* My dear Isaac

Dear Isaac is my nephew with an unidentified auditory processing disorder and dyscalculia, all mixed with a heavy dose of emotional distress.  He’s a bright, creative youngster with strengths in science and art.  But as a third grader, he still can’t add or subtract single digit numbers without his fingers.  If we hadn’t used Alan Walker’s multiplication methodology, Isaac wouldn’t have learned any multiplication facts.  After his initial refusal to engage with the Walker approach to memorization, Isaac cut his losses and became proud of his new knowledge.

After my initial assessment, I estimated that it would take six months to correct fundamental math reasoning errors.  That was an accurate estimate.  Isaac has made solid progress in solving problems.  You would be so proud if you could see him working on multiple-step word problems!

Sadly, dear Isaac is now burned to a crisp at school and when it’s time for homework.  He doesn’t act out at school but his teacher reports that he is frequently inattentive and withdrawn.  The school year has been too long and taxing.  Isaac feels stupid, is depressed, and his teacher flat out refuses to lessen the homework load.  Oh dear!

A predictable conundrum for him (and me!) is dealing with his errors.  He has made too many and now wants to be error-free for life.  If only!  He is reluctant to accept alternative methods of calculation when he feels especially low.  We had a difficult session this past week when he refused to write multiple digit addition problems vertically instead of horizontally.

After staring at his horizontally-written problem, Isaac screamed, “I can’t do this!   I thought you were going to help!”

“Write it vertically, Isaac.”

“I’m going to do it MY way!”  

“Go ahead.”  [I walk across the room because I know he’s going to implode if I stay close.  Or I might just bite my hand off.]

Repeat above scenario 3 times.

Finally, amidst tears and growls, Isaac rewrites the problem vertically and gets so much praise from me that we are back on track.  I remain at his side as his sense of humor returns and he completes all the dreaded homework in record time.

Here’s the adorable Isaac, taking aim at homework with a tripod?

Isaac 3



* What is homework?

To my nephew, Isaac, homework is “tor-tradition,” meaning torture + tradition.  See?  He has some math sense and lots of common sense.  Poor Isaac.  As third grade has shifted into hypermode to prepare for the end of grade tests, he has been left dangling. There’s not enough time for remediation after school, not with the tor-traditional piles of homework.  Fortunately, he has a flexible teacher who is now willing to let him move through the multiplication.com system of learning his times tables.  After months of trying more traditional (and yes, torturous approaches), I switched to Alan Walker‘s language- and association-based approach.  It has paid off bigtime for Isaac, providing him with a dose of much-needed confidence.  But is it too little and too late?


Isaac would benefit from a formal educational and psychological evaluation.  He appears to have serious weaknesses in auditory processing skills, along with attention, working memory, and long term memory issues.  His success in reading fluency camouflages many of his weaknesses.  Sometimes both teachers, parents, and kids think everything is fine if you can read above grade level.  Ouch.  Try giving Isaac multistep directions and watch the confusion.  And like many twice exceptional kids, Isaac’s mental energy has been fried to a crisp after half a day of school.  His teachers report that he spends his afternoons in silence, never responding and apparently inattentive.  At home, he screams and bangs his head when it’s time for homework.  Torture indeed.

I don’t think it’s too late for this sweet kiddo.  He is eager to learn, responds well to instruction in incremental steps, and has enough curiosity for an entire classroom.  And he can do a perfect Patrick or Spongebob imitation.  Isaac can go far, especially if a certain tortuous traditions can be axed.

* Multiplying fractions with visuals

In a previous post, I expressed my concern about the use of some models and processes for representing multiplication of decimals and fractions. A recent issue of Teaching Children Mathematics provides a better description of why my students and I were both confused by the representations being used by some classroom teachers.  As we plowed through classwork and homework, kids were increasingly anxious as I tried to inject some meaning into the process.  “My teacher says to color these boxes!” the kids would exclaim in desperation.  “We aren’t doing it that way!”  When asked what the boxes meant, the kids had no idea.

In an excellent article by Webel, Krupa, and McManus, the authors describe the confusion that results from using visual representations without context.  They emphasize that the visual should clearly depict the differing sizes of the two fractions.  The authors express concern that visuals may be reduced to a series of steps which are no more helpful than blindly following an algorithm.

As I’ve taught students in 4th and 5th grade, all three of these pitfalls have been evident.  Here are some graphics to demonstrate the authors’ perspective (and my students’ anxiety).

Let’s say we want to solve this problem (borrowed from the article):  I have 1/4 of a gallon of orange juice in my fridge.  Then I drank 1/4 of it.  What fraction of the gallon of OJ is left in the fridge?

Here are three possible visuals for this problem.  Multiplying fractions

A and B are very similar and bring back bad memories!  Use a grid to color across and down.  The double-shaded area represents the answer.  Ta-da!  Only it doesn’t answer the question!  In the OJ problem, we need to know what’s LEFT.  C is a superior representation of the problem but a teacher must guide kids into how we arrive at 16 squares in the grid.  That discussion follows logically when you focus on the problem, which has 2 distinct units.

For fragile learners, A and B have great appeal.  Kids can memorize the coloring process without difficulty.  Unfortunately, they are not even close to understanding what multiplication represents.  The “talking it through” process can be very frustrating because they know they’re BEHIND and want to speed along to catch up.  And as we know, THIS ISN’T THE WAY MY TEACHER DOES IT!

* Math Wars

I appreciate a column in yesterday’s News and Observer, “Say No to Math Wars,” written by Paola Sztajn, a professor of elementary education at N.C State University.  She reminds us that we already fought this particular math war in the 1990’s, although I recall that we skirmished 40 years before that.  Being old has its advantages…. What math war is this?  The battle between proficiency/ fluency and a deeper understanding of math concepts.

It’s easier to measure fluency, for sure.  How many math problems can a student solve per minute?   Got it.  It’s harder to determine whether kids grasp concepts because in early elementary grades, they can often memorize enough processes to fake it.  Hence, we get the math disabled kids showing up in 3rd and 4th grade with a staggering number of gaps and misconceptions.

I agree with Sztajn that understanding should precede fluency.  I agree that we don’t need to beat this dead horse any longer.  On the other hand-and you knew that was coming, right?- we need better teacher education so that kids do better with the deeper understanding part.  We also need deep curricula but some reasonable way to shorten its breadth.  Right now, classes are racing through new topics at a rate that leaves our weakest kids far behind.   Research tells us that solid math instruction is time-consuming.  For kids with dyscalculia or those at risk of math failure, every lesson that sails over their heads is another nail in their math coffin.  That sounds dramatic and believe me, the cost of missing lessons is exponential, not additive.accounting-

* Helping student relearn math, part 2

red-sandstone-286091_640Although I am re-retiring from teaching this fall, I had committed myself to bringing two students to grade level this summer, if at all possible.  One student is Will, a rising 6th grader who needs support to untangle the math confusion in his head, sorting out what needs to be unlearned and relearned.  Unless this venture is successful, Will has another dreadful year of anxiety and distress ahead of him.  He has a jumble of partially learned processes, partially learned concepts, and a mass of incorrect assumptions.  He also has some dreadful habits: tuning out math instruction since it represents both chaos and failure, reluctance to admit that he doesn’t understand something, and no experience in explaining his reasoning.  None of this is a surprise.  It’s amazing that Will has the resilience to tackle math instruction at all, given his prior experiences.  He does ask really good questions when I allow him the time to process information.  His willingness to say, “I don’t get it,” is much improved.

How did he get here?  There are hereditary factors, but the particular math program used at his school was probably the worst possible match for his style of learning.  With topic shifts every two weeks or less, along with continual testing which triggered anxiety, he did his best to memorize procedures while missing the concepts altogether.  There was little hands-on activity and even less visual support.  By third grade, he was starting to hit a wall with working memory; there was too much to hold together.  With tutoring support in 4th grade, he scraped through the EOGs, but fifth grade triggered a complete collapse.

I had previously conceptualized this as untangling skeins of yarn, but currently, it feels more like digging through layers of rock.  Every time I think we have hit the core, I glimpse a deeper layer of misunderstanding.

As we’ve dug along, I have searched for novel ways to reintroduce concepts.  Specifically, I needed to reduce Will’s anxiety associated with fractions.  One tool that has been particularly useful in digging through fractions is a ruler.  Will does understand linear measurement and feels competent in using a ruler.  Our work with a ruler has revealed his misunderstandings of whole numbers, his inability to understand number lines, his misconceptions of fractional parts, and his misunderstanding of place value.  In fact, the more we work on fractions, the more layers of difficulty are exposed.  Fortunately, there are important benchmarks that he does possess.  Unfortunately, they are mixed with fragments of “math muck,” for lack of a better way to describe it.  In all of this, I have seen that Will has an intuitive grasp of math which is better than mine.  I am encouraged that if we can keep digging in small steps, he will surge forward before he goes to 6th grade.  Stay tuned!

* Final thoughts on student using multiplication.com

multiplication.com 2My special math student, Khalil, finished 4th grade having memorized his multiplication and division facts to 11, thanks to the unique language-based approach by Alan Walker at multiplication.com.  Here are the highlights of Khalil’s performance:

  1. This was the first time Khalil had ever memorized any math facts.  It was the first time he had demonstrated fluency with math facts.  On his last quiz, Khalil scored 95% (with a couple of self-recognized errors) in under three minutes.
  2. Khalil’s dramatic improvement in learning boosted his confidence in math overall.
  3. He began to solve word problems using multiplication and division.
  4. The memorization process in his brain changed significantly.  He started memorizing facts without referring to the associated stories.  His memory appeared more typical as repeated practice actually helped him recall facts, which had not been the case prior to using this strategy.
  5. He was eager to learn these facts, tracking his progress on a multiplication.com times table chart.
  6. This process ensured that he understood the commutative property of multiplication.  Khalil LOVED the commutative property!
  7. Khalil both learned from and enjoyed the multiplication.com games.  We both hummed the tune from “Sketch’s World” uncontrollably!

Khalil’s math problems are severe enough for him to be identified with dyscalculia.  However, if you have students who struggle with only specific facts, multiplication.com could be an effective strategy to try this summer.  The site could also keep your kids fluent in multiplication facts simply by playing games.  I’ve heard that Sketch’s World has a catchy tune.

If you haven’t checked out multiplication.com lately, you’re in for a treat!  The site has just released a student management system for tracking class progress, along with new resources for teaching individual times tables.  Each fact also features improved videos of the multiplication stories.  I’m not sure how long they can keep this FREE, so sign up while you have a chance!  Definitely sign up for their monthly newsletter.  It’s packed with great teaching ideas and materials.

No, I don’t get a commission.  Maybe I should ask….

* Check out Math = Love to change mindsets

One of the most challenging aspects of reteaching math to struggling students (and helping them unlearn misconceptions) is their negative mindset.  By the time I am involved, these young kids are convinced they are stupid and cannot learn anything “math.”  They hate math.  They dread math, which can manifest as withdrawal or acting out.  It’s hard to find a place to start when kids are almost phobic about math.  For encouragement and strategies, check out the math teacher who writes Math = Love.  Sara Hagan is a young teacher from Oklahoma who freely shares her successes, which are many, along with the glitches that occur during the learning process.

One area that Sarah seems to excel is in transforming the mindset of students.  As a high school teacher, can you imagine the challenge of transforming students’ mindsets towards algebra?  How do you get from “I hate algebra!” to “I can do this!”?  Sarah says that she takes students who hate math and shows them the fun that it can be.  Wow!   Check out her site to see that mindsets can be changed!

You’ll find that Math = Love is also a treasure trove of downloadable posters.  Follow the link below for her blog on effective nurturing of a growth mindset, along with some super FREE downloads.

Math = Love: Growth Mindset and SBG Bulletin Board Downloads

* Update #2 on student using multiplication.com

16Wow.  I am amazed.  I admit it: I was a bit skeptical about the process of learning “stories” for each multiplication fact, but Khalil is making tremendous progress!  First of all, he could not multiply ANY numbers when we started, not even by ones or zeroes.  Yesterday he completed a 40 problem online quiz for the zeroes, ones, and twos times tables in just over a minute with 100% accuracy!  And he only sees me twice a week!  (And we have to complete his homework, too!)  This is a kid who has never had any math fluency.  In three years of trying to learn addition and subtraction facts, he averaged about 12 problems per minute.

Here’s the coolest part to me.  Of course, Khalil has been asking for a copy of all his quizzes because he is thrilled to experience success.  So yesterday he showed his mother the super results and I finally told her how he is learning these facts.  I knew that if I had some initial doubts, she would most likely think this process is crazy.   Her first reaction was, “How come you just don’t say, two times nine is eighteen?”  And Khalil answered, “THIS IS THE WAY I LEARN!   IT HELPS ME!”  Go Khalil!  And a big thanks to multiplication.com.  The best is yet to come for Khalil!

* Teaching: tyranny of the urgent

Alarm-Clock-Simple--4926-largeTutoring students in a one-to-one setting without typical classroom constraints has its advantages.  I enjoy being able to select appropriate materials, tailor activities to student interests, and address skills without the pressure of teaching the core curriculum.  On the other hand, I am frequently in the same battle as resource teachers and other specialists.  Homework and projects routinely impact my valuable time with students.  You know that I am not keen on homework, if you’ve been following this blog.  After an hour or more of tutoring, I don’t want my students to face a stack of homework, so I typically assist them to complete it as quickly as possible during our session.  But the disconnect between students’ skills and their homework drives me NUTS!

Here’s what happened today:  I was teaching a fourth grader who is struggling with math.  I wanted to continue our work on place value and rounding numbers.  Instead, I checked his homework and took a deep breath.  It was algebra (or “algebraic,” as he told me).  Knowing that he works much better on frustrating tasks with me than his parents (it was that way with my own kiddos), I decided to bite the bullet.  Here is a sample problem:  Sue had 5 times more pencils than Nate.  Together they had 18 pencils.  How many pencils does Sue have?  How many more pencils does she have than Nate?  My student was required to model the problem using symbols and write three or four equations to demonstrate how he solved it.

I imagine some kids in his class are totally ready for that problem.  But my student was not.  He had no idea where to start, was dealing with abstract procedures that made no sense to him, and didn’t have sufficient opportunities to work with manipulatives (and perhaps understand) what “5 times more” actually means.  This is a student who does not know when to add or subtract.  Not only did we lose valuable instructional time on the skills which match his current math understandings, but he needed two brain breaks in order to survive that portion of our session.  And what does he know after our “guided practice?”  Not a lot.

I was facing the dilemma described in an interesting article called “The Hard Part” (thank you, Tony’s mom!).  In his column in the Huffington Post, Peter Greene writes about teaching: “The hard part of teaching is coming to grips with this:  There is never enough.  There is never enough time. There are never enough resources. There is never enough you.”  Indeed!

I do understand that the classroom teacher has her own constraints.  She is required to teach “algebraic” for a short period of time and then assess, assess, and reassess.  How can she “individualize” the above assignment for my student when it is totally inappropriate for his current level of functioning?  He needs more opportunities to model multiplication, much less solving problems with variables.  His dilemma reminds me of my post from yesterday on “How The Brain Learns Mathematics” by David Sousa.  Sousa describes prerequisite skills for learning mathematics successfully, including the ability to visualize and manipulate mental pictures and the ability to reason deductively and inductively.  My 4th grader is particularly weak in those skills.  When will he have time to catch up?  Isn’t that what summers are for?

* Math Struggles 201: Brain-based teaching

Sousa mathI am enjoying another Corwin Press book on brain-based teaching: How The Brain Learns Mathematics.  In David Sousa’s chapter on identifying math difficulties, he first suggests that teachers analyze the type of math instruction being provided and consider other environmental issues before determining that a child has an actual disability.  He describes some of the pitfalls of current math instruction.  Sousa’s book provides excellent strategies to support students at different developmental levels. It also describes effective assessment for determining a student’s present level of performance.

Sousa reviews research that describes a continuum of learning preferences for quantitative versus qualitative reasoning.  Some mathematical behaviors associated with quantitative reasoning include a proclivity for linear thinking, an emphasis upon the components of problems rather than broader concepts, and difficulty with multi-step problems.  Qualitative reasoning is characterized by emphasis upon broader concepts, difficulties with precise calculations, multiple approaches to problem solving, and an enjoyment of geometry.

From Sousa’s description of this continuum, I immediately thought of dyslexic readers, who often have a better grasp of the “gestalt” and experience more difficulty with the “smaller” components of language.  However, Sousa cites research indicating that dyscalculia and dyslexia are not genetically linked, although kids may certainly have both impairments.

Research supports the use of a concrete-pictorial-abstract approach to math instruction, which allows students at all levels (including middle and high school) the opportunity to interact with math via manipulatives first.  (Check out Nerd in the Brain‘s website for her awesome use of math manipulatives.)  During the concrete stage, it is important to link math to real-world problems.  Students then transition to pictures which assist students in visualizing the math process.  The last step is the use of symbols as a more efficient means of representing mathematical operations.  Sousa emphasizes that without a concrete link between symbols and real-world problems, students will simply memorize material and procedures without understanding.

Depending upon the current bandwagon, I’ve seen math instruction stall at the concrete level or actually start at the symbolic level.  Over the years, regular classroom math instruction most often whisks students into the symbolic level too quickly.  I have also been guilty of rushing students through the concrete level because they may “waste time” or become distracted by playing with materials.  It’s a challenge to make math instruction efficient as well as cognitively appropriate.  I have found that an effective assessment eliminates some of that pressure because I can better target my instruction.  For kids who absolutely cannot touch manipulatives without building towers, there are some good online manipulatives which allow them to experiment without getting mentally lost.  Glencoe has a super set of free manipulatives, with creative work mats that kids truly enjoy.  The National Library of Virtual Manipulatives is also excellent (and free) but doesn’t have the flexibility of Glencoe’s site for creating real-world problems.