Professor Homunculus sez:

And I like this comment because I don’t get it.

Once kids can competently assemble mathematical values and operations instruction symbols into correct problem-concordant statements, and then they become intimately familiar and comfortable with this, they can then more readily learn how to connect and apply these processes to new areas, such translating English language statements into these familiar expressions.

Suppose somebody was hired to translate English documents into French. Wouldn’t it be necessary for them to be able to write in French?

In learning how to do word problems per se, it is critical to focus on the translation skill set by starting with and reinforcing simple things. We can see a problem, 30% of 9 is __? What needs to be taught is “of” always means “times”, and you have kids write
30% x 9 = .3 x 9, not just the answer 3.

Similarly, “x per y” means “x divided by”. More than/longer than/older than/faster than is plus, less than/shorter than/younger than/slower than is minus. It’s not enough for teachers to present these translations, kids need to WRITE simple translation statements, and a goodly number of them, repeatedly. When the eyes see something, and the ears hear, these trigger neural processing. When the student’s brain commands the hand to write, this involves different neural processing. Then when the student’s eyes see what his brain has commanded his hand to do, that’s a new feedback process. If it is incorrect, and then is corrected, this triggers additional feedback processing. Repetition “embeds” the skill.

Word problems are hard for even bright students. Many teachers say that you can start kids early on word problems without their having to write equations. This is sensible, but they should stress equation writing as well early on. Why? Once we learn a method, we apply it every time we can: if it works, we use it. But what happens is, we eventually encounter problems that need different methods. S, we’ve solved simple problems that could have been done algebraically, but we used our earlier methods, since they worked, and now we are stuck because the earlier methods don’t work, and we haven’t developed the necessary algebra skills that we should have, starting with easier problems that didn’t require algebra, but were algebraically soluble. We have to make a jump now, rather than a small-step transition. Too many kids can’t and then they decide they’ve taken enough math.

Let’s take a simple problem. The Neon gets 3 miles per gallon more than the Camry. The Neon gets 28 miles per gallon. How many miles per gallon does the Camry get?

This doesn’t require algebra. Kids can mentally translate the information without equations in their minds, and should be encouraged to do so. But suppose they also understand this:

N mi/g = 3 mi/g + C mi/g and N mi/g = 28 mi/g

I’ve used slashes here, but in a classroom presentation or textbook mi would be a numerator, and g would be a denominator, with a horizontal division bar: a crucial element in teaching applied mathematics. I did so many homework assignments, and have seen so many homework assignments done by other people that didn’t have units. I’ve seen tests that specified the units, so students’ answers didn’t require the students to state them. Bad practices.

Units are CRITICAL GIVEN INFORMATION. For example, if you have a problem that gives information in miles and minutes, but asks for miles per hour in the answer, and the student gets 3 miles in 4 minutes, the following makes sense: 3/4 mi/min x 60 min/hr. The minutes cancel out => mi/hr. But 3/4 mi /min / 60 min/hr generates mi-hr /min^2, which doesn’t make sense.

Looked at another way, when units are given in a problem, THAT’S PART OF THE GIVEN INFORMATION. Don’t teach, or allow, kids to “slough this information off.”

Finally, answer checking is given short-shrift in mathematics education. Every teacher gives lip service to answer checking, but how many teachers DUN students who show correct answers without showing answer checks? This is a major teaching mistake. Whether one uses an alternative method or reversing operations or substituting an answer value for the original equation’s variable, more skill is gained through the extra work, work-product quality improves, and self-confidence is gained. So teachers, take points off if answer checks are not shown. If students don’t know how to answer check, teach them how to do this. You’ll be pleased at the improved performances of your students.

Myself, I never hated it but I thought of it as something a little magical and something I couldn’t do because I wasn’t a math person. Several years of being a retail clerk cured me of that and homeschooling my daughter is opening me up to math even more.

Growing up, I had a Chinese-American acquaintance who was in a different league from those of us who were merely good in math. His dad, a radio and TV repair shop owner, used to devise hard problems for his son to do, and he had his son do all C problems, most of which the teachers did not assign.

My friend didn’t put a special effort into all subjects,but he worked exceptionally hard in math.

I know an Indian-American family in which the mother has a math degree from one of the Indian Institutes of Technology. Two sons are math aces, the elder having been invited to the USAMO summer team-tryout camp, the other very likely to do so in a few years. They’ve done an amazing amount of math at home.

The immigrant parents in both instances have viewed mathematics as a skill that opens doors of opportunity in this country. To take advantage, you have to make a beyond-ordinary time and effort investments.

For many kids, I don’t know if it is “healthy” to do 2+ hours of math nightly, in addition to other homework. One of the advantages of home education was that my kids did 3 hours +/- of math every morning, bright and fresh. They weren’t toiling at night. We had shorter-than-school-days, 5 hours, compensated by Saturday mornings and 44-48 week schedules. Actually, they didn’t have a fixed math schedule each day, but set their own topic-study and exercise-completion goals each day, and when they felt they had achieved what they planned, or didn’t but felt they were getting tired and losing concentration, they stopped for the day.

In this process, they debunked the old theory that kids only have 45 minute attention spans. I had a 45-minute math attention span growing up. But this was because school entrained me to have this limit. In my first college term, I had no capacity to focus on calculus textbook reading and problem sets for 2 hours, much less the 3 to 4 hours that earning an A required. But as a former high school runner, I applied the same concept of progressive training, this time to get my brain “in shape”, to develop mental stamina. It worked.

This idea was reinforced by a friend who was a competitive swimmer. In late elementary school he spent 60 minutes swimming every afternoon. This increased to two hours in high school. In college he had a one hour workout every morning and two and a half hours in the afternoon, swimming a minimum of 5000 meters (sprint interval days) and a maximum of 10,000 meters (endurance days). Ten thousand meters is about 6 1/2 miles. (He also earned one of the first biophysics B.S.s in the country more than 30 years ago: working with biophysics researchers, he created his own novel major. He worked as hard in math and science courses as he did in the pool.)

You see this phenomenon in soccer, tennis and basketball. These really talented kids, the ones whose parents don’t have to pay for college, because their children receive full-ride scholarships at NCAA Division I schools, aren’t just playing seasonal interscholastic sports, they’re playing year-round for club teams, and they aren’t putting in 50 minutes a day, they’re doing two and a half hours a day. Vic Bradenton’s tennis school in Florida for future pros has four hours of daily training. The onsite high school academic program is molded to fit around the athletic program.

We all have known some really talented musicians. They’re good because they play a lot, every day.

Should schools have double-periods for math? I think so, at least for kids who have a natural interest in the subject. I don’t mean covering twice as many topical units in a year in a blitz acceleration scheme. I mean going into topics in greater depth and exploring them. I mean giving kids more time to do exercises during class with a math-knowledgeable resource person present to help them overcome confusion and misunderstanding. I mean giving kids enough time to enjoy “Aha! Now, I get it” opportunities, which is what makes taking on the challenge of learning mathematics satisfying.

Brian sez:

Wow, these are phenomenal comments and thoughts, Mark. I really appreciate what you have added. I am 100% with you. I wish I had had some encouragement like that when I was younger, myself. I still have a bad attention span, and am doing my best to correct that, now in my fifties. I also believe that there is no age limit for when that cannot be corrected.

Your last paragraph is so insightful. Unfortunately, when schools do get extra resources or time, they tend to squander it on exactly what you mention, they don’t go into depth, help students discover meaning or develop insights - they “cover twice as many topical units in a year in a blitz acceleration scheme.” I don’t know why administrators so often suffer from arrested-development of the skills they need most.

Your comments are always welcome. I believe they can be a great inspiration and motivation to other parents. I know they are to me. Hotcha!

I’m new here, so I don’t know if you’ve mentioned hotmath.com. For people whose kids are taking math in school, or even home-schooling, using standard high school textbooks, Hotmath provides solutions to all the leading textbooks’ odd-numbered problems in high school math. There was initially some controversy, because although college textbook publishers had been publishing student solutions manuals for many years, this notion of giving younger students this kind of information was foreign to the secondary education establishment. But Hotmath gained legitimacy, as textbook publishers did not object, and many teachers submitted solutions to aid the enterprise. Upshot: we’re not going back.

Actually, with teachers being given solutions manuals to help them with problems they don’t get, the take-home lesson of students being given access to this information is obvious.

I’ll be passing this book on to my son, reading relevant chapters as he progresses through math. Thanks for turning me on to this book.