Knowledge Analysis Part I



In this entry we’ll focus on how to find: (1) exactly what new knowledge students must learn---so you can design instruction that will clearly communicate the new information, and (2) what pre-skills (background knowledge) students need, so that they will understand what you are talking about and will learn the new information.  Here’s a guideline for effective instruction.


Don’t Assume Your Students Know Anything.


Of course your students know plenty.  But it's safest to assume that you must be ready to teach every little thing.  Let’s say you are taking a trip to the jungles of South America.  You go to a physician to get immunization shots.  Which assumption do you want your physician to make?

1. “I’m sure you’re immune.  You don't need any shots.”  [What if you aren’t immune?]

2. “You may be immune to some diseases, but I can’t be sure.  To be safe, I’ll give you a battery of shots.  None will cause any harm.”  [The worst that happens is you get a shot you don’t need.]

Of course, you’d rather the doctor made an error on the safe side.  


The same goes with teaching.  For example,

1. You're about to teach long division.  Don’t assume your students are firm on all of the PRE-SKILLS (background knowledge) for long division: subtraction, estimation (27 goes into 99 how many times?), multiplication.  

2. You're about to teach students to read connected text (sentences).  Don’t assume they are firm on all of the PRE-SKILLS involved in reading connected text; e.g., letter-sound correspondence; how to sound out words.  

3. You're about to teach students to analyze poems.  Don’t assume they are firm on all of the PRE-SKILLS of poetry analysis, such as figures of speech, rhyme schemes, biographical influences, cultural context of the poem.  

4. You're about to teach students how to comprehend historical political documents.  Don’t assume they know the political philosophies of the writers of those documents, or the meanings of the words in the documents. 

So, when you are preparing to teach something new, identify the PRE-SKILLS---the elementary facts, concepts, rule-relationships, and steps in cognitive routines---that are part of the new thing you want to teach.  That way, you can (1) assess your students on these pre-skills; (2) review and firm up these pre-skills; or even (3) reteach these pre-skills if students’ knowledge is weak.

Here’s another guideline for effective teaching.


Prepare to Prevent or Correct Misunderstandings and Errors.


Imagine you are going in for surgery.  Which do you want?

1. A surgeon who knows every tiny step in the operation; who knows just about everything that can go wrong; and who is prepared to prevent or to correct things that go wrong.

2. A surgeon who doesn’t know every tiny step in the operation; who therefore cannot know just about everything that can go wrong; and who therefore is not prepared to prevent or to correct things that go wrong.           

                    “Whooops! This is NOT good.”


I bet you chose number 1.


This applies to teaching.  Imagine that students have learned how to pronounce sounds (mmmmm) and they know which sounds go with which letters.  So, now you are going to BUILD on those PRE-SKILLS, and teach students to USE those pre-skills and the NEW skills you will teach them, to sound out words.  What kinds of misunderstandings might students have, and what sorts of errors might students make, when you teach them the NEW skills in the cognitive routine for sounding out words?  Here’s the procedure the teacher uses.



Boys and girls.  I’ll show you how to sound out words.






When I touch under a letter I say the sound.
[Teacher moves her finger to the right, under the word.] 

Okay, your turn.  Sound it out!



What do you think the class will do now?

Remember that almost everything in the above was NEW to the children.  The only knowledge they already had (essential pre-skills) was pronunciation and letter-sound correspondence.  Notice what the teacher did NOT teach.

1. Always start with the letter on the left.  [She never EXPLICITLY stated that rule; for example, “First I put my finger on the ball.”]

2. Keep on saying a sound before you say the next one.  Don’t stop in between sounds.  [She did not EXPLICITLY teach that rule. “I do not stop between sounds.”]

3. First say the sounds slowly.  Then say them fast.  [She DID that, but she did not EXPLICITLY state THAT she was doing it, and she did not state a rule.  “First, I’ll say it slowly.  Then I’ll say it fast.” And she did not pause between saying it slowly (sssaaammm)  and then saying it fast (sam) so her students could hear the difference.

Let’s finish the instruction…


Boys and girls.  I’ll show you how to sound out words.






When I touch under a letter I say the sound.
[Teacher moves her finger to the right, under the word.]

Okay, your turn.  Sound it out!










The students used their pre-skills (they said the correct sounds), but they did not learn the new skills (the steps in the sounding out routine) because the teacher did not communicate the information clearly.

Okay, so there are two kinds of skills to think about when you design instruction:  

1. Pre-skills (background knowledge, knowledge elements) that students need in order to understand what you are doing (saying, showing) and to learn the general idea (e.g., concept, cognitive routine) from the models and examples you are presenting.

2. New skills that you are supposed to teach---during the phase of acquisition.

Here’s another example of instruction that does not consider the pre-skills and does not effectively teach new knowledge, either.


What Happens When Students’ Pre-Skills Are Weak and When 
The Teacher Does Not Communicate the New Information Clearly

Remember: Students need certain pre-skills (such as concepts, rules, and knowledge of steps in earlier-learned cognitive routines) in order to understand what the teacher is talking about, and in order to learn the new material.  Also, the teacher must communicate the NEW information clearly.  Let’s see what happens when students’ pre-skills are weak and when the teacher does not communicate the NEW information clearly.

Just so we’re looking at the same thing, here’s the solution to an equation with one parenthesis and one unknown. [Students have already worked on this KIND of problem.]

                 2(X + 4) = 26

                   2X + 8 = 26

             2X + 8 – 8 = 26 – 8

                        2X = 18

                    2X/2 = 18/2

                        X = 9

And here’s an example of the NEW kind of problem that the teacher will start teaching.  It has two parentheses and one unknown.  A few NEW steps are added to the earlier routine used with one parenthesis and one unknown.  So, students need to (1) USE their pre-skills for solving an equation with one parenthesis and one unknown; AND (2) learn NEW skills (such as simplifying the TWO parentheses one at a time, and then simplifying the WHOLE expression). Like this…

       2(X + 4) + 4(X – 6) = 80      [Two parentheses]

2X + 8 + 4(X – 6) = 80      [simplify one parenthesis]

  2X + 8 + 4X -24 = 80      [simplify the second  parenthesis—a new step]

               6X – 16 = 80    [simplify the whole expression—a new step]

       6X – 16 + 16 = 80 + 16  [isolate X]

                      6X = 96

                   6X/6 = 96/6  [reduce 6X to X]

                        X = 16

Now let’s see how the teacher teaches this new cognitive routine?  

The solution in the example below is taken from http://www.webmath.com/cgi-bin/gopoly.cgi?lhs=2+%28X+%2B+4%29+%2B+4+%28X+-+6%29&rhs=80&variable=X&back=solve.html 

[Comments in boldface and brackets tell you when students are weak on a pre-skill and when the teacher is not effectively teaching new knowledge.]



Boys and girls.  I’ll show you how to solve equations with TWO parentheses and one unknown.


2(X + 4) + 4(X – 6) = 80


I don’t know how to solve equations with ONE parenthesis! 

[Pre-skill; e.g., X (4 + 3) = 35. Solving equations with one parenthesis and one unknown is an ELEMENT---and therefore a PRE-SKILL---in solving equations with TWO parentheses and one unknown.  The teacher should have assessed knowledge of this and firmed it up as needed.]


What’s an unknown?

[Pre-skill.  Students need to know what “unknown” means (variable to be determined: a variable in an equation whose values are solutions of the equation), and students need to know that X, Y, etc., are examples of unknowns.  This knowledge is needed to understand what the teacher is talking about when teaching the NEW knowledge.  The teacher should have at least reviewed vocabulary words.]


Boys and girls. Here’s an equation with two parentheses and one unknown.

2(X + 4) – 4(X – 6) = 80


I don’t like the looks of this!

[Pre-skill. This should not LOOK so different from equations with ONE parenthesis. The teacher should have prepared students by showing earlier equations (with one parenthesis) NEXT TO the new type of equation, and pointed out the similarities (one unknown, parentheses, two sides) and the small difference (a second parenthesis).]


First, let's work on the left hand side of your equation: 2 (X + 4) + 4 (X - 6)


Why do we work on the left side first?

[Pre-skill. This knowledge is needed to learn the routine for solving this new kind of equation. It should have been learned earlier, and it should have been assessed, reviewed, and firmed up before the teacher began the new instruction.]


We multiply 2 by each term in X + 4, term by term. This is the distributive property of multiplication.


The what?

[Pre-skill. Since the teacher is USING this word, students should have learned it already. The teacher should have reviewed and firmed it up before starting the instruction.

New skill. There are two parenthesis  2 (X + 4) and 4 (X -6). The teacher began by simplifying 2 (X + 4). To avoid confusion, she might have said there are TWO parentheses, and that she was going to start with the one on the LEFT.]


2 x X = 2X and 2 x 4 = 8

So  we have 2X + 8.


I’m lost.

[Pre-skill. Students need to know how to multiply terms in a parenthesis by one unknown--- X(a +/- b)]---to solve the new kind of equation. Therefore, the teacher should have assessed students to see that they were firm on the X(a +/- b) routine BEFORE she began the new instruction.]


Now we multiply 4 by X-6.  We multiply 4 by each term in X - 6, term by term. 

4 x X = 4X and 4 x -6 = -24.
So we have 4X - 24.
Now, the equation looks like this...
2X + 8 + 4X – 24 = 80 

This is the result of simplifying the TWO parentheses.  This is new to students.]

Now, we simplify.

That’s 6X – 16 = 80   
[This, too, is new. It follows from simplifying the two parentheses.]


What did she just do?!
[New skill.  This was new knowledge.  But the teacher did not EXPLAIN what she was doing.  She did not tell students that you simplify each parenthesis 2 (X + 4)  and 4 (X – 6)] 
And then simplify the whole thing.
2X + 8 + 4X – 24 = 80 
6X – 16 = 80

She did not model how to do the above; she did not then lead students through it; and she did not then test/check (immediate acquisition test) to see if they learned it. But she is going on, anyway.]

Okay, so now we solve 6X – 16 = 80.
Move the -16 to the right hand side by adding 16 to both sides, like this:

To the left hand side: -16 + 16 = 0

To the right hand side: 80 + 16 = 96 …
[New skill.  The teacher has moved to the next new step before the students were firm on the prior ones.]


Get me outta here.


Hey, Dude, this is lame.


You might say that what we have here is a failure to communicate. The teacher knows her math.  And she’s working hard.  But she is not ensuring that students have the essential pre-skills (knowledge elements), and she is not ensuring that students are learning the new knowledge.  The result is that students are lost.  What if the teacher keeps going, anyway?  Do you think the students will suddenly know what the teacher is talking about?  NO.  Okay, what if she stops every time students don’t get it, and teaches them?  This is not a good idea!  Different students are weak on different PRE-SKILLS.  Also, a few seconds of reteaching is not enough.  Besides, the lesson will take 7 hours! 

The main reason why the teacher is not (1) assessing and reviewing the needed pre-skills, and why she is not(2)  teaching each new skill effectively (for example, modeling a small amount of new information, leading students through the new information, testing/checking to see if students have learned the new information, and immediately correcting any errors), is that

The teacher does not know exactly what the pre-skills
and the new knowledge are.

Let’s sum up.  If you know the pre-skills (knowledge elements) and the new knowledge needed to achieve an instructional objective, then you can:

1. Make sure that you TEACH the pre-skills (knowledge elements) that students will need later.

2. ASSESS your students’ pre-skills and provide firming-up practice or even reteaching if pre-skills are weak, before you teach something new that requires these pre-skills

3. ANTICIPATE ERRORS and therefore provide pre-corrections (e.g., reminders) and error corrections immediately.

4. DETERMINE whether errors (a) are just a simple mistake, or (b) mean that firming-up (practice right then) is needed, or (c) mean that some reteaching is needed (and therefore maybe you need to revise they WAY you teach), or whether (d) some students need more intensive instruction (more assistance, attention to even smaller skills).

And here is a rule.  Do not try to teach or reteach pre-skills at the same time you are teaching something new that REQUIRES those pre-skills. 

Is it a good idea to teach kids how to paddle and kick and breathe, AND (at the same time) teach them to use these skills to swim in the ocean?  Is it a good idea to learn how to open a parachute at the same time you are learning the whole routine for sky diving, while you are falling to earth?  In other words, make sure students are firm on the pre-skills before you teach something that requires those pre-skills.  Otherwise, you will have a disaster on your hands. 

But how do you find out what the needed pre-skills and new knowledge are?  The answer is knowledge analysis, which we’ll examine in Part II.

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