Amy blew up ten balloons, and gave six to Carlos……..MEANS 10-6……
After many years of teaching elementary math, and trying to get better at it each year, I finally realized that NO text book I have ever seen teaches the MOST important math skill!
So what IS this skill? It is the ability to look at an English-language sentence (or sentence written in whatever language a student speaks), and be able to TRANSLATE that sentence or phrase into math language. So many students are unable to do even the first step of a story problem because they do not have this skill. Unfortunately, the text books are written by people who have never lacked this skill, and these writers don’t understand that this is a skill which for most people, must be LEARNED.
A college professor I know recently told me that lack of this exact skill is one of the main causes of why many students entering college need to take remedial math courses.
It is clear to me that if elementary teachers were to spend only between five and ten minutes DAILY teaching this specific skill, far more students would feel really competent in math. To this end, I have gone through our Grade 3 math book, and have prepared a page in my sidebar entitled Example Mathematical Expressions for Third Graders. Or, find it HERE.
In my opinion, there are two ways this could be taught to insure one hundred percent participation. Five expressions could be written on the chalkboard, or prepared in advance on a photocopied paper, with spaces left for answers. Children should be given only ONE problem at a time to do. Feedback should be immediate, BEFORE moving on to the next problem.
An easier way, and a way that is more fun for many kids, is to take small individual chalk boards. Have each child write their answer on their chalk board, and hold it up as soon as they are finished. The teacher can say, “Yes!” or “Try again.” Once every child has tried for a minute to get the answer, the teacher needs to put the answer on the board, and explain it. Then move on to the next example. Do about five examples a day, and the children can keep the little chalk boards in their desks. Using these little chalk boards can be really FUN for kids!
Comments?
–Eileen
,
American Education Issues,
American Math Education,
Boys,
British Education,
British Math Education,
Calculus,
California,
California Education,
Children with Math Anxiety,
Chinese Education,
Chinese Educational Issues,
Chinese Math Education,
Colorado,
Colorado Education,
Curriculum,
Curriculum Content and Issues,
Education Issues,
Educational Issues,
Elementary,
Elementary Math Curriculaum,
Elementary Math Curriculum,
Elementary School,
Elementary School Issues,
England,
Europe,
European Education,
European Math Education,
France,
French Education,
French Math Education,
Fun with Math,
German Educational Issues,
German Math Education,
Germany,
Girls,
Grade 1,
Grade 2,
Grade 3,
Grade 4,
Grade 5,
Grade 6,
Grade 7,
Grade 8,
High School Math,
Hong Kong,
Hong Kong Education,
Ideas,
India,
Indian Education,
Indian Math Education,
Innovative Projects,
Interesting Story Problems,
International Math Rankings,
International Schools,
Italian Education,
Italy,
Japan,
Japanese Education,
Japanese Education Issues,
Japanese Math Education,
JapaneseMath Education,
Making Math Fun,
Math,
Math Achievement,
Math Anxiety,
Math Curriculum,
Math Exams,
Math Homework,
Math Problems,
Middle Eastern Math Education,
Middle School Math Curriculum,
New York,
New York Education,
New Zealand,
New Zealand Education,
North American Math Education,
Pakistani Math Education,
Parenting,
Parenting Issues,
Parenting Skills,
Parents,
Parents Helping Children with Math,
Schools,
Schools and Math,
Secondary Math Curriculum,
Singapore,
Singapore Education,
Singaporean Math Education,
Solutions for Math Anxiety,
South Africa,
South African Education,
South African Math Education,
South American Math Education,
Spain,
Spainish Education,
Spanish Math Education,
Student Problems,
Students,
Students' Concerns,
Students' Dilemmas,
Students' Worries,
Teaching Math,
Testing in Math,
Texas,
Texas Education,
Thai Education,
Thailand,
Third Graders,
Third World,
Thoughts,
U.S. Education,
United States of America,
Universities in America,
Universities in England,
Universities in Europe,
University Entrance Requirements,
education,
elementary math textbooks,
innovative solutions,
math education,
school,
teacher,
teachers,
think outside the box,
young children
This entry was posted
on July 8, 2009 at 12:14 pm and is filed under American Education, American Education Issues, American Math Education, Boys, British Education, British Math Education, Calculus, California, California Education, Children with Math Anxiety, Chinese Education, Chinese Educational Issues, Chinese Math Education, Colorado, Colorado Education, Curriculum, Curriculum Content and Issues, Education Issues, Educational Issues, Elementary, Elementary Math Curriculaum, Elementary Math Curriculum, Elementary School, Elementary School Issues, England, Europe, European Education, European Math Education, France, French Education, French Math Education, Fun with Math, German Educational Issues, German Math Education, Germany, Girls, Grade 1, Grade 2, Grade 3, Grade 4, Grade 5, Grade 6, Grade 7, Grade 8, High School Math, Hong Kong, Hong Kong Education, Ideas, India, Indian Education, Indian Math Education, Innovative Projects, Interesting Story Problems, International Math Rankings, International Schools, Italian Education, Italy, Japan, Japanese Education, Japanese Education Issues, Japanese Math Education, JapaneseMath Education, Making Math Fun, Math, Math Achievement, Math Anxiety, Math Curriculum, Math Exams, Math Homework, Math Problems, Middle Eastern Math Education, Middle School Math Curriculum, New York, New York Education, New Zealand, New Zealand Education, North American Math Education, Pakistani Math Education, Parenting, Parenting Issues, Parenting Skills, Parents, Parents Helping Children with Math, Schools, Schools and Math, Secondary Math Curriculum, Singapore, Singapore Education, Singaporean Math Education, Solutions for Math Anxiety, South Africa, South African Education, South African Math Education, South American Math Education, Spain, Spainish Education, Spanish Math Education, Student Problems, Students, Students' Concerns, Students' Dilemmas, Students' Worries, Teaching Math, Testing in Math, Texas, Texas Education, Thai Education, Thailand, Third Graders, Third World, Thoughts, U.S. Education, United States of America, Universities in America, Universities in England, Universities in Europe, University Entrance Requirements, education, elementary math textbooks, innovative solutions, math education, school, teacher, teachers, think outside the box, young children.
You can subscribe via RSS 2.0 feed to this post's comments.
You can comment below, or link to this permanent URL from your own site.
July 8, 2009 at 4:15 pm
Absolutely true. I volunteered in my son’s 2nd grade classroom tutoring math this year and 85% of the students couldn’t do this at all the none of them could do it without making regular errors in subtraction or division where order matters. What we’ve done to work on it with our son is to make a short list (6-8) of word problems and have him first go through and turn them into equations. We check those first and then he solves the equations. I will say that I think part of the problem has been the way in which the math facts are introduced in his curriculum (everyday mathematics). They learn fact families (e.g.2,3,6 is a multiplication/division family and the facts are 2×3=6, 3×2=6, 6/2=3 and 6/3=2). This is great for recognizing similarities between problems, but it seems to cause confusion when they have to figure out the order.
July 8, 2009 at 5:56 pm
Singapore math is especially fun in this regard. They’ll ask questions like:
Mitchell jogged 5.85 km on Saturday. He jogged 1.7 km less on Sunday than on Saturday. What was the total distance he jogged on the two days?
I have, so many times, seen the answer to “how many miles he jogged on SUNDAY” written as the answer rather than the total. I keep going, “Your math is right, but your answer is wrong because you answered the wrong question!”
July 13, 2009 at 5:19 am
I would like to suggest that it is most important that the students understand the story in their story problems. If they focus on understanding what is happening,not just taking numbers out of it and putting them in the right order, they have the opportunity to make sense of the math they are doing. Once they have a conceptual understanding of the story, then they can more easily represent it symbolically with a number sentence.
July 14, 2009 at 8:00 pm
Good post! I think your idea is a little broader than just translating from English to math. So if I may, I’ll make a few comments and also ask a few questions on your exercises.
Problems like 6, 15, 16, and 17 seem to fit exactly into translating from words to math. The answers you are looking for, I presume, are 2 ´ 15, 3 ´ 19, 2 ´ $850, and 3 ´ 12. Is this what you had in mind? (Possibly I’m misinterpreting something.)
And for problem 21, it would seem that the answer would be (8 ´ 250) / 1000. Do third graders understand the numbers written this way? And can they actually work out the answer? That seems more like fifth or sixth grade work to me, so possibly I’m not understanding it quite right.
Problems one and two could be translated directly into 5 + 8, and 10 – 6. However these two examples (and others) suggest something else to me. For number one, “Dan chose five papayas and eight coconuts”, the teacher could ask the class “Is that a problem? Can we solve it?” I would expect some kids would say, “Sure, it’s a problem. You add. The answer is 13.” Other kids might suggest that it doesn’t seem quite like a problem. The teacher could then go ahead to explain that, no, it’s not quite a problem. Here are two different problems:
pb A: Dan chose five papayas and eight coconuts. How many pieces of fruit did he have then?
pb B: Dan chose five papayas and eight coconuts. How many more coconuts than papayas did he choose?
Could we have a problem C? a problem that starts out the same, “Dan chose five papayas and eight coconuts” yet is somehow different than either problem A or problem B. I’m not sure.
It seems to me that at all levels of math, at least beyond some starting point, it would be beneficial for students to give some thought to the problems, other than just trying to solve them. I think there is always a tendency to do problems according to their superficial appearance. This is certainly understandable. If you don’t know how to do a problem you have to try something, even though it might be wrong. So if it looks like an addition problem, why not add?
So here is a suggestion. Spend a part of math time now and then “building problems”. For example present to the class the statement: “Amy blew up ten balloons and gave six to Carlos.” and ask, “Is that a good problem?” Then proceed to develop the idea that maybe it’s not a good problem yet. The problem, as stated, does not contain a question. We certainly might guess what the question is, but so far the question is not stated. Maybe the problem is this:
“Amy blew up ten balloons and gave six to Carlos. Now how many balloons does Amy have?”
This is certainly what we would expect. But does it have to be that way. Is this also a good problem? If not, why not?
“Amy blew up ten balloons and gave six to Carlos. How many balloons do the two children have altogether?”
Something very much like this is in the problem in the comment above by Mrs. C. The problem could be built up in steps.
Mitchell jogged 5.85 km on Saturday.
Ask the class “Is this a good problem”. I presume most children will say it’s not complete. Now build it up with one more step:
Mitchell jogged 5.85 km on Saturday. He jogged 1.7 km less on Sunday than on Saturday.
Again ask the class, “Is it now a good problem?” I presume there will be a natural tendency to say yes it is. But we can point out that so far we have not put in a question.
As Mrs. C implies, the question that will be inferred is not the only possible question. It may be quite natural to complete the problem this way:
Mitchell jogged 5.85 km on Saturday. He jogged 1.7 km less on Sunday than on Saturday. How far did he job on Sunday?
But it doesn’t have to be that way. In this case it’s
Mitchell jogged 5.85 km on Saturday. He jogged 1.7 km less on Sunday than on Saturday. What was the total distance he jogged on those two days?
My thought is that occasionally “building problems” one step at a time like this would be worthwhile. I do something like occasionally in my algebra classes. When studying written problems I have started out with:
A train leaves Chicago at 9:00 am going east at 50 miles per hours.
and ask the class if this is a good problem. There is general agreement that obviously it is not. So I add:
A second train leaves Cincinnati at 10:00 am going 60 miles per hour.
and again ask if this is a good problem (a “well-posed problem” if I have mentioned that term).
I conclude from the few times I have done this that students do have an intuitive sense of when a problem is well posed, and when it is incomplete.
A “well posed problem”, by the way, is a term used occasionally in math. Is there a place in math teaching for poorly posed problems? Is there a place for trick problems? Is there a place for unsolvable problems?
My guess is that students at all levels would profit by thinking about bad problems. But they would have to be presented carefully, so as not to cause confusion.
Consider this. I remember doing rows of problems like this, probably in first grade.
9 7 13 5 11 17 8 7 4 5
- 3 - 6 -8 – 6 – 9 – 8 – 9 – 7 – 3 - 8
If you look closely you will see a few problems in this set that perhaps don’t fit in first grade. What about 5 – 6. Does that belong? What will a first grader do with that problem? Or 8 – 9, or 5 – 8. Do problems like these belong?
I don’t know. I didn’t get problems like this in the first grade. What I have in mind is that after the basic idea of subtraction is understood, an exercise like this perhaps could serve a valuable purpose. Until we get into signed numbers (which was ninth grade for me, apparently earlier now days) problems like these would probably be interpreted as typographical errors, or perhaps as trick questions. Would it be desirable to label these problems as “not well-posed”? Or to simply label them as “bad problems”? Would it be desirable to remind kids that we can only take a small number from a big number, not the other way, and instruct kids to simply cross out the problems that are impossible to do.
The advantage I envision of doing something like this is to focus attention on the nature of problems. I would think it would be beneficial to have kids simply cross out the problems that can’t be done. It might or might not make sense to tell the kids that later we’ll used signed numbers and then we can do problems like this.
The idea is that if we train kids right from the start to be on the look out for “bad problems” we are doing them a favor. We are laying a foundation for thinking about the problems that they are doing. (Can we lay claim to “critical thinking”?)
I took a tangent away from the idea of translating words to math. I agree that is a very important skill that needs to be developed. I wonder if “building problems” and thinking about problems would help in this.
July 14, 2009 at 8:03 pm
As I thought might happen my example of subtraction problems in my post above didn’t come out as it did in my writing. All those numbers, if spaced right, are simple subtraction problems. Among the normal problems, like 9 – 3, are impossible problems, like 5 – 8. Hope that makes some sense.
September 11, 2009 at 5:26 am
Without a doubt, the single most useful skill I learned in my college education was “degree of freedom analysis” which is, largely, reading a story problem and writing down in mathematical terms what the sentences tell you (“knowns”) what they don’t tell you (“variables”), and then writing out the formulas that connect them. As a teaching assistant for thermodynamics in grad school I became painfully aware of how the inability to translate a paragraph of the English language into math was a major hindrance in American education. Chemical engineering thermodynamics tends to entail relatively simple math and relatively difficult conceptualization, making English to math translation very important. Unfortunately, many engineering colleges are focusing on using computer programs rather than learning mathematical literacy. This only results in confirming the trite computer programing axiom “garbage in garbage out”.