Guest blog: Allison and Paul Friedmann on Math Basics

This is a long blog. 1. New study.

2. The Friedmanns.

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1. New study: Basic Math Skills Linked to PSAT Success

New research from Western University provides brain imaging evidence that students well-versed in very basic single digit arithmetic (5+2=7 or 7-3=4) are better equipped to score higher on the Preliminary Scholastic Aptitude Test (PSAT), an examination sat by millions of students in the United States each year in preparation for college admission tests.

Well duh. I'd expect Einstein to be faster both at 5+2 and better at PSAT math puzzles. So what?

Students that scored better on the PSAT math tests generated greater positive activation in a brain region in the left side of the brain, called the supramarginal gyrus, which is known to be linked to fact retrieval, while students that scored lower, produced more activity in an area in the right side of the brain, called the intraparietal sulcus, which is involved in quantity processing and more effortful problem solving.

Ah. To be better in more complex math, you use recall of basic facts.

"Essentially, this means that those high school students that don't do so well on the PSAT use more problem solving strategies when they are doing very elementary sums and subtractions," offers Ansari. "If you are a high school student and you are using brain circuits that we know are associated with fact retrieval and fluency, we see evidence that you are also going to score better on the math portion of the college admission test.

There is a clear link between fluency and high level abilities -- being fluent at basic math counts."

This new knowledge is important from a math education perspective, concludes Ansari, because traditionally, debate rages between the "drill and kill" style approach versus more conceptual, problem-solving based pedagogy, but it is now clear that both methods are important in elementary education. These findings suggest that the way in which the brain is organized for single digit arithmetic calculation predicts performance on more complex math skills, illustrating the critical role that arithmetic fluency plays in building mathematical proficiency among students.

That is important, I think. We wonder: is it worth getting kids to memorize math facts? Particularly with the pervasiveness of calculators. And part of the answer, it seems, is: automatic recall frees up your brain for other math thinking.

This sets up other questions. What is the optimal mix of getting kids to automatically recall math versus getting them to think about it? And in what sequence, with what interrelation?

I hope to bring various viewpoints on this question over the coming weeks.

2. The Friedmanns.

A few days ago you saw a blog from Paul Friedmann, who teaches 7th grade math at Edward Brooke Charter School here in Boston, about teacher retention. His wife Allison teaches 3rd grade at the same school. Paul chimed in, too.

I asked:

Hi you two. Age-old question of teaching basic math facts versus teaching "true understanding."

Let's imagine a scale of 0 to 100. I'll write things out in the extremes in order to try to allow you to "pick your spot on the continuum."

Let's further suppose we're answering it for a 3rd grade teacher who controls her curriculum. About half her kids are on grade level, and half can't quickly/accurately do 349 + 175 or 41 - 29.

On this 0 to 100 scale

0 is: the most old fashioned version of math one can imagine. It's in a log cabin somewhere. Drills. Learn math entirely sequentially. When you have instant recall of, say, 4 + 9, you move to single digit subtraction. Then mult. Then div. Then 2 digits, same deal. Etc. You do NOT focus with kids on what this means. The belief is the kids can learn "meaning" later.

100 is: the most new math you can imagine. Focus on math process, ideas, and visualization over automaticity of basic facts and formulas. Kids will come to understand all the "basic stuff" by understanding the math ideas.

Here are two examples from an article to illustrate what "new math" might mean.

Example 1 from article: 349 + 175

"When we learned how to add 349 + 175, we stacked up the numbers, added the ones, carried the tens, added those, and so on in order to get the answer (524). With Investigations, third-graders, for instance, explore different methods for arriving at the answer. They may add the hundreds, tens, and ones separately (300 + 100, 40 + 70, 9 + 5) or break the numbers into rounded chunks (350 + 175 = 525 – 1 = 524)."

Example 2 from article: 41 - 29

Students might use objects like cubes or tiles (known as manipulatives) during a subtraction lesson, or they might use the hundreds board, a grid with 100 numbered squares, to figure out the answer to a problem like 41 – 29: The kids put a finger on 41 and then count back to 29. “They can count by tens, by ones, or count forward from twenty-nine to forty-one,” says Scoggin, now a consultant. “It’s fun—like counting spaces on a board game.”

So Allison. And Paul. Where are you on the continuum, when we're talking about kids who have been "taught" basics for at least a couple years but don't have mastery of them?

Allison, who teaches 3rd grade at Edward Brooke, writes:

I think I am exactly at a 50 on your 0 to 100 scale.

I do both conceptual and computational/fact fluency work with my students.

Let me help you understand our school. (Her school is terrific, and all the teachers row in the same direction).

We believe:

-You can have time to teach concepts and get mastery of basics. Students can master the basic concept and the computation. You might expect that there isn't time, but I believe that if kids really master the concept, then the computation learning goes quickly. You just have to ground it in the conceptual base that the kids have. So if you taught the concept of "you can trade in 10 ones for a 10 rod," then you build in a thought pattern in their head that goes, "I am trading in one ten rod for ten ones" at first and then they do this a hundred times and they don't need the internal monologue anymore.

I think the reason that we see so many kids lacking mastery who were taught, in their former schools, with programs like TERC is that TERC helps kids explore concepts and never routinizes the process.

The exploration is great. It builds number sense and a conceptual understanding. But I want my kids to eventually be able to do computations without much thought, freeing up brain space for problem solving. That level of mastery takes repetition, which TERC doesn't build into their program. (MG note: The local district has used TERC Investigations for many years. Background here).

-I would not start on third grade skills with the "behind" kids you are talking about. I would go back as far as you need to to get 80% of the class understanding. In your example, just 50% are understanding.

You don't need to go back far enough to have 100% of kids understanding because the 20% who need even more remedial work will be your tutoring group and may be retained anyway.

Q: What did you do back in the day when Brooke admitted a few new 5th graders each year?

A: They'd invariably be way way behind the kids who'd been with us from Grade K. With the new kids, we used to go all the way back to the concept of the difference between the value and number. That is essentially place value.

And the next big concept was the concept that you can only count things together if they are equal size. Both of these concepts help with basic computations with whole numbers as well as fractions and finally with variables when you start recombining variables and constants etc. in 7th grade. But really those concepts are like first grade concepts. So we went really far back.

We still taught addition and subtraction conceptually and just sacrificed our 5th grade math MCAS scores for those handful of newly arrived kids. Our "homegrown" students were doing so well on the 5th grade MCAS that we didn't feel much "test pressure" related to the brand new kids. We knew they'd flourish in later grades.

Q: Tell me more about how Brooke introduces concepts, skills, fluency.

A: Our school teaches conceptual basics first to mastery and then builds fluency to mastery, and finally cycles back to the conceptual with application questions. When they gain fluency, students can handle much bigger numbers in such questions.

At Brooke, we do this by putting the conceptual a year before they need a skill. Then they do the computational fluency in the next year. For instance, kids learn the concept of multiplication in 2nd grade. To move to third grade, they must show that they can turn multiplication number sentences into a variety of drawings, translate drawings into pictures, use manipulatives to model multiplication, multiplication sentences into repeated addition, use skip counting to solve a multiplication problem, draw arrays, turn story problems into pictures and number sentences etc. Then they have to prove to us in third that they have retained these skills.

As soon as they prove mastery on each, they begin memorizing their multiplication facts and working on computations. And while they are working on computational fluency, we pepper them with story problems the whole way, along with applications like finding factors and multiples etc. Every operation is taught this way.

Q: So what's the short list of topics?

A: It looks like this.

K - addition and subtraction concepts,

1st - addition and subtraction fluency and application,

2nd- multiplication and division concepts,

3rd - mult and div fluency and fraction concepts,

4th fraction fluency and decimal concepts,

5th - decimal fluency, exponent concepts,

6th exponent fluency and negative integers concepts

Q: What would you do if you were a solo teacher in a more traditional school? Lots of your kids would arrive to your classroom with low skills.

A: It's been a long time since I've taught at a crazy school. I forget how lucky I am to be able to rely on competent colleagues.

In a crazy school, I would still use the 50/50 split to catch kids up. I honestly don't believe that teaching conceptually is slower. I really believe that you don't end up needing as much cyclical review and with a concept behind them, kids master computational fluency faster to make up the time.

And even if kids don't cover as much content in the year, what they have learned will stick better with a concept behind it. And that is particularly important if the next teacher isn't going to continue what you are doing.

This doesn't answer your question exactly and you probably wouldn't want to post it because it goes against everything that no excuses charters seek to do....

Q: Don't worry about me. What's on your mind?

A: When I taught science in Chicago Public Schools in a school that was actually so crazy that it was shut down soon after I left, I thought a lot about my kids' life trajectories. I quickly realized that I was not going to change most of my kids' life trajectories in 45 minutes a day. The high school I was in admitted freshman classes of 450 students and usually graduated 40-50 of them. I was only thinking about college for a handful of kids.

So I thought about what I could give the rest of the kids that would make their life better.

Q: And what was that?

A: Seven things.

1) Habits of learning that would transfer to being able to hold a job down later in life

2) Understanding that gaining knowledge can be rewarding (which led to teaching more depth and less breadth because really understanding something is so much more rewarding)

3) An understanding of the scientific method and statistics so that they would not believe crazy advertisements that use pseudo research (We did a lot of experiments and analyzed a lot of data. We evaluated advertisements and research)

4) The ability to structure thinking around cause and effect (Isn't that really what science is all about anyway? And isn't cause and effect empowering in our own lives?)

5) In my biology class - An understanding of their body so that they would be able to advocate for themselves in a doctor's office (We spent longer on anatomy and less time at the cellular level)

6) In my environmental science class - An understanding of toxicity so that they could advocate for the community about issues of environmental justice

7) Phonics - Sounds ridiculous for a science class, but I had a couple of kids who couldn't sound out basic words. I passed them in biology for sitting through class and then also attending daily reading sessions with me after school. They had basic reading skills by the end of the year, and even though we had only contracted for them to sit through biology class, they actually learned a fair amount of the biology because they believed that what I was teaching was going to empower them. In one of their words, "You're for real."

Q: Got it. And if you taught math in that school instead of science, would the list be the same?

A: #1 and 2 above would be the same.


3) The ability to stick with a hard problem and explore it. The idea that grit leads to feeling good about oneself.

4) Mental math and estimation - If you need to do complicated calculations, you can just as easily pull out your cell phone to use the calculator on it as you can pull out a piece of paper to do the calculation by hand. The ability to manipulate numbers in your head is useful when you don't want to pull anything out.

5) Application questions - If you aren't going to college, you will use math to solve real world problems

6) Anything to do with money - Isn't that where we do most of our math? Isn't that where people with less education often get scammed?

7) Statistics for the same reason as above (I was happy to see that the Common Core has stronger focus on data and statistics than the Mass standards did. I think this makes sense to be literate human beings)

Q: Paul, what would you say?

A: I agree with everything my genius wife said above. I don't find that there is any value in teaching just computation skills, broken apart from application.

I also think the idea (that many charters adopt) is that kids can't do higher order thinking without complete mastery of the foundational skills to be small minded and sells kids short.

I often start units or sections of units with classes where kids tackle debatable questions or a new concept without any of the i do, we do, you we do stuff. It's fascinating what kind of discoveries and connections kids can make (even the weakest kids in the class). Of course, they are not operating from a point of zero background knowledge. And, I don't assume that they then have mastery of what they "discovered."

We then go through a very scaffolded series of lessons to get them there. And sure, some kids who don't have absolute fluency with numbers have more trouble along the way. However, it gives them the motivation to fix their foundational errors (which the teacher needs to help them with). And, the follow up conversations when they get something wrong is much easier when it's calculation errors, not conceptual. I guess I would say (like Allison) that they need the conceptual foundations before higher order thinking, but the fluency can happen along side that work. I want kids working on word problems from the beginning, otherwise I find there is little impetus for the kids to learn the skill.

Also, when kids are tackling just calculations in class, I find that it is very easy for the teacher to spend 90% of the lesson on calculation mastery and 10% on application. But the MCAS is 90% conceptual and application.....

Q: And if you didn't teach at Brooke, and at a school where your 7th graders arrived unable to multiply, then what?

A: I think I would prioritize some important concepts that link to the basics they are missing. So if kids are struggling with the multiplication algorithm, I would just teach the skill. If they didn't understand what multiplication was and how to apply it, I would prioritize that. I would happily skip the conceptual work I do around the fraction division algorithm if I could get kids to master it AND also get why it makes sense that dividing by a number less than one yields a larger quotient. Because at the end of the day, you can always look up the algorithm or use a calculator.

In this fictitious situation, I would make sure that my kids understood the 4 operations, place value, fractions and proportionality (ratios, rates percentages), backwards and forwards, with perhaps some basic work on solving and applying equations. Otherwise they are screwed when it comes to algebra, geometry, stats and probability.

What would be important to me would be that they could understand how to apply those skills to word problems, whether their answers seemed reasonable, etc.

Q: What's the hardest part of math teaching?

A: I always find that the hardest part of math is getting kids to use their basic skills to solve problems. An example I can think of relates to missing angle problems - kids might know the information about angles, equations, the 4 operations, etc. but if they can't put those together and use them to think, who cares? Because that kid is not going to be successful at the next level, whether that is 8th grade or high school or college.

Math should be, in my opinion, about being able to use number to think and solve problems, not to complete worksheets of practice. I guess it comes down to the idea that I would want them to leave my class with a foundation to build on that would actually support the next level of math. Pure basic calculations is insufficient for that purpose. I just don't think the basics are worthwhile if they are not, to some extent, connected to the bigger patterns of math and can be applied in context.