Tips for tutoring 8th grade math

The 8th grader that I have been tutoring in math has been assigned to “advanced geometry” for next year. This is apparently the highest classification available in the local high school. Her father brought me a bottle of Champagne and said that all of the credit for this was due me. Of course it would have been rude to contradict him so I gratefully accepted credit for my student’s hard work.

Now that I have demonstrated the ability to claim success in this domain it is time to share what I have learned.

Basically the American K-12 math curriculum is so dull that it takes an almost inhuman effort to stay awake and focused while solving the pointless and repetitive assigned problems. As with flying, the crew concept improves performance. One crew member (the student) solves the problems while the other crew member monitors and offers reminders to (1) slow down, (2) write everything down, (3) make the smallest change to an equation at a time (e.g., don’t add 4 to both sides and divide by -3 in one step; that’s two operations and therefore one should rewrite the equation twice). The student will be trying to escape the pain and boredom by doing multiple steps in his or her head. This leads to errors that wouldn’t be made if the steps were simpler and the result of each step written down.

A lot of these problems are basically arithmetic, despite the fact that the subject is called “math”. The school expects exact answers from a calculator, but of course it is easy to be way off by pressing the wrong key. So I worked with this 8th grader on estimating techniques so that if there were a huge discrepancy between the mental or pencil/paper estimate and the calculated result it would be noticed before handing in the work.

In a competitive and lucrative marketplace for textbooks I would have thought that a great book full of real-world examples would be out there, but apparently our local school system didn’t find one. I’m wondering if the way that we’re teaching math actually is the right way. Can that be true? Just tell students “this is an abstract subject and if you want to get a good job one day you need to do everything we assign”?

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24 Comments

  1. G C

    April 19, 2017 @ 1:43 pm

    1

    Good points. I find my third grader requires frequent reminders not to skip steps (which cause mistakes), and doesn’t like to practice much. Unfortunately, in my experience with math, practice solving problems, combined with feedback from grading, is the best way to learn the material. All the problems are now “word” problems, which is certainly different from when I learned math, 40 years ago, when I didn’t see a “word” problem until probably junior high.

  2. Brian

    April 19, 2017 @ 2:05 pm

  3. Ivan

    April 19, 2017 @ 2:21 pm

    3

    I do not think there’s a magic bullet here.

    I also have my doubts wrt tutoring value, but it may be culturally reasonable to try that if the environment discourages learning, who knows, it may work.

    I had a friend at my college who had to change his major and the college after struggling for a year because he could not grasp the prerequisite math. How is it related to tutoring ? In the old country, you had to take entrance exams to be admitted. For a technical college, it was usually written math(algebra+geometry), oral math, oral physics or chemistry and an essay. My friend was heavily tutored in math and physics and managed to be admitted based on his entrance exam results, but could not survive later on.

    Back in the day in the old country, as far as I recall, the amorphous “math” was split into arithmetic (grades 1-4), algebra (grades 5-10), geometry (grades 6-10). Three dimensional geometry started I believe in grade 9. Introductory calculus at the end of the 10th grade was very light: limits and differentiation. There was no attempt to introduce secondary school students to more abstract notions like sets and other things of that nature.

    Funnily enough, the arithmetic textbook at the time was heavily based on a 200 year old Арифметика ( https://en.wikipedia.org/wiki/Leonty_Magnitsky ), with slightly rephrased problems. My grandfather owned a pre-revolution edition of that book, so I could compare both.

    There was very little attempt to “excite” a student about a math subject on teachers’ part. Some students due to their predisposition liked the stuff, some not so much. However, almost everyone tried to do whatever one could. I believe primarily because of still existing parental authority and peer re-enforcement — all your friend acted more or less similarly. Of course, there were students who refused to do much no matter what, but it was a tiny minority. In my class year, I remember only two out of maybe a hundred.

    Upon graduating, roughly 5-10% went to college, maybe 15-20% percent to vocational schools (for which 8 grade certificate was sufficient), with the bulk of the students going into blue collar occupations.

    Other people experience may vary depending on time and location.

  4. Brian Gulino

    April 19, 2017 @ 3:27 pm

    4

    My experience, both as family tutor and college teacher, is that the textbooks are scandalously bad. They are loaded with meaningless pictures and photos which do nothing to increase understanding, but the students have to wade through it anyway. One effective technique I learned was to recommend a none textbook book on whatever subject my students need to study. What did you think of your student’s textbook?

  5. George A.

    April 19, 2017 @ 4:15 pm

    5

    As a kid, I too struggled with Math and Chemistry. I found those classes dull because all that I saw was numbers and steps to memorize. However, History and Physics were the most interesting subject for me. It wasn’t until later on when I started matured enough and realized the connection between Math and Physics is when Math begun to get interesting — that was the aha moment for me and is when I started to excel in school.

    With today’s XBox and VR technology, I would expect schools to be creative and create a learning environment around those technologies. Use VR to link a math equation to real word use-case. Have students play with equations (plug different numbers, figure out and fill in missing values, etc.) and see what impact that has on the object in the VR. I would think this would not only give students a better understanding of the subject matter, but it will make them more interested and even creative. The money spent on such technology is a drop in the bucket compared to the overall school budget.

    @Philg: How do you compare this tutoring experience to when you were in school at the same age?

  6. GermanL

    April 19, 2017 @ 4:22 pm

    6

    I felt I was pretty good at math until I went to my expensive liberal arts college and had to take The Calculus (as I quickly learned, college prep schools in Florida were nowhere near as good as the ones the New England – MA, CT, NY, RI, etc – that many of the rich kids I went with attended).

    We had to use (and buy for $100!) the horrible Calculus textbook that the professor/chair of the department had published. It was a behemoth, like 1000 pages, and I don’t even remember we ever referred to it in class. The American professor teaching the class was so opaque about the concepts (it was obvious he did not want to be there) that I couldn’t follow or keep interest to be honest. And I was someone who quite enjoyed algebra, trigonometry, and especially geometry. I later learned that 90% of the class had already taken Calculus in high school, and were just taking it as filler in the freshman year, so nobody needed good lessons anyway. I didn’t flunk, but my grade was poor IIRC. So I retook The Calculus the following summer at my local state school. It was taught by a Bulgarian professor and to students who probably weren’t cut out for New England elite colleges (perhaps neither was I!). But anyway, the lessons were very clear even though not the most engaging and I aced the test. What’s sad is that I’m sure that Bulgarian professor made less money than the professor I had in the fancy schmancy college.

    I have to say though, all my other classes in college were superb and much better than what I saw at the state school. With the exception of The Calculus.

  7. GermanL

    April 19, 2017 @ 4:29 pm

    7

    btw – my apologies for the grammar mistakes. Spanish was my first language, then English, then French, and now Deutsch. I also get daily exposure to Russian. My brain is basically fried by the end of the day…

  8. dean

    April 19, 2017 @ 5:00 pm

    8

    Do not be suckered into paying private school tuition (after paying school taxes) for sake of calculus. Here in fly-over deplore-land K10 – 12 students take Calculus in public schools and receive advanced placement credits that are accepted by hard technical colleges (catch: public schools are funded mostly exclusively by real estate taxes, but cost per student is 1/4 of what philg reported was in Cambridge, MA). I observed local HS graduates with calculus placement credits having no problem to take advance linear algebra, differential equations, partial differential equations and signal processing classes in hard technical colleges from tough domestic and foreign professors.
    The thing about mathematical education tooling that surprised me ws lack of good mathematical programs for young children on Apple IPad platform. Programs that I downloaded do not go beyond idiotic icon graphics a-la postmodern social realism with minimal content.

  9. John R. Klein

    April 19, 2017 @ 5:00 pm

    9

    Philip,

    At the risk of getting attacked by some of the readers of your blog who are in the teaching profession, let me say that part of the problem, besides the hawking of text books by publishers (which is an aspect of capitalism that in no way benefits the consumer: why edition 10 when edition 2 sufficed? Built in obsolescence…), is the fact that american teachers rarely are trained in the structure of mathematics.

    Most of the K8 curriculum is (ideally) supposed to teach kids how to compute inside the real numbers (despite the fact that the real numbers are never defined at all, as say limits of Cauchy sequences). I am not saying that kids should be taught Cauchy sequences, but I am saying that K-8 teachers should at least have an idea that the reals exist as a nontrivial construction. Instead, the kids are handed a calculator and are taught “alternative” inefficient algorithms for doing division (cf. http://everydaymath.uchicago.edu/parents/understanding-em/alternative-algorithms/ ).

    Regarding high school, in my day (the old country of Brooklyn in the 1960s), there was an attempt to teach plane geometry in a somewhat rigorous way. I challenge you to find a modern text that treats geometry rigorously.

    Ivan: The Soviet system was structured in a completely different way and one should first realize that there was respect/interest for mathematics and science among the intelligentsia. Such a respect for really doesn’t exist in the US, except among the rarified few.

  10. Ivan

    April 19, 2017 @ 5:55 pm

    10

    @dean

    Right, I do not believe the secondary/high school you attend matters that much if at all. Maybe peer pressure in this country inhibits learning somewhat, I do not know, hence a private/charter school option that presumably eliminates this sort of negative influence.

    I attended three different schools, all of them in rural/semi-rural areas, mainly peasant/blue color workers environment. City schools were reputed to be better for seemingly obvious reasons. However, at the college, there was no observed correlation, based on admittedly anecdotal evidence, between the secondary school attended and college grades. The main filter was entrance exams, except for some affirmative action quota students(former servicemen, Central Asian republics students, etc). Even those had to score some minimal passing points.

    As I mentioned, there were no miracle handbooks or teachers, essentially no calculus before the college, but those who could master what one could call “historical” math learning process, progressing from the multiplication table to algebra and geometry and getting intuitive feeling for numbers, series and geo structures, did not have much trouble learning rather advanced calculus at college.

  11. Anonymous

    April 20, 2017 @ 1:34 am

    11

    “Real world examples” don’t exist for the students. In reality the number of people who will use geometry or calculus is frighteningly small, and they use it mostly to make money off of those who don’t. Knowing calculus and geometry is surprisingly useful outside of their actual usage. The real power is knowing what they are capable of, not actually doing it.

    In a world where facts are malleable and truth doesn’t exist. Math above basic arithmetic is suspect; just like thinking for yourself.

  12. mishka

    April 20, 2017 @ 8:56 am

    12

    > The real power is knowing what they are capable of, not actually doing it.

    > In a world where facts are malleable and truth doesn’t exist. Math above basic
    > arithmetic is suspect; just like thinking for yourself.

    +1

  13. Henry

    April 20, 2017 @ 10:23 am

    13

    At my kids’ high school, in supposedly one of the best school districts in the state, the “honors” freshman math class uses the most amazingly scatterbrained ‘textbook’ I have ever seen. It’s not even a textbook, its some sort of binder of ‘text’ that supposedly a prestigious prep school uses. Inspecting it finds a seemingly random and incoherent sequence of problems from algebra, geometry, and I don’t know what. Here you have to learn to factor using some obscure trick, then the next problem is some random piece of geometry. I don’t know what they think this will teach kids, except that ‘math’ is a random incoherent set of concepts.

  14. dean

    April 20, 2017 @ 11:05 am

    14

    Ivan #9 – 100%. Eons ago I went to a better specialized city high school probably in the same country you did and studied ‘mathematical analysis’ i.e. advanced calculus with proofs there but not sure how it advantaged me. I had to retake it in college in the old country anyway, except it was boring second time around. Not sure how intimate understanding of calculus proof machinery, smooth and discrete functions and mastery of boundary problems helped me per se.
    Henry #12 – to save your kids time in college make sure that they take advanced placement classes. They come with optional standard subject (advance placement) tests, separate from SAT subject tests. If test grade is high then these classes will count for college credits for analogous college classes. There are hosts of rigorous and interesting mathematical printed books for high school students, you can raise this issue with the teacher and maybe learn philosophy behind his choice of text.

  15. Ivan

    April 20, 2017 @ 11:10 am

    15

    @Henry

    You are correct about “incoherent sequence of problems from algebra, geometry, and I don’t know what”. That was my experience with “math” textbooks too when we went through the local school system “in supposedly one of the best school district”.

    I started to augment my daughter’s “math” menu as early as in the primary school when I believe number related intuition is built(or not), just like a foreign language study during childhood results in a much better outcome than after the age of puberty. Fortunately, she had some natural ability and interest, so I did not have to work too hard. In the high school she was completely self-sufficient. She might not have been if she relied on the textbooks of the sort you mentioned.

    It is also possible but much harder in my opinion for math inclined kids to catch up in the college despite the previously wasted 12 years. I heard they have some remedial math courses there nowadays.

  16. Ivan

    April 20, 2017 @ 11:39 am

    16

    @dean/Henry

    I concur wrt AP.

    We had, as I recall, three levels: “normal”, “college track”, and AP starting at the middle school. The primary school did not have any differentiation by interest/ability, and interestingly enough middle school study level placement was rather arbitrary to the degree that one might need to convince the primary school principal to place his/her child on a more advanced track rather than the default “normal”.

    I was not too happy about the AP math extbooks either, but that was then, now the situation may or may not be better and depend on the school district.

  17. Alex

    April 20, 2017 @ 2:05 pm

    17

    My son has been home-schooled. We decided to enroll him in our state’s “virtual academy” this year, to start preparing him for college. He’s nominally a 10th-grader, although his performance on standardized tests is usually above grade level. When we enrolled him, we were told that he would have to take “Algebra I” because he did not have a transcript showing he had completed this level of math education. When we asked that he be allowed to write the Algebra II test, to demonstrate that he had already mastered the material, we were told that it was not permitted. They would only allow him to write the test for Algebra I. Don’t ask me to explain the reasoning behind that decision.

    Fortunately, we had the option of not enrolling him in their mathematics program, so now he does virtual academy courses in other subjects, and continues his home-schooling math curriculum. We use a program called Saxon Math, which is popular among home-schoolers. The text books seem to be a lot better than the public school textbooks that our other kids used.

  18. J. Peterson

    April 20, 2017 @ 2:28 pm

    18

    I recall (well into grad school) the main things that made math useful to me were symbolic math software and Numerical Recipes.

    The first removed the drudgery and errors of grinding through the algebra. The second provided a well constructed box of tools to solve specific problems.

  19. dean

    April 20, 2017 @ 2:56 pm

    19

    Alex #16, it does not look like it makes sense to homeschool advanced student in high school, unless your local high schools are violent environments. High school students can take their own set of advanced classes similar to college environment, together with other advance students and not in their age group. It may make sense to either complete school requirements early because some colleges accept students after 11th grade, or take Advanced Placement classes and exams for college credits and participate in extra-curricular activities (my local school used to have great selection of after school sport and other programs that are too important for personal development). I believe my students were done with HS requirements by the end of 10th grade. I think homeschoolers also can participate in after school activities. I think homeschoolers who do not forefeit use of prime real-estate that is already paid for by their parents’

  20. Ivan

    April 20, 2017 @ 3:41 pm

    20

    “it does not look like it makes sense to homeschool advanced student in high school, unless your local high schools are violent environments. ”

    That’s my belief as well. Perhaps, filling some voids in the primary school years which were more of a void than substance would make sense, but by the middle school, kids should be able to navigate on their own, maybe with a bit of occasional help. At least, that’s my impression based on my own and other people experince in various geographical locales.

    Besides, homeschooled kids may be lacking some important social survival skills due to less exposure and interaction with their peers. On the other hand, some parents may consider it a benefit rather than a flaw.

  21. Alex

    April 20, 2017 @ 4:10 pm

    21

    Dean (#17) and Ivan (#18), I would not argue that there aren’t some disadvantages to home-schooling at the high school level (like not having physics or chemistry labs). However, many home-schoolers find that those disadvantages are off-set by the advantages, like working at your own pace and having more schedule flexibility. And, by most accounts that I’ve seen, home-schooled students are successful in gaining admittance to competitive schools and performing at a high level in college.

    Regardless, in our case the decision to home-school is based on special needs that our son has, and his preference. He gets plenty of exposure to peers in other venues, and will have further opportunities when he starts taking advanced classes at the local community college next year.

  22. Ed

    April 20, 2017 @ 5:21 pm

    22

    For calculus, I highly recommend “Calculus made easy” by Silvanus P. Thomson. Real-world problems right from the very start. It has been through a huge number of re-prints.

    https://www.amazon.co.uk/Calculus-Made-Silvanus-Phillips-Thompson/dp/1456531980

    “Mathematics for the million” also does a nice job of motivating the development of mathermatics. https://www.amazon.co.uk/Mathematics-Million-Master-Magic-Numbers/dp/0850363802

  23. Tom

    April 20, 2017 @ 6:07 pm

    23

    Perhaps there’s a niche for ‘Russian Math Clinic – when AP is weak and laughable’?

    I think I asked this question previously but let’s do it again: A service/tool like Wolfram Cloud is now available for about 10 bucks a month and can solve many of the routine symbolic math problems for you, and boost the solution of others. (Or pay a bit more and get Mathematica.) The underlying Wolfram Language can apparently be taught to kids.

    So, should parents or even the educational system transition to using this to do math instead of the usual pen and paper approach?

  24. tekumse

    April 20, 2017 @ 11:10 pm

    24

    There are in fact quite a few Russian Math Schools around Boston:
    http://www.russianschool.com/our-locations

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