Wednesday, September 10, 2014

Is the whole world "failing" with science education?

I decided to get some "perspective' on US educational reform of science education by looking into German science education reform. And guess what the Germans say about science education in Germany...

Sputnik Moment 1957
Historic low teacher efficacy
Waning student motivation
System is failing

In other words, same shit, different language?

Considering what we know about growth mindsets now-a-days, I am beginning to suspect that education researchers have GOT to change their narrative on all this. Politicians are not saavy enough to engage teachers in change. All they can do is blame and try to legislate excellent teaching... which is absurd!

But these parallel conversations in a country that outperforms the USA makes me wonder about the bias of the tools used for comparison: namely TIMMS, PISA, and even the OECD nations international research apparatus.

We have a system and tools that appear to be very good at focusing people on negatives through peer comparisons. How about instead helping countries identify their strengths and their cultural funds of knowledge, and helping them use that to grow?

Politicians can't do this; they haven't the perspective or the expertise. The narrative shift has to come from the only international educational commonality: education research.

Friday, March 1, 2013

Gr 06 Deriving a Formula for the Area of a Circle

Yesterday we focused on the names of parts of the area of a circle. Today we are focusing on one of those areas: the quadrant.





  1. Ask students to title and label their math notebook page. Then they construct a 4-cm radius circle around point A.
  2. Students then construct a quadrant of their circle using two radii that are 90-deg apart.
  3. Verify with students that each radius is 4-cm long, then ask them to build a square out of that radius. They simply add lines perpendicular to the ones they have. But then verify with them the length of those new sides.
  4. Define this shape with them as a radius-square because it is built on the radius of a circles.
  5. Ask students, "how would I find the area of this square?" They'll be quick to offer up that it's 16 sq units. But ask how they did it and they'll say 4x4. Write that down. Then ask them what parts of the circle these two 4's are. The kids will say they are the radius.
  6. Then replace those numbers with r's to represent radius and explain to them (I know, this is the short version) that in math we can write A=rxr as A=r².

  7. Now comes the fun part. Ask them how many radius-squares they think they can fit inside a circle! I prepared a circle and some radius squares for them to cut out. They need to use up a radius square completely before moving on to the next one. I tell them to use the blue one only if the have to.
    f
    f
  8. When they are done, the should have an answer: that they can easily fit three of them inside a circle plus a little bit more. And that will make some students say, "PI!" And they are right!

  9. So if you wanted to, you could have them use up some of that blue square to fill in any spaces, but it's really hard to do. It is often good enough to stop here and formalize then the relationship.
  10. And that relationship is that it takes about three radius-squares to fill a circle. More precisely, it takes pi radius-squares to do it. Or
    A= π x r².

Gr 06 Introducing Parts of the Area of a Circle

Today was a very busy day. We are transitioning from talking about the circumference of a circle to talking about the inside of a circle, so I wanted to introduce some vocabulary that we might use to help us navigate what we're doing.

I used a foldable to introduce the vocabulary for Sector, Segment, Semicircle, and Quadrant.






  1. Fold an 8.5" x 11" piece of paper once length-wise and twice width-wise. Cut the length wise fold from the outside of the paper to the first fold perpendicular to it. You will have four flaps that you can title like this:

  2. Under the segment flap, have the kids construct a circle around point A with a radius of 3 cm. Have them construct and label two points on the circle and make a chord. Then they dark-color the perimeter of that piece of the circle, shade it in lightly in the same color and use their geometry vocabulary to describe what they see. Derive with them the definition of a segment:

  3. Under the semicircle flap, have them construct a similar circle around another point, but this time add points on the fold that passes through the center point and construct a chord at the diameter of the circle. Then they dark-color the perimeter of half of the circle, shade it in lightly in the same color and use their geometry vocabulary to describe what they see. Derive with them the definition of a semicircle:

  4. Under the sector flap, have the kids construct a similar circle around another point. Have them construct and label two points near each other on the circle and draw line segments between each point and the center. Then they dark-color the perimeter of that piece of the circle, shade it in lightly in the same color and use their geometry vocabulary to describe what they see. Derive with them the definition of a sector:

  5. Under the quadrant flap, have the kids construct a similar circle around another point. Have them construct and label one point on the fold going through the center point and another point that is at the top of the circle 90-deg. Ask them to draw line segments between each point and the center point. They dark-color the perimeter of that piece of the circle, shade it in lightly in the same color and use their geometry vocabulary to describe what they see. Derive with them the definition of a quadrant:

For homework, I merely asked them to take it home and to share with their families what they made today. In class, our goal is to try and use the vocabulary to talk about things we are seeing... so this is kind of like a take-home word wall.


Sunday, February 24, 2013

Preparing for the area of a circle, two conceptual area models...

So, I'm getting ready to tackle the area of a circle next week with my students. So far, there are two conceptual area models I have come across:

How Many Radius-Squares Can Fit inside a Circle?
Okay, so this is the model that is used by the singapore math folks. I used it last year and it was pretty successful.
  1. Make a circle of radius r centimeters.
  2. Make a square of side-length r centimeters.
  3. How many of these radius-squares can you fit inside the circle? Let's experiment to find out.
  4. Use scissors to cut up area-squares to fit them inside the circle. Three radius-squares will completely fill-up the first three quadrants of the circle. But all those left-over bits of the radius-squares can be cut-up and will fit into the last quadrant and there will be leftover space. So approximately three radius-squares plus a little more will fill the circle. That little bit more is about 0.14 radius-squares.
  5. The area is about 3.14 radius-squares or...
  6. A= π×r×r or A= π×r².
A Sliced-Up Circle Approximates a Rectangle
And this model is one used by Waldorf and most other math curricula.
  1. Create a circle of radius rCut the circle into three segments and glue them together, so that they kind of form a parallelogram.
  2. Repeat, but cut the circle into 4 pieces.
  3. Repeat, but cut the circle into 8 pieces.
  4. Repeat, but cut the circle into 16 pieces.
  5. Now, at this point, students can see that the shape is REALLY close to a parallelogram. If we were to continue repeating this proces, the parallelogram would get steeper and would become a rectangle. 
  6. The height of that rectangle will eventually be the radius, r. The base of that rectangle will be the circumference divided by two (or πr).
  7. So the area of the rectangle will be base x height or...
    A= πr•r or A= πr
    ².
I think that for 6th graders, the more direct way of doing it is the Singapore math method and I'll probably use it. But the other one might be a great extension for my higher-skills kids.

Thursday, February 21, 2013

Gr 06 Introduction to Pi foldable

The day before this activity, my students measured circular objects and their diameters to try and answer the question "how many diameters will fit along the outside of a circular object." We were looking for some kind of relationship between the two things. Kind of like with radius and diameter. We know that r= d÷2 and d= 2×r. So it seems like there should be some kind of relationship between diameter and circumference, too!

What we discovered was that circumference was about 3 times larger than the diameter. Or the diameter was about circumference divided by 3.

So today we examined that relationship a little more by making these foldables. We did the inside first. A copy of the cut-outs is available here: [link]
  1. Students had to color and then cut out the circles and the strip. The circles' circumferences and the strip had to be the same color; and the diameters had to all be the same color, but one different from the circumference.
  2. Students then drew a "starting line" on the left of their page. They glued the strip down touching the line. I explained that this line was the same length as a circle's circumference "unrolled" into a straight line.
  3. Then we glued down the circles to see how many diameters could fit along that circumference. We drew an ending line where the diameters stopped and could tell the circumference was longer. We added a second line to show where the circumference stopped and looked at that distance in-between them. I asked students to estimate about how much of a diameter that extra bit was.
  4. I told students that mathematicians have been looking at the extra little bit for thousands of years because it turns out, it's not such a simple number to get. I shared this YouTube video about the history of pi [link]. I prefer to use ViewPure to show less clutter on the screen (it strips out all the adds and the sometimes questionable video links that follow a YouTube video).
  5. Then we go back and add the information on the above the circles, that pi is a symbole used to represent the circumference divided by diameter (the idea of how many diameters fit on a circumference); and that two estimates of pi (mentioned in the video) are 3.14 and 22/7. Any time we use those numbers (3.14 or 22/7), we are estimating the number of diameters on a cirucmference... just like we did when we estimated that there were only three. But 3.14 and 22/7 are better estimates -- they are not precise measurements.
  6. We next add that + ~ 0.14 to the diagram and we go to the cover next to learn a few ways that people write the pi symbol (I have learned that students need to be taught some method of writing pi, or they end up with REALLY sloppy approximations). So they can choose any of the representations on the cover to write (and maybe there are other variations, but these are pretty common).

Wednesday, February 20, 2013

Gr 06 Parts of a circle Foldable

This was heavily adapted from some Waldorf materials I found online. I don't have the time to let the kids explore "circle" like Waldorf does. I'm sure they'd benefit from it. Nonetheless, my students appeared to like the process of creating and learning parts of circles this way.

Here's how I used it:
  1. First we folded the paper (fold a large + sign first, then fold the corners in). This is a picture of the finished product, but at this point, there shouldn't be any writing on it.
  2. Draw a center point on the fold underneath each flap. Each point should be a different color and should get a different label (A, B, C, D are fine, but kids could use whatever letters or colors they wanted).
  3. Point A: Ask kids to use a ruler to mark all the points they can find that are exactly 2 cm from point A. When they think they know what shape it's making, they can use a colored pencil to sketch it in. Ask them what it is that defines this shape (all the points that are the same distance from the center point). Then record the definition with them. I spiraled it in to show it was revolving around the center, but that's optional and it was kind of hard for the kids. Close the flap, place the title on it (Definition of a circle), and move on to the flap for point B.
  4. Point B: Write the title "Radius and Diameter" and ask students to construct another r= 2 cm circle around point B just like they did for point A before. Draw a line segment from the center to a point on the circle and ask them to describe what you have drawn. Ask them it if stays the same no matter what point on the circle you draw a line segment to. Formalize this distance with them as the definition of the radius. Then do a similar process with the diameter.
  5. Point C: Write the title "Circumference (C) and Arc" and ask students to construct another r= 2 cm circle around point C. Draw a point on the circle and a series of arrows that goes all around the circle and ends at the same point again. Ask students to describe what you have drawn using their geometry vocabulary, then formalize the definition of circumference with them. Now draw two new points on the circle and shade in the distance on the circle between them. Ask students to describe what you've drawn using their geometry vocabulary, then formalize the definition of Arc with them.
  6. Point D: Write the title "Chord and Tangent" and ask students to construct another r= 2 cm circle around point D. Draw two points on the circle and connect them with a straight line segment. Ask students to describe what you've drawn with their geometry vocabulary, then formalize the definition of chord with them. Do the same for tangent. Now what's cool is that students may recognize that the diameter of a circle is a special case of a chord. And if a tangent dips into the circle, it will no longer be a tangent because it will connect two points on the circle.

Tuesday, February 19, 2013

Gr 06 Introduction to Geometry Vocabulary Foldable

Just starting in on exploring geometry with my students this year and trying out foldables to help them organize and illustrate their vocabulary and concepts. These are the teacher versions made in my own notebook; my students made theirs in their own notebooks. It's really just note-taking, but they find it more entertaining to use colors and foldables and it makes a great reference.

This foldable was created on a single 8.5" x 11" piece of paper, folded in the middle. It was taped into the notebooks no the left side only so the tape acted as a hinge and all 4 pages can be accessed.

PAGE 1

PAGES 2 and 3

PAGE 4