By Sara Finegan

         An inordinate number of kids are growing up without learning their multiplication facts.  Some of this is due to the prevalent use of calculators.  Some is due to the fact that there’s been an overall rejection of rote learning – throwing the baby out with the bathwater, in this case, I think.  Some of it is due to low numeracy as a result of math disabilities. 

         Many kids with learning disabilities really struggle with numbers and their values.  And, I think, it also has to do with processing memory and retention of information that isn’t used on a regular basis.

         I think there’s a place for rote learning of math facts in a demanding classroom.   The only way to learn multiplication facts and what happens when you add ten to a number is to recite them over and over in some way or another. 

         The way I use is skip-counting.  Skip-counting can be done starting at any age, and should continue to be done if not daily, then at least several times a week once the kids know the routine.

 I cannot stress enough the importance of visual cuing when it comes to math.  In this case, a giant number line, or individual number lines to 100 for each student are  in order.  You can also use individual multiplication charts or a giant one; the point is that the kids have to have the numbers 1-100 in front of them when they are skip-counting, at least during the first few months.

           A lot of teachers practice skip-counting with their students through 3, 4, and 5 multiplication facts, and then stop.  As a result, many kids know their multiplication facts through five, and can’t go higher.  Don’t let this happen.

         It can be a part of your daily routine to start with 3, and have the kids skip-count to 60.  Move to 4, then 5, and 6.  Once they know those, move to 7 and 8.  Practice them religiously.  I like to point at the numbers on a chart or number line with a pointer or a yardstick at first, but later it becomes a favorite reward for a student to be allowed to stand at the front of the class and cue the numbers. 

         It doesn’t take long for kids to internalize the facts as they continue to skip-count regularly.  But here’s the deal:  You can’t stop here. 

         One mistake I’ve seen over and over again is special educators working on skip-counting with their students and then never extending the skill into actual math problems.  And when I say “actual math problems,” I don’t mean a standard multiplication worksheet.  I mean math problems that require critical thinking.

         There’s no point in rote learning unless the information learned by rote is applied in complex situations.  Skip-counting alone, in a vacuum, has little meaning.  We need our students to be able to use the math facts to solve word problems, to reason through division problems, to figure out the value of x, and to  calculate prices and amounts. 

          So teach the multiplication facts by rote, but then require your students to use them, and use them in a variety of ways.  Only in this way will they truly be learning.

          In a demanding classroom, rote memorization has a place in supporting mastery of facts to be used in deeper-level situations.

critical thinking · demanding classroom · high expectations · learning disabilities · learning disabled · low numeracy · mastery · math fluency · math language · multiplication · Nimble with Numbers · processing · rigor · rigorous instruction · rote learning · skip counting · special education · standards · visual cuing

By Sara Finegan

         I’m teaching a general education math class this year that includes many of my own kids with learning disabilities as well as kids who do not have IEPs.  We’re  about six days behind the other class at the same grade level, and both of us finished the first two units well before the district’s first math benchmark assessment. 

          Greg Roy, my general education colleague, has, as I have said before on this blog, taught me a great deal about math and math instruction.   He and I disagree about mastery, by the way, but that doesn’t impair our ability to collaborate.   He moves at his speed, and I move at a different one,  but we’re both moving forward.

          Anyway, Greg’s a proponent of math routines, and he does them at the beginning of the day as well as at the beginning of a math class.  Routines are great, for several reasons. 

          • First, they provide kids with ongoing practice of previously-learned skills.  My routines always have problems from lessons we have worked on since the beginning of the year – usually types of problems the kids struggled with mightily, so that the skills will become firmly embedded. 

(A recent math routine had a word problem involving adding negative numbers, 2 fraction simplifications, 1 changing from mixed fraction to improper fraction, 1 changing improper fraction to a mixed number, and 2 dealing with multiplying decimals by 10, because my kids bombed that area on the last unit test for some reason, as well as a problem from yesterday’s lesson.)

         The kids get to use their notes and math encyclopedias on the routines, and we go over the results really quickly.  The whole thing takes about 15 minutes each day. 

         • The second benefit of a daily routine is that they give you a snapshot of each student at that moment.  This is particularly helpful as you plan your instruction.  I can tell at a glance how many of the kids in my class have mastered yesterday’s lesson or skill, and what the next steps might be.  Because I make my kids show their work on each routine, I can pinpoint where in the process of problem-solving they are struggling. 

          • Finally they give kids a terrific sense of competence.  As they experience over a series of weeks and months the increasing ease of solving problems using skills they’ve learned, they realize that they can do it, that they are actually good at it. 

         The other day, a parent of one of my general education math students told me that her daughter, who has always struggled with math, is coming home saying “hey, mom!  I’m smart at math!”  This student is the first to raise her hand in my class, the first to want to show the class how she solved a problem, the first to ask questions, and the first to finish her work these days.  It’s not because of me; it’s because every day, when she starts learning math, she is reminded that she’s already got terrific skills.

          Many special educators neglect daily routines, or don’t make their routines as comprehensive as they should be.  We need to recognize that our kids should have multiple opportunities, over a long period of time, to practice newly-mastered skills, or those skills will evaporate like steam after a shower. 

          We need to challenge our kids daily, while at the same time reminding them that they have ability already, or they will never challenge themselves, and never remind themselves of how far they’ve come.

          We need to push, push, push our kids, using all of their strengths, innate and learned, or they will never catch up to their general education peers. 

          In a demanding classroom, mastery is not something to be admired and then ignored; it’s about skill sets we use, reuse, and apply in newer ways on a regular basis.

demanding classroom · high expectations · learning disabled · mastery · math instruction · proficiency · rigor · rigorous instruction · routines · skills practice · special education · standards · student monitoring · supports

By Sara Finegan

         There has long been a debate about the issue of teaching to mastery.  When districts and states set up learning modules on a schedule, or a series of standards to meet each year, the inclination of many teachers is to get through as many of them as possible. Some schools and districts place a great deal of pressure on teachers to move forward, and to keep moving through the year.

         I’m not averse to having a set of outcomes to work toward and achieve each year for each grade level. I am against moving forward before our kids really have become proficient at new skills, strategies, and knowledge.

         In a demanding classroom, we don’t get stuck in a routine of doing the same work over and over, and we don’t adhere to other people’s schedules about when learning should be accomplished. Instead, we focus on cementing new skills, step-by-step, concept-by-concept, so that when the foundation of math, science, and other learning is complete, there are as few weak spots as possible.

         If you think about it, if we move kids forward before they really get the previous unit or skill, you are building a house of cards on quicksand. Nothing is going to really stick and the child is going to be aware on a pretty consistent basis that he or she is missing something.

         And what are we teaching kids about learning if we do it this way? It seems to me that we are saying to our children: Learning isn’t about mastering information and strategies; it’s about zipping through lessons to completion rather than to skill.

         Given that one of the major issues for kids with special needs is that they rush through work, getting it done rather than getting it right, aren’t we re-enforcing their own poor learning habits when we teach to completion over quality?

         Many will argue that teaching to mastery takes too much time, and that we don’t have the extra hours or days to ensure that all of our students become proficient at each new lesson.

         I disagree. It isn’t necessarily so. It all depends on how we teach the new information or skills, and what kind and what quality of practice we give our students.

         It also depends on the manner in which we release responsibility back to kids as they work. If we jump too quickly from “I show you” to “you do it,” mastery will take much longer than if we move, increment by increment, from “I show you,” to “I show you again,” to “we do it together,” to “we do it together more,” to “you and a partner do it,, to “try it again, and I’ll be right here,” to “hey, try it and I’ll step back a bit,” to “hey, you can do this!”

         All of that, by the way, doesn’t take place in math, for example, for weeks and weeks; it’s really a matter of days.

demanding classroom · high expectations · learning disabled · mastery · modules · proficiency · rigor · rigorous instruction · special education · standards

 By Sara Finegan 

         Last year I started a program I called “Nimble with Numbers.”  Before each math lesson, and sometimes before we started reading and writing instruction, we’d do math problems.  But we never used paper.  We used language and our heads, and worked through them out loud. 

         It became one of my students’ favorite activities, and we continue it to this day.

          Nimble with Numbers involves what I think of as math fluency – the ability to work with numbers in creative ways to get to an answer.  Here’s an example of what we do:.

         Using the document camera, I will write a couple of numbers – with no operation expressions.  And I will ask out loud for an answer.  This is what it looks like:

 Me:   (writing the numbers 36 and 72).  What’s the difference between 36 and 72?

Students:  Oh!  Oh!  Let me!

Me:  Ok……..Brandon.  Give it a whack.

Brandon:  Well, 36 to 46 is ten.  And 46 to 56 is another ten.  56 to 66 is ten.

 Me:  Ok, so how many do we have so far.

 Brandon:   Thirty.  Ten and ten and ten.

Me:    Ok.

 Brandon:    And 66 to 70 is …67, 68. 69….four.

Me:     Four.  Ok.  So what do we have now?

 Brandon:  Thirty-four. 

Me:    Ok.

Brandon:    And 70 to 72 is two more.  So thirty six. 

Me:    So what is the answer?

 Brandon:   The difference between 36 and 72 is thirty six.

         We do this with multiplication, addition, and subtraction, always using the proper vocabulary (difference, sum, product). 

          What I have discovered is that TALKING our way through math problems embeds the skills and opens up the synapses for math reasoning in a way that nothing else has in my classroom.    The ability to explain our reasoning is an added benefit:  the real gift for my kids has been the development of fluency in their approach to math problem-solving by moving to friendly numbers and taking things step by step.

Another example:

Me:    Ok.  How many legs do 456 elephants have?  (Writing 456.)

 Kids:   Me!  Oh me!  Me! Me! Me!

 Me:     Hmmm.  Mariana.

 Mariana:   Well, 456 is really four hundred plus fifty plus six.

Me:   Uhuh. 

 Mariana:   And an elephant has four legs.

 Me:    Yep.

 Mariana:   Four legs times 400 elephants is……….four times four is 16.  So since it’s four times 400, we’re going to add two zeros to that number in the ones and tens places.  So that’s 1600.

 Me:     Uhuh.

 Mariana:   Then, four times 50 is……well, four times five is 20.  But 50 is five tens, not five ones.  So it’s 200.  

Me:     Uhuh.  What do we have so far? 

 Mariana:  Sixteen hundred and two hundred is eighteen hundred.

 Me:    Ok.

 Mariana:  So four times six elephants is 24 elephants.  So four hundred and twenty six elephants have eighteen hundred and twenty-four legs.

 Me:     Applause. 

         In a demanding classroom, the kids do the work.  The teacher calls on people and facilitates.

demanding classroom · high expectations · learning disabled · math fluency · math language · Nimble with Numbers · rigor · rigorous instruction · special education · standards

By Sara Finegan

         If we knew as much about math disabilities as we do about reading disabilities, we’d be in far better shape in this country. Great Britain, Australia, and New Zealand educators and researchers have been exploring all aspects of difficulty with numbers for a long time. I don’t notice a whole lot of interest among American researchers, and as a result, we are unfocused and less-than experts when it comes to math instruction in the special ed classroom.

          I’m the first to admit that I don’t know a lot, and that I am slow to educate myself. This is probably because I wasn’t much good with math myself as a student, and I still have the residue of my frustration and insecurity flowing through my veins.

         Still, I’ve come to actually like teaching math, and have lately been having more success than failure in developing strategies that work in my classroom. In my school district, kids take benchmark assessments every 6-8 weeks in math to ascertain their progress and status with regard to the standards.

         For the first time last year, most of my students scored at Basic level or above after the first math benchmark exam.  We also made our way all the way through the fourth and fifth grade math standards – in our own time, and in our own way.  This year, 12 of 14 of my students are in general education math at their own grade level.  I’m not sure yet how well it’s working, but at least we’re giving it a try.

My two mentors. 

         I owe pretty much everything about my approach to math to two skilled colleagues. 

          Greg Roy, the sixth grade teacher at the elementary school that has been my home away from home for the last five years, introduced me to “friendly numbers” and “decomposing big numbers” and base tens. Leading our staff development in math, he taught me how to think about numbers and operations in a new way.

          He continues to give me simple explanations for concepts that seem too complicated for me to teach. And he trusts me enough now to have handed over half of the sixth grade general education class to me for math this year.

         Leatrice Roberts was for many years a District Math Resource Teacher in San Diego Unified School District until budget cuts eliminated her position. She’s now in her own classroom and doesn’t have as much time to help guide my thinking about math, but I can always reach her by email. And before the financial meltdown forced our district to wipe out the entire math department, she spent hours in my classroom learning about learning disabilities and brainstorming with me about ways to teach new concepts and skills.

         If not for Leatrice, I would not have overcome my fear of teaching kids about decimals: she walked in and did a one-hour lesson that opened up all of the doors for us and jump-started a comprehensive unit in which the kids became inspired by math.

Just another brick in the wall.

         I envision (and I’m sure this is not an original train of thought here) math as a long brick wall. (when I was in school, I think I bruised my head on it more times than not). Each brick is an important skill. The more loose or missing bricks we have, the more unsteady the wall. Most students in Special Day Classes are missing a lot of bricks, and the ones that are in place are precarious, at best.

 So what are these bricks?

         The bottom row is all about math facts.

• Knowing what makes ten and what ten minus any number is.
• Knowing what makes twenty and what twenty minus any number is.
• Knowing what makes one hundred.
• Multiplication facts.

         Math facts fit into an area of math reasoning called numeracy, which has to do with understanding the value , use, and place of numbers.

         Low numeracy is a highly-prevalent math disability that is pretty much unspoken in the U.S. In England, they know all about it and are developing ways to help kids and adults with coping strategies.

         We don’t really know why so many people don’t grasp the basic math facts. And it really doesn’t matter why, now, does it? What’s important is how we teach those facts or, if a student cannot internalize them, what strategies we can teach him or her to use to move on: what mortar, so to speak can we use to strengthen this level of the math wall?

         Another row of the brick wall has to be the language of math. There’s a whole vocabulary of math that one must know in order to be able to do things like set up a numerical equation from a word problem, or just figure out what a word problem is asking.

         This vocabulary is part of math reasoning, and you can bet that math reasoning is yet another skill set that is tenuous, at best, in kids with learning disabilities.  I’m just starting to explore the language of math. Well, not the language itself, but how to blend it into my instruction and get my students to embed it in their skill sets.

         Part of the reason that understanding the language of math is important is that it helps us to visualize what’s going on in a problem and grasp what the question is all about. Consider this word problem:

Ms. Finegan bought 92 Halloween pencils – the kind with sparkly purple and orange ghosts on them, and erasers in the shape of skulls. She wants to share them equally with her 18 students. How many will each student receive?

         If you don’t know that the words “share equally” mean that each student is going to get the same number of pencils, you aren’t going to be able to draw a picture or diagram of the problem, which I think is really important, though not crucial, to developing math reasoning.

         And if you don’t know that “share equally” implies division, you, like most of my students last year, you aren’t going to have the foggiest idea how to proceed with this problem whether you can draw or not.

         So many of our students not only don’t know the basic math vocabulary but have receptive and/or expressive language disorders that sometimes math instruction seems incredibly daunting. It’s much easier to throw a worksheet in front of students and teach them the rote solving strategies we learned growing up than it is to force the language through the lesson, to talk and talk, and model, model, model. But it’s that talking and talking that is going to develop math students.

For my next few posts about math instruction, I’m going to focus on these basic numeracy skills. We’ll move on to higher-level reasoning later in the year.

demanding classroom · high expectations · learning disabled · low numeracy · math disabilities · math language · number sense · rigor · rigorous instruction · special education · standards