This document discusses the stepping stones and hurdles students may face when learning addition and subtraction in grades 3-4. It identifies 5 key stepping stones: 1) moving from concrete to abstract understanding of place value, 2) learning a wider range of strategies, 3) developing reasoning and logic skills, 4) solving word problems, and 5) learning algorithms. The document emphasizes that students progress at different paces and teachers must be patient and provide differentiated instruction to meet students' individual needs.
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A fun and interactive way to cover Common Core Math Standards in Primary Grades! Everything from ten frames to number sentences to addition and subtraction is covered in ONE user friendly page.
Teachers need to little or no prep to get this started in their own classrooms.
Your Math Students: Engaging and Understanding Every DayDreamBox Learning
The most important and challenging aspect of daily planning is to regularly—and yes, that means every day—create, adapt, locate, and consider mathematical tasks that are appropriate to the developmental learning needs of each student. A concern Francis (Skip) Fennell often shares with teachers is that many of us can find or create a lot of “fun” tasks that are, for the most part, worthless in regards to learning mathematics. Mathematical
tasks should provide a level of demand on the part of the student that ensures a focus on understanding and involves them in actually doing mathematics.
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A fun and interactive way to cover Common Core Math Standards in Primary Grades! Everything from ten frames to number sentences to addition and subtraction is covered in ONE user friendly page.
Teachers need to little or no prep to get this started in their own classrooms.
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The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
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• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
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This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
2. Stepping Stones/Hurdles at this level:
• These are stages that children move through – some more quickly
than others.
Just like taking that first step, some children do it much earlier,
some children may take a little longer to let go of your hand or that
table edge they are clinging to. We see a lot of this with the use of
strategies , for example, with addition, where children use their
fingers instead of taking risks and trying new ways.
• However, some stones may become hurdles and children may stay at
that stage for a length of time. Teachers and parents will have to be
patient and show children different ways to manage these
difficulties. When we move children on before they are ready, we can
create problems for later.
• Students are monitored and assessed regularly – and it’s not just the
NAPLAN! Data from these assessments are used to determine at
what stage a particular student is at and the program is
differentiated to meet his/her needs.
3. Stepping Stone 1:
MOVING FROM CONCRETE TO ABSTRACT
UNDERSTANDING
• Place Value – where the place of a digit indicates its value
and understanding the significance of a “zero”, and the
use of a Base 10 system
• How do you visualise 19, 057? Large numbers are
difficult too visualise
• What does 50 metres look like? What is your reference
point? For example: think of the length of a swimming
pool
• What do you mean by 2/3? 6/10? What does 5/7 + 3/8
mean?
4. PLACE VALUE ACTIVITIES –
I had a group of students who were stuck at a particular point and
I knew I couldn’t go forward without re-establishing what they
had already learnt about counting by 10s, using materials to
represent the different elements of the place value system and that
you could rename a number in many ways e.g. 11 is 1 ten and 1
one as well as 11 ones, 345 is 3 hundreds, 4 tens and 5 ones as well
as 34 tens and 5 ones, etc. This is important groundwork which
would help students understand how to use conventional
algorithms appropriately. (I’ll talk about algorithms later.)
5. STEPPING STONE 2:
WIDER RANGE OF STRATEGIES TO LEARN TO
WORK WITH:
• Assessment conducted at beginning of the year – to establish what
strategies students were using and grouping students for explicit
teaching, then reassessing them to determine if they have moved
on. Students could be in different groupings based on what they
need to learn or work on.
• Range of strategies for addition – building on from what students
have learnt in the Early Years: counting all, counting on , counting
in groups, doubles, making to 10, making to 100, etc and then
generalising from known strategies e.g. I know that 2 x 4 is 8
therefore 2 x 400 must be 800, even though I didn’t learn the 2
times table to 400 but because I understand how the place value
system works.
• Subtraction is the inverse operation to addition – if you “master”
addition, the theory is that you also “master” subtraction. But
students seem to find subtraction quite difficult to visualise.
• Fact families – number facts to 20. (This helps with recognising
when a calculation may be inaccurate.) Knowing these also help
with faster calculations without reaching for the calculator!
6. STEPPING STONE 3:
REASONING AND LOGIC
Because you cannot see something, you need other ways to be sure that your
calculations are logical and reasonable
e.g:
• knowing that when you add 2 numbers together, the total has to be larger
than either of the original numbers (5 + 3 = 8) but this does not always hold
true for subtraction e.g. 11 – 3 = 8.
• knowing that when you add or subtract a “0”, you get the original number.
• when you add 2 odd numbers, you will always get an even number, etc.
• Estimating skills are very important – they help to check if your answer is
accurate. e.g. 456 x 3 would be round about but less than 1500 because 456 is
less than 500 and if I wrote that 25.2 x 1.3 = 327.6 , I should question that
because 25 x 2 would only give me 50. Estimating is especially important
when children are using calculators because it is very easy to punch in the
wrong digits or put in too many 0s, etc and children tend to think the
calculator is always right!
7. STEPPING STONE 4:
PROBLEM SOLVING
Now that students have mastered each particular process, do
they necessarily know when to use each one?
Sample problems:
• I went to the market and bought 5 apples and 6 bananas. How
many things did I buy? Pretty straight forward : 5 + 6 = 11
• Kate and Jane participated in a marathon. Jane ran ½ as fast as
Kate. If Kate ran at 7 km/h, what was Jane’s speed? What does half
as fast mean? Was Kate faster or Jane faster? What does km/h
mean? What process do I need to use to find the answer? Can you
imagine a student who comes from a non-English speaking
background or is a little weak or slow in reading having to deal
with this? He/she may be a Maths wiz but might find it difficult to
work this out.
8. Then you get something like this:
• During the 100 meter dash in the 1988 Olympic Games in Seoul,
Florence Griffith-Joyner was timed at 0.91 seconds for 10 meters.
At that speed, could she pass a car traveling 35 km per hour in
a school zone? First response will probably be “Huh???”
SEVERAL STEPS FOR PROBLEM SOLVING:
• Interpreting the problem – what information is essential? What
do I need to find out?
• Strategies for solution – what processes can I use? Which is the
most efficient way of doing this?
• Estimating the answer – is my answer logical?
9. STEPPING STONE 5:
ALGORITHMS –
• These are merely conventional ways of setting out our calculations so
that everyone can understand them. In Maths, symbols are part of a
universal language and we have to learn to follow the conventions.
• Students often have a very strong understanding of how to add 2-digit
numbers in their heads or even horizontally –
e.g. 27 + 45, I know that 2 and 4 make 6, and the 7 and 5 make 12 so it
must be 72 but we then impose a structure upon their reasoning and
throw them off track with the “ do the left column first and then you
carry that number over there and then you add those numbers
together”.
• If students how a strong understanding of how the place value system
works, then using the algorithm makes sense to them and they are not
then merely a series of steps.
10. With subtraction, it gets even more complicated. You borrow but
you never return anything! Or you “decompose” or whatever
method your own teacher taught you……remember the frustration
when you didn’t get the “right answer”? How come when you
borrow a number from the left hand column and put it in the right,
it’s not 1 + 5 but it becomes 15?
1
6 5
- 3 7
____
We introduce the algorithms only after we have done the
groundwork with understanding place value and developing strong
estimating skills.
11. Last Stepping Stone:
THE MATHS CURRICULUM & THE REST OF THE
CURRICULUM!
There is a set quantity of material in the curriculum to be “covered”
at any grade level . As I mentioned at the beginning of the evening,
there is the continuum of logical progression for developing
understanding of concepts but we know that in life, nothing ever
develops in the way it should – some students fly through the
stages, some stumble and others can get quite stuck at a particular
point!
While we try to teach students at their pace, at the back of our
minds is the pressure of teaching them to meet the requisite
standards because they are in Grade 3 or 4 or 5 or whatever and we
don’t want to put a “D” on the reports. And it is understandable
when parents and we teachers get frustrated when a child doesn’t
move forward at the pace we expect.