Guest Post – Follow-Up on Homework in Elementary Schools

After I published the post called “What About Homework” I got an e-mail from a friend who is a principal in Washington County in southeastern Utah. With his permission, I am posting his e-mail here, with emphasis added where needed.

“Very interesting that you should bring up the homework controversy as I read Visible Learning for Teachers by John Hattie (maybe you’re familiar with his work?) and he does list homework in his compilation of research.  If you’ve read him, you’ll know that his studies include an estimated 240 million students, but in order to reach “effect sizes” he uses over 161 studies of 100,000 students or more (so his results are quite believable).  So… an “effect size” of 0.40 is considered “worthwhile” (whether it’s an intervention or a program, etc.) and anything above 0.40 is even more worthwhile.  It’s a very interesting read to see all of our practices and how “effective” his analysis of research shows they are.  It’s been fascinating, actually.

Ok… that being said… you’d be interested to know how homework rates?  In high school the effect size is 0.50…. so it’s worthwhile, actually pretty good.  But in elementary…not so good.  The effect size is -0.08…. That’s negative 8 hundredths!   That means it does more harm than even close to being good.  That made me wonder.  We haven’t changed much at our school, but we’ve discussed this research including the homework. Homework Monster

I just thought I’d pass that along.

Kelly Mitchell
Principal, Washington Elementary”

Thanks, Kelly. That ought to give us something to think about.

Homework Monster downloaded from http://lifeandtimesofbasil.blogspot.ca/2012/02/homework-monster.html
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What About Homework?

Back when I was a youngster, lo, these many years ago, homework was the bane of my existence. In fact, I tell people, truthfully, that I only did homework for two classes in high school – Trigonometry and Medieval Literature. That I graduated at all is a miracle. That I got college scholarships is even more of a mystery. Now I believe that homework itself is a mystery. At least in elementary school.

How much homework is too much? Should we send homework at all? What kind of work should be homework? Should there be policies regarding homework in elementary schools? See what I mean? Mysteries. When I talked with the LEA (Local Education Agency – that is fancy talk for school districts and charter schools) mathematics supervisors a few months ago it became clear that most LEAs haven’t thought a lot about homework policies or practices for quite a while.

The research on homework is an extremely mixed bag. In the Winter 2012 edition of the Harvard Graduate School of Education magazine Ed. the following quote appears, “As Cathy Vatterott, the author of Rethinking Homework, points out, ‘Homework has generated enough research so that a study can be found to support almost any position, as long as conflicting studies are ignored.’ ”

The same article quotes Alfie Kohn, the author of The Homework Myth, as saying “The fact that there isn’t anything close to unanimity among experts belies the widespread assumption that homework helps.” Mr. Kohn believes strongly that schools should never assign homework.

However, in the same article, Howard Gardner, a professor in the Harvard Graduate School of Education says, ““America and Americans lurch between too little homework in many of our schools to an excess of homework in our most competitive environments — Li’l Abner vs. Tiger Mother,” he says. “Neither approach makes sense. Homework should build on what happens in class, consolidating skills and helping students to answer new questions.”

Educational Leadership, the journal of the Association for Supervision and Curriculum Development, notes the following recommendations for homework:

Research provides strong evidence that, when used appropriately, homework benefits student achievement. To make sure that homework is appropriate, teachers and principals should follow these guidelines:

  • Assign purposeful homework. Legitimate purposes for homework include practicing a skill or process that students can do independently but not fluently, elaborating on information that has been addressed in class to deepen students’ knowledge, and providing opportunities for students to explore topics of their own interest. Teachers should never assign homework that introduces a new skill or asks students to practice concepts they still find difficult to understand.
  • Design homework to maximize the chances that students will complete it. For example, ensure that homework is at the appropriate level of difficulty. Students should be able to complete homework assignments independently with relatively high success rates, but they should still find the assignments challenging enough to be interesting.
  • Involve parents in appropriate ways (for example, as a sounding board to help students summarize what they learned from the homework) without requiring parents to act as teachers or to police students’ homework completion. If parents feel they must teach their children a concept the homework, at least in the younger grades, is probably not appropriate.
  • Carefully monitor the amount of homework assigned so that it is appropriate to students’ age levels and does not take too much time away from other home activities. The authors note that many research studies recommend the “10 minutes times the grade” rule. In other words, in first grade all homework combined should not take more than 10 minutes per night. In second grade the time allowed could raise to 20 minutes, and so on. However, the studies also note that positive effects of homework in junior high top out at 90 minutes. More time spent on homework over that time limit actually had a negative effect on achievement. In high school the limit seems to be 1 1/2 to 2 hours a night, depending on the study.

The point here is that elementary teachers should be very careful with what they assign as homework and with how much homework they assign. Parents should be made aware of the purpose of the homework assignments, the length of time the student should spend, and what the expectations are. Parents should feel free to call a halt to homework assignments if their child is getting frustrated, spending an inordinate amount of time on homework, or obviously doesn’t understand what to do. Sending a note or an e-mail to the teacher is entirely appropriate in those cases and teachers should respond positively.

As well, there are many technological solutions to help solve the homework mystery. In Cache School District, for example, many elementary teachers use an app called Educreations on their iPads to record lessons. They then post them either on the Educreations website or on their teacher blogs. I saw an outstanding example of that approach last October while visiting McKenzie Sorensen’s  fifth grade class at Summit Elementary in Smithfield. Here is my description of the event:

Ms. Sorensen was using her iPad and Educreations to teach long division with decimals. Her iPad  was connected to the projector and she was not only recording her voice but was using the iPad as a kind of smart board. She was writing on the iPad as she taught. She also recorded student comments, their answers to questions, and so on. Mr. Pugmire, the principal, was standing right next to me and told me she would post that lesson on her blog that night so students could refer back to it, so that those who missed class could see and hear what happened, and so that parents could understand what their students were doing and help them with their homework. It was a good thing, too, because she was teaching division with decimals in a mathematically correct and conceptual way, very different from how most of us were taught. Instead of just telling the students to move the decimal point she emphasized that they were changing the place value of the digits in both the divisor and the dividend by multiplying them by a power of ten. She stressed that they could have done the problem without changing the place value, but doing so made the problem easier to do because they didn’t have to worry about keeping track of the decimal point.  They were also able to preserve place value, which doesn’t often happen in long division.

Other teachers in Cache record short messages for parents and/or students about what they taught that day and how they taught it. Such messages can help parents understand the strategies and teaching methods they use to help students understand concepts.

Other teachers I have talked to record a short video on their smart phone at the end of the day and explain the homework for that night. They post that on their website and parents know to check for it. Others simply send an e-mail message. Others direct the parents to websites such as LearnZillion that have short videos made by master teachers explaining mathematical concepts. Jordan District, for example, has linked both the Utah Core Standards and their textbook to LearnZillion videos. Many textbook publishers also have websites for parents that can help. Contact your LEA to see if their adopted textbook has such a site.

So, to recap. The research isn’t clear about whether homework in the elementary grades has any correlation to student achievement. One researcher, Duke University Professor Harris Cooper, found that for elementary school students, “the average correlation between time spent on homework and achievement … hovered around zero.” So some elementary schools are eliminating homework other than nightly reading and citing research to do so. Still, it is hard in an era of “raise the bar” academic standards to think about forgoing homework altogether. Those elementary schools that do give homework should think very carefully about the recommendations given above. It is hard to justify homework outside of the given recommendations.

By the way, I am not endorsing Educreations. There are other apps like Doceri and Explain Everything that do the same thing. They are called interactive white board or screencasting apps. Happy mathematics learning to all!

Sources:

http://www.gse.harvard.edu/news-impact/2012/01/are-you-down-with-or-done-with-homework/#ixzz333M9Gvtk Illustration from the same source. 

Marzano, Robert J. and Pickering, Debra J. “Special Topic: The Case for and Against Homework. Educational Leadership. March 2007, Volume 64, Number 6, Pages 74-79 Accessed 5/28/2014 at http://www.ascd.org/publications/educational-leadership/mar07/vol64/num06/The-Case-For-and-Against-Homework.aspx

Cooper, Harris. “Does Homework Improve Academic Achievement? If So, How Much is Best?” Accessed 5/29/2014 at http://www.sedl.org/pubs/sedl-letter/v20n02/homework.html

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Association of State Supervisors of Mathematics

A few weeks ago I attended the annual meeting of the Association of State Supervisors of Education (ASSM), followed by the annual conference of the National Council of Supervisors of Mathematics (NCSM). These were amazing meetings and I learned a great deal. At the ASSM meeting, I, with my colleagues who lead mathematics in other states, heard from the likes of Steve Leinwand, Cathy Seeley, Phil Daro, Dan Meyer, Tim Kanold, Linda Gojak, Bill McCallum, Uri Treisman, and so on. These are some of the brightest minds in our field. I learned a great deal. I, with many of my colleagues, also tweeted a great deal. Tweeting is my way of taking notes at these meetings. I found an application called Storify (http://www.storify.com) that allows the user to insert tweets, videos, blog posts, and so on and put them into a story format. I decided to do that with the tweets from the ASSM meeting. If you would like a small sample of what was discussed and the powerful ideas that were presented at ASSM go here: https://storify.com/MathDavidUtah/association-of-state-supervisors-of-mathematics

Dan Meyer also wrote about his presentation at ASSM in his blog. I’m sure he didn’t develop the whole idea just for us. However, his presentation called “The Future of Textbooks, If There Is a Future for Textbooks” was really thought provoking. Taking digital textbooks out of airplane mode was his main theme. If they don’t do anything a smartphone in airplane mode would do, if they don’t connect to anything else, then they are just a textbook.

More later.

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Random Comments About Learning Mathematics

I Googled “Math Quotes Story Problems” today and got this:

Story ProblemsAccessed at http://www.pinterest.com/pin/200339883394174796/. No, I don’t have a pinterest account. That is just where Google took me. Really.

I Googled story problems for a reason. I went to see a new doctor this morning because my old one either retired or went crazy. I’m not sure which. You know that new doctors ask a lot of questions. One she asked me was what I do for a living. Now, that is a loaded question. Normally when I tell people that I am the Elementary Mathematics Specialist at the Utah State Office of Education I get a lot of unwanted sympathy.

“Oh, I’m so sorry! I hate math.”
“Oooo, I never liked math. How do you do it?”
“Bummer, man.” – okay, so they would have said that in the 60’s but I’m old enough to remember it.

When I tell them I love what I do they can hardly believe it. They look at me really funny when I tell them they should give math a second chance.

Another reaction I get is some kind of math story. My new doctor this morning told me hers. She said that she was always really good at math in high school and that was tough. People made fun of her because of her math achievement and especially because she was a girl. That all seemed to be socially unacceptable. But she hung in there and took math all the way through calculus in high school and did very well.

Things changed in college, though. This was in the days before AP tests so she had to retake calculus. She thought it would be a breeze because she got A’s in it in high school. No sweat. However, every problem was a story problem! Every one of them! She wasn’t used to that. In high school she just had to do sets of 20 to 40 problems and then, maybe, one story problem. She had no idea how to interpret all these story problems and she was intimidated. So, she talked to her engineer dad. He just laughed and said, “Listen, dear. Life is a story problem.” To an engineer math is for solving problems. I’m sure they have very little use for sets of problems with no connection to reality.

Another story – this morning my wife told me she had read my post on Strategies in Addition and Subtraction. She said it made a lot of sense to her. She struggled with math in elementary school until she got a teacher in sixth grade who took a special interest in her and taught her ways of dealing with numbers and operations. She was still put in the low group in seventh grade math but she quickly found out that, because of what her sixth grade teacher taught her, she could figure out what to do pretty easily. She couldn’t understand why other people were having so much trouble with the math. She is sure that her sixth grade teacher taught her some strategies that made sense to her. Even today she does math in her own way and is very, very quick and accurate.

What is the moral of this story problem? We have to teach in context, in problem solving situations, in the real world or as close to it as we can get. Dan Meyer in his blog  http://blog.mrmeyer.com/ talked about Fake World contexts not long ago. You need to read what he said, but one quote that will stick with me for a long time is “You have to know what your students worship.” Meaning, of course, that in order to really dive into the mathematics and be fully involved with problems students have to want to solve them. The problems have to raise their curiosity and perplex them a little. What a great concept.

We also have to make sure our students understand the mathematics they are learning. That is vitally important, and it doesn’t matter whether they will be engineers, coders, research scientists, mathematicians on the cutting edge of some obscure but important specialization or just people trying to balance their checkbook. Although, if students understand mathematical concepts and are engaged in math because they really want to solve those story problems, we may actually have more scientists, technologists, engineers, and mathematicians come out of our schools. That would be a great learning outcome.

 

 

 

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Math Journaling Webinar

Just a quick note. Yesterday the Utah State Office of Education and the Utah Education Network hosted a Core Academy in Action Math Webinar on Journaling in Mathematics. It was presented by Tamara Shaw, a former math coach and current elementary teacher in the Weber County School District. The webinar was recorded and is now available along with the PowerPoint Tamara used in her presentation.

Click here for the link to the webinar

Click here for the link to the PowerPoint

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Strategies for Addition and Subtraction

steve-jobs-150x150“Let’s go invent tomorrow rather than worrying about what happened yesterday.” -Steve Jobs

Sometimes we hear from parents and some teachers that they would love us to go back to teaching math the way they learned it. They say they can’t help their students because the work doesn’t look the same, it’s all in a new language. The kids complain their parents are telling them to do the math in a different way than their teacher did and it is confusing. Why is all this happening?

The Utah Core State Standards in Mathematics put a premium on understanding the why (conceptual understanding) as well as the how (procedural understanding, skill, and fluency) and the application of mathematics. All three aspects are equally as important. The standards don’t try to reinvent mathematics. But they may and do take a different path than they did in the past. Why? Because of a great deal of evidence that our past methods of teaching and learning math failed a great many of our students.

Marilyn Burns, in her book Math: Facing an American Phobia says, “Even in the face of widespread failure in learning mathematics, we seem to want to cling to educational methods with a nostalgia for them that has long outlasted their usefulness and has perpetuated failure. The way we’ve traditionally been taught mathematics has created a recurring cycle of math phobia, generation to generation that has been difficult to break. We start young children with counting and move them along through arithmetic, then on to algebra, geometry, trigonometry, and so on. The ‘and so on’ depends on whether or not the student sticks with math, which means not falling off the ladder of math progress in school. But an alarming percentage of the people in our country have fallen off the ladder and feel like mathematical failures. And once people fall off the ladder, there seems to be no way for them to get back on.

. . . Children must be helped to learn mathematics in a better way than we were, so that mathematical limits do not shut them out of certain life choices and career options.”

I agree with Marilyn, and with countless other mathematics educators and mathematicians, that we must “help children learn mathematics in a better way than we were…” That is what the Utah Core State Standards are constructed to do.

One giant step in helping children learn mathematics in the core is the evidence-based method of building understanding of and fluency in algorithms by using math strategies that are based on sound mathematical principles such as place value, properties of operations (e.g., associative, commutative, distributive, identity), the relationship between operations, and mental math. Make no mistake, great math teachers have been using these strategies for many years. They are not new.

The standards make heavy use of strategies instead of algorithms in the early grades when students are building number sense and making sense of operations like addition, subtraction, multiplication, and division. That the standards mention strategies eighteen times in grades K-3 but just seven times in grades 4 and 5 and only twice in grades 6 – 8 should tell us something about their use. They build understanding. Students then, at an appropriate time, with the correct understanding, begin using and become fluent with the standard algorithms. We want them to know those algorithms, but we are firm in our assertion that students must understand them before they use them. At each step of the game students are required to explain and justify their reasoning mathematically. Of course, as I said in my last post, many of us still use strategies we learned or devised for solving problems mentally. See that post for details.

In the document Progressions for the Common Core State Standards in Mathematics: K-5 Operations and Algebraic Thinking (2011) the authors note three levels of strategies for adding and subtracting one digit numbers used in the standards. They are:

Level One: Direct Modeling by Counting All or Taking Away

Level Two: Counting On

Level Three: Convert to an Easier Equivalent Problem

For more information on these levels, including definitions and examples, click here and scroll to page 36.

The standards also make good use of mental math strategies in the early grades. Those strategies, which are part of the Level Two and Level Three strategies, include:

Counting on: 8 + 4 = □ (8 …9, 10, 11, 12)
Counting back: 12 – 4 = □ (12…11, 10, 9, 8)
Making tens: 5 + 7 = □ (5 = 2 + 3 so 3 + 7 = 10 therefore 10 + 2 = 12)
Doubles: 6 + 6 = □
Doubles plus/minus one: 6 + 7 = □ (6 + 6 + 1 or 7 + 7 – 1)
Decomposing a number leading to a ten: 15 – 7 = □, so 15 – 5 = 10, therefore 10 – 2 = 8)
Working knowledge of fact families/related facts/number bonds: 3 + 9 = 12 so 12 – 9= □

When students begin to face addition and subtraction problems where regrouping is necessary they first explore different strategies of solving such problems rather than immediately diving into the confusing world of what has been called carrying, borrowing, regrouping, decomposing, and so on. Students can use their understanding of place value and the associative and commutative properties of addition to solve problems such as 47 + 33. Instead of carrying a one (which is actually a ten) students write the problem in expanded form as (40 + 7) + (30 + 3) (though they probably won’t use the parentheses). They can then use the properties of operations and come up with (40 + 30) + (7 + 3). Then they add 40 + 30 = 70 and 7 + 3 = 10 and then finally add 70 +10 = 80.

I am sure some of you are thinking that is not very efficient. Remember, at this point in a student’s learning we are not looking for fluency, we are looking for understanding. How does this help them understand? Because they recognize principles they have learned before. Place value tells then that 40 and 30 both have digits in the tens place so it makes sense to group them together. The same with 7 and 3; they have digits in the ones place. In addition, they are making sense of the problem and persisting in solving it, thinking analytically and reasoning mathematically, looking for patterns in their reasoning, and so on, all of which are contained in the Standards for Mathematical Practice in our standards.

Another strategy is to start adding at the left in a vertical problem rather than at the right, as follows:

two digit addition

Van de Walle (2013) lists the following advantages of using strategies, including those students come up with on their own:

  • Students make fewer errors. When students use strategies they understand they make fewer errors than they do when using algorithms they have been taught without understanding.
  • Less reteaching is required. Though developing student understanding through strategies is slower than simply teaching an algorithm students, through constructive struggling (read the article), gain “meaningful and well-integrated networks of ideas that are robust and long lasting.”
  • Students develop number sense. Through using number rich strategies that are based in sound mathematical principles students gain an appreciation and understanding of the base ten number system rather than simply using an algorithm they don’t understand.
  • Strategies are the basis for mental computation and estimation. When students record their thinking as they are working through their strategies for solving a problem they will find that they can more easily do the steps mentally.
  • Flexible methods are often faster than standard algorithms. In one of my Elementary Principals Math and Science Leadership Academy meetings I asked the principals to solve a problem similar to this one: 4005 – 3997 = ? and to raise their hands when they were done. Some of the principals immediately wrote the problem down and began using the standard algorithm to subtract using regrouping. Others raised their hands almost immediately. When I asked one of the principals who was done quickly to share her method she said, “Well, I simply looked at the numbers and realized that 3997 is 3 away from 4000 and then just added 5. So the answer is 8.” When the only tool you have is a hammer…
  • Algorithm invention is itself a significantly important process of “doing mathematics”. When students come up with their own means of solving a problem they are “intimately involved in the process of sense making.” They develop a confidence in their ability to do mathematics. And when they can explain what they did that confidence grows and leads to greater success.

At the end of this progression students become fluent in the standard algorithm. By then, they will understand it. We will make sure they understand it by having them explain their reasoning when they use it. That reasoning was developed through the strategies.

Sources cited:

Van de Walle, J. A., Karp, K.S., & Bay-Williams, J. M. (2013). Elementary and Middle School Mathematics: Teaching Developmentally, Pearson Education

Operations and Algebraic Thinking Progression,   http://ime.math.arizona.edu/progressions/, accessed on 3/27/2014

Burns, Marilyn. (1998) Math: Facing an American Phobia. Math Solutions

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Place Value Computation Strategies

Alice Through the Looking Glass SmallCan you do Division?  Divide a loaf by a knife – what’s the answer to that?  ~Lewis Carroll, Through the Looking Glass

Once students are fluent in place value they are ready to start using those concepts in computation. The Utah Core Standards are pretty explicit in how computation instruction is sequenced. If you read the standards carefully you will note that they speak of having students use strategies to learn how to compute. Then, when they understand what they are doing, they can be taught the standard algorithms. What is the difference between a strategy and an algorithm? Here are the definitions given in the Numbers and Operations in Base-Ten progression document:

Computation strategy. Purposeful manipulations that may be
chosen for specific problems, may not have a fixed order, and
may be aimed at converting one problem into another.

Computation algorithm. A set of predefined steps applicable
to a class of problems that gives the correct result in every case
when the steps are carried out correctly.

When most of us were in school algorithms were the instructional recipe of the day. We were taught the steps for solving addition, subtraction, multiplication, and division problems. Most likely, we were taught them without understanding, meaning that we were just taught the steps and not why they were used or how they worked. That is a main cause of much of the confusion from and dislike of mathematics in general. Unless you have a very logical and organized brain it is difficult to remember all those steps. Now, that is not to say the algorithms are bad. They are not. They are efficient means of solving problems and, as the definition says, you get the correct result every time you follow all the steps correctly. We want students to be fluent in those algorithms.

But that fluency comes at the end of an instructional sequence where students use strategies to solve problems before they move to the algorithm. Why do we do that? It leads to better understanding. Let’s explore why.

Van de Walle (2013) lists three significant differences between strategies and standard algorithms:

  1. “Strategies are number oriented rather than digit oriented.” When students add 43 + 25 using the standard algorithm they tend to think of 4 + 2 instead of 40 + 20. In some ways, you could say the standard algorithm unteaches place value. This is especially true of the division algorithm. Students (and adults) in a problem such as 345 ÷ 7 will think “How many time does 7 go into 3? It doesn’t, so how many times does 7 go into 34 ?” In reality, we are speaking of how many groups of 7 are there in 345. That is what 345 ÷ 7 means. To use a place value strategy students might do something like this (thinking through the process): Well, I know that 300 ÷ 7 is about 40 because 4 times 7 is 28 so 40 times 7 would be 280. So there are 40 groups of 7 in 345. So I subtract 345 – 280 which means I have 65 left. Now I can use my math facts. The closest multiple of 7 to 65 is 63, which is 9 7. So there are 9 more groups of 7 in 345. 65 – 63 is 2. So my remainder is 2. Now I add 40 + 9 and that equals 49. So the answer is 49 R2.” In long division that would look like this:

Long Division

That looks a lot like the standard algorithm, but the computation was done with place value strategies. Another place value strategy for division is called partial quotient division. Take a look at the video on this page to see what it looks like.

2. “Strategies are left handed rather than right handed”. Often, when students use a strategy to compute, they will begin with the largest values rather than the smallest. In my last post I gave this example:

Addition

A student when explaining this strategy might say, “First, I added the hundreds. 400 + 200 = 600, so I wrote that down. Then I added the tens. 50 + 60 = 110, so I wrote that down. I made sure to put the hundreds digit in the hundreds place. Then I added the ones. 7 + 3 = 10, so I wrote that down. I put the tens digit in the tens place. Then I knew I had to add those all together to get my answer. 600 + 100 = 700, 0 + 10 + 10 = 20, and 0 + 0 + 0 = 0. So my answer is 720.”

3. “Strategies are a range of flexible options rather than ‘one right way’.” Students tend to change their strategies based on the numbers they are using. For example, try adding 465+230 and 526 + 98. Did you use the same strategy? I would have computed these two problems like this:

465 + 230 is pretty straightforward. 400 + 200 = 600, 60 + 30 = 90, 5 + 0 = 5. So the answer is 695.

526 + 98 = well, it’s easier to add 526 + 100. That is 626. Then I have to take the 2 I added to 98 back off. So the answer is 624.

Of course, when students are first exposed to a new concept the strategies they use should include using concrete objects, such as base ten blocks, counters, linking cubes, and so on. Using those manipulatives gives them the opportunity to explore concepts with things they can actually touch and manipulate. Students may then move to the representational phase, which means they will use pictures, diagrams, visual models, and so on to take the place of the concrete objects. Then they will move to using abstract numbers, like those we have been using above.

This is true even when students are learning algebra. I was watching a seventh grade class not long ago working on linear equations. The teacher allowed the students to use whatever strategies they felt comfortable with. None of the students chose to use concrete objects. But some used tables, some used graphs, and others used equations. When the students explained what they had done they were asked to tell how and why they got their answers. Those using equations explained what each part of the equation meant and why they had written it as they did. Then the teacher tied all the strategies together to show how each illustrated the same thing.

The point is that students should never use a strategy nor an algorithm unless they can explain what it means, what they did, why they did it, and why they feel their answer is appropriate.

I am going to try to write posts in this blog every two weeks rather than once a month. Just so I don’t leave you hanging, next time we will talk about computational strategies for addition and subtraction and how students can learn the standard algorithm as the core requires.

Works cited:

Van de Walle, J. A., Karp, K.S., & Bay-Williams, J. M. (2013). Elementary and Middle School Mathematics: Teaching Developmentally, Pearson Education

Numbers and Operations in Base-Ten Progression,   http://ime.math.arizona.edu/progressions/, accessed on 2/25/2014

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