I’ve kind of been on a number sense kick lately. I know part of my focus is the time of the year (my honors class is working with rational numbers and my general classes are exploring ratios and percentages), but I think all of my experiences from last year (my first year teaching) are colliding with the experiences of this year in surprising ways. I’m recognizing topics with a higher conceptual load and topics that require a solid understanding of number.

A prime example occurred in my honors class the other week. Students were working with positive and negative decimals. The basic skills of adding, subtracting, multiplying, and dividing were already in the minds of students. The focus of the unit was applying integer rules to problems with negative decimals.

During a review day, students encountered problems with subtraction. I kept hearing students use phrases like, “You make 9 into 8 and the 0 into 10,” or, “You make the 6 into 5 and the 8 becomes 18.” Students were getting correct answers, but I was wondering if they knew why they needed to use a specific process when borrowing.

During the next problem (13.27 – 8.19), a student made a statement, “You make the 2 into 1 and the 7 becomes 17.” I stopped him after this statement and told the class, “2 is not becoming 1 and 7 is not becoming 17. There’s no 2 or 7 in this problem at all, even though we do have those digits in the given number. What is actually happening in this problem?”

Silence. Some students looked at me like I was crazy. Other students looked like they were convinced I was pulling a prank of some sort. After some wait time, I made the statement, “Think about place value. Where is the 2 located in the number? Where is the 7 located?”

Silence, but now I could see some wheels turning in the minds of students. Finally, a student offered, “2 tenths is becoming 1 tenth.” YES!

The other half of the borrowing process required some explanation. Students seemed unfamiliar with the idea of 1 tenth being 10 hundredths, so I needed to go down the path of equivalent fractions for a bit before we discussed adding 10 hundredths to 7 hundredths.

I was surprised at the length of this discussion (it took about 15 minutes). In all of my classes, students are perfectly capable of reading decimals correctly; however, I wonder how many students were taught to explain algorithms with place value.

For example, how many students read a multiplication problem with decimals and work through it without any thought of place value? Consider a problem like 3.7 times 9.8. How many students begin the solving process thinking without place value (7 times 8) versus the number of students who use place value (7 tenths times 8 tenths)? Do these differences in think explain many of the errors we see when students place a decimal point in their final product? The same idea applies to division. How many students blindly divide 1 by 8 over viewing the problem as a series of place value statements? I think many students read the problem as, “0 groups of 8 fit into 1, 1 group of 8 fits into 10, etc.” instead of viewing the problem as, “0 whole groups of 8 fit into 1, but 1 tenth of a group of 8 does… so does 12 hundredths of a group… so does 125 thousandths of a group.”

Where and when should these understandings of algorithms using place value develop? Is it a matter of consistently using mathematical vocabulary? Is it a matter of comparing algorithms to manipulatives?

I don’t know if there’s any solid answers to those research grade questions, but I do know one better question I will keep using to further the algorithm understanding of my students.

WHAT IS ACTUALLY HAPPENING?

[cross posted to trigotometry]

Fascinating insights here. I think it is important to continually push kids to think about the meaning of the symbols we use. A correct understanding of place value is something that develops over time and lots of careful thought and discussion. It sounds like you have found a good way to have those conversations with your students. Thanks for the reminder… I need to have conversations like this with my students.

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Thanks Andy! Place value is definitely a long term goal in any classroom. I’m hoping to have more discussions as the year unfolds. I think vocabulary goes hand-in-hand with this idea. It’s easy for students and myself to get lazy with our terminology. I sometimes take for granted the equivalence among operations with decimals and whole numbers that even allows to use the standard algorithms in the first place. It’s true a wonderful aspect of mathematics that we can easily overlook. I think if teachers work to hold their language to the highest specificity, students will follow suit with time and practice.

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An algorithm is like a formula but with more steps, usually of the same actions. Almost every time it is easier to carry out an algorithm than to deal with the problem using basic methods. This what makes it an algorithm.When one uses an algorithm all that is required is that one believes in it. This can be achieved by two methods:

A. repeated use seems to be giving right answers, and

B. understanding how the algorithm works, that is how it is derived from basic methods.

Method B is the currently favored method, but once one has built faith in it one does not need to think about why it works. Once they “get it” they should be quite content to “forget the details”.

There is too much of this in school math. I am thinking of the roots of a quadratic formula, differentiation from first principles, adding fractions, multiplying fractions, and on and on…….

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Thanks for your thoughts. I understand the idea of faith in an algorithms over the course of time, but we are referring to long periods of use and mastery of said algorithms. 6th graders are only a couple academic years removed from learning the very algorithms we are discussing and many are prone to making mistakes when borrowing (always making a 10, etc.) or multiplying (moving the decimal point, etc.) simply because they trust that incorrect algorithm that worked for them so many times in the past. My thoughts were more related to the idea of place value in the process of using algorithms. Students (and even teachers with years of experience) can easily forget to consider the quantities they are working with when following an algorithm. Sometimes, this forgetfulness can be the product of years of faith in an algorithm. For students, it’s often that they never learned to think about the quantities being manipulated by an algorithm in the first place.

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One of the perennial problems is that students are very hazy about checking “the answer”, so they think everything is ok when they get almost any “answer”. So any faith the acquire may be misplaced.

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For a problem like 3.7 x 9.8, I would wonder about at least three solution strategies:

1. The standard algorithm for multiplication;

2. Partial products as in

my previous post, where, instead of decomposing two digit whole numbers by place value at the tens and ones place, you decompose into whole number part and the remainder, i.e., 3 & 7/10 multiplied by 9 & 8/10;3. Observing 3.7=37/10 and 9.8=98/10, and re-writing the problem as (37/10)(98/10) = (37×98)/100.

I might also look for a “friendly number” approach. For example, following strategy 3:

37/10 x 98/10 = (37×98)/100.

And 37×98 = 37(100-2) = 3700-74 = 3626.

So the product is 3626/100 = 36.26.

(For manipulatives – as you may know – there are Digi-Blocks…)

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And I would be despairing if no-one figured out that the answer should be somewhere between 35 and 40.

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The sanity check of “a bit less than 40” is definitely helpful once one converts from decimals to fractions and back! (Another good check is to observe that the hundredths place in the answer should be 6…)

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Estimation is a tool I often have students use when working with decimals (like recognizing a product should be between 35 and 40). Once again, my point is questioning how students are reasoning and thinking about an algorithm. A student may be fluent with an algorithm, but I wonder how many students are actually using mathematically accurate descriptions of what is happening when they are thinking through a problem.

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I like the equivalent fraction strategy Maya! Perhaps that approach could foster a deeper understanding of the equivalence between fractions and decimals for students? Partial products is also another intriguing strategy, since drawings of partial products are basically a visualization of the standard algorithm.

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(My response here is unnecessarily long; please skip over what is unhelpful!)

My sequencing of these topics is to begin with bases places & faces [base 10; place value; ‘face value’ as a synonym for digit, which conveniently rhymes], then through the four operations, then some peculiarities of multiplication [lcm, gcf, primes], then fractions [lcm comes up when adding fractions with different denominators; gcf comes up when simplifying a fraction: i.e., divide numerator and denominator by their gcf], and then decimals.

So:In practice, by the time decimals are introduced, there is already a setup around the base-10 number system in which to situate them. Specifically, we consider some small but meaningful items (how do you read a number name, e.g., 102.73? for this number: ‘one hundred two AND seventy three hundredths’; note that the decimal point corresponds to the word ‘and’, & wedon’tsay, e.g.,’a hundred and two point seven three’ for a few reasons) and then we build on pre-existing knowledge around the expanded form of, e.g., 102, to consider the meaning of 102.7 or 102.73.In particular, we use the base-10 number system to show why the terms “tenths place” or “hundredths place” make sense, and even to look at conventions around exponent notation. In the context of your problem, we would read 3.7 x 9.8 as “three and seven tenths times nine and eight tenths” (at least until we’ve made sense of it; at some point — pun intended — we may fall back on the abbreviated ‘point such-and-such’ convention). This means we might decompose the former factor as 3.7 = 3 + 7/10, or even re-write it again as 37/10 (as suggested in my previous response). In this way, the full arsenal of whole number tools is available: The problem reduces to adding fractions, which reduces again to whole number operations.

Alternatively, we may use the partial products algorithm as previously described with the 3 & 0.7 and 9 & 0.8. This can be done using the diagram from my earlier post, or written out “horizontally” using the distributive law. Again, I think it is easiest to convert 0.7=7/10 and 0.8=8/10 (the latter could be reduced to 4/5, but one finds it not necessary here). Now the problem reduces to adding and multiplying fractions; again, this has already been covered.

I suppose in this case we’d have as partial products:

3×9 = 27; 3×8/10 = 2.4; (7/10)9 = 6.3; (7/10)(8/10) = 0.56.

Now: 27 + 2.4 + 6.3 + 0.56 =

27 + 2 + 6 + 0.4 + 0.3 + 0.56 =

= 35 + 1.26 = 36.26,

where we have also made use of commutative and associative laws of addition in our decomposing and recomposing of the numbers.

Finally: I agree with your last sentence, and I think it is very helpful to present the partial products algorithm alongside the standard algorithm, and to see how they connect with one another. (Hopefully something here was of use!)

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