Ladies and Gentlemen,
It is my honor to introduce Dorothea Steinke, an ABE/GED math teacher with a BA in education. She has been researching and teaching the Part-Whole concept in math since 1993. She currently teaches GED math and developmental-level math at Front Range Community College, Westminster. She has recently completed a Student Learning Research Project on number sense in developmental math classes at her campus.
Dorothea will be asking and answering questions that list subscribers have on "Number Sense in Adult Learners". This discussion will focus on number sense and looking at strategies for developing this fundamental math concept in adults.
Please post your questions or comments for Dorothea to the list by emailing firstname.lastname@example.org and remember to change the SUBJECT LINE in your email if it doesn't relate to the topic being discussed.
Dorothea's article, "Using Whole-Part Thinking in Math", appears on page 1 of the Focus on Basics issue at the following link: http://www.ncsall.net/fileadmin/resources/fob/2008/fob_9a.pdf
Welcome Dorothea Steinke!
I am excited to have you as a guest moderator because as an ABE/GED/Developmental Math instructor I feel I spend a great deal of the time teaching fractions. These concepts seem so foreign to most of the adults that I teach. Students will avoid problems that have fractions in them even if they know the underlying concept being introduced in the lesson. I am sure I am not alone in this observation and look forward to discussing how to help learners achieve a better understanding of fractions so that they aren't avoiding the problems that have fractions in them.
Brooke Istas, Moderator
LINCS Math and Numeracy List
Number Sense - Monday
Number sense. Like explicit material, we know it when we see it.
There are many different definitions of it. The more important point is - How do students acquire it? Or, more relevant to this discussion, what did our adult students miss?
Brooke had suggested an article I wrote as background reading to this discussion. Here's the link again, in case you missed it.
In the article I mention work by Dr. Leslie Steffe and his colleagues at the University of Georgia in the 1980s. Over several years of observing how children learn, they came up with a description of three stages of number sense. They did this based on the children's behavior when answering math questions involving small numbers (less than 20) and no paper and pencil.
Dr. Steffe and Dr. Paul Cobb (now at Vanderbilt University) wrote the book that started me thinking about math their way. The book: Construction of Arithmetical Meanings and Strategies (1988). Our adult students exhibit the same three stages of number sense.
On page 3 of the Focus on Basics article, I describe the behavior difference between those adults who think of numbers in terms of EITHER parts OR whole (no part-whole thinking) and those who think of numbers in terms of BOTH parts AND whole at the same time. I'll repeat that description here.
Let's suppose the question is in the form: 72 - 8 = ?
(The question mark represents the number that must be found.)
EITHER / OR thinking
When I think of parts and whole in either/or terms, I have trouble with "mental math." If I try to subtract eight from 72 in my head, I repeatedly guess what the number might be: 61? 65? I have to add eight to every number I guess until I find the correct number. I use this same "guess and check" strategy with word problems. It takes me a long time to find the right answer.
BOTH / AND thinking
If I have a sense of part-whole coexistence, I do the same problem (72 - 8 = ?) one of several ways: I may mentally break 72 into 60 and 12 and keep track of all the parts. I think "12 minus 8 is 4", and add the four to the remaining 60 for my answer. I may think, "8 and 2 is 10. 72 minus 10 is 62. Add back the extra two I took off. I have 64". Again, I keep track of all the parts. Whichever way I do the problem, I know the answer is right and I can tell you why.
The person with EITHER / OR thinking is at Steffe & Cobb's Stage 2. The person with BOTH / AND thinking is at Steffe and Cobb's Stage 3. Stage 3 has what we consider "number sense."
Those of us who drive a car use Stage 3 "both-and" thinking whenever we are behind the wheel. We have to be aware of what our vehicle is doing at the same time as we are aware of the traffic around us. Many of our students don’t use "both-and" thinking in math.
The Focus on Basics article gives some suggestions for helping Stage 2 "either-or" thinkers move toward Stage 3.
For now, I'd like to address Stage 1 thinkers.
Stage 1 behaviors that Steffe and Cobb saw in children are (and this is my summary):
Must count from 1 every time when adding two groups of visible objects
Can ONLY count visible objects
Stage 1 number sense may sound like young children. True. However, Stage 1 thinkers are also a part of the population of adults we see. In fact, a research project I did in 2010 at the school where I teach suggests that about 9% of the students in our basic math (whole numbers, fractions, decimals) and pre-algebra classes are Stage 1 thinkers.
These students can get a high enough score on the standardized placement test to get into our classes. They know the rules for manipulating number symbols. However, their understanding of number relationships appears to be at the pre-school or primary-grade level.
That understanding is:
Each number is a separate object with no size relationship between the numbers.
At Stage 1, the relationship of the numbers is much like that of fruit. The names all belong to the same category. That's it.
I'm talking normal, healthy adults here. These are not people with developmental delays.
These are the people who are "off" by 1 when subtracting. When given the question 14 - 6, they count off six numbers "14 - 13 - 12 - 11 - 10 - 9" and give the answer as 9. They count the digits instead of the spaces between the numbers on a number line.
What Stage 1 number sense adults (and elementary children) seem to lack is the physical sense of "and 1 more" - that same-sized 1 between any counting number and the next one higher or lower. One way to address this is walking a large number line and being aware of the same-sized space between the numbers. Be sure the number line starts at ZERO!
You can mark off floor tiles (if yours are the same size). You can draw equally-spaced chalk lines on the parking lot. You can use oversize post-it notes spaced about a foot apart (from front edge to front edge). And you may need to use cross-body patterning. (I'll talk about the why and how of that next time.)
When working the number line, be sure the student is focused on the SPACE BETWEEN the numbers, and is aware that the space is the SAME SIZE between successive numbers.
I have had up to three Stage 1 people in a math class of 24. How often have you seen the Stage 1 behaviors I mentioned?
I love these videos that show "math sense" and "makes no sense." I don't know what they are good for, though--students usually don't get them, and math teachers don't need them. Still, a laugh is a laugh...
Making Change: http://www.youtube.com/watch?v=nmykxv1yIic
Comparing Fractions: http://www.youtube.com/watch?v=am8bwMwQICo
Division with Remainders: http://www.youtube.com/watch?v=HGWwFLJxpdA
Fractions, Distance and Time: http://www.youtube.com/watch?v=4zS1Bou4sjU
Here's a great routine from Abbott and Costello: http://www.youtube.com/watch?v=rLprXHbn19I.
The article clearly described the author uses in her teaching. The part-whole approach may be the key to help my students conquer the math that they dread. The concept of stressing the symbol = is related to equality between the parts of the problem should help them "see" the problems differently. The description of using a name and titles for one person is very concrete and equally familiar to students.. btec/GED/SFCC
I saw it twice last week from students at the Transition (pre- pre-algebra) level; they counted everything. No, they didn't know their times tables; they used a chart, and that included for the ones times tables. What a complete exercise in futility "teaching" them long division is!!
Now, I would say that the lack of number sense is more like the pornography that is being read behind the guise of a more socially acceptable paperback. Let me manipulate these symbols -- but please don't ask me why!
And right now, I'm trying to work up (and I've committed to it, so it's gotta happen ;)) an hour's presentation on the fun of positive and negative integers. I absolutely positively want to start with that critical concept that our friend the equals sign no longer means "mess with this puzzle, do the dance steps you're told, and voila! stick the ANSWER on the other side of the door!" and graduate to "lots of different ways to describe what amounts to *exactly* the same amount.
I'm thinking of starting with good ol' positive numbers and just having people find as many ways as they can to get an answer of ten... and then instead of stopping there (as I was originally taught with that activity), to go on to show that we could match *any* combination on any side of the equals sign and it would still be a statement of absolute truth. I think I shall have small manipulatives handy but not make a big deal of them... but use them to show that no matter how I arrange 'em, they're still ten. (I might even tell the story of my friend who is horribly math phobic and desperately wanted to help her friend work at the produce stand... but could not ever remember what 7 + 4 was... and we played with it with manipulatives and counting and all sorts of things... and what made the light go on was when she broke down the 7 into 3 and four... because 7 was just too big for her to hold in working memory to add 4 to... ) Then I'm going to get out that number line... and see if I can invent a drama, and do some storytelling for the verbal learners... you know, those oppositional negatives that decided to form their *own* team...
For an interesting way to have students work on the concepts of positive and negative integers, check out the article, "Seeing is Believing" from Focus on Basics. It's written by three thoughtful and brilliant math adult educators who organize the Math Exchange Group in NYC.
Here's the link: http://www.ncsall.net/index.php?id=316
When teaching integers, I give the students the rules, but then I also try to make sure they understand what is really happening in the problem. For the students who say, "Just tell me the rules," it works, but then they can take a step back and wrap their brain around the situation in each problem. Using the idea of money really helps too!
Here is the addition part of a handout I use with them:
INTEGERS - ADDITION
Rule: POSITIVE + POSITIVE = POSITIVE
(+12) + (+10) = (+22)
Think of the problem like this:
(+12) You have $12 dollars in the bank.
+ and then
(+10) You deposit $10.
(+22) You have $22 in the bank.
Rule: NEGATIVE + NEGATIVE = NEGATIVE
(-2) + (-8) = (-10)
Think of the problem like this:
(-2) You borrowed $2 dollars from your friend.
+ and then
(-8) You borrowed $8 more from your friend.
(-10) You owe your friend $10.
Rule: POSITIVE + NEGATIVE = TAKES THE SIGN OF THE LARGER NUMBER
(+5) + (-15) = (-10)
Think of the problem like this:
(+5) You have $5 dollars in the bank.
+ and then
(-15) You wrote a check for $15 (overdrawing your account!)
(-10) You now owe the bank $10 (plus service fees, HA!)
Rule: NEGATIVE + POSITIVE = TAKES THE SIGN OF THE LARGER NUMBER
(-20) + (+2) = (-18)
Think of the problem like this:
(-20) You owe your friend $20.
+ and then
(+2) You pay her back $2.
(-18) You still owe her $18.
Hope this helps a little.
Kaye, great idea to use money to teach negative/positive functions.
I like using money, too. And I'm finally "getting" the idea of the "equal" sign. I'll use the "+ ...and then and =..now" in my integer lessons. A real addition to the "rules" approach which I sometimes mix up if I'm not thinking or haven't worked it for a while-yeah, I'm somewhat challenged.
My plan is to start the chat with different colored legos (because we have them; otherwise I'd use chips ;)) and use that approach as well as a big fat number line going up and down.
Right now, I'm thinking:start with 10 of the square lego blocks (the ones with four nubs on 'em), green. Talk about my biggest big idea of math, that it's not that the rules change, but that we expand our understanding -- the same way a kid first thinks all furry things are dogs.
Then I'll have them figure out different ways to express 10, and talk about and show about the "different names for the same thing."
Then it'll be a big ol' verticle number line of "elevation"and a big ol' picture of a mountain and the ocean and sea level, next to each other... for that negative concept.
Then turn it over sideways to show it as a number line like they're probably more accustomed to seeing.
Then... a coordinate grid.
Then it will be storytelling time... and the drama of the Positives and Negatives. Then break out the yellow "negative" legos...
The trick will be to make sure the connections being made don't just fade away and turn into symbols when I get into the adding and subtracting, but I think I'll be able to manage it by having them mess with the legos and the number lines.
But that's for tomorrow! Now it's time to ride home and fiddle with actionscript :)
I know a math teacher who worked in a city with a serious gang problem. She capitalized on this when she taught addition of positive and negative numbers. She divided her class up into gangs - half had a plus sign drawn on their foreheads, the others had minus signs. The scenario went: whenever two of you from opposite gangs meet, you kill each other. Then she had them get into different gang configurations and find out who was left after the rumble; positives, or negatives, and how many. This relies upon a certain context, but it might be tweaked into something you can use.
When teaching part-whole thinking, do you use manipulatives (otherthan fingers) for hands-on learners?? If so, which ones do you find work the best??
Cassi Birk email@example.com
I use the Singapore math bar model. The bar model is drawing of bars (or rectangles) to denote part-part-whole. It is a great visual tool!
I don't use manipulatives. I use pictures to show my students the part-whole thinking.
Sandra E Garcia
Several of you are mentionig manipulatives. I'm responding to several comments at once.
In the Wednesday posting I offer comments about manipulatives. Here is the key idea.
It is important to show BOTH Parts AND Whole at the same time when using visuals with EITHER / OR thinkers.
Manipulatives can actually get in the way of moving students from Stage 2 to Stage 3.
Read Wednesday's post and get back to me with more comments and questions.
Yes, I most definitely have several learners at stage one. We have done a lot of measuring with rulers, and I was surprised how many students did not know to start at 0; we've also spent quite a bit of time talking about the size of whatever's being used as the unit; whether it's an inch or a paddle or a pace. Part of the challenge is that in our current system we have an ABE course that is roughly equivalent to grades K-2, while the next is grades 3-5, which is a LOT to cover considering that when children do this their brains absorb and memorize much more easily than we do as adults.
I'm going to defer a more detailed response to your question to my Wednesday post titled "BOTH Parts AND Whole." I have no preference for any particular kind of manipulative. Last week I used the plastic tabs that close bread bags when working with some elementary-age children.
I'm unclear on which behavior you are referring to when you mention using "fingers." We all continue to use fingers to some extent when we keep track of how many times we have counted.
If you mean people who prefer hands-on/tactile/kinesthetic explanations of number relationships, that is one thing.
If you are seeing people who have trouble making sense of numbers unless they have some object or symbol to count, that is another matter. Do these people "see" the spaces on a number line or do they count only the digits?
Now that I'm reading this, I think I see a correlation between students who must use fingers or counters in order to add or subtract and those who can only remember their times tables when said in order from the beginning (often the same student). It seems like math is all just big meaningless lists of numbers in the student's head. Almost like needing to look at the
house number on every house on the street as you go by (502, 504, 506.) instead of being able to say, "Hmm, I'm at 502 and I need 613, so I need to go the next block and cross the street." I guess I always assumed that once you learned your multiplication tables, you could use them, but now I see why that might not be true.
Sue -Your example makes a good point on the difference between concepts and skills. If the conceptual understanding of number relationships isn't there, the process it rote. This goes as far down as addition. If the "and one more" concept isn't there (the concept between Stage 1 and Stage 2), addition only makes sense when there are objects or symbols to count.
Number Sense Tuesday - March 22, 2011
If you had a chance to read the Focus on Basics article, you may have noticed that I start talking to people about part-whole thinking with a very concrete, very familiar example: the human face.
Whatever we know, we learned it by starting from something physical. Then we moved on to recognizing it in pictures - even drawing pictures ourselves. Finally, we arrived at an abstract mental representation of that original physical reality. That is a quick summary of Steffe and Cobb's 3 Stages: Stage 1 - concrete; Stage 2 - representational; Stage 3 - abstract.
Yesterday I mentioned "cross-body patterning" as a way to help people with Stage 1 number sense move on to Stage 2. I'm saying this based on two different sets of information.
1) Long-standing research insights with elementary-age children that those who cannot cross the midline (i.e., comfortably move an arm across the midline of the body to the other side) do less well in school than those who can.
2) Anecdotal evidence from music teachers regarding helping students develop a sense of steady beat.
For, you see, the steady melody beat of simple children's songs (think TWINKLE, TWINKLE) is another model of the EQUAL DISTANCE CONCEPT that takes students from Stage 1 number sense to Stage 2 number sense. The number line model (Monday's discussion) shows equal distance between the numbers in SPACE. The quarter notes in the melody of TWINKLE, TWINKLE (or the Alphabet Song) show equal distance in TIME. Both provide a physical experience of the concept of EQUAL DISTANCE which is what Stage 1 thinkers lack.
The late Eloise Ristad described how she worked with her piano students with poor rhythm. When I have students with a ragged sense of rhythm, I ask them to show me how they crawl. It's pretty undignified, but you might try it. See if you crawl with right hand and right knee together, or whether you have a cross-crawl pattern of right hand with left knee and vice versa. People with a strong sense of rhythm usually have a strong cross-crawl pattern. People with fuzzy rhythm, on the other hand, are apt to crawl by shifting weight from the right hand and right knee to the left hand and left knee. pp. 29-30, A Soprano on Her Head, 1982. For a picture of cross crawling versus same-side crawling, go to: http://numberworks4all.com/Crawling.html
I do NOT ask students to crawl. We sit face to face in chairs. I first ask a student if he/she is physically comfortable raising one arm over his/her head. (Some of us have physical limitations that are not visible.) I do the action with the student, but as though I were a mirror* for him/her. Next, I ask the student to cross that raised hand to the opposite knee. I do the action as I say it.
Some time back, I got to this point with one math student. Instead of crossing to the opposite knee, her natural reaction was to put her hand on the knee on the same side of her body. It felt strange to her to put her hand on her opposite-side knee. It took concentration.
Working with her also meant doing each movement at a slower speed than I would for myself. Some of our learners have a slower-than-average body speed. Average adult body speed is 72 to 80 beats per minute. Children usually grow into that adult body speed around age 8. Some adults, however, have a body speed as low as 60 beats per minute. Try it for yourself. Tap to the ticks of the second hand on a clock as you say (or sing) the words to the Alphabet Song (or Twinkle, Twinkle). Did a speed of one beat per second feel too fast or too slow for you?
My math student was able to do a steady-beat cross-body pattern** of left hand to right knee, then right hand to left knee when I was modeling it and she was watching me. We were going at her slightly slower speed. As soon as I stopped modeling, even though I was saying the words of a chant*** in rhythm, her hand-to-knee motion became unrhythmic. We worked on it, and I told her how doing a physical action could help her body to understand what "equal distance" means.
It is true that many good mathematicians can't keep a steady beat. They may be strong visual learners who "get it" from the picture. They may have a strong sense of space and sequence. People learn in different ways.
However, my experience with adults is that if they lack the EQUAL DISTANCE concept with numbers, they also have weak rhythm and, often, trouble crossing the mid-line of the body.
Our students may need to start from the physical to establish the concept. Once they have the EXPERIENCE of the concept, then they can apply the concept to content.
Not every ABE/GED/developmental math student who lacks the equal-distance concept is willing to try this. Can you think of a student who might be willing to try a new way of experiencing number relationships? What variations on this suggestion would you try?
*Your pattern should be the mirror of your student's pattern if you try this.
You are facing the student. His/her right hand moves across as your left hand moves across.
**The pattern is actually a four-count pattern at a steady speed.
1) Raise right hand
2) Cross right hand to left knee
3) Raise left hand
4) Cross left hand to right knee
REPEAT at the students comfortable body speed for a minute or more.
***Try the words to Jingle Bells
I find the idea of equivalent math expressions using easy subtraction with adding back to be a concept I definitely will try for some of my lower stage 1 math students. Virtually all students can subtract ten from a number, so the idea of 72-8 as 72-10, then add back the two to get 62+2=64 seems like it would be a much simpler method for answering the problem.The idea of counting spaces on a graph, rather that points from beginning to end (which of course is always off by one) or spaces between lines on a ruler rather than lines from beginning to end of a measurement (also off by one) both seem inordinately difficult for stage 1 math students to unlearn and relearn. Floor tiles serve as an immediate example, but the lesson is often forgotten by the next class session. I am still somewhat unclear on how the discovery of a weak sense of rhythm or poor cross-line body patterning can be used to resolve problems with the concept of Equal Distance. Are you suggesting the student practice these cross-line exercises to improve their math conceptualization of Equal Distance? I fear many of my students would feel they would be made fun of and would refuse to do these activities. Have I misunderstood your point?
Wow--this is interesting. In another life, making home visits to enhance infant/ child development, I was to encourage giving floor time. The caregiver was to set out items to entice crawling so the cross pattern would evolve.
Dean S sandra ostheller
I don't use manipulatives. I use pictures to show my students the part-whole thinking.
Sandra E Garcia
I don't think bringing in crossing the midline or music is going to work today, with this student... but let me tell you that it's hard to teach you graphing lines when you are starting your counting with 1... and yes, when she made a number line for interval notation, she put negative 15 on the left side of zero .... and negative five on the right side of zero. She tells me that she's always been very good at math, and it's just that this teacher has made her lose her common sense and to always second guess herself -- and I am absolutely positive that there's truth in that statement. When we teach them from beyond their sense of the numbers, then our rules contradict "good sense" so we just memorize those new rules even when they seem to be nonsense. It makes **very** good sense to keep "math class rules" away from "real world rules" then because you *will* second guess things. But... when there's down time... I think I will try it...
I find the idea of equivalent math expressions using easy subtraction with adding back to be a concept I definitely will try for some of my lower stage 1 math students. Virtually all students can subtract ten from a number, so the idea of 72-8 as 72-10, then add back the two to get 62+2=64 seems like it would be a much simpler method for answering the problem.
I realized while imagining using manipulatives that yes, they often reinforce the idea that you can't really have a problem and a solution at the same time. If I rearrange manipulatives to "make 10," I only have the ten one time. I'm thinking of having *lots* of manipulatives. And, by the way, I watched the video about 'making change.' I was wondering whether math teachers would really "get" that an awful lot of my students really, honestly *do* think and act like the guy with the calculator when it comes to math. The movies make him look bad enough that I don't think I'd want to inflict it on my students -- they're fragile enough! The integer song, though, might get some air time. Looks like I'm going to want to set things up for a multimedia presentation ;)
I've just shown a woman with pennies why if a third of x is 10, x is 30. We weren't getting *anywhere* at first... then I told her about these people I know online from whom I was learning some new ideas... and talked about wholes and parts. Yes, indeed, she said, she had been trying to put the numbers together, despite being on opposite sides of that equal sign. Yes, it made things clearer and she made the connection -- entirely because she had a better grasp that the 10 was the *answer* already and not part of the question. She was able to transfer the concept and figure out that if 1/5 of something was 7, to figure out the whole thing you'd multiply by 5. I'm hoping that she was understanding beyond the procedure.
I didn't keep going to the next problems where 7/2 x = ... oh, some fraction or other. She had, at the beginning, said that looking at that problem made her think she should make it into a mixed number. The whole concept of "reading" a math problem adn figuring out what it means is new to her; however, the relative speed with which she embraced the "whole and part" idea makes me think that she is perfectly capable of doing it -- unless we speed along so fast that she doesn't have time for it.
Alas, yesterday's ladies aren't here today... I hope they'll come back tomorrow. (They're in the next higher class.) Now, I just have to figure out how to put lessons together while I'm riding my bicycle, since I really wanted to do a long ride on my day off Friday :)
Susan - Thanks for your comments. I'm going to use them as a sounding board to clarify the gulf between the 3 stages of number sense. And you have not missed my point about rhythm improving number sense.
If your students are truly Stage 1, then 72 and 8 , 62 and 2 an 64 are all totally different items with no size relationship between them, They may be able to use the strategy you are suggesting, but it will be all rote.
At Stage 2, when a student has a sense of the "equal distance of 1" between counting numbers, they understand the size relationship between the counting numbers. They are comfortable with addition and multiplication - putting pieces together.
Subtraction needs Stage 3 thinking. In subtraction, the WHOLE is given and one PART is given. I can find the other part because I understand that the parts are always there inside of and at the same time as the whole (both/and thinking).
The kind of thinking you describe in equivalent math expressions is just the kind of question I use to distinguish Stage 2 from Stage 3 people when I interview them. Stage 2 thinkers can learn the process you describe. True Stage 1 students likely will have no idea what you are talking about.
As far as counting spaces instead of lines - You are so correct about Stage 1 having trouble grasping the difference. That is why physical actions come into play.
Lakoff and Nunez wrote a book in 2000 that gives their take on how we develop math concepts physically.
Where mathematics comes from : how the embodied mind brings mathematics into being
They mention crawling as forward motion along a line. They say it is our physical model for the number line.
The concept of Equal Distance can be experienced in space (moving forward along a line) or in time (rhythm). It is the same concept experienced two different ways. Yes, rhythm - a steady beat - is another avenue for developing the Equal Distance concept.
As far as working with a student on crossing the midline, it has to be started one-on-one in a way that the student feels safe. If you have developed a relationship of trust with a student, then that student is open to a new way of approaching numbers.
I begin by telling a student that there may be something that he/she missed back as a pre-schooler. Most important, I tell the person, "It is not your fault that you missed it." I tell him/her, "If your body can do the concept, your mind can understand it."
After we have done about a minute of the pattern of crossing the midline to a steady beat (use a chant or jump-rope rhyme), I will ask the student, "Does it feel different to cross your hand to the other side of the body? Does it feel more difficult for one hand than the other?" Generally it does. That is the point at which the student buys in to the activity.
The student need not become proficient in rhythm to do cross-body work. The point of the exercise is experiencing the concept, which then translates to number relationships. It takes a couple weeks before the idea settles in.
That means doing the exercise briefly as a warm-up at each class session for half a dozen sessions. You can call it Jazzercize. You can call it karate training (step forward with one foot, cross the opposite arm to the side you step forward with). You can involve the entire class as a "brain stimulating" exercise (see John Ratey and SPARK http://www.johnratey.com/newsite/index.html about the connection between brain development and exercise).
You don't need to be a musician to use rhythm. You should be able to keep a steady beat to music yourself. Do you tap your toe or sway along with your favorite songs? That's all you need. You may want to use a recorded song with a slower beat to do the exercise. Or you may want to count to a beat of 8 (repeatedly) at a medium to slow speed when working with a student.
Yes, some students are unwilling to try crossing-the-midline exercises. I let them be. An unwilling student is not going to learn anything.
My mind was reeling with thoughts of what I've done in the past and ideas of how I can do it better in the future as I read Ms. Steinke's article. Situations and specific students popped into my mind--both present and past. I don't have it all--but I have a glimpe--hopefully a bit more! I'll re-read this article and solidify some of those ideas.
I've finished the second article today (Tuesday) and am quite taken with the idea of numeracy being math "common sense." That's just the term I've never quite gotten to my lips!
Jim and Jan Hollis
John Ratey's article on Physical Activity and Cognitive Ability really caught my attention. One setting where I work has a large percentage of special education students, many with diagnoses of ADHD and Impulse Control Disorder. In the past we have taken extremely low focus students on a walk between classes to calm them down and make them more receptive to learning. Perhaps a cross-side exercise for all at the beginning of class would improve ability to focus and retain information. Applied to the GED setting, perhaps students could be encouraged to take an exercise break when they feel overwhelmed by new information. Maybe even doing exercise before taking practice tests or the real GED would help improve scores? WDYT?
A colleague at the community college where I teach told a story about one of her classes being sluggish one day. She got them all up dancing the macarena. She said they were more alert and she felt the class went much better after that. She had done this before hearing my reasons for using rhythm with adults.
I told her she was doing everything right. 1) Physical activity gets the blood flowing so the brain gets more blood. 2) Rhythm reinforces basic concepts. 3) Cross-body patterns (which the macarena uses) improve rhythm sense.
I'm hoping to see other comments on Susan's message.
I liked his article as well. I'm not totally sure I understand the cross training exercise, but when I see students getting antsy or unfocused, I tell them to take a little walk up and down the hall to clear their minds for a few minutes and it really helps them come back focused and ready to go back to work.
BOTH Parts and Whole Wednesday
To those of us who think of numbers this way, it seems incredible that there are adults who don't. (Ask me how many months it took my husband, an engineer, to accept this idea.)
If you looked at the drawings in the Focus on Basics article http://www.ncsall.net/fileadmin/resources/fob/2008/fob_9a.pdf
you may have noticed that I showed two circles, both with 8 dots in them. In one circle, the 8 is undivided. In the other, the 8 has been shown as 3, 4 and 1.
It is important to show BOTH Parts AND Whole at the same time when using visuals with EITHER / OR thinkers. Here's why.
Let say you are using chips or bottle caps or some small physical objects to demonstrate 9 + 6. If you show the student the parts, he/she will agree that there are 9 objects and 6 objects. Once you put those objects together, the student will agree that there are 15 objects. However, the 9 and 6 have disappeared for the student. The 9 and 6 have turned into the 15. Only the whole, the 15, exists. The parts are gone.
That's why I use paper plates to hold all the objects, the whole. If the student can tell you there are 15 objects on the plate, then you can split the objects into 3 or more small groups. Then your question to the student is, "Are there still 15 objects on the plate if I put them in groups of 3, 3, 4, and 5?"
This is why I begin the first time I meet a groups of students with 'different name, same person." I have many names and nicknames, but my physical appearance remains the same no matter which of my names you use when getting my attention.
The true meaning of the equal sign is one of the overlooked concepts in elementary school math. By third grade (if not earlier), textbooks assume that children understand the meaning of =. Textbooks assume students understand that = means 'what is on one side is the same amount as what is on the other."
Understanding the equal sign is necessary in order to understand the BOTH / AND nature of number sentences. Without that BOTH / AND understanding, students struggle with place value, fractions, percents - the list goes on.
Look for an explanation of = in the adult math materials you use. Is it there? If you asked your students to give five different names for 12 using three or more numbers each time (they can repeat numbers) and only the addition operation, could they do it? Try that yourself. Did it change your perception of the meaning of = ?
There is another way we experience the BOTH / AND of parts and whole. When talking about the 'equal distance" concept the concept between Stage 1 and Stage 2) I said the steady melody beat of simple children's songs was a physical experience of the concept.
Music rhythm is also a model for the BOTH / AND of parts and whole (the concept between Stage 2 and Stage 3). How often do you find yourself wiggling, tapping, or stomping along to the beat of a song? You are feeling equal GROUPS of beats. Those same-sized groups are the whole. The notes of the melody (the smaller pieces) are the parts. The melody exists within and at the same time as that grouping beat.
Parts and whole exist at the same time in a physical experience in music. When I can get a student to understand the BOTH / AND idea in music (the beat AND the words at the same time), then they can transfer the concept to numbers, often in a matter of a minute or two. They have experienced the concrete. Then the representational and abstract start to make sense.
Full disclosure: I have a masters degree in piano performance.
Teachers and Staff:
The website I found helpful for Numeracy for Students and Teachers to give students to comprehend is http://amby.com/educate/math This website has a list of math lessons and exercises for students to learn and master.
Thanks for your support, and have a blessed day!
Thanks, Debbie. This is a site that GED students could use for extra practice on their own.
This discussion is making so much sense now. Two weeks ago I was working with 2 students regarding changing mixed numbers to improper fractions and back again. I knew they could "follow" the formula but I could tell they didn't have a clue why they were doing it. When I had them draw pictures representing the mixed numbers and then showed them how the improper fraction was the same but just the pieces, I could instantly see they did not grasp the concept. They are those level 1 students you are talking about. We worked on this for a number of days and though there was a little improvement, I know their understanding is not solid. They either see the whole or the part, but they can't see both in the same picture. I'm going to try some of the suggestions you gave earlier such as regarding to rhythm etc. and see if that helps before we attack it again. It was definitely an "aha" moment for me.
Hi, I am new to the discussion board. Just wondering what the rhythm is that you spoke about. I, too, am teaching a student how to change mixed numbers to improper fractions and I drew the pictures similar to what you mentioned and there was some understanding. Anyway, what about the rhythm?
Welcome aboard, Rita. Look at the TUESDAY posting - Concepts in Action. Moving to a steady beat in music is feeling equal spaces of time. Moving along a number line marked in equal spaces is feeling equal spaces in distance. Both experiences - in space and in time - are the same concept, equal distance.
That is the concept that students at Stage 1 are missing. To repeat, at Stage 1, a person thinks of each number as an individual amount with no size relationship between the numbers. Practicing cross-body movements at a steady beat (at the student's steady beat which may be slower than yours) is another way to experience the equal distance concept. If a person can experience a concept in his/her body, he/she has a physical reality that means something. Then you (the teacher) can relate that experience to the meaning of the math symbols on the page.
I relate this to money....a dollar can be a piece of paper, 4 quarters, 10 dimes....etc.
Using money is the best way for most of my students to "get" it. After all, it's so normal--something they do understand, something they can visualize or even see.
Jim and Jan Hollis
I would agree that multiplication and division of fractions is easier for the student to do than addition and subtraction. However, I find many students (especially those with aspergers or some form of autism) refuse to cancel out common factors in numerator and denominator.e.g. 4/10 x 15/12 becomes 60/120 = 1/2 instead of cancelling 4 into 12 and 5 into both 10 and 15 and getting 3/6 = 1/2. Even a more basic problem like 4/5 x 15/12 becomes 60/60 = 1 instead of cancelling to get 3/3 = 1. How can I more effectively present cancellation? If the expanded product is too large, they often give up on reducing it.
When presenting division of fractions, try using the initials KFC. They all know the chicken franchise, but now the letters mean KEEP the first fraction (K), FLIP the second fraction (F), and CHANGE the operation sign from division to multiplication (C).
When you do present the concept of common denominators, how do you get the student to choose the lowest common denominator. If one divides into the other, it's simple. 1/4 + 1/2 = 1/4 +2/4 = 3/4. Or if they have no factors in common, you multiply the denominators to find the common denominator. e.g. 1/3 + 1/4 = 4/12 + 3/12 = 7/12. But when some factors are the same and some different, the factor tree is the best solution I have for lcm. e.g. 1/6 +1/9 = 3/18 +2/18 = 5/18. Most of my students use 54 and reduce later (or not). 9/54 +6/54 = 15/54 = 5/18. Do you have a better approach for lcm? Only the better math students catch on to factoring each denominator for lcm
I like to teach prime factorizations to help with finding common denominators. Then 1/6 + 1/9 becomes 1/(2 x 3) + 1/(3 x 3). The LCM is the product that has both (2 x 3) and (3 x 3) for factors, i. e., (2 x 3 x 3) = 18. An added benefit is that it is immediately clear what to multiply the numerator by to get an equivalent fraction.
The need for *lowest* common denominator is arguable.
If, in learning how to do that, the student learns about prime factors and divisibility and the like, then it's worth doing.
On the other hand, if figuring out the fractions is what needs to be happening, we're leaping into cognitive overload... which of course they're accustomed to, so they'll do their best to Learn The Procedure.
I first make sure the language isn't confusing them. If they actually know "least' means "smallest," they sometimes try to make the denominator smaller... and find the greatest common factor, instead. I explain that the common denominator could be *huge* so it's easiest to work with a smaller one. I'm not teaching, but tutoring, so I can't always build the comprehension.) I tell 'em that we need a number big enough for groups of each of these to fit in... so it has to be at *least* as big as the biggest one. If that one doesn't work, they try that big number times two and check... times three and check... and depending on their verbal comprehension I may suggest looking at the smaller numbers for factors to multiply by.
Sometimes their teachers really want them to do the factor tree thing. I've figured out a reasonably visually/kinesthetically memorable version of "plucking" the common factors first, then going pack and picking up all those single leftovers.
However, LCD is not really necessary and I've read cogent arguments that understanding "common multiple" is ample, so I do tell students that sometimes they *have* to multiply the denominators... and they always *can,* but the numbers might be bigger and they'll have to reduce.
I've had European students who have learned a handy-dandy procedure that's embedded into motor memory. You mulitply the denominator of the first number by the numerator of the second number, put that on the top of the fraction bar. Then put plus and multiply the numerator of the first number with the denominator of the second number adn put that as the second number. Then multiply the two denominators for the new denominator. I'm afraid it's generally done with absolutely no connection to meaning, but boy howdy do the students get the right answer. (They tend to have learned those divisibility rules, too.) I put together a sort of compromise method that I describe here: http://www.resourceroom.net/math/denominators.asp I've got a Camtasia of it ... somewhere online... but it evades me. That includes the California school system that declares it's best to delay LCD until after you understand what the heck denominators are in the first place.)
For reducing, some of my guys are more comfortable with "canceling out" if they rewrite the number as its factors and then cancel the ones that are the same.
I start teaching divisibility rules for 2, 3, 5, 6, 9 and 10 early in the year so when we get to reducing fractions they know the 'tricks'
and can put them to use.
This helps for most students, but not all.....knowing multiplication tables helps!
One way I have students work with divisibility rules is to have them insert a digit anywhere into a particular three-digit number to make a number divisible by 72. For example, insert a digit into 718 to make a number divisible by 72. Using divisibility rules for 9 and 8, it is clear that inserting a 2 fills the bill: 7128 / 72 = 99.
When I taught sixth grade, Patrick O'Brien understood what thirds were. None of the other 22 students had a clue. They could do their homework about fractions, but they had no idea what a third of 30 was (or, they didn't until we were done ;)). If we hadn't taken a "side trip" down the conceptual path, 22/23 of them would have happily reproduced the procedure.
Stage 1 students are the ones who count the marks on a number line instead of the spaces. And, when you show them they need to count the distance between the numbers (the spaces), they don't get it. Try starting with a number line task - like subtracting - and see what happens. You may need to go way back there to give them a sense of whole number relationships before you begin to think about fractions. Number sense has to build from a correct understanding of adding whole numbers. If a step is missed along the way, cracks appear later on.
Read Thursday's posting for more ideas about fractions.
Q. Can you think of a student who might be willing to try a new way of experiencing number relationships? What variations on this suggestion would you try?
A. At this time I don't have that kind of a relationship with the students. I am only in the classroom a couple days a week and they don't feel they know me very well.
Q. When teaching part-whole thinking, do you use manipulatives (other than fingers) for hands-on learners? If so, which ones do you find work the best?
A. I am new to teaching and have only seen the fractions pie shapes to explain the concept. I have tried to draw pictures. I’ve seen students using their fingers to count. I have worked with a ruler with a student who was working on measurement too.
Q. I have had up to three Stage 1 people in a math class of 24. How often have you seen the Stage 1 behaviors I mentioned?
A. Like I've said I'm new to teaching and at this point I haven't worked with too many low level students though I have worked with a couple of people with learning disabilities. We worked with money, giving change, and one was very able to give answers but I couldn't get him to let the other 2 try and they were perfectly content to let him give all the answers.
FRACTIONS and the Properties of 1 - Thursday, March 24, 2011
At the moment I’m looking at a set of math texts for Grades 3, 4, and 5. It seems to me that the texts teach fractions primarily as a part of something.
Grade 3 - "A fraction is a number that names part of a whole."
Grade 4 - "A fraction is a number that names part of a whole or part of a set. In a fraction, the numerator tells the number of equal parts. The denominator tells the number of equal parts in all.”
Grade 5 - "A fraction is a number that names equal parts of a whole or parts of a set. A fraction represents division."
Defining fractions as "part of a whole" tells students to think about each fraction as a separate number. This definition misses the RELATIONSHIP concept of fractions: the denominator is the number of pieces or items I have; the numerator is the number of pieces or items I care about. Yes, the later texts bring in "part of a set", but the language to me seems to focus on the part.
We all tend to revert to the first meaning we learned for a new idea. In grade school students learn first that a fraction is a part of something. Period. How many of your students are stuck there?
Fractions compare parts to whole. I know I'm sliding into the definition of ratio at this point. My perspective is that all fractions are ratios that compare part to whole. (Ratios can also compare part to part; fractions only compare part to whole.)
How much emphasis does your math text put on comparing part of a group of items to the entire group of items (i.e., the number of apples in your refrigerator COMPARED to all the pieces of fruit in your refrigerator)? Does it only speak of fractions as a piece of an item (i.e., half an apple)?
Before touching fractions, I want my students to understand the real meaning of = and to have a sense of part-whole coexistence. Fractions are "the number of parts we care about" compared to "the number of parts in the whole." If students grasp part-whole relationships with whole numbers, then the definition of fractions as "parts compared to whole" makes sense.
Understanding fractions as relationship or comparison comes first before tackling equivalent fractions. For this next concept, students need to understand the properties of 1.
1 is such a simple number. Students gloss over the definitions in the book. They seem to know that 17 x 1 = 17 and 17 (divided by) 1 = 17. When pressed, they can also tell you that 17 (divided by) 17 = 1. [The standard division sign wasn't recognized by Microsoft Outlook.]
In my classes, I go over the Properties of 1 in excruciating detail, for they are key to understanding equivalent fractions.
(I suggest you write out these next examples by hand in standard fraction format, i.e. the numerator directly above the denominator. It will make the relationships easier to see.)
Equivalent fractions look so different. How can 3/6 = 4/8 ? Why is the common denominator 24 and how did I get there?
We get there by using the idea that "any number divide by itself is 1" and "multiplying by 1 does not change the value or amount of what you have."
Let’s start with…
3/6 x 4/4 = 12/24
4/4 is a "different name" for 1. I multiplied 3/6 by 1 (with a different name). Multiplying by 1 does not change how much I have. 12/24 is a different name for 3/6.
I remind my students about "different name, same amount" (the meaning of =). I tell them they still have the same amount of pizza, just cut in smaller slices. Here is where having them draw pictures of the equivalency helps.
The several college-level basic math texts I have seen all introduce multiplication and division of fractions prior to addition and subtraction. When the multiplication process is more secure, students can understand that the "multiplying" they do to solve fraction addition and subtraction (finding common denominators) is only one step in the process. That clears up a lot of cobwebs.
Once past the hurdle of equivalent fractions, proportions make sense. Percent problems can be set up as proportions (one template for setting up all percent problems, not three). In geometry, similar figures have sides that are proportional. You can probably think of more examples for applying proportions.
It all hinges on understanding the use of the properties of 1 to make equivalent fractions.
Look closely at the procedure for finding common denominators that is presented in the materials you use. Does it give a complete explanation of how 3/6 can be written as 12/24, or does it offer only a procedural shortcut? What other examples and materials do you use to teach equivalent fractions?
I am not a math teacher by trade, so all of this information is really opening my eyes. I especially agree with the idea of starting with multiplication/division of fractions before moving to addition/subtraction. Just the other day, one of my students told me he thought it was backwards in the text since "multiplication of fractions is about 100 times easier to figure out." I'm starting to realize we all need to really think about how many different fractions can be equal. I feel a fraction challenge coming on!
Cassi - If you feel a fraction challenge, think about your students. Easing into fractions from the "Properties of 1" has made a huge difference for many of the adults I teach. For many, no one ever explained to them that equivalent fractions are the same amount of "stuff" cut into smaller pieces.
Many of my students have difficulty with sequencing. This may invlove solving math problems, reading, and/or writing. Will teaching part-whole concepts help resolve problems with sequencing?
Jeanne Van Lengen-Taylor
According to Dr. Ruby Payne, it will help. If you understand the part-whole of a situation, you can plan. You understand cause and effect. You understand that actions have consequences. Sequencing is part of planning.
There are all kinds of "but.." and "what if"... that go with the above statement. I have had students with brain damage (either from an accident or from substance abuse) who made no progress at all in understanding math over several months of instruction. They had a grasp of what they knew before the accident, but nothing beyond.
While I'm getting a lot of ideas as I read the articles and the discussions (and as a someone with somewhat limited math literacy) I am reminded of how I have to remember to share "how" I do "it" with my students. When I began ABE a long time ago--I thought it was wrong to admit that I dropped off the fractions when setting up a fraction/division problem. One day it dawned on me how crazy it was. If a procedure helped me--why wouldn't it help my students. Since that day, I make it a point to "share" and encourage my students to do the same. I learn, the student learns, and often the whole class learns as well.
This whole discussion certainly undergirds the life-long learning principle for everybody--student and teacher alike. I want my students to know I'm still willing. AS for the cross-side exercise idea, it works great and can be a source bonding between students. Of course, all do not take part, but they often laugh. I start early to let them know that they should find and practice techniques that help them re-focus--stretching, walking around, using and exercise hand ball, unwrapping candy or gum, doodling, sitting back and visualizing something. It's not a huge step for them to take something useful for them personally into the GED test room.
While I know this discussion is almost over and this will be my last post, I want to say thank you all for your help. I've had some eye trouble this week and still have some reading to do--some I'm really looking forward to from your comments--and I have it all printed out. I've lurked some each day; it's energized me and made me eager to get into the meat.
Jim and Jan Hollis
I have been following the discussion.The description of fractions comparing part to the whole is a very usable concept. The importance of student comfort levels, trust and a non-judgmental setting are also key to learning. The idea of encouraging the cross-body movement would seem to aid students manage frustration when the learning becomes stressful. Movement could be a warm-up and a break to relax activity. Haven't seen the videos yet. julie/btec
Making Meaning with Numbers - Friday, March 25, 2011
What I have tried to do this week is peel back the onion of symbol manipulation of numbers to get at the core of what I consider number sense.
The key to number sense is understanding the part-whole relationship of quantities.
Once students have that understanding (the concept), then the abstract symbols of math language on paper make sense. Students can do the "skills" of the basic operations and fractions more easily. They know the "why", not just the rules.
Symbol manipulation is a skill. Of course students need those skills in math to answer practical questions about physical relationships in the world - how many more or less, how far, how fast. Unfortunately, students too often believe that the skills are all that math is. Standardized tests tend to reinforce that belief. How have you attempted to counteract a student's belief that math is only skills?
For a moment I'm going to draw a larger picture of part-whole thinking.
I mentioned in an earlier posting that we use part-whole thinking when we are behind the wheel of a car. Every day we use objects around the house that are whole and have parts. The house itself, for example, has a roof, floors, walls. Make your own list.
Part-whole thinking is also the key to good reading comprehension. Those of us who read easily understand the part-whole relationship of sounds to syllables to words to phrases to sentences to paragraphs and on up. Recognizing the pieces and the big picture at the same time helps us make sense of what we read.
Knowing that the parts and whole of something exist at the same time is how we make sense of our world. We all do it every day. How can we most effectively open students' eyes to the same concept in number relationships?
I was delighted to read the idea of using nicknames to explain how the different sides of an (equal) equation are the same thing. I independently came up with the same idea when I was teaching equivalent fractions several years ago. Students seemed to get it in a very visceral way. I'm teaching math again this next session and I'm going to extend this part/whole idea way past fractions. So excited to see the results ...
I'd like to point out that your students are probably NOT the only ones in your agency who are still at level 2 (or 2.5?). I just spent a very productive 15-20 minutes helping two fellow staff members (not math teachers, luckily!) begin to understand the WHY part of doing fractions.
(Why is 4/8 the same as 1/2?) They both caught on fairly quickly, but I definitely found myself fighting the "rote memorization monster." The one with more procedural knowledge kept making up problems with inappropriate fractions. For example, she wanted to show the other co-worker how to borrow when subtracting, and she had trouble coming up with a fraction bigger than 3/8. It was a great concept to work on, but I was struck by the fact that even though she can add, subtract, multiply, and divide fractions, she still has a very hard time telling which ones are bigger or smaller. In fact, my co-worker who declared herself math-phobic had better math sense than the other (especially when it related to measuring cups or money).
I often find this with my students, too--the ones who know (or half-know) the procedure sometimes take longer to understand the concept and the reasons. They're also less happy to spend the time to figure it out, since they "know it already." How do you help a student move to a place where they want to know reasons and make sense of things when they already have the rules memorized?
Rachel - A very good point. How do you get people to put the rules aside and think about relationships? I sneak into it in the first lesson, using their nicknames to demonstrate the meaning of the equal sign. Somewhere in that lesson I ask them about learning to spell versus learning to read with understanding. If you can only spell the words, do you know what the sentences mean? Do you understand the story? The rules and number manipulation (writing the algorithms) are to number sense in math like spelling is to reading. I want them to understand number relationships and concepts, not just how to "spell".
This brings the idea home for some.
This is something we talk about in our developmental ed conversations, and, I suspect, one (of many) reasons math courses are "gatekeeper" courses. Many students arrive at the community college prepared to Complete Assignments, Do Butt Time, and Get a Passing Grade. The first place that really falls apart is in Math. In my experience, the best way to get students embracing the idea of thinking of wholes and parts is finding where they are and making sure they know what it is and have some success with it. The lady who figured out the "third of a is 10, so a is 30" is motivated to apply that to math now. A fair number of my regulars say "I just have to know why or I can't remember it." It's making the "why" part actually make sense to their framework -- the wholes and parts they already think about -- that's the challenge! Knowing whether to use a number line or manipulatives or stories to connect with what they already know...
I have gotten a lot of ideas from the articles and discussions. I would like to put in my two-cents worth on using proportions and the concept of parts-to-whole with teaching percents. I used the percent proportion for teaching percents to middle school students for years. Now, I teach GED math classes. I don't know how many students have had some "aha" moments or expressed some relief when they saw this method of tackling percents. I feel they have a powerful solving problem tool, and that they understand why they are setting up the proportion. The anxiety of "Do I move the decimal?" and "Do I multiply or divide?" has vanished.
I think another reason GED students have a hard time with fractions is because they don't see relevance in their lives. Unless they sew, cook or are in construction, most people have a hard time figuring out why it is so important for them to learn fractions. Coming up with concrete examples of how and why franctions are important helps break down the resistance my students have in learning fractions.
I was working on reducing the improper fraction 30/15 with a student just last night, and I asked him if he cooked, did carpentry or made quilts, so I could relate this to him with something familiar. He laughed and told me "If it ain't got nothin' to do with huntin', fishin', or drinki'’ beer, then I do't know nothin' about it!" Ok, so I came back with "one cooler holds 15 cans of beer, you have 30 cans in the back of your truck. How many coolers do you need?" He had the answer instantly! Lol
Concrete examples are definitely helpful!
Kaye Mallory, AEL/GED Instructor
North Central Missouri College
I love it, Kaye. That's meeting the student where he's at and making it relevant!!!!!!
...and this supports my hypothesis that our students have more connection with fractions as division of groups, as opposed to pieces of one thing. The quilting, construction and cooking are where you run into the latter type more often. I think starting with the division and then scaffolding over to "in between two wholes" is very useful.
... and you'd want to extend that to say "okay, and what fraction of the beer is in each cooler? ... Half of it, eh? I was inspired by my lunch routine to add a multisensory experience to my negativity presentation. I made myself a cup of piping hot tea. Temperature oh, probably 200 degrees -- close to that boiling point of 212! Too hot for moi. So I *added* water from the fountain. What happened to my temperature? We'll taste the addition...
I tell my students there are several reasons why learning about fractions is important - besides figuring out how much pizza is left over.
1) It's the foundation for ratio, proportion and probability.
2) It's used in algebra equations. They need to know how to multiply, divide and cross-cancel.
3) It's used in standard (English) measurement - inches, tools, screws, etc.
4) It can be used for dimensional conversions (unit conversions) - inches to centimeters, centimeters to millimeters, etc. It solves the problem of "do I divide or multiply?"
In preparation for fractions, I have my class start working on their multiplication facts at the beginning of the session, one fact per week. We take our time, play card games, and have fun. By the time we get to fractions, after place value, operations and decimals, they're halfway comfortable with their facts. As we start fractions, they suddenly realize how important it is to be able to get those multiples quickly, and their motivation increases. I use lots of pizza circles, giant inches and graph paper to make fractions as visual and tangible as I can. At the end of the unit on fractions, I tie it back in to decimals, and they get a new perspective on decimals as a part of the whole. This leads nicely into percent!
I think fractions are extremely important, even though it may seem like we don't use them in real life that much. It's really worth taking the time to explore fractions thoroughly and get those skills down solid.
Another very important reason for fractions -- the ability to manage personal finances. In my work as a financial educator /coach, I see too many people who don't have a sense that their resources / income is the whole -- and that the parts need to add to the whole -- and that debt means that you are assigning major parts of your future earnings to somebody else. With almost no exception, people who are in financial difficulty also say that they hate math -- and probing will get to fundamental weaknesses not only in adding and subtracting along the number line (I loved that example of teaching number line using owing money to friends) -- but also in fractions, percents, decimals. We have to do solid work in examining what resources constitute the "whole" of the person's resources -- and than look at the parts -- and understand the relationships. I include in every financial counseling session, deliberate use of the words "whole" and "part" as we unravel people's financial profile -- as well as provide parents with hints on how to pass on concepts to children. Also find that Key Curriculum's "Enpowering Mathematics" series very useful in helping adults grasp the pattern making / relationship aspect of numbers. I have found this discussion this week very useful -- with many ideas.
Thank you, everyone who contributed.
Good Evening Everyone!
This is Dorothea's final day as Guest Facilitator. I would like to thank her for her time and for the invaluable information she has given to each of us. I would also like to thank all of the subscribers for their input and questions this past week; hearing from the field is a great way for all of us to develop professionally, so thank-you. Please feel free to carry on with the discussion but please note that Dorothea will no longer be with us as a guest. I will be preparing a user friendly transcript that will be posted on the LINCS website for public use; but it may take a few weeks before it is available. I will post the link once it is accessible so that others may benefit from this great discussion! Once again, thank you Dorothea! It has been a delight having you as a guest on the LINCS Math and Numeracy List!
Brooke Istas, Moderator
LINCS Math and Numeracy List