StickingPointsInMathAndAlgebra < Kitchen

 <font size=+2>Sticking points in math</font>

__Topics your child is likely to have trouble with in math, arranged by grade and/or topic.__

__This topic is currently under construction!__

<font size = +1>First and Second grades</font>

*Memorization of addition and subtraction facts* is sometimes tough for young kids. :Liping Ma suggests having kids this age memorize sums and differences up to 20; i.e. they should have calculations such as 17-9=8 at their fingertips. This makes subtraction with borrowing go more smoothly.

*Borrowing in subtraction* is difficult for many children, and to a lesser extent so is *carrying in addition*. Both are symptomatic of problems understanding *place value*.

<font size = +1>Third</font>

Multiplication itself is a fairly easy operation for kids to understand, but *memorizing the multiplication tables* is a critical thing to do, and difficult for some children. 

Kids are introduced to *fractions* this year, and often find the basic idea very confusing. They may need help understanding the equivalence of fractions (e.g., 1/2 = 2/4) and simple rules such as 3 * 1/3 = 1. 

<font size = +1>Fourth grade</font>

This is a big year in mathematics education in most places, when children are learning the algorithms which will allow them to do most mathematical computations by hand.

*Long division* is a difficult (but necessary!) algorithm for most children to master to automaticity. This can take a long time to master and needs frequent practice until it's mastered.

A sizeable number of kids have trouble with the *multidigit multiplication algorithm* during this year. A common problem is to use amounts carried over to the next column incorrectly. For example, in 34 x 34, the '1' from the '16' must be carried over and added to the result of 3x4 to make 13; but not infrequently a child will add the '1' to the 3 and then multiply by 4, to get 16 (incorrectly).   If you're not sure why your kid is making errors, look over their shoulder and watch while they do the computation.


<font size = +1>Fifth grade</font>

*Addition and subtraction of fractions* is difficult for a lot of kids; many will naturally want to add and subtract the numerators and denominators independently. Multiplication and division of fractions will be much easier. Paradoxically, it's easier to _understand_ why addition and subtraction of fractions is done the way it is, than to understand the reasoning behind the multiplication and division operations for fractions.

Related to the above, kids often have trouble both understanding and finding the *least common denominators* for two fractions to be added or subtracted.

Understanding the meaning of *place value in decimal representations* will often be difficult.

Purely computationally, kids will mix up the steps of *adding and subtracting decimal numbers* with the very different algorithm for *multiplying* decimal numbers.

Long division for decimal numbers seems not to be difficult provided the child can do long division.

Manipulation of *negative numbers* can be confusing but isn't usually a big hurdle.

*Story problems* are difficult in every grade because of the need to transform words into expressions, solve them, and transform back to words. Some of the problem is just due to attention problems in young children; every word problem contains those three steps at a minimum, and so it requires good executive functioning just to carry a word problem through to the end. Extensive practice is required, and the use of simple dimensional analysis to aid understanding can really be helpful (but must also be drilled). 

*Adding and subtracting mixed fractions* is hard for kids. Take the opportunity to teach 'borrowing from the whole number' when doing subtractions such as

<verbatim>2 1/2 - 1 3/4</verbatim>,

since it strengthens a kid's understanding of borrowing in general.

Cross-multiplication, cancellation in fraction multiplication  and other *computational tricks* are sometimes taught in this grade without sufficient explanation as to why they work, and sometimes the kids will simply 'learn them wrong' because they don't understand them. Keep an eye out for incorrectly applied tricks, and keep offering explanations as to why they work (deeper understanding will sometimes take a while to sink in).

<font size = +1>Pre-algebra</font>

The meaning of *negative exponents* are sometimes misunderstood. Kids will make mistakes simplifying expressions containing them. *Zero exponents* take a little getting used to. *Fractional exponents* are also very confusing.

Kids have to learn to get comfortable with moving back and forth between operations with negative numbers, and the familiar whole-number operations. *Subtracting integers* is particularly tricky. They'll have trouble seeing at first, for example, that
6 - (-3) is the same as 6 + 3.

Exponents in general are hard to understand; particularly fractional exponents (such as 2<SUP>2/3</SUP>) and their meaning. Exponent rules such as x<SUP>a+b</SUP> = x<SUP>a</SUP>x<SUP>b</SUP> and (x<SUP>a</SUP>)<SUP>b</SUP>=x<SUP>a*b</SUP> are hard to remember, and frequently get confused with the erroneous "distributive law for powers" (see the algebra section below). 

The *distributive law* can be difficult to recognize in new applications. In pre-algebra, they'll apply this law (which they may know in their bones in the arithmetic setting from doing mental multiplication) to symbols, and it won't seem familiar. Treat it as a new topic in every new situation, but help the kid make the connections back to familiar settings.

*Computational conversions between decimals, percents and fractions* may not have been so hard for your kid to learn in fifth grade -- but the application of these conversions in story problems are often a sticking point. Does your kid really understand that 8% sales tax means that a little cost -- and only a little -- is being added onto the price of something? A lot of 8th graders still can't compute sales tax.

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from Catherine:

http://www.kitchentablemath.net/twiki/pub/Bloggers/StickingPointsInAreaAndPerimeterLogPage/stickingpointperimeter.jpg%BR%

Christopher and his friend next door have a terrible time determining the perimeter of a figure like this. I'm writing this on March 9, 2006. Christopher is 3/5 of the way through 6th grade and he still can handle these figures. ([[http://www.kitchentablemath.net/twiki/bin/view/Kitchen/StickingPointsInAreaAndPerimeter][sticking points in area and perimeter]])
%BLACK%

<font size = +1>Algebra</font>

*Logarithms* are EXTREMELY confusing. This is when many children just plain give up on math. Get them all the help they need with logs; don't let your kid slide over them. Expect the inflexible knowledge stage for this topic to last a very long time.

*Incorrectly distributing powers, square roots, and reciprocals.* A very common -- and persistent -- error among algebra students are incorrect applications of the distributive law. For example:
(a+b)<SUP>n</SUP> = a<SUP>n</SUP> + b<SUP>n</SUP>, for values of n other than 1, is a VERY COMMON ERROR that gets entrenched in a kid's head very easily (probably because it makes hard problems so much easier -- too bad it isn't true!). To fix this, point it out explicitly every time you see the student make the error; show examples with specific numbers to help him see why the 'rule' he are trying to use is wrong. 

*Factoring polynomial expressions*.



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