offer some remediation tools.

Reading Dorothea Steinke’s 2008 *Focus on Basics* article, “Using Part-Whole Thinking in Math” will give participants some useful background information on the concepts that will be discussed this week.

Cheers, Susan Finn Miller

Moderator, Assessment & College and Career Standards CoPs

## Comments

Welcome all.

I have three goals for our discussion this week:

1) Raise awareness of early math concepts that adults lack, and why

2) Introduce a way to identify students who are lacking these concepts

3) Suggest ways to lead students to see math as number relationships, rather than a mechanical process

(On that last one, I’d like to hear your ideas, too.)

Here is the plan:

Days 1 and 2 – a bit of background, introducing the assessment tool, and giving examples to review

Day 3 – the mechanics of administering the assessment (it is SO easy) and evaluating your students’ results

Days 4 and 5 – suggestions for remediation with students lacking the concepts

I’ll be breaking up my daily posts into shorter chunks so you can more easily find the information when you come back to it later.

This topic is an outcome of a project that started when I read Piaget’s statement that “Children do not understand number until they are 7 or 8 years old.” That is true, when you understand that Piaget means the mental capacity to think of numbers in relationship to one another. I’ll say more later on about when that capacity arises in normal children’s normal brain development, and how the timing of that brain development connects to difficulties our adult education students have in math.

The first point that needs to be made is that this evaluation instrument is research-based.

In the 1980s a group of math educators led by Dr. Leslie Steffe at the University of Georgia was working on a model to describe how young children grew into understanding number relationships. Their objective: To discover how children learn, rather than to take a set curriculum and look at how teachers teach. The researchers interviewed a lot of children over many years.

Their model got simpler as it evolved, until Steffe and his colleague Paul Cobb had a 3 Stages model. They published a very detailed description of children’s counting behaviors in 1988. If you have read the suggested background article, some of what I’ll be saying about their 3 Stages model will be familiar.

Using Part-Whole Thinking in Math – Focus on Basics, May 2008

Link: www.ncsall.net/fileadmin/resources/fob/2008/fob_9a.pdf

In early 2009, I had a student who was confused about putting fractions and decimals on an empty number line. So I asked that student to put a few whole numbers on a number line. The way she placed the whole numbers made me realize that she was at Steffe & Cobbs’ Stage 1 in thinking about the physical relationship or whole numbers. The assessment tool that I developed, and we’ll be working on during this discussion, grew out of what I saw with that one student. This is what the tool looks like.

Number Sense Assessment tool - link

https://drive.google.com/file/d/0B5XYMKmeShFAM0Mta2FOOTVqM2M/view?usp=sharing

Notice that the directions are very simple: only four sentences written right on the assessment. The directions can be said in any language. I was able to use the test with a whole class (up to 30) of adult developmental math students in under 10 minutes, even allowing for the slowest performers to finish.

Back to Steffe and Cobb’s 3 Stages model, which is what the assessment is based on.

At Stage 1, children and adults can count with one-to-one-correspondence. That is, they say the words in order and say one word for each object counted. However, for Stage 1 thinkers, each number is a separate object. Numbers are a label, like a house number. There is an order, but not a specific size. (Here is a graphic that shows what that “separate object” thinking can be compared to.)

Stage 1 NO Equal Distance Thinking link

https://drive.google.com/file/d/0B5XYMKmeShFAMldRZEFiR1lfU1E/view?usp=sharing

That is why young children – and some adults – count from one when adding two groups of objects. If they have a pile of 7 and a pile of 4, they count the 7, count the 4, put the piles together and COUNT ALL. Stage 1 adults are less obvious. Besides using fingers, I have seen adults make tally marks on paper for each number (smaller than 20 usually) that needs to be added, then count all the marks to find the total.

This is in contrast to Stage 2 thinking. At Stage 2, I know that all the numbers exist, even if I can’t see them. I know that all the whole numbers are the same distance apart on the number line.

Stage 2 EQUAL DISTANCE between whole numbers link

https://drive.google.com/file/d/0B5XYMKmeShFAdTlVWHF3UGg1NzQ/view?usp=sharing

I'm having to split up the length of the comments.

Stage 1 number line examples are at this link:

https://drive.google.com/drive/folders/0B5XYMKmeShFAcnNIbmY1VENMNDg?usp=sharing

Look for the SPACING of the numbers in the examples:

Stage 1 example 1 – Clearly evenly spaced regardless of the size relationship of the numbers

Stage 1 example 2 – Evenly spaced between 4 of the 5 numbers

Stage 1 example 3 – Given numbers (1, 2, 5, 12, 17) evenly spaced

All the numbers are almost equally spaced apart, regardless of their true size relationship. What Stage 1 lacks is the understanding that there is an “equal sized 1” between all the counting numbers.

By the way, these samples are all from adults.

For those of you who want to get deeper into using the Number Line Assessment tool with your students, now is the time to look through the BLUE packet of assessments (LINK below). These are all results from adults in a workforce HSE class. Pick out the assessments that have almost equal-sized spaces between all the numbers.

BLUE packet link:

https://drive.google.com/file/d/0B5XYMKmeShFAbjFSb1hqdlFYRkE/view?usp=sharing

The packets on GoogleDocs are the results that were reported in the MPAEA Journal of Adult Education in July 2015. That article is available in ERIC.

Evaluating Number Sense in Workforce Students – July 2015

LINK: http://files.eric.ed.gov/fulltext/EJ1072923.pdf

One more comment: There is a Stage 0. (Steffe and Cobb did not identify it.) These are people who cannot count with one-to-one correspondence. They may include adults who are developmentally disabled or who have suffered traumatic brain injury. Very rarely, these people may be totally normal in other respects but cannot tell you which is larger, 4 or 7. Once in a while, someone will put the numbers on the line in the order in which they are in the box on the assessment – from left to right, 17 12 5 2 1. Further testing (a one-on-one interview) is needed to tell if the person is Stage 0 in number sense or if something else is going on.

Now I’ll skip to Stage 3 briefly. These are the people who have the correct sense of number relationships. They think about the PARTS (the individual numbers) in relationship to the size of the WHOLE line right from the start of the assessment. Some even put a mark for 10 on the line. AND they have NO ERASURES. These people get the placement right ON THE FIRST TIME (or close to right –more on that tomorrow). Pick out the assessments from the BLUE packet that are

obviouslyStage 3. Leave the questionable ones until after tomorrow’s discussion. Here are a couple Stage 3 examples not from the packet.Stage 3 examples link:

https://drive.google.com/drive/folders/0B5XYMKmeShFAZmxZZEVUem1XUjA?usp=sharing

This is what you are seeing in the examples (left to right):

Stage 3 10 is marked – This qualifies the line for Stage 3, even though other spacing is off.

Stage 3 Allow SNARC (I’ll explain this acronym tomorrow.)

Stage 3 Correct spacing; 10 and 15 are marked

In summary: Every number line in your packet will have the numbers in the correct order. You are looking for the

distance relationshipsbetween the numbers. If you can, do this with a colleague. See where you agree or disagree.Now it’s your turn to ask questions. You might post your comments with the part of Day 1 that the question is about.

When we did this with some teachers, we had some people erasing... and noting that once they realized what the task really was they 'got it.' To me, that summarizes what we're trying to get our students to do :)

Welcome back.

Yesterday I gave you the indicators for Stage 1 and Stage 3 number line assessments. Today we look at some Stage 2 results, and how to tell the difference between Stage 2 and Stage 3.

To recap:

Stage 1 has no sense that there is an “equal sized 1” between whole numbers on the number line. Each number is a separate object with no size relationship.

Stage 2 has that understanding. Click on the link for a graphic that clarifies the conceptual difference between Stage 1 and Stage 2.

https://drive.google.com/file/d/0B5XYMKmeShFAVGVVSmFvTkJaNk0/view?usp=sharing

Stage 2 “counts on from” when adding. For example, to add 4 + 5, Stage 2 will say (or use fingers to show) 4... 5 6 7 8 9

That means Stage 2 understands addition and multiplication. Both operations put PARTS together to form the WHOLE.

What Stage 2 lacks is the understanding that the PARTS and the WHOLE exist at the SAME TIME. Stage 2 is likely to “add up to” when solving a subtraction problem. This is using addition to find an answer. It does not show PART-WHOLE thinking.

Stage 3 does understand PART-WHOLE coexistence. Stage 3 understands subtraction, division, place value and fraction relationships, and beyond. Click on the link for a graphic that clarifies the conceptual difference in part-whole thinking between Stage 2 and Stage 3.

https://drive.google.com/file/d/0B5XYMKmeShFAbGl4SzUwNkxjMWM/view?usp=sharing

Continue on to more DAY 2 posts for examples of assessments in Stage 2, and telling the difference between Stage 2 and Stage 3.

There is a greater variety in Stage 2 assessments than in Stage 1 or Stage 3. Part of the reason for that is that Stage 2 has a sense of “equal distance of 1” between the numbers, but the distance seems to be a personal, internal size. It is not the size of the “1” on the number line. That is why it is important to maintain the full-page-sized assessment. If the line gets too short, Stage 2 people with a smaller, personal “1” will look like Stage 3 people.

The easiest Stage 2 result to spot is when 12 is left of the point where 10 should be on the line.

example 1 https://drive.google.com/file/d/0B5XYMKmeShFAd092S1dvRDdkbTQ/view?usp=sharing

example 2 https://drive.google.com/file/d/0B5XYMKmeShFANjlWOV82Z2hEYWc/view?usp=sharing

A while back, I gave the assessment to 311 Algebra 1 students at a community college. Of those results, 81 (over 25%) were Stage 2. About a quarter of those Stage 2 results had 12 left of center.

Another major clue for Stage 2 is when there is excess space between numbers.

The person may write the 17 and 12 first, then skip to the other end to write the 1, 2, and 5. This leaves too much space between 5 and 12 (click the lnk).

https://drive.google.com/file/d/0B5XYMKmeShFAaXNaWWQyYzc5RWs/view?usp=sharing

The person may write the 17 first, then go left to right from 0 for all the other numbers. This leaves too much space between 12 and 17.

https://drive.google.com/file/d/0B5XYMKmeShFAbHhuRWJCc3YwRlk/view?usp=sharing

Sometimes the person will finish placing the numbers, look at the result, and try to fix the results. These are also Stage 2. Recall that Stage 3 (Steffe and Cobb’s statement) has the answer right ON THE FIRST TRY. Therefore, any erasure or crossing out and rewriting indicates the assessment is Stage 2. Sometimes it is hard to see the erasures. Look carefully, and always take the first attempt at placing a number as the true result.

example 1 https://drive.google.com/file/d/0B5XYMKmeShFAdXp2UlU3c0lZVzA/view?usp=sharing

example 2 https://drive.google.com/file/d/0B5XYMKmeShFASGRJeWl0cDhaS28/view?usp=sharing

Look at the examples in the BLUE packet again. Pick out the ones that follow these “rules”:

12 left of center

too much space between 5 and 12

too much space between 12 and 17

erasures or corrections

You will still have a few assessments that don’t seem to fit Stage 2 or Stage 3. Check the next post for DAY 2 about the SNARC effect.

SNARC is the acronym for “spatial-numerical association of response codes.” In our left-to-right reading direction, smaller numbers appear to have more space between (numbers to the left), and larger numbers to the right appear to be have less space between. (The effect is reversed for people whose language reads right to left, i.e. Arabic and Hebrew.)

What this means for the Number Line Assessment is that the spaces between the numbers seem to be smaller as one goes farther to the right on the number line (even when we know that all the spaces between whole numbers are same-sized 1.) This affects the assessment because even Stage 3 thinkers may have a little too much space between 17 and 20.

How much is too much? What distinguishes Stage 2 from Stage 3? Here is where judgment and experience with many, many number line results comes in. Look at these three examples.

https://drive.google.com/file/d/0B5XYMKmeShFAT3dlb3NzUVpIQzg/view?usp=sharing

In the

first example on the link, notice that the 12 is slightly farther left, and that the spaces between 17 and 20 are larger than the spaces the person had marked between the smaller numbers. Those indicate Stage 2 for me.In the

third example, notice that the line marked for 10 is slightly longer than the others. The person may have marked the 10 line first, and placed the other numbers in relationship to the 10 and the end points. This small indication of marking the central point as 10 leads me to call this example a Stage 3. There is nothing on the markings that specifically show Stage 2.The

middle exampleis the one on the fence. The number placement is the same as the third example – which I consider Stage 3. Is the slight leftward tendency of the numbers due to the SNARC effect, or does this person’s internal size of “1” just happen to match the size of “1” (equal spaces) on the assessment? This is where I say “2.5” and look for confirmation of Stage 2 or Stage 3 in the classroom. (For example, does the person have trouble understanding numerator/denominator relationships in fractions?)In short, don’t lose sleep over these questionable results. When you have looked at enough examples, your first impression about Stage of number sense will be right about 80% of the time. When you have a quick tool that can tell you where 80% of your students are conceptually in math, you save yourself a lot of time in planning appropriate instruction.

Go back and review your Stage 2 and Stage 3 number lines. If you are doing the activity with a colleague, get together to decide which ones are surely Stage 2, surely Stage 3, or Stage 2.5. The link below will give you the “answer key” – the exact location for each of the numbers. Use it as a guide, not an absolute.

https://drive.google.com/file/d/0B5XYMKmeShFALTF5M2E3c3BXVW8/view?usp=sharing

Now that you have the full story and some practice with the BLUE packet, try one of the other packets (RED, GREEN, or YELLOW). Let me know what you find.

RED packet link:https://drive.google.com/file/d/0B5XYMKmeShFAbnQ4MTRRT0s3S1E/view?usp=sharing

GREEN packet link: https://drive.google.com/file/d/0B5XYMKmeShFAS0p4ZVFMVDVOSXM/view?usp=sharing

YELLOW packet link: https://drive.google.com/file/d/0B5XYMKmeShFAVHN4d1NWRWlnMjA/view?usp=sharing

I'm here to answer questions you may have on the assessments in the packets. Post here on the comments, or contact me through LINCS e-mail.

Now that you have had a chance to look at examples of assessment results, let’s talk about how this works when you give it to your students. The instructions for what to say and what to do is at this link:

https://drive.google.com/file/d/0B5XYMKmeShFAbW9sVHZKZ3o1NFk/view?usp=sharing

The key is to tell students up front why you are asking them to do this simple task: “This will tell me how you think about numbers so I know where we need to start with math.” For many, the reaction will be, “This is it?” Just read the directions on the assessment, and do not give any clues as to what they are supposed to do beyond the directions.

Other teachers have told me that some students asked for a ruler – definitely no! I have seen a couple adults try to copy off another person’s results. Watch for that as your students take the assessment. Such a person may be Stage 1 or even Stage 0 (no one-to-one-correspondence in counting) if they have no sense of what a number line is all about.

If students start to ask questions (for example: can we make more marks? [than for the given numbers]), say something like "it's up to you." or just shrug your shoulders. (Experience has shown that students who need to make extra marks are less likely to be Stage 3.)

Allow no more than 7 minutes for students to finish. (Many will be done in 2-3 minutes, most will be done in 5 minutes.) Even if one or two students are still working, pick up the papers after 7 minutes.

GROUPING THE RESULTS

Of course, the assessment is only one tool in uncovering what students need to learn in math. Many Stage 2 students will be fluent in the processes of basic operations and early Algebra. In that study of 311 adult students in Algebra 1 (developmental math) at a community college, over 25% were Stage 2.

What would you think about sharing the results from your students (anonymously of course) with other teachers? Could that give a more general picture of adult education students’ grasp or lack of the “equal distance” and “part-whole” concepts? How would a much larger pool of results compare with the group that I reported on in the MPAEA Journal of Adult Education? (Here’s the link for the article again.)

http://files.eric.ed.gov/fulltext/EJ1072923.pdf

Tomorrow: Remediation. How do you re-introduce primary-grade-level concepts to adults?

Remediation. How do you re-introduce primary-grade-level concepts to adults?

Before answering that question, let me address the more important one:

Why didn’t they get the concepts in early elementary school?

BRAIN DEVELOPMENT

To answer, we have to look at normal children’s normal brain development. Researchers studying children’s ability to pay attention found that not until about age 8 could the children pay attention to more than one thing at the same time (almost like adults). Some children turn 8 during 2nd grade, but not until 3rd grade is everyone in the class 8 years old (for school districts with a late August or early September Kindergarten entry cutoff – currently the majority of school districts in the country). Understanding

Part-Whole Coexistenceis all about being able to keep two things in mind at the same time.That means, introducing “fact families” and the idea of “reversibility” of addition and subtraction before 3rd grade may be counter productive for many children. Children younger than 8 may learn rules about fact families but likely do not have the brain development to understand WHY the fact families work. (Younger second graders, those with late spring and summer birthdays, are especially vulnerable.) They think of math in terms of rules and process, rather than in terms of number relationships. They continue this thinking pattern into adulthood.

Also, many of our adult students did not grow up in “normal” circumstances. A big project at Harvard is documenting that children who grow up in conditions of “toxic stress”– including growing up in poverty – have delays in normal brain development. That means the physical brain growth to be able to think of

Part-Whole Coexistenceoccurs even later for these children.Ask your adults to think about these questions (the adults do not have to answer in class) – Were you among the youngest in your class? Growing up, was your family under a lot of stress? Did you always know there would be enough to eat? Was there a serious illness in your house (cancer, for example) when you were about that age? Was there a lot of fighting in your family when you were a child?

Explain the brain development I described above. You immediately remediate math anxiety for several of your students. They

aresmart. Theycanunderstand math. The way the curriculum is structured in the early grades did not match their normal brain development. With the number sense assessment, you (the teacher) have a better idea of what they missed and will help them get past that hurdle.There is another whole category of people who have trouble with math. These people have developmental disabilities (Down Syndrome, Turner Syndrome, fragile “X” syndrome, Williams Syndrome) or traumatic brain injury. People with developmental disabilities may have near-normal language skills, but lack the brain connections to understand number relationships. People with traumatic brain injury may be able to solve math problems that they could solve prior to their injury, but may not be able to learn new concepts. These two groups make up a very small portion (some say 5%) of the general population. The remediation tasks I will be suggesting may not make sense to people in these two groups.

Look for today’s second post for tools to help the majority of adults (not the 5%) who think they can’t understand math when they really can.

Dorothea Steinke

REMEDIATION – Part-Whole (Stage 2 to Stage 3)

You can teach adults the concepts they are missing by using concrete examples that have nothing to do with math (or seemingly so). The more personal the better. And the more physical the better.

Brooke Istas just posted another topic:What Math Looks Like in the Mind. That article seems to support what I’m going to say about the rhythm/math concept connection (see item 3 below if you want to skip ahead.)1) Before teaching part-whole, the true meaning of the EQUALS SIGN has to be addressed. Because this symbol is introduced before children can understand the “same as” relationship (two things at the same time again), many adults still view = as “turns into” (an operation symbol) instead of “same amount as” (a relationship symbol).

I use people’s names and nicknames. On the first day of class, I ask if anyone has a name they prefer to be called by. I write both names on the board (for several students). Then I write = between the names. This leads to a discussion that both names mean the same person. Then I switch to equivalent expressions like 16 – 4 = 6 x 2. (Notice that I use two different operations.) The comment is: both expressions mean the same amount (like the names). The mantra becomes, “Different name, same amount.”

How do you think this would work with your class? Give some feedback when you try it.

2) For part-whole coexistence itself, I start with a picture of a person’s face. Specifically, a face covered by a Halloween ghost costume. Only the eyes are showing. The question: What can’t you see because of the costume (or mask)? How do you know those parts are still there? (We expect a normal face to have them.)

Move on to objects in the room: What are the parts of the chair? the clock? the computer? Do the parts disappear when we talk about the whole object? Does the whole object disappear when we talk about the parts? The punch line: The PARTS and the WHOLE are always there AT THE SAME TIME.

The next step is breaking small-ish numbers into three or more parts: How many ways can you make 8 with three or more numbers? How about 14? Do you still have the entire amount when you have the small parts?

Again, try something like this with your students. Give some feedback here. How did your students react? Have used ideas like this before now? How did they work?

3) There is one more “non-math” way to reinforce part-whole coexistence. This only works one-on-one with a cooperative student. It uses music rhythm. Only attempt this when you have developed a positive, trusting relationship with the student. Why? Because the student will only appreciate the rhythm/math link of the part-whole coexistence concept

afterthey have had the breakthrough because of the rhythm experience.Think about this: The melody rhythm of a song (like “My country ‘tis of Thee”) is individual notes, the parts. The grouping rhythm (or meter) is the whole. Think of the “1 to the measure” beat of a fast waltz. Both rhythms exist AT THE SAME TIME in our bodies. (I have posted a short video of how I use music rhythm to drive home the “coexistence” of parts and whole.) If you use this approach be sure to use music in TRIPLE METER (3 beats to the bar – On Top of Spaghetti, Happy Birthday). Our bodies are used to moving in groups of 2. Moving in groups of 3 awakens a different understanding of parts and whole at the same time.

Watch this short video to see one way to present rhythm as parts and whole.

https://drive.google.com/file/d/0B5XYMKmeShFAXzRIUU10WkdlRUE/view?usp=sharing

This is from the articleWhat Math Looks Like in the Mindthat Brooke Istas just posted in the Math and Numeracy group:“Math, like everything else in my life, is something I interface with nonvisually,” said Scott Blanks, the senior director of programs for Lighthouse for the Blind and Visually Impaired. “I’m congenitally blind, totally blind from birth, so I have zero visual experience or memory.”Blanks says that, for him, working out math problems in Braille—which is how he learned many mathematical concepts in school—is a key part of his numeric thinking. “It didn’t matter whether it was an algebraic equation, or a ‘a train traveling at...’ story problem, having the information under my fingers was key to maximum comprehension,” he said. ...“There is rarely a tactile element in these [real life] situations,” he added. “Rather, my initial contact with the math scenario occurs auditorily.”In other words, if you can’t see it, you can HEAR math and FEEL math.

But (my two cents here) you have to feel that steady beat. More on this tomorrow when we’ll deal with Stage 1 to Stage 2 Remediation (equal distance).

Dorothea Steinke

I was at our state Bicycle Summit Monday... playing catch-up at work Tuesday... and Wednesdays I never have breathing room (We've had big staff cuts - so everybody's twice as busy...)

I started working with a student who is blind yesterday... I don't know his history but I know that when we were talking about lines and slopes he gestured the space....

Now, he's in our Intermediate Algebra course ... I work *mainly* with people in our "Math Literacy," Pre-Algebra and Transitions (Pre-pre-Algebra) courses. The Transitions teacher uses that numberline assessment and just a few other questions (mostly about how they feel/think about math in general, and a couple of problems from https://mathreasoninginventory.com/ -- almost nobody can tell us what 4 and a half times two is...) We see similar patterns in these students who struggle with number sense. We're having much more success than the previous version of the course which had been "practice arithmetic procedures and hope you figure it out" (I loathed and despised trying to drag students through long division who didn't have the concepts behind "borrowing" in subtraction)... and for one year was "here, use ALEKS to practice those procedures -- it's adaptive!" which was even less successful. (Basically, now it's "life" getting in the way that's the main issue...)

I've had lots more success w/ algebra when going through that drill for the meaning of 'equals,' and framing things in 'parts and wholes.' Spending lots of extra time with the concept that notzero/notzero = 1 makes a huge difference with fractions, too.

Hi Sue, Thanks for your comments here. It's interesting to hear how this type of assessment is working in the field. I realize I'm showing my ignorance here, but what is ALEKS?

Cheers, Susan Finn Miller

Moderator, Assessment & College and Career Standards CoPs

ALEKS

is the software package that's supposed to be the greatest thing ... "adaptive," etc.

Fortunatley, our instructors use it in higher level courses now for the practice element but some schools use the "emporium" model and take students who are behind in math and plop them in front of ALEKS and have it assess them and then throw problems at them... without instruction.

They do have nicely worked examples, but it is all about procedure, not concept. If there's good teaching backing it up it can work (tho' for some students it's still just an overload).

I was invited to try a very early version of what became ALEKS in 2000. I pretended to be a student who did not understand subtraction. For a question like 531 - 374 my "student" gave the answer 243. In other words, this "student" took the difference between the digits in each column, no matter if the larger digit was on the bottom. All ALEKS did was give me another problem and tell me to try again. I have complained about this ever since. I am not aware that a correction to the program has ever been made.

Dorothea Steinke

The idea that whole numbers on a number line are all the same distance apart seems like a no brainer for us. How could some adults not understand that?

It goes back to physical experience. Lakoff and Núñez in 2000 wrote a book

Where Mathematics Come From: How The Embodied Mind Brings Mathematics Into Being. Their premise: physical experience is basic to math understanding. In another book,Philosophy in the Flesh, Lakoff says, “Thought requires a body—not in the trivial sense that you need a physical brain to think with, but in the profound sense that the very structure of our thoughts comes from the nature of the body.”(Esther Thelen, who coined the term “embodied cognition,” was the first to promote the idea that intelligence is both made in and realized through physicalactionson the world.)One of the early concepts Lakoff and Núñez mentioned in

Where Mathematics Come Fromwas “motion along a line” as a precursor to understanding addition. When do we first move along a line? When we crawl as infants. Efficient crawling moves in equal increments. Efficient crawling is moving opposite hand and knee forward at the same time. This is also known as “cross-crawling.”However, not all babies cross- crawl. Some crawl with left-hand-and knee, then right-hand-and knee moving forward. (See the graphic of the two different crawling styles at this link.)

https://drive.google.com/file/d/0B5XYMKmeShFAX3JBSF9HNmFrWVk/view?usp=sharing

Some babies scoot on their bottoms, some roll over and over, some pull themselves up and walk without crawling first. And one study of infants found that the percent of babies who do not crawl correctly nearly doubled in the U.S. since parents have been putting babies on their backs to sleep to prevent SIDS (Sudden Infant Death Syndrome).

When the physical sense of the concept has been missed at such an early developmental level, how can we get back to it?

1) Get people to FEEL THE SPACES between the numbers on a number line.

This will seem like “Kindergarten” work, and many adults will balk. Until they realize they can’t do it.

1st: Ask the person to trace the spaces BETWEEN the numbers on a number line. (See the example at this link.) Be sure they keep a STEADY SPEED (use that metronome app again if you need to).

https://drive.google.com/file/d/0B5XYMKmeShFAdHFDY3B3Wi15amc/view?usp=sharing

2nd: Once you are certain the person is aware of the SPACES – the equal distance – then ask them to show a simple addition problem on a number line, 7 + 5 = ? for example. If the person is Stage 1, they are likely to draw the 7 from zero, draw the 5 from zero, and then (counting on their fingers, or counting the digits above the number line) draw the 12 from zero. This is true of children and of adults. (See the example at this link.)

https://drive.google.com/file/d/0B5XYMKmeShFAT3Noc0FKTFRTSUk/view?usp=sharing

Your goal is to have them draw, trace (feel) and recognize that adding is placing one of the Parts after the other on the line to arrive at the Whole. (See the example of Stage 2 thinking about the 7 + 5 = ? problem.)

https://drive.google.com/file/d/0B5XYMKmeShFANFRjRVNJS2lfUUU/view?usp=sharing

2) WALKING the number line.

Use chalk number lines outdoors, great big Post-it notes on the floor (with the numbers in sequence on separate notes), or carpet or tile squares. For 7 + 5 = ? have the person start by standing on zero, walk the 7 spaces, then walk the 5 more, and tell you how many SPACES they walked. Again, the goal is to have the person recognize the

spaceordistancebetween the numbers.If the person’s addition seems OK, try subtraction. Start at some number in the ‘teens and subtract to a single digit answer, like 15 – 8 for example. The Stage 1 person will be off by 1 in their answer because the/she counts the digits, not the distance. (I’ve had this “off by 1” result with Stage 1 adults.) The person will say “1” when he/she is standing on the 15, rather than advancing across the space to the line for 14 to say “1”. Making the person aware of the deep level of his/her lack of understanding may make him/her ready for the following experience of “equal distance.”

How do we give students the “equal distance experience” of crawling without asking them to get down on the ground and crawl? Remember, crawling is the experience of equal distance along a line, per Lakoff and Núñez. Music teachers have found that “cross-crawling” is linked to being able to keep a steady beat – an equal distance in time – in music performance. So we are back to rhythm again.

3) Help people feel/acquire a steady beat in their bodies. This again is a one-on-one activity. It does NOT involve crawling. It does involve crossing the mid-line of the body. Cross-crawling is an activity on both sides of the mid-line at the same time. So before continuing, check this short video that shows how some adults may avoid crossing the mid-line (not acceptable) when doing the exercise described later.

https://drive.google.com/file/d/0B5XYMKmeShFATjMzUXduR2JxTXc/view?usp=sharing

Again, only attempt this exercise

afteryou have developed a positive relationship with the person. Also, before attempting this activity, be sure the person is physically able to move his/her arms in the way described in the video.https://drive.google.com/file/d/0B5XYMKmeShFATWhQRVY1S3d3LXc/view?usp=sharing

Notice that there are 3 steps in training the person to move his/her arms across the body mid-line at a steady speed: 1) Instructor leads the action; 2) Student leads the action, with the instructor guiding to maintain a steady beat; 3) Student leads the action with the instructor mirroring the correct action AND counting at the steady beat.

This is not a “one and done” activity for a Stage 1 person. It will need to be practiced (for a few weeks, 5 minutes at a time) until the person feels comfortable with a steady beat at his/her own speed.

Then it is time to go back to tracing those single-digit addition problems on the number line. See if the person understands those number relationships differently. Show the person what he/she had done earlier. What is the person’s reaction?

What

whole bodyphysical experiences have you tried with your students in the classroom? How have they worked? What were the students’ reactions?This is my final "topic" post for this discussion. I've given you a lot of information to process. Let me know your questions and your own classroom solutions.

Dorothea

Dorothea,

Thank you for sharing this assessment and all of the related background information! The assessment is great for ABE settings because it is quick and simple and gives the instructor information about students working at a wide range of levels. Also, this sense of the number line is foundational to many other concepts, so it makes sense to assess it and work with gaps as soon as possible.

I am wondering about the students who ask for a ruler (see your post about how to give the assessment). Just by asking for a ruler, students are (perhaps) indicating that there is some order to their plan to solve the problem at hand. What if an instructor were to assess students first with your method and then with the option of using a ruler? This may be diverging from your original intent with the assessment, but it may make some other interesting things about the students' thinking become evident, especially if you were able to follow up with individual interviews as to why a student's first line (no ruler) looks different from the second line (ruler). Some things that come to mind are...

Again, I believe this is a tangent from your original intent, but just because of the other interesting points that may come up, I believe I will try this both ways with my current (small) class.

Hi Amy. I had not thought of the things you brought up. I'm mentioning them out of order, based on experience giving the test. I've seen about 1500 examples now from 3rd grade to adult (that includes an entire middle school not in my state).

Students using the ruler would have to convert the 20 spaces they wanted to some fraction of an inch. (The line is about 9 inches long.) That would prove interesting because I suspect the people who felt they needed a ruler would be Stage 2. Stage 3 people have a sense where the middle of the line is for 10, or they might fold the paper to make a crease where 10 is (one student did this). The students asking for the ruler may assume that the equal spaces will be 1 inch long. Are you willing to report back when you try this?

Students who do not remember number lines I think would struggle with or without a ruler. The ruler would be a distraction to the perfectionist Stage 3 student who wanted to get the line in

exactlythe right place.Your first comment about lacking confidence and not even trying makes me think of Stage 1 students. Those people

doput marks on the paper as quickly as possible, especially with the hand indication (in the directions for the assessment) that the mark for each number is supposed to go up and down across the number line. (Of course, not everybody does put a mark for each number. Some just write the digits.)Most of the time, the comment about the assessment is, "This is it?" Students can't believe it is so simple.

Do let us know what your current class thinks about the assessment, and how they feel about the two different ways, first without the ruler and then on a separate paper with the ruler.

Dorothea

I've seen a lot of students who are simply so afraid of math because it's math that they would respond as a "Stage 1" -- but, if the task were in a different context (now that might be fun to dream up) where ... it was, say, a craft situation that they had to figure out (not A MATH CLASS THING!!! A MATH TEST!!) ... they'd look and think and might just come up with something better.

This week I've been working w/ a student w/ visual impairments... and I noticed the difference between when we were just "working on problems" on the computer and ... when the problems were A QUIZ. Welp, since I was also trying to help five other students, I told him I'd be back in a few minutse... and happily, when I came back ... he'd relaxed a bit and the problem that he was doing All The Wrong Things with ... was suddenly achievable. He noted "I guess I needed a ten minute nap!"

Still, many of the students who are that anxious have been anxious for years... so they didn't get out of Stage 1.

We want to thank Dorothea Steinke for sharing her expertise with our communities. Over the past week, Dorothea raised awareness of the early math concepts that many adults lack and explained why this is the case. She explained a process and a tool than can identify learners who need to develop these concepts, and she suggested ways to lead students to see math as number relationships, rather than a mechanical process, which is key to learners' moving forward to reaching their goals.

Dorothea, thank you kindly for so generously sharing your time and your expertise with all of us and for providing practical guidance to teachers on how to identify and address adult learners' needs.

Cheers, Susan Finn Miller

Moderator, Assessment & CCS CoPs

I would like to echo Susan's appreciation of Dorothea for sharing this information in a way that was easily understood. I loved all the examples.