Three problems

KitchenTableMath supplies urges readers to compare and contrast three problems from the last page of three different fifth-grade textbooks:

Primary Mathematics 5B (U.S. Edition):

18. A fish tank is 2/5 full after Sara poured 14 gal of water into it. What is the full capacity of the tank in gallons?

Saxon Homeschool Math 6/5 3rd Edition:

Change each of these base 10 numbers to base 5:
a. 31
b. 51
c. 10
d. 100
e. 38
f. 86

Math Trailblazers Grade 5:

4. Write a paragraph comparing two pieces of work in your portfolio that are alike in some way. For example, you can compare two labs or your solutions to two problems you solved. One piece should be new and one should be from the beginning of the year. Use these questions to help you write your paragraph:

Which two pieces did you choose to compare?

How are they alike? How are they different?

Do you see any improvement in the newest piece of work as compared to the older work? Explain.

If you could redo the older piece of work, how would you improve it?

How could you improve the newer piece of work?

Of course Trailblazers must have math-ish problems on earlier pages, but it is a constructivist curriculum.

Via Instructivist.

About Joanne


  1. wayne martin says:

    I would like to see all text books put on-line, either at the school WEB-site, at the publisher’s WEB-site, or perhaps on each State Department of Education’s WEB-site.

    There is no excuse that parents and taxpayers shouldn’t be able to see the text books that government schools choose for their kids.

    Being able to quickly, and easily review competing texts should be a mandate of school boards everywhere.

    There is a lot of evidence that text books are full of errors and “agenda”. Allowing the public to see what is going on could provide the stimulus for a generalized clean up of text books.

  2. As much as we all want to jump on the last “problem” as an example of What’s Wrong with Math Teaching Today, let’s be objective and remember that context is everything. What were the math problems that appeared earlier in the text? Were they problems like the first two mentioned above? Where they more rigorous, less rigorous, or what? Just because a student might be making a portfolio doesn’t necessarily mean the curriculum is any better or worse than anything else, and it’s not necessarily informative to compare just what happen to be the last problems in their respective books.

  3. wayne martin says:

    > What were the math problems that appeared earlier in the text?

    A good point. The following links provide hints about this text book:

    Math Trailblazers:

    The following proving insights about the Author–

    Phillip Wagreich:

    The Chicago Secondary Mathematics Improvement Project is designed to help Chicago Schools: * implement Chicago’s new graduation requirement of three years of high school level mathematics; * provide a quality mathematics education for all students; and * align curriculum and teaching practices with city, state and national goals and assessment practices.

    Institute For Math and Science Education:

  4. Richard Nieporent says:

    I am sorry Robert, but I don’t see how anyone can justify assigning a “problem” like that in a math class. The purpose of math is to learn how to solve problems not how to compare them. Also, what is this about a portfolio? You put together a portfolio in an art class not in a math class. If schools could be sued for malpractice this would be prima facie evidence for it.

  5. Wayne Martin, bravo on that suggestion.

    “Public education” without transparency is a fraud.


    As for this particular instance, I wonder whether the publishing industry’s need for text book obsolesence doesn’t tend to inspire folly within the pages of math books.

    A science or social studies text can naturally become outdated rather quickly, no?

    “Updating” an English textbook can be marginally justified by the discovery of new literary voices or new works by some old voices.

    I’m not a mathematician but I wonder if anything much ever gets discovered in the field that is relevent enough to K-12 mathematics to necessitate new books… Which leaves only the method of presenting the concepts as justification for asking schools to spend the millions to replace their books.

  6. Richard — You can put a portfolio together any place there is a body of work from which to select. Just because it’s done in an art class doesn’t mean it’s bad pedagogy elsewhere. I’m not defending whatever it is that the Math Trailblazers curriculum does — I’d never heard of it before this article, though I’m curious to check out the links that Wayne provided. But the logic of portfolios are done in art class > math is not art > math shouldn’t have portfolios doesn’t really fly.

  7. Walter E. Wallis says:

    Paper? Paper? We don’t need any stinkin’ paper.
    Hundred dollar laptop and recursive lessons, variable snap tests. 59 cents to correct an error.

  8. Well, it appears the glowing reviews of Saxon may be correct. I didn’t learn to work with bases until Honors Algebra II (algebra II/trig mix). I agree that the Trailblazers problem is unreal but Primary 5B seems to be on course for preparing a 7th grade algebra one entry (which is considered advances).

  9. Richard Nieporent says:

    But the logic of portfolios are done in art class > math is not art > math shouldn’t have portfolios doesn’t really fly.

    Are you serious, Robert? Can you explain what a math portfolio is? Art is judged by its aesthetics. The purpose of an art portfolio is to show you best work in a number of different subjects such as landscapes, portraits, etc. and different media such as pen and ink, water colors, etc. How do you do something comparable in math? I hope you are not going to tell me that you select your best homework papers. This one shows my long division problems. And in this one I have solved these really nice algebra problems. That would be meaningless because math is judged by the accuracy of the answers. Nobody wants to look at homework papers, they just want to know your grades.

  10. Robert,

    I am pretty familiar with Trailblazers and yes, that problem is rather typical for the curriculum.

    Trailblazers teaches alternative algorithms (in a rather rote way, no less) and does not teach the standard long division algorithm at all (something the middle school teachers just love) unless the teachers stop and supplement.

    Unfortunately, curriculums like Trailblazers and Everyday Math are more aligned with many state tests which is why they are so attractive to schools. It’s a place where long essays now stand in for the old “show your work.” Calculators now save them from the “redundancy” of “kill and drill.”

    Mastery is not necessary because the student will be exposed again next year through something called “spiraling.” Unfortunately, the middle schools aren’t interested so much in that idea as they start to introduce something far less gentle known as “tracking.”

    Trailblazers requires lots of discovery and group work. Our fifth graders spent a couple of weeks with manipulatives on subractions. Yes, subraction. With manipulatives. In the fifth grade. A good portion of these students do not have their times tables memorized. But they can always spiral around and learn them again in the 6th grade. Until then calculators can help out with those pesky single-digit multiplication problems.

  11. wayne martin says:

    Someone should do a study of Math test scores as a function of the text book/method. I can’t believe that such a study hasn’t been done already, but I’ve never seen a reference to such a study.

  12. Robert, Richard is right. In many classes children now have ‘portfolios’. The math portfolios usually consist of tangental non-math work similar to what’s being asked for in the problem. Many math portfolios are distinguishable from english porfolios only in that the grammar and spelling tend to be better in the math portfolios.

    It is vile what the schools are doing to our children. My daughter and son grew up with us, we augmented their limited public schooling with all manner of additional education.

    My foster daughter did not grow up with us. Like my other children, she is in the ‘accelerated’ or ‘advanced placement’ program at her high school. The sheer volume of things she simply does not know is astounding. She is nigh on incapable of simple arithmetic without a calculator.

    And she gets good grades.

    Her math portfolio is a work of art–and it contains works of art. Sketches of famous mathematicians, posters detailing the lives of Newton, Galileo.

    And she’s a sophomore. In AP college prep classes.

  13. In middle school, my son had great SAT score but so-so grade in math. This is causing a lot of problem in high school (another story that I would not go into.) I asked him why he could not get good grade in math back then, he said it was a art class rather than a math class. Now I believe him.


  1. says:

    Not the bonus essay question…

    Actually, this, in and of itself, doesn’t strike me as an unreasonable thing to ask a student: Write a paragraph comparing two pieces of work in your portfolio that are alike in some way. For example, you can compare two……