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 (10)

S Jones's picture

  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 ...  

donnawparrish's picture

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.

Marti Reppetto's picture

 

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 Reppetto's picture

 

N (squared)  - (N-1)N

                            2

ecappleton's picture

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

...

Marti Reppetto's picture

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.

 

Mary Jo Chmielewski's picture

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.

 

Patricia Helmuth's picture

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. 

 

Sarah Sirois's picture

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 

Duane Dorion's picture

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.