Maybe I’ve seen one too many deconstructed Caesar salad or peanut butter and jam sandwich on TV. Or maybe I’ve heard “This workbook covers the curriculum” one too many times¹.
Whatever my reason, I wanted to take a closer look at a learning outcome from the WNCP Math 8 curriculum document:It is expected that students will demonstrate an understanding of multiplying and dividing positive fractions and mixed numbers, concretely, pictorially, and symbolically [C, CN, ME, PS]
“It is expected that students will”
It’s about students’ learning. Worked examples on the whiteboard or in a textbook may be evidence of the teacher’s or publisher’s learning.
“demonstrate an understanding of”
Not will be able to. Students need to make sense of mathematics. Justifications and explanations are required for answers and methods.
“multiplying and dividing positive fractions and mixed numbers”
This is a topic. Curriculum is more than a collection of these.
“concretely, pictorially, and symbolically”
No longer just suggested, the use of concrete materials (i.e., manipulatives) is prescribed² as is having students draw to represent their thinking (diagrams not decorations).
[C, CN, ME, PS]
From K to 12, seven processes are to be integrated within the learning of mathematics. The ‘C’, for example, means that students should be provided with opportunities to communicate their learning– to write about and discuss mathematical ideas.
¹ To my US reader(s)– in my province, curriculum is different than recommended learning resource (i.e., the textbook). In theory, the textbook is not the course. In practice…
² For many teachers, this is probably the biggest change to the curriculum. Earlier this year, I created the posters below. My intent was to generate conversations among teachers, not to teach the concept. Plus, I got to be artsy-fartsy. Enjoy.
CPS Poster Algebra Tiles
CPS Poster Counters
CPS Poster Pattern Blocks
CPS Poster Toothpicks
Chris Hunter 
@Chris Hunter – Comments were closed on Dan’s blog, so I’ll continue here (my final remarks, promise)
Regarding Gates, you still haven’t commented on his having funded the Shell Centre and Marilyn Burns, which I hope you’ll agree are progressive in math ed.
Regarding JUMP math,that’s a common misconception. The Introductory Fractions Unit that you linked to is one small component of the curriculum. Yes, incremental practice and development are part of the curriculum, but look closer and you’ll find that it’s actually quite conceptual (teachers can register for free and download all of the Teachers Resources on the site). You say that it’s a extreme form of current math pedagogy, but look at the “research” page on the JUMP site and you’ll see that is uses guided discovery (again, I really recommend a closer look). This is a VERY common (and incorrect) first impression of JUMP math See (http://jumpmath1.org/supporting_research). Given that it passed an independent and large-scale randomized controlled field trial and was well received by experts in education research, this is nothing to dismiss so quickly.
The most salient example is the Sample Problem Solving Lesson. See…
Click to access Sample%20Problem%20Solving%20Lesson.pdf
If you have time to spare, watch http://www.youtube.com/watch?v=hmpVOUrLqq8
This is a direct quote from the teacher’s guide 3:
If your students are confident and engaged, try skipping steps when teaching new material, and challenge your students to figure out the steps themselves. But if students struggle, go back to teaching in small steps.
This is how the Grade 8 teacher’s guide suggest developing the Exterior Angle Theorem:
Exterior angles. Draw a triangle on the board, extend one of the arms
beyond the vertex and mark the exterior angle as shown in the margin. Ask
students if anyone knows what “exterior” means (outer, on the outside).
Explain that this angle is called an exterior angle of the triangle because it is
outside the triangle. Then mark the measures of ∠a and ∠b in this triangle
(e.g., ∠a = 50°, ∠b = 57°) and ask students to find the measure of ∠c. (73°)
Ask them to explain how they got the answer. ASK: What do you know
about ∠c and ∠x? (They are supplementary angles; they add to 180°) Have
students find the measure of ∠x. (107°) Start a table with headings ∠a, ∠b,
∠x and fill in the first row. Repeat with several other triangles labelled the
same way. Then ask students to look for a pattern and have them formulate
a conjecture about the sizes of the angles.
Ask students to pair up and to improve the conjecture they have written,
using the words “exterior” and “opposite.” (Explain that we are not using or
referring to opposite angles here. Instead, we are using opposite to mean
“across from,” as in “the door is opposite the window,” or “the opposite
sides of a rectangle are equal.”) Students can improve their conjecture
again in groups of four. Have all groups share their conjectures.
Finally, add a row to the table with variables a and b for ∠a and ∠b, and
have students find the measure of angle c (180° − a − b). Then ask them to
write the equation showing that ∠c and ∠x are supplementary angles, and
to find the measure of the angle c this way. (c + x = 180°, so x = 180° − c
= 180° − (180° − a − b) = 180° − 180° + a + b = a + b.)
Point out that students have now proved their conjecture using logic, so they can call this conjecture the Exterior Angle Theorem.
Doug, we’re in agreement with respect to the Shell Center and Marilyn Burns.
I decided to take a closer look at the JUMP Math lesson on multiplying and dividing integers since this was the subject of the Khan video. I don’t have the most recent version of 8:2. Mine is copyright 2006.
Here’s the gist of the JUMP Math explanation:
– you can think of multiplication as repeated addition (positive times negative)
– multiplication commutes (negative times positive)
– the product of two negative numbers is a positive because of the distributive law
– rewrite multiplication statements as division statements
I’d like to see more than “Mathematicians defined negative multiplications so that it would also commute” and “Mathematicians defined negative multiplication to satisfy the distributive law” to have students make sense of –ve times +ve and –ve times –ve. I’d like to see the rules for dividing integers as more than the result of the rules for multiplying integers.
To bring this back to the topic of my post, the prescribed learning outcome in my curriculum is “It is expected that students will demonstrate an understanding of multiplication and division of integers, concretely, pictorially, and symbolically [C, CN, PS, R, V]” What I have in front of me doesn’t cut it. To be fair, maybe this lesson has been improved in the current version.
My concerns with the JUMP Math lesson aside, it’s still better than the Khan video. At least John Mighton had the good sense to not start with a negative times a negative and pick two different numbers.
Chris, thank you for your timely reply.
Regarding the JUMP lesson in question, I fully agree with you in that it could be better taught. And I really support the multiple representations you advocate. In my experience (decades teaching various math and science courses) no product, curriculum, service, or teacher seems to understand ALL of the conceptual and pedagogical techniques fully (on either side of the “math wars”). This is where the collaborative aspect comes into play. With social media and other communication methods, we can all collaborate of, culling the best of the best. On the whole, though, I think you’ll find the JUMP is a pretty conceptually deep curriculum and has a lot to offer (no one expects a curriculum to be followed exactly – customization is key). Of course, we’re in agreement as well that the Khan video satirized was poorly done.
I guess my purpose with all this rambling is that, regarding Gates, Khan, JUMP and anyone else, really black-and-white categorization may be a little counter-productive. Some other Khan videos (that I like) actually taught ME something years of teaching hadn’t.
See
http://www.khanacademy.org/math/algebra/exponents-radicals/v/negative-exponent-intuition
http://www.khanacademy.org/math/algebra/logarithms/v/logarithmic-scale
If we use resources as tools, picking the best, and productively critiquing the poor, a lot of progress can be made.
My final remarks on Khan… I can’t say that I learned anything from Khan’s video on negative exponents. It’s a pretty standard way of introducing the topic. I do something similar to Khan with my students to introduce negative exponents. Here’s where Khan and I differ: in the video, Khan shows you why a^-b is equal to 1/a^b whereas, in my class, I’d have students look for a pattern and develop the rule. (Also, I’d use numbers first and generalize to ‘a’ later.)
The difference is pedagogical; it’s not in the explanation, but in who is giving the explanation. This is my greatest criticism of the KA as the future of math education. If you ask yourself “Who’s doing the math?” and “Which approach is better?” I think that there are clear answers. It is black-and-white.
Of course, I often hold a view that is more gray than black-and-white. Shades of gray might make for popular novels but don’t necessarily make for good blog posts.
Now that I’ve admitted Gates’ funding of the Shell Center is a good thing and my MS Word is back up and working, I’ll risk asking this: that bracelet thing is ridiculous, right?
Agreed. Khan’s recent video “Why Negative Times Negative is Positive”, though seems to say that he’s willing to learn from his criticizers. But he has a long way to go. FWIW, I wouldn’t use KA or most ed tech until at least a decade or so into the future.
I don’t necessarily want to promote one side or the other, but maybe some perspective might be informative here – from what I can tell, Khan’s goal is in getting students to use math in often a practical way. See http://www.khanacademy.org/about/blog/post/6844033473/bringing-creativity-to-class-time-by-sal-khan
The whole idea, about blogging being binary, is sadly, what’s wrong with social media and ed tech today. It’s just SO binary. I’d prefer to read/watch things where people qualify their opinions/research more than is done currently.
Regarding the bracelets, yeah ridiculous if being used to solely evaluate teachers, though that may never have been the intention. The idea was that combined with videotapes where teachers were at their pedagogical peak, the bracelets could provide another metric to ensure some engagement was present. I’m assuming they’d add/average out the waves to get some sort of mean engagement. Even so, I would have recommended Gates spend that much money elsewhere. Again, there’s a lot I disagree with, but that doesn’t mean the 101qs community or Mathalicious couldn’t seek funding in the next round of grants (there’s some precedence with supporting good ed practices).
*THAT’s what a meant by gray blogging – I hope it catches on.*