Streamlining and upgrading math instruction

A federal panel examining K-8 mathematics education in the USA has made some forthright recommendations, according to this article in the NYT today. Unlike many federal panels, this one has an uncommon amount of common sense in its conclusions. For example, this finding that is striking in the way it refrains from choosing sides in the math wars:

Parents and teachers in school districts across the country have fought passionately over the relative merits of traditional, or teacher-directed, instruction, in which students are told how to solve problems and then are drilled on them, as opposed to reform or child-centered instruction, which emphasizes student exploration and conceptual understanding. The panel said both methods have a role.

“There is no basis in research for favoring teacher-based or student-centered instruction,” said Dr. Larry R. Faulkner, the chairman of the panel, at a briefing for reporters on Wednesday. “People may retain their strongly held philosophical inclinations, but the research does not show that either is better than the other.” […]

“To prepare students for algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency and problem-solving skills,” the report said. “Debates regarding the relative importance of these aspects of mathematical knowledge are misguided. These capabilities are mutually supportive.”

Say what? An appeal to actual research rather than anecdotes and personal biases when thinking about effective math teaching? Amazing. And this shocking discovery:

[T]he panel found that it is important for students to master their basic math facts by heart.

“For all content areas, practice allows students to achieve automaticity of basic skills — the fast, accurate, and effortless processing of content information — which frees up working memory for more complex aspects of problem solving,” the report said.

Dr. Faulkner, a former president of the University of Texas at Austin, said the panel “buys the notion from cognitive science that kids have to know the facts.”

“In the language of cognitive science, working memory needs to be predominately dedicated to new material in order to have a learning progression, and previously addressed material needs to be in long-term memory,” he said.

Why, it’s almost as if they think that mastery precedes creativity or something. And finally:

The report makes a plea for shorter and more accurate math textbooks. Given the shortage of elementary teachers with a solid grounding in math, the report recommends further research on the use of math specialists to teach several different elementary grades, as is done in many top-performing nations.

The article goes on to give some of the panel’s recommended benchmarks for mathematical skills in grades 3-7. There’s also a link to the panel’s report.

All I can say is that I hope math educators, prospective teachers (especially prospective elementary school teachers), curriculum designers, ed schools, school boards, and everybody else who is a stakeholder with some influence in this process are listening. We’ve got 2 years until our oldest starts kindergarten and she needs teachers and curricula who get math right.

[h/t God Plays Dice]


Filed under Early education, Education, High school, Math, Teaching

17 responses to “Streamlining and upgrading math instruction

  1. OMG – Whoever wrote this report I am IN LOVE!

    Daily, I watch the algebra students on my campus struggle to learn factoring, do any problems involving fractions, and they can’t even complete the square… why? Because most of them are reaching for their calculators to do ANY step. What’s half of ten … pull out the calculator … then square that … consult the calculator … ARGH!

    The semesters that I got the best results in beginning algebra were the semesters where I BANNED all calcualtors and forced them to learn the arithmetic. By the time we would get to factoring, they would have learned their multiplication tables once and for all. But with the push for a lot of application problems in classes, it is not so easy to continue to implement the calculator ban for beginning algebra. I wholeheartedly approach this BALANCED view that “develop conceptual understanding, computational fluency and problem-solving skills” are all important. They are.

  2. coderprof

    Given the number of elementary ed students that I see who don’t really get math (thankfully the PRAXIS filters out some – but some get through with weak skills), I’m very supportive of math ed specialists at the elementary level. More than any single curriculum change, this seems to have the potential to improve math education where it is currently in the worse shape.

  3. @Maria: Yesterday I had a calculus student who, on a quiz, couldn’t correctly multiply 3 times 5/3 without a calculator. In CALCULUS. And sadly I think we’ve all got stories like this.

    @Coderprof: That part struck me as being particularly insightful as well. We have math specialists teach the math courses from middle school on up, so why not in elementary school? Particularly since, unfortunately, elementary education majors are not well-known for their skill in or enjoyment of mathematics? Which is sad — I see from my daughter’s preschool teachers that it takes a huge amount of intelligence and skill, and an enjoyment of a wide range of topics, to teach little kids effectively, and yet so many students major in el. ed. to get away from hard work or because they just want to babysit.

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  6. Agreed on the comment about math specs at the early levels. Although it would be awfully nice if it weren’t necessary.


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  8. elementaryteacher

    Well, I am originally from Colorado. In Colorado, to get ANY elementary or secondary certification in ANY subject, you have to pass a competency exam that goes through Algebra. The reason is that ANY teacher can be called upon to teach math, and must be competent.

    Even though it’s a good idea, I don’t think it’s very likely that math will be taught by specialists at the elementary level, because it would be too difficult to implement, both logistically, and cost-wise. The only way I can see to do it is if team-teaching were implemented at every level. But that would mean that half of elementary teachers would have to be math specialists. The only other way would be to have a math specialist be hired in the place of one other teacher, who would teach math in all six grades. The classroom teachers would then have to split the one class that the math teacher would normally be teaching (for example, Grade Six), and teach one Grade Six class in addition to their first or second grade class (to replace the hour the math teacher was taking their class). This would mean the Sixth grade (for example) would have a different teacher every hour (an unacceptable situation in an elementary school).

    Dedicated Elementary Teacher Overseas

  9. I have a question for elementaryteacher – don’t the teachers have a planning period? A lunch period? Couldn’t this be taken into account for scheduling math instruction by a math specialist? Does every elementary teacher also teach PE? Art? Music?

    I don’t know – these are just the questions that come to mind when I read your response. I won’t claim to know how scheduling works at the El Ed level. I’m just concerned that “it can’t be done” is easy to say. There has to be a solution.

    Lastly – passing a competency test does not ensure that one understands the concepts well enough to teach them.

  10. X Teacher

    My offhand reaction would be that a lot of this could be done with math specialists giving instruction to elementary teachers. Or, just hand out the Burns books to the teachers. The problem isn’t just math. It’s also science. Most of the people who end up in elementary teaching are much more adept at verbal matters than at math and science. As such, they often do not have a clue regarding the thought processes behind math and science. At the elementary school level, this is what you are trying to teach.

    Knowing the multiplication tables, and figuring out the commutative and associative processes that are behind the multiplication tables, go hand in hand. At the elementary school level, memorization should probably precede principles. Or, use the principles to show the why behind the facts, if students need it.

    Many adults forget how strong memorization is as a part of a child’s learning. A child doesn’t care about Chomsky’s grammar rules for learning the language: the child simply learns/memorizes what words mean and how to use them. If you don’t know facts, you have nothing with which to reason. Facts are the building blocks of reasoning.

    This reminds me of the phonetics/whole language conflict. A lot of the phonetics/ whole language brouhaha came about because adults applied adult mentality to children. Adults do not like memorizing and repetitive drills, so the whole language people reasoned that children do not like them. Not so. Young children love drilling and repeating. They repeat to reinforce what is relatively new for them: the five year old who wants the same story read over and over again. So it goes with phonetics.

    Whole language is not complete nonsense. First, children need to have stories read to them, not just exposure to sounds, when learning to read. Agreed. Second, ADULTS can use whole language skills to discern what words and sentences mean when many letters are cut out of words. However, to initially learn how sounds and words connect, phonetics are necessary for CHILDREN.

    So it goes with multiplication. Show children the why, with rows of objects etc. various manipulatives, while they memorize.

    As an elementary student, I learned the multiplication and division facts. When in high school I was exposed to the “new math,” with commutative and associative principles, I used said principles to faster multiply/estimate in my head. 71X 52 ~ 70*50= 7X5X10X10= 35*100 plus a little more, say 3600. With the principles and proofs, math became fun for the first time for me.

  11. When math specs give instruction to elementary teachers, something is often lost in the transition. Activities that involve blocks or drawing or coloring to teach math become activities that incorporate coloring or blocks or drawing, but no or little math. We really need math-strong teachers teaching the little ones. It would be best if it could be the regular classroom teacher, but frequently that is not possible.


  12. X Teacher

    When math specs give instruction to elementary teachers, something is often lost in the transition. Activities that involve blocks or drawing or coloring to teach math become activities that incorporate coloring or blocks or drawing, but no or little math.

    Good point. While doing the activity, the teacher needs to reinforce the math behind the activity, to point out that this is not mere coloring or blocks. Perhaps one way is to go back to classic pedagogy.

    The classic lesson plan is three fold.
    1) Say what you are going to do.
    2) Do it.
    3) State what you have just done. Sum up.

    This reinforces to the students why they are doing something. By the way, this is also the classic outline for giving a speech: state what points you are going to give in the speech, elaborate on the points, and in closing state what points you have made.

    A way to keep elementary teachers on track, as opposed to getting lost in the coloring/block activity, might be to have them make a point at the beginning of the lesson about the math behind it, to have the teacher reinforce the math point during the activity, and then to have the teacher sum up the math point of the activity after students have completed the activity.

  13. X Teacher,

    you are assuming that lower elementary teachers can follow your plan, with understanding. Unfortunately, I don’t think this is a safe assumption.


  14. X Teacher

    I am implying an admitted leap of faith combined with memorization. While the teacher may not initially understand the reason behind the activity, if forced to STATE the reason for the activity to the class, and is given a script for doing so, perhaps after sufficient repetition the teacher may understand it. Another leap of faith: while the teacher may not understand the reason behind the activity, if the teacher STATES the reason for the activity to the class there exists the possibility that the mere stating of the reason for the activity will enable this to sink into students’ heads.

    As I see it, the real problem begins in 3rd grade on up, when multiplication comes on the scene, because there are a lot of teachers who neither learned multiplication nor the principles behind it.

    No easy answer, I admit.

  15. elementaryteacher

    In reply to Jackie,

    Yes, elementary teachers DO have a planning period. But during the planning period, the children are in another class (like P.E.). The math specialist would have to take time out of the class time in which the teacher DOES have the children, and since they don’t want to give teachers an additional FREE hour during the day, NOR hire more teachers, that is why the only solution I can see (at least in private schools) that would be acceptable to management is that the additional free hour would be used to teach the math specialist’s home room class (such as Grade 6, for example) in a non-math subject. So, the math specialist would be teaching (for example) Grade 3, 4, 5, and 6 math (6 being his home room class); the Grade 3 teacher might then teach one hour of Grade 6 Social Studies; the Grade 4 teacher might teach one hour of Grade 6 spelling/English; and the Grade 5 teacher might teach one hour of Grade 6 Reading. As you can see, this would be FAR more difficult for everyone than the present system of each classroom teacher teaching all the academic subjects for each grade. THAT’s probably why it’s not done, and not going to get done (even though I agree it would improve math scores).

    The problem with each teacher teaching all of their own academic subjects is, as you say, weakness in some area or other (most commonly math, but not always–sometimes teachers are better in math, but poor in teaching reading, or English, or writing skills). As you say, passing a competency exam in math does not make one a good math teacher. But even so, I find in our school, many teachers have entirely skipped the STORY PROBLEM sections and only done the CALCULATION sections (which is one of the biggest problems as far as I’m concerned). Our school is now going to be trying to address this issue.

    Dedicated Elementary Teacher Overseas

  16. elementaryteacher

    I think this whole issue could be better addressed at the University/Teacher Certification Level.

    I am teaching in elementary (third grade), but my certification (in Colorado) was Secondary Social Studies. I used to be math phobic, but after about three years of teaching it, found that I now enjoy teaching math quite well, and my students understand my teaching. After teaching for a year, and having a lot of problems, I had a chance to come back to the United States, and observe in the Colorado school I went to as a child. I found they were using a really good system in the third grade classroom there.

    I teach eight-year-old students, in Grade 3.

    I present the lesson, and we do some examples in class. If it is confusing, or if our foreign parents are unlikely to understand the calculation method, I have the students take an ink pen and copy two or three examples off the board into their workbooks, so parents and students can look at those examples again when they get home.

    The next morning, I check to make sure every student has done their homework. If they have, they get A+ at the top (as I walk around the room), and that’s so even if it is all wrong (I’ll explain in a moment). If they have not done it, or not finished it, they get an “F” and a “homework alert” (a paper attached to their unfinished work saying they didn’t do their homework, and their parents have to sign it that night). They also get an F if they did all their work by showing answers only, but not their carry numbers, or cross-outs for borrowing (even if all the answers are correct). This ensures that they DO the work and don’t just use a calculator. (It also enables them to see just where they made their mistakes in class, when we correct).

    This pretty well ensures that EVERY student regularly comes to school having done his homework, and is READY TO LEARN FROM THEIR MISTAKES. I have each student get out an ink pen which we keep for making corrections (all the work having been done in pencil). The A+ (even for wrong work) keeps the students from erasing and changing their work just to have an A. (These grades are not actually recorded, but the students THINK they are. The parents know better, but work with me in keeping this fact secret.)

    I then tell them the answers, having them put a check mark next to correct answers. Incorrect answers, I have them draw one line through, and MARK THE CORRECT ANSWER NEXT TO IT. This makes it easy for them (and for their parents) to see WHAT to study before the next test. Then we go back through and work on the board as many of the problems as time allows (a good half hour’s worth each day).

    I find it SO gratifying when students say, “OH, NOW I see what I did wrong…..” because they have their OWN work to look at as we work the problem on the board. If they don’t see, I ask the student what answer they got, and proceed to show them (on the board) where they made their mistake. Common mistakes in Grade 3 are writing the carry number, but forgetting to add it in to the next column, or subtracting one column, and adding another (instead of subtracting again). If I was not having students correct their own work, and working out with them on the board WHERE they made their mistakes, I would be a FAR less effective teacher of math than I am today. But I don’t see any other teachers I know using this method (at least in my overseas school).

    The biggest strength of this method is that it MOTIVATES the students. When I was a kid, and the teacher corrected a math paper, even if she marked the places we made mistakes, NO ONE looked at that! We only looked at our GRADES!!! So this way, students DO look at their work, and DO learn from their mistakes, and DO enjoy making progress in math.

    Dedicated Elementary Teacher Overseas (in the Middle East)

  17. elementaryteacher

    I forgot to add my main point in the last post about WHY these issues could be better addressed at the teacher training/certification level. With a Secondary Social Studies certification I had NO training in how to teach ANY elementary-school subjects. It seems to me all the issues mentioned above should be addressed by having prospective teachers take a course in HOW TO TEACH MATH, and that this course they take SHOULD BE TAUGHT BY A MATH SPECIALIST. There should be projects required, where the teachers (as students in this class) are REQUIRED TO TEACH math lessons in front of the math specialist (or even on video that the math specialist teacher can view privately) in order to pass the course. Such a course could also be required of all current elementary teachers during the summer for example.

    Dedicated Elementary Teacher Overseas (in the Middle East)