Henri's Math Education Blog

"There is no one way"

Friday, November 9, 2018

On the desire to push kids ahead

This is a guest post by Rachel Chou, Math Department Chair at Menlo School in Silicon Valley. I wrote about this topic on my Web site, under the title Hyper-Acceleration.  You’ll see that Rachel addresses a particularly acute version of this problem, given the fact that she is in a private school in a region where hyper-acceleration is deeply embedded in the culture of privileged parents. Even if the situation at your school is not as extreme, you might get some ideas from Rachel on how to discuss this non-confrontationally with pushy parents or the administrators they pressure.

-- Henri

On the desire to push kids ahead

– by Rachel Chou

“We were disappointed to hear that the 6th grade wasn’t leveling.  We had hoped that Monica would be on the same path as her older brother Brian.”  This was a comment made to me in passing, by two parents new to the 6-12 independent school where I work.  These two lovely parents, were already concerned about whether or not their 11-year-old daughter would be exposed to Calculus by the 11th grade.  6th grade had only begun two weeks prior.  Their feelings and sentiments are normal though.  Many parents at independent schools were themselves both serious and successful students.  At any given point, if their children are not on a particular path to “be the best,” to be “ahead of everyone else,” they start to get nervous.  These parents are well-meaning.  They love their children, want the best for their children, and worry when a child gets “off-plan” on the way to this measure of success.  While the parents that I interact with are almost always kind, well-meaning, and have their children’s best interests at heart, their concerns and desires are often short-sighted.

Should we choose a school because it brags that it “gets all kids through” Algebra 1 in the 7th grade?

At my independent school, as a matter of philosophy and practice, we do not have students study a traditional Algebra 1 course in 7th grade followed by Geometry in 8th grade.  I’m particularly unimpressed with middle schools that brag, "We get all 7th graders through Algebra 1!"  Normal or even advanced adolescent cognitive development does not leave a child ready to deeply understand the topics in a traditional Algebra 1 sequence.  When middle schools brag about this, it means that they are teaching a course titled Algebra 1, but that they are necessarily altering the content and cognitive demand of the tasks being included, in an effort to make the material accessible to younger students.  Or worse, they are not altering the cognitive demand of the tasks being included, and instead are running a course that is predicated on teaching students to be docile memorizers of routines, but not mathematically thoughtful and powerful thinkers. I don’t want my own child in a class where she is learning to do Algebra.  I want her in a class in which she is being exposed to meaningful learning experiences, which have an effect on her overall ability to deeply understand Algebra.

Absolutely no one choose a pediatrician because he brags, "I get all babies walking by 12 months old!"  We accept that the normal age at which babies learn to walk is somewhere between 9 and 18 months of age, and the children who gain this skill on the latter end of the scale are no less athletic or physically capable children. We also certainly do not believe that when a child begins to walk is any reflection of the pediatrician caring for him.

Meanwhile, the parents who might think that their child is particularly advanced might choose a middle school because it is bragging that it gets all of its 7th graders thru Algebra 1; however, this is also very short-sighted.  Remember that this course must be targeted at the average cognitive maturity of a 7th grader.  If I thought my child was unusually precocious, such a course would necessarily defeat my purposes.

“But I just don’t want my child to repeat a class!”

At our school, half of the incoming 9th grade students come from outside middle schools.  Many of these middle schools do allow the acceleration mentioned above. We tell parents that we base placement on what a student knows and can do and not on the names of courses they may have taken.  In practice, while 20 to 30 of our incoming ninth graders have taken a course called Geometry, we place only 2 – 4 of them in Honors Algebra 2 as freshmen.  Parents often worry if their children have taken Geometry in middle school, that they will be "repeating" something in our Geometry class, and yet, exactly no one, has ever called my history colleagues asking if their children can skip 11th grade US History for the reason that "Their child studied US History in 8th grade."  We somehow accept that kids can look at the problems of history with a more thoughtful lens when they have reached an older age.  The same is true for Geometry topics.  As a matter of fact, students who have studied geometry in their 8th grade years report little to no advantage over their peers who haven’t.  Often, they are at a disadvantage because their ability to handle algebraic abstraction is less well-developed specifically because they studied Algebra 1 when they were not yet cognitively mature enough to understand it.

When might we allow some acceleration?

It may sound somewhat contradictory that we allow advanced placement in our high school and not our middle school, but this is a thoughtful choice, not an accident.  It is not our department’s belief that all students can acquire mathematical understandings or power at the exact same rate, but it is our belief that picking children out too early for advanced acceleration provides students with very little gain, and can lead to unwanted consequences.  I offer the following analogy: The tallest child in the class in 6th grade is not always the tallest child in the class in 10th grade.  (They haven’t gotten any shorter!  Their friends just grew taller!)  We accept and understand that children hit their physical growth spurts at very different ages.  This is also true of cognitive growth spurts, though it is far less obvious to non-teachers.  A child might be particularly mathematically precocious in sixth grade, and might be closer to the average by tenth grade.  This child has not grown less capable or intelligent, but rather her friends have simply caught up!  The point here is that there is a real danger in separating out children at a young age for active acceleration.  They might handle the extra challenges in the 6th grade, but if they end up in a situation in which they are hitting Precalculus in the tenth grade, they are often not ready for the cognitive demands of the class.  And all a school has succeeded in doing in this case is accelerating a child to a place of frustration.  These students often report that they “loved math when they were little” but that “they no longer do now.”  That can’t be our goal as parents or as educators.

On trusting the knowledge of practicing educators.

Not surprisingly, when I speak to parents at open house events and inform them that it is common for 20 to 30 of our admitted students to seek an Algebra 2 placement, but only a small few actually are placed there, the parents immediately assume that their child will be one of these “select few.”  This makes lots of sense.  First, as parents we are biologically predisposed to believe in our kids and think they are generally awesome.  Second, there is the issue of sample size.  The average parent at my school has a sample size of 1 to 4.  The parents with 3 or 4 kids will think their best math student is "incredibly brilliant" and needs more than what the grade-level curriculum has to offer.  But the educator who is making placement decisions typically will have a sample size of about 2,000 students. Math educators have a far better ability to understand where a particular student fits in terms of their cognitive maturity as compared to their peers, and parents would be wise to heed the advice of the math educators caring for their children.

Why are we trying to race toward calculus?

Back to the story of the two parents concerned about their 6th grade daughter’s eventual 11th grade math placement.  Another point worth mentioning is the odd belief in this country that K-12 mathematics is a race toward studying calculus.  First, there is so much interesting mathematics for kids to study that is not a pre-requisite for studying calculus.  Students are typically quite drawn to discrete math topics such as probability, combinatorics, elementary graph theory, number theory, sorting algorithms, and the list goes on.  If we view a traditionally taught calculus course as the only end goal of a K-12 math sequence, we might leave such interesting and mind-stretching topics out.  Why?  Instead of lobbying your local school or your private school to accelerate your child toward Calculus, consider advocating that math curriculums include more depth, more open-ended tasks, and more discrete math topics.

Second, and possibly conversely, why are we waiting so long to get to calculus exposure?  Students need not have formally studied continuous math topics such as advanced trigonometry, exponential and logarithmic functions, polynomials and the like to appreciate calculus concepts.  A thoughtful geometry teacher might guide her students to find a “slope-computing formula” (the derivative!) of the function which represents the top-half of a circle, by applying understandings of how a tangent line intersects the radius of a circle.  Similarly, included in either a geometry course or an introductory programming course, students might write code to compute the area under curves by breaking the area into skinny slices.  How many of us actually integrate by hand, unless we are ourselves teaching calculus, or training students for the next integration bee?

No child’s mathematical journey is the same.  As teachers, in any single classroom, we are charged with furthering the mathematical growth of a wide variety of needs.  The thoughtful teacher sets up her curriculum so that students can learn, grow, and deepen their mathematical thoughtfulness and creativity at varying levels within the same classroom.  It is time to stop believing that the best thing to do for our children is to advocate that they advance through a sequence of math courses at an accelerated pace.

Wednesday, November 7, 2018

More Catchphrases

Last summer, I wrote a post about catchphrases for math teachers. Some of those were created by other people, but most were my own. It was a fun way to think about what ideas I consider important enough to summarize in a hopefully memorable slogan. Since then, I have remembered three more of my mantras, which are mostly aimed at younger teachers.  I will share them in this short post, one per paragraph.

- As I mentioned in my first catchphrase post, nothing works in all classroom situations, and teachers need to be eclectic. Too often, we are pressured to adopt one or another currently fashionable article of faith. Doing so rigidly is never a good idea. My advice: trust your intuition, avoid dogma, be flexible, be kind. Of course, your intuition will improve as you get more experience, but even if your intuition is “wrong”, at least it is yours. Your students deserve getting the real you, not a poor imitation. If you are true to all four of these admonitions, you are sure to get better at this job.

- Some teachers are reluctant to miss school, even when they are sick, or when they have an opportunity to attend a workshop or conference. They feel that even missing a day or two would be a betrayal of their students. To them, I say: You are as important as your students. Your getting healthy is of course in your students’ best interest: it will allow you to be at your best when you return. Valuing your own professional development may seem selfish, but if you have a chance of learning something useful to your teaching, keep in mind that it will help not only your current students, but the students in your future — probably many many more than are in your classes right now. And really, are you so great that missing you for a couple of days  is going to permanently damage your students? Didn’t you manage to survive a number of less-than-perfect subs when you were their age?

- A balancing act faces all teachers: how much time should you spend grading? how much time should you spend planning? In the first year or two (or three) of one’s career it is difficult to think clearly about this. The main thing is to survive the day, the week, the semester. But maintaining an unrealistic workload is not sustainable in the long run: it will push you out of the profession sooner rather than later. If you want to stay in the classroom over the long run, you’ll need to balance the different parts of the job, and accept that perfection is not going to happen. (See my post on Growth Mindset for Teachers.) That said, here is a piece of advice from an old-timer: When grading, you are working for one student. When planning, you are working for the whole class. Keep that in mind when you are budgeting your time. Don’t grade more than you need to. Can students correct their own or each other’s work sometimes? When grading, do you really need to write a lot? Do your students read what you write? Do they heed what you write? I’m not suggesting you should be irresponsible, just that you should be efficient. I found that it was enough to circle mistakes on quizzes and tests, and to ask students to turn in quiz corrections. (I already know how to solve the problems: they are the ones who need to figure them out. And grading corrections is fast.)

Well, that’s all for now. If you have favorite catchphrases about our line of work, please share them in the comments!

-- Henri

Monday, October 15, 2018

Spiraling Out of Control?

In most math curricula, students work on a single topic at a time. (When I taught elementary school, decades ago, I noticed that if we’re working on subtraction, it must be November! But the same applies at all grade levels.) The idea is that is that by really focusing on the topic, you are helping students really learn it, before you move on to the next unit. Unfortunately, that is not how retention happens. It is much more effective, when learning a new concept, to see it again a few weeks later, and again some time after that. Thus the concept of spiraling. Years ago, the Saxon books distributed homework on any one topic across the year, typically with one or two exercises per topic on any given day. Some more recent curricula do facilitate that sort of homework spiraling by including review homework in addition to homework on the current topic after each lesson. The algebra textbook I coauthored in the 1990’s is spiraled throughout: not just in the homework, but in the makeup of each chapter and many lessons. This idea was so important to us, that there is an image of a spiral at the start of each chapter! (If you have the book, check that out! Or just look at it online.)

In this post, I want to argue that while I agree with the fundamental underlying idea of a spiraled curriculum,  there is such a thing as overdoing the spiral. I will end with specific recommendations for better spiraling.

Impact on Learning 

Too much spiraling can lead to atomized, shallow learning. If there is too much jumping around between topics in a given week, or in a given homework assignment, it is difficult to get into any of the topics in depth. Extreme spiraling makes more sense in a shallow curriculum that prioritizes remembering micro-techniques. In a program that prioritizes understanding, you need to dedicate a substantial amount of time to the most important topics. This means approaching them in multiple representations, using various learning tools, and applying them in different contexts. This cannot be done if one is constantly switching among multiple topics.

In particular, in homework or class work, it is often useful to assign nonrandom sets of exercises, which are related, and build on each other. For example, “Find the distance from (p, q) to (0, 0) where p and q are whole numbers between 0 and 10.” (This assignment is taken from my Geometry Labs.) At first sight, this is unreasonable: there are 121 such points. But as students work on this and enter their answers on a grid, they start seeing that symmetry cuts that number way down. In fact, the distances for points that lie on the same line through the origin can easily be obtained as they are all multiples of the same number. (For example, on the 45° line, they’re all multiples of the square root of two.) Nonrandom sets of problems can deepen understanding, but they are not possible in an overly spiraled homework system.

Impact on Teaching

The main problem with hyper-spiraling is the above-described impact on learning. But do not underestimate its  impact on the teacher. For example, some spiraling advocates suggest homework schemes such as “half the exercises on today’s material, one quarter on last week, one quarter on basics.” Frankly, it is not fair to make such demands on already-overworked teachers. Complicated schemes along these lines take too much time and energy to implement, and must be re-invented every time one makes a change in textbook or sequencing. Those sorts of systems are likely to be abandoned after a while, except by teachers who do not value sleep.

Another problem for teachers is that it makes using a hyper-spiraled curriculum difficult to use, because it is difficult to find where a given concept or technique is taught. (In the case of Algebra: Themes, Tools, Concepts we tried to compensate for that by offering an Index of Selected Topics and Tools. We also included notes in the margin of the Teachers’ Edition: “What this Lesson is About”. But even with all that, a hyper-spiraled approach makes extreme and unrealistic demands on teachers’ planning time. In fact, some hyper-spiraled curricula lack even those organizational features. Without them, a teacher needs to spend the whole summer working through the curriculum in order to be ready to teach it. This can be fun if the curriculum is well designed (e.g. the Exeter curriculum), but no one should feel guilty if they’re not up to that level of workaholism. 

Spiraling Made Easy and Effective

So, you ask, what do I suggest? In the decades following the publication of my overly-spiraled book, I developed an approach to spiraling that:

  • is unit-based, and allows for going in depth into each topic
  • is easy to implement and does not make unrealistic demands on the teacher
  • is transparent and does not hide what lessons are about (most of the time)

I have written a fair amount about this, under the heading extending exposure. The ingredients of this teacher-friendly approach are:

Implementing these policies does not require more prep time, or more classroom time, and it creates a non-artificial, organic way to implement “constant forward motion, eternal review”. It helps all students with the benefits of spiraling, but without the possible disadvantages. You really should try it! Read an overview of this approach on my Web site: Reaching the Full Range

-- Henri