Monday, January 11, 2010

Thoughts on Teaching Fractions and Student Work Ethic

I've come to believe that part of the issue with kids making errors or having trouble learning fractions is not due to the content or the teaching but due to some children's internal thought processes or work ethic issues. Other times the teaching method, especially if it moves along too swiftly before the last concept was mastered can set kids up for confusion and then failure. Once confusion begins, enthusiasm and fun go out the window. Children also may have negative self-talk that begins to erode their self-esteem (at least regarding learning or school lessons). The seeds of poor self-esteem can sometimes be hidden, masked, by children well so that parents and teachers may not even suspect it. I feel that not addressing this if it rears its head with fractions can have more serious implications for the child that can affect their career path in their adult years.

Teaching fractions, it seems to me, is the first complex math. I hear this from my friends whose kids go to school as well as from some homeschooling families. "They always did well with math until they hit fractions!" Kids have been doing addition, subtraction, multiplication and division for years, it is kind of routine. They may forget some things but there is so much review over the years that the kids usually do fine. However with fractions the learning is sped up suddenly. The learner, especially if enrolled in a school or when sticking closely to a homeschool curriculum, must keep up with the speed of the lessons, if they don't, they falter.

If a child does not grasp one concept about fractions and the content moves on to the next, they are off track and can get into a mess quickly. For example if they have not done enough reducing of fractions to make it automatic, they may go on to correctly multiply fractions but forget to do the reducing part of the problem.

If the child does not have their multiplication facts memorized yet, if trying to find lowest common denominators can be a nightmare.

If the child is of the mindset to do a problem qucikly and move on, their lack of thoroughness in checking that they are really finished can cause them to make an error. For example after multiplying a fraction they may either forget to reduce it (even though the directions clearly state to) or they may reduce it once but not take care to check to see it actually can be reduced a second time.

When the child finds out their answer is incorrect, it should be made clear if the issue was they didn't reduce or they made an operations error or just an arithmetic calculation mistake. If you don't tell them WHY it was wrong they don't know which issue to focus on re-learning the correct operation or whether to pay more attention to reducing or whatever the issue is. Throwing a grade at a child doesn't help them understand what they did wrong. If they don't know which thing they are doing wrong they don't know how to fix it.

Moving too quickly from addition to subtraction to multiplication to division of fractions can be tricky if all the previous concepts were not mastered. If the learner gets flubbed up somewhere along the way a general sense of confusion clouds them and they sometimes suddenly feel like they don't know what of fractions they are doing right and which they have a wrong sense about. They can feel frozen and then fear can set in. They can also begin to feel self-doubt even about operations they actually ARE doing correctly.

I suspect some kids goof up on fractions because it is the first time they are combining processes to do one operation. There can be multiple steps to one math problem. Some kids seem to be thrown off track with this. I speak of converting a mixed number into a fraction then doing an operation with that then reducing it once or twice. Somewhere along the way some kids forget one of the steps. One of my sons struggles to remember each of the steps in the process. This type of thinking is perhaps more complex than they are used to from the math work of year's past.

The closest math problem that I found that comes to teaching multiple step processes are complicated word problems but seldom are those done in typical homeschool math curriculums. The Singapore "Challenging Math Word Problems" books (sold by grade level as supplemental workbooks) are a great example of complicated word problems that use multiple steps. Most homeschool curriculums don't get as detailed as those (nor do the regular Singapore curriculums). I have no clue what the public schools do but I have little confidence that their math curriculum is on par with that Singapore Math workbook. (I own those books and have used them just a little with my second child, I didn't know about them before. We have not been using them lately as they are one of the non-core lessons that seems to fall off the agenda. As I confess this here I am kicking myself for dropping the ball; perhaps if I had used these for years this thought process would be ingrained and this struggle with fractions would be a non-issue.)

I wonder but do not know if kids who suffer with a visual processing disorder are more challenged by multiple step fractions math problems. My child with this challenge is the one struggling with fractions work so this is an idea in the back of my mind that is percolating. I suspect that kids with auditory processing disorders who are being taught using auditory methods will struggle (including the teacher or homeschool parent verbally explaining the math operation process). I recall Linda Kane, a neurodevelopmental specialist who I heard speak at a homeschool conference in 2009, suggest that when teaching math operations that the child struggles with to try this teaching method. The adult is to be silent and ask the student to watch. The adult does the math problem entirely with pencil on paper (or on a white board or chalk board). Repeat this up to five times at which point the student will often declare "I get it!" without a single word having been said by the adult. I tried this with success, it worked especially well for large multiplication problems and long division.

I note also kids seem to think of fractions as one big thing, so if they understand 90% of what they learned but struggle with the latest fractions operation/concept they think to themselves, "I stink at fractions, I can't do them." This all or nothing thinking messes them up because they lose confidence in the fractions work they did master and still can do correctly.

Again I want to stress, as I have this issue with one of my sons, that I feel there is a work ethic component to fraction work. This will also apply to algebra and geometry, and all higher level maths I suspect. If a child has this issue they should be made aware of it and should work in their own minds as well as with adult supervision on correcting it. This is when the learner rushes through a problem in order to get it done, or doing half the problem and not being thorough. It's good if they do know how to do the operation but if the problem calls for it to be converted to a mixed number they MUST follow through and do that part. If a fraction needs to be reduced they MUST do that part too. It is critical to be thorough when doing math operations. If the learner has to learn self-discipline to stay on task and do each problem carefully and wholly, then so be it.

This issue of their mindset of rushing or doing it half-way can also be an issue when doing multiple choice tests or other school lessons. They may see the first correct answer and choose it rather than reading all the choices and reading that the direction said "pick the BEST answer". This can cause standardized tests to have a falsely lower score than the child's actual aptitude. This can cause lower grades than the child actually can do correctly. It seems to me that some kids just have to focus and learn these skills with more effort than some other learners do. Maybe some of them will 'get this' by practicing 'test taking' skills, doing those workbooks, but perhaps if this was addressed strictly through discipline with math work it would cure it?

Children enrolled in school will have to have remedial help either at home from parents or with paid tutors for the actual math challenges they encounter. For homeschooled kids I feel that curriculums should be paused if they rush too quicky and the parent should do extra work to master the concepts before moving on to the next concept. If this happens too frequently with the homeschool math curriculum perhaps the parent should think about looking for a better product for that child. Since each child is unique, different curriculums are suited to different kids. A struggle with one company's product doesn't mean the product is necessarily a bad product.

No matter whether a child is schooled or homeschooled if your child struggles with fractions or algebra I wonder if these underlying 'work ethic' issues are the cause. Closely working with your child or closely going over their assignments to notice a pattern, along with talking to them will reveal where the issue lies. The problem may not be with the school teacher or the homeschool curriculum but it may be that your child (like one of mine) is using what I can only describe as sloppy work habits or poor worth ethic principles.

Addressing the 'work ethic' issue should be a top priority, because if the sloppiness continues it can impede their mastery of concepts and their grades of course. This issue in the middle school years could mean the difference between how the student fares in the high school years, what math courses they qualify to take in eighth grade or in ninth grade, or the higher grades in high school. Higher level  math in high school then could impact what college degrees they qualify to pursue as well as what colleges they can gain entry to, which can affect their career in their adult lives.

If my son doesn't clean up his act he'll not be able to do the higher level maths that are required to be completed in order to gain access to pursue the engineering degree he insists he wants to seek. I am not putting down the colleges with their strict admissions policies when I say that. The world needs engineers with a good work ethic, that don't rush, that respect and follow their authority figures, that can take direction from higher-up's and that can read and follow directions and rules when left to work on projects with autonomy. People with poor self-discipline seldom can be trusted to work with autonomy (in any career field). The world's engineers need to be good at math and science, they need to be able to do detailed math with many steps and following through to completion. Would you want to live in a world where our buildings (especially skyscrapers) and bridges were designed and built with sloppy math calculations?

I'm being optimistic at this point in time, that if the bar is raised for 'quality control' in our homeschool, if good work ethic habits are taught and used, and if attention to the right issues is done, that my son can succeed. I have this same confidence in other young people as well.

This same issue may be a challenge you are facing with your homeschooled or schooled child. Don't give up on them. Then again, a teacher or homeschool parent can sometimes only do so much without the learner's buy in. The old adge "you can lead a horse to water but you can't make him drink" is true. Perhaps not all kids will be able to fulfill their dreams or desires and some parents may be disappointed to find out their child will not excel to the level they had hoped and assumed they were capable of. The student must want to learn and the student must use the tools they have been taught, self-discipline and a good work ethic are both entirely in the control of the individual, there is no way of getting around that.

I feel that for some families, homeschool and schooled kids alike, fractions might be the first red flag warning that is raised in middle school, of a need for more thorough school work habits and the need to learn more self-discipline. I see no harm in watching for this and if the problem is found, to address it sooner rather than later.


LivingByLearning said...

Actually, fractions make their lovely appearance in 3rd Grade, both in the spiraling public school version and in Singapore. In simple terms, it appears in even lower grades.

My son is doing fractions with Singapore 4A. He's just finished the unit, but I'm going to have him do more review problems daily even as he moves on to Geometry.

I noticed that even though he has learned how to convert mixed & improper fractions, as well as doing math facts with fractions, he still has conceptual problems.

He's shaky on remembering what a fraction describes, i.e. what is the function of a denominator or numerator. Although he can follow a formula, it's too easy to make a mistake when he isn't confident about what these fractions represent.

Again, I'm glad that as homeschoolers we can slow down and ensure that he's mastered this before moving on. Otherwise, a shaky math foundation doesn't support a strong building says Confucious!

Michelle said...

My son has been doing fractions for several years and I thought he had them down well. Then at the beginning of his Algebra book, there is a fraction review section. He missed every single one.

I think the trouble is that the first time he learned the rote process, but didn't vizualize what was happening. Fractions are very hard to vizualize, much harder than simple addition or even mutiplication and division. I re-taught him the process and strategies, using visual examples to explain why and he's doing the problems correctly again, but I'm still pretty sure he doesn't understand what is really happening, why the denominators have to be the same when adding or subtracting fractions, why you can just invert the second term in a division problem and so on. It's just much harder to vizualize. I didn't get it really until I was teaching my own kids. (I'm not sure I should admit that!)

christinemm said...

My older son used Math U See and fractions begin in grade 5 or at least they did when he was using the curriculum.

I recall the fraction work in grade 2 and 3 Singapore Math that my younger son used was very simplistic.

Doing all the fraction work in a short time, every component of fraction work, not just bits or easy problems, is what I'm referencing here.

I wonder how my younger son with his very different mind will fare with ALL the fractions work.

blissful_e said...

I enjoyed reading this post, and I thought I would chime in since I have an engineering degree.

Engineering training, at any accredited university, requires a very strong work ethic, as well as a true enjoyment of getting tricky multi-phase math problems just right.

Dropout rates among would-be engineers, at least at my uni, are shockingly high. I am so impressed that you are thinking about these issues now rather than waiting for a Freshman 'weed-out' class.

Hurrah for homeschooling!

Lion said...

Permit me to give a small “Singapore Math” primer that might be useful:

Problems Involving Fraction, Ratio and Percentage. Discussion with examples:

Grade 3 examples, solutions with bar models are from a Grade 3 child.

Tom spent 3/8 of his money on a pair of jeans and 5/16 of it on a shirt. What fraction of his money had he spent?

Ted painted 1/5 of a room red and 3/10 of it yellow. What fraction of the room did he paint? Give your answer in its simplest form.)

Tracy spent 1/6 of her money on a pen and 5/12 of it on a writing pad. What fraction of her money did she spend?

In this case, the child decides not to use bar model as she considered it straightforward.

Shez said...

have a look at You have to register, but it's free. The author of this curriculum is brilliant. He breaks everything down into tiny, tiny steps. The teacher's manuals are all free downloads. I used the introductory fractions with my kids and some of the other stuff.

Your visual learning will love this program.

I've found that Shira is an auditory learner. It's not enough though for me to explain everything in tiny steps though. I have to help her develop a dialogue that she can say OUT LOUD while she does the problem. It's amazing. As soon as I twigged about the out loud dialogue, our troubles disappeared.

I love those Singapore challenges. I use them as a treat for work well done in my homeschool. As you've probably gathered, we're math geeks.