<p>Yes, addition becomes procedural knowledge. Below is an extract from the book "Smart Thinking by Art Markman", that clearly explains the process :</p>

<blockquote>
  <p>When you are learning to do addition, the two procedures you have for
  adding compete with each other. One of those procedures requires some
  effort. You start with the bigger number and then count up. Adding two
  and four means starting with four and then counting to five and six.
  The other procedure is effortless. You try to remember the answer. If
  you finish counting before you pull up an answer from memory, then the
  counting procedure wins. If you are confident you have pulled up the
  right answer from memory before you finish counting, then the habit
  wins. After you solve the problem (by either method), you store
  another memory that 2 + 4 = 6. So, each attempt at an addition problem
  provides memories that will make it faster for you to remember the
  correct answer in the future. </p>
  
  <p>The difficulty with math is there are lots of similar facts. You are
  learning 2 + 4 = 6, but at the same time, you are also encountering
  problems like 2 + 7 = 9 and 2 + 5 = 7. Sometimes, when you see 2 + 4,
  you will also recall some of those similar problems. When you retrieve
  these conflicting answers, you are going to be uncertain about which
  answer is correct. So you will finish carrying out your counting
  procedure before you have an answer from memory. Once you have a lot
  of examples of addition problems in your memory, most of what you pull
  out of memory when you see 2 + 4 will be other situations in which you
  also saw 2 + 4. At that point, you retrieve information from memory
  faster than you can count, and so you have a habit.</p>
  
  <p>Chapter references:</p>
  
  <p>Logan, G. D. (1988). Toward an Instance Theory of Automaticity.
  Psychological Review 95: 492–527. </p>
  
  <p>Schneider, W., and Shiffrin, R. M. (1977). Controlled and Automatic
  Human Information Processing: 1. Detection, Search, and Attention.
  Psychological Review 84 (1): 1–66. </p>
  
  <p>Shiffrin, R. M., and Schneider, W. (1977). Controlled and Automatic
  Human Information Processing: 2. Perceptual Learning, Automatic
  Attending, and a General Theory. Psychological Review 84: 127–190.</p>
</blockquote>

<p>The two competing systems mentioned above, are actually <a href="https://en.wikipedia.org/wiki/Dual_process_theory#Systems" rel="nofollow">system 1 and system 2</a> that are explained in detail by Daniel Kahneman, in his book Thinking, fast and slow.</p>

CC BY-SA 3.0
2015-11-04T17:20:26.930Z
Great reference! Where does Markman get the evidence for this. Is there a study in particular that he cites?
Seanny123
CC BY-SA 3.0
2015-11-04T17:35:15.237Z
@Seanny123 there is no direct reference, but I have added the relevant chapter references.
DesignerAnalyst
CC BY-SA 3.0
2015-11-04T17:18:31.650Z
2015-11-04T17:34:50.950Z
2015-11-04T17:34:50.950Z
DesignerAnalyst
<p>The work of Siegler in "The Perils of Averaging Data Over Strategies: An Example from Children's Addition", examines 5 different strategies associated with addition:</p>

<ul>
<li><strong>Retrieval</strong> Where the answer was retrieved from memory </li>
<li><strong>Min</strong> Where the smallest addend was used for counting from the starting point of the largest addend</li>
<li><strong>Decomposition</strong> Where the complicated problem was reduced into simpler known problems</li>
<li><strong>Count All</strong> Where both the addends were counted</li>
<li><strong>Guess</strong> Where the answer was guessed</li>
</ul>

<p>In the table below, which I've copied from the paper, you can see the progression of strategies from grade to grade.</p>

<p><a href="https://i.sstatic.net/aoF6f.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/aoF6f.png" alt="enter image description here"></a></p>

<p>There does seem to be a progression towards retrieval, decomposition and min-count, away from count-all and guessing. However, it is not a completely straight-forward progression.</p>

CC BY-SA 3.0
2015-11-12T02:04:31.030Z
2015-11-12T02:04:31.030Z
Seanny123
<p>How do children progress from counting to addition?</p>

<p>I understand that an grasp of counting is achieved before an understanding of cardinality (size of a set of objects). Thus, there does seem to be some division between procedural understanding and conceptual understanding of counting. However, how does this carry on to addition? Is addition also typically learned procedurally rather than conceptually?</p>

<cognitive-psychology><reference-request><developmental-psychology>
Children's progression from counting to addition
Seanny123
# Children's progression from counting to addition

(Asked by: username_0 on 2015-11-04)

How do children progress from counting to addition?

I understand that an grasp of counting is achieved before an understanding of cardinality (size of a set of objects). Thus, there does seem to be some division between procedural understanding and conceptual understanding of counting. However, how does this carry on to addition? Is addition also typically learned procedurally rather than conceptually?

**Tags:** cognitive-psychology, reference-request, developmental-psychology


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## 2 answers

### Answer by username_0 on 2015-11-12. Score: 2 | Accepted answer

The work of Siegler in "The Perils of Averaging Data Over Strategies: An Example from Children's Addition", examines 5 different strategies associated with addition:

* **Retrieval** Where the answer was retrieved from memory
* **Min** Where the smallest addend was used for counting from the starting point of the largest addend
* **Decomposition** Where the complicated problem was reduced into simpler known problems
* **Count All** Where both the addends were counted
* **Guess** Where the answer was guessed

In the table below, which I've copied from the paper, you can see the progression of strategies from grade to grade.

[![enter image description here](https://i.sstatic.net/aoF6f.png)](https://i.sstatic.net/aoF6f.png)

There does seem to be a progression towards retrieval, decomposition and min-count, away from count-all and guessing. However, it is not a completely straight-forward progression.


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### Answer by username_1 on 2015-11-04. Score: 2

Yes, addition becomes procedural knowledge. Below is an extract from the book "Smart Thinking by Art Markman", that clearly explains the process :

> When you are learning to do addition, the two procedures you have for
> adding compete with each other. One of those procedures requires some
> effort. You start with the bigger number and then count up. Adding two
> and four means starting with four and then counting to five and six.
> The other procedure is effortless. You try to remember the answer. If
> you finish counting before you pull up an answer from memory, then the
> counting procedure wins. If you are confident you have pulled up the
> right answer from memory before you finish counting, then the habit
> wins. After you solve the problem (by either method), you store
> another memory that 2 + 4 = 6. So, each attempt at an addition problem
> provides memories that will make it faster for you to remember the
> correct answer in the future.
>
> The difficulty with math is there are lots of similar facts. You are
> learning 2 + 4 = 6, but at the same time, you are also encountering
> problems like 2 + 7 = 9 and 2 + 5 = 7. Sometimes, when you see 2 + 4,
> you will also recall some of those similar problems. When you retrieve
> these conflicting answers, you are going to be uncertain about which
> answer is correct. So you will finish carrying out your counting
> procedure before you have an answer from memory. Once you have a lot
> of examples of addition problems in your memory, most of what you pull
> out of memory when you see 2 + 4 will be other situations in which you
> also saw 2 + 4. At that point, you retrieve information from memory
> faster than you can count, and so you have a habit.
>
> Chapter references:
>
> Logan, G. D. (1988). Toward an Instance Theory of Automaticity.
> Psychological Review 95: 492–527.
>
> Schneider, W., and Shiffrin, R. M. (1977). Controlled and Automatic
> Human Information Processing: 1. Detection, Search, and Attention.
> Psychological Review 84 (1): 1–66.
>
> Shiffrin, R. M., and Schneider, W. (1977). Controlled and Automatic
> Human Information Processing: 2. Perceptual Learning, Automatic
> Attending, and a General Theory. Psychological Review 84: 127–190.

The two competing systems mentioned above, are actually [system 1 and system 2](https://en.wikipedia.org/wiki/Dual_process_theory#Systems) that are explained in detail by Daniel Kahneman, in his book Thinking, fast and slow.

> **Comments:**
> **username_0**: Great reference! Where does Markman get the evidence for this. Is there a study in particular that he cites?  
> **username_1**: @username_0 there is no direct reference, but I have added the relevant chapter references.  

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