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C. and his friend J. were assigned a "digit word problem" for homework. Example:
The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number.
source: purplemath
Neither of them had a clue how to do it. (I think digit problems had been taught in class that day for the first time. They'd both seen 1 or 2 worked examples.)
I tried to teach it by writing the number 23 and then, below that, writing a variable for each digit, then asking them something wildly imprecise like, "What would you have to do to these variables to get them to equal 23?"
Actually, as I think about it now, I don't know what words a person educated in mathematics would use to express this, so maybe you all can fill me in.
In any case, they did get the idea they were to let x equal 2 and y equal 3, and they were to write an equation using these values for x and y that would equal 23.
Looking at this arrangement of numerals and variables didn't help.
J. said, "x + y?"
"That's 5," I said.
"x times y?"
"That's 6."
They were stumped.
This is a classic case of inflexible thinking, for two reasons.
First of all, they've seen "plus" and "times" more often than they've seen 10x + y. Confronting a novel problem, their brains went straight to the more familiar instead of the less familiar.
Of course, that's what a brain ought to do: When you hear hoof beats on the bridge don't think of zebras.
The problem was that, pace Willingham, once they figured out the hoof beats weren't made by horses they didn't think of zebras next.
Both boys know how to set up word problems using letter variables; both boys are proficient (or close to) at solving number and consecutive integer problems. They can set up and solve coin problems, distance problems, and age problems; they can also set up and solve simple linear function problems.
They've been writing algebraic expressions for at least a couple of years.
And they were stopped cold by a simple digit problem.
I wish I'd taken notes on what I did next. I think I said something like, "What is the 2?" (That's imprecise, too, right? What's the correct language there? "What value does 2 represent"?)
I don't think I had to go as far as to remind them of the base-10 place system, but, on the other hand, maybe I did and I'm repressing it.
In any case, the moment it dawned on them that the key was recognizing the 2 as 20 they could both set up and solve digit problems rapidly and efficiently.
Knowledge transfer and generalization are core, unsolved problems in education.
update: Looking at this now, I think I should have had them start with the "3" and write an equation using y alone, then had them write an equation using x and 2 (or 20). If I wasn't simply going to show them a worked example myself (should I have? I don't know) I needed to break this down into the smallest, simplest possible steps. And I should have started with the ones digit, not the tens.
I think.
cumulative practice and problem solving
Speaking of which, I keep promising to write a post about one of the three most valuable papers I've ever read on the subject of teaching math:
The Effects of Cumulative Practice on Mathematics Problem Solving
Kristin H. Mayfield and Philip N. Chase
Here is the introduction:
Over 35 years of international comparisons of mathematics achievement have indicated problems with the performance of students from the United States. According to the latest international study, the average score of U.S. students was below the international average, and the top 10% of U.S. students performed at the level of the average student in Singapore, the world leader (Wingert, 1996). In addition, recent tests administered by the U.S. National Assessment of Educational Progress revealed that 70% of fourth graders could not do arithmetic with whole numbers and solve problems that required one manipulation. Moreover, 79% of eighth graders and 40% of 12th graders could not compute with decimals, fractions,
and percentages, could not recognize geometric figures, and could not solve simple equations; and 93% of 12th graders failed to perform basic algebra manipulations and solve problems that required multiple manipulations (Campbell, Voelkl, & Donahue, 1997).
These statistics reveal students’ deficits in the fundamental skills of mathematics as well as mathematical reasoning and problem solving. Indeed, poor problem-solving skills have been targeted by the National Council of Teachers of Mathematics (Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1980; Carpenter, Kepner, Corbitt, Lindquist, & Reys, 1980; Kouba et al., 1988; National Council of Teachers of Mathematics, 1989, 2000). Thus, it seems appropriate that current behavior-analytic research in mathematics education should address problem-solving skills as well as basic mathematics skills (e.g., Wood, Frank, & Wacker, 1998).
I'm hoping I can get to this later on today.
But first -- must write chickens chapter!
30 comments:
The trick is to know how to define the equations. This type of problem throws almost everyone. Once you've seen it, you'll never have a problem again.
If
T = ten's digit
O = one's digit
Then the original number is O + 10*T
So you have
(1) T + O = 7
and
(2) O + 10*T + 27 = T +10*O
Two equations in two unknowns
(2) 9*O - 9*T = 27
(1) O = 7-T
Plug (1) into (2) and solve
9(7-T) - 9*T = 27
63 - 9*T - 9*T = 27
18*T = 36
T = 2
so
O = 5
and the original number is 25
My motto is: "Find the equations and let the math do the thinking."
"... should address problem-solving skills ..."
OK, but who gets to define what these skills are? They surely aren't vague things like guess and check, draw pictures, work backwards, or do a dance.
I solved the problem almost the same way as SteveH, but to show off multiple ways of solving I'll write up how I solved my system of two equations.
9O - 9T = 27 => O - T = 3
and
O + T = 7
Summing these two equations gets you:
2O + (T-T) = 10 -> 2O = 10 -> O = 5
Thus 5 - T = 3 -> T = 2.
Original number = 25.
I can't find the official name of the solving by summation technique.
-m
Different set of search terms found a name for it.
"Addition / Elimination" method.
http://www.richland.edu/james/lecture/m116/systems/elimination.html
-m
I'm not very happy with this problem because you can exhaustively search it.
Two digit numbers whose digits add up to 7 gives this (number and reverse):
70 07
61 16
52 25
43 34
34 43
25 52 <--- difference is 27
16 61
I wonder what the point of the question is/was. This *does* allow for the great "I guessed and checked" response to how one solved the problem ...
-Mark Roulo
I used to love that technique. We would put the equations on top of each other and subtract (or add) to eliminate a variable.
O - T = 3
O + T = 7
----------
2*O = 10
O = 5
"I'm not very happy with this problem because you can exhaustively search it."
Actually, isn't that how they hope you will solve it? Don't they think that this shows more understanding than using equations?
Unfortunately, brute force or guess and check will fail miserably when you get to more difficult problems.
"... should address problem-solving skills ..."
OK, but who gets to define what these skills are?
I think they are trying to sneakily redefine content as pedagogy. What a math problem is and what consitutes a solution to a math problem is what mathematicians are experts in. From what I have seen once you agree that the teaching of such skills is a pedagogical issue, then the introduction of non-math problems or non-math solutions is justified as "pedagogy." Viola, no more content.
The trick is to know how to define the equations. This type of problem throws almost everyone. Once you've seen it, you'll never have a problem again.
Oh, that's interesting.
This is where afterschooling in a subject you're learning yourself has SEVERE drawbacks.
I have no idea what kings of things throw almost everyone.
If I'd known that, I probably would have just shown him a worked example.
If
T = ten's digit
O = one's digit
Then the original number is O + 10*T
Right.
This is what I was trying to get them to come up with.
I'd like to repeat that they had seen one or two worked examples in class already. I wouldn't have taken a "come up with it" approach if they hadn't.
I was trying to see if they could reconstruct what they'd seen in class based on knowing how to set up equations for other problems and a weak memory of what they'd seen in class.
I often use this approach with myself.
Don't know if it's a good idea or not, but it seems to be....
"Unfortunately, brute force or guess and check will fail miserably when you get to more difficult problems."
That's what I don't like about it. I expect (because I am a pessimist) that a lot of the kids will still be trying to exhaustively list all the combinations even for variations where that won't work.
I'd be fine with sneaking this problem in with a bunch of others for which you had to do a moderate amount of algebra. In that case, the alert students would benefit from seeing the shortcut.
-Mark Roulo
"I think they are trying to sneakily redefine content as pedagogy."
Well, they are redefining math. It's about academic turf. They decide what K-12 math is, and they come right out and say so.
This appeared at MathNotations last September.
- - - - - -
Interview with Prof. Lynn Arthur Steen - Part II
18. Here’s an innocent little question, Prof. Steen! The current conflicts in mathematics education are usually referred to as the Math Wars. In your opinion, what were the major contributing factors in spawning this conflict and how would you resolve it?
There are many factors involved. I think I can identify a few, but I have no confidence that I could resolve any of them.
One is the natural tendency of parents to want their children to go through the same education that they received—even when, as often is the case with mathematics, they admit that it was a painful and unsuccessful ordeal. This makes many parents critical of any change, most especially if it introduces approaches that they do not understand and which therefore leaves them unable to help their children with homework.
Another source were scientists and mathematicians who pretty much breezed through school mathematics and who were increasingly frustrated with graduates (often their own children) who did not seem to know what these scientists knew (or thought they knew) when they had graduated from high school. Our weak performance on international tests appeared to provide objective confirmation of these concerns, and they came to pubic notice just as the NCTM standards became widely known in the early to mid-1990s. Even though very few students had gone through an education influenced by these standards, the confluence of events led many to believe that the standards contributed to the decline.
A third source can be traced to the way in which the NCTM Standards upset the caste system in mathematics education. Mathematicians are accustomed to a hierarchy of status and influence with internationally recognized researchers at the top, ordinary college teachers in the middle, below them high school teachers, and at the very bottom teachers in elementary grades. The gradient is determined by level of mathematics education and research. So it came as somewhat of a shock to research mathematicians when the organization representing elementary and secondary school teachers, seemingly without notice or permission, deigned to issue "standards" for mathematics. Mathematicians would say, and did say, "we define mathematics, not you."
- - - - - -
Here's the summary:
1. Parents want what they had when they were growing up. [i.e. parents are stupid]
2. Some parents are too good in math to understand. [i.e. parents are stupid]
3. We define K-12 mathematics, not you. [i.e. professors of mathematics and engineering are stupid]
Prof. Steen won't solve the problem by saying that others are stupid and that ed schools get to decide what K-12 math is.
Beware of K-16 education.
One way to increase the level of mathematics education is to provide students as early as the 3rd or 4th grade with a successful experience with actual algebra.
In other words, these children should be shown how they can understand and solve equations such as 4x + 3 = 3x + 9 and how to solve verbal problems such as "Four times a number, increased by 2, is the same as twice a number, increased by 6. Find the number."
Obviously, this goal is not going to be accomplished if we try to teach these algebraic concepts via the traditional abstract instructional methods-- indeed those approaches are not very successful even with older students.
With a program called Hands-On Equations 3rd, 4th and 5th graders can indeed experience success wit the above concepts in a very short period of time. The students use pawns and numbered cubes on a flat laminate balance to transform the abstract concepts into a physical representation they can understand and solve.
The website www.borenson.com provides examples of verbal problems and how they can be set up and solved using the game pieces (go to the news/blog section of the website). You can also find video demonstrations on the same website of young students solving equations that many 9th graders cannot do.
It is not the teaching of decimals and computation that will raise the level of mathematics education in the United States. These are needed but these things do not inspire students to excel.
Provide the students with an opportunity to do real algebra, even the "simple" types of problems noted above and see a whole change in student self-perception. By the way, the news/blog section noted above recently posted a couple of "inspirational stories" by a teacher using the program.
In "Three Shinning Stars" she relates how three of her weakest 6th grade students in mathematics became the top achievers in algebra and began to teach the "gifted" students how to solve the equations.
Making Algebra Child's Play workshops are offered throughout the United States to help teachers become familiar with the Hands-On Equations program, and how to use it.
I don't know where to put this...Does anybody have experience with CPM in the high school level? THANKS!!!
"CPM in the high school level?"
High school? Yikes. It's awful for middle school. You would be better off with "check-book" math. At least that would give you real skills.
"...to provide students as early as the 3rd or 4th grade with a successful experience with actual algebra."
I'm a proponent of early algebra (simple concepts) instead of the early guess and check approach to solving equations, but I don't think this is the idea behind the program.
Early algebra isn't going to fix the fact that most reform math programs expect no mastery of skills at any one time. Early algebra is not going to ensure that kids know their times tables cold by the end of third grade, or that they know how to easily manipulate fractions by the end of fifth or even sixth grade.
"Obviously, this goal is not going to be accomplished if we try to teach these algebraic concepts via the traditional abstract instructional method -"
What is "the traditional abstract instructional method"?
"Obviously"?
That isn't obvious.
"It is not the teaching of decimals and computation that will raise the level of mathematics education in the United States. These are needed but these things do not inspire students to excel."
This is vague, self-serving, and unsupported.
"Hands-On Equations"
Patented?
It's one thing to come to KTM to advocate real learning in math, but quite another to come and push your products. $35 each for what seems like very thin books? You can buy training too? You will have to do better than this.
I guess I'm still a little naive, even after all of these years. People actually look at the educational market as a hot way to make money or a name for themselves. Just provide a "supplement" product. Supplementation is a hot market. I'll bet they make more money from training.
Products might have interesting parts or ideas to them, but then marketing hype and money take over. Like MMR, this program positions itself as THE product that will change everything. They have the answers. They have the solution. If you do this everything will be fine.
Everything won't be fine.
What is "the traditional abstract instructional method"?
You know, that stuff that mathematicians call their field. There's all sorts of ways to avoid exposing people to math, I suppose teaching them that math is an experimental science done with physical objects is as good as any.
Perhaps there's even a market for delta epsilon manipulatives.
You know what I've come to love about KTM? Catherine can end her post with the cryptic, "Must write chickens chapter!", and everybody knows exactly what she's talking about.
"traditional abstract instructional method"
Math is entirely an abstraction. That is precisely why it is useful; it's an abstraction of concrete, but often intractable, problems.
One of the techniques for teaching this abstraction is to provide examples that allow the student to access the abstraction by analogy. This is, in fact, what people are referring to when they talk about "real world math". For those who might be a bit hard of thinking, "analogy" is basic form of abstraction.
How, exactly, do you propose to teach an abstraction without abstraction?
"Perhaps there's even a market for delta epsilon manipulatives."
There is, but it is a very small market.
The market for that joke is a very small one, too. But it made me laugh.
8-)
I wish I knew what the chickens chapter was about????
>I wish I knew what the chickens chapter was about????
I hope it is about chicken camp! (Hint, hint).
You know what I've come to love about KTM? Catherine can end her post with the cryptic, "Must write chickens chapter!", and everybody knows exactly what she's talking about.
Oh, that's funny!
I never thought of that.
I do like non sequiturs, though, I must say.
Maybe especially in a blog.
Why is that?
Because you're right ---- there's something fun or entertaining (or something) about a serial form where people have been around each other long enough that an out-of-left-field comment "works."
No idea what I'm talking about.
One of the techniques for teaching this abstraction is to provide examples that allow the student to access the abstraction by analogy.
YES!
oh boy, the chickens chapter is depressing
I probably have to stop eating chickens as a result of writing it
also eggs
very, very upsetting stuff
What "kings" of things?
I meant "kinds"
jeez
I haven't gotten around to FINALLY writing the cumulative practice post, have I?
This really has to be done.
Shucks, sounds like the chickens chapter is about the poultry industry, not about training chickens. The latter would be definitely upbeat, and humorous to boot. A friend trained a chicken to type for a TV commercial, and (at another time) to run a dog agility course. It was hilarious when we (the dog club) did a demo at a local fair, and the star attraction was the grand finale -- a chicken running the course, including jumps, teeter-totter, weave poles and tunnels.
Karen Pryor has suggested that the world would be a better place if heavily populated by chicken trainers ;-)
Marilyn Breland and Bob Bailey started "chicken camps" so I was hoping Catherine was writing about it....
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