Week 2: Chapters 4 - 5, Question #5 Look at the six questions that Boaler uses to create rich mathematical tasks.

Look at the six questions that Boaler uses to create rich mathematical tasks. Which question appeals to you? Why? Which question is the most challenging for you? Why?

Challenge: Use at least one of the questions to revise or create an activity for your class. Share the activity in the discussion board.

Comments

  One of the most common utterances of my students is, "So I am making this harder than it should be."  Yes, these are students who don't use number sense.   

The overwhelming majority of my students have long abandoned number sense.   Some of them never developed it.   Others have it -- but don't see how it connects to This Math Stuff (and there's a ton of research that supports this -- students believe their natural thinking of numbers is basically unrelated to Math Course Problems).   Others drive me nuts because it's there, they're using it ... and then anxiety kicks in and it flies out the window... 

I wish our lower courses infused number sense.  I think we tend to conflate it with "knowing the facts."   I'm thinking it would make sense to spend time with number sense and use it to build more fluency.   

Just some starting thoughts... I remember Sebastian Thrun for claiming that MOOCs were going to bridge the equity gap in math, and then deciding that no, poor people had too many other problems for that to happen ...  

I am reminded of the student that walked into our learning center for required remediation after having failed Math 20 three times. He was very secure with the idea that all he had to do was memorize and retain long enough to pass a test. I almost bit a hole in my tongue to keep from saying, "I can't help you, then." Well, I couldn't help him.  He used up all his financial aid and had no college course to his credit.

Number sense is often missing from the learners we deal with.  It takes a long time to teach that to students who have struggled with math for years, but I agree with you that it is vitally important. For students to memorize rule after rule without having connections is a waste of time.

 

Figure Squares on bottom (N) Total Squares N squared   Total Missing from Square 1 1 1 1   0 2 2 3 4   1 3 3 6 9   3 4 4 10 16   6 5 5 15 25   10 6 6 21 36   15 7 7 28 49   21 8 8 36 64   28 9 9 45 81   36 10 10 55 100   45 11 11 66 121   55 12 12 78 144   66 13 13 91 169   78 14 14 105 196   91 15 15 120 225   105 16 16 136 256   120 17 17 153 289   136 18 18 171 324   153 19 19 190 361   171 20 20 210 400   190 21 21 231 441   210 22 22 253 484   231 23 23 276 529   253 24 24 300 576   276 25 25 325 625   300 26 26 351 676   325 27 27 378 729   351 28 28 406 784   378 29 29 435 841   406 30 30 465 900   435 31 31 496 961   465 32 32 528 1024   496 33 33 561 1089   528 34 34 595 1156   561 35 35 630 1225   595 36 36 666 1296   630 37 37 703 1369   666 38 38 741 1444   703 39 39 780 1521   741 40 40 820 1600   780 41 41 861 1681   820 42 42 903 1764   861 43 43 946 1849   903 44 44 990 1936   946 45 45 1035 2025   990 46 46 1081 2116   1035 47 47 1128 2209   1081 48 48 1176 2304   1128 49 49 1225 2401   1176 50 50 1275 2500   1225 51 51 1326 2601   1275 52 52 1378 2704   1326 53 53 1431 2809   1378 54 54 1485 2916   1431 55 55 1540 3025   1485 56 56 1596 3136   1540 57 57 1653 3249   1596 58 58 1711 3364   1653 59 59 1770 3481   1711 60 60 1830 3600   1770 61 61 1891 3721   1830 62 62 1953 3844   1891 63 63 2016 3969   1953 64 64 2080 4096   2016 65 65 2145 4225   2080 66 66 2211 4356   2145 67 67 2278 4489   2211 68 68 2346 4624   2278 69 69 2415 4761   2346 70 70 2485 4900   2415 71 71 2556 5041   2485 72 72 2628 5184   2556 73 73 2701 5329   2628 74 74 2775 5476   2701 75 75 2850 5625   2775 76 76 2926 5776   2850 77 77 3003 5929   2926 78 78 3081 6084   3003 79 79 3160 6241   3081 80 80 3240 6400   3160 81 81 3321 6561   3240 82 82 3403 6724   3321 83 83 3486 6889   3403 84 84 3570 7056   3486 85 85 3655 7225   3570 86 86 3741 7396   3655 87 87 3828 7569   3741 88 88 3916 7744   3828 89 89 4005 7921   3916 90 90 4095 8100   4005 91 91 4186 8281   4095 92 92 4278 8464   4186 93 93 4371 8649   4278 94 94 4465 8836   4371 95 95 4560 9025   4465 96 96 4656 9216   4560 97 97 4753 9409   4656 98 98 4851 9604   4753 99 99 4950 9801   4851 100 100 5050 10000   4950

 

Marti,

I like the expression you came up with! You were using a negative space way of looking at the figure, right? I'm imagining the n^2 as the enclosing square formed by squaring the base. (n-1)n/2 is the triangle that is being taken away from that square.

This is the task, right?

 

I wonder what equivalent expressions there could be based on different ways of seeing the growth. For example, I see it growing from the bottom, so the total squares are:

1

1 + 2

1 + 2 + 3

1 + 2 + 3 + 4

...

Yes.  In fact I ordered the numbers the way you did which made me think of trying to subtract 1 from a set of squares and try to average them out.  I was trying all sorts of things to the point where I am not exactly sure how I stumbled on the formula.  I just know that the three sided square added up to a total of 6 and the 4 sided square added up to 10...like so.

4 + 1

3 + 2

2 + 3

1  This being the extra leftover.

 

I personally think all six questions are appealing and challenging all at the same time. While I see and understand the need for conceptual understanding, multiple representations, inquiry based...... lessons, the bottom line in my class is that they must do well on the Accuplacer test which does not support any of these findings as far as what good math teaching looks like. I know it's our reality, but it makes creating this balance a challenge. I have used a lot of Robert Kaplinsky's ideas and lessons. http://staging.robertkaplinsky.com/lessons/ These lessons create a culture of inquiry and also encourages a productive struggle. For instance, the lesson on the 100 x 100 In and Out Burger - I teach this lesson before introducing anything about slope or y-intercept. Once the students have figured out various ways of demonstrating their solutions, we see if it would be possible to graph our findings. This leads naturally to the idea of how come the line can't start at zero? Or why are we increasing by 90cents each time we move up on the graph. The concepts suddenly have meaning not to mention it comes with a lot of excitement from the students. I would highly recommend Robert Kaplinsky's website as he also goes into why it's so important to anticipate students' questions beforehand and even has a handout for students to keep track of their work as they are attempting different strategies to solve a problem.

On another note, I was thrilled to hear that Jo Boaler would write her daughter's teachers a note after they had proved they knew the concept of any math problem. I have done the same thing for my children and it actually motivated my children to teach me their own understanding of a concept so they could move onto something else and not spend hours laboriously doing 30 addition problems.

 

I found Mary Jo's comment about homework interesting, and applaud her (and Jo Boaler) for monitoring their children's conceptual understanding of math concepts and supporting them by sending notes to school that "opt-out" of pages of tedious and redundant math problems.

I admit, though, that I do send home self-study guides with my adult students to support or extend what we do in class, or for students that ask for additional practice that they can do at home. I give them similar advice on moving through those booklets: do a few problems and when you're sure you understand it, move onto the next page. So, this leads me to a few questions:

  • Is it ever appropriate to give these workbooks (prepared for adult education self-study) to my students to take home?
  • What would meaningful tasks (that Jo Boaler refers to on page 46) look like for adult ed students? (Maybe something like Which One Doesn't Belong?)

I find that most students, even those who ask for those booklets, spend little (if any) time working through them. 

 

HI Mary Jo,

I enjoyed reading your post. I too like the work of Robert Kaplinsky and use some of his tasks with my students. I understand your frustration with the Accuplacer test.  I  believe however that if we help students develop number sense and conceptual understanding, they will be able to make better sense of the problems they encounter on the Accuplacer and other standardized tests and thus develop greater confidence in solving them. Teaching strategies that are useful in any problem is key. Sometimes I think we worry too much about teaching to a test and should focus instead on helping students develop mathematical thinking and the mindset that Boaler speaks of.

 

Just my thoughts,

Sarah 

The question that appeals to me the most is asking problem before teaching the method.  I can think of a few of my classes where I can use this concept where it should work extremely well.  A few of them are In geometry and another would be with exponent rules.  When you have two like bases and your multiplying exponents you add the exponents.  This is why we come up with feet squared.  Or when you multiply three numbers it will become the cube of something.  This all stems from the formula for exponents where when you have like bases and your are multiplying you are adding the exponents.  In geometry, I can see where this can be utilized for finding area, perimeter, and volume for sure.  Or showing why a triangle has the formula of 1/2BH.