Learning Math Facts

As I continue my investigation into cognitive science, I continue to run into the concept of “distributed practice,” sometimes called “spaced” practice.  One of my passions for a long time has been helping students learn their math facts.  Over the years, I’ve tried several approaches to teaching students their facts, with mixed results.  What follows is a suggestion based both on my recent experiences and my readings in cognitive science.

This will be a whole-class activity.  Begin with a numbered set of math fact drill cards (I have a homemade set to offer. See below).  Select 5 cards, one for each day of the week.  On Monday, present the first fact card and ask students to repeat the fact both ways (6+5=11 and 5+6=11).  After this introduction, find 5 or 6 opportunities on Monday to quiz the class on the first fact.  On Tuesday, present a new fact.  On Tuesday, your 5 or 6 later quizzes should include both Monday’s and Tuesday’s facts.  Repeat Wednesday through Friday.  In like manner, keep introducing the “fact of the day” through the next week.  At this point you will have introduced 10 facts and will be quizzing 10 facts.  On the second Friday, give an assessment to see which of the 10 facts have been learned.  Learned facts go into a separate stack to be randomly quizzed from time to time (perhaps once a week), just to keep students “sharp.”  

Keep adding daily cards to the “quiz stack” until you reach 10 facts.  You now have 3 stacks:  1.  new cards; 2. quiz stack; 3.  mastered cards.  Avoid letting the quiz stack get beyond about 12 cards.  Stop introducing until you can reduce the quiz stack to less than 10 cards.  Stop introducing if you can’t transfer cards from the quiz stack to the mastered stack.  An easy assessment would be done orally, giving just 10-20 problems combining recently added facts and those previously mastered- no fancy form needed.  Just use lined paper.

Because the introductions and daily quizzing would be very quick, it’d make a good transition activity.  I’ve written the above as succinctly as possible, so if anything is not clear, feel free to contact me:  rhowell46@sbcglobal.net

Also, if you’d like a set of homemade flash card masters for addition and multiplication, send me an email and I’ll send you .pdf format masters.  They include numbered back-to-back masters which can be copied on to 8.5x11 cardstock to make serviceable flash cards for this activity.  If you’d like, of course, you could copy off additional sets for students who want them.

Again, if you have any questions, don’t hesitate to contact me.

More Neuroscience

This would be my normal end-of-the-week post.  I'm headed out of town for some fishing and won't be home 'til the weekend.

I am not a neuroscientist.  I am, nonetheless, fascinated by what neuroscientists have learned about our brains and how they learn.  Currently, I’m reading Make It Stick, by Peter C. Brown.  The aim of the book is to apply what is known in neuroscience to the domain of education.  it’s already had a huge impact on me.

No doubt you know the old Chinese saying:  “Give a man a fish, and he'll eat for a day; teach a man to fish, and he'll feed himself for the rest of his life.”  Make it Stick lays a huge emphasis on the importance of self-testing as a technique for learning new material.  Depending on the material, self-testing can take many forms.  If learning materials, let’s say a textbook, offer study aids like key concepts or chapter questions, students who want to learn the material need to define those concepts and answer those questions - from memory first, then study after.  Doing so will allow the student to consolidate what they know and accurately assess what they don’t.  While there is no fault in taking notes as one reads, it’s a fallacy to read through notes and assume that one is learning the material.  Because the notes are familiar, the student is more likely to think that the material is learned, however, research has shown that memory from this study approach diminishes to 10%-20% in as little as two weeks.  Notes, tied to prior knowledge will have a greater likelihood of consolidating new learning.

I would guess that students would need to be guided over and over to embrace this technique.  I would also guess that a teacher must make very explicit to students that they are learning a study technique which has been shown by research to be more effective than simple re-reading or reading one’s notes.  It is the repeated retrieval of information from memory that strengthens both the memory, it’s relationship to other known information, and the neural pathways to that information.  So, I’d ask students to answer questions.  What they can’t answer, I’d ask for them to go back to the material and find the answer.  Then, several days later, I’d ask them to answer the questions again.  It’s likely that, bit by bit, they’d really LEARN the material.  And, over time, with persistence of perhaps several teacher at succeeding grades, students would embrace a study skill that will feed them for the rest of their lives.

By the way, I’ve discovered that the above information about studying is not just for our students.  It’s making a profound and immediate impact on my ability to understand and assimilate all the new material I’m reading.  I suspect it might do the same for you.  Besides, we teach best what we know best.

Brain Apps.

This entry is a "bonus."  Thanks to the rainy weather, I'm reading a lot and getting inspired as I do:

I’m loving my reading of Make It Stick, by Peter Brown, and I encourage every educator to read it.  Understand, it’s not a warm and fuzzy read.  It is literally what I call, “…two feet on the floor, at the kitchen table…” reading.  It’s been fun for me, though, because with it, I’m practicing what science tells me is good learning:  I’m drafting review questions, studying it in spaced sessions, returning to what I’ve done previously, tying what I’m learning to what I already know, and, here, writing about what I’m learning.  No doubt, the book will spawn many blog entries.

Brown says that the result of deep and purposeful learning is what he typifies as “brain apps.”  His example is driving a car, but driving a golf ball would apply as well.  At first, all the many skills of driving are nearly overwhelming.  Perhaps this is why young drivers have a comparatively poor record with insurance companies.  But, with thousands of hours of actually driving a car, skills like speed control, lane placement, awareness of dangers, and braking become semi-automatic.  I’ve driven cars and trucks with a manual transmissions for fifty years and I seldom think about what gear I’m in or how to manage a clutch.   But, it’s recently become obvious to me that an automatic transmission is a challenge to me because, for someone who’s used to a stick shift, an automatic is NOT automatic.

Think of what this means for classroom education.  So many of the skills that seems so obviously “easy” to us are, to young students,  like the 16 year-old learning to keep a car in the center of the lane.  Even when they “get” a skill like “borrowing” in subtraction, it is, at first, an isolated and uncomfortable operation, not the seamlessly integrated part of a more general “Subtraction Brain App” that it will someday become.  In time, countless subtraction situations, practice with operations, and the arithmetic facts that support the computations, will coalesce into the “idea” of subtraction which will be more like the refined skill which we attempt to teach.

My point?  I think we need to patiently disaggregate our understanding of the things we teach, attempting to remember how we arrived at our present skill.  Regardless of any pressure we feel to show that our kids are meeting the current set of standards that some political or administrative entity has drafted for us, it’s important to understand how learning takes place.   We can provided distributed practice.  We can look for simple links to students’ lives and experiences to tie current learning to their prior knowledge.  Most importantly, we can provided reassurance that learning is long range, not immediate.

Any attempts to hurry up the process or learning are likely to lead to some short-term successes - think “cramming” for the test - but long-term failures when, without measured repetition, the learning never achieves the status of “Brain App.” 

Administrators!

Recently, I had the opportunity to visit with a school administrator.  Understand at the outset, I was never the darling of the administrators that I worked under.  So, it comes as no surprise to me that the conversation did nothing to change my opinion of administrators.

The topic of our conversation was one of the school’s programs that I felt was being poorly implemented because of lack of training.  I had every confidence in the program itself if teachers simply had sufficient background in the procedures of the program, their purpose, and the research that supported the materials and routines.

The administrator’s response was predictable.  Staff was “studying” all the school’s programs to see which programs teachers supported.  After due study teachers would recommend which programs should be continued and which abandoned.  Nowhere did I hear of any research into the effectiveness of the programs.  Nothing was said about contacting experts, training, reading supporting research, or asking for presentations from representatives.  To my credit, I remained passive.

Teachers are very busy people and so mired down in the “day to day” that it is the natural way of most programs over time that they get reduced to their “manageable” bare bones.  Teachers seldom have time to read the research supporting new or even adopted programs. Over my years in the classroom, I’ve seen this time and again, particularly with math and science programs. And yet, it is precisely those materials which make clear the purpose of the prescript routines, their research bases, and the assumptions which underlie their choice.